diff --git a/data/tlg1799/tlg001/tlg1799.tlg001.perseus-eng1.xml b/data/tlg1799/tlg001/tlg1799.tlg001.perseus-eng1.xml index 296a25b04..725e47433 100644 --- a/data/tlg1799/tlg001/tlg1799.tlg001.perseus-eng1.xml +++ b/data/tlg1799/tlg001/tlg1799.tlg001.perseus-eng1.xml @@ -82,7 +82,7 @@ EUCLID AND THE TRADITIONS ABOUT HIM.

As in the case of the other great mathematicians of Greece, so in Euclid's case, we have only the most meagre particulars of the life and personality of the man.

Most of what we have is contained in the passage of Proclus' summary relating to him, which is as followsProclus, ed. Friedlein, p. 68, 6-20.:

-

Not much younger than these (sc. Hermotimus of Colophon and Philippus of Medma) is Euclid, who put together the Elements, collecting many of Eudoxus' theorems, perfecting many of Theaetetus', and also bringing to irrefragable demonstration the things which were only somewhat loosely proved by his predecessors. This man livedThe word ge/gone must apparently mean flourished, as Heiberg understands it (Litterargeschichtliche Studien über Euklid, 1882, p. 26), not was born, as Hankel took it : otherwise part of Proclus' argument would lose its cogency. in the time of the first Ptolemy. For Archimedes, who came immediately after the first (Ptolemy)So Heiberg understands e)pibalw\n tw=| prw/tw| (sc. *ptolemai/w|). Friedlein's text has kai\ between e)pibalw\n and tw=| prw/tw|; and it is right to remark that another reading is kai\ e)n tw=| prw/tw| (without e)pibalw/n) which has been translated in his first book, by which is understood On the Sphere and Cylinder I., where (1) in Prop. 2 are the words let BC be made equal to D by the second (proposition) of the first of Euclid's (books), +

Not much younger than these (sc. Hermotimus of Colophon and Philippus of Medma) is Euclid, who put together the Elements, collecting many of Eudoxus' theorems, perfecting many of Theaetetus', and also bringing to irrefragable demonstration the things which were only somewhat loosely proved by his predecessors. This man livedThe word ge/gone must apparently mean flourished, as Heiberg understands it (Litterargeschichtliche Studien über Euklid, 1882, p. 26), not was born, as Hankel took it : otherwise part of Proclus' argument would lose its cogency. in the time of the first Ptolemy. For Archimedes, who came immediately after the first (Ptolemy)So Heiberg understands e)pibalw\n tw=| prw/tw| (sc. *ptolemai/w|). Friedlein's text has kai\ between e)pibalw\n and tw=| prw/tw|; and it is right to remark that another reading is kai\ e)n tw=| prw/tw| (without e)pibalw/n) which has been translated in his first book, by which is understood On the Sphere and Cylinder I., where (1) in Prop. 2 are the words let BC be made equal to D by the second (proposition) of the first of Euclid's (books), and (2) in Prop. 6 the words For these things are handed down in the Elements (without the name of Euclid). Heiberg thinks the former passage is referred to, and that Proclus must therefore have had before him the words by the second of the first of Euclid: a fair proof that they are genuine, though in themselves they would be somewhat suspicious., makes mention of Euclid: and, further, they say that Ptolemy once asked him if there was in geometry any shorter way than that of the elements, and he answered that there was no royal road to geometryThe same story is told in Stobaeus, Ecl. (II. p. 228, 30, ed. Wachsmuth) about Alexander and Menaechmus. Alexander is represented as having asked Menaechmus to teach him geometry concisely, but he replied : O king, through the country there are royal roads and roads for common citizens, but in geometry there is one road for all.. He is then younger than the pupils of Plato but older than Eratosthenes and Archimedes; for the latter were contemporary with one another, as Eratosthenes somewhere says.

@@ -93,8 +93,8 @@ To get out of the difficulty he saysibid. p. 70, 19 sqq. that, if one should ask him what was the aim (skopo/s) of the treatise, he would reply by making a distinction between Euclid's intentions (1) as regards the subjects with which his investigations are concerned, (2) as regards the learner, and would say as regards (1) that the whole of the geometer's argument is concerned with the cosmic figures. This latter statement is obviously incorrect. It is true that Euclid's Elements end with the construction of the five regular solids; but the planimetrical portion has no direct relation to them, and the arithmetical no relation at all; the propositions about them are merely the conclusion of the stereometrical division of the work.

One thing is however certain, namely that Euclid taught, and founded a school, at Alexandria. This is clear from the remark of Pappus about ApolloniusPappus, VII. p. 678, 10-12, susxola/sas toi=s u(po\ *eu)klei/dou maqhtai=s e)n *)alecandrei/<*> plei=ston xro/non, o(/qen e)/sxe kai\ th\n toiau/thn e)/cin ou)k a)maqh=. : he spent a very long time with the pupils of Euclid at Alexandria, and it was thus that he acquired such a scientific habit of thought.

It is in the same passage that Pappus makes a remark which might, to an unwary reader, seem to throw some light on the personality of Euclid. He is speaking about Apollonius' preface to the first book of his Conics, where he says that Euclid had not completely worked out the synthesis of the three- and four-line locus, - which in fact was not possible without some theorems first discovered by himself. Pappus says on thisPappus, VII. pp. 676, 25-678, 6. Hultsch, it is true, brackets the whole passage pp. 676, 25-678, 15, but apparently on the ground of the diction only.: Now Euclid— regarding Aristaeus as deserving credit for the discoveries he had already made in conics, and without anticipating him or wishing to construct anew the same system (such was his scrupulous fairness and his exemplary kindliness towards all who could advance mathematical science to however small an extent), being moreover in no wise contentious and, though exact, yet no braggart like the other [Apollonius] —wrote so much about the locus as was possible by means of the conics of Aristaeus, without claiming completeness for his demonstrations. It is however evident, when the passage is examined in its context, that Pappus is not following any tradition in giving this account of Euclid: he was offended by the terms of Apollonius' reference to Euclid, which seemed to him unjust, and he drew a fancy picture of Euclid in order to show Apollonius in a relatively unfavourable light.

-

Another story is told of Euclid which one would like to believe true. According to StobaeusStobaeus, l.c., some one who had begun to read geometry with Euclid, when he had learnt the first theorem, asked Euclid, ’But what shall I get by-learning these things?’ Euclid called his slave and said ’Give him threepence, since he must make gain out of what he learns.’

+ which in fact was not possible without some theorems first discovered by himself. Pappus says on thisPappus, VII. pp. 676, 25-678, 6. Hultsch, it is true, brackets the whole passage pp. 676, 25-678, 15, but apparently on the ground of the diction only.: Now Euclid— regarding Aristaeus as deserving credit for the discoveries he had already made in conics, and without anticipating him or wishing to construct anew the same system (such was his scrupulous fairness and his exemplary kindliness towards all who could advance mathematical science to however small an extent), being moreover in no wise contentious and, though exact, yet no braggart like the other [Apollonius] —wrote so much about the locus as was possible by means of the conics of Aristaeus, without claiming completeness for his demonstrations. It is however evident, when the passage is examined in its context, that Pappus is not following any tradition in giving this account of Euclid: he was offended by the terms of Apollonius' reference to Euclid, which seemed to him unjust, and he drew a fancy picture of Euclid in order to show Apollonius in a relatively unfavourable light.

+

Another story is told of Euclid which one would like to believe true. According to StobaeusStobaeus, l.c., some one who had begun to read geometry with Euclid, when he had learnt the first theorem, asked Euclid, ’But what shall I get by-learning these things?’ Euclid called his slave and said ’Give him threepence, since he must make gain out of what he learns.’

In the middle ages most translators and editors spoke of Euclid as Euclid of Megara. This description arose out of a confusion between our Euclid and the philosopher Euclid of Megara who lived about 400 B.C. The first trace of this confusion appears in Valerius Maximus (in the time of Tiberius) who saysVIII. 12, ext. 1. that Plato, on being appealed to for a solution of the problem of doubling the cubical altar, sent the inquirers to Euclid the geometer. There is no doubt about the reading, although an early commentator on Valerius Maximus wanted to correct Eucliden into Eudoxum, @@ -106,25 +106,25 @@

Another idea, that Euclid was born at Gela in Sicily, is due to the same confusion, being based on Diogenes Laertius' descriptionDiog. L. II. 106, p. 58 ed. Cobet. of the philosopher Euclid as being of Megara, or, according to some, of Gela, as Alexander says in the *diadoxai/.

In view of the poverty of Greek tradition on the subject even as early as the time of Proclus (410-485 A.D.), we must necessarily take cum grano the apparently circumstantial accounts of Euclid given by Arabian authors; and indeed the origin of their stories can be explained as the result (1) of the Arabian tendency to romance, and (2) of misunderstandings.

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We readCasiri, Bibliotheca Arabico-Hispana Escurialensis, I. p. 339. Casiri's source is alQifti (d. 1248), the author of the Ta'rīkh al-H<*>ukamā, a collection of biographies of philosophers, mathematicians, astronomers etc. that Euclid, son of Naucrates, grandson of ZenarchusThe Fihrist says son of Naucrates, the son of Berenice (?) - (see Suter's translation in Abhandlungen zur Gesch. d. Math. VI. Heft, 1892, p. 16)., called the author of geometry, a philosopher of somewhat ancient date, a Greek by nationality domiciled at Damascus, born at Tyre, most learned in the science of geometry, published a most excellent and most useful work entitled the foundation or elements of geometry, a subject in which no more general treatise existed before among the Greeks: nay, there was no one even of later date who did not walk in his footsteps and frankly profess his doctrine. Hence also Greek, Roman and Arabian geometers not a few, who undertook the task of illustrating this work, published commentaries, scholia, and notes upon it, and made an abridgment of the work itself. For this reason the Greek philosophers used to post up on the doors of their schools the well-known notice: ’Let no one come to our school, who has not first learned the elements of Euclid.’ - The details at the beginning of this extract cannot be derived from Greek sources, for even Proclus did not know anything about Euclid's father, while it was not the Greek habit to record the names of grandfathers, as the Arabians commonly did. Damascus and Tyre were no doubt brought in to gratify a desire which the Arabians always showed to connect famous Greeks in some way or other with the East. Thus Nas<*>īraddīn, the translator of the Elements, who was of T<*>ūs in Khurāsān, actually makes Euclid out to have been Thusinus - alsoThe same predilection made the Arabs describe Pythagoras as a pupil of the wise Salomo, Hipparchus as the exponent of Chaldaean philosophy or as the Chaldaean, Archimedes as an Egyptian etc. (H<*>ăjī Khalfa, Lexicon Bibliographicum, and Casiri).. The readiness of the Arabians to run away with an idea is illustrated by the last words of the extract. Everyone knows the story of Plato's inscription over the porch of the Academy: let no one unversed in geometry enter my doors +

We readCasiri, Bibliotheca Arabico-Hispana Escurialensis, I. p. 339. Casiri's source is alQifti (d. 1248), the author of the Ta'rīkh al-H<*>ukamā, a collection of biographies of philosophers, mathematicians, astronomers etc. that Euclid, son of Naucrates, grandson of ZenarchusThe Fihrist says son of Naucrates, the son of Berenice (?) + (see Suter's translation in Abhandlungen zur Gesch. d. Math. VI. Heft, 1892, p. 16)., called the author of geometry, a philosopher of somewhat ancient date, a Greek by nationality domiciled at Damascus, born at Tyre, most learned in the science of geometry, published a most excellent and most useful work entitled the foundation or elements of geometry, a subject in which no more general treatise existed before among the Greeks: nay, there was no one even of later date who did not walk in his footsteps and frankly profess his doctrine. Hence also Greek, Roman and Arabian geometers not a few, who undertook the task of illustrating this work, published commentaries, scholia, and notes upon it, and made an abridgment of the work itself. For this reason the Greek philosophers used to post up on the doors of their schools the well-known notice: ’Let no one come to our school, who has not first learned the elements of Euclid.’ + The details at the beginning of this extract cannot be derived from Greek sources, for even Proclus did not know anything about Euclid's father, while it was not the Greek habit to record the names of grandfathers, as the Arabians commonly did. Damascus and Tyre were no doubt brought in to gratify a desire which the Arabians always showed to connect famous Greeks in some way or other with the East. Thus Nas<*>īraddīn, the translator of the Elements, who was of T<*>ūs in Khurāsān, actually makes Euclid out to have been Thusinus + alsoThe same predilection made the Arabs describe Pythagoras as a pupil of the wise Salomo, Hipparchus as the exponent of Chaldaean philosophy or as the Chaldaean, Archimedes as an Egyptian etc. (H<*>ăjī Khalfa, Lexicon Bibliographicum, and Casiri).. The readiness of the Arabians to run away with an idea is illustrated by the last words of the extract. Everyone knows the story of Plato's inscription over the porch of the Academy: let no one unversed in geometry enter my doors ; the Arab turned geometry into Euclid's geometry, and told the story of Greek philosophers in general and their Academies.

-

Equally remarkable are the Arabian accounts of the relation of Euclid and ApolloniusThe authorities for these statements quoted by Casiri and H<*>ājī Khalfa are al-Kindi's tract de instituto libri Euclidis (al-Kindī died about 873) and a commentary by Qād<*>īzāde ar-Rūmī (d. about 1440) on a book called Ashkāl at-ta' sīs (fundamental propositions) by Ashraf Shamsaddīn as-Samarqandī (c. 1276) consisting of elucidations of 35 propositions selected from the first books of Euclid. Nas<*>īraddīn likewise says that Euclid cut out two of 15 books of elements then existing and published the rest under his own name. According to Qād<*>īzāde the king heard that there was a celebrated geometer named Euclid at Tyre: Nas<*>īraddīn says that he sent for Euclid of T<*>ūs.. According to them the Elements were originally written, not by Euclid, but by a man whose name was Apollonius, a carpenter, who wrote the work in 15 books or sectionsSo says the Fihrist. Suter (op. cit. p. 49) thinks that the author of the Fihrist did not suppose Apollonius of Perga to be the writer of the Elements, as later Arabian authorities did, but that he distinguished another Apollonius whom he calls a carpenter. +

Equally remarkable are the Arabian accounts of the relation of Euclid and ApolloniusThe authorities for these statements quoted by Casiri and H<*>ājī Khalfa are al-Kindi's tract de instituto libri Euclidis (al-Kindī died about 873) and a commentary by Qād<*>īzāde ar-Rūmī (d. about 1440) on a book called Ashkāl at-ta' sīs (fundamental propositions) by Ashraf Shamsaddīn as-Samarqandī (c. 1276) consisting of elucidations of 35 propositions selected from the first books of Euclid. Nas<*>īraddīn likewise says that Euclid cut out two of 15 books of elements then existing and published the rest under his own name. According to Qād<*>īzāde the king heard that there was a celebrated geometer named Euclid at Tyre: Nas<*>īraddīn says that he sent for Euclid of T<*>ūs.. According to them the Elements were originally written, not by Euclid, but by a man whose name was Apollonius, a carpenter, who wrote the work in 15 books or sectionsSo says the Fihrist. Suter (op. cit. p. 49) thinks that the author of the Fihrist did not suppose Apollonius of Perga to be the writer of the Elements, as later Arabian authorities did, but that he distinguished another Apollonius whom he calls a carpenter. Suter's argument is based on the fact that the Fihrist's article on Apollonius (of Perga) says nothing of the Elements; and that it gives the three great mathematicians, Euclid, Archimedes and Apollonius, in the correct chronological order.. In the course of time some of the work was lost and the rest became disarranged, so that one of the kings at Alexandria who desired to study geometry and to master this treatise in particular first questioned about it certain learned men who visited him and then sent for Euclid who was at that time famous as a geometer, and asked him to revise and complete the work and reduce it to order. Euclid then re-wrote it in 13 books which were thereafter known by his name. (According to another version Euclid composed the 13 books out of commentaries which he had published on two books of Apollonius on conics and out of introductory matter added to the doctrine of the five regular solids.) To the thirteen books were added two more books, the work of others (though some attribute these also to Euclid) which contain several things not mentioned by Apollonius. According to another version Hypsicles, a pupil of Euclid at Alexandria, offered to the king and published Books XIV. and XV., it being also stated that Hypsicles had discovered the books, by which it appears to be suggested that Hypsicles had edited them from materials left by Euclid.

We observe here the correct statement that Books XIV. and XV. were not written by Euclid, but along with it the incorrect information that Hypsicles, the author of Book XIV., wrote Book XV. also.

The whole of the fable about Apollonius having preceded Euclid and having written the Elements appears to have been evolved out of the preface to Book XIV. by Hypsicles, and in this way; the Book must in early times have been attributed to Euclid, and the inference based upon this assumption was left uncorrected afterwards when it was recognised that Hypsicles was the author. The preface is worth quoting:

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Basilides of Tyre, O Protarchus, when he came to Alexandria and met my father, spent the greater part of his sojourn with him on account of their common interest in mathematics. And once, when examining the treatise written by Apollonius about the comparison between the dodecahedron and the icosahedron inscribed in the same sphere, (showing) what ratio they have to one another, they thought that Apollonius had not expounded this matter properly, and accordingly they emended the exposition, as I was able to learn from my father. And I myself, later, fell in with another book published by Apollonius, containing a demonstration relating to the subject, and I was greatly interested in the investigation of the problem. The book published by Apollonius is accessible to all— for it has a large circulation, having apparently been carefully written out later—but I decided to send you the comments which seem to me to be necessary, for you will through your proficiency in mathematics in general and in geometry in particular form an expert judgment on what I am about to say, and you will lend a kindly ear to my disquisition for the sake of your friendship to my father and your goodwill to me. +

Basilides of Tyre, O Protarchus, when he came to Alexandria and met my father, spent the greater part of his sojourn with him on account of their common interest in mathematics. And once, when examining the treatise written by Apollonius about the comparison between the dodecahedron and the icosahedron inscribed in the same sphere, (showing) what ratio they have to one another, they thought that Apollonius had not expounded this matter properly, and accordingly they emended the exposition, as I was able to learn from my father. And I myself, later, fell in with another book published by Apollonius, containing a demonstration relating to the subject, and I was greatly interested in the investigation of the problem. The book published by Apollonius is accessible to all— for it has a large circulation, having apparently been carefully written out later—but I decided to send you the comments which seem to me to be necessary, for you will through your proficiency in mathematics in general and in geometry in particular form an expert judgment on what I am about to say, and you will lend a kindly ear to my disquisition for the sake of your friendship to my father and your goodwill to me.

The idea that Apollonius preceded Euclid must evidently have been derived from the passage just quoted. It explains other things besides. Basilides must have been confused with basileu/s, and we have a probable explanation of the Alexandrian king, and of the learned men who visited Alexandria. It is possible also that in the Tyrian of Hypsicles' preface we have the origin of the notion that Euclid was born in Tyre. These inferences argue, no doubt, very defective knowledge of Greek: but we could expect no better from those who took the Organon of Aristotle to be instrumentum musicum pneumaticum, and who explained the name of Euclid, which they variously pronounced as Uclides or Icludes, to be compounded of Ucli a key, and Dis a measure, or, as some say, geometry, so that Uclides is equivalent to the key of geometry!

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Lastly the alternative version, given in brackets above, which says that Euclid made the Elements out of commentaries which he wrote on two books of Apollonius on conics and prolegomena added to the doctrine of the five solids, seems to have arisen, through a like confusion, out of a later passageHeiberg's Euclid, vol. V. p. 6. in Hypsicles' Book XIV.: And this is expounded by Aristaeus in the book entitled ’Comparison of the five figures,’ and by Apollonius in the second edition of his comparison of the dodecahedron with the icosahedron. +

Lastly the alternative version, given in brackets above, which says that Euclid made the Elements out of commentaries which he wrote on two books of Apollonius on conics and prolegomena added to the doctrine of the five solids, seems to have arisen, through a like confusion, out of a later passageHeiberg's Euclid, vol. V. p. 6. in Hypsicles' Book XIV.: And this is expounded by Aristaeus in the book entitled ’Comparison of the five figures,’ and by Apollonius in the second edition of his comparison of the dodecahedron with the icosahedron. The doctrine of the five solids in the Arabic must be the Comparison of the five figures in the passage of Hypsicles, for nowhere else have we any information about a work bearing this title, nor can the Arabians have had. The reference to the two books of Apollonius on conics will then be the result of mixing up the fact that Apollonius wrote a book on conics with the second edition of the other work mentioned by Hypsicles. We do not find elsewhere in Arabian authors any mention of a commentary by Euclid on Apollonius and Aristaeus: so that the story in the passage quoted is really no more than a variation of the fable that the Elements were the work of Apollonius.

@@ -133,13 +133,13 @@ CHAPTER II. EUCLID'S OTHER WORKS. -

In giving a list of the Euclidean treatises other than the Elements, I shall be brief: for fuller accounts of them, or speculations with regard to them, reference should be made to the standard histories of mathematicsSee, for example, Loria, Le scienze esatte nell' antica Grecia, 1914, pp. 245-268; 1. L. Heath, History of Greek Mathematics, 1921, I. pp. 421-446. Cf. Heiberg, Litterargeschichtliche Studien über Euklid, pp. 36-153; Euclidis opera omnia, ed. Heiberg and Menge, Vols. VI.—VIII..

+

In giving a list of the Euclidean treatises other than the Elements, I shall be brief: for fuller accounts of them, or speculations with regard to them, reference should be made to the standard histories of mathematicsSee, for example, Loria, Le scienze esatte nell' antica Grecia, 1914, pp. 245-268; 1. L. Heath, History of Greek Mathematics, 1921, I. pp. 421-446. Cf. Heiberg, Litterargeschichtliche Studien über Euklid, pp. 36-153; Euclidis opera omnia, ed. Heiberg and Menge, Vols. VI.—VIII..

I will take first the works which are mentioned by Greek authors.

I. The Pseudaria.

I mention this first because Proclus refers to it in the general remarks in praise of the Elements which he gives immediately after the mention of Euclid in his summary. He saysProclus, p. 70, 1-18.: But, inasmuch as many things, while appearing to rest on truth and to follow from scientific principles, really tend to lead one astray from the principles and deceive the more superficial minds, he has handed down methods for the discriminative understanding of these things as well, by the use of which methods we shall be able to give beginners in this study practice in the discovery of paralogisms, and to avoid being misled. This treatise, by which he puts this machinery in our hands, he entitled (the book) of Pseudaria, enumerating in order their various kinds, exercising our intelligence in each case by theorems of all sorts, setting the true side by side with the false, and combining the refutation of error with practical illustration. This book then is by way of cathartic and exercise, while the Elements contain the irrefragable and complete guide to the actual scientific investigation of the subjects of geometry.

The book is considered to be irreparably lost. We may conclude however from the connexion of it with the Elements and the reference to its usefulness for beginners that it did not go outside the domain of elementary geometryHeiberg points out that Alexander Aphrodisiensis appears to allude to the work in his commentary on Aristotle's Sophistici Elenchi (fol. 25 b): Not only those (e)/legxoi) which do not start from the principles of the science under which the problem is classed...but also those which do start from the proper principles of the science but in some respect admit a paralogism, e.g. the Pseudographemata of Euclid. - Tannery (Bull. des sciences math. et astr. 2^{e} Série, VI., 1882, I^{e\re} Partie, p. 147) conjectures that it may be from this treatise that the same commentator got his information about the quadratures of the circle by Antiphon and Bryson, to say nothing of the lunules of Hippocrates. I think however that there is an objection to this theory so far as regards Bryson; for Alexander distinctly says that Bryson's quadrature did not start from the proper principles of geometry, but from some principles more general..

+ Tannery (Bull. des sciences math. et astr. 2^{e} Série, VI., 1882, I^{e\re} Partie, p. 147) conjectures that it may be from this treatise that the same commentator got his information about the quadratures of the circle by Antiphon and Bryson, to say nothing of the lunules of Hippocrates. I think however that there is an objection to this theory so far as regards Bryson; for Alexander distinctly says that Bryson's quadrature did not start from the proper principles of geometry, but from some principles more general..

2. The Data.

The Data (dedome/na) are included by Pappus in the Treasury of Analysis (to/pos a)naluo/menos), and he describes their contentsPappus, VII. p. 638. They are still concerned with elementary geometry, though forming part of the introduction to higher analysis. Their form is that of propositions proving that, if certain things in a figure are given (in magnitude, in species, etc.), something else is given. The subjectmatter is much the same as that of the planimetrical books of the Elements, to which the Data are often supplementary. We shall see this later when we come to compare the propositions in the Elements which give us the means of solving the general quadratic equation with the corresponding propositions of the Data which give the solution. The Data may in fact be regarded as elementary exercises in analysis.

It is not necessary to go more closely into the contents, as we have the full Greek text and the commentary by Marinus newly edited by Menge and therefore easily accessibleVol. VI. in the Teubner edition of Euclidis opera omnia by Heiberg and Menge. A translation of the Data is also included in Simson's Euclid (though naturally his text left much to be desired)..

@@ -154,7 +154,7 @@ (lo/gw|): thus to divide a triangle into triangles would be to divide it into like figures, to divide a triangle into a triangle and a quadrilateral would be to divide it into unlike figures.

-

The treatise is lost in Greek but has been discovered in the Arabic. First John Dee discovered a treatise De divisionibus by one Muhammad BagdadinusSteinschneider places him in the 10th c. H. Suter (Bibliotheca Mathematica, IV_{3}, 1903, pp. 24, 27) identifies him with Abū (Bekr) Muh. b. 'Abdalbāqī al-Baġdādī, Qād|ī (Judge) of Māristān (circa 1070-1141), to whom he also attributes the Liber judei (? judicis) super decimum Euclidis translated by Gherard of Cremona. and handed over a copy of it (in Latin) in 1563 to Commandinus, who published it, in Dee did not himself translate the tract from the Arabic; he in 1570De superficierum divisionibus liber Machometo Bagdadino adscriptus, nunc primum Ioannis Dee Londinensis et Federici Commandini Urbinatis opera in lucem editus, Pisauri, 1570, afterwards included in Gregory's Euclid (Oxford, 1703).. Dee did not himself translate the tract from the Arabic; he found it in Latin in a MS. which was then in his own possession but was about 20 years afterwards stolen or destroyed in an attack by a mob on his house at MortlakeR. C. Archibald, Euclid's Book on the Division of Figures with a restoration based on Woepck's text and on the Practica geometriae of Leonardo Pisano, Cambridge, 1915, pp. 4-9.. Dee, in his preface addressed to Commandinus, says nothing of his having translated the book, but only remarks that the very illegible MS. had caused him much trouble and (in a later passage) speaks of the actual, very ancient, copy from which I wrote out... +

The treatise is lost in Greek but has been discovered in the Arabic. First John Dee discovered a treatise De divisionibus by one Muhammad BagdadinusSteinschneider places him in the 10th c. H. Suter (Bibliotheca Mathematica, IV_{3}, 1903, pp. 24, 27) identifies him with Abū (Bekr) Muh. b. 'Abdalbāqī al-Baġdādī, Qād|ī (Judge) of Māristān (circa 1070-1141), to whom he also attributes the Liber judei (? judicis) super decimum Euclidis translated by Gherard of Cremona. and handed over a copy of it (in Latin) in 1563 to Commandinus, who published it, in Dee did not himself translate the tract from the Arabic; he in 1570De superficierum divisionibus liber Machometo Bagdadino adscriptus, nunc primum Ioannis Dee Londinensis et Federici Commandini Urbinatis opera in lucem editus, Pisauri, 1570, afterwards included in Gregory's Euclid (Oxford, 1703).. Dee did not himself translate the tract from the Arabic; he found it in Latin in a MS. which was then in his own possession but was about 20 years afterwards stolen or destroyed in an attack by a mob on his house at MortlakeR. C. Archibald, Euclid's Book on the Division of Figures with a restoration based on Woepck's text and on the Practica geometriae of Leonardo Pisano, Cambridge, 1915, pp. 4-9.. Dee, in his preface addressed to Commandinus, says nothing of his having translated the book, but only remarks that the very illegible MS. had caused him much trouble and (in a later passage) speaks of the actual, very ancient, copy from which I wrote out... (in ipso unde descripsi vetustissimo exemplari). The Latin translation of this tract from the Arabic was probably made by Gherard of Cremona (1114-1187), among the list of whose numerous translations a liber divisionum occurs. The Arabic original cannot have been a direct translation from Euclid, and probably was not even a direct adaptation of it; it contains mistakes and unmathematical expressions, and moreover does not contain the propositions about the division of a circle alluded to by Proclus. Hence it can scarcely have contained more than a fragment of Euclid's work.

But Woepcke found in a MS. at Paris a treatise in Arabic on the division of figures, which he translated and published in 1851Fournal Asiatique, 1851, p. 233 sqq.. It is expressly attributed to Euclid in the MS. and corresponds to the description of it by Proclus. Generally speaking, the divisions are divisions into figures of the same kind as the original figures, e.g. of triangles into triangles; but there are also divisions into unlike @@ -163,47 +163,47 @@ Unfortunately the proofs are given of only four propositions (including the two last mentioned) out of 36, because the Arabic translator found them too easy and omitted them. To illustrate the character of the problems dealt with I need only take one more example: To cut off a certain fraction from a (parallel-) trapezium by a straight line which passes through a given point lying inside or outside the trapezium but so that a straight line can be drawn through it cutting both the parallel sides of the trapezium. The genuineness of the treatise edited by Woepcke is attested by the facts that the four proofs which remain are elegant and depend on propositions in the Elements, and that there is a lemma with a true Greek ring: to apply to a straight line a rectangle equal to the rectangle contained by AB, AC and deficient by a square. Moreover the treatise is no fragment, but finishes with the words end of the treatise, - and is a well-ordered and compact whole. Hence we may safely conclude that Woepcke's is not only Euclid's own work but the whole of it. A restoration of the work, with proofs, was attempted by OfterdingerL. F. Ofterdinger, Beiträge zur Wiederherstellung der Schrift des Euklides über die Theilung der Figuren, Ulm, 1853., Who however does not give Woepcke's props. 30, 31, 34, 35, 36. We have now a satisfactory restoration, with ample notes and an introduction, by R. C. Archibald, who used for the purpose Woepcke's text and a section of Leonardo of Pisa's Practica geometriae (1220). There is a remarkable similarity between the propositions of Woepcke's text and those of Leonardo, suggesting that Leonardo may have had before him a translation (perhaps by Gherard of Cremona) of the Arabic tract..

+ and is a well-ordered and compact whole. Hence we may safely conclude that Woepcke's is not only Euclid's own work but the whole of it. A restoration of the work, with proofs, was attempted by OfterdingerL. F. Ofterdinger, Beiträge zur Wiederherstellung der Schrift des Euklides über die Theilung der Figuren, Ulm, 1853., Who however does not give Woepcke's props. 30, 31, 34, 35, 36. We have now a satisfactory restoration, with ample notes and an introduction, by R. C. Archibald, who used for the purpose Woepcke's text and a section of Leonardo of Pisa's Practica geometriae (1220). There is a remarkable similarity between the propositions of Woepcke's text and those of Leonardo, suggesting that Leonardo may have had before him a translation (perhaps by Gherard of Cremona) of the Arabic tract..

4.The Porisms.

It is not possible to give in this place any account of the controversies about the contents and significance of the three lost books of Porisms, or of the important attempts by Robert Simson and Chasles to restore the work. These may be said to form a whole literature, references to which will be found most abundantly given by Heiberg and Loria, the former of whom has treated the subject from the philological point of view, most exhaustively, while the latter, founding himself generally on Heiberg, has added useful details, from the mathematical side, relating to the attempted restorations, etc.Heiberg, Euklid-Studien, pp. 56-79, and Loria, op. cit., pp. 253-265. It must suffice here to give an extract from the only original source of information about the nature and contents of the Porisms, namely PappusPappus, ed. Hultsch, VII. pp. 648-660. I put in square brackets the words bracketed by Hultsch.. In his general preface about the books composing the Treasury of Analysis (to/pos a)naluo/menos) he says:

-

“After the Tangencies (of Apollonius) come, in three books, the Porisms of Euclid, [in the view of many] a collection most ingeniously devised for the analysis of the more weighty problems, [and] although nature presents and unlimited number of such porismsI adopt Heiberg's reading of a comma here instead of a full stop., [they have added nothing to what was written originally by Euclid, except that some before my time have shown their want of taste by adding to a few (of the propositions) second proofs, each (proposition) admitting of a definite number of demonstrations, as we have shown, and Euclid having given one for each, namely that which is the most lucid. These porisms embody a theory subtle, natural, necessary, and of considerable generality, which is fascinating to those who can see and produce results].

-

“Now all the varieties of porisms belong, neither to theorems nor problems, but to a species occupying a sort of intermediate position [so that their enunciations can be formed like those of either theorems or problems], the result being that, of the great number of geometers, some regarded them as of the class of theorems, and others of problems, looking only to the form of the proposition. But that the ancients knew better the difference between these three things is clear from the definitions. For they said that a theorem is that which is proposed with a view to the demonstration of the very thing proposed, a problem that which is thrown out with a view to the construction of the very thing proposed, and a porism that which is proposed with a view to the producing of the very thing proposed. [But this definition of the porism was changed by the more recent writers who could not produce everything, but used these elements and proved only the fact that that which is sought really exists, but did not produce itHeiberg points out that Props. 5-9 of Archimedes' treatise On Spirals are porisms in this sense. To take Prop. 5 as an example, DBF is a tangent to a circle with centre K. It is then possible, says Archimedes, to draw a straight line KHF, meeting the circumference in H and the tangent in F, such that +

“After the Tangencies (of Apollonius) come, in three books, the Porisms of Euclid, [in the view of many] a collection most ingeniously devised for the analysis of the more weighty problems, [and] although nature presents and unlimited number of such porismsI adopt Heiberg's reading of a comma here instead of a full stop., [they have added nothing to what was written originally by Euclid, except that some before my time have shown their want of taste by adding to a few (of the propositions) second proofs, each (proposition) admitting of a definite number of demonstrations, as we have shown, and Euclid having given one for each, namely that which is the most lucid. These porisms embody a theory subtle, natural, necessary, and of considerable generality, which is fascinating to those who can see and produce results].

+

“Now all the varieties of porisms belong, neither to theorems nor problems, but to a species occupying a sort of intermediate position [so that their enunciations can be formed like those of either theorems or problems], the result being that, of the great number of geometers, some regarded them as of the class of theorems, and others of problems, looking only to the form of the proposition. But that the ancients knew better the difference between these three things is clear from the definitions. For they said that a theorem is that which is proposed with a view to the demonstration of the very thing proposed, a problem that which is thrown out with a view to the construction of the very thing proposed, and a porism that which is proposed with a view to the producing of the very thing proposed. [But this definition of the porism was changed by the more recent writers who could not produce everything, but used these elements and proved only the fact that that which is sought really exists, but did not produce itHeiberg points out that Props. 5-9 of Archimedes' treatise On Spirals are porisms in this sense. To take Prop. 5 as an example, DBF is a tangent to a circle with centre K. It is then possible, says Archimedes, to draw a straight line KHF, meeting the circumference in H and the tangent in F, such that

where c is the circumference of any circle. To prove this he assumes the following construction. E being any straight line greater than c, he says: let KG be parallel to DF, and let the line GH equal to E be placed verging to the point B. Archimedes must of course have known how to effect this construction, which requires conics. But that it is possible requires very little argument, for if we draw any straight line BHG meeting the circle in H and KG in G, it is obvious that as G moves away from C, HG becomes greater and greater and may be made as great as we please. The later writers would no doubt have contented themselves with this consideration without actually constructing HG. and were accordingly confuted by the definition and the whole doctrine. They based their definition on an incidental characteristic, thus: A porism is that which falls short of a locustheorem in respect of its hypothesisAs Heiberg says, this translation is made certain by a preceding passage of Pappus (p. 648, 1-3) where he compares two enunciations, the latter of which falls short of the former in hypothesis but goes beyond it in requirement. E.g. the first enunciation requiring us, given three circles, to draw a circle touching all three, the second may require us, given only two circles (one less datum), to draw a circle touching them and of a given size (an extra requirement).. Of this kind of porisms loci are a species, and they abound in the Treasury of Analysis; but this species has been collected, named and handed down separately from the porisms, because it is more widely diffused than the other species]. But it has further become characteristic of porisms that, owing to their complication, the enunciations are put in a contracted form, much being by usage left to be understood; so that many geometers understand them only in a partial way and are ignorant of the more essential features of their contents.

-

“[Now to comprehend a number of propositions in one enunciation is by no means easy in these porisms, because Euclid himself has not in fact given many of each species, but chosen, for examples, one or a few out of a great multitudeI translate Heiberg's reading with a full stop here followed by pro\s a)rxh=| de\ o(/mws [pro\s a)rxh\n (dedome/non) Hultsch] tou= prw/tou bibli/ou..... But at the beginning of the first book he has given some propositions, to the number of ten, of one species, namely that more fruitful species consisting of loci.] Consequently, finding that these admitted of being comprehended in one enunciation, we have set it out thus: +

“[Now to comprehend a number of propositions in one enunciation is by no means easy in these porisms, because Euclid himself has not in fact given many of each species, but chosen, for examples, one or a few out of a great multitudeI translate Heiberg's reading with a full stop here followed by pro\s a)rxh=| de\ o(/mws [pro\s a)rxh\n (dedome/non) Hultsch] tou= prw/tou bibli/ou..... But at the beginning of the first book he has given some propositions, to the number of ten, of one species, namely that more fruitful species consisting of loci.] Consequently, finding that these admitted of being comprehended in one enunciation, we have set it out thus:

If, in a system of four straight linesThe four straight lines are described in the text as (the sides) u(pti/ou h)\ parupti/ou, i.e. sides of two sorts of quadrilaterals which Simson tries to explain (see p. 120 of the Index Graecitatis of Hultsch's edition of Pappus). which cut each other two and two, three points on one straight line be given while the rest except one lie on different straight lines given in position, the remaining point also will lie on a straight line given in position.In other words (Chasles, p. 23; Loria, p. 256), if a triangle be so deformed that each of its sides turns about one of three points in a straight line, and two of its vertices lie on two straight lines given in position, the third vertex will also lie on a straight line..

-

“This has only been enunciated of four straight lines, of which not more than two pass through the same point, but it is not known (to most people) that it is true of any assigned number of straight lines if enunciated thus: -

If any number of straight lines cut one another, not more than two (passing) through the same point, and all the points (of intersection situated) on one of them be given, and if each of those which are on another (of them) lie on a straight line given in position—

- or still more generally thus: if any number of straight lines cut one another, not more than two (passing) through the same point, and all the points (of intersection situated) on one of them be given, while of the other points of intersection in multitude equal to a triangular number a number corresponding to the side of this triangular number lie respectively on straight lines given in position, provided that of these latter points no three are at the angular points of a triangle (sc. having for sides three of the given straight lines)—each of the remaining points will lie on a straight line given in positionLoria (p. 256, n. 3) gives the meaning of this as follows, pointing out that Simson was the discoverer of it: If a complete n-lateral be deformed so that its sides respectively turn about n points on a straight line, and (n-1) of its n(n-1)/2 vertices move on as many straight lines, the other (n-1)(n-2)/2 of its vertices likewise move on as many straight lines: but it is necessary that it should be impossible to form with the (n-1) vertices any triangle having for sides the sides of the polygon. +

“This has only been enunciated of four straight lines, of which not more than two pass through the same point, but it is not known (to most people) that it is true of any assigned number of straight lines if enunciated thus: +

If any number of straight lines cut one another, not more than two (passing) through the same point, and all the points (of intersection situated) on one of them be given, and if each of those which are on another (of them) lie on a straight line given in position—

+ or still more generally thus: if any number of straight lines cut one another, not more than two (passing) through the same point, and all the points (of intersection situated) on one of them be given, while of the other points of intersection in multitude equal to a triangular number a number corresponding to the side of this triangular number lie respectively on straight lines given in position, provided that of these latter points no three are at the angular points of a triangle (sc. having for sides three of the given straight lines)—each of the remaining points will lie on a straight line given in positionLoria (p. 256, n. 3) gives the meaning of this as follows, pointing out that Simson was the discoverer of it: If a complete n-lateral be deformed so that its sides respectively turn about n points on a straight line, and (n-1) of its n(n-1)/2 vertices move on as many straight lines, the other (n-1)(n-2)/2 of its vertices likewise move on as many straight lines: but it is necessary that it should be impossible to form with the (n-1) vertices any triangle having for sides the sides of the polygon. .

-

“It is probable that the writer of the Elements was not unaware of this but that he only set out the principle; and he seems, in the case of all the porisms, to have laid down the principles and the seed only [of many important things], the kinds of which should be distinguished according to the differences, not of their hypotheses, but of the results and the things sought. [All the hypotheses are different from one another because they are entirely special, but each of the results and things sought, being one and the same, follow from many different hypotheses.]

-

“We must then in the first book distinguish the following kinds of things sought:

-

“At the beginning of the bookReading, with Heiberg, tou= bibli/ou [tou= z Hultsch]. is this proposition: -

I. ’If from two given points straight lines be drawn meeting on a straight line given in position, and one cut off from a straight line given in position (a segment measured) to a given point on it, the other will also cut off from another (straight line a segment) having to the first a given ratio.’

+

“It is probable that the writer of the Elements was not unaware of this but that he only set out the principle; and he seems, in the case of all the porisms, to have laid down the principles and the seed only [of many important things], the kinds of which should be distinguished according to the differences, not of their hypotheses, but of the results and the things sought. [All the hypotheses are different from one another because they are entirely special, but each of the results and things sought, being one and the same, follow from many different hypotheses.]

+

“We must then in the first book distinguish the following kinds of things sought:

+

“At the beginning of the bookReading, with Heiberg, tou= bibli/ou [tou= z Hultsch]. is this proposition: +

I. ’If from two given points straight lines be drawn meeting on a straight line given in position, and one cut off from a straight line given in position (a segment measured) to a given point on it, the other will also cut off from another (straight line a segment) having to the first a given ratio.’

-

“Following on this (we have to prove) +

“Following on this (we have to prove)

II. that such and such a point lies on a straight line given in position;

- III. that the ratio of such and such a pair of straight lines is given;” + III. that the ratio of such and such a pair of straight lines is given;” etc. etc. (up to XXIX.).

The three books of the porisms contain 38 lemmas; of the theorems themselves there are 171.

Pappus further gives lemmas to the Porisms (pp. 866-918, ed. Hultsch).

With Pappus' account of Porisms must be compared the passages of Proclus on the same subject. Proclus distinguishes two senses in which the word po/risma is used. The first is that of corollary where something appears as an incidental result of a proposition, obtained without trouble or special seeking, a sort of bonus which the investigation has presented us withProclus, pp. 212, 14; 301, 22.. The other sense is that of Euclid's Porismsibid. p. 212, 12. The term porism is used of certam problems, like the Porisms written by Euclid. -. In this senseibid. pp. 301, 25 sqq. porism is the name given to things which are sought, but need some finding and are neither pure bringing into existence nor simple theoretic argument. For (to prove) that the angles at the base of isosceles triangles are equal is a matter of theoretic argument, and it is with reference to things existing that such knowledge is (obtained). But to bisect an angle, to construct a triangle, to cut off, or to place—all these things demand the making of something; and to find the centre of a given circle, or to find the greatest common measure of two given commensurable magnitudes, or the like, is in some sort between theorems and problems. For in these cases there is no bringing into existence of the things sought, but finding of them, nor is the procedure purely theoretic. For it is necessary to bring that which is sought into view and exhibit it to the eye. Such are the porisms which Euclid wrote, and arranged in three books of Porisms. +. In this senseibid. pp. 301, 25 sqq. porism is the name given to things which are sought, but need some finding and are neither pure bringing into existence nor simple theoretic argument. For (to prove) that the angles at the base of isosceles triangles are equal is a matter of theoretic argument, and it is with reference to things existing that such knowledge is (obtained). But to bisect an angle, to construct a triangle, to cut off, or to place—all these things demand the making of something; and to find the centre of a given circle, or to find the greatest common measure of two given commensurable magnitudes, or the like, is in some sort between theorems and problems. For in these cases there is no bringing into existence of the things sought, but finding of them, nor is the procedure purely theoretic. For it is necessary to bring that which is sought into view and exhibit it to the eye. Such are the porisms which Euclid wrote, and arranged in three books of Porisms.

Proclus' definition thus agrees well enough with the first, older, definition of Pappus. A porism occupies a place between a theorem and a problem: it deals with something already existing, as a theorem does, but has to find it (e.g. the centre of a circle), and, as a certain operation is therefore necessary, it partakes to that extent of the nature of a problem, which requires us to construct or produce something not previously existing. Thus, besides III. I of the Elements and X. 3, 4 mentioned by Proclus, the following propositions are real porisms: III. 25, VI. 11-13, VII. 33, 34, 36, 39, VIII. 2, 4, X. 10, XIII. 18. Similarly in Archimedes On the Sphere and Cylinder I. 2-6 might be called porisms.

The enunciation given by Pappus as comprehending ten of Euclid's propositions may not reproduce the form of Euclid's enunciations; but, comparing the result to be proved, that certain points lie on straight lines given in position, with the class indicated by II. above, where the question is of such and such a point lying on a straight line given in position, and with other classes, e.g. (V.) that such and such a line is given in position, (VI.) that such and such a line verges to a given point, (XXVII.) that there exists a given point such that straight lines drawn from it to such and such (circles) will contain a triangle given in species, we may conclude that a usual form of a porism was to prove that it is possible to find a point with such and such a property or a straight line on which lie all the points satisfying given conditions etc.

-

Simson defined a porism thus: Porisma est propositio in qua proponitur demonstrare rem aliquam, vel plures datas esse, cui, vel quibus, ut et cuilibet ex rebus innumeris, non quidem datis, sed quae ad ea quae data sunt eandem habent relationem, convenire ostendendum est affectionem quandam communem in propositione descriptamThis was thus expressed by Chasles: Le porisme est une proposition dans laquelle on demande de démontrer qu'une chose ou plusieurs choses sont données, qui, ainsi que l'une quelconque d'une infinité d'autres choses non données, mais dont chacune est avec des choses données dans une même relation, ont une certaine propriété commune, décrite dans la proposition. +

Simson defined a porism thus: Porisma est propositio in qua proponitur demonstrare rem aliquam, vel plures datas esse, cui, vel quibus, ut et cuilibet ex rebus innumeris, non quidem datis, sed quae ad ea quae data sunt eandem habent relationem, convenire ostendendum est affectionem quandam communem in propositione descriptamThis was thus expressed by Chasles: Le porisme est une proposition dans laquelle on demande de démontrer qu'une chose ou plusieurs choses sont données, qui, ainsi que l'une quelconque d'une infinité d'autres choses non données, mais dont chacune est avec des choses données dans une même relation, ont une certaine propriété commune, décrite dans la proposition. .

From the above it is easy to understand Pappus' statement that loci constitute a large class of porisms. A locus is well defined by Simson thus: Locus est proposition in qua propositum est datam esse demonstrare, vel invenire lineam aut superficiem cuius quodlibet punctum, vel superficiem in qua quaelibet linea data lege descripta, communem quandam habet proprietatem in propositione descriptam. @@ -212,7 +212,7 @@

A difficult point, however, arises on the passage of Pappus, which says that a porism is that which, in respect of its hypothesis, falls short of a locus-theorem (topikou= qewrh/matos). Heiberg explains it by comparing the porism from Apollonius' Plane Loci just given with Pappus' enunciation of the same thing, to the effect that, if from two given points two straight lines be drawn meeting in a point, and these straight lines have to one another a given ratio, the point will lie on either a straight line or a circumference of a circle given in position. Heiberg observes that in this latter enunciation something is taken into the hypothesis which was not in the hypothesis of the enunciation of the porism, viz. that the ratio of the straight lines is the same. - I confess this does not seem to me satisfactory: for there is no real difference between the enunciations, and the supposed difference in hypothesis is very like playing with words. Chasles says: Ce qui constitue le porisme est ce qui manque à l' hypothèse d'un théorème local (en d'autres termes, le porisme est inférieur, par l'hypothèse, au théorème local; c'est-à-dire que quand quelques parties d'une proposition locale n'ont pas dans l'énoncé la détermination qui leur est propre, cette proposition cesse d'être regardée comme un theéorème et devient un porisme). + I confess this does not seem to me satisfactory: for there is no real difference between the enunciations, and the supposed difference in hypothesis is very like playing with words. Chasles says: Ce qui constitue le porisme est ce qui manque à l' hypothèse d'un théorème local (en d'autres termes, le porisme est inférieur, par l'hypothèse, au théorème local; c'est-à-dire que quand quelques parties d'une proposition locale n'ont pas dans l'énoncé la détermination qui leur est propre, cette proposition cesse d'être regardée comme un theéorème et devient un porisme). But the subject still seems to require further elucidation.

While there is so much that is obscure, it seems certain (1) that the Porisms were distinctly part of higher geometry and not of elementary geometry, (2) that they contained propositions belonging to the modern theory of transversals and to projective geometry. It should be remembered too that it was in the course of his researches on this subject that Chasles was led to the idea of anharmonic ratios.

Lastly, allusion should be made to the theory of ZeuthenDie Lehre von den Kegelschnitten im Altertum, chapter VIII. on the subject of the porisms. He observes that the only porism of which Pappus gives the complete enunciation, If from two given points straight lines be drawn meeting on a straight line given in position, and one cut off from a straight line given in position (a segment measured) towards a given point on it, the other will also cut off from another (straight line a segment) bearing to the first a given ratio, @@ -220,8 +220,8 @@ and that this extended porism can be used for completing Apollonius' exposition of that locus. Zeuthen concludes that the Porisms were in part byproducts of the theory of conics and in part auxiliary means for the study of conics, and that Euclid called them by the same name as that applied to corollaries because they were corollaries with respect to conics. But there appears to be no evidence to confirm this conjecture.

5. The Surface-loci (to/poi pro\s e)pifanei/a|).

The two books on this subject are mentioned by Pappus as part of the Treasury of AnalysisPappus, VII. p. 636.. As the other works in the list which were on plane subjects dealt only with straight lines, circles, and conic sections, it is a priori likely that among the loci in this treatise (loci which are surfaces) were included such loci as were cones, cylinders and spheres. Beyond this all is conjecture based on two lemmas given by Pappus in connexion with the treatise.

-

(1) The first of these lemmasibid. VII. p. 1004. and the figure attached to it are not satisfactory as they stand, but a possible restoration is indicated by TanneryBulletin des sciences math. et astron., 2^{o} Série, VI. 149.. If the latter is right, it suggests that one of the loci contained all the points on the elliptical parallel sections of a cylinder and was therefore an oblique circular cylinder. Other assumptions with regard to the conditions to which the lines in the figure may be subject would suggest that other loci dealt with were cones regarded as containing all points on particular elliptical parallel sections of the conesFurther particulars will be found in The Works of Archimedes, pp. lxii—lxiv, and in Zeuthen, Die Lehre von den Kegelschnitten, p. 425 sqq..

-

(2) In the second lemma Pappus states and gives a complete proof of the focus-and-directrix property of a conic, viz. that the locus of a point whose distance from a given point is in a given ratio to its distance from a fixed line is a conic section, which is an ellipse, a parabola or a hyperbola according as the given ratio is less than, equal to, or greater than unityPappus, VII. pp. 1006-1014, and Hultsch's Appendix, pp. 1270-3.. Two conjectures are possible as to the application of this theorem in Euclid's Surface-loci. (a) It may have been used to prove that the locus of a point whose distance from a given straight line is in a given ratio to its distance from a given plane is a certain cone. (b) It may have been used to prove that the locus of a point whose distance from a given point is in a given ratio to its distance from a given plane is the surface formed by the revolution of a conic about its major or conjugate axisFor further details see The Works of Archimedes, pp. lxiv, lxv, and Zeuthen, l. c.. Thus Chasles may have been correct in his conjecture that the Surface-loci dealt with surfaces of revolution of the second degree and sections of the sameAperçu historique, pp. 273-4..

+

(1) The first of these lemmasibid. VII. p. 1004. and the figure attached to it are not satisfactory as they stand, but a possible restoration is indicated by TanneryBulletin des sciences math. et astron., 2^{o} Série, VI. 149.. If the latter is right, it suggests that one of the loci contained all the points on the elliptical parallel sections of a cylinder and was therefore an oblique circular cylinder. Other assumptions with regard to the conditions to which the lines in the figure may be subject would suggest that other loci dealt with were cones regarded as containing all points on particular elliptical parallel sections of the conesFurther particulars will be found in The Works of Archimedes, pp. lxii—lxiv, and in Zeuthen, Die Lehre von den Kegelschnitten, p. 425 sqq..

+

(2) In the second lemma Pappus states and gives a complete proof of the focus-and-directrix property of a conic, viz. that the locus of a point whose distance from a given point is in a given ratio to its distance from a fixed line is a conic section, which is an ellipse, a parabola or a hyperbola according as the given ratio is less than, equal to, or greater than unityPappus, VII. pp. 1006-1014, and Hultsch's Appendix, pp. 1270-3.. Two conjectures are possible as to the application of this theorem in Euclid's Surface-loci. (a) It may have been used to prove that the locus of a point whose distance from a given straight line is in a given ratio to its distance from a given plane is a certain cone. (b) It may have been used to prove that the locus of a point whose distance from a given point is in a given ratio to its distance from a given plane is the surface formed by the revolution of a conic about its major or conjugate axisFor further details see The Works of Archimedes, pp. lxiv, lxv, and Zeuthen, l. c.. Thus Chasles may have been correct in his conjecture that the Surface-loci dealt with surfaces of revolution of the second degree and sections of the sameAperçu historique, pp. 273-4..

6. The Conics.

Pappus says of this lost work: The four books of Euclid's Conics were completed by Apollonius, who added four more and gave us eight books of Conics Pappus, VII. p. 672.. @@ -233,11 +233,11 @@

Euclid still used the old names for the conics (sections of a rightangled, acute-angled, or obtuse-angled cone), but he was aware that an ellipse could be obtained by cutting a cone in any manner by a plane not parallel to the base (assuming the section to lie wholly between the apex of the cone and its base) and also by cutting a cylinder. This is expressly stated in a passage from the Phaenomena of Euclid about to be mentionedPhaenomena, ed. Menge, p. 6: If a cone or a cylinder be cut by a plane not parallel to the base, the section is a section of an acute-angled cone, which is like a shield (qureo/s). .

7. The Phaenomena.

-

This is an astronomical work and is still extant. A much interpolated version appears in Gregory's Euclid. An earlier and better recension is however contained in the MS. Vindobonensis philos. Gr. 103, though the end of the treatise, from the middle of prop. 16 to the last (18), is missing. The book, now edited by MengeEuclidis opera omnia, vol. VIII., 1916, pp. 2-156.,consists of propositions in spheric geometry. Euclid based it on Autolycus' work peri\ kinoume/nhs sfai/ras, but also, evidently, on an earlier textbook of Sphaerica of exclusively mathematical content. It has been conjectured that the latter textbook may have been due to EudoxusHeiberg, Euklid-Studien, p. 46; Hultsch, Autolycus, p. XII; A. A. Björnbo, Studien über Menelaos' Sphärik (Abhandlungen zur Geschichte der mathematischen Wissenschaften, XIV. 1902), p. 56sqq..

+

This is an astronomical work and is still extant. A much interpolated version appears in Gregory's Euclid. An earlier and better recension is however contained in the MS. Vindobonensis philos. Gr. 103, though the end of the treatise, from the middle of prop. 16 to the last (18), is missing. The book, now edited by MengeEuclidis opera omnia, vol. VIII., 1916, pp. 2-156.,consists of propositions in spheric geometry. Euclid based it on Autolycus' work peri\ kinoume/nhs sfai/ras, but also, evidently, on an earlier textbook of Sphaerica of exclusively mathematical content. It has been conjectured that the latter textbook may have been due to EudoxusHeiberg, Euklid-Studien, p. 46; Hultsch, Autolycus, p. XII; A. A. Björnbo, Studien über Menelaos' Sphärik (Abhandlungen zur Geschichte der mathematischen Wissenschaften, XIV. 1902), p. 56sqq..

8. The Optics.

This book needs no description, as it has been edited by Heiberg recentlyEuclidis opera omnia, vol. VII. (1895).,both in its genuine form and in the recension by Theon. The Catoptrica published by Heiberg in the same volume is not genuine, and Heiberg suspects that in its present form it may be Theon's. It is not even certain that Euclid wrote Catoptrica at all, as Proclus may easily have had Theon's work before him and inadvertently assigned it to EuclidHeiberg, Euclid's Optics, etc. p. l..

9. Besides the above-mentioned works, Euclid is said to have written the Elements of MusicProclus, p. 69, 3. (ai( kata\ mousikh\n stoixeiw/seis). Two treatises are attributed to Euclid in our MSS. of the Musici, the katatomh\ kano/nos, Sectio canonis (the theory of the intervals), and the ei)sagwgh\ a(rmonikh/ (introduction to harmony)Both treatises edited by Jan in Musici Scriptores Graeci, 1895. pp. 113-166, 167-207, and by Menge in Euclidis opera omnia, vol. VIII., 1916, pp. 157-183, 185-223..The first, resting on the Pythagorean theory of music, is mathematical, and the style and diction as well as the form of the propositions mostly agree with what we find in the Elements. Jan thought it genuine, especially as almost the whole of the treatise (except the preface) is quoted in extenso, and Euclid is twice mentioned by name, in the commentary on Ptolemy's Harmonica published by Wallis and attributed by him to Porphyry. Tannery was of the opposite opinionComptes rendus de l' Acad. des inscriptions et belles-lettres, Paris, 1904, pp. 439-445. Cf. Bibliotheca Mathematica, vI3, 1905-6, p. 225, note I..The latest editor, Menge, suggests that it may be a redaction by a less competent hand from the genuine Euclidean Elements of Music. The second treatise is not Euclid's, but was written by Cleonides, a pupil of AristoxenusHeiberg, Euklid-Studien, pp. 52-55; Jan, Musici Scriptores Graeci, pp. 169-174..

-

Lastly, it is worth while to give the Arabians' list of Euclid's works. I take this from Suter's translation of the list of philosophers and mathematicians in the Fihrist, the oldest authority of the kind that we possessH. Suter, Das Mathematiker-Verzeichniss im Fihrist in Abhandlungen zur Geschichte der Mathematik, VI., 1892, pp. 1-87 (see especially p. 17). Cf. Casiri, 1. 339, 340, and Gartz, De interpretibus et explanatoribus Euclidis Arabicis, 1823, pp. 4, 5..To the writings of Euclid belong further [in addition to the Elements]: the book of Phaenomena; the book of Given Magnitudes [Data]; the book of Tones, known under the name of Music, not genuine; the book of Division, emended by Thābit; the book of Utilisations or Applications [Porisms], not genuine; the book of the Canon; the book of the Heavy and Light; the book of Synthesis, not genuine; and the book of Analysis, not genuine. +

Lastly, it is worth while to give the Arabians' list of Euclid's works. I take this from Suter's translation of the list of philosophers and mathematicians in the Fihrist, the oldest authority of the kind that we possessH. Suter, Das Mathematiker-Verzeichniss im Fihrist in Abhandlungen zur Geschichte der Mathematik, VI., 1892, pp. 1-87 (see especially p. 17). Cf. Casiri, 1. 339, 340, and Gartz, De interpretibus et explanatoribus Euclidis Arabicis, 1823, pp. 4, 5..To the writings of Euclid belong further [in addition to the Elements]: the book of Phaenomena; the book of Given Magnitudes [Data]; the book of Tones, known under the name of Music, not genuine; the book of Division, emended by Thābit; the book of Utilisations or Applications [Porisms], not genuine; the book of the Canon; the book of the Heavy and Light; the book of Synthesis, not genuine; and the book of Analysis, not genuine.

It is to be observed that the Arabs already regarded the book of Tones (by which must be meant the ei)sagwgh\ a(rmonikh/) as spurious. The book of Division is evidently the book on Divisions (of figures). The next book is described by Casiri as liber de utilitate suppositus. Suter gives reason for believing the Porisms to be meantSuter, op. cit. pp. 49, 50. Wenrich translated the word as utilia. @@ -250,7 +250,7 @@ CHAPTER III. GREEK COMMENTATORS ON THE ELEMENTS OTHER THAN PROCLUS.

That there was no lack of commentaries on the Elements before the time of Proclus is evident from the terms in which Proclus refers to them; and he leaves us in equally little doubt as to the value which, in his opinion, the generality of them possessed. Thus he says in one place (at the end of his second prologue)Proclus, p. 84, 8.:

-

Before making a beginning with the investigation of details, I warn those who may read me not to expect from me the things which have been dinned into our ears ad nauseam (diateqru/lhtai) by those who have preceded me, viz. lemmas, cases, and so forth. For I am surfeited with these things and shall give little attention to them. But I shall direct my remarks principally to the points which require deeper study and contribute to the sum of philosophy, therein emulating the Pythagoreans who even had this common phrase for what I mean ’a figure and a platform, but not a figure and sixpencei.e. we reach a certain height, use the platform so attained as a base on which to build another stage, then use that as a base and so on..’ +

Before making a beginning with the investigation of details, I warn those who may read me not to expect from me the things which have been dinned into our ears ad nauseam (diateqru/lhtai) by those who have preceded me, viz. lemmas, cases, and so forth. For I am surfeited with these things and shall give little attention to them. But I shall direct my remarks principally to the points which require deeper study and contribute to the sum of philosophy, therein emulating the Pythagoreans who even had this common phrase for what I mean ’a figure and a platform, but not a figure and sixpencei.e. we reach a certain height, use the platform so attained as a base on which to build another stage, then use that as a base and so on..’

In another placeProclus, p. 200, 10. he says: Let us now turn to the elucidation of the things proved by the writer of the Elements, selecting the more subtle of the comments made on them by the ancient writers, while cutting down their interminable diffuseness, giving the things which are more systematic and follow scientific methods, attaching more importance to the working-out of the real subject-matter than to the variety of cases and lemmas to which we see recent writers devoting themselves for the most part.

@@ -262,7 +262,7 @@

Proclus alludes to Heron twice as Heron mechanicusProclus, p. 305, 24; p. 346, 13., in another placeibid. p. 41, 10. he associates him with Ctesibius, and in the three other passagesibid. p. 196, 16; p. 323, 7; p. 429, 13. where Heron is mentioned there is no reason to doubt that the same person is meant, namely Heron of Alexandria. The date of Heron is still a vexed question. In the early stages of the controversy much was made of the supposed relation of Heron to Ctesibius. The best MS. of Heron's Belopoeica has the heading *hrwnos *kthsi*bi/ou belopoii+ka/, and an anonymous Byzantine writer of the tenth century, evidently basing himself on this title, speaks of Ctesibius as Heron's kaqhghth/s, master or teacher. - We know of two men of the name of Ctesibius. One was a barber who lived in the time of Ptolemy Euergetes II, i.e. Ptolemy VII, called Physcon (died 117 B.C.), and who is said to have made an improved water-organAthenaeus, Deipno-Soph. iv., c. 75, p. 174 b—c.. The other was a mechanician mentioned by Athenaeus as having made an elegant drinking-horn in the time of Ptolemy Philadelphus (285-247 B.C.)ibid. XI., c. 97, p. 497 b—c.. MartinMartin, Recherches sur la vie et les ouvrages d'Héron d'Alexandrie, Paris, 1854, p. 27. took the Ctesibius in question to be the former and accordingly placed Heron at the beginning of the first century B.C., say 126-50 B.C. But Philo of ByzantiumPhilo, Mechan. Synt., p. 50, 38, ed. Schöne., who repeatedly mentions Ctesibius by name, says that the first mechanicians had the advantage of being under kings who loved fame and supported the arts. Hence our Ctesibius is more likely to have been the earlier Ctesibius who was contemporary with Ptolemy II Philadelphus.

+ We know of two men of the name of Ctesibius. One was a barber who lived in the time of Ptolemy Euergetes II, i.e. Ptolemy VII, called Physcon (died 117 B.C.), and who is said to have made an improved water-organAthenaeus, Deipno-Soph. iv., c. 75, p. 174 b—c.. The other was a mechanician mentioned by Athenaeus as having made an elegant drinking-horn in the time of Ptolemy Philadelphus (285-247 B.C.)ibid. XI., c. 97, p. 497 b—c.. MartinMartin, Recherches sur la vie et les ouvrages d'Héron d'Alexandrie, Paris, 1854, p. 27. took the Ctesibius in question to be the former and accordingly placed Heron at the beginning of the first century B.C., say 126-50 B.C. But Philo of ByzantiumPhilo, Mechan. Synt., p. 50, 38, ed. Schöne., who repeatedly mentions Ctesibius by name, says that the first mechanicians had the advantage of being under kings who loved fame and supported the arts. Hence our Ctesibius is more likely to have been the earlier Ctesibius who was contemporary with Ptolemy II Philadelphus.

But, whatever be the date of Ctesibius, we cannot safely conclude that Heron was his immediate pupil. The title Heron's (edition of) Ctesibius's Belopoeica does not, in fact, justify any inferenee as to the interval of time between the two works.

We now have better evidence for a terminus post quem. The Metrica of Heron, besides quoting Archimedes and Apollonius, twice refers to the books about straight lines (chords) in a circle @@ -271,17 +271,17 @@ : and, even if this Posidonius lived before Archimedes, as the context seems to imply, it is certain that another work of Heron's, the Definitions, owes something to Posidonius of Apamea or Rhodes, Cicero's teacher (135-51 B.C.). This brings Heron's date down to the end of the first century B.C., at least.

We have next to consider the relation, if any, between Heron and Vitruvius. In his De Architectura, brought out apparently in 14 B.C., Vitruvius quotes twelve authorities on machinationes including Archytas (second), Archimedes (third), Ctesibius (fourth) and Philo of Byzantium (sixth), but does not mention Heron. Nor is it possible to establish inter-dependence between Vitruvius and Heron; the differences between them seem on the whole more numerous and important than the resemblances (e.g. Vitruvius uses 3 as the value of p, while Heron always uses the Archimedean value 3 1/7). The inference is that Heron can hardly have written earlier than the first century A.D.

The most recent theory of Heron's date makes him later than Claudius Ptolemy the astronomer (100-178 A.D.). The arguments are mainly these. (1) Ptolemy claims as a discovery of his own a method of measuring the distance between two places (as an arc of a great circle on the earth's surface) in the case where the places are neither on the same meridian nor on the same parallel circle. Heron, in his Dioptra, speaks of this method as of a thing generally known to experts. (2) The dioptra described in Heron's work is a fine and accurate instrument, much better than anything Ptolemy had at his disposal. (3) Ptolemy, in his work *peri\ r(opw=n, asserted that water with water round it has no weight and that the diver, however deep he dives, does not feel the weight of the water above him. Heron, strangely enough, accepts as true what Ptolemy says of the diver, but is dissatisfied with the explanation given by some, - namely that it is because water is uniformly heavy—this seems to be equivalent to Ptolemy's dictum that water in water has no weight—and he essays a different explanation based on Archimedes. (4) It is suggested that the Dionysius to whom Heron dedicated his Definitions is a certain Dionysius who was praefectus urbi in 301 A.D.

+ namely that it is because water is uniformly heavy—this seems to be equivalent to Ptolemy's dictum that water in water has no weight—and he essays a different explanation based on Archimedes. (4) It is suggested that the Dionysius to whom Heron dedicated his Definitions is a certain Dionysius who was praefectus urbi in 301 A.D.

On the other hand Heron was earlier than Pappus, who was writing under Diocletian (284-305 A.D.), for Pappus alludes to and draws upon the works of Heron. The net result, then, of the most recent research is to place Heron in the third century A.D. and perhaps little earlier than Pappus. HeibergHeronis Alexandrini opera, vol. v. (Teubner, 1914), p. IX. accepts this conclusion, which may therefore, perhaps, be said to hold the field for the present.Fuller details of the various arguments will be found in my History of Greek Mathematics, 1921, vol. II., pp. 298-306..

-

That Heron wrote a systematic commentary on the Elements might be inferred from Proclus, but it is rendered quite certain by references to the commentary in Arabian writers, and particularly in an-Nairīzī's commentary on the first ten Books of the Elements. The Fihrist says, under Euclid, that Heron wrote a commentary on this book [the Elements], endeavouring to solve its difficultiesDas Mathematiker- Verzeichniss im Fihrist (tr. Suter), p. 16. +

That Heron wrote a systematic commentary on the Elements might be inferred from Proclus, but it is rendered quite certain by references to the commentary in Arabian writers, and particularly in an-Nairīzī's commentary on the first ten Books of the Elements. The Fihrist says, under Euclid, that Heron wrote a commentary on this book [the Elements], endeavouring to solve its difficultiesDas Mathematiker- Verzeichniss im Fihrist (tr. Suter), p. 16. ; and under Heron, He wrote: the book of explanation of the obscurities in Euclidibid. p. 22..... - An-Nairīzī's commentary quotes Heron by name very frequently, and often in such a way as to leave no doubt that the author had Heron's work actually before him. Thus the extracts are given in the first person, introduced by Heron says + An-Nairīzī's commentary quotes Heron by name very frequently, and often in such a way as to leave no doubt that the author had Heron's work actually before him. Thus the extracts are given in the first person, introduced by Heron says (Dixit Yrinus or Heron ); and in other places we are told that Heron says nothing, or is not found to have said anything, - on such and such a proposition. The commentary of an-Nairīzī is in part edited by Besthorn and Heiberg from a Leiden MS. of the translation of the Elements by al-Hajjāj with the commentary attachedCodex Leidensis 399, 1. Euclidis Elementa ex interprctatione al-Hadschdschadschii cum commentariis al-Narizii. Five parts carrying the work to the end of Book iv. were issued in 1893, 1897, 1900, 1905 and 1910 respectively.. But this MS. only contains six Books, and several pages in the first Book, which contain the comments of Simplicius on the first twenty-two definitions of the first Book, are missing. Fortunately the commentary of an-Nairīzī has been discovered in a more complete form, in a Latin translation by Gherardus Cremonensis of the twelfth century, which contains the missing comments by Simplicius and an-Nairīzī's comments on the first ten Books. This valuable work has recently been edited by CurtzeAnaritii in decem libros priores elementorum Euclidis commentarii ex interpretatione Gherardi Cremonensis...edidit Maximilianus Curtze (Teubner, Leipzig, 1899)..

-

Thus from the three sources, Proclus, and the two versions of an-Nairīzī, which supplement one another, we are able to form a very good idea of the character of Heron's commentary. In some cases observations given by Proclus without the name of their author are seen from an-Nairīzī to be Heron's; in a few cases notes attributed by Proclus to Heron are found in an-Nairīzī without Heron's name; and, curiously enough, one alternative proof (of I. 25) given as Heron's by Proclus is introduced by the Arab with the remark that he has not been able to discover who is the author.

+ on such and such a proposition. The commentary of an-Nairīzī is in part edited by Besthorn and Heiberg from a Leiden MS. of the translation of the Elements by al-Hajjāj with the commentary attachedCodex Leidensis 399, 1. Euclidis Elementa ex interprctatione al-Hadschdschadschii cum commentariis al-Narizii. Five parts carrying the work to the end of Book iv. were issued in 1893, 1897, 1900, 1905 and 1910 respectively.. But this MS. only contains six Books, and several pages in the first Book, which contain the comments of Simplicius on the first twenty-two definitions of the first Book, are missing. Fortunately the commentary of an-Nairīzī has been discovered in a more complete form, in a Latin translation by Gherardus Cremonensis of the twelfth century, which contains the missing comments by Simplicius and an-Nairīzī's comments on the first ten Books. This valuable work has recently been edited by CurtzeAnaritii in decem libros priores elementorum Euclidis commentarii ex interpretatione Gherardi Cremonensis...edidit Maximilianus Curtze (Teubner, Leipzig, 1899)..

+

Thus from the three sources, Proclus, and the two versions of an-Nairīzī, which supplement one another, we are able to form a very good idea of the character of Heron's commentary. In some cases observations given by Proclus without the name of their author are seen from an-Nairīzī to be Heron's; in a few cases notes attributed by Proclus to Heron are found in an-Nairīzī without Heron's name; and, curiously enough, one alternative proof (of I. 25) given as Heron's by Proclus is introduced by the Arab with the remark that he has not been able to discover who is the author.

Speaking generally, the comments of Heron do not seem to have contained much that can be called important. We find

(1) A few general notes, e.g. that Heron would not admit more than three axioms.

(2) Distinctions of a number of particular cases of Euclid's propositions according as the figure is drawn in one way or in another.

@@ -291,7 +291,7 @@

(4) Heron supplies certain converses of Euclid's propositions, e.g. converses of II. 12, 13, VIII. 27.

(5) A few additions to, and extensions of, Euclid's propositions are also found. Some are unimportant, e.g. the construction of isosceles and scalene triangles in a note on I. 1, the construction of two tangents in III. 17, the remark that VII. 3 about finding the greatest common measure of three numbers can be applied to as many numbers as we please (as Euclid tacitly assumes in VII. 31). The most important extension is that of III. 20 to the case where the angle at the circumference is greater than a right angle, and the direct deduction from this extension of the result of III. 22. Interesting also are the notes on I. 37 (on I. 24 in Proclus), where Heron proves that two triangles with two sides of one equal to two sides of the other and with the included angles supplementary are equal, and compares the areas where the sum of the two included angles (one being supposed greater than the other) is less or greater than two right angles, and on I. 47, where there is a proof (depending on preliminary lemmas) of the fact that, in the figure of the proposition, the straight lines AL, BK, CF meet in a point. After iv. 16 there is a proof that, in a regular polygon with an even number of sides, the bisector of one angle also bisects its opposite, and an enunciation of the corresponding proposition for a regular polygon with an odd number of sides.

Van PeschDe Procli fontibus, Lugduni-Batavorum, 1900, gives reason for attributing to Heron certain other notes found in Proclus, viz. that they are designed to meet the same sort of points as Heron had in view in other notes undoubtedly written by him. These are (a) alternative proofs of I. 5, I. 17, and I. 32, which avoid the producing of certain straight lines, (b) an alternative proof of 1.9 avoiding the construction of the equilateral triangle on the side of BC opposite to A; (c) partial converses of I. 35-38, starting from the equality of the areas and the fact of the parallelograms or triangles being in the same parallels, and proving that the bases are the same or equal, may also be Heron's. Van Pesch further supposes that it was in Heron's commentary that the proof by Menelaus of I. 25 and the proof by Philo of I. 8 were given.

-

The last reference to Heron made by an-Nairīzī occurs in the note on VIII. 27, so that the commentary of the former must at least have reached that point.

+

The last reference to Heron made by an-Nairīzī occurs in the note on VIII. 27, so that the commentary of the former must at least have reached that point.

II. Porphyry.

The Porphyry here mentioned is of course the Neo-Platonist who lived about 232-304 A.D. Whether he really wrote a systematic commentary on the Elements is uncertain. The passages in Proclus which seem to make this probable are two in which he mentions him (1) as having demonstrated the necessity of the words not on the same side in the enunciation of I. 14Proclus, pp. 297, 1-298, 10., and (2) as having pointed out the necessity of understanding correctly the enunciation of I. 26, since, if the particular injunctions as to the sides of the triangles to be taken as equal are not regarded, the student may easily fall into erroribid. p. 352, 13, 14 and the pages preceding,. These passages, showing that Porphyry carefully anaiysed Euclid's enunciations in these cases, certainly suggest that his remarks were part of a systematic commentary. Further, the list of mathematicians in the Fihrist gives Porphyry as having written a book on the Elements. @@ -303,8 +303,8 @@

Two other references to Porphyry found in Proclus cannot have anything to do with commentaries on the Elements. In the first a work called the *summikta/ is quoted, while in the second a philosophical question is raised.

III. Pappus.

The references to Pappus in Proclus are not numerous; but we have other evidence that he wrote a commentary on the Elements. Thus a scholiast on the definitions of the Data uses the phrase as Pappus says at the beginning of his (commentary) on the 10th (book) of EuclidEuclid's Data, ed. Menge, p. 262.. - Again in the Fihrist we are told that Pappus wrote a commentary to the tenth book of Euclid in two partsFihrist (tr. Suter), p. 22.. Fragments of this still survive in a MS. described by WoepckeMémoirés présentés à l'académie des sciences, 1856, XIV. pp. 658-719., Paris. No. 952. 2 (supplément arabe de la Bibliothèque impériale), which contains a translation by Abū `Uthmān (beginning of 10th century) of a Greek commentary on Book X. It is in two books, and there can now be no doubt that the author of the Greek commentary was PappusWoepcke read the name of the author, in the title of the first book, as B. los (the dot representing a missing vowel). He quotes also from other MSS. (e.g. of the Ta)rīkh alHukamā and of the Fihrist) where he reads the name of the commentator as B. lis, B.n.s or B.l.s. Woepcke takes this author to be Valens, and thinks it possible that he may be the same as the astrologer Vettius Valens. This Heiberg (Euklid-Studien, pp. 169, 170) proves to be impossible, because, while one of the mss. quoted by Woepcke says that B.n.s, le RoÛmi - (late-Greek) was later than Claudius Ptolemy and the Fihrist says B.l.s, le RoÛmi + Again in the Fihrist we are told that Pappus wrote a commentary to the tenth book of Euclid in two partsFihrist (tr. Suter), p. 22.. Fragments of this still survive in a MS. described by WoepckeMémoirés présentés à l'académie des sciences, 1856, XIV. pp. 658-719., Paris. No. 952. 2 (supplément arabe de la Bibliothèque impériale), which contains a translation by Abū `Uthmān (beginning of 10th century) of a Greek commentary on Book X. It is in two books, and there can now be no doubt that the author of the Greek commentary was PappusWoepcke read the name of the author, in the title of the first book, as B. los (the dot representing a missing vowel). He quotes also from other MSS. (e.g. of the Ta)rīkh alHukamā and of the Fihrist) where he reads the name of the commentator as B. lis, B.n.s or B.l.s. Woepcke takes this author to be Valens, and thinks it possible that he may be the same as the astrologer Vettius Valens. This Heiberg (Euklid-Studien, pp. 169, 170) proves to be impossible, because, while one of the mss. quoted by Woepcke says that B.n.s, le RoÛmi + (late-Greek) was later than Claudius Ptolemy and the Fihrist says B.l.s, le RoÛmi wrote a commentary on Ptolemy's Planisphaerium, Vettius Valens seems to have lived under Hadrian, and must therefore have been an eldercontemporary of Ptolemy. But Suter shows (Fihrist, p. 22 and p. 54, note 92) that Banos is only distinguished from Babos by the position of a certain dot, and Balos may also easily have arisen from an original Babos (there is no P in Arabic), so that Pappus must be the person meant. This is further confirmed by the fact that the Fihrist gives this author and Valens as the subjects of two separate paragraphs, attributing to the latter astrological works only.. Again Eutocius, in his note on Archimedes, On the Sphere and Cylinder I. 13, says that Pappus explained in his commentary on the Elements how to inscribe in a circle a polygon similar to a polygon inscribed in another circle; and this would presumably come in his commentary on Book XII., just as the problem is solved in the second scholium on Eucl. XII. I. Thus Pappus' commentary on the Elements must have been pretty complete, an additional confirmation of this supposition being forthcoming in the reference of Marinus (a pupil and follower of Proclus) in his preface to the Data to the commentaries of Pappus on the bookHeiberg, Euklid-Studien, p. 173; Euclid's Data, edd. Menge, pp. 256, lii..

The actual references to Pappus in Proclus are as follows:

@@ -322,7 +322,7 @@ rectilineal angles savour of Pappus.

2. On 1. 9 Proclus saysProclus, p. 272, 10. that Others, starting from the Archimedean spirals, divided any given rectilineal angle in any given ratio. We cannot but compare this with Pappus iv. p. 286, where the spiral is so used; hence this note, including remarks immediately preceding about the conchoid and the quadratrix, which were used for the same purpose, may very well be due to Pappus.

-

3. The subject of isoperimetric figures was a favourite one with Pappus, who wrote a recension of Zenodorus' treatise on the subjectPappus, v. pp. 304-350; for Zenodorus' own treatise see Hultsch's Appendix, pp. 1189 —1211.. Now on I. 35 Proclus speaksProclus, pp. 396-8. about the paradox of parallelograms having equal area (between the same parallels) though the two sides between the parallels may be of any length, adding that of parallelograms with equal perimeter the rectangle is greatest if the base be given, and the square greatest if the base be not given etc. He returns to the subject on 1. 37 about trianglesibid. pp. 403-4.. Compareibid. pp. 236-7. also his note on 1. 4. These notes may have been taken from Pappus.

+

3. The subject of isoperimetric figures was a favourite one with Pappus, who wrote a recension of Zenodorus' treatise on the subjectPappus, v. pp. 304-350; for Zenodorus' own treatise see Hultsch's Appendix, pp. 1189 —1211.. Now on I. 35 Proclus speaksProclus, pp. 396-8. about the paradox of parallelograms having equal area (between the same parallels) though the two sides between the parallels may be of any length, adding that of parallelograms with equal perimeter the rectangle is greatest if the base be given, and the square greatest if the base be not given etc. He returns to the subject on 1. 37 about trianglesibid. pp. 403-4.. Compareibid. pp. 236-7. also his note on 1. 4. These notes may have been taken from Pappus.

4. Again, on 1. 21, Proclus remarks on the paradox that straight lines may be drawn from the base to a point within a triangle which are (1) together greater than the two sides, and (2) include a less angle, provided that the straight lines may be drawn from points in the base other than its extremities. The subject of straight lines satisfying condition (1) was treated at length, with reference to a variety of cases, by PappusPappus, 111. pp. 104-130., after a collection of paradoxes by Erycinus, of whom nothing more is known. Proclus gives Pappus' first case, and adds a rather useless proof of the possibility of drawing straight lines satisfying condition (2) alone, adding that the proposition stated has been proved by me without using the parallels of the commentatorsProclus, p. 328, 15.. By the commentators @@ -332,8 +332,8 @@ Proclus, p. 165, 24; cf. pp. 328, 329. is mentioned in the notes on 1. Def. 24-29 and I. 21. As Pappus wrote on Zenodorus' work in which the term occurredSee Pappus, ed. Hultsch, pp. 1154, 1206., Pappus may be responsible for these notes.

IV. Simplicius.

According to the FihristFihrist (tr. Suter), p. 21., Simplicius the Greek wrote a commentary to the beginning of Euclid's book, which forms an introduction to geometry. - And in fact this commentary on the definitions, postulates and axioms (including the postulate known as the ParallelAxiom) is preserved in the Arabic commentary of an-NairĩzĩAn-Nairĩzĩ, ed. Besthorn-Heiberg, pp. 9-41, 119-133, ed. Curtze, pp. 1-37, 65-73. The Codex Leidensis, from which Besthorn and Heiberg's edition is taken, has unfortunately lost some leaves, so that there is a gap from Def. 1 to Def. 23 (parallels). The loss is, however, made good by Curtze's edition of the translation by Gherard of Cremona.. On two subjects this commentary of Simplicius quotes a certain .Aganis, - the first subject being the definition of an angle, and the second the definition of parallels and the parallel-postulate. Simplicius gives word for word, in a long passage placed by an-Nairīzī after 1. 29, an attempt by Aganis + And in fact this commentary on the definitions, postulates and axioms (including the postulate known as the ParallelAxiom) is preserved in the Arabic commentary of an-NairĩzĩAn-Nairĩzĩ, ed. Besthorn-Heiberg, pp. 9-41, 119-133, ed. Curtze, pp. 1-37, 65-73. The Codex Leidensis, from which Besthorn and Heiberg's edition is taken, has unfortunately lost some leaves, so that there is a gap from Def. 1 to Def. 23 (parallels). The loss is, however, made good by Curtze's edition of the translation by Gherard of Cremona.. On two subjects this commentary of Simplicius quotes a certain .Aganis, + the first subject being the definition of an angle, and the second the definition of parallels and the parallel-postulate. Simplicius gives word for word, in a long passage placed by an-Nairīzī after 1. 29, an attempt by Aganis to prove the parallel-postulate. It starts from a definition of parallels which agrees with Geminus' view of them as given by ProclusProclus, p. 177, 21., and is closely connected with the definition given by Posidoniusibid. p. 176, 7.. Hence it has been assumed that Aganis is none other than Geminus, and the historical importance of the commentary of Simplicius has been judged accordingly. But it has been recently shown by Tannery that the identification of Aganis with Geminus is practically impossibleBibliotheca Mathematica, 113, 1900, pp. 9-11. In the translation of Besthorn-Heiberg Aganis is called by Simplicius in one place philosophus Aganis, @@ -341,7 +341,7 @@ in Gherard's version he is socius Aganis and socius noster Aganis. These expressions seem to leave no doubt that Aganis was a contemporary and friend, if not master, of Simplicius; and it is impossible to suppose that Simplicius (fl. about 500 A.D.) could have used them of a man who lived four and a half centuries before his time. A phrase in Simplicius' word-forword quotation from Aganis leads to the same conclusion. He speaks of people who objected even in ancient times - (iam antiquitus) to the use by geometers of this postulate. This would not have been an appropriate phrase had Geminus been the writer. I do not think that this difficulty can he got over by Suter's suggestionZeitschrift für Math. u. Physik, XLIV., hist.-litt. Abth. p. 61. that the passages in question may have been taken out of Heron's commentary, and that an-Nairĩzĩ may have forgotten to name the author; it seems clear that Simplicius is the person who described Aganis. + (iam antiquitus) to the use by geometers of this postulate. This would not have been an appropriate phrase had Geminus been the writer. I do not think that this difficulty can he got over by Suter's suggestionZeitschrift für Math. u. Physik, XLIV., hist.-litt. Abth. p. 61. that the passages in question may have been taken out of Heron's commentary, and that an-Nairĩzĩ may have forgotten to name the author; it seems clear that Simplicius is the person who described Aganis. Hence we are driven to suppose that Aganis was not Geminus, but some unknown contemporary of SimpliciusThe above argument seems to me quite insuperable. The other arguments of Tannery do not, however, carry conviction to my mind. I do not follow the reasoning based on Aganis' definition of an angle. It appears to me a pure assumption that Geminus would have seen that Posidonius' definition of parallels was not admissible. Nor does it seem to me to count for much that Proclus, while telling us that Geminus held that the postulate ought to be proved and warned the unwary against hastily concluding that two straight lines approaching one another must necessarily meet (cf. a curve and its asymptote), gives no hint that Geminus did try to prove the postulate. It may well be that Proclus omitted Geminus' proof (if he wrote one) because he preferred Ptolemy's attempt which he gives (pp. 365-7). Considerable interest will however continue to attach to the comments of Simplicius so fortunately preserved.

Proclus tells us that one Aegaeas (? Aenaeas) of Hierapolis wrote an epitome of the ElementsProclus, p. 361, 21.; but we know nothing more of him or of it.

@@ -349,17 +349,17 @@ CHAPTER IV. - PROCLUS AND HIS SOURCESMy task in this chapter is made easy by the appearance, in the nick of time, of the dissertation De Procli fontibus by J. G. van Pesch (Lugduni-Batavorum, Apud L. van Nifterik, mdcccc). The chapters dealing directly with the subject show a thorough acquaintance on the part of the author with all the literature bearing on it; he covers the whole field and he exercises a sound and sober judgment in forming his conclusions. The same cannot always be said of his only predecessor in the same inquiry, Tannery (in La Géométrie grecque, 1887), who often robs his speculations of much of their value through his proneness to run away with an idea; he does so in this case, basing most of his conclusions on an arbitrary and unwarranted assumption as to the significance of the words oi( peri/ tina (e.g. *h(/rwna, *poseidw/nion etc.) as used in Proclus.. + PROCLUS AND HIS SOURCESMy task in this chapter is made easy by the appearance, in the nick of time, of the dissertation De Procli fontibus by J. G. van Pesch (Lugduni-Batavorum, Apud L. van Nifterik, mdcccc). The chapters dealing directly with the subject show a thorough acquaintance on the part of the author with all the literature bearing on it; he covers the whole field and he exercises a sound and sober judgment in forming his conclusions. The same cannot always be said of his only predecessor in the same inquiry, Tannery (in La Géométrie grecque, 1887), who often robs his speculations of much of their value through his proneness to run away with an idea; he does so in this case, basing most of his conclusions on an arbitrary and unwarranted assumption as to the significance of the words oi( peri/ tina (e.g. *h(/rwna, *poseidw/nion etc.) as used in Proclus..

It is well known that the commentary of Proclus on Eucl. Book I. is one of the two main sources of information as to the history of Greek geometry which we possess, the other being the Collection of Pappus. They are the more precious because the original works of the forerunners of Euclid, Archimedes and Apollonius are lost, having probably been discarded and forgotten almost immediately after the appearance of the masterpieces of that great trio.

Proclus himself lived 410-485 A.D., so that there had already passed a sufficient amount of time for the tradition relating to the pre-Euclidean geometers to become obscure and defective. In this connexion a passage is quoted from SimpliciusSimplicius on Aristotle's Physics, ed. Diels, pp. 54-69. who, in his account of the quadrature of certain lunes by Hippocrates of Chios, while mentioning two authorities for his statements, Alexander Aphrodisiensis (about 220 A.D.) and Eudemus, says in one placeibid. p. 68, 32., As regards Hippocrates of Chios we must pay more attention to Eudemus, since he was nearer the times, being a pupil of Aristotle.

The importance therefore of a critical examination of Proclus' commentary with a view to determining from what original sources he drew need not be further emphasised.

-

Proclus received his early training in Alexandria, where Olympiodorus was his instructor in the works of Aristotle, and mathematics was taught him by one HeronCf. Martin, Recherches sur la vie et les ouvrages d'Héron d'Alexandrie, pp. 240-2. (of course a different Heron from the mechanicus Hero +

Proclus received his early training in Alexandria, where Olympiodorus was his instructor in the works of Aristotle, and mathematics was taught him by one HeronCf. Martin, Recherches sur la vie et les ouvrages d'Héron d'Alexandrie, pp. 240-2. (of course a different Heron from the mechanicus Hero of whom we have already spoken). He afterwards went to Athens where he was imbued by Plutarch, and by Syrianus, with the Neo-Platonic philosophy, to which he then devoted heart and soul, becoming one of its most prominent exponents. He speaks everywhere with the highest respect of his masters, and was in turn regarded with extravagant veneration by his contemporaries, as we learn from Marinus his pupil and biographer. On the death of Syrianus he was put at the head of the Neo-Platonic school. He was a man of untiring industry, as is shown by the number of books which he wrote, including a large number of commentaries, mostly on the dialogues of Plato. He was an acute dialectician, and pre-eminent among his contemporaries in the range of his learningZeller calls him Der Gelehrte, dem kein Feld damaligen Wissens verschlossen ist. ; he was a competent mathematician; he was even a poet. At the same time he was a believer in all sorts of myths and mysteries and a devout worshipper of divinities both Greek and Oriental.

-

Though he was a competent mathematician, he was evidently much more a philosopher than a mathematicianVan Pesch observes that in his commentaries on the Tïmaeus (pp. 671-2) he speaks as no real mathematician could have spoken. In the passage referred to the question is whether the sun occupies a middle place among the planets. Proclus rejects the view of Hipparchus and Ptolemy because o( qeourgo/s +

Though he was a competent mathematician, he was evidently much more a philosopher than a mathematicianVan Pesch observes that in his commentaries on the Tïmaeus (pp. 671-2) he speaks as no real mathematician could have spoken. In the passage referred to the question is whether the sun occupies a middle place among the planets. Proclus rejects the view of Hipparchus and Ptolemy because o( qeourgo/s (sc. the Chaldean, says Zeller) thinks otherwise, whom it is not lawful to disbelieve. - Martin says rather neatly, Pour Proclus, les Éléments d'Euclide ont l'heureuse chance de n'e=tre contredits ni par les Oracles chaldaïques, ni par les spéculations des pythagoriciens anciens et nouveaux...... + Martin says rather neatly, Pour Proclus, les Éléments d'Euclide ont l'heureuse chance de n'e=tre contredits ni par les Oracles chaldaïques, ni par les spéculations des pythagoriciens anciens et nouveaux...... . This is shown even in his commentary on Eucl. I., where, not only in the Prologues (especially the first), but also in the notes themselves, he seizes any opportunity for a philosophical digression. He says himself that he attaches most importance to the things which require deeper study and contribute to the sum of philosophyProclus, p. 84, 13. ; alternative proofs, cases, and the like (though he gives many) have no attraction for him; and, in particular, he attaches no value to the addition of Heron to 1. 47ibid. p. 429, 12., which is of considerable mathematical interest. Though he esteemed mathematics highly, it was only as a handmaid to philosophy. He quotes Plato's opinion to the effect that mathematics, as making use of hypotheses, falls short of the non-hypothetical and perfect scienceibid. p. 31, 20. ... Let us then not say that Plato excludes mathematics from the sciences, but that he declares it to be secondary to the one supreme scienceibid. p. 32, 2.. @@ -383,12 +383,12 @@

There is in fact no satisfactory evidence that Proclus did actually write any more commentaries than that on Book 1.True, a Vatican Ms. has a collection of scholia on Books I. (extracts from the extant commentary of Proclus), II., V., VI., X. headed *ei)s ta\ *eu)klei/dou stoixei=a prolambano/mena e)k tw=n *pro/klou .spora/dhn kai\ kat) e)pitomh/n. Heiberg holds that this title itself suggests that the authorship of Proclus was limited to the scholia on Book I.; for prolambano/mena e)k tw=n *pro/klou suits extracts from Proclus' prologues, but hardly scholia to later Books. Again, a certain scholium (Heiberg in Hermes XXXVIII., 1903, p. 341, No. 17) purports to quote words from the end of a scholium of Proclus on X. 9. The words quoted are from the scholium X. No. 62, one of the Scholia Vaticana. But none of the other, older, sources connect Proclus' name with X. No. 62; it is probable therefore that a Byzantine, who had in his Ms. of Euclid the collection of Schol. Vat. and knew that those on Book I. came from Proclus, himself attached Proclus' name to the others. The contrary view receives support from two facts pointed out by Heiberg, viz. (1) that the scholiast's copy of Proclus was not so much better than our MSS. as to suggest that the scholiast had further commentaries of Proclus which have vanished for usWhile one class of scnona (Schol. Vat.) have some better readings than our MSS. of Proclus have, and partly fill up the gaps at 1. 36, 37 and 1. 41-43, the other class (Schol. Vind.) derive from an inferior Proclus MS. which also had the same lacunae.; (2) that there is no trace in the scholia of the notes which Proclus promised in the passages quoted above.

Coming now to the question of the sources of Proclus, we may say that everything goes to show that his commentary is a compilation, though a compilation in the better sense - of the termKnoche, Untersuchungen über des Proklus Diadochus Commentar zu Euklid's Elementen (1862), p. 11.. He does not even give us to understand that we shall find in it much of his own; let us, + of the termKnoche, Untersuchungen über des Proklus Diadochus Commentar zu Euklid's Elementen (1862), p. 11.. He does not even give us to understand that we shall find in it much of his own; let us, he says, now turn to the exposition of the theorems proved by Euclid, selecting the more subtle of the comments made on them by the ancient writers, and cutting down their interminable diffuseness...Proclus, p. 200, 10-13. -: not a word about anything of his own. At the same time, he seems to imply that he will not necessarily on each occasion quote the source of each extract from an earlier commentary; and, in fact, while he quotes the name of his authority in many places, especially where the subject is important, in many others, where it is equally certain that he is not giving anything of his own, he mentions no authority. Thus he quotes Heron by name six times; but we now know, from the commentary of an-Nairīzī, that a number of other passages, where he mentions no name, are taken from Heron, and among them the not unimportant addition of an alternative proof to 1. 19. Hence we can by no means conclude that, where no authority is mentioned, Proclus is giving notes of his own. The presumption is génerally the other way; and it is often possible to arrive at a conclusion, either that a particular note is not Proclus' own, or that it is definitely attributable to someone else, by applying the ordinary principles of criticism. Thus, where the note shows an unmistakable affinity to another which Proclus definitely attributes to some commentator by name, especially when both contain some peculiar and distinctive idea, we cannot have much doubt in assigning both to the same commentatorInstances of the application of this criterion will be found in the discussion of Proclus' indebtedness to the commentaries of Heron, Porphyry and Pappus.. Again, van Pesch finds a criterion in the form of a note, where the explanation is so condensed as to be only just intelligible; the note is that in which a converse of 1. 32 is provedVan Pesch attributes this converse and proof to Pappus, arguing from the fact that the proof is followed by a passage which, on comparison with Pappus' note on the postulate that all right angles are equal, he feels justified in assigning to Pappus. I doubt if the evidence is sufficient. the proposition namely that a rectilineal figure which has all its interior angles together equal to two right angles is a triangle.

+: not a word about anything of his own. At the same time, he seems to imply that he will not necessarily on each occasion quote the source of each extract from an earlier commentary; and, in fact, while he quotes the name of his authority in many places, especially where the subject is important, in many others, where it is equally certain that he is not giving anything of his own, he mentions no authority. Thus he quotes Heron by name six times; but we now know, from the commentary of an-Nairīzī, that a number of other passages, where he mentions no name, are taken from Heron, and among them the not unimportant addition of an alternative proof to 1. 19. Hence we can by no means conclude that, where no authority is mentioned, Proclus is giving notes of his own. The presumption is génerally the other way; and it is often possible to arrive at a conclusion, either that a particular note is not Proclus' own, or that it is definitely attributable to someone else, by applying the ordinary principles of criticism. Thus, where the note shows an unmistakable affinity to another which Proclus definitely attributes to some commentator by name, especially when both contain some peculiar and distinctive idea, we cannot have much doubt in assigning both to the same commentatorInstances of the application of this criterion will be found in the discussion of Proclus' indebtedness to the commentaries of Heron, Porphyry and Pappus.. Again, van Pesch finds a criterion in the form of a note, where the explanation is so condensed as to be only just intelligible; the note is that in which a converse of 1. 32 is provedVan Pesch attributes this converse and proof to Pappus, arguing from the fact that the proof is followed by a passage which, on comparison with Pappus' note on the postulate that all right angles are equal, he feels justified in assigning to Pappus. I doubt if the evidence is sufficient. the proposition namely that a rectilineal figure which has all its interior angles together equal to two right angles is a triangle.

It is not safe to attribute a passage to Proclus himself because he uses the first person in such expressions as I say or I will prove - —for he was in the habit of putting into his own words the substance of notes borrowed from others—nor because, in speaking of an objection raised to a particular proposition, he uses such expressions as perhaps someone may object + —for he was in the habit of putting into his own words the substance of notes borrowed from others—nor because, in speaking of an objection raised to a particular proposition, he uses such expressions as perhaps someone may object (i)/sws d) a)/n tines e)nstai=en...): for sometimes other words in the same passage indicate that the objection had actually been taken by someoneVan Pesch illustrates this by an objection refuted in the note on 1. 9, p. 273, 11 sqq. After using the above expression to introduce the objection, Proclus uses further on (p. 273, 25) the term they say (fasi/n).. Speaking generally, we shall not be justified in concluding that Proclus is stating something new of his own unless he indicates this himself in express terms.

As regards the form of Proclus' references to others by name, van Pesch notes that he very seldom mentions the particular work from which he is borrowing. If we leave out of account the references to Plato's dialogues, there are only the following references to books: the Bacchae of PhilolausProclus, p. 22, 15., the Symmikta of Porphyryibid. p. 56, 25., Archimedes On the Sphere and Cylinderibid. p. 71, 18., Apollonius On the cochliasibid. p. 105, 5., a book by Eudemus on The Angleibid. p. 125, 8., a whole book of Posidonius directed against Zeno of the Epicurean sectibid. p. 200, 2., Carpus' Astronomyibid. p. 241, 19., Eudemus' History of Geometryibid. p. 352, 15., and a tract by Ptolemy on the parallel-postulateibid. p. 362, 15..

@@ -421,26 +421,26 @@

We cannot attribute to Eudemus the beginning of the note on 1. 47 where Proclus says that if we listen to those who like to recount ancient history, we may find some of them referring this theorem to Pythagoras and saying that he sacrificed an ox in honour of his discoveryibid. p. 426, 6-9.. As such a sacrifice was contrary to the Pythagorean tenets, and Eudemus could not have been unaware of this, the story cannot rest on his authority. Moreover Proclus speaks as though he were not certain of the correctness of the tradition; indeed, so far as the story of the sacrifice is concerned, the same thing is told of Thales in connexion with his discovery that the angle in a semicircle is a right angleDiogenes Laertius, I. 24, p. 6, ed. Cobet., and Plutarch is not certain whether the ox was sacrificed on the discovery of I. 47 or of the problem about application of areasPlutarch, non posse suaviter vivi secundum Epicurum, 11; Symp. VIII, 2.. Plutarch's doubt suggests that he knew of no evidence for the story beyond the vague allusion in the distich of Apollodorus Logisticus (the calculator -) cited by Diogenes Laertius alsoDiog. Laert. VIII. 12, p. 207, ed. Cobet: *(hni/ka *puqago/rhs to\ periklee\s eu(/reto gra/mma, kei=n) e)f) o)/w| kleinh\n h)/gage bouqusi/hn. See on this subject Tannery, La Géoinétrie grecque, p. 105.; and Proclus may have had in mind this couplet with the passages of Plutarch.

-

We come now to the question of the famous historical summary given by ProclusProclus, pp. 64-70.. No one appears to maintain that Eudemus is the author of even the early part of this summary in the form in which Proclus gives it. It is, as is well known, divided into two distinct parts, between which comes the remark, Those who compiled historiesThe plural is well explained by Tannery, La Géoinétrie grecque, pp. 73, 74. No doubt the author of the summary tried to supplement Eudemus by means of any other histories which threw light on the subject. Thus e.g. the allusion (p. 64, 21) to the Nile recalls Herodotus. Cf. the expression in Proclus, p. 64, 19, para\ tw=n pollw=n i/sto/rhtai. bring the development of this science up to this point. Not much younger than these is Euclid, who put together the Elements, collecting many of the theorems of Eudoxus, perfecting many others by Theaetetus, and bringing to irrefragable demonstration the things which had only been somewhat loosely proved by his predecessors. - Since Euclid was later than Eudemus, it is impossible that Eudemus can have written this. Yet the style of the summary after this point does not show any such change from that of the former portion as to suggest different authorship. The author of the earlier portion recurs frequently to the question of the origin of the elements of geometry in a way in which no one would be likely to do who was not later than Euclid; and it must be the same hand which in the second portion connects Euclid's Elements with the work of Eudoxus and TheaetetusTannery, La Géométrie grecque, p. 75..

+) cited by Diogenes Laertius alsoDiog. Laert. VIII. 12, p. 207, ed. Cobet: *(hni/ka *puqago/rhs to\ periklee\s eu(/reto gra/mma, kei=n) e)f) o)/w| kleinh\n h)/gage bouqusi/hn. See on this subject Tannery, La Géoinétrie grecque, p. 105.; and Proclus may have had in mind this couplet with the passages of Plutarch.

+

We come now to the question of the famous historical summary given by ProclusProclus, pp. 64-70.. No one appears to maintain that Eudemus is the author of even the early part of this summary in the form in which Proclus gives it. It is, as is well known, divided into two distinct parts, between which comes the remark, Those who compiled historiesThe plural is well explained by Tannery, La Géoinétrie grecque, pp. 73, 74. No doubt the author of the summary tried to supplement Eudemus by means of any other histories which threw light on the subject. Thus e.g. the allusion (p. 64, 21) to the Nile recalls Herodotus. Cf. the expression in Proclus, p. 64, 19, para\ tw=n pollw=n i/sto/rhtai. bring the development of this science up to this point. Not much younger than these is Euclid, who put together the Elements, collecting many of the theorems of Eudoxus, perfecting many others by Theaetetus, and bringing to irrefragable demonstration the things which had only been somewhat loosely proved by his predecessors. + Since Euclid was later than Eudemus, it is impossible that Eudemus can have written this. Yet the style of the summary after this point does not show any such change from that of the former portion as to suggest different authorship. The author of the earlier portion recurs frequently to the question of the origin of the elements of geometry in a way in which no one would be likely to do who was not later than Euclid; and it must be the same hand which in the second portion connects Euclid's Elements with the work of Eudoxus and TheaetetusTannery, La Géométrie grecque, p. 75..

If then the summary is the work of one author, and that author not Eudemus, who is it likely to have been? Tannery answers that it is Geminusibid. pp. 66-75.; but I think, with van Pesch, that he has failed to show why it should be Geminus rather than another. And certainly the extracts which we have from Geminus' work suggest that the sort of topics which it dealt with was quite different; they seem rather to have been general questions of the content of mathematics, and even Tannery admits that historical details could only have come incidentally into the workibid. p. 19..

Could the author have been Proclus himself? Circumstances which seem to suggest this possibility are (1) that, as already stated, the question of the origin of the Elements is kept prominent, (2) that there is no mention of Democritus, whom Eudemus would not be likely to have ignored, while a follower of Plato would be likely enough to do him the injustice, following the example of Plato who was an opponent of Democritus, never once mentions him, and is said to have wished to burn all his writingsDiog. Laertius, IX. 40, p. 237, ed. Cobet., and (3) the allusion at the beginning to the inspired Aristotle (o( daimonios *)aristote/lhs)Proclus, p. 64, 8., though this may easily have been inserted by Proclus in a quotation made by him from someone else. On the other hand there are considerations which suggest that Proclus himself was not the writer. (1) The style of the whole passage is not such as to point to him as the author. (2) If he wrote it, it is hardly conceivable that he would have passed over in silence the discovery of the analytical method, the invention of Plato to which he attached so much importanceProclus, p. 211, 19 sqq.; the passage is quoted above, p. 36..

There is nothing improbable in the conjecture that Proclus quoted the summary from a compendium of Eudemus' history made by some later writer: but as yet the question has not been definitely settled. All that is certain is that the early part of the summary must have been made up from scattered notices found in the great work of Eudemus.

-

Proclus refers to another work of Eudemus besides the history, viz. a book on The Angle (bibli/on peri\ gwni/as)ibid. p. 125, 8.. Tannery assumes that this must have been part of the history, and uses this assumption to confirm his idea that the history was arranged according to subjects, not according to chronological orderTannery, La Géométrie grecque, p. 26.. The phraseology of Proclus however unmistakably suggests a separate work; and that the history was chronologically arranged seems to be clearly indicated by the remark of Simplicius that Eudemus also counted Hippocrates among the more ancient writers +

Proclus refers to another work of Eudemus besides the history, viz. a book on The Angle (bibli/on peri\ gwni/as)ibid. p. 125, 8.. Tannery assumes that this must have been part of the history, and uses this assumption to confirm his idea that the history was arranged according to subjects, not according to chronological orderTannery, La Géométrie grecque, p. 26.. The phraseology of Proclus however unmistakably suggests a separate work; and that the history was chronologically arranged seems to be clearly indicated by the remark of Simplicius that Eudemus also counted Hippocrates among the more ancient writers (e)n toi=s palaiote/rois)Simplicius, ed. Diels, p. 69, 23..

The passage of Simplicius about the lunes of Hippocrates throws considerable light on the style of Eudemus' history. Eudemus wrote in a memorandum-like or summary manner (to\n u(pomnhmatiko\n tro/pon tou= *eu)dh/mou)ibid. p. 60, 29. when reproducing what he found in the ancient writers; sometimes it is clear that he left out altogether proofs or constructions of things by no means easyCf. Simplicius, p. 63, 19 sqq.; p. 64, 25 sqq.; also Usener's note de supplendis Hippocratis quas omisit Eudemus constructionibus - added to Diels' preface, pp. XXIII—XXVI..

+ added to Diels' preface, pp. XXIII—XXVI..

Geminus.

The discussions about the date and birthplace of Geminus form a whole literature, as to which I must refer the reader to Manitius and TittelManitius, Gemini elementa astronomiae (Teubner, 1898), pp. 237-252; Tittel, art. Geminos - in Pauly-Wissowa's Real-Encyclopädie der classischen Altertumswissenschaft, vol. VII.. 1910.. Though the name looks like a Latin name (Gem&icaron;nus), Manitius concluded that, since it appears as *gemi=nos in all Greek MSS. and as *gemei=nos in some inscriptions, it is Greek and possibly formed from gem as *)ergi=nos is from e)rg and *)aleci=nos from a)lec (cf. also *)ikti=nos, *krati=nos). Tittel is equally positive that it is Gem&icaron;nus and suggests that *gemi=nos is due to a false analogy with *)aleci=nos etc. and *gemei=nos wrongly formed on the model of *)antwnei=nos, *)agrippei=na. Geminus, a Stoic philosopher, born probably in the island of Rhodes, was the author of a comprehensive work on the classification of mathematics, and also wrote, about 73-67 B.C., a not less comprehensive commentary on the meteorological textbook of his teacher Posidonius of Rhodes.

+ in Pauly-Wissowa's Real-Encyclopädie der classischen Altertumswissenschaft, vol. VII.. 1910.. Though the name looks like a Latin name (Gemǐnus), Manitius concluded that, since it appears as *gemi=nos in all Greek MSS. and as *gemei=nos in some inscriptions, it is Greek and possibly formed from gem as *)ergi=nos is from e)rg and *)aleci=nos from a)lec (cf. also *)ikti=nos, *krati=nos). Tittel is equally positive that it is Gemǐnus and suggests that *gemi=nos is due to a false analogy with *)aleci=nos etc. and *gemei=nos wrongly formed on the model of *)antwnei=nos, *)agrippei=na. Geminus, a Stoic philosopher, born probably in the island of Rhodes, was the author of a comprehensive work on the classification of mathematics, and also wrote, about 73-67 B.C., a not less comprehensive commentary on the meteorological textbook of his teacher Posidonius of Rhodes.

It is the former work in which we are specially interested here. Though Proclus made great use of it, he does not mention its title, unless we may suppose that, in the passage (p. 177, 24) where, after quoting from Geminus a classification of lines which never meet, he says, these remarks I have selected from the filokali/a of Geminus, filokali/a is a title or an alternative title. Pappus however quotes a work of Geminus on the classification of the mathematics (e)n tw=| peri\ th=s tw=n maqhma/twn ta/cews)Pappus, ed. Hultsch, p. 1026, 9., while Eutocius quotes from the sixth book of the doctrine of the mathematics - (e)n tw=| e(/ktw| th=s tw=n maqhma/twn qewri/as)Apollonius, ed. Heiberg, vol. II. p. 170.. TanneryTannery, La Géométrie grecque, pp. 18, 19. pointed out that the former title corresponds well enough to the long extractProclus, pp. 38, 1-42, 8. which Proclus gives in his first prologue, and also to the fragments contained in the Anonymi variae collectiones published by Hultsch at the end of his edition of HeronHeron, ed. Hultsch, pp. 246, 16-249, 12.; but it does not suit most of the óther passages borrowed by Proclus. The correct title was therefore probably that given by Eutocius, The Doctrine, or Theory, of the Mathematics; and Pappus probably refers to one particular portion of the work, say the first Book. If the sixth Book treated of conics, as we may conclude from Eutocius, there must have been more Books to follow, because Proclus has preserved us details about higher curves, which must have come later. If again Geminus finished his work and wrote with the same fulness about the other branches of mathematics as he did about geometry, there must have been a considerable number of Books altogether. At all events it seems to have been designed to give a complete view of the whole science of mathematics, and in fact to be a sort of encyclopaedia of the subject.

+ (e)n tw=| e(/ktw| th=s tw=n maqhma/twn qewri/as)Apollonius, ed. Heiberg, vol. II. p. 170.. TanneryTannery, La Géométrie grecque, pp. 18, 19. pointed out that the former title corresponds well enough to the long extractProclus, pp. 38, 1-42, 8. which Proclus gives in his first prologue, and also to the fragments contained in the Anonymi variae collectiones published by Hultsch at the end of his edition of HeronHeron, ed. Hultsch, pp. 246, 16-249, 12.; but it does not suit most of the óther passages borrowed by Proclus. The correct title was therefore probably that given by Eutocius, The Doctrine, or Theory, of the Mathematics; and Pappus probably refers to one particular portion of the work, say the first Book. If the sixth Book treated of conics, as we may conclude from Eutocius, there must have been more Books to follow, because Proclus has preserved us details about higher curves, which must have come later. If again Geminus finished his work and wrote with the same fulness about the other branches of mathematics as he did about geometry, there must have been a considerable number of Books altogether. At all events it seems to have been designed to give a complete view of the whole science of mathematics, and in fact to be a sort of encyclopaedia of the subject.

I shall now indicate first the certain, and secondly the probable, obligations of Proclus to Geminus, in which task I have only to follow van Pesch, who has embodied the results of Tittel's similar inquiry alsoVan Pesch, De Procli fontibus, pp. 97-113. The dissertation of Tittel is entitled De Gemini Stoici studiis mathematicis (1895).. I shall only omit the passages as regards which a case for attributing them to Geminus does not seem to me to have been made out.

First come the following passages which must be attributed to Geminus, because Proclus mentions his name:

(1) (In the first prologue of ProclusProclus, pp. 38, 1-42, 8, except the allusion in p. 41, 8-10, to Ctesibius and Heron and their pneumatic devices (qanmatopoii+kh/), as regards which Proclus' authority may be Pappus (VIII. p. 1024, 24-27) who uses very similar expressions. Heron, even if not later than Geminus, could hardly have been included in a historical work by him. Perhaps Geminus may have referred to Ctesibius only, and Proclus may have inserted and Heron @@ -448,7 +448,7 @@

(2) (in the note on the definition of a straight line) on the classification of lines (including curves) as simple (straight or circular) and mixed, composite and incomposite, uniform (o(moiomerei=s) and non-uniform (a)nomoiomerei=s), lines about solids and lines produced by cutting solids, including conic and spiric sectionsProclus, pp. 103, 21-107, 10; pp. 111, 1-113, 3.;

(3) (in the note on the definition of a plane surface) on similar distinctions extended to surfaces and solidsibid. pp. 117, 14-120, 12, where perhaps in the passage pp. 117, 22-118, 23 we may have Geminus' own words.;

-

(4) (in the note on the definition of parallels) on lines which do not meet (a)su/mptwtoi) but which are not on that account parallel, e.g. a curve and its asymptote, showing that the property of not meeting does not make lines parallel—a favourite observation of Geminus—and, incidentally, on bounded lines or those which enclose a figure and those which do notibid. pp. 176, 18-177, 25; perhaps also p. 175. The note ends with the words These things too we have selected from Geminus' *filokali/a for the elucidation of the matters in question. +

(4) (in the note on the definition of parallels) on lines which do not meet (a)su/mptwtoi) but which are not on that account parallel, e.g. a curve and its asymptote, showing that the property of not meeting does not make lines parallel—a favourite observation of Geminus—and, incidentally, on bounded lines or those which enclose a figure and those which do notibid. pp. 176, 18-177, 25; perhaps also p. 175. The note ends with the words These things too we have selected from Geminus' *filokali/a for the elucidation of the matters in question. Tannery (p. 27) takes these words coming at the end of the commentary on the definitions as referring to the whole of the portion of the commentary dealing with the definitions. Van Pesch properly regards them as only applying to the note on parallels. This seems to me clear from the use of the word too (tosau=ta kai/).;

(5) (in the same note) the definition of parallels given by PosidoniusProclus, p. 176, 5-17.;

(6) on the distinction between postulates and axioms, the futility of trying to prove axioms, as Apollonius tried to prove Axiom 1, and the equal incorrectness of assuming what really requires proof, as Euclid did in the fourth postulate [equality of right angles] and in the fifth postulate [the parallel-postulate]ibid. pp. 178-182, 4; pp. 183, 14-184, 10; cf. p. 188, 3-11. @@ -479,7 +479,7 @@ as used in the title of Euclid's Porisms, as distinct from the other meaning of corollary ibid. pp. 301, 21-302, 13.;

(14) a note on the Epicurean objection to I. 20 as being obvious even to an assibid. pp. 322, 4-323, 3.;

-

(15) a passage on the properties of parallels, with allusions to Apollonius' Conics, and the curves invented by Nicomedes, Hippias and PerseusPróclus, pp. 355, 20-356, 16.;

+

(15) a passage on the properties of parallels, with allusions to Apollonius' Conics, and the curves invented by Nicomedes, Hippias and PerseusPróclus, pp. 355, 20-356, 16.;

(16) a passage on the parallel-postulate regarded as the converse of I. 17ibid. p. 364, 9-12; pp. 364, 20-365, 4..

Of the authors to whom Proclus was indebted in a less degree the most important is Apollonius of Perga. Two passages allude to his Conicsibid. p. 71, 19; p. 356, 8, 6., one to a work on irrationalsibid. p. 74, 23, 24., and two to a treatise On the cochlias (apparently the cylindrical helix) by Apolloniusibid. pp. 105, 5, 6, 14, 15.. But more important for our purpose are six references to Apollonius in connexion with elementary geometry.

(1) He appears as the author of an attempt to explain the idea of a line (possessing length but no breadth) by reference to daily experience, e.g. when we tell someone to measure, merely, the length of a road or of a wallibid. p. 100, 5-19.; and doubtless the similar passage showing how we may in like manner get a notion of a surface (without depth) is his alsoibid. p. 114, 20-25..

@@ -491,7 +491,7 @@

(6) his solution of the problem in I. 23ibid. pp. 335, 16-336, 5..

HeibergPhilologus, vol. XLIII. p. 489. conjectures that Apollonius departed from Euclid's method in these propositions because he objected to solving problems of a more general, by means of problems of a more particular, character. Proclus however considers all three solutions inferior to Euclid's; and his remarks on Apollonius' handling of these elementary matters generally suggest that he was nettled by criticisms of Euclid in the work containing the things which he quotes from Apollonius, just as we conclude that Pappus was offended by the remarks of Apollonius about Euclid's incomplete treatment of the three- and four-line locusSee above, pp. 2, 3.. If this was the case, Proclus can hardly have got his information about these things at second-hand; and there seems to be no reason to doubt that he had the actual work of Apollonius before him. This work may have been the treatise mentioned by Marinus in the words Apollonius in his general treatise - (*)apollw/nios e)n th=| kaqo/lou pragmatei/a|)Marinus in Euclidis Data, ed. Menge, p. 234, 16.. If the notice in the FihristFihrist, tr. Suter, p. 19. stating, on the authority of Thābit b. Qurra, that Apollonius wrote a tract on the parallel-postulate be correct, it may have been included in the same work. We may conclude generally that, in it, Apollonius tried to remodel the beginnings of geometry, reducing the number of axioms, appealing, in his definitions of lines, surfaces etc., more to experience than to abstract reason, and substituting for certain proofs others of a more general character.

+ (*)apollw/nios e)n th=| kaqo/lou pragmatei/a|)Marinus in Euclidis Data, ed. Menge, p. 234, 16.. If the notice in the FihristFihrist, tr. Suter, p. 19. stating, on the authority of Thābit b. Qurra, that Apollonius wrote a tract on the parallel-postulate be correct, it may have been included in the same work. We may conclude generally that, in it, Apollonius tried to remodel the beginnings of geometry, reducing the number of axioms, appealing, in his definitions of lines, surfaces etc., more to experience than to abstract reason, and substituting for certain proofs others of a more general character.

The probabilities are that, in quoting from the tract of Ptolemy in which he tried to prove the parallel-postulate, Proclus had the actual work before him. For, after an allusion to it as a certain bookProclus, p. 191, 23. he gives two long extractsibid. pp. 362, 14-363, 18; pp. 365, 7-367, 27., and at the beginning of the second indicates the title of the tract, in the (book) about the meeting of straight lines produced from (angles) less than two right angles, as he has very rarely done in other cases.

@@ -503,17 +503,17 @@ (*poseidw/nio/s fhsi to\n *zh/nwna sukofantei=n)ibid. p. 218, 1.. It is not necessary to suppose that Proclus had the original work of Zeno before him, because Zeno's arguments may easily have been got from Posidonius' reply; but he would appear to have quoted direct from the latter at all events.

The work of Carpus mechanicus (a treatise on astronomy) quoted from by Proclusibid. pp. 241, 19-243, 11. must have been accessible to him at first-hand, because a portion of the extract from it about the relation of theorems and problemsibid. pp. 242, 22-243, 11. is reproduced word for word. Moreover, if he were not using the book itself, Proclus would hardly be in a position to question whether the introduction of the subject of theorems and problems was opportune in the place where it was found (ei) me\n kata\ kairo\n h(\ mh/, parei/sqw pro\s to\ paro/n)Proclus, p. 241, 21, 22..

-

It is of course evident that Proclus had before him the original works of Plato, Aristotle, Archimedes and Plotinus, as well as the *summikta/ of Porphyry and the works of his master Syrianus (o( h(me/teros kaqhgemw/n)ibid. p. 123, 19., from whom he quotes in his note on the definition of an angle. Tannery also points out that he must have had before him a group of works representing the Pythagorean tradition on its mystic, as distinct from its mathematical, side, from Philolaus downwards, and comprising the more or less apocryphal i(ero\s lo/gos of Pythagoras, the Oracles (lo/gia), and Orphic versesTannery, La Géométrie grecque, pp. 25, 26..

+

It is of course evident that Proclus had before him the original works of Plato, Aristotle, Archimedes and Plotinus, as well as the *summikta/ of Porphyry and the works of his master Syrianus (o( h(me/teros kaqhgemw/n)ibid. p. 123, 19., from whom he quotes in his note on the definition of an angle. Tannery also points out that he must have had before him a group of works representing the Pythagorean tradition on its mystic, as distinct from its mathematical, side, from Philolaus downwards, and comprising the more or less apocryphal i(ero\s lo/gos of Pythagoras, the Oracles (lo/gia), and Orphic versesTannery, La Géométrie grecque, pp. 25, 26..

Besides quotations from writers whom we can identify with more or less certainty, there are many other passages which are doubtless quoted from other commentators whose names we do not know. A list of such passages is given by van PeschVan Pesch, De Procli fontibus, p. 139., and there is no need to cite them here.

Van Pesch also gives at the end of his workibid. p. 155. a convenient list of the books which, as the result of his investigation, he deems to have been accessible to and directly used by Proclus. The list is worth giving here, on the same ground of convenience. It is as follows: Eudemus: history of geometry. Geminus: the theory of the mathematical sciences. Heron: commentary on the Elements of Euclid. Porphyry: commentary on the Elements of Euclid. Pappus: commentary on the Elements of Euclid. Apollonius of Perga: a work relating to elementary geometry. Ptolemy: on the parallel-postulate. Posidonius: a book controverting Zeno of Sidon. Carpus: astronomy. Syrianus: a discussion on the angle. Pythagorean philosophical tradition. Plato's works. Aristotle's works. Archimedes' works. Plotinus: Enneades.

Lastly we come to the question what passages, if any, in the commentary of Proclus represent his own contributions to the subject. As we have seen, the onus probandi must be held to rest upon him who shall maintain that a particular note is original on the part of Proclus. Hence it is not enough that it should be impossible to point to another writer as the probable source of a note; we must have a positive reason for attributing it to Proclus. The criterion must therefore be found either (1) in the general terms in which Proclus points out the deficiencies in previous commentaries and indicates the respects in which his own will differ from them, or (2) in specific expressions used by him in introducing particular notes which may indicate that he is giving his own views. Besides indicating that he paid more attention than his predecessors to questions requiring deeper study (to\ pragmateiw=des) and pursued clear distinctions - (to\ eu)diai/reton metadiw/kontas)Proclus, p. 84, 13, p. 432, 14, 15.— by which he appears to imply that his predecessors had confused the different departments of their commentaries, viz. lemmas, cases, and objections (e)nsta/seis)Cf. ibid. p. 289, 11-15; p. 432, 15-17.—Proclus complains that the earlier commentators had failed to indicate the ultimate grounds or causes of propositionsibid. p. 432, 17.. Although it is from Geminus that he borrowed a passage maintaining that it is one of the proper functions of geometry to inquire into causes (th\n ai)ti/an kai\ to\ dia\ ti/)ibid. p. 202, 9-25., yet it is not likely that Geminus dealt with Euclid's propositions one by one; and consequently, when we find Proclus, on I. 8, 16, 17, 18, 32, and 47See Proclus, p. 270, 5-24 (I. 8); pp. 309, 3-310, 8 (I. 16); pp. 310, 19-311, 23 (I. 17); pp. 316, 14-318, 2 (I. 18); p. 384, 13-21 (I. 32); pp. 426, 22-427, 8 (I. 47)., endeavouring to explain causes, we have good reason to suppose that the explanations are his own.

+ (to\ eu)diai/reton metadiw/kontas)Proclus, p. 84, 13, p. 432, 14, 15.— by which he appears to imply that his predecessors had confused the different departments of their commentaries, viz. lemmas, cases, and objections (e)nsta/seis)Cf. ibid. p. 289, 11-15; p. 432, 15-17.—Proclus complains that the earlier commentators had failed to indicate the ultimate grounds or causes of propositionsibid. p. 432, 17.. Although it is from Geminus that he borrowed a passage maintaining that it is one of the proper functions of geometry to inquire into causes (th\n ai)ti/an kai\ to\ dia\ ti/)ibid. p. 202, 9-25., yet it is not likely that Geminus dealt with Euclid's propositions one by one; and consequently, when we find Proclus, on I. 8, 16, 17, 18, 32, and 47See Proclus, p. 270, 5-24 (I. 8); pp. 309, 3-310, 8 (I. 16); pp. 310, 19-311, 23 (I. 17); pp. 316, 14-318, 2 (I. 18); p. 384, 13-21 (I. 32); pp. 426, 22-427, 8 (I. 47)., endeavouring to explain causes, we have good reason to suppose that the explanations are his own.

Again, his remarks on certain things which he quotes from Pappus can scarcely be due to anyone else, since Pappus is the latest of the commentators whose works he appears to have used. Under this head come

(1) his objections to certain new axioms introduced by PappusProclus, p. 198, 5-15.,

(2) his conjecture as to how Pappus came to think of his alternative proof of I. 5ibid. p. 250, 12-19.,

(3) an addition to Pappus' remarks about the curvilineal angle which is equal to a right angle without being oneibid. p. 190, 9-23..

The defence of Geminus against Carpus, who combated his view of theorems and problems, is also probably due to Proclusibid. p. 243, 12-29., as well as an observation on I. 38 to the effect that I. 35-38 are really comprehended in VI. 1 as particular casesibid. pp. 405, 6-406, 9..

-

Lastly, we can have no hesitation in attributing to Proclus himself (1) the criticism of Ptolemy's attempt to prove the parallel-postulateibid. p. 368, 1-23., and (2) the other attempted proof given ín the same noteibid. pp. 371, 11-373, 2. (on I. 29) and assuming as an axiom that if from one point two straight lines forming an angle be produced ad infinitum the distance between them when so produced ad infinitum exceeds any finite magnitude (i.e. length), +

Lastly, we can have no hesitation in attributing to Proclus himself (1) the criticism of Ptolemy's attempt to prove the parallel-postulateibid. p. 368, 1-23., and (2) the other attempted proof given in the same noteibid. pp. 371, 11-373, 2. (on I. 29) and assuming as an axiom that if from one point two straight lines forming an angle be produced ad infinitum the distance between them when so produced ad infinitum exceeds any finite magnitude (i.e. length), an assumption which purports to be the equivalent of a statement in AristotleAristotle, de caelo, I. 5 (271 b 28-30).. It is introduced by words in which the writer appears to claim originality for his proof: To him who desires to see this proved (kataskeuazo/menon) let it be said by us (lege/sqw par) h<*>mw=n) etc.Proclus, p. 371, 10. Moreover, Philoponus, in a note on Aristotle's Anal. post. I. 10, says that the geometer (Euclid) assumes this as an axiom, but it wants a great deal of proof, insomuch that both Ptolemy and Proclus wrote a whole book upon itBerlin Aristotle, vol. IV. p. 214 a 9-12..

@@ -521,7 +521,7 @@ CHAPTER V. - THE TEXTThe material for the whole of this chapter is taken from Heiberg's edition of the Elements, introduction to vol. v., and from the same scholar's Litterargeschichtliche Studien über Euklid, p. 174 sqq. and Paralipomena zu Euklid in Hermes, XXXVIII., 1903.. + THE TEXTThe material for the whole of this chapter is taken from Heiberg's edition of the Elements, introduction to vol. v., and from the same scholar's Litterargeschichtliche Studien über Euklid, p. 174 sqq. and Paralipomena zu Euklid in Hermes, XXXVIII., 1903..

It is well known that the title of Simson's edition of Euclid (first brought out in Latin and English in 1756) claims that, in it, the errors by which Theon, or others, have long ago vitiated these books are corrected, and some of Euclid's demonstrations are restored ; and readers of Simson's notes are familiar with the phrases used, where anything in the text does not seem to him satisfactory, to the effect that the demonstration has been spoiled, or things have been interpolated or omitted, by Theon or some other unskilful editor. Now most of the MSS. of the Greek text prove by their titles that they proceed from the recension of the Elements by Theon; they purport to be either from the edition of Theon @@ -532,25 +532,25 @@ Thus we are more fortunate than Simson, since our judgment of Theon's recension can be formed on the basis, not of mere conjecture, but of the documentary evidence afforded by a comparison of the Vatican MS. just mentioned with what we may conveniently call, after Heiberg, the Theonine MSS.

The MSS. used for Heiberg's edition of the Elements are the following:

(1) P = Vatican MS. numbered 190, 4to, in two volumes (doubtless one originally); 10th c.

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This is the MS. which Peyrard was able to use; it was sent from Rome to Paris for his use and bears the stamp of the Paris Imperial Library on the last page. It is well and carefully written. There are corrections some of which are by the original hand, but generally in paler ink, others, still pretty old, by several different hands, or by one hand with different ink in different places (P m. 2), and others again by the latest hand (P m. rec.). It contains, first, the Elements I.—XIII. with scholia, then Marinus' commentary on the Data (without the name of the author), followed by the Data itself and scholia, then the Elements XIV., XV. (so called), and lastly three books and a part of a fourth of a commentary by Theon ei)s tou\s proxei/rous kano/nas *ptolemai/ou.

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This is the MS. which Peyrard was able to use; it was sent from Rome to Paris for his use and bears the stamp of the Paris Imperial Library on the last page. It is well and carefully written. There are corrections some of which are by the original hand, but generally in paler ink, others, still pretty old, by several different hands, or by one hand with different ink in different places (P m. 2), and others again by the latest hand (P m. rec.). It contains, first, the Elements I.—XIII. with scholia, then Marinus' commentary on the Data (without the name of the author), followed by the Data itself and scholia, then the Elements XIV., XV. (so called), and lastly three books and a part of a fourth of a commentary by Theon ei)s tou\s proxei/rous kano/nas *ptolemai/ou.

The other MSS. are Theonine.

(2) F = MS. XXVIII, 3, in the Laurentian Library at Florence, 4to; 10th c.

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This MS. is written in a beautiful and scholarly hand and contains the Elements I.—XV., the Optics and the Phaenomena, but is not well preserved. Not only is the original writing renewed in many places, where it had become faint, by a later hand of the 16th c., but the same hand has filled certain smaller lacunae by gumming on to torn pages new pieces of parchment, and has replaced bodily certain portions of the MS., which had doubtless become illegible, by fresh leaves. The larger gaps so made good extend from Eucl. VII. 12 to IX. 15, and from XII. 3 to the end; so that, besides the conclusion of the Elements, the Optics and Phaenomena are also in the later hand, and we cannot even tell what in addition to the Elements I.—XIII. the original MS. contained. Heiberg denotes the later hand by f and observes that, while in restoring words which had become faint and filling up minor lacunae the writer used no other MS., yet in the two larger restorations he used the Laurentian MS. XXVIII, 6, belonging to the 13th—14th c. The latter MS. (which Heiberg denotes by f) was copied from the Viennese MS. (V) to be described below.

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This MS. is written in a beautiful and scholarly hand and contains the Elements I.—XV., the Optics and the Phaenomena, but is not well preserved. Not only is the original writing renewed in many places, where it had become faint, by a later hand of the 16th c., but the same hand has filled certain smaller lacunae by gumming on to torn pages new pieces of parchment, and has replaced bodily certain portions of the MS., which had doubtless become illegible, by fresh leaves. The larger gaps so made good extend from Eucl. VII. 12 to IX. 15, and from XII. 3 to the end; so that, besides the conclusion of the Elements, the Optics and Phaenomena are also in the later hand, and we cannot even tell what in addition to the Elements I.—XIII. the original MS. contained. Heiberg denotes the later hand by f and observes that, while in restoring words which had become faint and filling up minor lacunae the writer used no other MS., yet in the two larger restorations he used the Laurentian MS. XXVIII, 6, belonging to the 13th—14th c. The latter MS. (which Heiberg denotes by f) was copied from the Viennese MS. (V) to be described below.

(3) B = Bodleian MS., D'Orville X. 1 inf. 2, 30, 4to; A.D. 888.

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This MS. contains the Elements I.—XV. with many scholia. Leaves 15-118 contain I. 14 (from about the middle of the proposition) to the end of Book VI., and leaves 123-387 (wrongly numbered 397) Books VII.—XV. in one and the same elegant hand (9th c.). The leaves preceding leaf 15 seem to have been lost at some time, leaves 6 to 14 (containing Elem. I. to the place in I. 14 above referred to) being carelessly written by a later hand on thick and common parchment (13th c.). On leaves 2 to 4 and 122 are certain notes in the hand of Arethas, who also wrote a two-line epigram on leaf 5, the greater part of the scholia in uncial letters, a few notes and corrections, and two sentences on the last leaf, the first of which states that the MS. was written by one Stephen clericus in the year of the world 6397 (= 888 A.D.), while the second records Arethas' own acquisition of it. Arethas lived from, say, 865 to 939 A.D. He was Archbishop of Caesarea and wrote a commentary on the Apocalypse. The portions of his library which survive are of the greatest interest to palaeography on account of his exact notes of dates, names of copyists, prices of parchment etc. It is to him also that we owe the famous Plato MS. from Patmos (Cod. Clarkianus) which was written for him in November 895See Pauly-Wissowa, Real-Encyclopädie der class. Altertumswissenschaft, vol. II., 1896, p. 675..

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This MS. contains the Elements I.—XV. with many scholia. Leaves 15-118 contain I. 14 (from about the middle of the proposition) to the end of Book VI., and leaves 123-387 (wrongly numbered 397) Books VII.—XV. in one and the same elegant hand (9th c.). The leaves preceding leaf 15 seem to have been lost at some time, leaves 6 to 14 (containing Elem. I. to the place in I. 14 above referred to) being carelessly written by a later hand on thick and common parchment (13th c.). On leaves 2 to 4 and 122 are certain notes in the hand of Arethas, who also wrote a two-line epigram on leaf 5, the greater part of the scholia in uncial letters, a few notes and corrections, and two sentences on the last leaf, the first of which states that the MS. was written by one Stephen clericus in the year of the world 6397 (= 888 A.D.), while the second records Arethas' own acquisition of it. Arethas lived from, say, 865 to 939 A.D. He was Archbishop of Caesarea and wrote a commentary on the Apocalypse. The portions of his library which survive are of the greatest interest to palaeography on account of his exact notes of dates, names of copyists, prices of parchment etc. It is to him also that we owe the famous Plato MS. from Patmos (Cod. Clarkianus) which was written for him in November 895See Pauly-Wissowa, Real-Encyclopädie der class. Altertumswissenschaft, vol. II., 1896, p. 675..

(4) V = Viennese MS. Philos. Gr. No. 103; probably 12th c.

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This MS. contains 292 leaves, Eucl. Elements I.—XV. occupying leaves 1 to 254, after which come the Optics (to leaf 271), the Phaenomena (mutilated at the end) from leaf 272 to leaf 282, and lastly scholia, on leaves 283 to 292, also imperfect at the end. The different material used for different parts and the varieties of handwriting make it necessary for Heiberg to discuss this MS. at some lengthHeiberg, vol. v. pp. xxix—xxxiii.. The handwriting on leaves 1 to 183 (Book I. to the middle of X. 105) and on leaves 203 to 234 (from XI. 31, towards the end of the proposition, to XIII. 7, a few lines down) is the same; between leaves 184 and 202 there are two varieties of handwriting, that of leaves 184 to 189 and that of leaves 200 (verso) to 202 being the same. Leaf 235 begins in the same handwriting, changes first gradually into that of leaves 184 to 189 and then (verso) into a third more rapid cursive writing which is the same as that of the greater part of the scholia, and also as that of leaves 243 and 282, although, as these leaves are of different material, the look of the writing and of the ink seems altered. There are corrections both by the first and a second hand, and scholia by many hands. On the whole, in spite of the apparent diversity of handwriting in the MS., it is probable that the whole of it was written at about the same time, and it may (allowing for changes of material, ink etc.) even have been written by the same man. It is at least certain that, when the Laurentian MS. XXVIII, 6 was copied from it, the whole MS. was in the condition in which it is now, except as regards the later scholia and leaves 283 to 292 which are not in the Laurentian MS., that MS. coming to an end where the Phaenomena breaks off abruptly in V. Hence Heiberg attributes the whole MS. to the 12th c.

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This MS. contains 292 leaves, Eucl. Elements I.—XV. occupying leaves 1 to 254, after which come the Optics (to leaf 271), the Phaenomena (mutilated at the end) from leaf 272 to leaf 282, and lastly scholia, on leaves 283 to 292, also imperfect at the end. The different material used for different parts and the varieties of handwriting make it necessary for Heiberg to discuss this MS. at some lengthHeiberg, vol. v. pp. xxix—xxxiii.. The handwriting on leaves 1 to 183 (Book I. to the middle of X. 105) and on leaves 203 to 234 (from XI. 31, towards the end of the proposition, to XIII. 7, a few lines down) is the same; between leaves 184 and 202 there are two varieties of handwriting, that of leaves 184 to 189 and that of leaves 200 (verso) to 202 being the same. Leaf 235 begins in the same handwriting, changes first gradually into that of leaves 184 to 189 and then (verso) into a third more rapid cursive writing which is the same as that of the greater part of the scholia, and also as that of leaves 243 and 282, although, as these leaves are of different material, the look of the writing and of the ink seems altered. There are corrections both by the first and a second hand, and scholia by many hands. On the whole, in spite of the apparent diversity of handwriting in the MS., it is probable that the whole of it was written at about the same time, and it may (allowing for changes of material, ink etc.) even have been written by the same man. It is at least certain that, when the Laurentian MS. XXVIII, 6 was copied from it, the whole MS. was in the condition in which it is now, except as regards the later scholia and leaves 283 to 292 which are not in the Laurentian MS., that MS. coming to an end where the Phaenomena breaks off abruptly in V. Hence Heiberg attributes the whole MS. to the 12th c.

But it was apparently in two volumes originally, the first consisting of leaves 1 to 183; and it is certain that it was not all copied at the same time or from one and the same original. For leaves 184 to 202 were evidently copied from two MSS. different both from one another and from that from which the rest was copied. Leaves 184 to the middle of leaf 189 (recto) must have been copied from a MS. similar to P, as is proved by similarity of readings, though not from P itself. The rest, up to leaf 202, were copied from the Bologna MS. (b) to be mentioned below. It seems clear that the content of leaves 184 to 202 was supplied from other MSS. because there was a lacuna in the original from which the rest of V was copied.

Heiberg sums up his conclusions thus. The copyist of V first copied leaves 1 to 183 from an original in which two quaterniones were missing (covering from the middle of Eucl. X. 105 to near the end of XI. 31). Noticing the lacuna he put aside one quaternio of the parchment used up to that point. Then he copied onwards from the end of the lacuna in the original to the end of the Phaenomena. After this he looked about him for another MS. from which to fill up the lacuna; finding one, he copied from it as far as the middle of leaf 189 (recto). Then, noticing that the MS. from which he was copying was of a different class, he had recourse to yet another MS. from which he copied up to leaf 202. At the same time, finding that the lacuna was longer than he had reckoned for, he had to use twelve more leaves of a different parchment in addition to the quaternio which he had put aside. The whole MS. at first formed two volumes (the first containing leaves 1 to 183 and the second leaves 184 to 282); then, after the last leaf had perished, the two volumes were made into one to which two more quaterniones were also added. A few leaves of the latter of these two have since perished.

(5) b = MS. numbered 18-19 in the Communal Library at Bologna, in two volumes, 4to; 11th c.

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This MS. has scholia in the margin written both by the first hand and by two or three later hands; some are written by the latest hand, Theodorus Cabasilas (a descendant apparently of Nicolaus Cabasilas, 14th c.) who owned the MS. at one time. It contains (a) in 14 quaterniones the definitions and the enunciations (without proofs) of the Elements I.—XIII. and of the Data, (b) in the remainder of the volumes the Proem to Geometry (published among the Variae Collectiones in Hultsch's edition of Heron, pp. 252, 24 to 274, 14) followed by the Elements I.—XIII. (part of XIII. 18 to the end being missing), and then by part of the Data (from the last three words of the enunciation of Prop. 38 to the end of the penultimate clause in Prop. 87, ed. Menge). From XI. 36 inclusive to the end of XII. this MS. appears to represent an entirely different recension. Heiberg is compelled to give this portion of b separately in an appendix. He conjectures that it is due to a Byzantine mathematician who thought Euclid's proofs too long and tiresome and consequently contented himself with indicating the course followedZeitschrift fiir Math. u. Physik, XXIX., hist.-litt. Abtheilung, p. 13.. At the same time this Byzantine must have had an excellent MS. before him, probably of the ante-Theonine variety of which the Vatican MS. 190 (P) is the sole representative.

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This MS. has scholia in the margin written both by the first hand and by two or three later hands; some are written by the latest hand, Theodorus Cabasilas (a descendant apparently of Nicolaus Cabasilas, 14th c.) who owned the MS. at one time. It contains (a) in 14 quaterniones the definitions and the enunciations (without proofs) of the Elements I.—XIII. and of the Data, (b) in the remainder of the volumes the Proem to Geometry (published among the Variae Collectiones in Hultsch's edition of Heron, pp. 252, 24 to 274, 14) followed by the Elements I.—XIII. (part of XIII. 18 to the end being missing), and then by part of the Data (from the last three words of the enunciation of Prop. 38 to the end of the penultimate clause in Prop. 87, ed. Menge). From XI. 36 inclusive to the end of XII. this MS. appears to represent an entirely different recension. Heiberg is compelled to give this portion of b separately in an appendix. He conjectures that it is due to a Byzantine mathematician who thought Euclid's proofs too long and tiresome and consequently contented himself with indicating the course followedZeitschrift fiir Math. u. Physik, XXIX., hist.-litt. Abtheilung, p. 13.. At the same time this Byzantine must have had an excellent MS. before him, probably of the ante-Theonine variety of which the Vatican MS. 190 (P) is the sole representative.

(6) p = Paris MS. 2466, 4to; 12th c.

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This manuscript is written in two hands, the finer hand occupying leaves 1 to 53 (recto), and a more careless hand leaves 53 (verso) to 64, which are of the same parchment as the earlier leaves, and leaves 65 to 239, which are of a thinner and rougher parchment showing traces of writing of the 8th—9th c. (a Greek version of the Old Testament). The MS. contains the Elements I.—XIII. and some scholia after Books XI., XII. and XIII.

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This manuscript is written in two hands, the finer hand occupying leaves 1 to 53 (recto), and a more careless hand leaves 53 (verso) to 64, which are of the same parchment as the earlier leaves, and leaves 65 to 239, which are of a thinner and rougher parchment showing traces of writing of the 8th—9th c. (a Greek version of the Old Testament). The MS. contains the Elements I.—XIII. and some scholia after Books XI., XII. and XIII.

(7) q = Paris MS. 2344, folio; 12th c.

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It is written by one hand but includes scholia by many hands. On leaves 1 to 16 (recto) are scholia with the same title as that found by Wachsmuth in a Vatican MS. and relied upon by him to prove that Proclus continued his commentaries beyond Book I.[ei)s t]a\ tou= *eu)klei/dou stoixei=a prolambano/mena e)k tw=n *pro/klou spora/dhn kai\ kat) e)pitomh/n. Cf. p. 32, note 8, above. Leaves 17 to 357 contain the Elements I.—XIII. (except that there is a lacuna from the middle of VIII. 25 to the e)/kqesis of IX. 14); before Books VII. and X. there are some leaves filled with scholia only, and leaves 358 to 366 contain nothing but scholia.

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(8) Heiberg also used a palimpsest in the British Museum (Add. 17211). Five pages are of the 7th—8th c. and are contained (leaves 49-53) in the second volume of the Syrian MS. Brit. Mus. 687 of the 9th c.; half of leaf 50 has perished. The leaves contain various fragments from Book X. enumerated by Heiberg, Vol. III., p. v, and nearly the whole of XIII. 14.

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Since his edition of the Elements was published, Heiberg has collected further material bearing on the history of the textHeiberg, Paralipomena zu Euklid in Hermes, XXXVIII., 1903, pp. 46-74, 161-201, 321-356.. Besides giving the results of further or new examination of MSS., he has collected the fresh evidence contained in an-Nairīzī's commentary, and particularly in the quotations from Heron's commentary given in it (often word for word), which enable us in several cases to trace differences between our text and the text as Heron had it, and to identify some interpolations which actually found their way into the text from Heron's commentary itself; and lastly he has dealt with some valuable fragments of ancient papyri which have recently come to light, and which are especially important in that the evidence drawn from them necessitates some modification in the views expressed in the preface to Vol. V. as to the nature of the changes made in Theon's recension, and in the principles laid down for differentiating between Theon's recension and the original text, on the basis of a comparison between P and the Theonine MSS. alone.

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It is written by one hand but includes scholia by many hands. On leaves 1 to 16 (recto) are scholia with the same title as that found by Wachsmuth in a Vatican MS. and relied upon by him to prove that Proclus continued his commentaries beyond Book I.[ei)s t]a\ tou= *eu)klei/dou stoixei=a prolambano/mena e)k tw=n *pro/klou spora/dhn kai\ kat) e)pitomh/n. Cf. p. 32, note 8, above. Leaves 17 to 357 contain the Elements I.—XIII. (except that there is a lacuna from the middle of VIII. 25 to the e)/kqesis of IX. 14); before Books VII. and X. there are some leaves filled with scholia only, and leaves 358 to 366 contain nothing but scholia.

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(8) Heiberg also used a palimpsest in the British Museum (Add. 17211). Five pages are of the 7th—8th c. and are contained (leaves 49-53) in the second volume of the Syrian MS. Brit. Mus. 687 of the 9th c.; half of leaf 50 has perished. The leaves contain various fragments from Book X. enumerated by Heiberg, Vol. III., p. v, and nearly the whole of XIII. 14.

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Since his edition of the Elements was published, Heiberg has collected further material bearing on the history of the textHeiberg, Paralipomena zu Euklid in Hermes, XXXVIII., 1903, pp. 46-74, 161-201, 321-356.. Besides giving the results of further or new examination of MSS., he has collected the fresh evidence contained in an-Nairīzī's commentary, and particularly in the quotations from Heron's commentary given in it (often word for word), which enable us in several cases to trace differences between our text and the text as Heron had it, and to identify some interpolations which actually found their way into the text from Heron's commentary itself; and lastly he has dealt with some valuable fragments of ancient papyri which have recently come to light, and which are especially important in that the evidence drawn from them necessitates some modification in the views expressed in the preface to Vol. V. as to the nature of the changes made in Theon's recension, and in the principles laid down for differentiating between Theon's recension and the original text, on the basis of a comparison between P and the Theonine MSS. alone.

The fragments of ancient papyri referred to are the following.

1. Papyrus Herculanensis No. 1061.Described by Heiberg in Oversigt over det kngl. danske Videnskabernes Selskabs Forhandlinger, 1900, p. 161.

@@ -598,12 +598,12 @@

(The discovery of the ancient papyrus showing readings agreeing with some, or with all, of the Theonine MSS. against P now makes it necessary to be very cautious in applying these criteria.)

It is of course the last class (d) of changes which we have to investigate in order to get a proper idea of Theon's recension.

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Heiberg first observes, as regards these, that we shall find that Theon, in editing the Elements, altered hardly anything without some reason, often inadequate according to our ideas, but still some reason which seemed to him sufficient. Hence, in cases of very slight differences where both the Theonine MSS. and P have readings good and probable in themselves, Heiberg is not prepared to put the differences down to Theon. In those passages where we cannot see the least reason why Theon, if he had the reading of P before him, should have altered it, Heiberg would not at once assume the superiority of P unless there was such a consistency in the differences as would indicate that they were due not to accident but to design. In the absence of such indications, he thinks that the ordinary principles of criticism should be followed and that proper weight should be attached to the antiquity of the sources. And it cannot be denied that the sources of the Theonine version are the more ancient. For not only is the British Museum palimpsest (L), which is intimately connected with the rest of our MSS., át least two centuries older than P, but the other Theonine MSS. are so nearly allied that they must be held to have had a common archetype intermediate between them and the actual edition of Theon; and, since they themselves are as old as, or older than P, their archetype must have been much older. Heiberg gives (pp. xlvi, xlvii) a list of passages where, for this reason, he has followed the Theonine MSS. in preference to P.

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Heiberg first observes, as regards these, that we shall find that Theon, in editing the Elements, altered hardly anything without some reason, often inadequate according to our ideas, but still some reason which seemed to him sufficient. Hence, in cases of very slight differences where both the Theonine MSS. and P have readings good and probable in themselves, Heiberg is not prepared to put the differences down to Theon. In those passages where we cannot see the least reason why Theon, if he had the reading of P before him, should have altered it, Heiberg would not at once assume the superiority of P unless there was such a consistency in the differences as would indicate that they were due not to accident but to design. In the absence of such indications, he thinks that the ordinary principles of criticism should be followed and that proper weight should be attached to the antiquity of the sources. And it cannot be denied that the sources of the Theonine version are the more ancient. For not only is the British Museum palimpsest (L), which is intimately connected with the rest of our MSS., at least two centuries older than P, but the other Theonine MSS. are so nearly allied that they must be held to have had a common archetype intermediate between them and the actual edition of Theon; and, since they themselves are as old as, or older than P, their archetype must have been much older. Heiberg gives (pp. xlvi, xlvii) a list of passages where, for this reason, he has followed the Theonine MSS. in preference to P.

It has been mentioned above that the copyist of P or rather of its archetype wished to give an ancient recension. Therefore (apart from clerical errors and interpolations) the first hand in P may be relied upon as giving a genuine reading even where a correction by the first hand has been made at the same time. But in many places the first hand has made corrections afterwards; on these occasions he must have used new sources, e.g. when inserting the scholia to the first Book which P alone has, and in a number of passages he has made additions from Theonine MSS.

We cannot make out any family tree for the different Theonine MSS. Although they all proceeded from a common archetype later than the edition of Theon itself, they cannot have been copied one from the other; for, if they had been, how could it have come about that in one place or other each of them agrees alone with P in preserving the genuine reading? Moreover the great variety in their agreements and disagreements indicates that they have all diverged to about the same extent from their archetype. As we have seen that P contains corrections from the Theonine family, so they show corrections from P or other MSS. of the same family. Thus V has part of the lacuna in the MS. from which it was copied filled up from a MS. similar to P, and has corrections apparently derived from the same; the copyist, however, in correcting V, also used another MS. to which he alludes in the additions to IX. 19 and 30 (and also on X. 23 Por.): in the book of the Ephesian (this) is not found. Who this Ephesian of the 12th c. was, we do not know.

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We now come to the alterations made by Theon in his edition of the Elements. I shall indicate classes into which these alterations may be divided but without details (except in cases where they affect the mathematical content as distinct from form or language pure and simple).Exhaustive details under all the different heads are given by Heiberg (Vol. v. pp. lii—lxxv)..

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We now come to the alterations made by Theon in his edition of the Elements. I shall indicate classes into which these alterations may be divided but without details (except in cases where they affect the mathematical content as distinct from form or language pure and simple).Exhaustive details under all the different heads are given by Heiberg (Vol. v. pp. lii—lxxv)..

I. Alterations made by Theon where he found, or thought he found, mistakes in the original.

1. Real blots in the original which Theon saw and tried to remove.

@@ -669,7 +669,7 @@ may also be mentioned, the proper distinction between the words having been ignored as it was by Theon also. But there are a number of imperfections in the ante-Theonine text which it would be unsafe to put down to the errors of copyists, those namely where the good MSS. agree and it is not possible to see any motive that a copyist could have had for altering a correct reading. In these cases it is possible that the imperfections are due to a certain degree of carelessness on the part of Euclid himself; for it is not possible Euclidem ab omni naevo vindicare, to use the words of SaccheriEuclides ab omni naevo vindicatus, Mediolani, 1733., and consequently Simson is not right in attributing to Theon and other editors all the things in Euclid to which mathematical objection can be taken. Thus, when Euclid speaks of the ratio compounded of the sides for the ratio compounded of the ratios of the sides, - there is no reason for doubting that Euclid himself is responsible for the more slip-shod expression. Again, in the Books XI.—XIII. relating to solid geometry there are blots neither few nor altogether unimportant which can only be attributed to Euclid himselfCf. especially the assumption, without proof or definition, of the criterion for equal solid angles, and the incomplete proof of XII. 17.; and there is the less reason for hesitation in so attributing them because solid geometry was then being treated in a thoroughly systematic manner for the first time. Sometimes the conclusion (sumpe/rasma) of a proposition does not correspond exactly to the enunciation, often it is cut short with the words kai\ ta\ e(ch=s and the rest + there is no reason for doubting that Euclid himself is responsible for the more slip-shod expression. Again, in the Books XI.—XIII. relating to solid geometry there are blots neither few nor altogether unimportant which can only be attributed to Euclid himselfCf. especially the assumption, without proof or definition, of the criterion for equal solid angles, and the incomplete proof of XII. 17.; and there is the less reason for hesitation in so attributing them because solid geometry was then being treated in a thoroughly systematic manner for the first time. Sometimes the conclusion (sumpe/rasma) of a proposition does not correspond exactly to the enunciation, often it is cut short with the words kai\ ta\ e(ch=s and the rest (especially from Book X. onwards), and very often in Books VIII., IX. it is omitted. Where all the MSS. agree, there is no ground for hesitating to attribute the abbreviation or omission to Euclid; though, of course, where one or more MSS. have the longer form, it must be retained because this is one of the cases where a copyist has a temptation to abbreviate.

Where the true reading is preserved in one of the Theonine MSS. alone, Heiberg attributes the wrong reading to a mistake which arose before Theon's time, and the right reading of the single MS. to a successful correction.

We now come to the most important question of the Interpolations introduced before Theon's time.

@@ -678,9 +678,9 @@ or that of X. 90 it is possible to prove more shortly (suntomw/teron). Now it is impossible to suppose that Euclid would have given one proof as that definitely accepted by him and then added another with the express comment that the latter has certain advantages over the former. Had he considered the two proofs and come to this conclusion, he would have inserted the latter in the received text instead of the former. These alternative proofs must therefore have been interpolated. The same argument applies to alternatives introduced with the words or even thus (h)\ kai\ ou(/tws), or even otherwise - (h)\ kai\ a)/llws). Under this head come the alternatives for the last portions of III. 7, 8; and Heiberg also compares the alternatives for parts of III. 31 (that the angle in a semicircle is a right angle) and XIII. 18, and the alternative proof of the lemma after X. 32. The alternatives to X. 105 and 106, again, are condemned by the place in which they occur, namely after an alternative proof to X. 115. The above alternatives being all admitted to be spurious, suspicion must necessarily attach to the few others which are in themselves unobjectionable. Heiberg instances the alternative proofs to III. 9, III. 10, VI. 30, VI. 31 and XI. 22, observing that it is quite comprehensible that any of these might have occurred to a teacher or editor and seemed to him, rightly or wrongly, to be better than the corresponding proofs in Euclid. Curiously enough, Simson adopted the alternatives to III. 9, 10 in preference to the genuine proofs. Since Heiberg's preface was written, his suspicion has been amply confirmed as regards III. 10 by the commentary of an-Nairīzī (ed. Curtze) which shows not only that this alternative is Heron's, but also that the substantive proposition III. 12 in Euclid is also Heron's, having been given by him to supplement III. II which must originally have been enunciated of circles touching one another + (h)\ kai\ a)/llws). Under this head come the alternatives for the last portions of III. 7, 8; and Heiberg also compares the alternatives for parts of III. 31 (that the angle in a semicircle is a right angle) and XIII. 18, and the alternative proof of the lemma after X. 32. The alternatives to X. 105 and 106, again, are condemned by the place in which they occur, namely after an alternative proof to X. 115. The above alternatives being all admitted to be spurious, suspicion must necessarily attach to the few others which are in themselves unobjectionable. Heiberg instances the alternative proofs to III. 9, III. 10, VI. 30, VI. 31 and XI. 22, observing that it is quite comprehensible that any of these might have occurred to a teacher or editor and seemed to him, rightly or wrongly, to be better than the corresponding proofs in Euclid. Curiously enough, Simson adopted the alternatives to III. 9, 10 in preference to the genuine proofs. Since Heiberg's preface was written, his suspicion has been amply confirmed as regards III. 10 by the commentary of an-Nairīzī (ed. Curtze) which shows not only that this alternative is Heron's, but also that the substantive proposition III. 12 in Euclid is also Heron's, having been given by him to supplement III. II which must originally have been enunciated of circles touching one another simply, i.e. so as to include the case of external as well as internal contact, though the proof covered the case of internal contact only. Euclid, in the 11th proposition, - says Heron, supposed two circles touching one another internally and wrote the proposition on this case, proving what it was required to prove in it. But I will show how it is to be proved if the contact be external.An-Nairīzī, ed. Curtze, p. 121.. + says Heron, supposed two circles touching one another internally and wrote the proposition on this case, proving what it was required to prove in it. But I will show how it is to be proved if the contact be external.An-Nairīzī, ed. Curtze, p. 121.. This additional proposition of Heron's is by way of adding another case, which brings us to that class of interpolation. It was the practice of Euclid and the ancients to give only one case (generally the most difficult one) and to leave the others to be investigated by the reader for himself. One interpolation of a second case (VI. 27) is due, as we have seen, to Theon. The two extra cases of XI. 23 were manifestly interpolated before Theon's time, for the preliminary distinction of three cases, (the centre) will either be within the triangle LMN, or on one of the sides, or outside. First let it be within, is a spurious addition (B and V only). Similarly an unnecessary case is interpolated in III. 11.

II. Lemmas.

@@ -710,7 +710,7 @@ an actual proof is nevertheless given. Clearly the proofs are interpolated; and there are other similar interpolations. There are also interpolations of intermediate steps in proofs, unnecessary explanations and so on, as to which I need not enter into details.

Lastly, following Heiberg's order, I come to

VII. Interpolated definitions, axioms etc.

-

Apart from VI. Def. 5 (which may have been interpolated by Theon although it is found written in the margin of P by the first hand), the definition of a segment of a circle in Book I. is interpolated, as is clear from the fact that it occurs in a more appropriate place in Book III. and Proclus omits it. VI. Def. 2 (reciprocal figures) is rightly condemned by Simson—perhaps it was taken from Heron—and Heiberg would reject VII. Def. 10, as to which see my note on that definition. Lastly the double definition of a solid angle (XI. Def. 11) constitutes a difficulty. The use of the word e)pifa/neia suggests that the first definition may have been older than Euclid, and he may have quoted it from older elements, especially as his own definition which follows only includes solid angles contained by planes, whereas the other includes other sorts (cf. the words grammw=n, grammai=s) which are also distinguished by Heron (Def. 22). If the first definition had come last, it could have been rejected without hesitation: but it is not so easy to reject the first part up to and including otherwise +

Apart from VI. Def. 5 (which may have been interpolated by Theon although it is found written in the margin of P by the first hand), the definition of a segment of a circle in Book I. is interpolated, as is clear from the fact that it occurs in a more appropriate place in Book III. and Proclus omits it. VI. Def. 2 (reciprocal figures) is rightly condemned by Simson—perhaps it was taken from Heron—and Heiberg would reject VII. Def. 10, as to which see my note on that definition. Lastly the double definition of a solid angle (XI. Def. 11) constitutes a difficulty. The use of the word e)pifa/neia suggests that the first definition may have been older than Euclid, and he may have quoted it from older elements, especially as his own definition which follows only includes solid angles contained by planes, whereas the other includes other sorts (cf. the words grammw=n, grammai=s) which are also distinguished by Heron (Def. 22). If the first definition had come last, it could have been rejected without hesitation: but it is not so easy to reject the first part up to and including otherwise (a)/llws). No difficulty need be felt about the definitions of oblong, rhombus, and rhomboid, @@ -728,7 +728,7 @@ CHAPTER VI. THE SCHOLIA. -

Heiberg has collected scholia, to the number of about 1500, in Vol. v. of his edition of Euclid, and has also discussed and classified them in a separate short treatise, in which he added a few others.Heiberg, Om Scholierne til Euklids Elementer, Kjøbenhavn, 1888. The tract is written in Danish, but, fortunately for those who do not read Danish easily, the author has appended (pp. 70-78) a résumé in French.

+

Heiberg has collected scholia, to the number of about 1500, in Vol. v. of his edition of Euclid, and has also discussed and classified them in a separate short treatise, in which he added a few others.Heiberg, Om Scholierne til Euklids Elementer, Kjøbenhavn, 1888. The tract is written in Danish, but, fortunately for those who do not read Danish easily, the author has appended (pp. 70-78) a résumé in French.

These scholia cannot be regarded as doing much to facilitate the reading of the Elements. As a rule, they contain only such observations as any intelligent reader could make for himself. Among the few exceptions are XI. Nos. 33, 35 (where XI. 22, 23 are extended to solid angles formed by any number of plane angles), XII. No. 85 (where an assumption tacitly made by Euclid in XII. 17 is proved), IX. Nos. 28, 29 (where the scholiast has pointed out the error in the text of IX. 19).

Nor are they very rich in historical information; they cannot be compared in this respect with Proclus' commentary on Book I. or with those of Eutocius on Archimedes and Apollonius. But even under this head they contain some things of interest, e.g. II. No. 11 explaining that the gnomon was invented by geometers for the sake of brevity, and that its name was suggested by an incidental characteristic, namely that from it the whole is known (gnwri/zetai), either of the whole area or of the remainder, when it (the gnw/mwn) is either placed round or taken away ; II. No. 13, also on the gnomon; IV. No. 2 stating that Book IV. was the discovery of the Pythagoreans; V. No. 1 attributing the content of Book V. to Eudoxus; X. No. 1 with its allusion to the discovery of incommensurability by the Pythagoreans and to Apollonius' work on irrationals; X. No. 62 definitely attributing X. 9 to Theaetetus; XIII. No. 1 about the Platonic @@ -753,9 +753,9 @@

F=Scholia in F by the first hand.

Vat.=Scholia of the Vatican MS. 204 of the 10th c., which has these scholia on leaves 198-205 (the end is missing) as an independent collection. It does not contain the text of the Elements.

V^{c}=Scholia found on leaves 283-292 of V and written in the same hand as that part of the MS. itself which begins at leaf 235.

-

Vat. 192=a Vatican MS. of the 14th c. which contains, after (1) the Elements I.—XIII. (without scholia), (2) the Data with scholia, (3) Marinus on the Data, the Schol. Vat. as an independent collection and in their entirety, beginning with 1. No. 88 and ending with XIII. No. 44.

-

The Schol. Vat., the most ancient and important collection of scholia, comprise those which are found in PBF Vat. and, from VII. 12 to IX. 15, in PB Vat. only, since in that portion of the Elements F was restored by a later hand without scholia; they also include 1. No. 88 which only happens to be erased in F, and IX. Nos. 28, 29 which may be left out because F. here has a different text. In F and Vat. the collection ends with Book X.; but it must also include Schol. PB of Books XI.—XIII., since these are found along with Schol. Vat. to Books I.—X. in several MSS. (of which Vat. 192 is one) as a separate collection. The Schol. Vat. to Books X.—XIII. are also found in the collection V^{c} (where, curiously enough, XIII. Nos. 43, 44 are at the beginning). The Schol. Vat. accordingly include Schol. PBV^{c} Vat. 192, and doubtless also those which are found in two of these sources. The total number of scholia classified by Heiberg as Schol. Vat. is 138.

-

As regards the contents of Schol. Vat. Heiberg has the following observations. The thirteen scholia to Book I. are extracts made from Proclus by a writer thoroughly conversant with the subject, and cleverly recast (with some additions). Their author does not seem to have had the two lacunae which our text of Proclus has (at the end of the note on I. 36 and the beginning of the next note, and at the beginning of the note on I. 43), for the scholia I. Nos. 125 and 137 seem to fill the gaps appropriately, at least in part. In some passages he had better readings than our MSS. have. The rest of Schol. Vat. (on Books II.—XIII.) are essentially of the same character as those on Book I., containing prolegomena, remarks on the object of the propositions, critical remarks on the text, converses, lemmas; they are, in general, exact and true to tradition. The reason of the resemblance between them and Proclus appears to be due to the fact that they have their origin in the commentary of Pappus, of which we know that Proclus also made use. In support of the view that Pappus is the source, heiberg places some of the Schol. Vat. to Book X. side by side with passages from the commentary of Pappus in the Arabic translation discovered by Woepcke;Om Scholierne til Euklids Elementer, pp. 11, 12: cf. Euklid-Stulien, pp. 170, 171; Woepcke, Mémoires présent. à l' Acad. des Sciences, 1856, XIV. p. 658 sqq.; he also refers to the striking confirmation afforded by the fact that XII. No. 2 contains the solution of the problem of inscribing in a given circle a polygon similar to a polygon inscribed in another circle, which problem Eutocius saysArchimedes, ed. Heiberg, III. P. 28, 19-22. that Pappus gave in his commentary on the Elements.

+

Vat. 192=a Vatican MS. of the 14th c. which contains, after (1) the Elements I.—XIII. (without scholia), (2) the Data with scholia, (3) Marinus on the Data, the Schol. Vat. as an independent collection and in their entirety, beginning with 1. No. 88 and ending with XIII. No. 44.

+

The Schol. Vat., the most ancient and important collection of scholia, comprise those which are found in PBF Vat. and, from VII. 12 to IX. 15, in PB Vat. only, since in that portion of the Elements F was restored by a later hand without scholia; they also include 1. No. 88 which only happens to be erased in F, and IX. Nos. 28, 29 which may be left out because F. here has a different text. In F and Vat. the collection ends with Book X.; but it must also include Schol. PB of Books XI.—XIII., since these are found along with Schol. Vat. to Books I.—X. in several MSS. (of which Vat. 192 is one) as a separate collection. The Schol. Vat. to Books X.—XIII. are also found in the collection V^{c} (where, curiously enough, XIII. Nos. 43, 44 are at the beginning). The Schol. Vat. accordingly include Schol. PBV^{c} Vat. 192, and doubtless also those which are found in two of these sources. The total number of scholia classified by Heiberg as Schol. Vat. is 138.

+

As regards the contents of Schol. Vat. Heiberg has the following observations. The thirteen scholia to Book I. are extracts made from Proclus by a writer thoroughly conversant with the subject, and cleverly recast (with some additions). Their author does not seem to have had the two lacunae which our text of Proclus has (at the end of the note on I. 36 and the beginning of the next note, and at the beginning of the note on I. 43), for the scholia I. Nos. 125 and 137 seem to fill the gaps appropriately, at least in part. In some passages he had better readings than our MSS. have. The rest of Schol. Vat. (on Books II.—XIII.) are essentially of the same character as those on Book I., containing prolegomena, remarks on the object of the propositions, critical remarks on the text, converses, lemmas; they are, in general, exact and true to tradition. The reason of the resemblance between them and Proclus appears to be due to the fact that they have their origin in the commentary of Pappus, of which we know that Proclus also made use. In support of the view that Pappus is the source, heiberg places some of the Schol. Vat. to Book X. side by side with passages from the commentary of Pappus in the Arabic translation discovered by Woepcke;Om Scholierne til Euklids Elementer, pp. 11, 12: cf. Euklid-Stulien, pp. 170, 171; Woepcke, Mémoires présent. à l' Acad. des Sciences, 1856, XIV. p. 658 sqq.; he also refers to the striking confirmation afforded by the fact that XII. No. 2 contains the solution of the problem of inscribing in a given circle a polygon similar to a polygon inscribed in another circle, which problem Eutocius saysArchimedes, ed. Heiberg, III. P. 28, 19-22. that Pappus gave in his commentary on the Elements.

But, on the other hand, Schol. Vat. contain some things which cannot have come from Pappus, e.g. the allusion in X. No. 1 to Theon and irrational surfaces and solids, Theon being later than Pappus; III. No. 10 about porisms is more like Proclus' treatment of the subject than Pappus', though one expression recalls that of Pappus about forming (sxhmati/zesqai) the enunciations of porisms like those of either theorems or problems.

The Schol. Vat. give us important indications as regards the text of the Elements as Pappus had it. In particular, they show that he could not have had in his text certain of the lemmas in Book X. For example, three of these are identical with what we find in Schol. Vat. (the lemma to X. 17=Schol. X. No. 106, and the lemmas to X. 54, 60 come in Schol. X. No. 328); and it is not possible to suppose. that these lemmas, if they were already in the text, would also be given as scholia. Of these three lemmas, that before X. 60 has already been condemned for other reasons; the other two, unobjectionable in themselves, must be rejected on the ground now stated. There were four others against which Heiberg found nothing to urge when writing his prolegomena to Vol. v., viz. the lemmas before X. 42, X. 14, X. 22 and X. 33. Of these, the lemma to X. 22 is not reconcilable with Schol. X. No. 161, which takes up the assumption in the text of Eucl. X. 22 as if no lemma had gone before. The lemma to X. 42, which, on account of the words introducing it (see p. 60 above), Heiberg at first hesitated to regard as an interpolation, is identical with Schol. X. No. 270. It is true that in Schol. X. No. 269 we find the words this lemma has been proved before (e)n toi=s e)/mprosqen), but it shall also be proved now for convenience' sake (tou= e(toi/mou e(/neka), and it is possible to suppose that before @@ -769,12 +769,12 @@ relating to a passage in the text immediately preceding, which second lemma belongs to Schol. Vat. and is taken from Pappus, the third in all probability came from Pappus also. The same is true of Schol. XII. No. 72 and XIII. No. 69, which are respectively identical with the propositions vulgo XI. 38 (Heiberg, App. to Book XI., No. 3) and XIII. 6; for both of these interpolations are older than Theon. Moreover most of the scholia which P in the first hand alone has are of the same character as Schol. Vat. Thus VII. No. 7 and XIII. No. I introducing Books VII. and XIII. respectively are of the same historical character as several of Schol. Vat.; that VII. No. 7 appears in the text of P at the beginning of Book VII. constitutes no difficulty. There are a number of converses, remarks on the relation of propositions to one another, explanations such as XII. No. 89 in which it is remarked that *f, *w in Euclid's figure to XII. 17 (Z, V in my figure) are really the same point but that this makes no difference in the proof. Two other Schol. P on XII. 17 are connected by their headings with XII. No. 72 mentioned above. XI. No. 10 (P) is only another form of XI. No. 11 (B); and B often, alone with P, has preserved Schol. Vat. On the whole Heiberg considers some 40 scholia found in P alone to belong to Schol. Vat.

-

The history of Schol. Vat. appears to have been, in its main outlines, the following. They were put together after 500 A.D., since they contain extracts from Proclus, to which we ought not to assign a date too near to that of Proclus' work itself; and they must at least be earlier than the latter half of the 9th c., in which B was written. As there must evidently have been several intermediate links between the archetype and B, we must assign them rather to the first half of the period between the two dates, and it is not improbable that they were a new product of the great development of mathematical studies at the end of the 6th c. (Isidorus of Miletus). The author extracted what he found of interest in the commentary of Proclus on Book I. and in that of Pappus on the rest of the work, and put these extracts in the margin of a MS. of the class of P. As there are no scholia to I. 1-22, the first leaves of the archetype or of one of the earliest copies must have been lost at an early date, and it was from that mutilated copy that partly P and partly a MS. of the Theonine class were taken, the scholia being put in the margin in both. Then the collection spread through the Theonine MSS., gradually losing some scholia which could not be read or understood, or which were accidentally or deliberately omitted. Next it was extracted from one of these MSS. and made into a separate work which has been preserved, in part, in its entirety (Vat. 192 etc.) and, in part, divided into sections, so that the scholia to Books X.—XIII. were detached (V^{c}). It had the same fate in the MSS. which kept the original arrangement (in the margin), and in consequence there are some MSS. where the scholia to the stereometric Books are missing, those Books having come to be less read in the period of decadence. It is from one of these MSS. that the collection was extracted as a separate work such as we find it in Vat. (10th c.).

+

The history of Schol. Vat. appears to have been, in its main outlines, the following. They were put together after 500 A.D., since they contain extracts from Proclus, to which we ought not to assign a date too near to that of Proclus' work itself; and they must at least be earlier than the latter half of the 9th c., in which B was written. As there must evidently have been several intermediate links between the archetype and B, we must assign them rather to the first half of the period between the two dates, and it is not improbable that they were a new product of the great development of mathematical studies at the end of the 6th c. (Isidorus of Miletus). The author extracted what he found of interest in the commentary of Proclus on Book I. and in that of Pappus on the rest of the work, and put these extracts in the margin of a MS. of the class of P. As there are no scholia to I. 1-22, the first leaves of the archetype or of one of the earliest copies must have been lost at an early date, and it was from that mutilated copy that partly P and partly a MS. of the Theonine class were taken, the scholia being put in the margin in both. Then the collection spread through the Theonine MSS., gradually losing some scholia which could not be read or understood, or which were accidentally or deliberately omitted. Next it was extracted from one of these MSS. and made into a separate work which has been preserved, in part, in its entirety (Vat. 192 etc.) and, in part, divided into sections, so that the scholia to Books X.—XIII. were detached (V^{c}). It had the same fate in the MSS. which kept the original arrangement (in the margin), and in consequence there are some MSS. where the scholia to the stereometric Books are missing, those Books having come to be less read in the period of decadence. It is from one of these MSS. that the collection was extracted as a separate work such as we find it in Vat. (10th c.).

II. The second great division of the scholia is Schol. Vind.

This title is taken from the Viennese MS. (V), and the letters used by Heiberg to indicate the sources here in question are as follows.

V^{a}=scholia in V written by the same hand that copied the MS. itself from fol. 235 onward.

q=scholia of the Paris MS. 2344 (q) written by the first hand.

-

l=scholia of the Florence MS. Laurent. XXVIII, 2 written in the 13th—14th c., mostly in the first hand, but partly in two later hands.

+

l=scholia of the Florence MS. Laurent. XXVIII, 2 written in the 13th—14th c., mostly in the first hand, but partly in two later hands.

V^{b}=scholia in V written by the same hand as the first part (leaves 1-183) of the MS. itself; V^{b} wrote his scholia after V^{a}.

q^{1}=scholia of the Paris MS. (q) found here and there in another hand of early date.

Schol. Vind. include scholia found in V^{a}q. 1 is nearly related to q; and in fact the three MSS. which, so far as Euclid's text is concerned, show no direct interdependence, are, as regards their scholia, derived from one original. Heiberg proves this by reference to the readings of the three in two passages (found in Schol. 1. No. 109 and X. No. 39 respectively). The common source must have contained, besides the scholia found in the three MSS. V^{a}ql, those also which are contained in two of them, for it is more unlikely that two of the three should contain common interpolations than that a particular scholium should drop out of one of them. Besides V^{a} and q, the scholia V^{b} and q^{1} must equally be referred to Schol. Vind., since the greater part of their scholia are found in 1. There is a lacuna in q from Eucl. VIII. 25 to IX. 14, so that for this portion of the Elements Schol. Vind. are represented by Vl only. Heiberg gives about 450 numbers in all as belonging to this collection.

@@ -783,20 +783,20 @@

The scholia to Book I. are here also extracts from Proclus, but more copious and more verbatim than in Schol. Vat. The author has not always understood Proclus; and he had a text as bad as that of our MSS., with the same lacunae. The scholia to the other Books are partly drawn (1) from Schol. Vat., the MSS. representing Schol. Vind. and Schol. Vat. in these cases showing nearly all possible combinations; but there is no certain trace in Schol. Vind. of the scholia peculiar to P. The author used a copy of Schol. Vat. in the form in which they were attached to the Theonine text; thus Schol. Vind. correspond to BF Vat., where these diverge from P, and especially closely to B. Besides Schol. Vat., the editors of Schol. Vind. used (2) other old collections of scholia of which we find traces in B and F; Schol. Vind. have also some scholia common with b. The scholia which Schol. Vind. have in common with BF come from two different sources, and were apparently afterwards introduced into the other MSS.; one result of this is that several scholia are reproduced twice.

But, besides the scholia derived from these sources, Schol. Vind. contain a large number of others of late date, characterised by incorrect language or by triviality of content (there are many examples in numbers, citations of propositions used, absurd a)pori/ai, and the like). Unlike Schol. Vat., these scholia often quote words from Euclid as a heading (in one case a heading is inserted in Schol. Vind. where a scholium without the heading is quoted from Schol. Vat., see V. No. 14). The explanations given often presuppose very little knowledge on the part of the reader and frequently contain obscurities and gross errors.

Schol. Vind. were collected for use with a MS. of the Theonine class; this follows from the fact that they contain a note on the proposition vulgo VII. 22 interpolated by Theon (given in Heiberg's App. to Vol. 11. p. 430). Since the scholium to VII. 39 given in V and p in the text after the title of Book VIII. quotes the proposition as VII. 39, it follows that this scholium must have been written before the interpolation of the two propositions vulgo VII. 20, 22; Schol. Vind. contain (VII. No. 80) the first sentence of it, but without the heading referring to VII. 39. Schol. VII. No. 97 quotes VII. 33 as VII. 34, so that the proposition vulgo VII. 22 may have stood in the scholiast's text but not the later interpolation vulgo VII. 20 (later because only found in B in the margin by the first hand). Of course the scholiast had also the interpolations earlier than Theon.

-

For the date of the collection we have a lower limit in the date (12th c.) of MSS. in which the scholia appear. That it was not much earlier than the 12th c. is indicated (1) by the poverty of its contents, (2) by the quality of the MS. of Proclus which was used in the compilation of it (the Munich MS. used by Friedlein with which the scholiast's excerpts are essentially in agreement belongs to the 11th— 12th c.), (3) by the fact that Schol. Vind. appear only in MSS. of the 12th c. and no trace of them is found in our MSS. belonging to the 9th—10th c. in which Schol. Vat. are found. The collection may therefore probably be assigned to the 11th c. Perhaps it may be in part due to Psellus who lived towards the end of that century: for in a Florence MS. (Magliabecch. XI, 53 of the 15th c.) containing a mathematical compendium intended for use in the reading of Aristotle the scholia 1. Nos. 40 and 49 appear with the name of Psellus attached.

+

For the date of the collection we have a lower limit in the date (12th c.) of MSS. in which the scholia appear. That it was not much earlier than the 12th c. is indicated (1) by the poverty of its contents, (2) by the quality of the MS. of Proclus which was used in the compilation of it (the Munich MS. used by Friedlein with which the scholiast's excerpts are essentially in agreement belongs to the 11th— 12th c.), (3) by the fact that Schol. Vind. appear only in MSS. of the 12th c. and no trace of them is found in our MSS. belonging to the 9th—10th c. in which Schol. Vat. are found. The collection may therefore probably be assigned to the 11th c. Perhaps it may be in part due to Psellus who lived towards the end of that century: for in a Florence MS. (Magliabecch. XI, 53 of the 15th c.) containing a mathematical compendium intended for use in the reading of Aristotle the scholia 1. Nos. 40 and 49 appear with the name of Psellus attached.

Schol. Vind. are not found without the admixture of foreign elements in any of our three sources. In 1 there are only very few such in the first hand. In q there are several new scholia in the first hand, for the most part due to the copyist himself. The collection of scholia on Book X. in q (Heiberg's q^{c}) is also in the first hand; it is not original, and it may perhaps be due to Psellus (Maglb. has some definitions of Book X. with a heading scholia of...Michael Psellus on the definitions of Euclid's 10th Element and Schol. X. No. 9), whose name must have been attached to it in the common source of Maglb. and q; to a great extent it consists of extracts from Schol. Vind. taken from the same source as Vl. The scholia q^{1} (in an ancient hand in q), confined to Book II., partly belong to Schol. Vind. and partly correspond to b^{1} (Bologna MS.). q^{a} and q^{b} are in one hand (Theodorus Antiochita), the nearest to the first hand of q; they are doubtless due to an early possessor of the MS. of whom we know nothing more.

-

V^{a} has, besides Schol. Vind., a number of scholia which also appear in other MSS., one in BFb, some others in P, and some in v (Codex Vat. 1038, 13th c.); these scholia were taken from a source in which many abbreviations were used, as they were often misunderstood by V^{a}. Other scholia in V^{a} which are not found in the older sources—some appearing in V^{a} alone—are also not original, as is proved by mistakes or corruptions which they contain; some others may be due to the copyist himself.

+

V^{a} has, besides Schol. Vind., a number of scholia which also appear in other MSS., one in BFb, some others in P, and some in v (Codex Vat. 1038, 13th c.); these scholia were taken from a source in which many abbreviations were used, as they were often misunderstood by V^{a}. Other scholia in V^{a} which are not found in the older sources—some appearing in V^{a} alone—are also not original, as is proved by mistakes or corruptions which they contain; some others may be due to the copyist himself.

V^{b} seldom has scholia common with the other older sources; for the most part they either appear in V^{b} alone or only in the later sources as v or F^{2} (later scholia in F), some being original, others not. In Book X. V^{b} has three series of numerical examples, (1) with Greek numerals, (2) alternatives added later, also mostly with Greek numerals, (3) with Arabic numerals. The last class were probably the work of the copyist himself. These examples (cf. p. 74 below) show the facility with which the Byzantines made calculations at the date of the MS. (12th c.). They prove also that the use of the Arabic numerals (in the East-Arabian form) was thoroughly established in the 12th c.; they were actually known to the Byzantines a century earlier, since they appear, in the first hand, in an Escurial MS. of the 11th c.

-

Of collections in other hands in V distinguished by Heiberg (see preface to Vol. v.), V^{1} has very few scholia which are found in other sources, the greater part being original; V^{2}, V^{3} are the work of the copyist himself; V^{4} are so in part only, and contain several scholia from Schol. Vat. and other sources. V^{3} and V^{4} are later than 13th —14th c., since they are not found in f (cod. Laurent. XXVIII, 6) which was copied from V and contains, besides V^{a} V^{b}, the greater part of V^{1} and VI. No. 20 of V^{2} (in the text).

+

Of collections in other hands in V distinguished by Heiberg (see preface to Vol. v.), V^{1} has very few scholia which are found in other sources, the greater part being original; V^{2}, V^{3} are the work of the copyist himself; V^{4} are so in part only, and contain several scholia from Schol. Vat. and other sources. V^{3} and V^{4} are later than 13th—14th c., since they are not found in f (cod. Laurent. XXVIII, 6) which was copied from V and contains, besides V^{a} V^{b}, the greater part of V^{1} and VI. No. 20 of V^{2} (in the text).

In P there are, besides P^{3} (a quite late hand, probably one of the old Scriptores Graeci at the Vatican), two late hands (P^{2}), one of which has some new and independent scholia, while the other has added the greater part of Schol. Vind., partly in the margin and partly on pieces of leaves stitched on.

Our sources for Schol. Vat. also contain other elements. In P there were introduced a certain number of extracts from Proclus, to supplement Schol. Vat. to Book I.; they are all written with a different ink from that used for the oldest part of the MS., and the text is inferior. There are additions in the other sources of Schol. Vat. (F and B) which point to a common source for FB and which are nearly all found in other MSS., and, in particular, in Schol. Vind., which also used the same source; that they are not assignable to Schol. Vat. results only from their not being found in Vat. Of other additions in F, some are peculiar to F and some common to it and b; but they are not original. F^{2} (scholia in a later hand in F) contains three original scholia; the rest come from V. B contains, besides scholia common to it and F, b or other sources, several scholia which seem to have been put together by Arethas, who wrote at least a part of them with his own hand.

Heiberg has satisfied himself, by a closer study of b, that the scholia which he denotes by b, b and b^{1} are by one hand; they are mostly to be found in other sources as well, though some are original. By the same hand (Theodorus Cabasilas, 15th c.) are also the scholia denoted by b^{2}, B^{2}, b^{3} and B^{3}. These scholia come in great part from Schol. Vind., and in making these extracts Theodorus probably used one of our sources, l, mistakes in which often correspond to those of Theodorus. To one scholium is attached the name of Demetrius (who must be Demetrius Cydonius, a friend of Nicolaus Cabasilas, 14th c.); but it could not have been written by him, since it appears in B and Schol. Vind. Nor are all the scholia which bear the name of Theodorus due to Theodorus himself, though some are so.

As B^{3} (a late hand in B) contains several of the original scholia of b^{2}, B^{3} must have used b itself as his source, and, as all the scholia in B^{3} are in b, the latter is also the source of the scholia in B^{3} which are found in other MSS. B and b were therefore, in the 15th c., in the hands of the same person; this explains, too, the fact that b in a late hand has some scholia which can only come from B. We arrive then at the conclusion that Theodorus Cabasilas, in the 15th c., owned both the MSS. B and b, and that he transferred to B scholia which he had before written in b, either independently or after other sources, and inversely transferred some scholia from B to b. Further, B^{2} are earlier than Theodorus Cabasilas, who certainly himself wrote B^{3} as well as b^{2} and b^{3}.

-

An author's name is also attached to the scholia VI. No. 6 and X. No. 223, which are attributed to Maximus Planudes (end of 13th c.) along with scholia on I. 31, X. 14 and X. 18 found in 1 in a quite late hand and published on pp. 46, 47 of Heibėrg's dissertation. These seem to have been taken from lectures of Planudes on the Elements by a pupil who used l as his copy.

+

An author's name is also attached to the scholia VI. No. 6 and X. No. 223, which are attributed to Maximus Planudes (end of 13th c.) along with scholia on I. 31, X. 14 and X. 18 found in 1 in a quite late hand and published on pp. 46, 47 of Heiberg's dissertation. These seem to have been taken from lectures of Planudes on the Elements by a pupil who used l as his copy.

There are also in l two other Byzantine scholia, written by a late hand, and bearing the names Ioannes and Pediasimus respectively; these must in like manner have been written by a pupil after lectures of Ioannes Pediasimus (first half of 14th c.), and this pupil must also have used l.

Before these scholia were edited by Heiberg, very few of them had been published in the original Greek. The Basel editio princeps has a few (V. No. I, VI. Nos. 3, 4 and some in Book X.) which are taken, some from the Paris MS. (Paris. Gr. 2343) used by Grynaeus, others probably from the Venice MS. (Marc. 301) also used by him; one published by Heiberg, not in his edition of Euclid but in his paper on the scholia, may also be from Venet. 301, but appears also in Paris. Gr. 2342. The scholia in the Basel edition passed into the Oxford edition in the text, and were also given by August in the Appendix to his Vol. II.

-

Several specimens of the two series of scholia (Vat. and Vind.) were published by C. Wachsmuth (Rhein. Mus. XVIII. p. 132 sqq.) and by Knoche (Untersuchungen über die neu aufgefundenen Scholien des Proklus, Herford, 1865).

+

Several specimens of the two series of scholia (Vat. and Vind.) were published by C. Wachsmuth (Rhein. Mus. XVIII. p. 132 sqq.) and by Knoche (Untersuchungen über die neu aufgefundenen Scholien des Proklus, Herford, 1865).

The scholia published in Latin were much more numerous. G. Valla (De expetendis et fugiendis rebus, 1501) reproduced apparently some 200 of the scholia included in Heiberg's edition. Several of these he obtained from two Modena MSS. which at one time were in his possession (Mutin. III B, 4 and II E, 9, both of the 15th c.); but he must have used another source as well, containing extracts from other series of scholia, notably Schol. Vind. with which he has some 87 scholia in common. He has also several that are new.

Commandinus included in his translation under the title Scholia antiqua the greater part of the Schol. Vat. which he certainly obtained from a MS. of the class of Vat. 192; on the whole he adhered closely to the Greek text. Besides these scholia Commandinus has the scholia and lemmas which he found in the Basel editio princeps, and also three other scholia not belonging to Schol. Vat., as well as one new scholium (to XII. 13) not included in Heiberg's edition, which are distinguished by different type and were doubtless taken from the Greek MS. used by him along with the Basel edition.

@@ -807,51 +807,51 @@ Isaak Monachus is doubtless Isaak Argyrus, 14th c.; and Dasypodius used a MS. in which, besides the passage in Hultsch's Variae Collectiones, there were a number of scholia marked in the margin with the name of Isaak (cf. those in b under the name of Theodorus Cabasilas). Whether the new scholia are original cannot be decided until they are published in Greek; but it is not improbable that they are at all events independent arrangements of older scholia. All but five of the others, and all but one of the Greek scholia to Book v., are taken from Schol. Vat.; three of the excepted ones are from Schol. Vind., and the other three seem to come from F (where some words of them are illegible, but can be supplied by means of Mut. III B, 4, which has these three scholia and generally shows a certain likeness to Isaak's scholia).

Dasypodius also published in 1564 the arithmetical commentary of Barlaam the monk (14th c.) on Eucl. Book II., which finds a place in Appendix IV. to the Scholia in Heiberg's edition.

Hultsch has some remarks on the origin of the scholiaArt. Eukleides - in Pauly-Wissowa's Real-Encyclopädie.. He observes that the scholia to Book I. contain a considerable portion of Geminus' commentary on the definitions and are specially valuable because they contain extracts from Geminus only, whereas Proclus, though drawing mainly upon him, quotes from others as well. On the postulates and axioms the scholia give more than is found in Proclus. Hultsch conjectures that the scholium on Book v., No. 3, attributing the discovery of the theorems to Eudoxus but their arrangement to Euclid, represents the tradition going back to Geminus, and that the scholium XIII., No. 1, has the same origin.

-

A word should be added about the numerical illustrations of Euclid's propositions in the scholia to Book X. They contain a large number of calculations with sexagesimal fractionsHultsch has written upon these in Bibliotheca Mathematica, V_{3}, 1904, pp. 225-233.; the fractions go as far as fourth-sixtieths (1/60^{4}). Numbers expressed in these fractions are handled with skill and include some results of surprising accuracyThus \sgrt{(27)} is given (allowing for a slight correction by means of the context) as 5 II' 46'' 10''', which gives for \sgrt{3} the value 1 43' 55'' 23''', being the same value as that given by Hipparchus in his Table of Chords, and correct to the seventh decimal place. Similarly \sgrt{8} is given as 2 49' 42'' 20''' 10'''', which is equivalent to\sgrt{2}=1.41421335. Hultsch gives instances of the various operations, addition, subtraction, etc., carried out in these fractions, and shows how the extraction of the square root was effected. Cf. T. L. Heath, Históry of Greek Mathematics, 1., pp. 59-63.

+ in Pauly-Wissowa's Real-Encyclopädie.. He observes that the scholia to Book I. contain a considerable portion of Geminus' commentary on the definitions and are specially valuable because they contain extracts from Geminus only, whereas Proclus, though drawing mainly upon him, quotes from others as well. On the postulates and axioms the scholia give more than is found in Proclus. Hultsch conjectures that the scholium on Book v., No. 3, attributing the discovery of the theorems to Eudoxus but their arrangement to Euclid, represents the tradition going back to Geminus, and that the scholium XIII., No. 1, has the same origin.

+

A word should be added about the numerical illustrations of Euclid's propositions in the scholia to Book X. They contain a large number of calculations with sexagesimal fractionsHultsch has written upon these in Bibliotheca Mathematica, V_{3}, 1904, pp. 225-233.; the fractions go as far as fourth-sixtieths (1/60^{4}). Numbers expressed in these fractions are handled with skill and include some results of surprising accuracyThus \sgrt{(27)} is given (allowing for a slight correction by means of the context) as 5 II' 46'' 10''', which gives for \sgrt{3} the value 1 43' 55'' 23''', being the same value as that given by Hipparchus in his Table of Chords, and correct to the seventh decimal place. Similarly \sgrt{8} is given as 2 49' 42'' 20''' 10'''', which is equivalent to\sgrt{2}=1.41421335. Hultsch gives instances of the various operations, addition, subtraction, etc., carried out in these fractions, and shows how the extraction of the square root was effected. Cf. T. L. Heath, Históry of Greek Mathematics, 1., pp. 59-63.

 CHAPTER VII. EUCLID IN ARABIA. -

We are told by [Hnull ]ājī KhalfaLexico&ndot; bibliogr. et encyclop. ed. Flügel, III. pp. 91, 92. that the Caliph al-Mansūr (754-775) sent a mission to the Byzantine Emperor as the result of which he obtained from him a copy of Euclid among other Greek books, and again that the Caliph al-Ma'mūn (813-833) obtained manuscripts of Euclid, among others, from the Byzantines. The version of the Elements by al-[Hnull ]ajjāj b. Yūsuf b. Matar is, if not the very first, at least one of the first books translated from the Greek into ArabicKlamroth, Zeitschrift der Deutschen Morgenländischen Gesellschaft, XXXV. p. 303.. According to the FihristFihrist (tr. Suter), p. 16. it was translated by al-[Hnull ]ajjāj twice; the first translation was known as Hārūni - (for Hārūn -), the second bore the name Ma'mūni - (for al-Ma'mūn -) and was the more trustworthy. Six Books of the second of these versions survive in a Leiden MS. (Codex Leidensis 399, 1) now in part published by Besthorn and HeibergCodex Leidensis 399, 1. Euclidis Elementa ex interpretatione al-Hadschdschadschii cum commentariis al-Narizii, Hauniae, part 1. i. 1893, part I. ii. 1897, part II. i. 1900, part II. ii. 1905, part III. i. 1910.. In the preface to this MS. it is stated that, in the reign of Hārūn ar-Rashīd (786-809), al-[Hnull ]ajjāj was commanded by Ya[hnull ]yā b. Khālid b. Barmak to translate the book into Arabic. Then, when al-Ma'mūn became Caliph, as he was devoted to learning, al-[Hnull ]ajjāj saw that he would secure the favour of al-Ma'mūn if he illustrated and expounded this book and reduced it to smaller dimensions. He accordingly left out the superfluities, filled up the gaps, corrected or removed the errors, until he had gone through the book and reduced it, when corrected and explained, to smaller dimensions, as in this copy, but without altering the substance, for the use of men endowed with ability and devoted to learning, the earlier edition being left in the hands of readers. +

We are told by [Hnull ]ājī KhalfaLexicoṅ bibliogr. et encyclop. ed. Flügel, III. pp. 91, 92. that the Caliph al-Mansūr (754-775) sent a mission to the Byzantine Emperor as the result of which he obtained from him a copy of Euclid among other Greek books, and again that the Caliph al-Ma'mūn (813-833) obtained manuscripts of Euclid, among others, from the Byzantines. The version of the Elements by al-[Hnull ]ajjāj b. Yūsuf b. Matar is, if not the very first, at least one of the first books translated from the Greek into ArabicKlamroth, Zeitschrift der Deutschen Morgenländischen Gesellschaft, XXXV. p. 303.. According to the FihristFihrist (tr. Suter), p. 16. it was translated by al-[Hnull ]ajjāj twice; the first translation was known as Hārūni + (for Hārūn +), the second bore the name Ma'mūni + (for al-Ma'mūn +) and was the more trustworthy. Six Books of the second of these versions survive in a Leiden MS. (Codex Leidensis 399, 1) now in part published by Besthorn and HeibergCodex Leidensis 399, 1. Euclidis Elementa ex interpretatione al-Hadschdschadschii cum commentariis al-Narizii, Hauniae, part 1. i. 1893, part I. ii. 1897, part II. i. 1900, part II. ii. 1905, part III. i. 1910.. In the preface to this MS. it is stated that, in the reign of Hārūn ar-Rashīd (786-809), al-[Hnull ]ajjāj was commanded by Ya[hnull ]yā b. Khālid b. Barmak to translate the book into Arabic. Then, when al-Ma'mūn became Caliph, as he was devoted to learning, al-[Hnull ]ajjāj saw that he would secure the favour of al-Ma'mūn if he illustrated and expounded this book and reduced it to smaller dimensions. He accordingly left out the superfluities, filled up the gaps, corrected or removed the errors, until he had gone through the book and reduced it, when corrected and explained, to smaller dimensions, as in this copy, but without altering the substance, for the use of men endowed with ability and devoted to learning, the earlier edition being left in the hands of readers.

-

The Fihrist goes on to say that the work was next translated by Ishāq b. Hunain, and that this translation was improved by Thābit b. Qurra. This Abū Ya`qūb Is[hnull ]āq b. [Hnull ]unain b. Is[hnull ]āq al-`Ibādī (d. 910) was the son of the most famous of Arabic translators, Hunain b. Ishāq al-`Ibādī (809-873), a Christian and physician to the Caliph alMutawakkil (847-861). There seems to be no doubt that Is[hnull ]āq, who must have known Greek as well as his father, made his translation direct from the Greek. The revision must apparently have been the subject of an arrangement between Is[hnull ]āq and Thābit as the latter died in 901 or nine years before Is[hnull ]āq. Thābit undoubtedly consulted Greek MSS. for the purposes of his revision. This is expressly stated in a marginal note to a Hebrew version of the Elements, made from Ishāq's, attributed to one of two scholars belonging to the same family, viz. either to Moses b. Tibbon (about 1244-1274) or to Jakob b. Machir (who died soon after 1306)Steinschneider, Zeitschrift für Math. u. Physik, XXXI., hist.-litt. Abtheilung, pp. 85, 86, 99.. Moreover Thābit observes, on the proposition which he gives as IX. 31, that he had not found this proposition and the one before it in the Greek but only in the Arabic; from which statement Klamroth draws two conclusions, (1) that the Arabs had already begun to interest themselves in the authenticity of the text and (2) that Thābit did not alter the numbers of the propositions in Ishāq's translationKlamroth, p. 279.. The Fihrist also says that Yu[hnull ]annā al-Qass (i.e. the Priest -) had seen in the Greek copy in his possession the proposition in Book I. which Thābit took credit for, and that this was confirmed by Na[znull ]īf, the physician, to whom Yuhannā had shown it. This proposition may have been wanting in Ishāq, and Thābit may have added it, but without claiming it as his own discoverySteinschneider, p. 88.. As a fact, I. 45 is missing in the translation by al-[Hnull ]ajjāj.

-

The original version of Is[hnull ]āq without the improvements by Thābit has probably not survived any more than the first of the two versions by al-Hajjāj; the divergences between the MSS. are apparently due to the voluntary or involuntary changes of copyists, the former class varying according to the degree of mathematical knowledge possessed by the copyists and the extent to which they were influenced by considerations of practical utility for teaching purposesKlamroth, p. 306.. Two MSS. of the Ishāq-Thābit version exist in the Bodleian Library (No. 279 belonging to the year 1238, and No. 280 written in 1260-1)These MSS. are described by Nicoll and Pusey, Catalogus cod. mss. orient. bibl. Bodleianae, pt. II. 1835 (pp. 257-262).; Books I.—XIII. are in the Is[hnull ]āq-Thābit version, the non-Euclidean Books XIV., XV. in the translation of Qustā b. L'ūqā al-Ba`labakkī (d. about 912). The first of these MSS. (No. 279) is that (O) used by Klamroth for the purpose of his paper on the Arabian Euclid. The other MS. used by Klamroth is (K) Kjobenhavn LXXXI, undated but probably of the 13th c., containing Books V.—XV., Books V.—X. being in the Is[hnull ]āq-Thābit version, Books XI.—XIII. purporting to be in al-Hajjāj's translation, and Books XIV., XV. in the version of Qus&tnull;ā b. Lūqā. In not a few propositions K and O show not the slightest difference, and, even where the proofs show considerable differences, they are generally such that, by a careful comparison, it is possible to reconstruct the common archetype, so that it is fairly clear that we have in these cases, not two recensions of one translation, but arbitrarily altered and shortened copies of one and the same recensionKlamroth, pp. 306-8.. The Bodleian MS. No. 280 contains a preface, translated by Nîcoll, which cannot be by Thābit himself because it mentions Avicenna (980-1037) and other later authors. The MS. was written at Marāġa in the year 1260-1 and has in the margin readings and emendations from the edition of Na&snull;ĩraddĩn a&tnull;-[Tnull ]ĩsī (shortly to be mentioned) who was living at Marāġa at the time. Is it possible that a&tnull;-[Tnull ]ūsī himself is the author of the prefaceSteinschneider, p. 98. Heiberg has quoted the whole of this preface in the Zeitschrift fűr Math. u. Physik, XXIX., hist.-litt. Abth. p. 16.? Be this as it may, the preface is interesting because it throws light on the liberties which the Arabians allowed themselves to take with the text. After the observation that the book (in spite of the labours of many editors) is not free from errors, obscurities, redundancies, omissions etc., and is without certain definitions necessary for the proofs, it goes on to say that the man has not yet been found who could make it perfect, and next proceeds to explain (1) that Avicenna cut out postulates and many definitions - and attempted to clear up difficult and obscure passages, (2) that Abū'l Wafā al-Būzjānĩ (939-997) introduced unnecessary additions and left out many things of great importance and entirely necessary, - inasmuch as he was too long in various places in Book VI. and too short in Book X. where he left out entirely the proofs of the apotomae, while he made an unsuccessful attempt to emend XII. 14, (3) that Abū Ja`far al-Khāzin (d. between 961 and 971) arranged the postulates excellently but disturbed the number and order of the propositions, reduced several propositions to one +

The Fihrist goes on to say that the work was next translated by Ishāq b. Hunain, and that this translation was improved by Thābit b. Qurra. This Abū Ya`qūb Is[hnull ]āq b. [Hnull ]unain b. Is[hnull ]āq al-`Ibādī (d. 910) was the son of the most famous of Arabic translators, Hunain b. Ishāq al-`Ibādī (809-873), a Christian and physician to the Caliph alMutawakkil (847-861). There seems to be no doubt that Is[hnull ]āq, who must have known Greek as well as his father, made his translation direct from the Greek. The revision must apparently have been the subject of an arrangement between Is[hnull ]āq and Thābit as the latter died in 901 or nine years before Is[hnull ]āq. Thābit undoubtedly consulted Greek MSS. for the purposes of his revision. This is expressly stated in a marginal note to a Hebrew version of the Elements, made from Ishāq's, attributed to one of two scholars belonging to the same family, viz. either to Moses b. Tibbon (about 1244-1274) or to Jakob b. Machir (who died soon after 1306)Steinschneider, Zeitschrift für Math. u. Physik, XXXI., hist.-litt. Abtheilung, pp. 85, 86, 99.. Moreover Thābit observes, on the proposition which he gives as IX. 31, that he had not found this proposition and the one before it in the Greek but only in the Arabic; from which statement Klamroth draws two conclusions, (1) that the Arabs had already begun to interest themselves in the authenticity of the text and (2) that Thābit did not alter the numbers of the propositions in Ishāq's translationKlamroth, p. 279.. The Fihrist also says that Yu[hnull ]annā al-Qass (i.e. the Priest +) had seen in the Greek copy in his possession the proposition in Book I. which Thābit took credit for, and that this was confirmed by Naẓīf, the physician, to whom Yuhannā had shown it. This proposition may have been wanting in Ishāq, and Thābit may have added it, but without claiming it as his own discoverySteinschneider, p. 88.. As a fact, I. 45 is missing in the translation by al-[Hnull ]ajjāj.

+

The original version of Is[hnull ]āq without the improvements by Thābit has probably not survived any more than the first of the two versions by al-Hajjāj; the divergences between the MSS. are apparently due to the voluntary or involuntary changes of copyists, the former class varying according to the degree of mathematical knowledge possessed by the copyists and the extent to which they were influenced by considerations of practical utility for teaching purposesKlamroth, p. 306.. Two MSS. of the Ishāq-Thābit version exist in the Bodleian Library (No. 279 belonging to the year 1238, and No. 280 written in 1260-1)These MSS. are described by Nicoll and Pusey, Catalogus cod. mss. orient. bibl. Bodleianae, pt. II. 1835 (pp. 257-262).; Books I.—XIII. are in the Is[hnull ]āq-Thābit version, the non-Euclidean Books XIV., XV. in the translation of Qustā b. L'ūqā al-Ba`labakkī (d. about 912). The first of these MSS. (No. 279) is that (O) used by Klamroth for the purpose of his paper on the Arabian Euclid. The other MS. used by Klamroth is (K) Kjobenhavn LXXXI, undated but probably of the 13th c., containing Books V.—XV., Books V.—X. being in the Is[hnull ]āq-Thābit version, Books XI.—XIII. purporting to be in al-Hajjāj's translation, and Books XIV., XV. in the version of Qusṭā b. Lūqā. In not a few propositions K and O show not the slightest difference, and, even where the proofs show considerable differences, they are generally such that, by a careful comparison, it is possible to reconstruct the common archetype, so that it is fairly clear that we have in these cases, not two recensions of one translation, but arbitrarily altered and shortened copies of one and the same recensionKlamroth, pp. 306-8.. The Bodleian MS. No. 280 contains a preface, translated by Nîcoll, which cannot be by Thābit himself because it mentions Avicenna (980-1037) and other later authors. The MS. was written at Marāġa in the year 1260-1 and has in the margin readings and emendations from the edition of Naṣĩraddĩn aṭ-[Tnull ]ĩsī (shortly to be mentioned) who was living at Marāġa at the time. Is it possible that aṭ-[Tnull ]ūsī himself is the author of the prefaceSteinschneider, p. 98. Heiberg has quoted the whole of this preface in the Zeitschrift fűr Math. u. Physik, XXIX., hist.-litt. Abth. p. 16.? Be this as it may, the preface is interesting because it throws light on the liberties which the Arabians allowed themselves to take with the text. After the observation that the book (in spite of the labours of many editors) is not free from errors, obscurities, redundancies, omissions etc., and is without certain definitions necessary for the proofs, it goes on to say that the man has not yet been found who could make it perfect, and next proceeds to explain (1) that Avicenna cut out postulates and many definitions + and attempted to clear up difficult and obscure passages, (2) that Abū'l Wafā al-Būzjānĩ (939-997) introduced unnecessary additions and left out many things of great importance and entirely necessary, + inasmuch as he was too long in various places in Book VI. and too short in Book X. where he left out entirely the proofs of the apotomae, while he made an unsuccessful attempt to emend XII. 14, (3) that Abū Ja`far al-Khāzin (d. between 961 and 971) arranged the postulates excellently but disturbed the number and order of the propositions, reduced several propositions to one etc. Next the preface describes the editor's own claimsThis seems to include a rearrangement of the contents of Books XIV., XV. added to the Elements. and then ends with the sentences, But we have kept to the order of the books and propositions in the work itself (i.e. Euclid's) except in the twelfth and thirteenth books. For we have dealt in Book XIII. with the (solid) bodies and in Book XII. with the surfaces by themselves.

-

After Thābit the Fihrist mentions Abū ’Uthmān ad-Dimashqī as having translated some Books of the Elements including Book X. (It is Abū ’Uthmān's translation of Pappus’ commentary on Book X. which Woepcke discovered at Paris.) The Fihrist adds also that Na[znull ]ĩf the physician told me that he had seen the tenth Book of Euclid in Greek, that it had 40 propositions more than the version in common circulation which had 109 propositions, and that he had determined to translate it into Arabic. +

After Thābit the Fihrist mentions Abū ’Uthmān ad-Dimashqī as having translated some Books of the Elements including Book X. (It is Abū ’Uthmān's translation of Pappus’ commentary on Book X. which Woepcke discovered at Paris.) The Fihrist adds also that Naẓīf the physician told me that he had seen the tenth Book of Euclid in Greek, that it had 40 propositions more than the version in common circulation which had 109 propositions, and that he had determined to translate it into Arabic.

-

But the third form of the Arabian Euclid actually accessible to us is the edition of Abū Ja`far Mu[hnull ]. b. Mu[hnull ]. b. al-[Hnull ]asan Na&snull;raddĩn aţ-Ţĩsī (whom we shall call aţ-Ţĩsī for short), born at Tĩs (in Khurāsān) in 1201 (d. 1274). This edition appeared in two forms, a larger and a smaller. The larger is said to survive in Florence only (Pal. 272 and 313, the latter MS. containing only six Books); this was published at Rome in 1594, and, remarkably enough, some copies of this edition are to be found with 12 and some with 13 Books, some with a Latin title and some withoutSuter, Die Mathematiker und Astronomen der Araber, p. 151. The Latin title is Euclidis elementorum geometricorum libri tredecim. Ex traditione doctissimi Nasiridini Tusini nunc primum arabice impressi. Romae in typographia Medicea MDXCIV. Cum licentia superiorum.. But the book was printed in Arabic, so that Kästner remarks that he will say as much about it as can be said about a book which one cannot readKästner, Geschichte der Mathematik, I. p. 367.. The shorter form, which however, in most MSS., is in 15 Books, survives at Berlin, Munich, Oxford, British Museum (974, 1334Suter has a note that this MS. is very old, having been copied from the original in the author's lifetime., 1335), Paris (2465, 2466), India Office, and Constantinople; it was printed at Constantinople in 1801, and the first six Books at Calcutta in 1824Suter, p. 151..

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A&tnull;-[Tnull ]ũsĩ's work is however not a translation of Euclid's text, but a re-written Euclid based on the older Arabic translations. In this respect it seems to be like the Latin version of the Elements by Campanus (Campano), which was first published by Erhard Ratdolt at Venice in 1482 (the first printed edition of EuclidDescribed by Kästner, Geschichte der Mathematik, I. pp. 289-299, and by Weissenborn, Die Ubersetzungen des Euklid durch Campano und Zamberti, Halle a. S., 1882, pp. 1-7. See also infra, Chapter VIII, p. 97.). Campanus (13th c.) was a mathematician, and it is likely enough that he allowed himself the same liberty as a&tnull;-[Tnull ]ĩsī in reproducing Euclid. Whatever may be the relation between Campanus’ version and that of Athelhard of Bath (about 1120), and whether, as Curtze thinksSonderabdruck des Jahresberichtes über die Fortschritte der klassischen Alterthumswissenschaft vom Okt. 1879-1882, Berlin, 1884., they both used one and the same Latin version of 10th—11th c., or whether Campanus used Athelhard's version in the same way as a&tnull;-[Tnull ]ĩsī used those of his predecessorsKlamroth, p. 271.,it is certain that both versions came from an Arabian source, as is evident from the occurrence of Arabic words in themCurtze, op. cit. p. 20; Heiberg, Euklid-Studien, p. 178.. Campanus’ version is not of much service for the purpose of forming a judgment on the relative authenticity of the Greek and Arabian tradition ; but it sometimes preserves traces of the purer source, as when it omits Theon's addition to VI. 33Heiberg's Euclid, vol. v. p. ci.. A curious circumstance is that, while Campanus’ version agrees with aţ-Ţĩsī's in the number of the propositions in all the genuine Euclidean Books except V. and IX., it agrees with Athelhard's in having 34 propositions in Book V. (as against 25 in other versions), which confirms the view that the two are not independent, and also leads, as Klamroth says, to this dilemma: either the additions to Book V. are Athelhard's own, or he used an Arabian Euclid which is not known to usKlamroth, pp. 273-4.. Heiberg also notes that Campanus’ Books XIV., XV. show a certain agreement with the preface to the Thābit-Is[hnull ]āq version, in which the author claims to have (1) given a method of inscribing spheres in the five regular solids, (2) carried further the solution of the problem how to inscribe any one of the solids in any other and (3) noted the cases where this could not be done.Heiberg, Zeitschrift fi<*>r Matk. u. Physik, XXIX., hist.-litt. Abtheilung, p. 21.

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With a view to arriving at what may be called a common measure of the Arabian tradition, it is necessary to compare, in the first place, the numbers of propositions in the various Books. [Hnull ]ājĩ Khalfa says that al-[Hnull ]ajjāj's translation contained 468 propositions, and Thābit's 478; this is stated on the authority of a&tnull;-[Tnull ]ĩsĩ, whose own edition contained 468Klamroth, p. 274; Steinschneider, Zeitschrift für Math. u. Physik, XXXI., hist.-litt. Abth. p. 98.. The fact that Thābit's version had 478 propositions is confirmed by an index in the Bodleian MS. 279 (called O by Klamroth). A register at the beginning of the Codex Leidensis 399, 1 which gives Is[hnull ]aq's numbers (although the translation is that of al-[Hnull ]ajjāj) apparently makes the total 479 propositions (the number in Book XIV. being apparently 11, instead of the 10 of OBesthorn-Heiberg read 11? +

But the third form of the Arabian Euclid actually accessible to us is the edition of Abū Ja`far Mu[hnull ]. b. Mu[hnull ]. b. al-[Hnull ]asan Naṣraddĩn aţ-Ţĩsī (whom we shall call aţ-Ţĩsī for short), born at Tĩs (in Khurāsān) in 1201 (d. 1274). This edition appeared in two forms, a larger and a smaller. The larger is said to survive in Florence only (Pal. 272 and 313, the latter MS. containing only six Books); this was published at Rome in 1594, and, remarkably enough, some copies of this edition are to be found with 12 and some with 13 Books, some with a Latin title and some withoutSuter, Die Mathematiker und Astronomen der Araber, p. 151. The Latin title is Euclidis elementorum geometricorum libri tredecim. Ex traditione doctissimi Nasiridini Tusini nunc primum arabice impressi. Romae in typographia Medicea MDXCIV. Cum licentia superiorum.. But the book was printed in Arabic, so that Kästner remarks that he will say as much about it as can be said about a book which one cannot readKästner, Geschichte der Mathematik, I. p. 367.. The shorter form, which however, in most MSS., is in 15 Books, survives at Berlin, Munich, Oxford, British Museum (974, 1334Suter has a note that this MS. is very old, having been copied from the original in the author's lifetime., 1335), Paris (2465, 2466), India Office, and Constantinople; it was printed at Constantinople in 1801, and the first six Books at Calcutta in 1824Suter, p. 151..

+

Aṭ-[Tnull ]ũsĩ's work is however not a translation of Euclid's text, but a re-written Euclid based on the older Arabic translations. In this respect it seems to be like the Latin version of the Elements by Campanus (Campano), which was first published by Erhard Ratdolt at Venice in 1482 (the first printed edition of EuclidDescribed by Kästner, Geschichte der Mathematik, I. pp. 289-299, and by Weissenborn, Die Ubersetzungen des Euklid durch Campano und Zamberti, Halle a. S., 1882, pp. 1-7. See also infra, Chapter VIII, p. 97.). Campanus (13th c.) was a mathematician, and it is likely enough that he allowed himself the same liberty as aṭ-[Tnull ]ĩsī in reproducing Euclid. Whatever may be the relation between Campanus’ version and that of Athelhard of Bath (about 1120), and whether, as Curtze thinksSonderabdruck des Jahresberichtes über die Fortschritte der klassischen Alterthumswissenschaft vom Okt. 1879-1882, Berlin, 1884., they both used one and the same Latin version of 10th—11th c., or whether Campanus used Athelhard's version in the same way as aṭ-[Tnull ]ĩsī used those of his predecessorsKlamroth, p. 271.,it is certain that both versions came from an Arabian source, as is evident from the occurrence of Arabic words in themCurtze, op. cit. p. 20; Heiberg, Euklid-Studien, p. 178.. Campanus’ version is not of much service for the purpose of forming a judgment on the relative authenticity of the Greek and Arabian tradition ; but it sometimes preserves traces of the purer source, as when it omits Theon's addition to VI. 33Heiberg's Euclid, vol. v. p. ci.. A curious circumstance is that, while Campanus’ version agrees with aţ-Ţĩsī's in the number of the propositions in all the genuine Euclidean Books except V. and IX., it agrees with Athelhard's in having 34 propositions in Book V. (as against 25 in other versions), which confirms the view that the two are not independent, and also leads, as Klamroth says, to this dilemma: either the additions to Book V. are Athelhard's own, or he used an Arabian Euclid which is not known to usKlamroth, pp. 273-4.. Heiberg also notes that Campanus’ Books XIV., XV. show a certain agreement with the preface to the Thābit-Is[hnull ]āq version, in which the author claims to have (1) given a method of inscribing spheres in the five regular solids, (2) carried further the solution of the problem how to inscribe any one of the solids in any other and (3) noted the cases where this could not be done.Heiberg, Zeitschrift fi<*>r Matk. u. Physik, XXIX., hist.-litt. Abtheilung, p. 21.

+

With a view to arriving at what may be called a common measure of the Arabian tradition, it is necessary to compare, in the first place, the numbers of propositions in the various Books. [Hnull ]ājĩ Khalfa says that al-[Hnull ]ajjāj's translation contained 468 propositions, and Thābit's 478; this is stated on the authority of aṭ-[Tnull ]ĩsĩ, whose own edition contained 468Klamroth, p. 274; Steinschneider, Zeitschrift für Math. u. Physik, XXXI., hist.-litt. Abth. p. 98.. The fact that Thābit's version had 478 propositions is confirmed by an index in the Bodleian MS. 279 (called O by Klamroth). A register at the beginning of the Codex Leidensis 399, 1 which gives Is[hnull ]aq's numbers (although the translation is that of al-[Hnull ]ajjāj) apparently makes the total 479 propositions (the number in Book XIV. being apparently 11, instead of the 10 of OBesthorn-Heiberg read 11? as the number, Klamroth had read it as 21 (p. 273).). I subjoin a table of relative numbers taken from Klamroth, to which I have added the corresponding numbers in August's and Heiberg's editions of the Greek text.

The numbers in the case of Heiberg include all propositions which he has printed in the text; they include therefore XIII. 6 and III. 12 now to be regarded as spurious, and X. 112-115 which he brackets as doubtful. He does not number the propositions in Books XIV., XV., but I conclude that the numbers in P reach at least 9 in XIV., and 9 in XV.

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The Fihrist confirms the number 109 for Book X., from which Klamroth concludes that Is[hnull ]āq's version was considered as by far the most authoritative.

+

The Fihrist confirms the number 109 for Book X., from which Klamroth concludes that Is[hnull ]āq's version was considered as by far the most authoritative.

In the text of O, Book IV. consists of 17 propositions and Book XIV. of 12, differing in this respect from its own table of contents; IV. 15, 16 in O are really two proofs of the same proposition.

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In al-[Hnull ]ajjāj's version Book I. consists of 47 propositions only, I. 45 being omitted. It has also one proposition fewer in Book III., the Heronic proposition III. 12 being no doubt omitted.

+

In al-[Hnull ]ajjāj's version Book I. consists of 47 propositions only, I. 45 being omitted. It has also one proposition fewer in Book III., the Heronic proposition III. 12 being no doubt omitted.

In speaking of particular propositions, I shall use Heiberg's numbering, except where otherwise stated.

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The difference of 10 propositions between Thābit-Is[hnull ]āq and a&tnull;-[Tnull ]ĩsī is accounted for thus:

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(1) The three propositions VI. 12 and X. 28, 29 which both Is[hnull ]āq and the Greek text have are omitted in a&tnull;-[Tnull ]ĩsī.

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(2) Is[hnull ]āq divides each of the propositions XIII. 1-3 into two, making six instead of three in a&tnull;-[Tnull ]ĩsī and in the Greek.

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(3) Is[hnull ]āq has four propositions (numbered by him VIII. 24, 25, IX. 30, 31) which are neither in the Greek Euclid nor in aţ-Ţĩsī.

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Apart from the above differences al-[Hnull ]ajjāj (so far as we know), Ishāq and a&tnull;-[Tnull ]ĩsī agree; but their Euclid shows many differences from our Greek text. These differences we will classify as followsSee Klamroth, pp. 275-6, 280, 282-4, 314-15, 326;Heiberg, vol. v. pp. xcvi, xcvii..

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The difference of 10 propositions between Thābit-Is[hnull ]āq and aṭ-[Tnull ]ĩsī is accounted for thus:

+

(1) The three propositions VI. 12 and X. 28, 29 which both Is[hnull ]āq and the Greek text have are omitted in aṭ-[Tnull ]ĩsī.

+

(2) Is[hnull ]āq divides each of the propositions XIII. 1-3 into two, making six instead of three in aṭ-[Tnull ]ĩsī and in the Greek.

+

(3) Is[hnull ]āq has four propositions (numbered by him VIII. 24, 25, IX. 30, 31) which are neither in the Greek Euclid nor in aţ-Ţĩsī.

+

Apart from the above differences al-[Hnull ]ajjāj (so far as we know), Ishāq and aṭ-[Tnull ]ĩsī agree; but their Euclid shows many differences from our Greek text. These differences we will classify as followsSee Klamroth, pp. 275-6, 280, 282-4, 314-15, 326;Heiberg, vol. v. pp. xcvi, xcvii..

1. Propositions.

The Arabian Euclid omits VII. 20, 22 of Gregory's and August's editions (Heiberg, App. to Vol. II. pp. 428-32); VIII. 16, 17; X. 7, 8, 13, 16, 24, 112, 113, 114, besides a lemma vulgo X. 13, the proposition X. 117 of Gregory's edition, and the scholium at the end of the Book (see for these Heiberg's Appendix to Vol. III. pp. 382, 408-416); XI. 38 in Gregory and August (Heiberg, App. to Vol. IV. p. 354); XII. 6, 13, 14; (also all but the first third of Book XV.).

The Arabian Euclid makes III. 11, 12 into one proposition, and divides some propositions (X. 31, 32; XI. 31, 34; XIII. 1-3) into two each.

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The order is also changed in the Arabic to the following extent. V. 12, 13 are interchanged and the order in Books VI., VII., IX.— XIII. is:

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The order is also changed in the Arabic to the following extent. V. 12, 13 are interchanged and the order in Books VI., VII., IX.— XIII. is:

VI. 1-8, 13, 11, 12, 9, 10, 14-17, 19, 20, 18, 21, 22, 24, 26, 23, 25, 27-30, 32, 31, 33.

VII. 1-20, 22, 21, 23-28, 31, 32, 29, 30, 33-39.

IX. 1-13, 20, 14-19, 21-25, 27, 26, 28-36, with two new propositions coming before prop. 30.

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X. 1-6, 9-12, 15, 14, 17-23, 26-28, 25, 29-30, 31, 32, 33— 111, 115.

+

X. 1-6, 9-12, 15, 14, 17-23, 26-28, 25, 29-30, 31, 32, 33— 111, 115.

XI. 1-30, 31, 32, 34, 33, 35-39.

XII. 1-5, 7, 9, 8, 10, 12, 11, 15, 16-18.

XIII. 1-3, 5, 4, 6, 7, 12, 9, 10, 8, 11, 13, 15, 14, 16-18.

@@ -868,77 +868,77 @@

The analyses and syntheses to XIII. 1-5 are also omitted in the Arabic.

Klamroth is inclined, on a consideration of all these differences, to give preference to the Arabian tradition over the Greek (1) on historical grounds, subject to the proviso that no Greek MS. as ancient as the 8th c. is found to contradict his conclusions, which are based generally (2) on the improbability that the Arabs would have omitted so much if they had found it in their Greek MSS., it being clear from the Fihrist that the Arabs had already shown an anxiety for a pure text, and that the old translators were subjected in this matter to the check of public criticism. Against the historical grounds, - Heiberg is able to bring a considerable amount of evidenceHeiberg in Zeitschrift für Math. u. Physik, XXIX., hist.-litt. Abth. p. 3 sqq.. First of all there is the British Museum palimpsest (L) of the 7th or the beginning of the 8th c. This has fragments of propositions in Book X. which are omitted in the Arabic; the numbering of one proposition, which agrees with the numbering in other Greek MS., is not comprehensible on the assumption that eight preceding propositions were omitted in it, as they are in the Arabic; and lastly, the readings in L are tolerably like those of our MSS., and surprisingly like those of B. It is also to be noted that, although P dates from the 10th c. only, it contains, according to all appearance, an ante-Theonine recension.

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Moreover there is positive evidence against certain omissions by the Arabians. A&tnull;-[Tnull ]ĩsī omits VI. 12, but it is scarcely possible that, if Eutocius had not had it, he would have quoted VI. 23 by that numberApollonius, ed. Heiberg, vol. 11. p. 218, 3-5.. This quotation of VI. 23 by Eutocius also tells against Is[hnull ]āq who has the proposition as VI. 25. Again, Simplicius quotes VI. 10 by that number, whereas it is VI. 13 in Is[hnull ]āq; and Pappus quotes, by number, XIII. 2 (Is[hnull ]āq 3, 4), XIII. 4 (Is[hnull ]āq 8), XIII. 16 (Is[hnull ]āq 19). On the other hand the contraction of III. 11, 12 into one proposition in the Arabic tells in favour of the Arabic.

+ Heiberg is able to bring a considerable amount of evidenceHeiberg in Zeitschrift für Math. u. Physik, XXIX., hist.-litt. Abth. p. 3 sqq.. First of all there is the British Museum palimpsest (L) of the 7th or the beginning of the 8th c. This has fragments of propositions in Book X. which are omitted in the Arabic; the numbering of one proposition, which agrees with the numbering in other Greek MS., is not comprehensible on the assumption that eight preceding propositions were omitted in it, as they are in the Arabic; and lastly, the readings in L are tolerably like those of our MSS., and surprisingly like those of B. It is also to be noted that, although P dates from the 10th c. only, it contains, according to all appearance, an ante-Theonine recension.

+

Moreover there is positive evidence against certain omissions by the Arabians. Aṭ-[Tnull ]ĩsī omits VI. 12, but it is scarcely possible that, if Eutocius had not had it, he would have quoted VI. 23 by that numberApollonius, ed. Heiberg, vol. 11. p. 218, 3-5.. This quotation of VI. 23 by Eutocius also tells against Is[hnull ]āq who has the proposition as VI. 25. Again, Simplicius quotes VI. 10 by that number, whereas it is VI. 13 in Is[hnull ]āq; and Pappus quotes, by number, XIII. 2 (Is[hnull ]āq 3, 4), XIII. 4 (Is[hnull ]āq 8), XIII. 16 (Is[hnull ]āq 19). On the other hand the contraction of III. 11, 12 into one proposition in the Arabic tells in favour of the Arabic.

Further, the omission of certain porisms in the Arabic cannot be supported; for Pappus quotes the porism to XIII. 17Pappus, V. p. 436, 5., Proclus those to II. 4, III. 1, VII. 2Proclus, pp. 303-4., and Simplicius that to IV. 15.

Lastly, some propositions omitted in the Arabic are required in later propositions. Thus X. 13 is used in X. 18, 22, 23, 26 etc.; X. 17 is wanted in X. 18, 26, 36; XII. 6, 13 are required for XII. 11 and XII. 15 respectively.

It must also be remembered that some of the things which were properly omitted by the Arabians are omitted or marked as doubtful in Greek MSS. also, especially in P, and others are rightly suspected for other reasons (e.g. a number of alternative proofs, lemmas, and porisms, as well as the analyses and syntheses of XIII. 1-5). On the other hand, the Arabic has certain interpolations peculiar to our inferior MSS. (cf. the definition VI. Def. 2 and those of proportion and ordered proportion).

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Heiberg comes to the general conclusion that, not only is the Arabic tradition not to be preferred offhand to that of the Greek MSS., but it must be regarded as inferior in authority. It is a question how far the differences shown in the Arabic are due to the use of Greek MSS. differing from those which have been most used as the basis of our text, and how far to the arbitrary changes made by the Arabians themselves. Changes of order and arbitrary omissions could not surprise us, in view of the preface above quoted from the Oxford MS. of Thābit-Is[hnull ]āq, with its allusion to the many important and necessary things left out by Abĩ'l Wafā and to the author's own rearrangement of Books XII., XIII. But there is evidence of differences due to the use by the Arabs of other Greek MSS. HeibergZeitschrift für Math. u. Physik, XXIX., hist.-litt. Abth. p. 6 sqq. is able to show considerable resemblances between the Arabic text and the Bologna MS. b in that part of the MS. where it diverges so remarkably from our other MSS. (see the short description of it above, p.49); in illustration he gives a comparison of the proofs of XII. 7 in b and in the Arabic respectively, and points to the omission in both of the proposition given in Gregory's edition as XI. 38, and to a remarkable agreement between them as regards the order of the propositions of Book XII. As above stated, the remarkable divergence of b only affects Books XI. (at end) and XII.; and Book XIII. in b shows none of the transpositions and other peculiarities of the Arabic. There are many differences between b and the Arabic, especially in the definitions of Book XI., as well as in Book XIII. It is therefore a question whether the Arabians made arbitrary changes, or the Arabic form is the more ancient, and b has been altered through contact with other MSS. Heiberg points out that the Arabians must be alone responsible for their definition of a prism, which only covers a prism with a triangular base. This could not have been Euclid's own, for the word prism already has the wider meaning in Archimedes, and Euclid himself speaks of prisms with parallelograms and polygons as bases (XI. 39; XII. 10). Moreover, a Greek would not have been likely to leave out the definitions of the Platonic +

Heiberg comes to the general conclusion that, not only is the Arabic tradition not to be preferred offhand to that of the Greek MSS., but it must be regarded as inferior in authority. It is a question how far the differences shown in the Arabic are due to the use of Greek MSS. differing from those which have been most used as the basis of our text, and how far to the arbitrary changes made by the Arabians themselves. Changes of order and arbitrary omissions could not surprise us, in view of the preface above quoted from the Oxford MS. of Thābit-Is[hnull ]āq, with its allusion to the many important and necessary things left out by Abĩ'l Wafā and to the author's own rearrangement of Books XII., XIII. But there is evidence of differences due to the use by the Arabs of other Greek MSS. HeibergZeitschrift für Math. u. Physik, XXIX., hist.-litt. Abth. p. 6 sqq. is able to show considerable resemblances between the Arabic text and the Bologna MS. b in that part of the MS. where it diverges so remarkably from our other MSS. (see the short description of it above, p.49); in illustration he gives a comparison of the proofs of XII. 7 in b and in the Arabic respectively, and points to the omission in both of the proposition given in Gregory's edition as XI. 38, and to a remarkable agreement between them as regards the order of the propositions of Book XII. As above stated, the remarkable divergence of b only affects Books XI. (at end) and XII.; and Book XIII. in b shows none of the transpositions and other peculiarities of the Arabic. There are many differences between b and the Arabic, especially in the definitions of Book XI., as well as in Book XIII. It is therefore a question whether the Arabians made arbitrary changes, or the Arabic form is the more ancient, and b has been altered through contact with other MSS. Heiberg points out that the Arabians must be alone responsible for their definition of a prism, which only covers a prism with a triangular base. This could not have been Euclid's own, for the word prism already has the wider meaning in Archimedes, and Euclid himself speaks of prisms with parallelograms and polygons as bases (XI. 39; XII. 10). Moreover, a Greek would not have been likely to leave out the definitions of the Platonic regular solids.

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Heiberg considers that the Arabian translator had before him a MS. which was related to b, but diverged still further from the rest of our MSS. He does not think that there is evidence of the existence of a redaction of Books I.—X. similar to that of Books XI., XII. in b; for Klamroth observes that it is the Books on solid geometry (XI.—XIII.) which are more remarkable than the others for omissions and shorter proofs, and it is a noteworthy coincidence that it is just in these Books that we have a divergent text in b.

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An advantage in the Arabic version is the omission of VII. Def. 10, although, as Iamblichus had it, it may have been deliberately omitted by the Arabic translator. Another advantage is the omission of the analyses and syntheses of XIII. 1-5; but again these may have been omitted purpōsely, as were evidently a number of porisms which are really necessary.

-

One or two remarks may be added about the Arabic versions as compared with one another. Al-[Hnull ]ajjāj's object seems to have been less to give a faithful reflection of the original than to write a useful and convenient mathematical text-book. One characteristic of it is the careful references to earlier propositions when their results are used. Such specific quotations of earlier propositions are rare in Euclid; but in al-[Hnull ]ajjāj we find not only such phrases as by prop. so and so, +

Heiberg considers that the Arabian translator had before him a MS. which was related to b, but diverged still further from the rest of our MSS. He does not think that there is evidence of the existence of a redaction of Books I.—X. similar to that of Books XI., XII. in b; for Klamroth observes that it is the Books on solid geometry (XI.—XIII.) which are more remarkable than the others for omissions and shorter proofs, and it is a noteworthy coincidence that it is just in these Books that we have a divergent text in b.

+

An advantage in the Arabic version is the omission of VII. Def. 10, although, as Iamblichus had it, it may have been deliberately omitted by the Arabic translator. Another advantage is the omission of the analyses and syntheses of XIII. 1-5; but again these may have been omitted purposely, as were evidently a number of porisms which are really necessary.

+

One or two remarks may be added about the Arabic versions as compared with one another. Al-[Hnull ]ajjāj's object seems to have been less to give a faithful reflection of the original than to write a useful and convenient mathematical text-book. One characteristic of it is the careful references to earlier propositions when their results are used. Such specific quotations of earlier propositions are rare in Euclid; but in al-[Hnull ]ajjāj we find not only such phrases as by prop. so and so, which was proved or which we showed how to do in prop. so and so, but also still longer phrases. Sometimes he kepeats a construction, as in I. 44 where, instead of constructing the parallelogram BEFG equal to the triangle C in the angle EBG which is equal to the angle D and placing it in a certain position, he produces AB to G, making BG equal to half DE (the base of the triangle CDE in his figure), and on GB so constructs the parallelogram BHKG by I. 42 that it is equal to the triangle CDE, and its angle GBH is equal to the given angle.

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Secondly, al-[hnull ]ajjāj, in the arithmetical books, in the theory of proportion, in the applications of the Pythagorean I. 47, and generally where possible, illustrates the proofs by numerical examples. It is true, observes Klamroth, that these examples are not apparently separated from the commentary of an-Nairīzī, and might not therefore have been due to al-[Hnull ]ajjāj himself; but the marginal notes to the Hebrew translation in Municn MS. 36 show that these additions were in the copy of al-[Hnull ]ajjāj used by the translator, for they expressly give these proofs in numbers as variants taken from al-[Hnull ]ajjājKlamroth, p. 310; Steinschneider, pp. 85-6..

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These characteristics, together with al-[Hnull ]ájjāj's freer formulation of the propositions and expansion of the proofs, constitute an intelligible reason why Is[hnull ]āq should have undertaken a fresh translation from the Greek. Klamroth calls Is[hnull ]āq's version a model of a good translation of a mathematical text; the introductory and transitional phrases are stereotyped and few in number, the technical terms are simply and consistently rendered, and the less formal expressions connect themselves as closely with the Greek as is consistent with intelligibility and the character of the Arabic language. Only in isolated cases does the formulation of definitions and enunciations differ to any considerable extent from the original. In general, his object seems to have been to get rid of difficulties and unevennesses in the Greek text by next devices, while at the same time giving a faithful reproduction of it.Klamroth, p. 290, illustrates is[hnull ]āq's method by his way of distinguishing e)farmo/zein (to be congruent with) and e)farmo/zesqai (to be applied to), the confusion of which by translators was animadverted on by Savile. Is[hnull ]āq avoided the confusion by using two entirely different words..

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There are curious points of contact between the versions of al-[Hnull ]ajjāj and T[hnull ]amacr;bit-Is[hnull ]āq. For example, the definitions and enunciations of propositions are often word for word the same. Presumably this is owing to the fact that Is[hnull ]āq found these definitions and enunciations already established in the schools in his time, where they would no doubt be learnt by heart, and refrained from translating them afresh, merely adopting the older version with some changesKlamroth, pp. 310-1.. Secondly, there is remarkable agreement between the Arabic versions as regards the figures, which show considerable variations from the figures of the Greek text, especially as regards the letters; this is also probably to be explained in the same way, all the later translators having most likely borrowed al-[hnull ]ajjāj's adaptation of the Greek figuresibid. p. 287.. Lastly, it is remarkable that the version of Books XI.—XIII. in the Kjfbenhavn MS. (K), purporting to be by al-[Hnull ]ajjāj, is almost exactly the same as the Thābit-Is[hnull ]āq version of the same Books in O. Klamroth conjectures that Is[hnull ]āq may not have translated the Books on solid geometry at all, and that Thābit took them from al-[Hnull ]ajjāj, only making some changes in order to fit them to the translation of Is[hnull ]āqibid. pp. 304-5..

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From the facts (1) that a&tnull;-[Tnull ]ĩsī's edition had the same number of propositions (468) as al-[hnull ]ajjāj's version, while Thābit-Is[hnull ]āq's had 478, and (2) that a&tnull;-[Tnull ]ĩsī has the same careful references to earlier propositions, Klamroth concludes that a&tnull;-[Tnull ]ĩsī deliberately preferred al-[hnull ]ajjāj's version to that of Is[hnull ]āqibid. p. 274.. Heiberg, however, points out (1) that a&tnull;-[Tnull ]ĩsī left out VI. 12 which, if we may judge by Klamroth's silence, al-[hnull ]ajjāj had, and (2) al-[hnull ]ajjāj's version had one proposition less in Books I. and III. than a&tnull;-[Tnull ]ĩsī has. Besides, in a passage quoted by [hnull ]ājī Khalfa[hnull ]ājī Khalfa, I. p. 383. from a&tnull;-[Tnull ]ĩsī, the latter says that he separated the things which, in the approved editions, were taken from the archetype from the things which had been added thereto, - indicating that he had compiled his edition from both the earlier translationsHeiberg, Zeitschrift für Math. u. Physik, XXIX., hist.-litt. Abth. pp. 2, 3..

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There were a large number of Arabian commentaries on, or reproductions of, the Elements or portions thereof, which will be found fully noticed by SteinschneiderSteinschneider, Zeitschrift für Math. u. Physik, XXXI., hist.-litt. Abth. pp. 86 sqq.. I shall mention here the commentators etc. referred to in the Fihrist, with a few others.

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1. Abĩ 'l `Abbās al-Fa[dnull ]l b. [hnull ]ātim an-Nairīzī (born at Nairīz, died about 922) has already been mentionedSteinschneider, p. 86, Fihrist (tr. Suter), pp. 16, 67; Suter, Die Mathematiker und Astronomen der Araber (1900), p. 45.. His commentary survives, as regards Books I.—VI., in the Codex Leidensis 399, I, now edited, as to four Books, by Besthorn and Heiberg, and as regards Books I.—X. in the Latin translation made by Gherard of Cremona in the 12th c. and now published by Curtze from a Cracow MSSupplementum ad Euclidis opera omnia, ed. Heiberg and Menge, Leipzig, 1899.. Its importance lies mainly in the quotations from Heron and Simplicius.

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2. Ahmad b. `Umar al-Karābīsī (date uncertain, probably 9th— 10th c.), who was among the most distinguished geometers and arithmeticiansFihrist, pp. 16, 38; Steinschneider, p. 87; Suter, p. 65.. +

Secondly, al-[hnull ]ajjāj, in the arithmetical books, in the theory of proportion, in the applications of the Pythagorean I. 47, and generally where possible, illustrates the proofs by numerical examples. It is true, observes Klamroth, that these examples are not apparently separated from the commentary of an-Nairīzī, and might not therefore have been due to al-[Hnull ]ajjāj himself; but the marginal notes to the Hebrew translation in Municn MS. 36 show that these additions were in the copy of al-[Hnull ]ajjāj used by the translator, for they expressly give these proofs in numbers as variants taken from al-[Hnull ]ajjājKlamroth, p. 310; Steinschneider, pp. 85-6..

+

These characteristics, together with al-[Hnull ]ájjāj's freer formulation of the propositions and expansion of the proofs, constitute an intelligible reason why Is[hnull ]āq should have undertaken a fresh translation from the Greek. Klamroth calls Is[hnull ]āq's version a model of a good translation of a mathematical text; the introductory and transitional phrases are stereotyped and few in number, the technical terms are simply and consistently rendered, and the less formal expressions connect themselves as closely with the Greek as is consistent with intelligibility and the character of the Arabic language. Only in isolated cases does the formulation of definitions and enunciations differ to any considerable extent from the original. In general, his object seems to have been to get rid of difficulties and unevennesses in the Greek text by next devices, while at the same time giving a faithful reproduction of it.Klamroth, p. 290, illustrates is[hnull ]āq's method by his way of distinguishing e)farmo/zein (to be congruent with) and e)farmo/zesqai (to be applied to), the confusion of which by translators was animadverted on by Savile. Is[hnull ]āq avoided the confusion by using two entirely different words..

+

There are curious points of contact between the versions of al-[Hnull ]ajjāj and T[hnull ]amacr;bit-Is[hnull ]āq. For example, the definitions and enunciations of propositions are often word for word the same. Presumably this is owing to the fact that Is[hnull ]āq found these definitions and enunciations already established in the schools in his time, where they would no doubt be learnt by heart, and refrained from translating them afresh, merely adopting the older version with some changesKlamroth, pp. 310-1.. Secondly, there is remarkable agreement between the Arabic versions as regards the figures, which show considerable variations from the figures of the Greek text, especially as regards the letters; this is also probably to be explained in the same way, all the later translators having most likely borrowed al-[hnull ]ajjāj's adaptation of the Greek figuresibid. p. 287.. Lastly, it is remarkable that the version of Books XI.—XIII. in the Kjfbenhavn MS. (K), purporting to be by al-[Hnull ]ajjāj, is almost exactly the same as the Thābit-Is[hnull ]āq version of the same Books in O. Klamroth conjectures that Is[hnull ]āq may not have translated the Books on solid geometry at all, and that Thābit took them from al-[Hnull ]ajjāj, only making some changes in order to fit them to the translation of Is[hnull ]āqibid. pp. 304-5..

+

From the facts (1) that aṭ-[Tnull ]ĩsī's edition had the same number of propositions (468) as al-[hnull ]ajjāj's version, while Thābit-Is[hnull ]āq's had 478, and (2) that aṭ-[Tnull ]ĩsī has the same careful references to earlier propositions, Klamroth concludes that aṭ-[Tnull ]ĩsī deliberately preferred al-[hnull ]ajjāj's version to that of Is[hnull ]āqibid. p. 274.. Heiberg, however, points out (1) that aṭ-[Tnull ]ĩsī left out VI. 12 which, if we may judge by Klamroth's silence, al-[hnull ]ajjāj had, and (2) al-[hnull ]ajjāj's version had one proposition less in Books I. and III. than aṭ-[Tnull ]ĩsī has. Besides, in a passage quoted by [hnull ]ājī Khalfa[hnull ]ājī Khalfa, I. p. 383. from aṭ-[Tnull ]ĩsī, the latter says that he separated the things which, in the approved editions, were taken from the archetype from the things which had been added thereto, + indicating that he had compiled his edition from both the earlier translationsHeiberg, Zeitschrift für Math. u. Physik, XXIX., hist.-litt. Abth. pp. 2, 3..

+

There were a large number of Arabian commentaries on, or reproductions of, the Elements or portions thereof, which will be found fully noticed by SteinschneiderSteinschneider, Zeitschrift für Math. u. Physik, XXXI., hist.-litt. Abth. pp. 86 sqq.. I shall mention here the commentators etc. referred to in the Fihrist, with a few others.

+

1. Abĩ 'l `Abbās al-Fa[dnull ]l b. [hnull ]ātim an-Nairīzī (born at Nairīz, died about 922) has already been mentionedSteinschneider, p. 86, Fihrist (tr. Suter), pp. 16, 67; Suter, Die Mathematiker und Astronomen der Araber (1900), p. 45.. His commentary survives, as regards Books I.—VI., in the Codex Leidensis 399, I, now edited, as to four Books, by Besthorn and Heiberg, and as regards Books I.—X. in the Latin translation made by Gherard of Cremona in the 12th c. and now published by Curtze from a Cracow MSSupplementum ad Euclidis opera omnia, ed. Heiberg and Menge, Leipzig, 1899.. Its importance lies mainly in the quotations from Heron and Simplicius.

+

2. Ahmad b. `Umar al-Karābīsī (date uncertain, probably 9th— 10th c.), who was among the most distinguished geometers and arithmeticiansFihrist, pp. 16, 38; Steinschneider, p. 87; Suter, p. 65..

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3. Al-'Abbās b. Sa`īd al-Jauharī (fl. 830) was one of the astronomical observers under al-Ma'mĩn, but devoted himself mostly to geometry. He wrote a commentary to the whole of the Elements, from the beginning to the end; also the Book of the propositions which he added to the first book of EuclidFihrist, pp. 16, 25; Steinschneider, p. 88; Suter, p. 12.. +

3. Al-'Abbās b. Sa`īd al-Jauharī (fl. 830) was one of the astronomical observers under al-Ma'mĩn, but devoted himself mostly to geometry. He wrote a commentary to the whole of the Elements, from the beginning to the end; also the Book of the propositions which he added to the first book of EuclidFihrist, pp. 16, 25; Steinschneider, p. 88; Suter, p. 12..

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4. Muh. b. `Īsā Abĩ `Abdallāh al-Māhānī (d. between 874 and 884) wrote, according to the Fihrist, (I) a commentary on Eucl. Book v., (2) On proportion, +

4. Muh. b. `Īsā Abĩ `Abdallāh al-Māhānī (d. between 874 and 884) wrote, according to the Fihrist, (I) a commentary on Eucl. Book v., (2) On proportion, (3) On the 26 propositions of the first Book of Euclid which are proved without reductio ad absurdumFihrist, pp. 16, 25, 58.. The work On proportion survives and is probably identical with, or part of, the commentary on Book v.Suter, p. 26, note, quotes the Paris MS. 2467, 16^{o} containing the work on proportion as the authority for this conjecture. He also wrote, what is not mentioned by the Fihrist, a commentary on Eucl. Book X., a fragment of which survives in a Paris MS.MS. 2457, 39^{o} (cf. Woepcke in Me/m. pre/s. agrave; l'acad. des sciences, XIV., 1856, p. 669).

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5. Abĩ Ja`far al-Khāzin (i.e. the treasurer +

5. Abĩ Ja`far al-Khāzin (i.e. the treasurer or librarian -), one of the first mathematicians and astronomers of his time, was born in Khurāsān and died between the years 961 and 971. The Fihrist speaks of him as having written a commentary on the whole of the ElementsFihrist, p. 17., but only the commentary on the beginning of Book X. survives (in Leiden, Berlin and Paris); therefore either the notes on the rest of the Books have perished, or the Fihrist is in errorSuter, p. 58, note b.. The latter would seem more probable, for, at the end of his commentary, al-Khāzin remarks that the rest had already been commented on by Sulaimān b. 'Usma (Leiden MS.)Steinschneider, p. 89. or 'Oqba (Suter), to be mentioned below. Al-Khāzin's method is criticised unfavourably in the preface to the Oxford MS. quoted by Nicoll (see p. 77 above).

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6. Abĩ'l Wafā al-Bĩzjānī (940-997), one of the greatest Arabian mathematicians, wrote a commentary on the Elements, but did not complete itFihrist, p. 17.. His method is also unfavourably regarded in the same preface to the Oxford MS. 280. According to Hājī Khalfa, he also wrote a book on geometrical constructions, in thirteen chapters. Apparently a book answering to this description was compiled by a gifted pupil from lectures by Abĩ 'l Wafā, and a Paris MS. (Anc. fonds 169) contains a Persian translation of this work, not that of Abĩ 'l Wafā himself. An analysis of the work was given by WoepckeWoepcke, Fournal Asiatique, Sér. v. T. v. pp. 218-256 and 309-359., and some particulars will be found in CantorGesch. d. Math. vol. 13, pp. 743-6.. Abĩ 'l Wafā also wrote a commentary on Diophantus, as well as a separate book of proofs to the propositions which Diophantus used in his book and to what he (Abĩ 'l Wafā) employed in his commentaryFihrist, p. 39; Suter, p. 71.. +), one of the first mathematicians and astronomers of his time, was born in Khurāsān and died between the years 961 and 971. The Fihrist speaks of him as having written a commentary on the whole of the ElementsFihrist, p. 17., but only the commentary on the beginning of Book X. survives (in Leiden, Berlin and Paris); therefore either the notes on the rest of the Books have perished, or the Fihrist is in errorSuter, p. 58, note b.. The latter would seem more probable, for, at the end of his commentary, al-Khāzin remarks that the rest had already been commented on by Sulaimān b. 'Usma (Leiden MS.)Steinschneider, p. 89. or 'Oqba (Suter), to be mentioned below. Al-Khāzin's method is criticised unfavourably in the preface to the Oxford MS. quoted by Nicoll (see p. 77 above).

+

6. Abĩ'l Wafā al-Bĩzjānī (940-997), one of the greatest Arabian mathematicians, wrote a commentary on the Elements, but did not complete itFihrist, p. 17.. His method is also unfavourably regarded in the same preface to the Oxford MS. 280. According to Hājī Khalfa, he also wrote a book on geometrical constructions, in thirteen chapters. Apparently a book answering to this description was compiled by a gifted pupil from lectures by Abĩ 'l Wafā, and a Paris MS. (Anc. fonds 169) contains a Persian translation of this work, not that of Abĩ 'l Wafā himself. An analysis of the work was given by WoepckeWoepcke, Fournal Asiatique, Sér. v. T. v. pp. 218-256 and 309-359., and some particulars will be found in CantorGesch. d. Math. vol. 13, pp. 743-6.. Abĩ 'l Wafā also wrote a commentary on Diophantus, as well as a separate book of proofs to the propositions which Diophantus used in his book and to what he (Abĩ 'l Wafā) employed in his commentaryFihrist, p. 39; Suter, p. 71..

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7. Ibn Rāhawaihi al-Arjānī also commented on Eucl. Book X.Fihrist, p. 17; Suter, p. 17..

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8. `Alī b. Ahmad Abĩ 'l-Qāsim al-An&tnull;ākī (d. 987) wrote a commentary on the whole bookFihrist, p. 17.; part of it seems to survive (from the 5th Book onwards) at Oxford (Catal. MSS. orient. II. 281)Suter, p. 64..

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9. Sind b. `Alī Abĩ '&tnull;-[Tnull ]aiyib was a Jew who went over to Islam in the time of al-Ma'mĩn, and was received among his astronomical observers, whose head he becameFihrist, p. 17, 29; Suter, pp. 13, 14. (about 830); he died after 864. He wrote a commentary on the whole of the Elements; Abĩ `Alī saw nine books of it, and a part of the tenthFihrist, p. 17.. +

7. Ibn Rāhawaihi al-Arjānī also commented on Eucl. Book X.Fihrist, p. 17; Suter, p. 17..

+

8. `Alī b. Ahmad Abĩ 'l-Qāsim al-Anṭākī (d. 987) wrote a commentary on the whole bookFihrist, p. 17.; part of it seems to survive (from the 5th Book onwards) at Oxford (Catal. MSS. orient. II. 281)Suter, p. 64..

+

9. Sind b. `Alī Abĩ 'ṭ-[Tnull ]aiyib was a Jew who went over to Islam in the time of al-Ma'mĩn, and was received among his astronomical observers, whose head he becameFihrist, p. 17, 29; Suter, pp. 13, 14. (about 830); he died after 864. He wrote a commentary on the whole of the Elements; Abĩ `Alī saw nine books of it, and a part of the tenthFihrist, p. 17.. His book On the Apotomae and the Medials, mentioned by the Fihrist, may be the same as, or part of, his commentary on Book X.

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10. Abĩ Yĩsuf Ya`qĩb b. Muh. ar-Rāzī wrote a commentary on Book X., and that an excellent one, at the instance of Ibn al`AmīdFihrist, p. 17; Suter, p. 66.. +

10. Abĩ Yĩsuf Ya`qĩb b. Muh. ar-Rāzī wrote a commentary on Book X., and that an excellent one, at the instance of Ibn al`AmīdFihrist, p. 17; Suter, p. 66..

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11. The Fihrist next mentions al-Kindī (Abĩ Yĩsuf Ya'qĩb b. Ishāq b. as-Sabbāh al-Kindī, d. about 873), as the author (1) of a work on the objects of Euclid's book, +

11. The Fihrist next mentions al-Kindī (Abĩ Yĩsuf Ya'qĩb b. Ishāq b. as-Sabbāh al-Kindī, d. about 873), as the author (1) of a work on the objects of Euclid's book, in which occurs the statement that the Elements were originally written by Apollonius, the carpenter (see above, p. 5 and note), (2) of a book on the improvement of Euclid's work, and (3) of another on the improvement of the 14th and 15th Books of Euclid. - He was the most distinguished man of his time, and stood alone in the knowledge of the old sciences collectively; he was called ’the philosopher of the Arabians’; his writings treat of the most different branches of knowledge, as logic, philosophy, geometry, calculation, arithmetic, music, astronomy and othersFihrist, p. 17, 10-15.. - Among the other geometrical works of al-Kindī mentioned by the FihristThe mere catalogue of al-Kindī's works on the various branches of science takes up four octavo pages (11-15) of Suter's translation of the Fihrist. are treatises on the closer investigation of the results of Archimedes concerning the measure of the diameter of a circle in terms of its circumference, on the construction of the figure of the two mean proportionals, on the approximate determination of the chords of the circle, on the approximate determination of the chord (side) of the nonagon, on the division of triangles and quadrilaterals and constructions for that purpose, on the manner of construction of a circle which is equal to the surface of a given cylinder, on the division of the circle, in three chapters etc.

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12. The physician Na[znull ]ĩf b. Yumn (or Yaman) al-Qass (the priest -) is mentioned by the Fihrist as having seen a Greek copy of Eucl. Book X. which had 40 more propositions than that which was in general circulation (containing 109), and having determined to translate it into ArabicFihrist, pp. 16, 17.. Fragments of such a translation exist at Paris, Nos. 18 and 34 of the MS. 2457 (952, 2 Suppl. Arab. in Woepcke's tract); No. 18 contains additions to some propositions of the 10th Book, existing in the Greek languageWoepcke, Mém. prés. à l'acad. des sciences, XIV. pp. 666, 668.. - Nazĩf must have died about 990Suter, p. 68..

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13. Yũ[hnull ]annã b. Yūsuf b. al-[Hnull ]ãrith b. al-Bitrĩq al-Qass (d. about 980) lectured on the Elements and other geometrical books, made translations from the Greek, and wrote a tract on the proof + He was the most distinguished man of his time, and stood alone in the knowledge of the old sciences collectively; he was called ’the philosopher of the Arabians’; his writings treat of the most different branches of knowledge, as logic, philosophy, geometry, calculation, arithmetic, music, astronomy and othersFihrist, p. 17, 10-15.. + Among the other geometrical works of al-Kindī mentioned by the FihristThe mere catalogue of al-Kindī's works on the various branches of science takes up four octavo pages (11-15) of Suter's translation of the Fihrist. are treatises on the closer investigation of the results of Archimedes concerning the measure of the diameter of a circle in terms of its circumference, on the construction of the figure of the two mean proportionals, on the approximate determination of the chords of the circle, on the approximate determination of the chord (side) of the nonagon, on the division of triangles and quadrilaterals and constructions for that purpose, on the manner of construction of a circle which is equal to the surface of a given cylinder, on the division of the circle, in three chapters etc.

+

12. The physician Naẓīf b. Yumn (or Yaman) al-Qass (the priest +) is mentioned by the Fihrist as having seen a Greek copy of Eucl. Book X. which had 40 more propositions than that which was in general circulation (containing 109), and having determined to translate it into ArabicFihrist, pp. 16, 17.. Fragments of such a translation exist at Paris, Nos. 18 and 34 of the MS. 2457 (952, 2 Suppl. Arab. in Woepcke's tract); No. 18 contains additions to some propositions of the 10th Book, existing in the Greek languageWoepcke, Mém. prés. à l'acad. des sciences, XIV. pp. 666, 668.. + Nazĩf must have died about 990Suter, p. 68..

+

13. Yũ[hnull ]annã b. Yūsuf b. al-[Hnull ]ãrith b. al-Bitrĩq al-Qass (d. about 980) lectured on the Elements and other geometrical books, made translations from the Greek, and wrote a tract on the proof of the case of two straight lines both meeting a third and making with it, on one side, two angles together less than two right anglesFihrist, p. 38; Suter, p. 60.. Nothing of his appears to survive, except that a tract on rational and irrational magnitudes, No. 48 in the Paris MS. just mentioned, is attributed to him.

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14. Abũ Mu[hnull ]. al-[Hnull ]asan b. 'Ubaidallãh b. Sulaimãn b. Wahb (d. 901) was a geometer of distinction, who wrote works under the two distinct titles A commentary on the difficult parts of the work of Euclid +

14. Abũ Mu[hnull ]. al-[Hnull ]asan b. 'Ubaidallãh b. Sulaimãn b. Wahb (d. 901) was a geometer of distinction, who wrote works under the two distinct titles A commentary on the difficult parts of the work of Euclid and The Book on ProportionFihrist, p. 26, and Suter's note, p. 60.. Suter thinks that another reading is possible in the case of the second title, and that it may refer to the Euclidean work on the divisions (of figures)Suter, p. 211, note 23..

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15. Qustã b. Lũqã al-Ba'labakkĩ (d. about 912), a physician, philosopher, astronomer, mathematician and translator, wrote on the difficult passages of Euclid's book +

15. Qustã b. Lũqã al-Ba'labakkĩ (d. about 912), a physician, philosopher, astronomer, mathematician and translator, wrote on the difficult passages of Euclid's book and on the solution of arithmetical problems from the third book of EuclidFihrist, p. 43. ; also an introduction to geometry, in the form of question and answerFihrist, p. 43; Suter, p. 41..

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16. Thãbit b. Qurra (826-901), besides translating some parts of Archimedes and Books V.mdashVII. of the Conics of Apollonius, and revising Ishãq's translation of Euclid's Elements, also revised the translation of the Data by the same Ishãq and the book On divisions of figures translated by an anonymous writer. We are told also that he wrote the following works: (I) On the Premisses (Axioms, Postulates etc.) of Euclid, (2) On the Propositions of Euclid, (3) On the propositions and questions which arise when two straight lines are cut by a third (or on the proof +

16. Thãbit b. Qurra (826-901), besides translating some parts of Archimedes and Books V.mdashVII. of the Conics of Apollonius, and revising Ishãq's translation of Euclid's Elements, also revised the translation of the Data by the same Ishãq and the book On divisions of figures translated by an anonymous writer. We are told also that he wrote the following works: (I) On the Premisses (Axioms, Postulates etc.) of Euclid, (2) On the Propositions of Euclid, (3) On the propositions and questions which arise when two straight lines are cut by a third (or on the proof of Euclid's famous postulate). The last tract is extant in the MS. discovered by Woepcke (Paris 2457, 32^{o}). He is also credited with an excellent work in the shape of an Introduction to the Book of Euclid, - a treatise on Geometry dedicated to Ismã'il b. Bulbul, a Compendium of Geometry, and a large number of other works for the titles of which reference may be made to Suter, who also gives particulars as to which are extantSuter, pp. 34-8..

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17. Abũ Sa'ĩd Sinãn b. Thãbit b. Qurra, the son of the translator of Euclid, followed in his father's footsteps as geometer, astronomer and physician. He wrote an improvement of the book of ...... on the Elements of Geometry, in which he made various additions to the original. - It is natural to conjecture that Euclid is the name missing in this description (by Ibn abĩ U&snull;aibi'a); Casiri has the name AqãtonFihrist (ed. Suter), p. 59, note 132; Suter, p. 52, note b.. The latest editor of the Ta'rĩkh al-[Hnull ]ukamã, however, makes the name to be Iflãton (=Plato), and he refers to the statement by the Fihrist and Ibn al-Qiftĩ attributing to Plato a work on the Elements of Geometry translated by Qust<*>ã. It is just possible, therefore, that at the time of Qus&tnull;ã the Arabs were acquainted with a book on the Elements of Geometry translated from the Greek, which they attributed to PlatoSee Suter in Bibliotheca Mathematica, IV_{3}, 1903-4, pp. 296-7, review of Julius Lippert's Ibn al-Qiftĩ. Ta'rĩch al-hukamã, Leipzig, 1903.. Sinãn died in 943.

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18. Abũ Sahl Wĩjan (or Waijan) b. Rustam al-Kũhĩ (fl. 988), born at Kũh in [Tnull ]abaristãn, a distinguished geometer and astronomer, wrote, according to the Fihrist, a Book of the Elements + a treatise on Geometry dedicated to Ismã'il b. Bulbul, a Compendium of Geometry, and a large number of other works for the titles of which reference may be made to Suter, who also gives particulars as to which are extantSuter, pp. 34-8..

+

17. Abũ Sa'ĩd Sinãn b. Thãbit b. Qurra, the son of the translator of Euclid, followed in his father's footsteps as geometer, astronomer and physician. He wrote an improvement of the book of ...... on the Elements of Geometry, in which he made various additions to the original. + It is natural to conjecture that Euclid is the name missing in this description (by Ibn abĩ Uṣaibi'a); Casiri has the name AqãtonFihrist (ed. Suter), p. 59, note 132; Suter, p. 52, note b.. The latest editor of the Ta'rĩkh al-[Hnull ]ukamã, however, makes the name to be Iflãton (=Plato), and he refers to the statement by the Fihrist and Ibn al-Qiftĩ attributing to Plato a work on the Elements of Geometry translated by Qust<*>ã. It is just possible, therefore, that at the time of Qusṭã the Arabs were acquainted with a book on the Elements of Geometry translated from the Greek, which they attributed to PlatoSee Suter in Bibliotheca Mathematica, IV_{3}, 1903-4, pp. 296-7, review of Julius Lippert's Ibn al-Qiftĩ. Ta'rĩch al-hukamã, Leipzig, 1903.. Sinãn died in 943.

+

18. Abũ Sahl Wĩjan (or Waijan) b. Rustam al-Kũhĩ (fl. 988), born at Kũh in [Tnull ]abaristãn, a distinguished geometer and astronomer, wrote, according to the Fihrist, a Book of the Elements after that of EuclidFihrist, p. 40.; the 1st and 2nd Books survive at Cairo, and a part of the 3rd Book at Berlin (5922)Suter, p. 75.. He wrote also a number of other geometrical works: Additions to the 2nd Book of Archimedes on the Sphere and Cylinder (extant at Paris, at Leiden, and in the India Office), On the finding of the side of a heptagon in a circle (India Office and Cairo), On two mean proportionals (India Office), which last may be only a part of the Additions to Archimedes' On the Sphere and Cylinder, etc.

-

19. Abũ Na&snull;r Mu[hnull ]. b. Mu[hnull ]. b. [Tnull ]arkhãn b. Uzlaġ al-Fãrãbĩ (870-950) wrote a commentary on the difficulties of the introductory matter to Books I. and V.Suter, p. 55. This appears to survive in the Hebrew translation which is, with probability, attributed to Moses b. TibbonSteinschneider, p. 92..

-

20. Abũ 'Alĩ al-Hasan b. al-Hasan b. al-Haitham (about 9651039), known by the name Ibn al-Haitham or Abũ 'Alĩ al-Basrĩ, was a man of great powers and knowledge, and no one of his time approached him in the field of mathematical science. He wrote several works on Euclid the titles of which, as translated by Woepcke from Usaibi'a, are as followsSteinschneider, pp. 92-3.: +

19. Abũ Naṣr Mu[hnull ]. b. Mu[hnull ]. b. [Tnull ]arkhãn b. Uzlaġ al-Fãrãbĩ (870-950) wrote a commentary on the difficulties of the introductory matter to Books I. and V.Suter, p. 55. This appears to survive in the Hebrew translation which is, with probability, attributed to Moses b. TibbonSteinschneider, p. 92..

+

20. Abũ 'Alĩ al-Hasan b. al-Hasan b. al-Haitham (about 9651039), known by the name Ibn al-Haitham or Abũ 'Alĩ al-Basrĩ, was a man of great powers and knowledge, and no one of his time approached him in the field of mathematical science. He wrote several works on Euclid the titles of which, as translated by Woepcke from Usaibi'a, are as followsSteinschneider, pp. 92-3.:

1. Commentary and abridgment of the Elements.

2. Collection of the Elements of Geometry and Arithmetic, drawn from the treatises of Euclid and Apollonius.

3. Collection of the Elements of the Calculus deduced from the principles laid down by Euclid in his Elements.

@@ -954,30 +954,30 @@

The last-named work (which Suter calls a commentary on the Postulates of Euclid) survives in an Oxford MS. (Catal. MSS. orient. I. 908) and in Algiers (1446, 1^{o}).

A Leiden MS. (966) contains his Commentary on the difficult places - up to Book V. We do not know whether in this commentary, which the author intended to form, with the commentary on the Musãdarãt, a sort of complete commentary, he had collected the separate memoirs on certain doubts and difficult passages mentioned in the above list.

+ up to Book V. We do not know whether in this commentary, which the author intended to form, with the commentary on the Musãdarãt, a sort of complete commentary, he had collected the separate memoirs on certain doubts and difficult passages mentioned in the above list.

A commentary on Book V. and following Books found in a Bodleian MS. (Catal. II. p. 262) with the title Commentary on Euclid and solution of his difficulties is attributed to b. Haitham; this might be a continuation of the Leiden MS.

The memoir on X. 1 appears to survive at St Petersburg, MS. de l'Institut des langues orient. 192, 5^{o} (Rosen, Catal. p. 125).

-

21. Ibn Sĩnã, known as Avicenna (980-1037), wrote a Compendium of Euclid, preserved in a Leiden MS. No. 1445, and forming the geometrical portion of an encyclopaedic work embracing Logic, Mathematics, Physics and MetaphysicsSteinschneider, p. 92; Suter, p. 89..

-

22. Ahmad b. al-Husain al-Ahwãzĩ al-Kãtib wrote a commentary on Book X., a fragment of which (some 10 pages) is to be found at Leiden (970), Berlin (5923) and Paris (2467, 18^{o})Suter, p. 57..

-

23. Na&snull;ĩraddĩn a&tnull;-[Tnull ]ũsĩ (1201-1274) who, as we have seen, brought out a Euclid in two forms, wrote: +

21. Ibn Sĩnã, known as Avicenna (980-1037), wrote a Compendium of Euclid, preserved in a Leiden MS. No. 1445, and forming the geometrical portion of an encyclopaedic work embracing Logic, Mathematics, Physics and MetaphysicsSteinschneider, p. 92; Suter, p. 89..

+

22. Ahmad b. al-Husain al-Ahwãzĩ al-Kãtib wrote a commentary on Book X., a fragment of which (some 10 pages) is to be found at Leiden (970), Berlin (5923) and Paris (2467, 18^{o})Suter, p. 57..

+

23. Naṣĩraddĩn aṭ-[Tnull ]ũsĩ (1201-1274) who, as we have seen, brought out a Euclid in two forms, wrote:

1. A treatise on the postulates of Euclid (Paris, 2467, 5^{o}).

2. A treatise on the 5th postulate, perhaps only a part of the foregoing (Berlin, 5942, Paris, 2467, 6^{o}).

3. Principles of Geometry taken from Euclid, perhaps identical with No. 1 above (Florence, Pal. 298).

4. 105 problems out of the Elements (Cairo). He also edited the Data (Berlin, Florence, Oxford etc.)Suter, pp. 150-1..

-

24. Muh. b. Ashraf Shamsaddĩn as-Samarqandĩ (fl. 1276) wrote Fundamental Propositions, being elucidations of 35 selected propositions of the first Books of Euclid, +

24. Muh. b. Ashraf Shamsaddĩn as-Samarqandĩ (fl. 1276) wrote Fundamental Propositions, being elucidations of 35 selected propositions of the first Books of Euclid, which are extant at Gotha (1496 and 1497), Oxford (Catal. I. 967, 2^{o}), and Brit. Mus.Suter, p. 157..

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25. Mũsã b. Mu[hnull ]. b. Ma[hnull ]mũd, known as Qã[dnull ]ĩzãde ar-Rũmĩ (i.e. the son of the judge from Asia Minor), who died between 1436 and 1446, wrote a commentary on the Fundamental Propositions +

25. Mũsã b. Mu[hnull ]. b. Ma[hnull ]mũd, known as Qã[dnull ]ĩzãde ar-Rũmĩ (i.e. the son of the judge from Asia Minor), who died between 1436 and 1446, wrote a commentary on the Fundamental Propositions just mentioned, of which many MSS. are extantibid. p. 175.. It contained biographical statements about Euclid alluded to above (p. 5, note).

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26. Abũ Dã'ũd Sulaimãn b. ’Uqba, a contemporary of al-Khãzin (see above, No. 5), wrote a commentary on the second half of Book X., which is, at least partly, extant at Leiden (974) under the title On the binomials and apotomae found in the 10th Book of Euclidibid. p. 56.. +

26. Abũ Dã'ũd Sulaimãn b. ’Uqba, a contemporary of al-Khãzin (see above, No. 5), wrote a commentary on the second half of Book X., which is, at least partly, extant at Leiden (974) under the title On the binomials and apotomae found in the 10th Book of Euclidibid. p. 56..

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27. The Codex Leidensis 399, 1 containing al-[Hnull ]ajjãj's translation of Books I.—VI. is said to contain glosses to it by Sa`ĩd b. Mas'ũd b. al-Qass, apparently identical with Abũ Nasr Gars al-Na'ma, son of the physician Mas'ũd b. al-Qass al-Bagdãdĩ, who lived in the time of the last Caliph al-Musta'sim (d. 1258)ibid. pp. 153-4, 227..

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28. Abũ Mu[hnull ]ammad b. Abdalbãqĩ al-Bag[dnull ]ãdĩ al-Fara[dnull ]ĩ (d. 1141, at the age of over 70 years) is stated in the Ta'rĩkh al-[Hnull ]ukamã to have written an excellent commentary on Book X. of the Elements, in which he gave numerical examples of the propositionsGartz, p. 14; Steinschneider, pp. 94-5.. This is published in Curtze's edition of an-Nairĩzĩ where it occupies pages 252-386Suter in Bibliotheca Mathematica, IV3, 1903, pp. 25, 295; Suter has also an article on its contents, Bibliotheca Mathematica, VII3, 1906-7, pp. 234-251..

-

29. Ya[hnull ]yã b. Mu[hnull ]. b. 'Abdãn b. 'Abdalwã[hnull ]id, known by the name of Ibn al-Lubũdĩ (1210-1268), wrote a Compendium of Euclid, and a short presentation of the postulatesSteinschneider, p. 94; Suter, p. 146..

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30. Abũ 'Abdallãh Mu[hnull ]. b. Mu'ãdh al-Jayyãnĩ wrote a commentary on Eucl. Book V. which survives at Algiers (1446, 3^{o})Suter, Nachträge und Berichtigungen, in Abhandlungen zur Gesch. der math. Wissenschaften, XIV., 1902, p. 170..

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31. Abũ Na&snull;r Mansũr b. 'Aliã b.'Irãq wrote, at the instance of Mu[hnull ]. b. A[hnull ]mad Abũ ’r-Rai[hnull ]ãn al-Bĩrũnĩ (973-1048), a tract on a doubtful (difficult) passage in Eucl. Book XIII. - (Berlin, 5925). Suter, p. 81, and Nachträge, p. 172..

+

27. The Codex Leidensis 399, 1 containing al-[Hnull ]ajjãj's translation of Books I.—VI. is said to contain glosses to it by Sa`ĩd b. Mas'ũd b. al-Qass, apparently identical with Abũ Nasr Gars al-Na'ma, son of the physician Mas'ũd b. al-Qass al-Bagdãdĩ, who lived in the time of the last Caliph al-Musta'sim (d. 1258)ibid. pp. 153-4, 227..

+

28. Abũ Mu[hnull ]ammad b. Abdalbãqĩ al-Bag[dnull ]ãdĩ al-Fara[dnull ]ĩ (d. 1141, at the age of over 70 years) is stated in the Ta'rĩkh al-[Hnull ]ukamã to have written an excellent commentary on Book X. of the Elements, in which he gave numerical examples of the propositionsGartz, p. 14; Steinschneider, pp. 94-5.. This is published in Curtze's edition of an-Nairĩzĩ where it occupies pages 252-386Suter in Bibliotheca Mathematica, IV3, 1903, pp. 25, 295; Suter has also an article on its contents, Bibliotheca Mathematica, VII3, 1906-7, pp. 234-251..

+

29. Ya[hnull ]yã b. Mu[hnull ]. b. 'Abdãn b. 'Abdalwã[hnull ]id, known by the name of Ibn al-Lubũdĩ (1210-1268), wrote a Compendium of Euclid, and a short presentation of the postulatesSteinschneider, p. 94; Suter, p. 146..

+

30. Abũ 'Abdallãh Mu[hnull ]. b. Mu'ãdh al-Jayyãnĩ wrote a commentary on Eucl. Book V. which survives at Algiers (1446, 3^{o})Suter, Nachträge und Berichtigungen, in Abhandlungen zur Gesch. der math. Wissenschaften, XIV., 1902, p. 170..

+

31. Abũ Naṣr Mansũr b. 'Aliã b.'Irãq wrote, at the instance of Mu[hnull ]. b. A[hnull ]mad Abũ ’r-Rai[hnull ]ãn al-Bĩrũnĩ (973-1048), a tract on a doubtful (difficult) passage in Eucl. Book XIII. + (Berlin, 5925). Suter, p. 81, and Nachträge, p. 172..

@@ -991,25 +991,25 @@

Magnus Aurelius Cassiodorus (b. about 475 A.D.) in the geometrical part of his encyclopaedia De artibus ac disciplinis liberalium literarum says that geometry was represented among the Greeks by Euclid, Apollonius, Archimedes, and others, of whom Euclid was given us translated into the Latin language by the same great man Boethius ; also in his collection of lettersCassiodorus, Variae, I. 45, p. 40, 12 ed. Mommsen. is a letter from Theodoric to Boethius containing the words, for in your translations...Nicomachus the arithmetician, and Euclid the geometer, are heard in the Ausonian tongue. The so-called Geometry of Boethius which has come down to us by no means constitutes a translation of Euclid. The MSS. variously give five, four, three or two Books, but they represent only two distinct compilations, one normally in five Books and the other in two. Even the latter, which was edited by Friedlein, is not genuineSee especially Weissenborn in Abhandlungen zur Gesch. d. Math. II. p. 18_{5} sq.; Heiberg in Philologus, XLIII. p. 507 sq.; Cantor, 1_{3}, p. 580 sq.,but appears to have been put together in the 11th c., from various sources. It begins with the definitions of Eucl. I., and in these are traces of perfectly correct readings which are not found even in the MSS. of the 10th c., but which can be traced in Proclus and other ancient sources; then come the Postulates (five only), the Axioms (three only), and after these some definitions of Eucl. II., III., IV. Next come the enunciations of Eucl. I., of ten propositions of Book II., and of some from Books III., IV., but always without proofs; there follows an extraordinary passage which indicates that the author will now give something of his own in elucidation of Euclid, though what follows is a literal translation of the proofs of Eucl. I. 1-3. This latter passage, although it affords a strong argument against the genuineness of this part of the work, shows that the Pseudoboethius had a Latin translation of Euclid from which he extracted the three propositions.

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Curtze has reproduced, in the preface to his edition of the translation by Gherard of Cremona of an-Nairĩzĩ's Arabic commentary on Euclid, some interesting fragments of a translation of Euclid taken from a Munich MS. of the 10th c. They are on two leaves used for the cover of the MS. (Bibliothecae Regiae Universitatis Monacensis 2^{o} 757) and consist of portions of Eucl. I. 37, 38 and II. 8, translated literally word for word from the Greek text. The translator seems to have been an Italian (cf. the words capitolo nono +

Curtze has reproduced, in the preface to his edition of the translation by Gherard of Cremona of an-Nairĩzĩ's Arabic commentary on Euclid, some interesting fragments of a translation of Euclid taken from a Munich MS. of the 10th c. They are on two leaves used for the cover of the MS. (Bibliothecae Regiae Universitatis Monacensis 2^{o} 757) and consist of portions of Eucl. I. 37, 38 and II. 8, translated literally word for word from the Greek text. The translator seems to have been an Italian (cf. the words capitolo nono used for the ninth prop. of Book II.) who knew very little Greek and had moreover little mathematical knowledge. For example, he translates the capital letters denoting points in figures as if they were numerals: thus ta\ *a*b*g, *d*e*z is translated que primo secundo et tertio quarto quinto et septimo, T becomes tricentissimo and so on. The Greek MS. which he used was evidently written in uncials, for *d*e*z*q becomes in one place quod autem septimo nono, showing that he mistook *d*e for the particle de/, and kai\ o( *s*t*u is rendered sicut tricentissimo et quadringentissimo, showing that the letters must have been written KAIOCTU.

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The date of the Englishman Athelhard (Æthelhard) is approximately fixed by some remarks in his work Perdifficiles Quaestiones Naturales which, on the ground of the personal allusions they contain, must be assigned to the first thirty years of the 12th c.Cantor, Gesch. d. Math. I3, p. 906. He wrote a number of philosophical works. Little is known about his life. He is said to have studied at Tours and Laon, and to have lectured at the latter school. He travelled to Spain, Greece, Asia Minor and Egypt, and acquired a knowledge of Arabic, which enabled him to translate from the Arabic into Latin, among other works, the Elements of Euclid. The date of this translation must be put at about 1120. MSS. purporting to contain Athelhard's version are extant in the British Museum (Harleian No. 5404 and others), Oxford (Trin. Coll. 47 and Ball. Coll. 257 of 12th c.), Nürnberg (Johannes Regiomontanus' copy) and Erfurt.

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Among the very numerous works of Gherard of Cremona (1114— 1187) are mentioned translations of 15 Books of Euclid - and of the DataBoncompagni, Della vita e delle opere di Gherardo Cremonese, Rome, 1851, p. 5.. Till recently this translation of the Elements was supposed to be lost; but Axel Anthon Björnbo has succeeded (1904) in discovering a translation from the Arabic which is different from the two others known to us (those by Athelhard and Campanus respectively), and which he, on grounds apparently convincing, holds to be Gherard's. Already in 1901 Björnbo had found Books X.—XV. of this translation in a MS. at Rome (Codex Reginensis lat. 1268 of 14th c.)Described in an appendix to Studien über Menelaos' Sphärik (Abhandlungen zur Gesckichse der mathematischen Wissenschaften, XIV., 1902).; but three years later he had traced three MSS. containing the whole of the same translation at Paris (Cod. Paris. 7216, 15th c.), Boulogne-sur-Mer (Cod. Bononiens. 196, 14th c.), and Bruges (Cod. Brugens. 521, 14th c.), and another at Oxford (Cod. Digby 174, end of 12th c.) containing a fragment, XI. 2 to XIV. The occurrence of Greek words in this translation such as rombus, romboides (where Athelhard keeps the Arabic terms), ambligonius, orthogonius, gnomo, pyramis etc., show that the translation is independent of Athelhard's. Gherard appears to have had before him an old translation of Euclid from the Greek which Athelhard also often followed, especially in his terminology, using it however in a very different manner. Again, there are some Arabic terms, e.g. meguar for axis of rotation, which Athelhard did not use, but which is found in almost all the translations that are with certainty attributed to Gherard of Cremona; there occurs also the The date of the Englishman Athelhard (Æthelhard) is approximately fixed by some remarks in his work Perdifficiles Quaestiones Naturales which, on the ground of the personal allusions they contain, must be assigned to the first thirty years of the 12th c.Cantor, Gesch. d. Math. I3, p. 906. He wrote a number of philosophical works. Little is known about his life. He is said to have studied at Tours and Laon, and to have lectured at the latter school. He travelled to Spain, Greece, Asia Minor and Egypt, and acquired a knowledge of Arabic, which enabled him to translate from the Arabic into Latin, among other works, the Elements of Euclid. The date of this translation must be put at about 1120. MSS. purporting to contain Athelhard's version are extant in the British Museum (Harleian No. 5404 and others), Oxford (Trin. Coll. 47 and Ball. Coll. 257 of 12th c.), Nürnberg (Johannes Regiomontanus' copy) and Erfurt.

+

Among the very numerous works of Gherard of Cremona (1114— 1187) are mentioned translations of 15 Books of Euclid + and of the DataBoncompagni, Della vita e delle opere di Gherardo Cremonese, Rome, 1851, p. 5.. Till recently this translation of the Elements was supposed to be lost; but Axel Anthon Björnbo has succeeded (1904) in discovering a translation from the Arabic which is different from the two others known to us (those by Athelhard and Campanus respectively), and which he, on grounds apparently convincing, holds to be Gherard's. Already in 1901 Björnbo had found Books X.—XV. of this translation in a MS. at Rome (Codex Reginensis lat. 1268 of 14th c.)Described in an appendix to Studien über Menelaos' Sphärik (Abhandlungen zur Gesckichse der mathematischen Wissenschaften, XIV., 1902).; but three years later he had traced three MSS. containing the whole of the same translation at Paris (Cod. Paris. 7216, 15th c.), Boulogne-sur-Mer (Cod. Bononiens. 196, 14th c.), and Bruges (Cod. Brugens. 521, 14th c.), and another at Oxford (Cod. Digby 174, end of 12th c.) containing a fragment, XI. 2 to XIV. The occurrence of Greek words in this translation such as rombus, romboides (where Athelhard keeps the Arabic terms), ambligonius, orthogonius, gnomo, pyramis etc., show that the translation is independent of Athelhard's. Gherard appears to have had before him an old translation of Euclid from the Greek which Athelhard also often followed, especially in his terminology, using it however in a very different manner. Again, there are some Arabic terms, e.g. meguar for axis of rotation, which Athelhard did not use, but which is found in almost all the translations that are with certainty attributed to Gherard of Cremona; there occurs also the 4 See Bibliotheca Mathematica, VI3, 1905-6, PP. 242-8.expression superficies equidistantium laterum et rectorum angulorum, - found also in Gherard's translation of an-Nairĩzĩ, where Athelhard says parallelogrammum rectangulum. + found also in Gherard's translation of an-Nairĩzĩ, where Athelhard says parallelogrammum rectangulum. The translation is much clearer than Athelhard's: it is neither abbreviated nor edited - as Athelhard's appears to have been; it is a word-for-word translation of an Arabic MS. containing a revised and critical edition of Thãbit's version. It contains several notes quoted from Thãbit himself (Thebit dixit), e.g. about alternative proofs etc. which Thãbit found in another Greek MS., - and is therefore a further testimony to Thãbit's critical treatment of the text after Greek MSS. The new editor also added critical remarks of his own, e.g. on other proofs which he found in other Arabic versions, but not in the Greek: whence it is clear that he compared the Thãbit version before him with other versions as carefully as Thãbit collated the Greek MSS. Lastly, the new editor speaks of Thebit qui transtulit hunc librum in arabicam linguam + as Athelhard's appears to have been; it is a word-for-word translation of an Arabic MS. containing a revised and critical edition of Thãbit's version. It contains several notes quoted from Thãbit himself (Thebit dixit), e.g. about alternative proofs etc. which Thãbit found in another Greek MS., + and is therefore a further testimony to Thãbit's critical treatment of the text after Greek MSS. The new editor also added critical remarks of his own, e.g. on other proofs which he found in other Arabic versions, but not in the Greek: whence it is clear that he compared the Thãbit version before him with other versions as carefully as Thãbit collated the Greek MSS. Lastly, the new editor speaks of Thebit qui transtulit hunc librum in arabicam linguam and of translatio Thebit, - which may tend to confirm the statement of al-Qiftĩ who credited Thãbit with an independent translation, and not (as the Fihrist does) with a mere improvement of the version of Is[hnull ]ãq b. Hunain.

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Gherard's translation of the Arabic commentary of an-Nairĩzĩ on the first ten Books of the Elements was discovered by Maximilian Curtze in a MS. at Cracow and published as a supplementary volume to Heiberg and Menge's EuclidAnaritii in decem libros priores Elementorum Euclidis Commentarii ex interpretatione Gherardi Cremonensis in codice Cracoviensi 569 servata edidit Maximilianus Curtze, Leipzig (Teubner), 1899.: it will often be referred to in this work.

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Next in chronological order comes Johannes Campanus (Campano) of Novara. He is mentioned by Roger Bacon (1214-1294) as a prominent mathematician of his timeCantor, II_{1}, p. 88., and this indication of his date is confirmed by the fact that he was chaplain to Pope Urban IV, who was Pope from 1261 to 1281Tiraboschi, Storia della letteratura italiana, IV. 145mdash;160.. His most important achievement was his edition of the Elements including the two Books XIV. and XV. which are not Euclid's. The sources of Athelhard's and Campanus' translations, and the relation between them, have been the subject of much discussion, which does not seem to have led as yet to any definite conclusion. Cantor (II_{1}, p. 91) gives referencesH. Weissenborn in Zeitschrift für Math. u. Physik, XXV., Supplement, pp. 143mdash;166, and in his monograph, Die Übersetzungen des Euklid durch Campano und Zamberti (1882); Max. Curtze in Philologische Rundschau (1881), 1. pp. 943-950, and in Fahresbericht über die Fortschritte der classischen Alterthumswissenschaft, XL. (188_{4}, III.) pp. 19mdash;22; Heiberg in Zeitschrift für Math. u. Physik, XXXV., hist.-litt. Abth., pp. 48mdash;58 and pp. 81mdash;6. and some particulars. It appears that there is a MS. at Munich (Cod. lat. Mon. 13021) written by Sigboto in the 12th c. at Prüfning near Regensburg, and denoted by Curtze by the letter R, which contains the enunciations of part of Euclid. The Munich MSS. of Athelhard and Campanus' translations have many enunciations textually identical with those in R, so that the source of all three must, for these enunciations, have been the same; in others Athelhard and Campanus diverge completely from R, which in these places follows the Greek text and is therefore genuine and authoritative. In the 32nd definition occurs the word elinuam, + which may tend to confirm the statement of al-Qiftĩ who credited Thãbit with an independent translation, and not (as the Fihrist does) with a mere improvement of the version of Is[hnull ]ãq b. Hunain.

+

Gherard's translation of the Arabic commentary of an-Nairĩzĩ on the first ten Books of the Elements was discovered by Maximilian Curtze in a MS. at Cracow and published as a supplementary volume to Heiberg and Menge's EuclidAnaritii in decem libros priores Elementorum Euclidis Commentarii ex interpretatione Gherardi Cremonensis in codice Cracoviensi 569 servata edidit Maximilianus Curtze, Leipzig (Teubner), 1899.: it will often be referred to in this work.

+

Next in chronological order comes Johannes Campanus (Campano) of Novara. He is mentioned by Roger Bacon (1214-1294) as a prominent mathematician of his timeCantor, II_{1}, p. 88., and this indication of his date is confirmed by the fact that he was chaplain to Pope Urban IV, who was Pope from 1261 to 1281Tiraboschi, Storia della letteratura italiana, IV. 145mdash;160.. His most important achievement was his edition of the Elements including the two Books XIV. and XV. which are not Euclid's. The sources of Athelhard's and Campanus' translations, and the relation between them, have been the subject of much discussion, which does not seem to have led as yet to any definite conclusion. Cantor (II_{1}, p. 91) gives referencesH. Weissenborn in Zeitschrift für Math. u. Physik, XXV., Supplement, pp. 143mdash;166, and in his monograph, Die Übersetzungen des Euklid durch Campano und Zamberti (1882); Max. Curtze in Philologische Rundschau (1881), 1. pp. 943-950, and in Fahresbericht über die Fortschritte der classischen Alterthumswissenschaft, XL. (188_{4}, III.) pp. 19mdash;22; Heiberg in Zeitschrift für Math. u. Physik, XXXV., hist.-litt. Abth., pp. 48mdash;58 and pp. 81mdash;6. and some particulars. It appears that there is a MS. at Munich (Cod. lat. Mon. 13021) written by Sigboto in the 12th c. at Prüfning near Regensburg, and denoted by Curtze by the letter R, which contains the enunciations of part of Euclid. The Munich MSS. of Athelhard and Campanus' translations have many enunciations textually identical with those in R, so that the source of all three must, for these enunciations, have been the same; in others Athelhard and Campanus diverge completely from R, which in these places follows the Greek text and is therefore genuine and authoritative. In the 32nd definition occurs the word elinuam, the Arabic term for rhombus, and throughout the translation are a number of Arabic figures. But R was not translated from the Arabic, as is shown by (among other things) its close resemblance to the translation from Euclid given on pp. 377 sqq. of the Gromatici Veteres and to the so-called geometry of Boethius. The explanation of the Arabic figures and the word elinuam in Def. 32 appears to be that R was a late copy of an earlier original with corruptions introduced in many places; thus in Def. 32 a part of the text was completely lost and was supplied by some intelligent copyist who inserted the word elinuam, @@ -1024,8 +1024,8 @@ I. Latin translations prior to 1533. -

1482. In this year appeared the first printed edition of Euclid, which was also the first printed mathematical book of any importance. This was printed at Venice by Erhard Ratdolt and contained Campanus' translationCurtze (An-Nairīzī, p. xiii) reproduces the heading of the first page of the text as follows (there is no title-page): Preclariffimũ opus elemento<*> Euclidis megarēfis [vmacr ]na cū cōmentis Campani pfpicaciffimi in artē geometriā incipit felicit', after which the definitions begin at once. Other copies have the shorter heading: Preclarissimus liber elementorum Euclidis perspicacissimi: in artem Geometrie incipit quam foelicissime. At the end stands the following: <*> Opus elementorū euclidis megarenfis in geometriā artē Jnid quoq<*> Campani pfpicaciffimi Cōmentationes finiũt. Erhardus ratdolt Augustensis impreffor folertiffimus. venetijs impreffit . Anno falutis . M.cccc.lxxxij . Octauis . Cale[ntilde] . Ju[ntilde] . Lector . Vale.. Ratdolt belonged to a family of artists at Augsburg, where he was born about 1443. Having learnt the trade of printing at home, he went in 1475 to Venice, and founded there a famous printing house which he managed for II years, after which he returned to Augsburg and continued to print important books until 1516. He is said to have died in 1528. KästnerKästner, Geschichte der Mathematik, I. p. 289 sqq. See also Weissenborn, Die Übersetzungen des Euklid durch Campano und Zamberti, pp. 1-7. gives a short description of this first edition of Euclid and quotes the dedication to Prince Mocenigo of Venice which occupies the page opposite to the first page of text. The book has a margin of 2 1/2 inches, and in this margin are placed the figures of the propositions. Ratdolt says in his dedication that at that time, although books by ancient and modern authors were printed every day in Venice, little or nothing mathematical had appeared: a fact which he puts down to the difficulty involved by the figures, which no one had up to that time succeeded in printing. He adds that after much labour he had discovered a method by which figures could be produced as easily as lettersMea industria non sine maximo labore effeci vt qua facilitate litterarum elementa imprimuntur ea etiam geometrice figure conficerentur. -. Experts are in doubt as to the nature of Ratdolt's discovery. Was it a method of making figures up out of separate parts of figures, straight or curved lines, put together as letters are put together to make words? In a life of Joh. Gottlob Immanuel Breitkopf, a contemporary of Kästner's own, this member of the great house of Breitkopf is credited with this particular discovery. Experts in that same house expressed the opinion that Ratdolt's figures were woodcuts, while the letters denoting points in the figures were like the other letters in the text; yet it was with carved wooden blocks that printing began. If Ratdolt was the first to print geometrical figures, it was not long before an emulator arose; for in the very same year Mattheus Cordonis of Windischgrätz employed woodcut mathematical figures in printing Oresme's De latitudinibusCurtze in Zeitschrift für Math. u. Physik, XX., hist.-litt. Abth. p. 58.. How eagerly the opportunity of spreading geometrical knowledge was seized upon is proved by the number of editions which followed in the next few years. Even the year 1482 saw two forms of the book, though they only differ in the first sheet. Another edition came out in 1486 (Ulmae, apud Io. Regerum) and another in 1491 (Vincentiae per Leonardum de Basilea et Gulielmum de Papia), but without the dedication to Mocenigo who had died in the meantime (1485). If Campanus added anything of his own, his additions are at all events not distinguished by any difference of type or otherwise; the enunciations are in large type, and the rest is printed continuously in smaller type. There are no superscriptions to particular passages such as Euclides ex Campano, Campanus, Campani additio, or Campani annotatio, which are found for the first time in the Paris edition of 1516 giving both Campanus' version and that of Zamberti (presently to be mentioned).

+

1482. In this year appeared the first printed edition of Euclid, which was also the first printed mathematical book of any importance. This was printed at Venice by Erhard Ratdolt and contained Campanus' translationCurtze (An-Nairīzī, p. xiii) reproduces the heading of the first page of the text as follows (there is no title-page): Preclariffimũ opus elemento<*> Euclidis megarēfis [vmacr ]na cū cōmentis Campani pfpicaciffimi in artē geometriā incipit felicit', after which the definitions begin at once. Other copies have the shorter heading: Preclarissimus liber elementorum Euclidis perspicacissimi: in artem Geometrie incipit quam foelicissime. At the end stands the following: <*> Opus elementorū euclidis megarenfis in geometriā artē Jnid quoq<*> Campani pfpicaciffimi Cōmentationes finiũt. Erhardus ratdolt Augustensis impreffor folertiffimus. venetijs impreffit . Anno falutis . M.cccc.lxxxij . Octauis . Cale[ntilde] . Ju[ntilde] . Lector . Vale.. Ratdolt belonged to a family of artists at Augsburg, where he was born about 1443. Having learnt the trade of printing at home, he went in 1475 to Venice, and founded there a famous printing house which he managed for II years, after which he returned to Augsburg and continued to print important books until 1516. He is said to have died in 1528. KästnerKästner, Geschichte der Mathematik, I. p. 289 sqq. See also Weissenborn, Die Übersetzungen des Euklid durch Campano und Zamberti, pp. 1-7. gives a short description of this first edition of Euclid and quotes the dedication to Prince Mocenigo of Venice which occupies the page opposite to the first page of text. The book has a margin of 2 1/2 inches, and in this margin are placed the figures of the propositions. Ratdolt says in his dedication that at that time, although books by ancient and modern authors were printed every day in Venice, little or nothing mathematical had appeared: a fact which he puts down to the difficulty involved by the figures, which no one had up to that time succeeded in printing. He adds that after much labour he had discovered a method by which figures could be produced as easily as lettersMea industria non sine maximo labore effeci vt qua facilitate litterarum elementa imprimuntur ea etiam geometrice figure conficerentur. +. Experts are in doubt as to the nature of Ratdolt's discovery. Was it a method of making figures up out of separate parts of figures, straight or curved lines, put together as letters are put together to make words? In a life of Joh. Gottlob Immanuel Breitkopf, a contemporary of Kästner's own, this member of the great house of Breitkopf is credited with this particular discovery. Experts in that same house expressed the opinion that Ratdolt's figures were woodcuts, while the letters denoting points in the figures were like the other letters in the text; yet it was with carved wooden blocks that printing began. If Ratdolt was the first to print geometrical figures, it was not long before an emulator arose; for in the very same year Mattheus Cordonis of Windischgrätz employed woodcut mathematical figures in printing Oresme's De latitudinibusCurtze in Zeitschrift für Math. u. Physik, XX., hist.-litt. Abth. p. 58.. How eagerly the opportunity of spreading geometrical knowledge was seized upon is proved by the number of editions which followed in the next few years. Even the year 1482 saw two forms of the book, though they only differ in the first sheet. Another edition came out in 1486 (Ulmae, apud Io. Regerum) and another in 1491 (Vincentiae per Leonardum de Basilea et Gulielmum de Papia), but without the dedication to Mocenigo who had died in the meantime (1485). If Campanus added anything of his own, his additions are at all events not distinguished by any difference of type or otherwise; the enunciations are in large type, and the rest is printed continuously in smaller type. There are no superscriptions to particular passages such as Euclides ex Campano, Campanus, Campani additio, or Campani annotatio, which are found for the first time in the Paris edition of 1516 giving both Campanus' version and that of Zamberti (presently to be mentioned).

1501. G. Valla included in his encyclopaedic work De expetendis et fugiendis rebus published in this year at Venice (in aedibus Aldi Romani) a number of propositions with proofs and scholia translated from a Greek MS. which was once in his possession (cod. Mutin. III B, 4 of the 15th c.).

1505. In this year Bartolomeo Zamberti (Zambertus) brought out at Venice the first translation, from the Greek text, of the whole of the Elements. From the titleThe title begins thus: Euclidis megaresis philosophi platonicj mathematicarum disciplinarum Janitoris: Habent in hoc volumine quicunque ad mathematicam substantiam aspirant: elementorum libros xiij cum expositione Theonis insignis mathematici. quibus multa quae deerant ex lectione graeca sumpta addita sunt nec non plurima peruersa et praepostere: voluta in Campani interpretatione: ordinata digesta et castigata sunt etc. For a description of the book see Weissenborn, p. 12 sqq., as well as from his prefaces to the Catoptrica and Data, with their allusions to previous translators who take some things out of authors, omit some, and change some, @@ -1039,12 +1039,12 @@ (i.e. of course Latin and Greek), as well as in medicine and the more sublime studies, had helped to make the edition more perfect. Though Zamberti is not once mentioned, this latter remark must have reference to Zamberti's statement that his translation was from the Greek text; and no doubt Zamberti is aimed at in the wish of Paciuolo's that others too would seek to acquire knowledge instead of merely showing off, or that they would not try to make a market of the things of which they are ignorant, as it were (selling) smoke Atque utinam et alii cognoscere vellent non ostentare aut ea quae nesciunt veluti fumum venditare non conarentur. . - Weissenborn observes that, while there are many trivialities in Paciuolo's notes, they contain some useful and practical hints and explanations of terms, besides some new proofs which of course are not difficult if one takes the liberty, as Paciuolo does, of divering from Euclid's order and assuming for the proof of a proposition results not arrived at till later. Two not inapt terms are used in this edition to describe the figures of III. 7, 8, the former of which is called the goose's foot (pes anseris), the second the peacock's tail (cauda pavonis) Paciuolo as the castigator of Campanus' translation, as he calls himself, failed to correct the mistranslation of V. Def. 5Campanus' translation in Ratdolt's edition is as follows: Quantitates quae dicuntur continuam habere proportionalitatem, sunt, quarum equè multiplicia aut equa sunt aut equè sibi sine interruptione addunt aut minuunt - (!), to which Campanus adds the note: Continuè proportionalia sunt quorum omnia multiplicia equalia sunt continuè proportionalia. Sed noluit ipsam diffinitionem proponere sub hac forma, quia tunc diffiniret idem per idem, aperte (? a parte) tamen rei est istud cum sua diffinitione convertibile. + Weissenborn observes that, while there are many trivialities in Paciuolo's notes, they contain some useful and practical hints and explanations of terms, besides some new proofs which of course are not difficult if one takes the liberty, as Paciuolo does, of divering from Euclid's order and assuming for the proof of a proposition results not arrived at till later. Two not inapt terms are used in this edition to describe the figures of III. 7, 8, the former of which is called the goose's foot (pes anseris), the second the peacock's tail (cauda pavonis) Paciuolo as the castigator of Campanus' translation, as he calls himself, failed to correct the mistranslation of V. Def. 5Campanus' translation in Ratdolt's edition is as follows: Quantitates quae dicuntur continuam habere proportionalitatem, sunt, quarum equè multiplicia aut equa sunt aut equè sibi sine interruptione addunt aut minuunt + (!), to which Campanus adds the note: Continuè proportionalia sunt quorum omnia multiplicia equalia sunt continuè proportionalia. Sed noluit ipsam diffinitionem proponere sub hac forma, quia tunc diffiniret idem per idem, aperte (? a parte) tamen rei est istud cum sua diffinitione convertibile. . Before the fifth Book he inserted a discourse which he gave at Venice on the 15th August, 1508, in S. Bartholomew's Church, before a select audience of 500, as an introduction to his elucidation of that Book.

1516. The first of the editions giving Campanus' and Zamberti's translations in conjunction was brought out at Paris (in officina Henrici Stephani e regione scholae Decretorum). The idea that only the enunciations were Euclid's, and that Campanus was the author of the proofs in his translation, while Theon was the author of the proofs in the Greek text, reappears in the title of this edition; and the enunciations of the added Books XIV., XV. are also attributed to Euclid, Hypsicles being credited with the proofsEuclidis Megarensis Geometricorum Elementorum Libri XV. Campani Galli transalpini in eosdem commentariorum libri XV. Theonis Alexandrini Bartholomaeo. Zamberto Veneto interprete, in tredecim priores, commentationum libri XIII. Hypsiclis Alexandrini in duos posteriores, eodem Bartholomaeo Zamberto Veneto interprete, commentariorum libri II. On the last page (261) is a similar statement of content, but with the difference that the expression ex Campani...deinde Theonis...et Hypsiclis...traditionibus. - For description see Weissenborn, p. 56 sqq.. The date is not on the title-page nor at the end, but the letter of dedication to François Briconnet by Jacques Lefèvre is dated the day after the Epiphany, 1516. The figures are in the margin. The arrangement of the propositions is as follows: first the enunciation with the heading Euclides ex Campano, then the proof with the note Campanus, and after that, as Campani additio, any passage found in the edition of Campanus' translation but not in the Greek text; then follows the text of the enunciation translated from the Greek with the heading Euclides ex Zamberto, and lastly the proof headed Theo ex Zamberto. There are separate figures for the two proofs. This edition was reissued with few changes in 1537 and 1546 at Basel (apud Iohannem Hervagium), but with the addition of the Phaenomena, Optica, Catoptrica etc. For the edition of 1537 the Paris edition of 1516 was collated with a Greek copy + For description see Weissenborn, p. 56 sqq.. The date is not on the title-page nor at the end, but the letter of dedication to François Briconnet by Jacques Lefèvre is dated the day after the Epiphany, 1516. The figures are in the margin. The arrangement of the propositions is as follows: first the enunciation with the heading Euclides ex Campano, then the proof with the note Campanus, and after that, as Campani additio, any passage found in the edition of Campanus' translation but not in the Greek text; then follows the text of the enunciation translated from the Greek with the heading Euclides ex Zamberto, and lastly the proof headed Theo ex Zamberto. There are separate figures for the two proofs. This edition was reissued with few changes in 1537 and 1546 at Basel (apud Iohannem Hervagium), but with the addition of the Phaenomena, Optica, Catoptrica etc. For the edition of 1537 the Paris edition of 1516 was collated with a Greek copy (as the preface says) by Christian Herlin, professor of mathematical studies at Strassburg, who however seems to have done no more than correct one or two passages by the help of the Basel editio princeps (1533), and add the Greek word in cases where Zamberti's translation of it seemed unsuitable or inaccurate.

We now come to

@@ -1053,47 +1053,47 @@

1533 is the date of the editio princeps, the title-page of which reads as follows: *e*u*k*l*e*i*d*o*u *s*t*o*i*x*e*i*w*n *b*i*b*l<*> *i*e<*><*> *e*k *t*w*n *q*e*w*n*o*s *s*u*n*o*u*s*i*w*n. *ei)s tou= au)tou= to\ prw=ton, e)chghma/twn *pro/klou bibl. d_. Adiecta praefatiuncula in qua de disciplinis Mathematicis nonnihil. BASILEAE APVD IOAN. HERVAGIVM ANNO M.D.XXXIII. MENSE SEPTEMBRI.

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The editor was Simon Grynaeus the elder (d. 1541), who, after working at Vienna and Ofen, Heidelberg and Tübingen, taught last of all at Basel, where theology was his main subject. His praefatiuncula - is addressed to an Englishman, Cuthbert Tonstall (14741559), who, having studied first at Oxford, then at Cambridge, where he became Doctor of Laws, and afterwards at Padua, where in addition he learnt mathematics—mostly from the works of Regiomontanus and Paciuolo—wrote a book on arithmeticDe arté supputandi libri quatuor. as a farewell to the sciences, +

The editor was Simon Grynaeus the elder (d. 1541), who, after working at Vienna and Ofen, Heidelberg and Tübingen, taught last of all at Basel, where theology was his main subject. His praefatiuncula + is addressed to an Englishman, Cuthbert Tonstall (14741559), who, having studied first at Oxford, then at Cambridge, where he became Doctor of Laws, and afterwards at Padua, where in addition he learnt mathematics—mostly from the works of Regiomontanus and Paciuolo—wrote a book on arithmeticDe arté supputandi libri quatuor. as a farewell to the sciences, and then, entering politics, became Bishop of London and member of the Privy Council, and afterwards (1530) Bishop of Durham. Grynaeus tells us that he used two MSS. of the text of the Elements, entrusted to friends of his, one at Venice by Lazarus Bayfius - (Lazare de Baïf, then the ambassador of the King of France at Venice), the other at Paris by Ioann. Rvellius + (Lazare de Baïf, then the ambassador of the King of France at Venice), the other at Paris by Ioann. Rvellius (Jean Ruel, a French doctor and a Greek scholar), while the commentaries of Proclus were put at the disposal of Grynaeus himself by Ioann. Claymundus - at Oxford. Heiberg has been able to identify the two MSS. used for the text; they are (1) cod. Venetus Marcianus 301 and (2) cod. Paris. gr. 2343 of the 16th c., containing Books I.—XV., with some scholia which are embodied in the text. When Grynaeus notes in the margin the readings from the other copy, + at Oxford. Heiberg has been able to identify the two MSS. used for the text; they are (1) cod. Venetus Marcianus 301 and (2) cod. Paris. gr. 2343 of the 16th c., containing Books I.—XV., with some scholia which are embodied in the text. When Grynaeus notes in the margin the readings from the other copy, this other copy is as a rule the Paris MS., though sometimes the reading of the Paris MS. is taken into the text and the other copy of the margin is the Venice MS. Besides these two MSS. Grynaeus consulted Zamberti, as is shown by a number of marginal notes referring to Zampertus or to latinum exemplar - in certain propositions of Books IX.—XI. When it is considered that the two MSS. used by Grynaeus are among the worst, it is obvious how entirely unauthoritative is the text of the editio princeps. Yet it remained the source and foundation of later editions of the Greek text for a long period, the editions which followed being designed, not for the purpose of giving, from other MSS., a text more nearly representing what Euclid himself wrote, but of supplying a handy compendium to students at a moderate price.

+ in certain propositions of Books IX.—XI. When it is considered that the two MSS. used by Grynaeus are among the worst, it is obvious how entirely unauthoritative is the text of the editio princeps. Yet it remained the source and foundation of later editions of the Greek text for a long period, the editions which followed being designed, not for the purpose of giving, from other MSS., a text more nearly representing what Euclid himself wrote, but of supplying a handy compendium to students at a moderate price.

1536. Orontius Finaeus (Oronce Fine) published at Paris (apud Simonem Colinaeum) demonstrations on the first six books of Euclid's elements of geometry, in which the Greek text of Euclid himself is inserted in its proper places, with the Latin translation of Barth. Zamberti of Venice, - which seems to imply that only the enunciations were given in Greek. The preface, from which Kästner quotesKästner, I. p. 260., says that the University of Paris at that time required, from all who aspired to the laurels of philosophy, a most solemn oath that they had attended lectures on the said first six Books. Other editions of Fine's work followed in 1544 and 1551.

+ which seems to imply that only the enunciations were given in Greek. The preface, from which Kästner quotesKästner, I. p. 260., says that the University of Paris at that time required, from all who aspired to the laurels of philosophy, a most solemn oath that they had attended lectures on the said first six Books. Other editions of Fine's work followed in 1544 and 1551.

1545. The enunciations of the fifteen Books were published in Greek, with an Italian translation by Angelo Caiani, at Rome (apud Antonium Bladum Asulanum). The translator claims to have corrected the books and purged them of six hundred things which did not seem to savour of the almost divine genius and the perspicuity of EuclidHeiberg, vol. V. p. cvii.

1549. Joachim Camerarius published the enunciations of the first six Books in Greek and Latin (Leipzig). The book, with preface, purports to be brought out by Rhaeticus (1514-1576), a pupil of Copernicus. Another edition with proofs of the propositions of the first three Books was published by Moritz Steinmetz in 1577 (Leipzig); a note by the printer attributes the preface to Camerarius himself.

1550. Ioan. Scheubel published at Basel (also per Ioan. Hervagium) the first six Books in Greek and Latin together with true and appropriate proofs of the propositions, without the use of letters - (i.e. letters denoting points in the figures), the various straight lines and angles being described in wordsKästner, I. p. 359..

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1557 (also 1558). Stephanus Gracilis published another edition (repeated 1573, 1578, 1598) of the enunciations (alone) of Books I.—XV. in Greek and Latin at Paris (apud Gulielmum Cavellat). He remarks in the preface that for want of time he had changed scarcely anything in Books I.—VI., but in the remaining Books he had emended what seemed obscure or inelegant in the Latin translation, while he had adopted in its entirety the translation of Book X. by Pierre Mondoré (Petrus Montaureus), published separately at Paris in 1551. Gracilis also added a few scholia. + (i.e. letters denoting points in the figures), the various straight lines and angles being described in wordsKästner, I. p. 359..

+

1557 (also 1558). Stephanus Gracilis published another edition (repeated 1573, 1578, 1598) of the enunciations (alone) of Books I.—XV. in Greek and Latin at Paris (apud Gulielmum Cavellat). He remarks in the preface that for want of time he had changed scarcely anything in Books I.—VI., but in the remaining Books he had emended what seemed obscure or inelegant in the Latin translation, while he had adopted in its entirety the translation of Book X. by Pierre Mondoré (Petrus Montaureus), published separately at Paris in 1551. Gracilis also added a few scholia.

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1564. In this year Conrad Dasypodius (Rauchfuss), the inventor and maker of the clock in Strassburg cathedral, similar to the present one, which did duty from 1571 to 1789, edited (Strassburg, Chr. Mylius) (1) Book I. of the Elements in Greek and Latin with scholia, (2) Book II. in Greek and Latin with Barlaam's arithmetical version of Book II., and (3) the enunciations of the remaining Books III.—XIII. Book I. was reissued with vocabula quaedam geometrica - of Heron, the enunciations of all the Books of the Elements, and the other works of Euclid, all in Greek and Latin. In the preface to (1) he says that it had been for twenty-six years the rule of his school that all who were promoted from the classes to public lectures should learn the first Book, and that he brought it out, because there were then no longer any copies to be had, and in order to prevent a good and fruitful regulation of his school from falling through. In the preface to the edition of 1571 he says that the first Book was generally taught in all gymnasia and that it was prescribed in his school for the first class. In the preface to (3) he tells us that he published the enunciations of Books III.—XIII. in order not to leave his work unfinished, but that, as it would be irksome to carry about the whole work of Euclid in extenso, he thought it would be more convenient to students of geometry to learn the Elements if they were compressed into a smaller book.

+

1564. In this year Conrad Dasypodius (Rauchfuss), the inventor and maker of the clock in Strassburg cathedral, similar to the present one, which did duty from 1571 to 1789, edited (Strassburg, Chr. Mylius) (1) Book I. of the Elements in Greek and Latin with scholia, (2) Book II. in Greek and Latin with Barlaam's arithmetical version of Book II., and (3) the enunciations of the remaining Books III.—XIII. Book I. was reissued with vocabula quaedam geometrica + of Heron, the enunciations of all the Books of the Elements, and the other works of Euclid, all in Greek and Latin. In the preface to (1) he says that it had been for twenty-six years the rule of his school that all who were promoted from the classes to public lectures should learn the first Book, and that he brought it out, because there were then no longer any copies to be had, and in order to prevent a good and fruitful regulation of his school from falling through. In the preface to the edition of 1571 he says that the first Book was generally taught in all gymnasia and that it was prescribed in his school for the first class. In the preface to (3) he tells us that he published the enunciations of Books III.—XIII. in order not to leave his work unfinished, but that, as it would be irksome to carry about the whole work of Euclid in extenso, he thought it would be more convenient to students of geometry to learn the Elements if they were compressed into a smaller book.

1620. Henry Briggs (of Briggs' logarithms) published the first six Books in Greek with a Latin translation after Commandinus, corrected in many places (London, G. Jones).

-

1703 is the date of the Oxford edition by David Gregory which, until the issue of Heiberg and Menge's edition, was still the only edition of the complete works of Euclid*e*u*k*l*e*i*d*o*u *t*a *s*w*z*o*m*e*n*a. Euclidis quae supersunt omnia. Ex recensione Davidis Gregorii M.D. Astronomiae Professoris Saviliani et R.S.S. Oxoniae, e Theatro Sheldoniano, An. Dom. MDCCIII.. In the Latin translation attached to the Greek text Gregory says that he followed Commandinus in the main, but corrected numberless passages in it by means of the books in the Bodleian Library which belonged to Edward Bernard (1638-1696), formerly Savilian Professor of Astronomy, who had conceived the plan of publishing the complete works of the ancient mathematicians in fourteen volumes, of which the first was to contain Euclid's Elements I.—XV. As regards the Greek text, Gregory tells us that he consulted, as far as was necessary, not a few MSS. of the better sort, bequeathed by the great Savile to the University, as well as the corrections made by Savile in his own hand in the margin of the Basel edition. He had the help of John Hudson, Bodley's Librarian, who punctuated the Basel text before it went to the printer, compared the Latin version with the Greek throughout, especially in the Elements and Data, and, where they differed or where he suspected the Greek text, consulted the Greek MSS. and put their readings in the margin if they agreed with the Latin and, if they did not agree, affixed an asterisk in order that Gregory might judge which reading was geometrically preferable. Hence it is clear that no Greek MS., but the Basel edition, was the foundation of Gregory's text, and that Greek MSS. were only referred to in the special passages to which Hudson called attention.

-

1814-1818. A most important step towards a good Greek text was taken by F. Peyrard, who published at Paris, between these years, in three volumes, the Elements and Data in Greek, Latin and FrenchEuclidis quae supersunt. Les Œuvres d'Euclide, en Grec, en Latin et en Français d'après un manuscrit très-ancien, qui était resté inconnu jusqu'à nos jours Par F. Peyrard. Ouvra<*>e approuvé par l'Institut de France (Paris, chez M. Patris).. At the time (1808) when Napoleon was having valuable MSS. selected from Italian libraries and sent to Paris, Peyrard managed to get two ancient Vatican MSS. (190 and 1038) sent to Paris for his use (Vat. 204 was also at Paris at the time, but all three were restored to their owners in 1814). Peyrard noticed the excellence of Cod. Vat. 190, adopted many of its readings, and gave in an appendix a conspectus of these readings and those of Gregory's edition; he also noted here and there readings from Vat. 1038 and various Paris MSS. He therefore pointed the way towards a better text, but committed the error of correcting the Basel text instead of rejecting it altogether and starting afresh.

-

1824-1825. A most valuable edition of Books I.—VI. is that of J. G. Camerer (and C. F. Hauber) in two volumes published at BerlinEuclidis elcmentorum libri sex priores graece et latine commentario e scriptis veterum ac recentiarum mathematicorum et Pfleidereri maxime illustrati (Berolini, sumptibus G. Reimeri). Tom. I. 1824; tom. II. 1825.. The Greek text is based on Peyrard, although the Basel and Oxford editions were also used. There is a Latin translation and a collection of notes far more complete than any other I have seen and well nigh inexhaustible. There is no editor or commentator of any mark who is not quoted from; to show the variety of important authorities drawn upon by Camerer, I need only mention the following names: Proclus, Pappus, Tartaglia, Commandinus, Clavius, Peletier, Barrow, Borelli, Wallis, Tacquet, Austin, Simson, Playfair. No words of praise would be too warm for this veritable encyclopaedia of information.

-

1825. J. G. C. Neide edited, from Peyrard, the text of Books I.—VI., XI. and XII. (Halis Saxoniae).

+

1703 is the date of the Oxford edition by David Gregory which, until the issue of Heiberg and Menge's edition, was still the only edition of the complete works of Euclid*e*u*k*l*e*i*d*o*u *t*a *s*w*z*o*m*e*n*a. Euclidis quae supersunt omnia. Ex recensione Davidis Gregorii M.D. Astronomiae Professoris Saviliani et R.S.S. Oxoniae, e Theatro Sheldoniano, An. Dom. MDCCIII.. In the Latin translation attached to the Greek text Gregory says that he followed Commandinus in the main, but corrected numberless passages in it by means of the books in the Bodleian Library which belonged to Edward Bernard (1638-1696), formerly Savilian Professor of Astronomy, who had conceived the plan of publishing the complete works of the ancient mathematicians in fourteen volumes, of which the first was to contain Euclid's Elements I.—XV. As regards the Greek text, Gregory tells us that he consulted, as far as was necessary, not a few MSS. of the better sort, bequeathed by the great Savile to the University, as well as the corrections made by Savile in his own hand in the margin of the Basel edition. He had the help of John Hudson, Bodley's Librarian, who punctuated the Basel text before it went to the printer, compared the Latin version with the Greek throughout, especially in the Elements and Data, and, where they differed or where he suspected the Greek text, consulted the Greek MSS. and put their readings in the margin if they agreed with the Latin and, if they did not agree, affixed an asterisk in order that Gregory might judge which reading was geometrically preferable. Hence it is clear that no Greek MS., but the Basel edition, was the foundation of Gregory's text, and that Greek MSS. were only referred to in the special passages to which Hudson called attention.

+

1814-1818. A most important step towards a good Greek text was taken by F. Peyrard, who published at Paris, between these years, in three volumes, the Elements and Data in Greek, Latin and FrenchEuclidis quae supersunt. Les Œuvres d'Euclide, en Grec, en Latin et en Français d'après un manuscrit très-ancien, qui était resté inconnu jusqu'à nos jours Par F. Peyrard. Ouvra<*>e approuvé par l'Institut de France (Paris, chez M. Patris).. At the time (1808) when Napoleon was having valuable MSS. selected from Italian libraries and sent to Paris, Peyrard managed to get two ancient Vatican MSS. (190 and 1038) sent to Paris for his use (Vat. 204 was also at Paris at the time, but all three were restored to their owners in 1814). Peyrard noticed the excellence of Cod. Vat. 190, adopted many of its readings, and gave in an appendix a conspectus of these readings and those of Gregory's edition; he also noted here and there readings from Vat. 1038 and various Paris MSS. He therefore pointed the way towards a better text, but committed the error of correcting the Basel text instead of rejecting it altogether and starting afresh.

+

1824-1825. A most valuable edition of Books I.—VI. is that of J. G. Camerer (and C. F. Hauber) in two volumes published at BerlinEuclidis elcmentorum libri sex priores graece et latine commentario e scriptis veterum ac recentiarum mathematicorum et Pfleidereri maxime illustrati (Berolini, sumptibus G. Reimeri). Tom. I. 1824; tom. II. 1825.. The Greek text is based on Peyrard, although the Basel and Oxford editions were also used. There is a Latin translation and a collection of notes far more complete than any other I have seen and well nigh inexhaustible. There is no editor or commentator of any mark who is not quoted from; to show the variety of important authorities drawn upon by Camerer, I need only mention the following names: Proclus, Pappus, Tartaglia, Commandinus, Clavius, Peletier, Barrow, Borelli, Wallis, Tacquet, Austin, Simson, Playfair. No words of praise would be too warm for this veritable encyclopaedia of information.

+

1825. J. G. C. Neide edited, from Peyrard, the text of Books I.—VI., XI. and XII. (Halis Saxoniae).

1826-9. The last edition of the Greek text before Heiberg's is that of E. F. August, who followed the Vatican MS. more closely than Peyrard did, and consulted at all events the Viennese MS. Gr. 103 (Heiberg's V). August's edition (Berlin, 1826-9) contains Books I.-XIII.

III. Latin versions or commentaries after 1533. -

1545. Petrus Ramus (Pierre de la Ramée, 1515-1572) is credited with a translation of Euclid which appeared in 1545 and again in 1549 at ParisDescribed by Boncompagni, Bullettino, II. p. 389.. Ramus, who was more rhetorician and logician than geometer, also published in his Scholae mathematicae (1559, Frankfurt; 1569, Basel) what amounts to a series of lectures on Euclid's Elements, in which he criticises Euclid's arrangement of his propositions, the definitions, postulates and axioms, all from the point of view of logic.

+

1545. Petrus Ramus (Pierre de la Ramée, 1515-1572) is credited with a translation of Euclid which appeared in 1545 and again in 1549 at ParisDescribed by Boncompagni, Bullettino, II. p. 389.. Ramus, who was more rhetorician and logician than geometer, also published in his Scholae mathematicae (1559, Frankfurt; 1569, Basel) what amounts to a series of lectures on Euclid's Elements, in which he criticises Euclid's arrangement of his propositions, the definitions, postulates and axioms, all from the point of view of logic.

1557. Demonstrations to the geometrical Elements of Euclid, six Books, by Peletarius (Jacques Peletier). The second edition (1610) contained the same with the addition of the Greek text of Euclid ; but only the enunciations of the propositions, as well as the definitions etc., are given in Greek (with a Latin translation), the rest is in Latin only. He has some acute observations, for instance about the angle of contact.

1559. Johannes Buteo, or Borrel (1492-1572), published in an appendix to his book De quadratura circuli some notes on the errors of Campanus, Zambertus, Orontius, Peletarius, Pena, interpreters of Euclid. Buteo in these notes proved, by reasoned argument based on original authorities, that Euclid himself and not Theon was the author of the proofs of the propositions.

-

1566. Franciscus Flussates Candalla (François de Foix, Comte de Candale, 1502-1594) restored +

1566. Franciscus Flussates Candalla (François de Foix, Comte de Candale, 1502-1594) restored the fifteen Books, following, as he says, the terminology of Zamberti's translation from the Greek, but drawing, for his proofs, on both Campanus and Theon (i.e. Zamberti) except where mistakes in them made emendation necessary. Other editions followed in 1578, 1602, 1695 (in Dutch).

1572. The most important Latin translation is that of Commandinus (1509-1575) of Urbino, since it was the foundation of most translations which followed it up to the time of Peyrard, including that of Simson and therefore of those editions, numerous in England, which give Euclid chiefly after the text of Simson. Simson's first (Latin) edition (1756) has ex versione Latina Federici Commandini @@ -1102,15 +1102,15 @@

He remarks in his preface that Orontius Finaeus had only edited six Books without reference to any Greek MS., that Peletarius had followed Campanus' version from the Arabic rather than the Greek text, and that Candalla had diverged too far from Euclid, having rejected as inelegant the proofs given in the Greek text and substituted faulty proofs of his own. Commandinus appears to have used, in addition to the Basel editio princeps, some Greek MS., so far not identified; he also extracted his scholia antiqua from a MS. of the class of Vat. 192 containing the scholia distinguished by Heiberg as Schol. Vat. - New editions of Commandinus' translation followed in 1575 (in Italian), 1619, 1749 (in English, by Keill and Stone), 1756 (Books I.—VI., XI., XII. in Latin and English, by Simson), 1763 (Keill). Besides these there were many editions of parts of the whole work, e.g. the first six Books.

+ New editions of Commandinus' translation followed in 1575 (in Italian), 1619, 1749 (in English, by Keill and Stone), 1756 (Books I.—VI., XI., XII. in Latin and English, by Simson), 1763 (Keill). Besides these there were many editions of parts of the whole work, e.g. the first six Books.

1574. The first edition of the Latin version by ClaviusEuclidis elemcntorum librit XV. Accessit XVI. de solidorum regularium comparatione. Omnes perspicuis demonstrationibus, accuratisque scholiis illustrati. Auctore Christophoro Clavio (Romae, apud Vincentium Accoltum), 2 vols. (Christoph Klau [?]. born at Bamberg 1537, died 1612) appeared in 1574, and new editions of it in 1589, 1591, 1603, 1607, 1612. It is not a translation, as Clavius himself states in the preface, but it contains a vast amount of notes collected from previous commentators and editors, as well as some good criticisms and elucidations of his own. Among other things, Clavius finally disposed of the error by which Euclid had been identified with Euclid of Megara. He speaks of the differences between Campanus who followed the Arabic tradition and the commentaries of Theon, by which he appears to mean the Euclidean proofs as handed down by Theon; he complains of predecessors who have either only given the first six Books, or have rejected the ancient proofs and substituted worse proofs of their own, but makes an exception as regards Commandinus, a geometer not of the common sort, who has lately restored Euclid, in a Latin translation, to his original brilliancy. Clavius, as already stated, did not give a translation of the Elements but rewrote the proofs, compressing them or adding to them, where he thought that he could make them clearer. Altogether his book is a most useful work.

1621. Henry Savile's lectures (Praelectiones tresdecim in principium Elementorum Euclidis Oxoniae habitae MDC.XX., Oxonii 1621), though they do not extend beyond I. 8, are valuable because they grapple with the difficulties connected with the preliminary matter, the definitions etc., and the tacit assumptions contained in the first propositions.

-

1654. André Tacquet's Elementa geometriae planae et solidae containing apparently the eight geometrical Books arranged for general use in schools. It came out in a large number of editions up to the end of the eighteenth century.

+

1654. André Tacquet's Elementa geometriae planae et solidae containing apparently the eight geometrical Books arranged for general use in schools. It came out in a large number of editions up to the end of the eighteenth century.

1655. Barrow's Euclidis Elementorum Libri XV breviter demonstrati is a book of the same kind. In the preface (to the edition of 1659) he says that he would not have written it but for the fact that Tacquet gave only eight Books of Euclid. He compressed the work into a very small compass (less than 400 small pages, in the edition of 1659, for the whole of the fifteen Books and the Data) by abbreviating the proofs and using a large quantity of symbols (which, he says, are generally Oughtred's). There were several editions up to 1732 (those of 1660 and 1732 and one or two others are in English).

1658. Giovanni Alfonso Borelli (1608-1679) published Euclides restitutus, on apparently similar lines, which went through three more editions (one in Italian, 1663).

-

1660. Claude François Milliet Dechales' eight geometrical Books of Euclid's Elements made easy. Dechales' versions of the Elements had great vogue, appearing in French, Italian and English as well as Latin. Riccardi enumerates over twenty editions.

+

1660. Claude François Milliet Dechales' eight geometrical Books of Euclid's Elements made easy. Dechales' versions of the Elements had great vogue, appearing in French, Italian and English as well as Latin. Riccardi enumerates over twenty editions.

1733. Saccheri's Euclides ab omni naevo vindicatus sive conatus geometricus quo stabiliuntur prima ipsa geometriae principia is important for his elaborate attempt to prove the parallel-postulate, forming an important stage in the history of the development of nonEuclidean geometry.

1756. Simson's first edition, in Latin and in English. The Latin title is

Euclidis elementorum libri priores sex, item undecimus et duodecimus, ex versione latina Federici Commandini; sublatis iis quibus olim libri hi a Theone, aliisve, vitiati sunt, et quibusdam Euclidis demonstrationibus restitutis. A Roberto Simson M.D. Glasguae, in aedibus Academicis excudebant Robertus et Andreas Foulis, Academiae typographi.

@@ -1119,14 +1119,14 @@ IV. Italian versions or commentaries. -

1543. Tartaglia's version, a second edition of which was published in 1565The title-page of the edition of 1565 is as follows: Euclide Megarense philosopho, solo introduttore delle scientie mathematice, diligentemente rassettato, et alla integrità ridotto, per il degno professore di tal scientie Nicolo Tartalea Brisciano. secondo le due tradottioni. con una ampla espositione dello istesso tradottore di nuouo aggiunta. talmente chiara, che ogni mediocre ingegno, sensa la notitia, ouer suffragio di alcun' altra scientia con facilità serà capace a poterlo intendere. In Venetia, Appresso Curtio Troiano, 1565., and a third in 1585. It does not appear that he used any Greek text, for in the edition of 1565 he mentions as available only the first translation by Campano, +

1543. Tartaglia's version, a second edition of which was published in 1565The title-page of the edition of 1565 is as follows: Euclide Megarense philosopho, solo introduttore delle scientie mathematice, diligentemente rassettato, et alla integrità ridotto, per il degno professore di tal scientie Nicolo Tartalea Brisciano. secondo le due tradottioni. con una ampla espositione dello istesso tradottore di nuouo aggiunta. talmente chiara, che ogni mediocre ingegno, sensa la notitia, ouer suffragio di alcun' altra scientia con facilità serà capace a poterlo intendere. In Venetia, Appresso Curtio Troiano, 1565., and a third in 1585. It does not appear that he used any Greek text, for in the edition of 1565 he mentions as available only the first translation by Campano, the second made by Bartolomeo Zamberto Veneto who is still alive, the editions of Paris or Germany in which they have included both the aforesaid translations, and our own translation into the vulgar (tongue).

1575. Commandinus' translation turned into Italian and revised by him.

1613. The first six Books reduced to practice - by Pietro Antonio Cataldi, re-issued in 1620, and followed by Books VII.—IX. (1621) and Book X. (1625).

+ by Pietro Antonio Cataldi, re-issued in 1620, and followed by Books VII.—IX. (1621) and Book X. (1625).

1663. Borelli's Latin translation turned into Italian by Domenico Magni.

1680. Euclide restituto by Vitale Giordano.

1690. Vincenzo Viviani's Elementi piani e solidi di Euclide (Book V. in 1674).

@@ -1137,48 +1137,48 @@ V. German. -

1558. The arithmetical Books VII.-IX. by ScheubelDas sibend acht und neunt buch des hochberümbten Mathematici Euclidis Megarensis... - durch Magistrum Fohann Scheybl, der löblichen universitet zu Tübingen, des Euclidis und Arithmetic Ordinarien, auss dem latein ins teutsch gebracht.... (cf. the edition of the first six Books, with enunciations in Greek and Latin, mentioned above, under date 1550).

-

1562. The version of the first six Books by Wilhelm Holtzmann (Xylander).Die sechs erste b'ücher Euclidis vom anfang oder grund der Geometrj...Auss Griechischer sprach in die Teütsch gebracht aigentlich erklärt...Demassen vormals in Teütscher sprach nie gesehen worden...Durch Wilhelm Holtzman genant Xylander von Augspurg. Getruckht zu Basel.. This work has its interest as the first edition in German, but otherwise it is not of importance. Xylander tells us that it was written for practical people such as artists, goldsmiths, builders etc., and that, as the simple amateur is of course content to know facts, without knowing how to prove them, he has often left out the proofs altogether. He has indeed taken the greatest possible liberties with Euclid, and has not grappled with any of the theoretical difficulties, such as that of the theory of parallels.

+

1558. The arithmetical Books VII.-IX. by ScheubelDas sibend acht und neunt buch des hochberümbten Mathematici Euclidis Megarensis... + durch Magistrum Fohann Scheybl, der löblichen universitet zu Tübingen, des Euclidis und Arithmetic Ordinarien, auss dem latein ins teutsch gebracht.... (cf. the edition of the first six Books, with enunciations in Greek and Latin, mentioned above, under date 1550).

+

1562. The version of the first six Books by Wilhelm Holtzmann (Xylander).Die sechs erste b'ücher Euclidis vom anfang oder grund der Geometrj...Auss Griechischer sprach in die Teütsch gebracht aigentlich erklärt...Demassen vormals in Teütscher sprach nie gesehen worden...Durch Wilhelm Holtzman genant Xylander von Augspurg. Getruckht zu Basel.. This work has its interest as the first edition in German, but otherwise it is not of importance. Xylander tells us that it was written for practical people such as artists, goldsmiths, builders etc., and that, as the simple amateur is of course content to know facts, without knowing how to prove them, he has often left out the proofs altogether. He has indeed taken the greatest possible liberties with Euclid, and has not grappled with any of the theoretical difficulties, such as that of the theory of parallels.

1651. Heinrich Hoffmann's Teutscher Euclides (2nd edition 1653), not a translation.

1694. Ant. Ernst Burkh. v. Pirckenstein's Teutsch Redender Euclides (eight geometrical Books), for generals, engineers etc. proved in a new and quite easy manner. Other editions 1699, 1744.

1697. Samuel Reyher's In teutscher Sprache vorgestellter Euclides (six Books), made easy, with symbols algebraical or derived from the newest art of solution.

-

1714. Euclidis XV Bücher teutsch, treated in a special and brief manner, yet completely, +

1714. Euclidis XV Bücher teutsch, treated in a special and brief manner, yet completely, by Chr. Schessler (another edition in 1729).

1773. The first six Books translated from the Greek for the use of schools by J. F. Lorenz. The first attempt to reproduce Euclid in German word for word.

-

1781. Books XI., XII. by Lorenz (supplementary to the preceding). Also Euklid's Elemente fünfzehn Bücher translated from the Greek by Lorenz (second edition 1798; editions of 1809, 1818, 1824 by Mollweide, of 1840 by Dippe). The edition of 1824, and I presume those before it, are shortened by the use of symbols and the compression of the enunciation and setting-out +

1781. Books XI., XII. by Lorenz (supplementary to the preceding). Also Euklid's Elemente fünfzehn Bücher translated from the Greek by Lorenz (second edition 1798; editions of 1809, 1818, 1824 by Mollweide, of 1840 by Dippe). The edition of 1824, and I presume those before it, are shortened by the use of symbols and the compression of the enunciation and setting-out into one.

-

1807. Books I.—VI., XI., XII. newly translated from the Greek, +

1807. Books I.—VI., XI., XII. newly translated from the Greek, by J. K. F. Hauff.

1828. The same Books by Joh. Jos. Ign. Hoffmann as guide to instruction in elementary geometry, followed in 1832 by observations on the text by the same editor.

1833. Die Geometrie des Euklid und das Wesen derselben by E. S. Unger; also 1838, 1851.

-

1901. Max Simon, Euclid und die sechs planimetrischen Bücher.

+

1901. Max Simon, Euclid und die sechs planimetrischen Bücher.

 VI. French. -

1564-1566. Nine Books translated by Pierre Forcadel, a pupil and friend of P. de la Ramée.

+

1564-1566. Nine Books translated by Pierre Forcadel, a pupil and friend of P. de la Ramée.

1604. The first nine Books translated and annotated by Jean Errard de Bar-le-Duc; second edition, 1605.

1615. Denis Henrion's translation of the 15 Books (seven editions up to 1676).

1639. The first six Books demonstrated by symbols, by a method very brief and intelligible, - by Pierre Hérigone, mentioned by Barrow as the only editor who, before him, had used symbols for the exposition of Euclid.

+ by Pierre Hérigone, mentioned by Barrow as the only editor who, before him, had used symbols for the exposition of Euclid.

1672. Eight Books rendus plus faciles - by Claude FranÇis Milliet Dechales, who also brought out Les élémens d'Euclide expliqués d'une manière nouvelle et très facile, which appeared in many editions, 1672, 1677, 1683 etc. (from 1709 onwards revised by Ozanam), and was translated into Italian (1749 etc.) and English (by William Halifax, 1685).

-

1804. In this year, and therefore before his edition of the Greek text, F. Peyrard published the Elements literally translated into French. A second edition appeared in 1809 with the addition of the fifth Book. As this second edition contains Books I.—VI. XI., XII. and X. I, it would appear that the first edition contained Books I.—IV., VI., XI., XII. Peyrard used for this translation the Oxford Greek text and Simson.

+ by Claude FranÇis Milliet Dechales, who also brought out Les élémens d'Euclide expliqués d'une manière nouvelle et très facile, which appeared in many editions, 1672, 1677, 1683 etc. (from 1709 onwards revised by Ozanam), and was translated into Italian (1749 etc.) and English (by William Halifax, 1685).

+

1804. In this year, and therefore before his edition of the Greek text, F. Peyrard published the Elements literally translated into French. A second edition appeared in 1809 with the addition of the fifth Book. As this second edition contains Books I.—VI. XI., XII. and X. I, it would appear that the first edition contained Books I.—IV., VI., XI., XII. Peyrard used for this translation the Oxford Greek text and Simson.

 VII. Dutch. -

1606. Jan Pieterszoon Dou (six Books). There were many later editions. Kästner, in mentioning one of 1702, says that Dou explains in his preface that he used Xylander's translation, but, having afterwards obtained the French translation of the six Books by Errard de Bar-le-Duc (see above), the proofs in which sometimes pleased him more than those of the German edition, he made his Dutch version by the help of both.

+

1606. Jan Pieterszoon Dou (six Books). There were many later editions. Kästner, in mentioning one of 1702, says that Dou explains in his preface that he used Xylander's translation, but, having afterwards obtained the French translation of the six Books by Errard de Bar-le-Duc (see above), the proofs in which sometimes pleased him more than those of the German edition, he made his Dutch version by the help of both.

1617. Frans van Schooten, The Propositions of the Books of Euclid's Elements ; the fifteen Books in this version enlarged by Jakob van Leest in 1662.

1695. C. J. Vooght, fifteen Books complete, with Candalla's 16th.

1702. Hendrik Coets, six Books (also in Latin, 1692); several editions up to 1752. Apparently not a translation. but an edition for school use.

-

1763. Pybo Steenstra, Books I.—VI., XI., XII., likewise an abberviated version, several times reissued until 1825.

+

1763. Pybo Steenstra, Books I.—VI., XI., XII., likewise an abberviated version, several times reissued until 1825.

 VIII. English. @@ -1187,10 +1187,10 @@

Faithfully (now first) translated into the Englishe toung, by H. Billingsley, Citizen of London. Whereunto are annexed certaine Scholies, Annotations, and Inuentions, of the best Mathematiciens, both of time past, and in this our age.

-

With a very fruitfull Preface by M. I. Dee, specifying the chiefe Mathematicall Sciēces, what they are, and whereunto commodious: where, also, are disclosed certaine new Secrets Mathematicall and Mechanicall, vntill these our daies, greatly missed.

+

With a very fruitfull Preface by M. I. Dee, specifying the chiefe Mathematicall Sciēces, what they are, and whereunto commodious: where, also, are disclosed certaine new Secrets Mathematicall and Mechanicall, vntill these our daies, greatly missed.

Imprinted at London by John Daye.

-

The Preface by the translator, after a sentence observing that without the diligent study of Euclides Elementes it is impossible to attain unto the perfect knowledge of Geometry, proceeds thus. Wherefore considering the want and lacke of such good authors hitherto in our Englishe tounge, lamenting also the negligence, and lacke of zeale to their countrey in those of our nation, to whom God hath geuen both knowledge and also abilitie to translate into our tounge, and to publishe abroad such good authors and bookes (the chiefe instrumentes of all learninges): seing moreouer that many good wittes both of gentlemen and of others of all degrees, much desirous and studious of these artes, and seeking for them as much as they can, sparing no paines, and yet frustrate of their intent, by no meanes attaining to that which they seeke: I haue for their sakes, with some charge and great trauaile, faithfully translated into our vulgare toũge, and set abroad in Print, this booke of Euclide. Whereunto I haue added easie and plaine declarations and examples by figures, of the definitions. In which booke also ye shall in due place finde manifolde additions, Scholies, Annotations, and Inuentions: which I haue gathered out of many of the most famous and chiefe Mathematiciēs, both of old time, and in our age: as by diligent reading it in course, ye shall well perceaue.... +

The Preface by the translator, after a sentence observing that without the diligent study of Euclides Elementes it is impossible to attain unto the perfect knowledge of Geometry, proceeds thus. Wherefore considering the want and lacke of such good authors hitherto in our Englishe tounge, lamenting also the negligence, and lacke of zeale to their countrey in those of our nation, to whom God hath geuen both knowledge and also abilitie to translate into our tounge, and to publishe abroad such good authors and bookes (the chiefe instrumentes of all learninges): seing moreouer that many good wittes both of gentlemen and of others of all degrees, much desirous and studious of these artes, and seeking for them as much as they can, sparing no paines, and yet frustrate of their intent, by no meanes attaining to that which they seeke: I haue for their sakes, with some charge and great trauaile, faithfully translated into our vulgare toũge, and set abroad in Print, this booke of Euclide. Whereunto I haue added easie and plaine declarations and examples by figures, of the definitions. In which booke also ye shall in due place finde manifolde additions, Scholies, Annotations, and Inuentions: which I haue gathered out of many of the most famous and chiefe Mathematiciēs, both of old time, and in our age: as by diligent reading it in course, ye shall well perceaue....

It is truly a monumental work, consisting of 464 leaves, and therefore 928 pages, of folio size, excluding the lengthy preface by Dee. The notes certainly include all the most important that had ever been written, from those of the Greek commentators, Proclus and the others whom he quotes, down to those of Dee himself on the last books. Besides the fifteen Books, Billingsley included the sixteenth added by Candalla. The print and appearance of the book are worthy of its contents; and, in order that it may be understood how no pains were spared to represent everything in the clearest and most perfect form, I need only mention that the figures of the propositions in Book XI. are nearly all duplicated, one being the figure of Euclid, the other an arrangement of pieces of paper (triangular, rectangular etc.) pasted at the edges on to the page of the book so that the pieces can be turned up and made to show the real form of the solid figures represented.

@@ -1203,7 +1203,7 @@

1685. William Halifax's version of Dechales' Elements of Euclid explained in a new but most easy method (London and Oxford).

1705. The English Euclide; being the first six Elements of Geometry, translated out of the Greek, with annotations and usefull supplements by Edmund Scarburgh (Oxford). A noteworthy and useful edition.

-

1708. Books I.—VI., XI., XII., translated from Commandinus' Latin version by Dr John Keill, Savilian Professor of Astronomy at Oxford.

+

1708. Books I.—VI., XI., XII., translated from Commandinus' Latin version by Dr John Keill, Savilian Professor of Astronomy at Oxford.

Keill complains in his preface of the omissions by such editors as Tacquet and Dechales of many necessary propositions (e.g. VI. 27-29), and of their substitution of proofs of their own for Euclid's. He praises Barrow's version on the whole, though objecting to the algebraical form of proof adopted in Book II., and to the excessive use of notes and symbols, which (he considers) make the proofs too short and thereby obscure; his edition was therefore intended to hit a proper mean between Barrow's excessive brevity and Clavius' prolixity.

Keill's translation was revised by Samuel Cunn and several times reissued. 1749 saw the eighth edition, 1772 the eleventh, and 1782 the twelfth.

@@ -1241,9 +1241,9 @@ IX. Spanish. -

1576. The first six Books translated into Spanish by Rodrigo Çamorano.

+

1576. The first six Books translated into Spanish by Rodrigo Çamorano.

1637. The first six Books translated, with notes, by L. Carduchi.

-

1689. Books I.—VI., XI., XII., translated and explained by Jacob Knesa.

+

1689. Books I.—VI., XI., XII., translated and explained by Jacob Knesa.

 @@ -1255,23 +1255,23 @@ XI. Swedish. -

1744. Mårten Strömer, the first six Books; second edition 1748. The third edition (1753) contained Books XI.—XII. as well; new editions continued to appear till 1884.

+

1744. Mårten Strömer, the first six Books; second edition 1748. The third edition (1753) contained Books XI.—XII. as well; new editions continued to appear till 1884.

1836. H. Falk, the first six Books.

-

1844, 1845, 1859. P. R. Bråkenhjelm, Books I.—VI., XI., XII.

+

1844, 1845, 1859. P. R. Bråkenhjelm, Books I.—VI., XI., XII.

1850. F. A. A. Lundgren.

-

1850. H. A. Witt and M. E. Areskong, Books I.—VI., XI., XII.

+

1850. H. A. Witt and M. E. Areskong, Books I.—VI., XI., XII.

 XII. Danish.

1745. Ernest Gottlieb Ziegenbalg.

-

1803. H. C. Linderup, Books I.—VI.

+

1803. H. C. Linderup, Books I.—VI.

 XIII. Modern Greek.

1820. Benjamin of Lesbos.

I should add a reference to certain editions which have appeared in recent years.

-

A Danish translation (Euklid's Elementer oversat af Thyra Eibe) was completed in 1912; Books I.—II. were published (with an Introduction by Zeuthen) in 1897, Books III.—IV. in 1900, Books V.—VI. in 1904, Books VII.—XIII. in 1912.

-

The Italians, whose great services to elementary geometry are more than once emphasised in this work, have lately shown a noteworthy disposition to make the ipsissima verba of Euclid once more the object of study. Giovanni Vacca has edited the text of Book I. (Il primo libro degli Elementi. Testo greco, versione italiana, introduzione e note, Firenze 1916.) Federigo Enriques has begun the publication of a complete Italian translation (Gli Elementi d' Euclide e la critica antica e moderna); Books I.—IV. appeared in 1925 (Alberto Stock, Roma).

+

A Danish translation (Euklid's Elementer oversat af Thyra Eibe) was completed in 1912; Books I.—II. were published (with an Introduction by Zeuthen) in 1897, Books III.—IV. in 1900, Books V.—VI. in 1904, Books VII.—XIII. in 1912.

+

The Italians, whose great services to elementary geometry are more than once emphasised in this work, have lately shown a noteworthy disposition to make the ipsissima verba of Euclid once more the object of study. Giovanni Vacca has edited the text of Book I. (Il primo libro degli Elementi. Testo greco, versione italiana, introduzione e note, Firenze 1916.) Federigo Enriques has begun the publication of a complete Italian translation (Gli Elementi d' Euclide e la critica antica e moderna); Books I.—IV. appeared in 1925 (Alberto Stock, Roma).

An edition of Book I. by the present writer was published in 1918 (Euclid in Greek, Book I., with Introduction and Notes, Camb. Univ. Press).

 @@ -1279,14 +1279,14 @@ CHAPTER IX. - § 1. ON THE NATURE OF ELEMENTS. + § 1. ON THE NATURE OF ELEMENTS.

It would not be easy to find a more lucid explanation of the terms element and elementary, and of the distinction between them, than is found in ProclusProclus, Comm. on Eucl. I., ed. Friedlein, pp. 72 sqq., who is doubtless, here as so often, quoting from Geminus. There are, says Proclus, in the whole of geometry certain leading theorems, bearing to those which follow the relation of a principle, all-pervading, and furnishing proofs of many properties. Such theorems are called by the name of elements; and their function may be compared to that of the letters of the alphabet in relation to language, letters being indeed called by the same name in Greek (stoixei=a).

The term elementary, on the other hand, has a wider application: it is applicable to things which extend to greater multiplicity, and, though possessing simplicity and elegance, have no longer the same dignity as the elements, because their investigation is not of general use in the whole of the science, e.g. the proposition that in triangles the perpendiculars from the angles to the transverse sides meet in a point.

-

“Again, the term element is used in two senses, as Menaechmus says. For that which is the means of obtaining is an element of that which is obtained, as the first proposition in Euclid is of the second, and the fourth of the fifth. In this sense many things may even be said to be elements of each other, for they are obtained from one another. Thus from the fact that the exterior angles of rectilineal figures are (together) equal to four right angles we deduce the number of right angles equal to the internal angles (taken together)to\ plh=qos tw=n e)nto\s o)rqai=s i)/swn. If the text is right, we must apparently take it as the number of the angles equal to right angles that there are inside, +

“Again, the term element is used in two senses, as Menaechmus says. For that which is the means of obtaining is an element of that which is obtained, as the first proposition in Euclid is of the second, and the fourth of the fifth. In this sense many things may even be said to be elements of each other, for they are obtained from one another. Thus from the fact that the exterior angles of rectilineal figures are (together) equal to four right angles we deduce the number of right angles equal to the internal angles (taken together)to\ plh=qos tw=n e)nto\s o)rqai=s i)/swn. If the text is right, we must apparently take it as the number of the angles equal to right angles that there are inside, i.e. that are made up by the internal angles, and vice versa. Such an element is like a lemma. But the term element is otherwise used of that into which, being more simple, the composite is divided; and in this sense we can no longer say that everything is an element of everything, but only that things which are more of the nature of principles are elements of those which stand to them in the relation of results, as postulates are elements of theorems. It is according to this signification of the term element that the elements found in Euclid were compiled, being partly those of plane geometry, and partly those of stereometry. In like manner many writers have drawn up elementary treatises in arithmetic and astronomy.

-

“Now it is difficult, in each science, both to select and arrange in due order the elements from which all the rest proceeds, and into which all the rest is resolved. And of those who have made the attempt some were able to put together more and some less; some used shorter proofs, some extended their investigation to an indefinite length; some avoided the method of reductio ad absurdum, some avoided proportion; some contrived preliminary steps directed against those who reject the principles; and, in a word, many different methods have been invented by various writers of elements.

-

It is essential that such a treatise should be rid of everything superfluous (for this is an obstacle to the acquisition of knowledge); it should select everything that embraces the subject and brings it to a point (for this is of supreme service to science); it must have great regard at once to clearness and conciseness (for their opposites trouble our understanding); it must aim at the embracing of theorems in general terms (for the piecemeal division of instruction into the more partial makes knowledge difficult to grasp). In all these ways Euclid's system of elements will be found to be superior to the rest; for its utility avails towards the investigation of the primordial figurestw=n a)rxikw=n sxhma/twn, by which Proclus probably means the regular polyhedra (Tannery, P. 143 n.)., its clearness and organic perfection are secured by the progression from the more simple to the more complex and by the foundation of the investigation upon common notions, while generality of demonstration is secured by the progression through the theorems which are primary and of the nature of principles to the things sought. As for the things which seem to be wanting, they are partly to be discovered by the same methods, like the construction of the scalene and isosceles (triangle), partly alien to the character of a selection of elements as introducing hopeless and boundless complexity, like the subject of unordered irrationals which Apollonius worked out at lengthWe have no more than the most obscure indications of the character of this work in an Arabic MS. analysed by Woepcke, Essai d'une restitution de travaux perdus d'Apollonius sur les quantités irrationelles d'après des indications tirées d'un manuscrit arabe in Mémoires présentés à l'académie des sciences, XIV. 658-720, Paris, 1856. Cf. Cantor, Gesch. d. Math. I_{3}, pp. 348-9: details are also given in my notes to Book X., and partly developed from things handed down (in the elements) as causes, like the many species of angles and of lines. These things then have been omitted in Euclid, though they have received full discussion in other works; but the knowledge of them is derived from the simple (elements). +

“Now it is difficult, in each science, both to select and arrange in due order the elements from which all the rest proceeds, and into which all the rest is resolved. And of those who have made the attempt some were able to put together more and some less; some used shorter proofs, some extended their investigation to an indefinite length; some avoided the method of reductio ad absurdum, some avoided proportion; some contrived preliminary steps directed against those who reject the principles; and, in a word, many different methods have been invented by various writers of elements.

+

It is essential that such a treatise should be rid of everything superfluous (for this is an obstacle to the acquisition of knowledge); it should select everything that embraces the subject and brings it to a point (for this is of supreme service to science); it must have great regard at once to clearness and conciseness (for their opposites trouble our understanding); it must aim at the embracing of theorems in general terms (for the piecemeal division of instruction into the more partial makes knowledge difficult to grasp). In all these ways Euclid's system of elements will be found to be superior to the rest; for its utility avails towards the investigation of the primordial figurestw=n a)rxikw=n sxhma/twn, by which Proclus probably means the regular polyhedra (Tannery, P. 143 n.)., its clearness and organic perfection are secured by the progression from the more simple to the more complex and by the foundation of the investigation upon common notions, while generality of demonstration is secured by the progression through the theorems which are primary and of the nature of principles to the things sought. As for the things which seem to be wanting, they are partly to be discovered by the same methods, like the construction of the scalene and isosceles (triangle), partly alien to the character of a selection of elements as introducing hopeless and boundless complexity, like the subject of unordered irrationals which Apollonius worked out at lengthWe have no more than the most obscure indications of the character of this work in an Arabic MS. analysed by Woepcke, Essai d'une restitution de travaux perdus d'Apollonius sur les quantités irrationelles d'après des indications tirées d'un manuscrit arabe in Mémoires présentés à l'académie des sciences, XIV. 658-720, Paris, 1856. Cf. Cantor, Gesch. d. Math. I_{3}, pp. 348-9: details are also given in my notes to Book X., and partly developed from things handed down (in the elements) as causes, like the many species of angles and of lines. These things then have been omitted in Euclid, though they have received full discussion in other works; but the knowledge of them is derived from the simple (elements).

Proclus, speaking apparently on his own behalf, in another place distinguishes two objects aimed at in Euclid's Elements. The first has reference to the matter of the investigation, and here, like a good Platonist, he takes the whole subject of geometry to be concerned with the cosmic figures, the five regular solids, which in Book XIII. are constructed, inscribed in a sphere and compared with one another. The second object is relative to the learner; and, from this standpoint, the elements may be described as a means of perfecting the learner's understanding with reference to the whole of geometry. For, starting from these (elements), we shall be able to acquire knowledge of the other parts of this science as well, while without them it is impossible for us to get a grasp of so complex a subject, and knowledge of the rest is unattainable. As it is, the theorems which are most of the nature of principles, most simple, and most akin to the first hypotheses are here collected, in their appropriate order; and the proofs of all other propositions use these theorems as thoroughly well known, and start from them. Thus Archimedes in the books on the sphere and cylinder, Apollonius, and all other geometers, clearly use the theorems proved in this very treatise as constituting admitted principlesProclus, pp. 70, 19-71, 21. @@ -1296,25 +1296,25 @@ Topics VIII. 14, 163 b 23. ; in general the first of the elements are, given the definitions, e.g. of a straight line and of a circle, most easy to prove, although of course there are not many data that can be used to establish each of them because there are not many middle terms Topics VIII. 3, 158 b 35. - ; among geometrical propositions we call those ’elements’ the proofs of which are contained in the proofs of all or most of such propositions + ; among geometrical propositions we call those ’elements’ the proofs of which are contained in the proofs of all or most of such propositions Metaph. 998 a 25 .; (as in the case of bodies), so in like manner we speak of the elements of geometrical propositions and, generally, of demonstrations; for the demonstrations which come first and are contained in a variety of other demonstrations are called elements of those demonstrations... the term element is applied by analogy to that which, being one and small, is useful for many purposes Metaph. 1014 a 35-b 5..

 - § 2. ELEMENTS ANTERIOR TO EUCLID'S. + § 2. ELEMENTS ANTERIOR TO EUCLID'S.

The early part of the famous summary of Proclus was no doubt drawn, at least indirectly, from the history of geometry by Eudemus; this is generally inferred from the remark, made just after the mention of Philippus of Medma, a disciple of Plato, that those who have written histories bring the development of this science up to this point. We have therefore the best authority for the list of writers of elements given in the summary. Hippocrates of Chios (fl. in second half of 5th c.) is the first; then Leon, who also discovered diorismi, put together a more careful collection, the propositions proved in it being more numerous as well as more serviceableProclus, p. 66, 20 w)/ste to\n *le/onta kai\ ta\ stoixei=a sunqei=nai tw=| te plh(qei kai\ th= xrei/a| tw=n deiknume/nwn e)pimele/steron.. Leon was a little older than Eudoxus (about 408-355 B.C.) and a little younger than Plato (428/7-347/6 B.C.), but did not belong to the latter's school. The geometrical text-book of the Academy was written by Theudius of Magnesia, who, with Amyclas of Heraclea, Menaechmus the pupil of Eudoxus, Menaechmus' brother Dinostratus and Athenaeus of Cyzicus consorted together in the Academy and carried on their investigations in common. Theudius put together the elements admirably, making many partial (or limited) propositions more generalProclus, p. 67, 14 kai\ ga\r ta\ stoixei=a kalw=s sune/tacen kai\ polla\ tw=n merikw=n [o(rikw=n (?) Friedlein] kaqolikw/tera e)poi/hsen.. Eudemus mentions no text-book after that of Theudius, only adding that Hermotimus of Colophon discovered many of the elementsProclus, p. 67, 22 tw=n stoixei/wn polla\ a)neu=re.. Theudius then must be taken to be the immediate precursor of Euclid, and no doubt Euclid made full use of Theudius as well as of the discoveries of Hermotimus and all other available material. Naturally it is not in Euclid's Elements that we can find much light upon the state of the subject when he took it up; but we have another source of information in Aristotle. Fortunately for the historian of mathematics, Aristotle was fond of mathematical illustrations; he refers to a considerable number of geometrical propositions, definitions etc., in a way which shows that his pupils must have had at hand some textbook where they could find the things he mentions; and this text-book must have been that of Theudius. Heiberg has made a most valuable collection of mathematical extracts from AristotleMathematisches zu Aristoteles in Abhandlungen zur Gesch. d. math. Wissenschaften, XVIII. Heft (1904), pp. 1-49., from which much is to be gathered as to the changes which Euclid made in the methods of his predecessors; and these passages, as well as others not included in Heiberg's selection, will often be referred to in the sequel.

 - § 3. FIRST PRINCIPLES: DEFINITIONS, POSTULATES, AND AXIOMS. + § 3. FIRST PRINCIPLES: DEFINITIONS, POSTULATES, AND AXIOMS.

On no part of the subject does Aristotle give more valuable information than on that of the first principles as, doubtless, generally accepted at the time when he wrote. One long passage in the Posterior Analytics is particularly full and lucid, and is worth quoting in extenso. After laying it down that every demonstrative science starts from necessary principlesAnal. post. 1. 6, 74 b 5., he proceedsibid. 1. 10, 76 a 31-77 a 4.:

-

“By first principles in each genus I mean those the truth of which it is not possible to prove. What is denoted by the first (terms) and those derived from them is assumed; but, as regards their existence, this must be assumed for the principles but proved for the rest. Thus what a unit is, what the straight (line) is, or what a triangle is (must be assumed); and the existence of the unit and of magnitude must also be assumed, but the rest must be proved. Now of the premisses used in demonstrative sciences some are peculiar to each science and others common (to all), the latter being common by analogy, for of course they are actually useful in so far as they are applied to the subject-matter included under the particular science. Instances of first principles peculiar to a science are the assumptions that a line is of such and such a character, and similarly for the straight (line); whereas it is a common principle, for instance, that, if equals be subtracted from equals, the remainders are equal. But it is enough that each of the common principles is true so far as regards the particular genus (subject-matter); for (in geometry) the effect will be the same even if the common principle be assumed to be true, not of everything, but only of magnitudes, and, in arithmetic, of numbers.

-

“Now the things peculiar to the science, the existence of which must be assumed, are the things with reference to which the science investigates the essential attributes, e.g. arithmetic with reference to units, and geometry with reference to points and lines. With these things it is assumed that they exist and that they are of such and such a nature. But, with regard to their essential properties, what is assumed is only the meaning of each term employed: thus arithmetic assumes the answer to the question what is (meant by) ’odd’ or ’even,’ ’a square’ or ’a cube,’ and geometry to the question what is (meant by) ’the irrational’ or ’deflection’ or (the so-called) ’verging’ (to a point); but that there are such things is proved by means of the common principles and of what has already been demonstrated. Similarly with astronomy. For every demonstrative science has to do with three things, (1) the things which are assumed to exist, namely the genus (subject-matter) in each case, the essential properties of which the science investigates, (2) the common axioms so-called, which are the primary source of demonstration, and (3) the properties with regard to which all that is assumed is the meaning of the respective terms used. There is, however, no reason why some sciences should not omit to speak of one or other of these things. Thus there need not be any supposition as to the existence of the genus, if it is manifest that it exists (for it is not equally clear that number exists and that cold and hot exist); and, with regard to the properties, there need be no assumption as to the meaning of terms if it is clear: just as in the common (axioms) there is no assumption as to what is the meaning of subtracting equals from equals, because it is well known. But none the less is it true that there are three things naturally distinct, the subject-matter of the proof, the things proved, and the (axioms) from which (the proof starts).

-

Now that which is per se necessarily true, and must necessarily be thought so, is not a hypothesis nor yet a postulate. For demonstration has not to do with reasoning from outside but with the reason dwelling in the soul, just as is the case with the syllogism. It is always possible to raise objection to reasoning from outside, but to contradict the reason within us is not always possible. Now anything that the teacher assumes, though it is matter of proof, without proving it himself, is a hypothesis if the thing assumed is believed by the learner, and it is moreover a hypothesis, not absolutely, but relatively to the particular pupil; but, if the same thing is assumed when the learner either has no opinion on the subject or is of a contrary opinion, it is a postulate. This is the difference between a hypothesis and a postulate; for a postulate is that which is rather contrary than otherwise to the opinion of the learner, or whatever is assumed and used without being proved, although matter for demonstration. Now definitions are not hypotheses, for they do not assert the existence or non-existence of anything, while hypotheses are among propositions. Definitions only require to be understood: a definition is therefore not a hypothesis, unless indeed it be asserted that any audible speech is a hypothesis. A hypothesis is that from the truth of which, if assumed, a conclusion can be established. Nor are the geometer's hypotheses false, as some have said: I mean those who say that ’you should not make use of what is false, and yet the geometer falsely calls the line which he has drawn a foot long when it is not, or straight when it is not straight.’ The geometer bases no conclusion on the particular line which he has drawn being that which he has described, but (he refers to) what is illustrated by the figures. Further, the postulate and every hypothesis are either universal or particular statements; definitions are neither +

“By first principles in each genus I mean those the truth of which it is not possible to prove. What is denoted by the first (terms) and those derived from them is assumed; but, as regards their existence, this must be assumed for the principles but proved for the rest. Thus what a unit is, what the straight (line) is, or what a triangle is (must be assumed); and the existence of the unit and of magnitude must also be assumed, but the rest must be proved. Now of the premisses used in demonstrative sciences some are peculiar to each science and others common (to all), the latter being common by analogy, for of course they are actually useful in so far as they are applied to the subject-matter included under the particular science. Instances of first principles peculiar to a science are the assumptions that a line is of such and such a character, and similarly for the straight (line); whereas it is a common principle, for instance, that, if equals be subtracted from equals, the remainders are equal. But it is enough that each of the common principles is true so far as regards the particular genus (subject-matter); for (in geometry) the effect will be the same even if the common principle be assumed to be true, not of everything, but only of magnitudes, and, in arithmetic, of numbers.

+

“Now the things peculiar to the science, the existence of which must be assumed, are the things with reference to which the science investigates the essential attributes, e.g. arithmetic with reference to units, and geometry with reference to points and lines. With these things it is assumed that they exist and that they are of such and such a nature. But, with regard to their essential properties, what is assumed is only the meaning of each term employed: thus arithmetic assumes the answer to the question what is (meant by) ’odd’ or ’even,’ ’a square’ or ’a cube,’ and geometry to the question what is (meant by) ’the irrational’ or ’deflection’ or (the so-called) ’verging’ (to a point); but that there are such things is proved by means of the common principles and of what has already been demonstrated. Similarly with astronomy. For every demonstrative science has to do with three things, (1) the things which are assumed to exist, namely the genus (subject-matter) in each case, the essential properties of which the science investigates, (2) the common axioms so-called, which are the primary source of demonstration, and (3) the properties with regard to which all that is assumed is the meaning of the respective terms used. There is, however, no reason why some sciences should not omit to speak of one or other of these things. Thus there need not be any supposition as to the existence of the genus, if it is manifest that it exists (for it is not equally clear that number exists and that cold and hot exist); and, with regard to the properties, there need be no assumption as to the meaning of terms if it is clear: just as in the common (axioms) there is no assumption as to what is the meaning of subtracting equals from equals, because it is well known. But none the less is it true that there are three things naturally distinct, the subject-matter of the proof, the things proved, and the (axioms) from which (the proof starts).

+

Now that which is per se necessarily true, and must necessarily be thought so, is not a hypothesis nor yet a postulate. For demonstration has not to do with reasoning from outside but with the reason dwelling in the soul, just as is the case with the syllogism. It is always possible to raise objection to reasoning from outside, but to contradict the reason within us is not always possible. Now anything that the teacher assumes, though it is matter of proof, without proving it himself, is a hypothesis if the thing assumed is believed by the learner, and it is moreover a hypothesis, not absolutely, but relatively to the particular pupil; but, if the same thing is assumed when the learner either has no opinion on the subject or is of a contrary opinion, it is a postulate. This is the difference between a hypothesis and a postulate; for a postulate is that which is rather contrary than otherwise to the opinion of the learner, or whatever is assumed and used without being proved, although matter for demonstration. Now definitions are not hypotheses, for they do not assert the existence or non-existence of anything, while hypotheses are among propositions. Definitions only require to be understood: a definition is therefore not a hypothesis, unless indeed it be asserted that any audible speech is a hypothesis. A hypothesis is that from the truth of which, if assumed, a conclusion can be established. Nor are the geometer's hypotheses false, as some have said: I mean those who say that ’you should not make use of what is false, and yet the geometer falsely calls the line which he has drawn a foot long when it is not, or straight when it is not straight.’ The geometer bases no conclusion on the particular line which he has drawn being that which he has described, but (he refers to) what is illustrated by the figures. Further, the postulate and every hypothesis are either universal or particular statements; definitions are neither (because the subject is of equal extent with what is predicated of it).

Every demonstrative science, says Aristotle, must start from indemonstrable principles: otherwise, the steps of demonstration would be endless. Of these indemonstrable principles some are (a) common to all sciences, others are (b) particular, or peculiar to the particular science; (a) the common principles are the axioms, most commonly illustrated by the axiom that, if equals be subtracted from equals, the remainders are equal. Coming now to (b) the principles peculiar to the particular science which must be assumed, we have first the genus or subject-matter, the existence of which must be assumed, viz. magnitude in the case of geometry, the unit in the case of arithmetic. Under this we must assume definitions of manifestations or attributes of the genus, e.g. straight lines, triangles, deflection etc. The definition in itself says nothing as to the existence of the thing defined: it only requires to be understood. But in geometry, in addition to the genus and the definitions, we have to assume the existence of a few primary things which are defined, viz. points and lines only: the existence of everything else, e.g. the various figures made up of these, as triangles, squares, tangents, and their properties, e.g. incommensurability etc., has to be proved (as it is proved by construction and demonstration). In arithmetic we assume the existence of the unit: but, as regards the rest, only the definitions, e.g. those of odd, even, square, cube, are assumed, and existence has to be proved. We have then clearly distinguished, among the indemonstrable principles, axioms and definitions. A postulate is also distinguished from a hypothesis, the latter being made with the assent of the learner, the former without such assent or even in opposition to his opinion (though, strangely enough, immediately after saying this, Aristotle gives a wider meaning to postulate which would cover hypothesis @@ -1332,18 +1332,18 @@ Similarly every demonstrative (science) investigates, with regard to some subject-matter, the essential attributes, starting from the common opinionsMetaph. 997 a 20-22.. We have then here, as Heiberg says, a sufficient explanation of Euclid's term for axioms, viz. common notions (koinai\ e)/nnoiai), and there is no reason to suppose it to be a substitution for the original term due to the Stoics: cf. Proclus' remark that, according to Aristotle and the geometers, axiom and common notion are the same thingProclus, p. 194, 8..

Aristotle discusses the indemonstrable character of the axioms in the Metaphysics. Since all the demonstrative sciences use the axiomsMetaph. 997 a 10., - the question arises, to what science does their discussion belongibid. 996 b 26.? The answer is that, like that of Being (ou)si/a), it is the province of the (first) philosopheribid. 1005 a 21—b 11.. It is impossible that there should be demonstration of everything, as there would be an infinite series of demonstrations: if the axioms were the subject of a demonstrative science, there would have to be here too, as in other demonstrative sciences, a subject-genus, its attributes and corresponding axiomsibid. 997 a 5-8.; thus there would be axioms behind axioms, and so on continually. The axiom is the most firmly established of all principlesibid. 1005 b 11-17.. It is ignorance alone that could lead any one to try to prove the axiomsibid. 1006 a 5.; the supposed proof would be a petitio principiiibid. 1006 a 17.. If it is admitted that not everything can be proved, no one can point to any principle more truly indemonstrableibid. 1006 a 10.. If any one thought he could prove them, he could at once be refuted; if he did not attempt to say anything, it would be ridiculous to argue with him: he would be no better than a vegetableibid. 1006 a 11-15.. The first condition of the possibility of any argument whatever is that words should signify something both to the speaker and to the hearer: without this there can be no reasoning with any one. And, if any one admits that words can mean anything to both hearer and speaker, he admits that something can be true without demonstration. And so onibid. 1006 a 18 sqq..

+ the question arises, to what science does their discussion belongibid. 996 b 26.? The answer is that, like that of Being (ou)si/a), it is the province of the (first) philosopheribid. 1005 a 21—b 11.. It is impossible that there should be demonstration of everything, as there would be an infinite series of demonstrations: if the axioms were the subject of a demonstrative science, there would have to be here too, as in other demonstrative sciences, a subject-genus, its attributes and corresponding axiomsibid. 997 a 5-8.; thus there would be axioms behind axioms, and so on continually. The axiom is the most firmly established of all principlesibid. 1005 b 11-17.. It is ignorance alone that could lead any one to try to prove the axiomsibid. 1006 a 5.; the supposed proof would be a petitio principiiibid. 1006 a 17.. If it is admitted that not everything can be proved, no one can point to any principle more truly indemonstrableibid. 1006 a 10.. If any one thought he could prove them, he could at once be refuted; if he did not attempt to say anything, it would be ridiculous to argue with him: he would be no better than a vegetableibid. 1006 a 11-15.. The first condition of the possibility of any argument whatever is that words should signify something both to the speaker and to the hearer: without this there can be no reasoning with any one. And, if any one admits that words can mean anything to both hearer and speaker, he admits that something can be true without demonstration. And so onibid. 1006 a 18 sqq..

It was necessary to give some sketch of Aristotle's view of the first principles, if only in connexion with Proclus' account, which is as follows. As in the case of other sciences, so the compiler of elements in geometry must give separately the principles of the science, and after that the conclusions from those principles, not giving any account of the principles but only of their consequences. No science proves its own principles, or even discourses about them: they are treated as self-evident....Thus the first essential was to distinguish the principles from their consequences. Euclid carries out this plan practically in every book and, as a preliminary to the whole enquiry, sets out the common principles of this science. Then he divides the common principles themselves into hypotheses, postulates, and axioms. For all these are different from one another: an axiom, a postulate and a hypothesis are not the same thing, as the inspired Aristotle somewhere says. But, whenever that which is assumed and ranked as a principle is both known to the learner and convincing in itself, such a thing is an axiom, e.g. the statement that things which are equal to the same thing are also equal to one another. When, on the other hand, the pupil has not the notion of what is told him which carries conviction in itself, but nevertheless lays it down and assents to its being assumed, such an assumption is a hypothesis. Thus we do not preconceive by virtue of a common notion, and without being taught, that the circle is such and such a figure, but, when we are told so, we assent without demonstration. When again what is asserted is both unknown and assumed even without the assent of the learner, then, he says, we call this a postulate, e.g. that all right angles are equal. This view of a postulate is clearly implied by those who have made a special and systematic attempt to show, with regard to one of the postulates, that it cannot be assented to by any one straight off. According then to the teaching of Aristotle, an axiom, a postulate and a hypothesis are thus distinguishedProclus, pp. 75, 10-77, 2..

We observe, first, that Proclus in this passage confuses hypotheses and definitions, although Aristotle had made the distinction quite plain. The confusion may be due to his having in his mind a passage of Plato from which he evidently got the phrase about not giving an account of the principles. The passage isRepublic, VI. 510 c. Cf. Aristotle, Nic. Eth. 1151 a 17.: I think you know that those who treat of geometries and calculations (arithmetic) and such things take for granted (u(poqe/menoi) odd and even, figures, angles of three kinds, and other things akin to these in each subject, implying that they know these things, and, though using them as hypotheses, do not even condescend to give any account of them either to themselves or to others, but begin from these things and then go through everything else in order, arriving ultimately, by recognised methods, at the conclusion which they started in search of. - But the hypothesis is here the assumption, e.g. ’that there may be such a thing as length without breadth, henceforward called a lineH. Jackson, Journal of Philology, vol. x. p. 144.,’ and so on, without any attempt to show that there is such a thing; it is mentioned in connexion with the distinction between Plato's ’superior’ and ’inferior’ intellectual method, the former of which uses successive hypotheses as stepping-stones by which it mounts upwards to the idea of Good.

+ But the hypothesis is here the assumption, e.g. ’that there may be such a thing as length without breadth, henceforward called a lineH. Jackson, Journal of Philology, vol. x. p. 144.,’ and so on, without any attempt to show that there is such a thing; it is mentioned in connexion with the distinction between Plato's ’superior’ and ’inferior’ intellectual method, the former of which uses successive hypotheses as stepping-stones by which it mounts upwards to the idea of Good.

We pass now to Proclus' account of the difference between postulates and axioms. He begins with the view of Geminus, according to which they differ from one another in the same way as theorems are also distinguished from problems. For, as in theorems we propose to see and determine what follows on the premisses, while in problems we are told to find and do something, in like manner in the axioms such things are assumed as are manifest of themselves and easily apprehended by our untaught notions, while in the postulates we assume such things as are easy to find and effect (our understanding suffering no strain in their assumption), and we require no complication of machineryProclus, pp. 178, 12-179, 8. In illustration Proclus contrasts the drawing of a straight line or a circle with the drawing of a single-turn spiral or of an equilateral triangle, the spiral requiring more complex machinery and even the equilateral triangle needing a certain method. For the geometrical intelligence will say that by conceiving a straight line fixed at one end but, as regards the other end, moving round the fixed end, and a point moving along the straight line from the fixed end, I have described the single-turn spiral; for the end of the straight line describing a circle, and the point moving on the straight line simultaneously, when they arrive and meet at the same point, complete such a spiral. And again, if I draw equal circles, join their common point to the centres of the circles and draw a straight line from one of the centres to the other, I shall have the equilateral triangle. These things then are far from being completed by means of a single act or of a moment's thought (p. 180, 8-21).. ...Both must have the characteristic of being simple and readily grasped, I mean both the postulate and the axiom; but the postulate bids us contrive and find some subject-matter (u(/lh) to exhibit a property simple and easily grasped, while the axiom bids us assert some essential attribute which is self-evident to the learner, just as is the fact that fire is hot, or any of the most obvious thingsProclus, p. 181,4-11..

-

Again, says Proclus, some claim that all these things are alike postulates, in the same way as some maintain that all things that are sought are problems. For Archimedes begins his first book on InequilibriumIt is necessary to coin a word to render a)nisorropiw=n, which is moreover in the plural. The title of the treatise as we have it is Equilibria of planes or cenires of gravity of planes in Book I and Equilibria of planes in Book II. with the remark ’I postulate that equal weights at equal distances are in equilibrium,’ though one would rather call this an axiom. Others call them all axioms in the same way as some regard as theorems everything that requires demonstrationProclus, p. 181, 16-23.. +

Again, says Proclus, some claim that all these things are alike postulates, in the same way as some maintain that all things that are sought are problems. For Archimedes begins his first book on InequilibriumIt is necessary to coin a word to render a)nisorropiw=n, which is moreover in the plural. The title of the treatise as we have it is Equilibria of planes or cenires of gravity of planes in Book I and Equilibria of planes in Book II. with the remark ’I postulate that equal weights at equal distances are in equilibrium,’ though one would rather call this an axiom. Others call them all axioms in the same way as some regard as theorems everything that requires demonstrationProclus, p. 181, 16-23..

Others again will say that postulates are peculiar to geometrical subject-matter, while axioms are common to all investigation which is concerned with quantity and magnitude. Thus it is the geometer who knows that all right angles are equal and how to produce in a straight line any limited straight line, whereas it is a common notion that things which are equal to the same thing are also equal to one another, and it is employed by the arithmetician and any scientific person who adapts the general statement to his own subjectibid. p. 182, 6-14..

@@ -1360,21 +1360,21 @@

- § 4. THEOREMS AND PROBLEMS. + § 4. THEOREMS AND PROBLEMS.

Again the deductions from the first principles, says Proclus, are divided into problems and theorems, the former embracing the generation, division, subtraction or addition of figures, and generally the changes which are brought about in them, the latter exhibiting the essential attributes of eachProclus, p. 77, 7-12..

-

“Now, of the ancients, some, like Speusippus and Amphinomus, thought proper to call them all theorems, regarding the name of theorems as more appropriate than that of problems to theoretic sciences, especially as these deal with eternal objects. For there is no becoming in things eternal, so that neither could the problem have any place with them, since it promises the generation and making of what has not before existed, e.g. the construction of an equilateral triangle, or the describing of a square on a given straight line, or the placing of a straight line at a given point. Hence they say it is better to assert that all (propositions) are of the same kind, and that we regard the generation that takes place in them as referring not to actual making but to knowledge, when we treat things existing eternally as if they were subject to becoming: in other words, we may say that everything is treated by way of theorem and not by way of problemibid. pp. 77, 15-78, 8. (pa/nta qewrhmatikw=s a)ll) ou) problhmatikw=s lamba/nesqai).

-

“Others on the contrary, like the mathematicians of the school of Menaechmus, thought it right to call them all problems, describing their purpose as twofold, namely in some cases to furnish (pori/sasqai) the thing sought, in others to take a determinate object and see either what it is, or of what nature, or what is its property, or in what relations it stands to something else.

+

“Now, of the ancients, some, like Speusippus and Amphinomus, thought proper to call them all theorems, regarding the name of theorems as more appropriate than that of problems to theoretic sciences, especially as these deal with eternal objects. For there is no becoming in things eternal, so that neither could the problem have any place with them, since it promises the generation and making of what has not before existed, e.g. the construction of an equilateral triangle, or the describing of a square on a given straight line, or the placing of a straight line at a given point. Hence they say it is better to assert that all (propositions) are of the same kind, and that we regard the generation that takes place in them as referring not to actual making but to knowledge, when we treat things existing eternally as if they were subject to becoming: in other words, we may say that everything is treated by way of theorem and not by way of problemibid. pp. 77, 15-78, 8. (pa/nta qewrhmatikw=s a)ll) ou) problhmatikw=s lamba/nesqai).

+

“Others on the contrary, like the mathematicians of the school of Menaechmus, thought it right to call them all problems, describing their purpose as twofold, namely in some cases to furnish (pori/sasqai) the thing sought, in others to take a determinate object and see either what it is, or of what nature, or what is its property, or in what relations it stands to something else.

In reality both assertions are correct. Speusippus is right because the problems of geometry are not like those of mechanics, the latter being matters of sense and exhibiting becoming and change of every sort. The school of Menaechmus are right also because the discoveries even of theorems do not arise without an issuing-forth into matter, by which I mean intelligible matter. Thus forms going out into matter and giving it shape may fairly be said to be like processes of becoming. For we say that the motion of our thought and the throwing-out of the forms in it is what produces the figures in the imagination and the conditions subsisting in them. It is in the imagination that constructions, divisions, placings, applications, additions and subtractions (take place), but everything in the mind is fixed and immune from becoming and from every sort of changeibid. pp. 78, 8-79, 2..

Now those who distinguish the theorem from the problem say that every problem implies the possibility, not only of that which is predicated of its subject-matter, but also of its opposite, whereas every theorem implies the possibility of the thing predicated but not of its opposite as well. By the subject-matter I mean the genus which is the subject of inquiry, for example, a triangle or a square or a circle, and by the property predicated the essential attribute, as equality, section, position, and the like. When then any one enunciates thus, To inscribe an equilateral triangle in a circle, he states a problem; for it is also possible to inscribe in it a triangle which is not equilateral. Again, if we take the enunciation On a given limited straight line to construct an equilateral triangle, this is a problem; for it is possible also to construct one which is not equilateral. But, when any one enunciates that In isosceles triangles the angles at the base are equal, we must say that he enunciates a theorem; for it is not also possible that the angles at the base of isosceles triangles should be unequal. It follows that, if any one were to use the form of a problem and say In a semicircle to describe a right angle, he would be set down as no geometer. For every angle in a semicircle is rightProclus, pp. 79, 11-80, 5..

Zenodotus, who belonged to the succession of Oenopides, but was a disciple of Andron, distinguished the theorem from the problem by the fact that the theorem inquires what is the property predicated of the subject-matter in it, but the problem what is the cause of what effect (ti/nos o)/ntos ti/ e)stin). Hence too Posidonius defined the one (the problem) as a proposition in which it is inquired whether a thing exists or not (ei) e)/stin h)\ mh/), the other (the theoremIn the text we have to\ de\ pro/blhma answering to to\ me\n without substantive: pro/blhma was obviously inserted in error.) as a proposition in which it is inquired what (a thing) is or of what nature (ti/ e)stin h)\ poi=o/n ti); and he said that the theoretic proposition must be put in a declaratory form, e.g., Any triangle has two sides (together) greater than the remaining side and In any isosceles triangle the angles at the base are equal, but that we should state the problematic propositi\on as if inquiring whether it is possible to construct an equilateral triangle upon such and such a straight line. For there is a difference between inquiring absolutely and indeterminately (a(plw=s te kai\ a)ori/stws) whether there exists a straight line from such and such a point at right angles to such and such a straight line and investigating which is the straight line at right anglesProclus, pp. 80, 15-81, 4..

-

That there is a certain difference between the problem and the theorem is clear from what has been said; and that the Elements of Euclid contain partly problems and partly theorems will be made manifest by the individual propositions, where Euclid himself adds at the end of what is proved in them, in some cases, ’that which it was required to do,’ and in others, ’that which it was required to prove,’ the latter expression being regarded as characteristic of theorems, in spite of the fact that, as we have said, demonstration is found in problems also. In problems, however, even the demonstration is for the purpose of (confirming) the construction: for wė bring in the demonstration in order to show that what was enjoined has been done; whereas in theorems the demonstration is worthy of study for its own sake as being capable of putting before us the nature of the thing sought. And you will find that Euclid sometimes interweaves theorems with problems and employs them in turn, as in the first book, while at other times he makes one or other preponderate. For the fourth book consists wholly of problems, and the fifth of theoremsProclus, p. 81, 5-22.. +

That there is a certain difference between the problem and the theorem is clear from what has been said; and that the Elements of Euclid contain partly problems and partly theorems will be made manifest by the individual propositions, where Euclid himself adds at the end of what is proved in them, in some cases, ’that which it was required to do,’ and in others, ’that which it was required to prove,’ the latter expression being regarded as characteristic of theorems, in spite of the fact that, as we have said, demonstration is found in problems also. In problems, however, even the demonstration is for the purpose of (confirming) the construction: for wė bring in the demonstration in order to show that what was enjoined has been done; whereas in theorems the demonstration is worthy of study for its own sake as being capable of putting before us the nature of the thing sought. And you will find that Euclid sometimes interweaves theorems with problems and employs them in turn, as in the first book, while at other times he makes one or other preponderate. For the fourth book consists wholly of problems, and the fifth of theoremsProclus, p. 81, 5-22..

-

Again, in his note on Eucl. 1. 4, Proclus says that Carpus, the writer on mechanics, raised the question of theorems and problems in his treatise on astronomy. Carpus, we are told, says that the class of problems is in order prior to theorems. For the subjects, the properties of which are sought, are discovered by means of problems. Moreover in a problem the enunciation is simple and requires no skilled intelligence; it orders you plainly to do such and such a thing, to construct an equilateral triangle, or, given two straight lines, to cut off from the greater (a straight line) equal to the lesser, and what is there obscure or elaborate in these things? But the enunciation of a theorem is a matter of labour and requires much exactness and scientific judgment in order that it may not turn out to exceed or fall short of the truth; an example is found even in this proposition (1. 4), the first of the theorems. Again, in the case of problems, one general way has been discovered, that of analysis, by following which we can always hope to succeed; it is this method by which the more obscure problems are investigated. But, in the case of theorems, the method of setting about them is hard to get hold of since ’up to our time,’ says Carpus, ’no one has been able to hand down a general method for their discovery. Hence, by reason of their easiness, the class of problems would naturally be more simple.’ After these distinctions, he proceeds: ’Hence it is that in the Elements too problems precede theorems, and the Elements begin from them; the first theorem is fourth in order, not because the fifthto\ pe/mpton. This should apparently be the fourth because in the next words it is implied that none of the first three propositions are required in proving it. is proved from the problems, but because, even if it needs for its demonstration none of the propositions which precede it, it was necessary that they should be first because they are problems, while it is a theorem. In fact, in this theorem he uses the common notions exclusively, and in some sort takes the same triangle placed in different positions; the coincidence and the equality proved thereby depend entirely upon sensible and distinct apprehension. Nevertheless, though the demonstration of the first theorem is of this character, the problems properly preceded it, because in general problems are allotted the order of precedenceProclus, pp. 241, 19-243, 11..’ +

Again, in his note on Eucl. 1. 4, Proclus says that Carpus, the writer on mechanics, raised the question of theorems and problems in his treatise on astronomy. Carpus, we are told, says that the class of problems is in order prior to theorems. For the subjects, the properties of which are sought, are discovered by means of problems. Moreover in a problem the enunciation is simple and requires no skilled intelligence; it orders you plainly to do such and such a thing, to construct an equilateral triangle, or, given two straight lines, to cut off from the greater (a straight line) equal to the lesser, and what is there obscure or elaborate in these things? But the enunciation of a theorem is a matter of labour and requires much exactness and scientific judgment in order that it may not turn out to exceed or fall short of the truth; an example is found even in this proposition (1. 4), the first of the theorems. Again, in the case of problems, one general way has been discovered, that of analysis, by following which we can always hope to succeed; it is this method by which the more obscure problems are investigated. But, in the case of theorems, the method of setting about them is hard to get hold of since ’up to our time,’ says Carpus, ’no one has been able to hand down a general method for their discovery. Hence, by reason of their easiness, the class of problems would naturally be more simple.’ After these distinctions, he proceeds: ’Hence it is that in the Elements too problems precede theorems, and the Elements begin from them; the first theorem is fourth in order, not because the fifthto\ pe/mpton. This should apparently be the fourth because in the next words it is implied that none of the first three propositions are required in proving it. is proved from the problems, but because, even if it needs for its demonstration none of the propositions which precede it, it was necessary that they should be first because they are problems, while it is a theorem. In fact, in this theorem he uses the common notions exclusively, and in some sort takes the same triangle placed in different positions; the coincidence and the equality proved thereby depend entirely upon sensible and distinct apprehension. Nevertheless, though the demonstration of the first theorem is of this character, the problems properly preceded it, because in general problems are allotted the order of precedenceProclus, pp. 241, 19-243, 11..’

Proclus himself explains the position of Prop. 4 after Props. 1-3 as due to the fact that a theorem about the essential properties of triangles ought not to be introduced before we know that such a thing as a triangle can be constructed, nor a theorem about the equality of sides or straight lines until we have shown, by constructing them, that there can be two straight lines which are equal to one anotheribid. pp. 233, 21-234, 6.. It is plausible enough to argue in this way that Props. 2 and 3 at all events should precede Prop. 4. And Prop. 1 is used in Prop. 2, and must therefore precede it. But Prop. I showing how to construct an equilateral triangle on a given base is not important, in relation to Prop. 4, as dealing with the production of triangles in general: for it is of no use to say, as Proclus does, that the construction of the equilateral triangle is common to the three species (of triangles)Proclus, p. 234, 21., @@ -1399,7 +1399,7 @@ are not problems in the proper sense (kuri/ws lego/mena problh/mata), but only equivocallyibid. pp. 221, 13-222,14..

 - § 5. THE FORMAL DIVISIONS OF A PROPOSITION. + § 5. THE FORMAL DIVISIONS OF A PROPOSITION.

Every problem, says Proclusibid. pp. 203, 1-204, 13; 204, 23-205, 8., and every theorem which is complete with all its parts perfect purports to contain in itself all of the following elements: enunciation (pro/tasis), setting-out (e)/kqesis), definition or specification (diorismo/s), construction or machinery (kataskeuh/), proof (a)po/deicis), conclusion (sumpe/rasma). Now of these the enunciation states what is given and what is that which is sought, the perfect enunciation consisting of both these parts. The setting-out marks off what is given, by itself, and adapts it beforehand for use in the investigation. The definition or specification states separately and makes clear what the particular thing is which is sought. The construction or machinery adds what is wanting to the datum for the purpose of finding what is sought. The proof draws the required inference by reasoning scientifically from acknowledged facts. The conclusion reverts again to the enunciation, confirming what has been demonstrated. These are all the parts of problems and theorems, but the most essential and those which are found in all are enunciation, proof, conclusion. For it is equally necessary to know beforehand what is sought, to prove this by means of the intermediate steps, and to state the proved fact as a conclusion; it is impossible to dispense with any of these three things. The remaining parts are often brought in, but are often left out as serving no purpose. Thus there is neither setting-out nor definition in the problem of constructing an isosceles triangle having each of the angles at the base double of the remaining angle, and in most theorems there is no construction because the setting-out suffices without any addition for proving the required property from the data. When then do we say that the setting-out is wanting? The answer is, when there is nothing given in the enunciation; for, though the enunciation is in general divided into what is given and what is sought, this is not always the case, but sometimes it states only what is sought, i.e. what must be known or found, as in the case of the problem just mentioned. That problem does not, in fact, state beforehand with what datum we are to construct the isosceles triangle having each of the equal angles double of the remaining angle, but (simply) that we are to find such a triangle.... When, then, the enunciation contains both (what is given and what is sought), in that case we find both definition and setting-out, but, whenever the datum is wanting, they too are wanting. For not only is the setting-out concerned with the datum, but so is the definition also, as, in the absence of the datum, the definition will be identical with the enunciation. In fact, what could you say in defining the object of the aforesaid problem except that it is required to find an isosceles triangle of the kind referred to? But that is what the enunciation stated. If then the enunciation does not include, on the one hand, what is given and, on the other, what is sought, there is no setting-out in virtue of there being no datum, and the definition is left out in order to avoid a mere repetition of the enunciation.

@@ -1410,16 +1410,16 @@ Similarly in VI. 28 the enunciation To a given straight line to apply a parallelogram equal to a given rectilineal figure and falling short by a parallelogrammic figure similar to a given one is at once followed by the necessary condition of possibility: Thus the given rectilineal figure must not be greater than that described on half the line and similar to the defect.

-

Tannery supposed that, in giving the other description of the diorismo/s as quoted above, Proclus, or rather his guide, was using the term incorrectly. The diorismo/s in the better known sense of the determination of limits or conditions of possibility was, we are told, invented by Leon. Pappus uses the word in this sense only. The other use of the term might, Tannery thought, be due to a confusion occasioned by the use of the same words (dei= dh/) in introducing the parts of a proposition corresponding to the two meanings of the word diorismo/sLa Géométrie grecque, p. 149 note. Where dei= dh\ introduces the closer description of the problem we may translate, it is then required +

Tannery supposed that, in giving the other description of the diorismo/s as quoted above, Proclus, or rather his guide, was using the term incorrectly. The diorismo/s in the better known sense of the determination of limits or conditions of possibility was, we are told, invented by Leon. Pappus uses the word in this sense only. The other use of the term might, Tannery thought, be due to a confusion occasioned by the use of the same words (dei= dh/) in introducing the parts of a proposition corresponding to the two meanings of the word diorismo/sLa Géométrie grecque, p. 149 note. Where dei= dh\ introduces the closer description of the problem we may translate, it is then required or thus it is required (to constructetc.): when it introduces the condition of possibility we may translate thus it is necessary etc. Heiberg originally wrote dei= de\ in the latter sense in 1. 22 on the authority of Proclus and Eutocius, and against that of the MSS. Later, on the occasion of XI. 23, he observed that he should have followed the MSS. and written dei= dh\ which he found to be, after all, the right reading in Eutocius (Apollonius, ed. Heiberg, II. p. 178). dei= dh\ is also the expression used by Diophantus for introducing conditions of possibility.. On the other hand it is to be observed that Eutocius distinguishes clearly between the two uses and implies that the difference was well known. See the passage of Eutocius referred to in last note.. The diorismo/s in the sense of condition of possibility follows immediately on the enunciation, is even part of it; the diorismo/s in the other sense of course comes immediately after the setting-out.

-

Proclus has a useful observation respecting the conclusion of a propositionProclus, p. 207, 4-25.. The conclusion they are accustomed to make double in a certain way: I mean, by proving it in the given case and then drawing a general inference, passing, that is, from the partial conclusion to the general. For, inasmuch as they do not make use of the individuality of the subjects taken, but only draw an angle or a straight line with a view to placing the datum before our eyes, they consider that this same fact which is established in the case of the particular figure constitutes a conclusion true of every other figure of the same kind. They pass accordingly to the general in order that we may not conceive the conclusion to be partial. And they are justified in so passing, since they use for the demonstration the particular things set out, not quâ particulars, but quâ typical of the rest. For it is not in virtue of such and such a size attaching to the angle which is set out that I effect the bisection of it, but in virtue of its being rectilineal and nothing more. Such and such size is peculiar to the angle set out, but its quality of being rectilineal is common to all rectilineal angles. Suppose, for example, that the given angle is a right angle. If then I had employed in the proof the fact of its being right, I should not have been able to pass to every species of rectilineal angle; but, if I make no use of its being right, and only consider it as rectilineal, the argument will equally apply to rectilineal angles in general. +

Proclus has a useful observation respecting the conclusion of a propositionProclus, p. 207, 4-25.. The conclusion they are accustomed to make double in a certain way: I mean, by proving it in the given case and then drawing a general inference, passing, that is, from the partial conclusion to the general. For, inasmuch as they do not make use of the individuality of the subjects taken, but only draw an angle or a straight line with a view to placing the datum before our eyes, they consider that this same fact which is established in the case of the particular figure constitutes a conclusion true of every other figure of the same kind. They pass accordingly to the general in order that we may not conceive the conclusion to be partial. And they are justified in so passing, since they use for the demonstration the particular things set out, not quâ particulars, but quâ typical of the rest. For it is not in virtue of such and such a size attaching to the angle which is set out that I effect the bisection of it, but in virtue of its being rectilineal and nothing more. Such and such size is peculiar to the angle set out, but its quality of being rectilineal is common to all rectilineal angles. Suppose, for example, that the given angle is a right angle. If then I had employed in the proof the fact of its being right, I should not have been able to pass to every species of rectilineal angle; but, if I make no use of its being right, and only consider it as rectilineal, the argument will equally apply to rectilineal angles in general.

- § 6. OTHER TECHNICAL TERMS. + § 6. OTHER TECHNICAL TERMS.

1. Things said to be given.

Proclus attaches to his description of the formal divisions of a proposition an explanation of the different senses in which the word given or datum (dedome/non) is used in geometry. Everything that is given is given in one or other of the following ways, in position, in ratio, in magnitude, or in species. The point is given in position only, but a line and the rest may be given in all the sensesProclus, p. 205, 13-15..

@@ -1442,7 +1442,7 @@

The term lemma, says Proclus Proclus, pp. 211, 1-212, 4., is often used of any proposition which is assumed for the construction of something else: thus it is a common remark that a proof has been made out of such and such lemmas. But the special meaning of - lemma in geometry is a proposition requiring confirmation. For when, in either construction or demonstration, we assume anything which has not been proved but requires argument, then, because we regard what has been assumed as doubtful in itsėlf and therefore worthy of investigation, we call it a lemma + lemma in geometry is a proposition requiring confirmation. For when, in either construction or demonstration, we assume anything which has not been proved but requires argument, then, because we regard what has been assumed as doubtful in itsėlf and therefore worthy of investigation, we call it a lemma It would appear, says Tannery (p. 151 n.), that Geminus understood a lemma as being simply lambano/menon, something assumed (cf. the passage of Proclus, p. 73, 4, relating to Menaechmus' view of elements): hence we cannot consider ourselves authorised in attributing to Geminus the more technical definition of the term here given by Proclus, according to which it is only used of propositions not proved beforehand. This view of a lemma must be considered as relatively modern. It seems to have had its origin in an imperfection of method. In the course of a demonstration it was necessary to assume a proposition which required proof, but the proof of which would, if inserted in the particular place, break the thread of the demonstration: hence it was necessary either to prove it beforehand as a preliminary proposition or to postpone it to be proved afterwards (w(s e(ch=s deixqh/setai). @@ -1455,7 +1455,7 @@ This passage and another from Diogenes Laertius (III. 24, p. 74 ed. Cobet) to the effect that He [Plato] explained (ei)shgh/sato) to Leodamas of Thasos the method of inquiry by analysis have been commonly understood as ascribing to Plato the invention of the method of analysis; but Tannery points out forcibly (pp. 112, 113) how difficult it is to explain in what Plato's discovery could have consisted if analysis be taken in the sense attributed to it in Pappus, where we can see no more than a series of successive, reductions of a problem until it is finally reduced to a known problem. On the other hand, Proclus' words about carrying up the thing sought to an acknowledged principle suggest that what he had in mind was the process described at the end of Book VI of the Republic by which the dialectician (unlike the mathematician) uses hypotheses as stepping-stones up to a principle which is not hypothetical, and then is able to descend step by step verifying every one of the hypotheses by which he ascended. This description does not of course refer to mathematical analysis, - but it may have given rise to the idea that analysis was Plato's discovery, since analysis and synthesis following each other are related in the same way as the upward and the downward progression in the dialectician's intellectual method. And it may be that Plato's achievement was to observe the importance, from the point of view of logical rigour, of the confirmatory synthesis following analysis, and to regulariśe in this way and elevate into a completely irrefragable method the partial and uncertain analysis upon which the works of his predecessors depended., + but it may have given rise to the idea that analysis was Plato's discovery, since analysis and synthesis following each other are related in the same way as the upward and the downward progression in the dialectician's intellectual method. And it may be that Plato's achievement was to observe the importance, from the point of view of logical rigour, of the confirmatory synthesis following analysis, and to regulariśe in this way and elevate into a completely irrefragable method the partial and uncertain analysis upon which the works of his predecessors depended., and by which the latter, too, is said to have discovered many things in geometry. The second is the method of division Here again the successive bipartitions of genera into species such as we find in the Sophist and Republic have very little to say to geometry, and the very fact that they are here mentioned side by side with analysis suggests that Proclus confused the latter with the philosophical method of Rep. VI., which divides into its parts the genus proposed for consideration and gives a starting-point for the demonstration by means of the elimination of the other elements in the construction of what is proposed, which method also Plato extolled as being of assistance to all sciences. The third is that by means of the reductio ad absurdum, which does not show what is sought directly; but refutes its opposite and discovers the truth incidentally. @@ -1475,7 +1475,7 @@ In fact the porism-corollary is with Euclid rather a modified form of the regular conclusion than a separate proposition.. Cf. the note on I. 15.

5. Objection.

-

The objection (e)/nstasis) obstructs the whole course of the argument by appearing as an obstacle (or crying ’halt,’ a)pantw=sa) either to the construction or to the demonstration. There is this difference between the objection and the case, that, whereas he who propounds the case has to prove the proposition to be true of it, he who makes the objection does not need to prove anything: on the contrary it is necessary to destroy the objection and to show that its author is saying what is falseProclus, p. 212, 18-23.. +

The objection (e)/nstasis) obstructs the whole course of the argument by appearing as an obstacle (or crying ’halt,’ a)pantw=sa) either to the construction or to the demonstration. There is this difference between the objection and the case, that, whereas he who propounds the case has to prove the proposition to be true of it, he who makes the objection does not need to prove anything: on the contrary it is necessary to destroy the objection and to show that its author is saying what is falseProclus, p. 212, 18-23..

That is, in general the objection endeavours to make it appear that the demonstration is not true in every case; and it is then necessary to prove, in refutation of the objection, either that the supposed case is impossible, or that the demonstration is true even for that case. A good instance is afforded by Eucl. 1. 7. The text-books give a second case which is not in the original text of Euclid. Proclus remarks on the proposition as given by Euclid that the objection may conceivably be raised that what Euclid declares to be impossible may after all be possible in the event of one pair of stiaight lines falling completely within the other pair. Proclus then refutes the objection by proving the impossibility in that case also. His proof then came to be given in the text-books as part of Euclid's proposition.

The objection is one of the technical terms in Aristotle's logic and its nature is explained in the Prior AnalyticsAnal. prior. II. 26, 69 a 37.. An objection is a proposition contrary to a proposition.... Objections are of two sorts, general or partial.... For when it is maintained that an attribute belongs to every (member of a class), we object either that it belongs to none (of the class) or that there is some one (member of the class) to which it does not belong. @@ -1496,12 +1496,12 @@ Or again, proof (leading) to the impossible differs from the direct (deiktikh=s) in that it assumes what it desires to destroy [namely the hypothesis of the falsity of the conclusion] and then reduces it to something admittedly false, whereas the direct proof starts from premisses admittedly trueibid. II. 14, 62 b 29..

Proclus has the following description of the reductio ad absurdum. Proofs by reductio as absurdum in every case reach a conclusion manifestly impossible, a conclusion the contradictory of which is admitted. In some cases the conclusions are found to conflict with the common notions, or the postulates, or the hypotheses (from which we started); in others they contradict propositions previously establishedProclus, p. 254, 22-27. - ...Every reductio ad absurdum assumes what conflicts with the desired result, then, using that as a basis, proceeds until it arrives at an admitted absurdity, and, by thus destroying the hypothesis, establishes the result originally desired. For it is necessary to understand generally that all mathematical arguments either proceed from the first principles or lead back to them, as Porphyry somewhere says. And those which proceed from the first principles are again of two kinds, for they start either from common notions and the clearness of the self-evident alone, or from results previously proved; while those which lead back to the principles are either by way of assuming the principles or by way of destroying them. Those which assume the principles are called analyses, and the opposite of these are syntheses— for it is possible to start from the said principles and to proceed in the regular order to the desired conclusion, and this process is synthesis—while the arguments which would destroy the principles are called reductiones ad absurdum. For it is the function of this method to upset something admitted as clearProclus, p. 255, 8-26.. + ...Every reductio ad absurdum assumes what conflicts with the desired result, then, using that as a basis, proceeds until it arrives at an admitted absurdity, and, by thus destroying the hypothesis, establishes the result originally desired. For it is necessary to understand generally that all mathematical arguments either proceed from the first principles or lead back to them, as Porphyry somewhere says. And those which proceed from the first principles are again of two kinds, for they start either from common notions and the clearness of the self-evident alone, or from results previously proved; while those which lead back to the principles are either by way of assuming the principles or by way of destroying them. Those which assume the principles are called analyses, and the opposite of these are syntheses— for it is possible to start from the said principles and to proceed in the regular order to the desired conclusion, and this process is synthesis—while the arguments which would destroy the principles are called reductiones ad absurdum. For it is the function of this method to upset something admitted as clearProclus, p. 255, 8-26..

8. Analysis and Synthesis.

It will be seen from the note on Eucl. XIII. I that the MSS. of the Elements contain definitions of Analysis and Synthesis followed by alternative proofs of XIII. 1-5 after that method. The definitions and alternative proofs are interpolated, but they have great historical interest because of the possibility that they represent an ancient method of dealing with these propositions, anterior to Euclid. The propositions give properties of a line cut in extreme and mean ratio, and they are preliminary to the construction and comparison of the five regular solids. Now Pappus, in the section of his Collection dealing with the latter subjectPappus, v. p. 410 sqq., says that he will give the comparisons between the five figures, the pyramid, cube, octahedron, dodecahedron and icosahedron, which have equal surfaces, not by means of the so-called analytical inquiry, by which some of the ancients worked out the proofs, but by the synthetical methodibid. pp. 410, 27-412, 2..... - The conjecture of Bretschneider that the matter interpolated in Eucl. XIII. is a survival of investigations due to Eudoxus has at first sight much to commend itBretschneider, p. 168. See however Heiberg's recent suggestion (Paralipomena zu Euklid in Hermes, XXXVIII., 1903) that the author was Heron. The suggestion is based on a comparison with the remarks on analysis and synthesis quoted from Heron by an-Nairĩzĩ (ed. Curtze, p. 89) at the beginning of his commentary on Eucl. Book II. On the whole, this suggestion commends itself to me more than that of Bretschneider.. In the first place, we are told by Proclus that Eudoxus greatly added to the number of the theorems which Plato originated regarding the section, and employed in them the method of analysisProclus, p. 67, 6.. + The conjecture of Bretschneider that the matter interpolated in Eucl. XIII. is a survival of investigations due to Eudoxus has at first sight much to commend itBretschneider, p. 168. See however Heiberg's recent suggestion (Paralipomena zu Euklid in Hermes, XXXVIII., 1903) that the author was Heron. The suggestion is based on a comparison with the remarks on analysis and synthesis quoted from Heron by an-Nairĩzĩ (ed. Curtze, p. 89) at the beginning of his commentary on Eucl. Book II. On the whole, this suggestion commends itself to me more than that of Bretschneider.. In the first place, we are told by Proclus that Eudoxus greatly added to the number of the theorems which Plato originated regarding the section, and employed in them the method of analysisProclus, p. 67, 6.. It is obvious that the section was some particular section which by the time of Plato had assumed great importance; and the one section of which this can safely be said is that which was called the golden section, namely, the division of a straight line in extreme and mean ratio which appears in Eucl. II. 11 and is therefore most probably Pythagorean. Secondly, as Cantor points outCantor, Gesch. d. Math. I_{3}, p. 241., Eudoxus was the founder of the theory of proportions in the form in which we find it in Euclid V., VI., and it was no doubt through meeting, in the course of his investigations, with proportions not expressible by whole numbers that he came to realise the necessity for a new theory of proportions which should be applicable to incommensurable as well as commensurable magnitudes. The golden section @@ -1510,17 +1510,17 @@ ones (a)/rrhtos) as well. Theorems about sections like those in Euclid's second Book are common to both [arithmetic and geometry] except that in which the straight line is cut in extreme and mean ratioibid. p. 60, 16-19..

The definitions of Analysis and Synthesis interpolated in Eucl. XIII. are as follows (I adopt the reading of B and V, the only intelligible one, for the second).

-

“Analysis is an assumption of that which is sought as if it were admitted <and the passage> through its consequences to something admitted (to be) true.

+

“Analysis is an assumption of that which is sought as if it were admitted <and the passage> through its consequences to something admitted (to be) true.

Synthesis is an assumption of that which is admitted <and the passage> through its consequences to the finishing or attainment of what is sought.

The language is by no means clear and has, at the best, to be filled out.

Pappus has a fuller accountPappus, VII. pp. 634-6.:

-

“The so-called a)naluo/menos (’Treasury of Analysis’) is, to put it shortly, a special body of doctrine provided for the use of those who, after finishing the ordinary Elements, are desirous of acquiring the power of solving problems which may be set them involving (the construction of) lines, and it is useful for this alone. It is the work of three men, Euclid the author of the Elements, Apollonius of Perga, and Aristaeus the elder, and proceeds by way of analysis and synthesis.

-

“Analysis then takes that which is sought as if it were admitted and passes from it through its successive consequences to something which is admitted as the result of synthesis: for in analysis we assume that which is sought as if it were (already) done (gegono/s), and we inquire what it is from which this results, and again what is the antecedent cause of the latter, and so on, until by so retracing our steps we come upon something already known or belonging to the class of first principles, and such a method we call analysis as being solution backwards (a)na/palin lu/sin).

-

“But in synthesis, reversing the process, we take as already done that which was last arrived at in the analysis and, by arranging in their natural order as consequences what were before antecedents, and successively connecting them one with another, we arrive finally at the construction of what was sought; and this we call synthesis.

+

“The so-called a)naluo/menos (’Treasury of Analysis’) is, to put it shortly, a special body of doctrine provided for the use of those who, after finishing the ordinary Elements, are desirous of acquiring the power of solving problems which may be set them involving (the construction of) lines, and it is useful for this alone. It is the work of three men, Euclid the author of the Elements, Apollonius of Perga, and Aristaeus the elder, and proceeds by way of analysis and synthesis.

+

“Analysis then takes that which is sought as if it were admitted and passes from it through its successive consequences to something which is admitted as the result of synthesis: for in analysis we assume that which is sought as if it were (already) done (gegono/s), and we inquire what it is from which this results, and again what is the antecedent cause of the latter, and so on, until by so retracing our steps we come upon something already known or belonging to the class of first principles, and such a method we call analysis as being solution backwards (a)na/palin lu/sin).

+

“But in synthesis, reversing the process, we take as already done that which was last arrived at in the analysis and, by arranging in their natural order as consequences what were before antecedents, and successively connecting them one with another, we arrive finally at the construction of what was sought; and this we call synthesis.

Now analysis is of two kinds, the one directed to searching for the truth and called theoretical, the other directed to finding what we are told to find and called problematical. (1) In the theoretical kind we assume what is sought as if it were existent and true, after which we pass through its successive consequences, as if they too were true and established by virtue of our hypothesis, to something admitted: then (a), if that something admitted is true, that which is sought will also be true and the proof will correspond in the reverse order to the analysis, but (b), if we come upon something admittedly false, that which is sought will also be false. (2) In the problematical kind we assume that which is propounded as if it were known, after which we pass through its successive consequences, taking them as true, up to something admitted: if then (a) what is admitted is possible and obtainable, that is, what mathematicians call given, what was originally proposed will also be possible, and the proof will again correspond in reverse order to the analysis, but if (b) we come upon something admittedly impossible, the problem will also be impossible.

-

The ancient Analysis has been made the subject of careful studies by several writers during the last half-century, the most complete being those of Hankel, Duhamel and Zeuthen; others by Ofterdinger and Cantor should also be mentionedHankel, Zur Geschichte der Mathematik in Alterthum und Mittelalter, 1874, pp. 137-150; Duhamel, Des'méthodes dans les sciences de raisonnement, Part I., 3 ed., Paris, 1885, pp. 39-68; Zeuthen, Geschichte der Mathematik im Altertnm und Mittelalter, 1896, pp. 92-104; Ofterdinger, Beiträge zur Geschichte der griechischen Mathematik, Ulm, 1860; Cantor, Geschichte der Mathematik, I_{3}, pp. 220-2..

+

The ancient Analysis has been made the subject of careful studies by several writers during the last half-century, the most complete being those of Hankel, Duhamel and Zeuthen; others by Ofterdinger and Cantor should also be mentionedHankel, Zur Geschichte der Mathematik in Alterthum und Mittelalter, 1874, pp. 137-150; Duhamel, Des'méthodes dans les sciences de raisonnement, Part I., 3 ed., Paris, 1885, pp. 39-68; Zeuthen, Geschichte der Mathematik im Altertnm und Mittelalter, 1896, pp. 92-104; Ofterdinger, Beiträge zur Geschichte der griechischen Mathematik, Ulm, 1860; Cantor, Geschichte der Mathematik, I_{3}, pp. 220-2..

The method is as follows. It is required, let us say, to prove that a certain proposition A is true. We assume as a hypothesis that A is true and, starting from this we find that, if A is true, a certain other proposition B is true; if B is true, then C; and so on until we arrive at a proposition K which is admittedly true. The object of the method is to enable us to infer, in the reverse order, that, since K is true, the proposition A originally assumed is true. Now Aristotle had already made it clear that false hypotheses might lead to a conclusion which is true. There is therefore a possibility of error unless a certain precaution is taken. While, for example, B may be a necessary consequence of A, it may happen that A is not a necessary consequence of B. Thus, in order that the reverse inference from the truth of K that A is true may be logically justified, it is necessary that each step in the chain of inferences should be unconditionally convertible. As a matter of fact, a very large number of theorems in elementary geometry are unconditionally convertible, so that in practice the difficulty in securing that the successive steps shall be convertible is not so great as might be supposed. But care is always necessary. For example, as Hankel saysHankel, p. 139., a proposition may not be unconditionally convertible in the form in which it is generally quoted. Thus the proposition The vertices of all triangles having a common base and constant vertical angle lie on a circle cannot be converted into the proposition that All triangles with common base and vertices lying on a circle have a constant vertical angle ; for this is only true if the further conditions are satisfied (1) that the circle passes through the extremities of the common base and (2) that only that part of the circle is taken as the locus of the vertices which lies on one side of the base. If these conditions are added, the proposition is unconditionally convertible. Or again, as Zeuthen remarksZeuthen, p. 103., K may be obtained by a series of inferences in which A or some other proposition in the series is only apparently used; this would be the case e.g. when the method of modern algebra is being employed and the expressions on each side of the sign of equality have been inadvertently multiplied by some composite magnitude which is in reality equal to zero.

@@ -1572,7 +1572,7 @@ - § 7. THE DEFINITIONS. + § 7. THE DEFINITIONS. General. Real and Nominal @@ -1590,7 +1590,7 @@ Anterior knowledge of two sorts is necessary: for it is necessary to presuppose, with regard to some things, that they exist; in other cases it is necessary to understand what the thing described is, and in other cases it is necessary to do both. Thus, with the fact that one of two contradictories must be true, we must know that it exists (is true); of the triangle we must know that it means such and such a thing; of the unit we must know both what it means and that it existsAnal. post. I. 1, 71 a II sqq.. What is here so much insisted on is the very fact which Mill pointed out in his discussion of earlier views of Definitions, where he says that the so-called real definitions or definitions of things do not constitute a different kind of definition from nominal definitions, or definitions of names; the former is simply the latter plus something else, namely a covert assertion that the thing defined exists. This covert assertion is not a definition but a postulate. The definition is a mere identical proposition which gives information only about the use of language, and from which no conclusion affecting matters of fact can possibly be drawn. The accompanying postulate, on the other hand, affirms a fact which may lead to consequences of every degree of importance. It affirms the actual or possible existence of Things possessing the combination of attributes set forth in the definition: and this, if true, may be foundation sufficient on which to build a whole fabric of scientific truthMill's System of Logic, Bk. I. ch. Viii.. This statement really adds nothing to Aristotle's doctrineIt is true that it was in opposition to the ideas of most of the Aristotelian logicians - (rather than of Aristotle himself) that Mill laid such stress on his point of view. Cf. his observation: We have already made, and shall often have to repeat, the remark, that the philosophers who overthrew Realism by no means got rid of the consequences of Realism, but retained long afterwards, in their own philosophy, numerous propositions which could only have a rational meaning as part of a Realistic system. It had been handed down from Aristotle, and probably from earlier times, as an obvious truth, that the science of geometry is deduced from definitions. This, so long as a definition was considered to be a proposition ’unfolding the nature of the thing,’ did well enough. But Hobbes followed and rejected utterly the notion that a definition declares the nature of the thing, or does anything but state the meaning of a name; yet he continued to affirm as broadly as any of his predecessors that the a)rxai/, principia, or original premisses of mathematics, and even of all science, are definitions; producing the singular paradox that systems of scientific truth, nay, all truths whatever at which we arrive by reasoning, are deduced from the arbitrary conventions of mankind concerning the signification of words. + (rather than of Aristotle himself) that Mill laid such stress on his point of view. Cf. his observation: We have already made, and shall often have to repeat, the remark, that the philosophers who overthrew Realism by no means got rid of the consequences of Realism, but retained long afterwards, in their own philosophy, numerous propositions which could only have a rational meaning as part of a Realistic system. It had been handed down from Aristotle, and probably from earlier times, as an obvious truth, that the science of geometry is deduced from definitions. This, so long as a definition was considered to be a proposition ’unfolding the nature of the thing,’ did well enough. But Hobbes followed and rejected utterly the notion that a definition declares the nature of the thing, or does anything but state the meaning of a name; yet he continued to affirm as broadly as any of his predecessors that the a)rxai/, principia, or original premisses of mathematics, and even of all science, are definitions; producing the singular paradox that systems of scientific truth, nay, all truths whatever at which we arrive by reasoning, are deduced from the arbitrary conventions of mankind concerning the signification of words. Aristotle was guilty of no such paradox; on the contrary, he exposed it as plainly as did Mill.: it has even the slight disadvantage, due to the use of the word postulate to describe the covert assertion in all cases, of not definitely pointing out that there are cases where existence has to be proued as distinct from those where it must be assumed. It is true that the existence of a definiend may have to be taken for granted provisionally until the time comes for proving it; but, so far as regards any case where existence must be proved sooner or later, the provisional assumption would be for Aristotle, not a postulate, but a hypothesis. In modern times, too, Mill's account of the true distinction between real and nominal definitions had been fully anticipated by SaccheriThis has been fully brought out in two papers by G. Vailati, La teoria Aristotelica della definizione (Riuista di Filosofia e scienze affini, 1903), and Di un' opera dimenticata del P. Gerolamo Saccheri (Logica Demonstrativa, @@ -1602,12 +1602,12 @@

Confusion between the nominal and the real definition as thus described, i.e. the use of the former in demonstration before it has been turned into the latter by the necessary proof that the thing defined exists, is according to Saccheri one of the most fruitful sources of illusory demonstration, and the danger is greater in proportion to the complexity of the definition, i.e. the number and variety of the attributes belonging to the thing defined. For the greater is the possibility that there may be among the attributes some that are incompatible, i.e. the simultaneous presence of which in a given figure can be proved, by means of other postulates etc. forming part of the basis of the science, to be impossible.

The same thought is expressed by Leibniz also. If, - he says, we give any definition, and it is not clear from it that the idea, which we ascribe to the thing, is possible, we cannot rely upon the demonstrations which we have derived from that definition, because, if that idea by chance involves a contradiction, it is possible that even contradictories may be true of it at one and the same time, and thus our demonstrations will be useless. Whence it is clear that definitions are not arbitrary. And this is a secret which is hardly sufficiently knownOpuscules et fragments inédits de Leibniz, Paris, Alcan, 1903, p. 431. Quoted by Vailati.. + he says, we give any definition, and it is not clear from it that the idea, which we ascribe to the thing, is possible, we cannot rely upon the demonstrations which we have derived from that definition, because, if that idea by chance involves a contradiction, it is possible that even contradictories may be true of it at one and the same time, and thus our demonstrations will be useless. Whence it is clear that definitions are not arbitrary. And this is a secret which is hardly sufficiently knownOpuscules et fragments inédits de Leibniz, Paris, Alcan, 1903, p. 431. Quoted by Vailati.. Leibniz' favourite illustration was the regular polyhedron with ten faces, the impossibility of which is not obvious at first sight.

It need hardly be added that, speaking generally, Euclid's definitions, and his use of them, agree with the doctrine of Aristotle that the definitions themselves say nothing as to the existence of the things defined, but that the existence of each of them must be proved or (in the case of the principles ) assumed. In geometry, says Aristotle, the existence of points and lines only must be assumed, the existence of the rest being proved. Accordingly Euclid's first three postulates declare the possibility of constructing straight lines and circles (the only lines - except straight lines used in the Elements). Other things are defined and afterwards constructed and proved to exist: e.g. in Book I., Def. 20, it is explained what is meant by an equilateral triangle; then (I. 1) it is proposed to construct it, and, when constructed, it is proved to agree with the definition. When a square is defined (I. Def. 22), the question whether such a thing really exists is left open until, in I. 46, it is proposed to construct it and, when constructed, it is proved to satisfy the definitionTrendelenburg, Elementa Logices Aristoteleae, § 50.. Similarly with the right angle (I. Def. 10, and I. 11) and parallels (I. Def. 23, and I. 27-29). The greatest care is taken to exclude mere presumption and imagination. The transition from the subjective definition of names to the objective definition of things is made, in geometry, by means of constructions (the first principles of which are postulated), as in other sciences it is made by means of experienceTrendelenburg, Erläuterungen zu den Elementen der aristotelischen Logik, 3 ed. p. 107. On construction as proof of existence in ancient geometry cf. H. G. Zeuthen, Die geometrische Construction als <quote>Existenzbeweis</quote> + except straight lines used in the <title>Elements). Other things are defined and afterwards constructed and proved to exist: e.g. in Book I., Def. 20, it is explained what is meant by an equilateral triangle; then (I. 1) it is proposed to construct it, and, when constructed, it is proved to agree with the definition. When a square is defined (I. Def. 22), the question whether such a thing really exists is left open until, in I. 46, it is proposed to construct it and, when constructed, it is proved to satisfy the definitionTrendelenburg, Elementa Logices Aristoteleae, § 50.. Similarly with the right angle (I. Def. 10, and I. 11) and parallels (I. Def. 23, and I. 27-29). The greatest care is taken to exclude mere presumption and imagination. The transition from the subjective definition of names to the objective definition of things is made, in geometry, by means of constructions (the first principles of which are postulated), as in other sciences it is made by means of experienceTrendelenburg, Erläuterungen zu den Elementen der aristotelischen Logik, 3 ed. p. 107. On construction as proof of existence in ancient geometry cf. H. G. Zeuthen, Die geometrische Construction als <quote>Existenzbeweis</quote> in der antiken Geometrie (in Mathematische Annalen, 47. Band)..

 @@ -1615,8 +1615,8 @@

We now come to the positive characteristics by which, according to Aristotle, scientific definitions must be marked.

First, the different attributes in a definition, when taken separately, cover more than the notion defined, but the combination of them does not. Aristotle illustrates this by the triad, into which enter the several notions of number, odd and prime, and the last in both its two senses (a) of not being measured by any (other) number (w(s mh\ metrei=sqai a)riqmw=|) and (b) of not being obtainable by adding numbers together - (w(s mh\ sugkei=sqai e)c a)riqmw=n), a unit not being a number. Of these attributes some are present in all other odd numbers as well, while the last [primeness in the second sense] belongs also to the dyad, but in nothing but the triad are they all presentAnal. post. II. 13, 96 a 33—b 1..” - The fact can be equally well illustrated from geometry. Thus, e.g. into the definition of a square (Eucl. I., Def. 22) there enter the several notions of figure, four-sided, equilateral, and right-angled, each of which covers more than the notion into which all enter as attributesTrendelenburg, Erläuterungen, p. 108..

+ (w(s mh\ sugkei=sqai e)c a)riqmw=n), a unit not being a number. Of these attributes some are present in all other odd numbers as well, while the last [primeness in the second sense] belongs also to the dyad, but in nothing but the triad are they all presentAnal. post. II. 13, 96 a 33—b 1..” + The fact can be equally well illustrated from geometry. Thus, e.g. into the definition of a square (Eucl. I., Def. 22) there enter the several notions of figure, four-sided, equilateral, and right-angled, each of which covers more than the notion into which all enter as attributesTrendelenburg, Erläuterungen, p. 108..

Secondly, a definition must be expressed in terms of things which are prior to, and better known than, the things definedTopics VI. 4, 141 a 26 sqq.. This is clear, since the object of a definition is to give us knowledge of the thing defined, and it is by means of things prior and better known that we acquire fresh knowledge, as in the course of demonstrations. But the terms prior and better known are, as usual susceptible of two meanings; they may mean (1) absolutely or logically prior and better known, or (2) better known relatively to us. In the absolute sense, or from the standpoint of reason, a point is better known than a line, a line than a plane, and a plane than a solid, as also a unit is better known than number (for the unit is prior to, and the first principle of, any number). Similarly, in the absolute sense, a letter is prior to a syllable. But the case is sometimes different relatively to us; for example, a solid is more easily realised by the senses than a plane, a plane than a line, and a line than a point. Hence, while it is more scientific to begin with the absolutely prior, it may, perhaps, be permissible, in case the learner is not capable of following the scientific order, to explain things by means of what is more intelligible to him. Among the definitions framed on this principle are those of the point, the line and the plane; all these explain what is prior by means of what is posterior, for the point is described as the extremity of a line, the line of a plane, the plane of a solid. @@ -1629,12 +1629,12 @@ Aristotle on unscientific definitions.

Aristotle distinguishes three kinds of definition which are unscientific because founded on what is not prior (mh\ e)k prote/rwn). The first is a definition of a thing by means of its opposite, e.g. of good by means of bad -; this is wrong because opposites are naturally evolved together, and the knowledge of opposites is not uncommonly regarded as one and the same, so that one of the two opposites cannot be better known than the other. It is true that, in some cases of opposites, it would appear that no other sort of definition is possible: e.g. it would seem impossible to define double apart from the half and, generally, this would be the case with things which in their very nature (kaq) au(ta/) are relative terms (pro/s ti le/getai), since one cannot be known without the other, so that in the notion of either the other must be comprised as wellTopics VI. 4, 142 a 22-31.. The second kind of definition which is based on what is not prior is that in which there is a complete circle through the unconscious use in the definition itself of the notion to be defined though not of the nameibid. 142 a 34—b 6.. Trendelenburg illustrates this by two current definitions, (1) that of magnitude as that which can be increased or diminished, which is bad because the positive and negative comparatives more +; this is wrong because opposites are naturally evolved together, and the knowledge of opposites is not uncommonly regarded as one and the same, so that one of the two opposites cannot be better known than the other. It is true that, in some cases of opposites, it would appear that no other sort of definition is possible: e.g. it would seem impossible to define double apart from the half and, generally, this would be the case with things which in their very nature (kaq) au(ta/) are relative terms (pro/s ti le/getai), since one cannot be known without the other, so that in the notion of either the other must be comprised as wellTopics VI. 4, 142 a 22-31.. The second kind of definition which is based on what is not prior is that in which there is a complete circle through the unconscious use in the definition itself of the notion to be defined though not of the nameibid. 142 a 34—b 6.. Trendelenburg illustrates this by two current definitions, (1) that of magnitude as that which can be increased or diminished, which is bad because the positive and negative comparatives more and less presuppose the notion of the positive great, (2) the famous Euclidean definition of a straight line as that which lies evenly with the points on itself (e)c i)/sou toi=s e)f) e(auth=s shmei/ois kei=tai), where lies evenly - can only be understood with the aid of the very notion of a straight line which is to be definedTrendelenburg, Erläuterungen, p. 115.. The third kind of vicious definition from that which is not prior is the definition of one of two coordinate species by means of its coordinate (a)ntidih|rhme/non), e.g. a definition of odd + can only be understood with the aid of the very notion of a straight line which is to be definedTrendelenburg, Erläuterungen, p. 115.. The third kind of vicious definition from that which is not prior is the definition of one of two coordinate species by means of its coordinate (a)ntidih|rhme/non), e.g. a definition of odd as that which exceeds the even by a unit (the second alternative in Eucl. VII. Def. 7); for odd and even are coordinates, being differentiae of numberTopics VI. 4, 142 b 7-10.. This third kind is similar to the first. Thus, says Trendelenburg, it would be wrong to define a square as a rectangle with equal sides. @@ -1647,7 +1647,7 @@ We seek the cause (to\ dio/ti) when we are already in possession of the fact (to\ o(/ti). Sometimes they both become evident at the same time, but at all events the cause cannot possibly be known [as a cause] before the fact is knownibid. II. 8, 93 a 17.. It is impossible to know what a thing is if we do not know that it isibid. 93 a 20. Trendelenburg paraphrases: The definition of the notion does not fulfil its purpose until it is made genetic. It is the producing cause which first reveals the essence of the thing.... . The nominal definitions of geometry have only a provisional significance and are superseded as soon as they are made genetic by means of construction. - E.g. the genetic definition of a parallelogram is evolved from Eucl. I. 31 (giving the construction for parallels) and I. 33 about the lines joining corresponding ends of two straight lines parallel and equal in length. Where existence is proved by construction, the cause and the fact appear togetherTrendelenburg, Erläuterungen, p. 110..

+ E.g. the genetic definition of a parallelogram is evolved from Eucl. I. 31 (giving the construction for parallels) and I. 33 about the lines joining corresponding ends of two straight lines parallel and equal in length. Where existence is proved by construction, the cause and the fact appear togetherTrendelenburg, Erläuterungen, p. 110..

Again, it is not enough that the defining statement should set forth the fact, as most definitions do; it should also contain and present the cause; whereas in practice what is stated in the definition is usually no more than a conclusion (sumpe/rasma). For example, what is quadrature? The construction of an equilateral right-angled figure equal to an oblong. But such a definition expresses merely the conclusion. Whereas, if you say that quadrature is the discovery of a mean proportional, then you state the reasonDe anima II. 2, 413 a 13-20,. This is better understood if we compare the statement elsewhere that the cause is the middle term, and this is what is sought in all casesAnal. post. II. 2, 90 a 6,, and the illustration of this by the case of the proposition that the angle in a semicircle is a right angle. Here the middle term which it is sought to establish by means of the figure is that the angle in the semi-circle is equal to the half of two right angles. We have then the syllogism: Whatever is half of two right angles is a right angle; the angle in a semi-circle is the half of two right angles; therefore (conclusion) the angle in a semi-circle is a right angleibid. II. 11, 94 a 28.. As with the demonstration, so it should be with the definition. A definition which is to show the genesis of the thing defined should contain the middle term or cause; otherwise it is a mere statement of a conclusion. Consider, for instance, the definition of quadrature @@ -1662,7 +1662,7 @@

It will be observed that what is here defined, quadrature or squaring (tetragwnismo/s), is not a geometrical figure, or an attribute of such a figure or a part of a figure, but a technical term used to describe a certain problem. Euclid does not define such things; but the fact that Aristotle alludes to this particular definition as well as to definitions of deflection (kekla/sqai) and of verging (neu/ein) seems to show that earlier text-books included among definitions explanations of a number of technical terms, and that Euclid deliberately omitted these explanations from his Elements as surplusage. Later the tendency was again in the opposite direction, as we see from the much expanded Definitions of Heron, which, for example, actually include a definition of a deflected line (keklasme/nh grammh/)Heron, Def. 12 (vol. IV. Heib. pp. 22-24).. Euclid uses the passive of kla=n occasionallye.g. in III. 20 and in Data 89., but evidently considered it unnecessary to explain such terms, which had come to bear a recognised meaning.

-

The mention too by Aristotle of a definition of verging (neu/ein) suggests that the problems indicated by this term were not excluded from elementary text-books before Euclid. The type of problem (neu=sis) was that of placing a straight line across two lines, e.g. two straight lines, or a straight line and a circle, so that it shall verge to a given point (i.e. pass through it if produced) and at the same time the intercept on it made by the two given lines shall be of given length. In general, the use of conics is required for the theoretical solution of these problems, or a mechanical contrivance for their practical solutionCf. the chapter on neu/seis in The Works of Archimedes, pp. c—cxxii.. Zeuthen, following Oppermann, gives reasons for supposing, not only that mechanical constructions were practically used by the older Greek geometers for solving these problems, but that they were theoretically recognised as a permissible means of solution when the solution could not be effected by means of the straight line and circle, and that it was only in later times that it was considered necessary to use conics in every case where that was possibleZeuthen, Die Lehre von den Kegelschnitten im Altertum, ch. 12, p. 262.. HeibergHeiberg, Mathematisches zu Aristoteles, p. 16. suggests that the allusion of Aristotle to neu/seis perhaps confirms this supposition, as Aristotle nowhere shows the slightest acquaintance with conics. I doubt whether this is a safe inference, since the problems of this type included in the elementary text-books might easily have been limited to those which could be solved by plane +

The mention too by Aristotle of a definition of verging (neu/ein) suggests that the problems indicated by this term were not excluded from elementary text-books before Euclid. The type of problem (neu=sis) was that of placing a straight line across two lines, e.g. two straight lines, or a straight line and a circle, so that it shall verge to a given point (i.e. pass through it if produced) and at the same time the intercept on it made by the two given lines shall be of given length. In general, the use of conics is required for the theoretical solution of these problems, or a mechanical contrivance for their practical solutionCf. the chapter on neu/seis in The Works of Archimedes, pp. c—cxxii.. Zeuthen, following Oppermann, gives reasons for supposing, not only that mechanical constructions were practically used by the older Greek geometers for solving these problems, but that they were theoretically recognised as a permissible means of solution when the solution could not be effected by means of the straight line and circle, and that it was only in later times that it was considered necessary to use conics in every case where that was possibleZeuthen, Die Lehre von den Kegelschnitten im Altertum, ch. 12, p. 262.. HeibergHeiberg, Mathematisches zu Aristoteles, p. 16. suggests that the allusion of Aristotle to neu/seis perhaps confirms this supposition, as Aristotle nowhere shows the slightest acquaintance with conics. I doubt whether this is a safe inference, since the problems of this type included in the elementary text-books might easily have been limited to those which could be solved by plane methods (i.e. by means of the straight line and circle). We know, e.g., from Pappus that Apollonius wrote two Books on plane neu/seisPappus VII. pp. 670-2.. But one thing is certain, namely that Euclid deliberately excluded this class of problem, doubtless as not being essential in a book of Elements.

 @@ -1970,7 +1970,7 @@

The full Greek expression would be h( u(po\ tw=n *b*a, *a*g periexome/nh gwni/a, the angle contained by the (straight lines) BA, AC. - But it was a common practice of Greek geometers, e.g. of Archimedes and Apollonius (and Euclid too in Books X.—XIII.), to use the abbreviation ai( *b*a*g for ai( *b*a, *a*g, the (straight lines) BA, AC. + But it was a common practice of Greek geometers, e.g. of Archimedes and Apollonius (and Euclid too in Books X.—XIII.), to use the abbreviation ai( *b*a*g for ai( *b*a, *a*g, the (straight lines) BA, AC. Thus, on periexome/nh being dropped, the expression would become first h( u(po\ tw=n *b*a*g gwni/a, then h( u(po\ *b*a*g gwni/a, and finally h( u(po\ *b*a*g, without gwni/a, as we regularly find it in Euclid.

 @@ -1979,7 +1979,7 @@ has been noticed above (note on Common Notion 4, pp. 224-5).

 -

Heiberg (Paralipomena su Euklid in Hermes, XXXVIII., 1903, p. 56) has pointed out, as a conclusive reason for regarding these words as an early interpolation, that the text of an-Nairīzī (Codex Leidensis 399, 1, ed. Besthorn-Heiberg, p. 55) does not give the words in this place but after the conclusion Q.E.D., which shows that they constitute a scholium only. They were doubtless added by some commentator who thought it necessary to explain the immediate inference that, since B coincides with E and C with F, the straight line BC coincides with the straight line EF, an inference which really follows from the definition of a straight line and Post. 1; and no doubt the Postulate that Two straight lines cannot enclose a space +

Heiberg (Paralipomena su Euklid in Hermes, XXXVIII., 1903, p. 56) has pointed out, as a conclusive reason for regarding these words as an early interpolation, that the text of an-Nairīzī (Codex Leidensis 399, 1, ed. Besthorn-Heiberg, p. 55) does not give the words in this place but after the conclusion Q.E.D., which shows that they constitute a scholium only. They were doubtless added by some commentator who thought it necessary to explain the immediate inference that, since B coincides with E and C with F, the straight line BC coincides with the straight line EF, an inference which really follows from the definition of a straight line and Post. 1; and no doubt the Postulate that Two straight lines cannot enclose a space (afterwards placed among the Common Notions) was interpolated at the same time.

 @@ -2905,7 +2905,7 @@ -

[I say that they are also in the same parallels.] Heiberg has proved (Hermes, XXXVIII., 1903, p. 50) from a recently discovered papyrus-fragment (Fayūm towns and their papyri, p. 96, No. IX.) that these words are an interpolation by some one who did not observe that the words And let AD be joined +

[I say that they are also in the same parallels.] Heiberg has proved (Hermes, XXXVIII., 1903, p. 50) from a recently discovered papyrus-fragment (Fayūm towns and their papyri, p. 96, No. IX.) that these words are an interpolation by some one who did not observe that the words And let AD be joined are part of the setting-out (e)/kqesis), but took them as belonging to the construction (kataskeuh/) and consequently thought that a diorismo/s or definition (of the thing to be proved) should precede. The interpolator then altered And into For @@ -3822,7 +3822,7 @@

I say further that the angle of the semicircle contained by the straight line BA and the circumference CHA is greater than any acute rectilineal angle, and the remaining angle contained by the circumference CHA and the straight line AE is less than any acute rectilineal angle.

For, if there is any rectilineal angle greater than the angle contained by the straight line BA and the circumference CHA, and any rectilineal angle less than the angle contained by the circumference CHA and the straight line AE, then into the space between the circumference and the straight line AE a straight line will be interposed such as will make an angle contained by straight lines which is greater than the angle contained by the straight line BA and the circumference CHA, and another angle contained by straight lines which is less than the angle contained by the circumference CHA and the straight line AE.

But such a straight line cannot be interposed;

-

therefore there will not be any acute angle contained by straight lines which is greater than the angle contained by the straight line BA and the circumference CHA, nor yet any acute angle contained by straight lines which is less than the angle contained by the circumference CHA and the straight line AE.—

+

therefore there will not be any acute angle contained by straight lines which is greater than the angle contained by the straight line BA and the circumference CHA, nor yet any acute angle contained by straight lines which is less than the angle contained by the circumference CHA and the straight line AE.—

PORISM.

From this it is manifest that the straight line drawn at right angles to the diameter of a circle from its extremity touches the circle. Q. E. D.

@@ -3972,7 +3972,7 @@ ABC will clearly be greater than a semicircle.

Therefore, given a segment of a circle, the complete circle has been described. Q. E. F. -

to describe the complete circle, prosanagra/yai to\n ku/klon, literally “to describe the circle on to it.’

+

to describe the complete circle, prosanagra/yai to\n ku/klon, literally “to describe the circle on to it.’

@@ -5901,7 +5901,7 @@

Let AB, CD be the two given numbers not prime to one another.

Thus it is required to find the greatest common measure of AB, CD.

-

If now CD measures AB—and it also measures itself—CD is a common measure of CD, AB.

+

If now CD measures AB—and it also measures itself—CD is a common measure of CD, AB.

And it is manifest that it is also the greatest; for no greater number than CD will measure CD.

But, if CD does not measure AB, then, the less of the numbers AB, CD being continually subtracted from the greater, some number will be left which will measure the one before it.

@@ -7011,8 +7011,8 @@

Therefore A, B are similar plane numbers; for their sides are proportional. Q. E. D.

25. For, since F......27. C to B. The text has clearly suffered corruption here. It is not necessary to infer from other facts that, as D is to E, so is A to C; for this is part of the hypotheses (ll. 6, 7). Again, there is no explanation of the statement (l. 25) that F by multiplying E has made C. It is the statement and explanation of this latter fact which are alone wanted; after which the proof proceeds as in l. 28. We might therefore substitute for ll. 25-28 the following.

-

“For, since E measures C the same number of times that D measures A [l. 8], that is, according to the units in F [l. 10], therefore F by multiplying E has made C.

-

And, since E by multiplying F, G,” +

“For, since E measures C the same number of times that D measures A [l. 8], that is, according to the units in F [l. 10], therefore F by multiplying E has made C.

+

And, since E by multiplying F, G,” etc. etc.

@@ -7308,10 +7308,10 @@

Let there be as many numbers as we please, B, C, D, E, beginning from the unit A and in continued proportion; I say that B, the least of the numbers B, C, D, E, measures E according to some one of the numbers C, D.

For since, as the unit A is to B, so is D to E, therefore the unit A measures the number B the same number of times as D measures E; therefore, alternately, the unit A measures D the same number of times as B measures E. [VII. 15]

-

But the unit A measures D according to the units in it; therefore B also measures E according to the units in D; so that B the less measures E the greater according to some number of those which have place among the proportional numbers.—

+

But the unit A measures D according to the units in it; therefore B also measures E according to the units in D; so that B the less measures E the greater according to some number of those which have place among the proportional numbers.—

PORISM. -

And it is manifest that, whatever place the measuring number has, reckoned from the unit, the same place also has the number according to which it measures, reckoned from the number measured, in the direction of the number before it.—

+

And it is manifest that, whatever place the measuring number has, reckoned from the unit, the same place also has the number according to which it measures, reckoned from the number measured, in the direction of the number before it.—

 Q. E. D. @@ -7793,7 +7793,7 @@ - PROPOSITIONS I—47. + PROPOSITIONS I—47. PROPOSITION 1.

Two unequal magnitudes being set out, if from the greater there be subtracted a magnitude greater than its half, and from that which is left a magnitude greater than its half, and if this process be repeated continually, there will be left some magnitude which will be less than the lesser magnitude set out.

@@ -7832,7 +7832,7 @@

Given two commensurable magnitudes, to find their greatest common measure.

Let the two given commensurable magnitudes be AB, CD of which AB is the less; thus it is required to find the greatest common measure of AB, CD.

Now the magnitude AB either measures CD or it does not.

-

If then it measures it—and it measures itself also—AB is a common measure of AB, CD.

+

If then it measures it—and it measures itself also—AB is a common measure of AB, CD.

And it is manifest that it is also the greatest; for a greater magnitude than the magnitude AB will not measure AB.

Next, let AB not measure CD.

@@ -7983,7 +7983,7 @@

To find two straight lines incommensurable, the one in length only, and the other in square also, with an assigned straight line.

Let A be the assigned straight line; thus it is required to find two straight lines incommensurable, the one in length only, and the other in square also, with A.

Let two numbers B, C be set out which have not to one another the ratio which a square number has to a square number, that is, which are not similar plane numbers; and let it be contrived that, as B is to C, so is the square on A to the square on D -

—for we have learnt how to do this— [X. 6, Por.] therefore the square on A is commensurable with the square on D. [X. 6]

+
—for we have learnt how to do this— [X. 6, Por.] therefore the square on A is commensurable with the square on D. [X. 6]

And, since B has not to C the ratio which a square number has to a square number, therefore neither has the square on A to the square on D the ratio which a square number has to a square number; therefore A is incommensurable in length with D. [X. 9]

Let E be taken a mean proportional between A, D; therefore, as A is to D, so is the square on A to the square on E. [V. Def. 9]

But A is incommensurable in length with D; therefore the square on A is also incommensurable with the square on E; [X. 11] therefore A is incommensurable in square with E.

@@ -8508,10 +8508,10 @@

If two rational straight lines commensurable in square only be added together, the whole is irrational; and let it be called binomial.

For let two rational straight lines AB, BC commensurable in square only be added together;

I say that the whole AC is irrational.

-

For, since AB is incommensurable in length with BC— for they are commensurable in square only— and, as AB is to BC, so is the rectangle AB, BC to the square on BC, therefore the rectangle AB, BC is incommensurable with the square on BC. [X. 11 +

For, since AB is incommensurable in length with BC— for they are commensurable in square only— and, as AB is to BC, so is the rectangle AB, BC to the square on BC, therefore the rectangle AB, BC is incommensurable with the square on BC. [X. 11 ]

But twice the rectangle AB, BC is commensurable with the rectangle AB, BC [X. 6 -], and the squares on AB, BC are commensurable with the square on BC—for AB, BC are rational straight lines commensurable in square only— [X. 15 +], and the squares on AB, BC are commensurable with the square on BC—for AB, BC are rational straight lines commensurable in square only— [X. 15 ] therefore twice the rectangle AB, BC is incommensurable with the squares on AB, BC. [X. 13 ]

And, componendo, twice the rectangle AB, BC together with the squares on AB, BC, that is, the square on AC [II. 4 @@ -8662,7 +8662,7 @@ and, componendo, the squares on AB<

A first bimedial straight line is divided at one point only.

Let AB be a first bimedial straight line divided at C, so that AC, CB are medial straight lines commensurable in square only and containing a rational rectangle;

I say that AB is not so divided at another point.

For, if possible, let it be divided at D also, so that AD, DB are also medial straight lines commensurable in square only and containing a rational rectangle.

-

Since, then, that by which twice the rectangle AD, DB differs from twice the rectangle AC, CB is that by which the squares on AC, CB differ from the squares on AD, DB, while twice the rectangle AD, DB differs from twice the rectangle AC, CB by a rational area—for both are rational— therefore the squares on AC, CB also differ from the squares on AD, DB by a rational area, though they are medial: which is absurd. [x. 26 +

Since, then, that by which twice the rectangle AD, DB differs from twice the rectangle AC, CB is that by which the squares on AC, CB differ from the squares on AD, DB, while twice the rectangle AD, DB differs from twice the rectangle AC, CB by a rational area—for both are rational— therefore the squares on AC, CB also differ from the squares on AD, DB by a rational area, though they are medial: which is absurd. [x. 26 ]

Therefore a first bimedial straight line is not divided into its terms at different points; therefore it is so divided at one point only.

@@ -8711,7 +8711,7 @@ and, componendo, the squares on AB<

Let AB be a major straight line divided at C, so that AC, CB are incommensurable in square and make the sum of the squares on AC, CB rational, but the rectangle AC, CB medial; [X. 39 ]

I say that AB is not so divided at another point.

For, if possible, let it be divided at D also, so that AD, DB are also incommensurable in square and make the sum of the squares on AD, DB rational, but the rectangle contained by them medial.

-

Then, since that by which the squares on AC, CB differ from the squares on AD, DB is also that by which twice the rectangle AD, DB differs from twice the rectangle AC, CB, while the squares on AC, CB exceed the squares on AD, DB by a rational area—for both are rational— therefore twice the rectangle AD, DB also exceeds twice the rectangle AC, CB by a rational area, though they are medial: which is impossible. [X. 26 +

Then, since that by which the squares on AC, CB differ from the squares on AD, DB is also that by which twice the rectangle AD, DB differs from twice the rectangle AC, CB, while the squares on AC, CB exceed the squares on AD, DB by a rational area—for both are rational— therefore twice the rectangle AD, DB also exceeds twice the rectangle AC, CB by a rational area, though they are medial: which is impossible. [X. 26 ]

Therefore a major straight line is not divided at different points; therefore it is only divided at one and the same point. Q. E. D.

@@ -8780,7 +8780,7 @@ and, componendo, the squares on AB< - PROPOSITIONS 48—84. + PROPOSITIONS 48—84. PROPOSITION 48.

To find the first binomial straight line.

@@ -8911,7 +8911,7 @@ and, componendo, the squares on AB<

And since, as FB is to BG, so is DB to BE, while, as FB is to BG, so is AB to DG, and, as DB is to BE, so is DG to BC, [VI. 1] therefore also, as AB is to DG, so is DG to BC. [V. 11]

Therefore DG is a mean proportional between AB, BC.

I say next that DC is also a mean proportional between AC, CB.

-

For since, as AD is to DK, so is KG to GC— for they are equal respectively— and, componendo, as AK is to KD, so is KC to CG, [V. 18] while, as AK is to KD, so is AC to CD, and, as KC is to CG, so is DC to CB, [VI. 1] therefore also, as AC is to DC, so is DC to BC. [V. 11]

+

For since, as AD is to DK, so is KG to GC— for they are equal respectively— and, componendo, as AK is to KD, so is KC to CG, [V. 18] while, as AK is to KD, so is AC to CD, and, as KC is to CG, so is DC to CB, [VI. 1] therefore also, as AC is to DC, so is DC to BC. [V. 11]

Therefore DC is a mean proportional between AC, CB. Being what it was proposed to prove.

 @@ -9390,8 +9390,8 @@ and, componendo, the squares on AB<

But DE is equal to the squares on AB, BC, and DH to twice the rectangle AB, BC; therefore DE is incommensurable with DH.

But, as DE is to DH, so is GD to DF; [VI. 1] therefore GD is incommensurable with DF. [X. 11]

And both are rational; therefore GD, DF are rational straight lines commensurable in square only; therefore FG is an apotome. [X. 73]

-

But DI is rational, and the rectangle contained by a rational and an irrational straight line is irrational, [deduction from X. 20] and its ’side’ is irrational.

-

And AC is the ’side’ of FE; therefore AC is irrational.

+

But DI is rational, and the rectangle contained by a rational and an irrational straight line is irrational, [deduction from X. 20] and its ’side’ is irrational.

+

And AC is the ’side’ of FE; therefore AC is irrational.

And let it be called a second apotome of a medial straight line. Q. E. D.

@@ -9560,7 +9560,7 @@ and, componendo, the squares on AB< - PROPOSITIONS 85—115. + PROPOSITIONS 85—115. PROPOSITION 85.

To find the first apotome.

@@ -11668,7 +11668,7 @@ and, componendo, the squares on AB< HISTORICAL NOTE.

I have already given, in the note to IV. 10, the evidence upon which the construction of the five regular solids is attributed to the Pythagoreans. Some of them, the cube, the tetrahedron (which is nothing but a pyramid), and the octahedron (which is only a double pyramid with a square base), cannot but have been known to the Egyptians. And it appears that dodecahedra have been found, of bronze or other material, which may belong to periods earlier than Pythagoras' time by some centuries (for references see Cantor's Geschichte der Mathematik I_{3}, pp. 175-6).

It is true that the author of the scholium No. I to Eucl. XIII. says that the Book is about the five so-called Platonic figures, which however do not belong to Plato, three of the aforesaid five figures being due to the Pythagoreans, namely the cube, the pyramid and the dodecahedron, while the octahedron and the icosahedron are due to Theaetetus. - This statement (taken probably from Geminus) may perhaps rest on the fact that Theaetetus was the first to write at any length about the two last-mentioned solids. We are told indeed by Suidas (S. V. *qeai/thtos) that Theaetetus first wrote on the ’five solids’ as they are called. + This statement (taken probably from Geminus) may perhaps rest on the fact that Theaetetus was the first to write at any length about the two last-mentioned solids. We are told indeed by Suidas (S. V. *qeai/thtos) that Theaetetus first wrote on the ’five solids’ as they are called. This no doubt means that Theaetetus was the first to write a complete and systematic treatise on all the regular solids; it does not exclude the possibility that Hippasus or others had already written on the dodecahedron. The fact that Theaetetus wrote upon the regular solids agrees very well with the evidence which we possess of his contributions to the theory of irrationals, the connexion between which and the investigation of the regular solids is seen in Euclid's Book XIII.

Theaetetus flourished about 380 B.C., and his work on the regular solids was soon followed by another, that of Aristaeus, an elder contemporary of Euclid, who also wrote an important book on Solid Loci, i.e. on conics treated as loci. This Aristaeus (known as the elder ) wrote in the period about 320 B.C. We hear of his Comparison of the five regular solids from Hypsicles (2nd cent. B.C.), the writer of the short book commonly included in the editions of the Elements as Book XIV. Hypsicles gives in this Book some six propositions supplementing Eucl. XIII.; and he introduces the second of the propositions (Heiberg's Euclid, Vol. v. p. 6) as follows:

@@ -12115,7 +12115,7 @@ and, componendo, the squares on AB<

I say next that it is also equiangular.

For, since the straight line NP has been cut in extreme and mean ratio at R, and PR is the greater segment, while PR is equal to PS, therefore NS has also been cut in extreme and mean ratio at P, and NP is the greater segment; [XIII. 5] therefore the squares on NS, SP are triple of the square on NP. [XIII. 4]

But NP is equal to NB, and PS to SV; therefore the squares on NS, SV are triple of the square on NB; hence the squares on VS, SN, NB are quadruple of the square on NB.

-

But the square on SB is equal to the squares on SN, NB; therefore the squares on BS, SV, that is, the square on BV —for the angle VSB is right—is quadruple of the square on NB; therefore VB is double of BN.

+

But the square on SB is equal to the squares on SN, NB; therefore the squares on BS, SV, that is, the square on BV —for the angle VSB is right—is quadruple of the square on NB; therefore VB is double of BN.

But BC is also double of BN; therefore BV is equal to BC.

And, since the two sides BU, UV are equal to the two sides BW, WC, and the base BV is equal to the base BC, therefore the angle BUV is equal to the angle BWC. [I. 8]

Similarly we can prove that the angle UVC is also equal to the angle BWC; therefore the three angles BWC, BUV, UVC are equal to one another.