book 1, triangles

book 2, quadratics

books 3 and 4, circles

book 5, theory of proportion

book 6, geometry and the theory of proportion

books 7, 8 and 9 ,number theory

book II identities

gemetrical solution of quadratics

application of areas

transformation of areas

VI. 27 note

VI. 28 note

VI. 29 note

Side and Diameter numbers

Book V notes

Book VII notes

Book VIII notes

Book IX notes

[p. 91]



Cicero is the first Latin author to mention Euclid1 ; but it is not likely that in Cicero's time Euclid had been translated into Latin or was studied to any considerable extent by the Romans; for, as Cicero says in another place2 , while geometry was held in high honour among the Greeks, so that nothing was more brilliant than their mathematicians, the Romans limited its scope by having regard only to its utility for measurements and calculations. How very little theoretical geometry satisfied the Roman agrimensores is evidenced by the work of Balbus de mensuris3 , where some of the definitions of Eucl. Book I. are given. Again, the extracts from the Elements found in the fragment attributed to Censorinus (fl. 238 A.D)4 are confined to the definitions, postulates, and common notions. But by degrees the Elements passed even among the Romans into the curriculum of a liberal education; for Martianus Capella speaks of the effect of the enunciation of the proposition “how to construct an equilateral triangle on a given straight line� among a company of philosophers, who, recognising the first proposition of the Elements, straightway break out into encomiums on Euclid5 . But the Elements were then (c. 470 A.D.) doubtless read in Greek; for what Martianus Capella gives6 was drawn from a Greek source, as is shown by the occurrence of Greek words and by the wrong translation of I. def. 1 (“punctum vero est cuius pars nihil est�). Martianus may, it is true, have quoted, not from Euclid himself, but from Heron or some other ancient source.

But it is clear from a certain palimpsest at Verona that some scholar had already attempted to translate the Elements into Latin. This palimpsest7 has part of the “Moral reflections on the Book of Job� by Pope Gregory the Great written in a hand of the 9th c. above certain fragments which in the opinion of the best judges date from the 4th c. Among these are fragments of Vergil and of Livy, as well as a geometrical fragment which purports to be taken from the 14th and 15th Books of Euclid. As a matter of fact it is from Books XII. and XIII. and is of the nature of a free rendering, or rather a new [p. 92] arrangement, of Euclid with the propositions in different order8 . The MS. was evidently the translator's own copy, because some words are struck out and replaced by synonyms. We do not know whether the translator completed the translation of the whole, or in what relation his version stood to our other sources.

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 letters9 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 genuine10 ,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.

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 ABG, [p. 93] DEZ 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 DEZTh becomes in one place “quod autem septimo nono,� showing that he mistook DE for the particle de, and kai ho STU is rendered “sicut tricentissimo et quadringentissimo,� showing that the letters must have been written KAIOCTU.

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.11 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 Data12 . 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.)13 ; 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 14 [p. 94] expression “superficies equidistantium laterum et rectorum angulorum,� 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� 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.

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 Euclid15 : 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 time16 , 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 128117 . 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 references18 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 [p. 95] 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,� which was known to him, and also the Arabic figures. Thus Athelhard certainly was not the first to translate Euclid into Latin; there must have been in existence before the 11th c. a Latin translation which was the common source of R, the passage in the Gromatici, and “Boethius.� As in the two latter there occur the proofs as well as the enunciations of I. 1mdash;3, it is possible that this translation originally contained the proofs also. Athelhard must have had before had before him this translation of the enunciations, as well as the Arabic source from which he obtained his proofs. That some sort of translation, or at least fragments of one, were available before Athelhard's time even in England is indicated by some old English verses19 : “The clerk Euclide on this wyse hit fonde Thys craft of gemetry yn Egypte londe Yn Egypte he tawghte hyt ful wyde, In dyvers londe on every syde. Mony erys afterwarde y understonde Yer that the craft com ynto thys londe. Thys craft com into England, as y yow say, Yn tyme of good kyng Adelstone's day,� which would put the introduction of Euclid into England as far back as 924-940 A.D.

We now come to the relation between Athelhard and Campanus. That their translations were not independent, as Weissenborn would have us believe, is clear from the fact that in all MSS. and editions, apart from orthographical differences and such small differences as are bound to arise when MSS. are copied by persons with some knowledge of the subject-matter, the definitions, postulates, axioms, and the 364 enunciations are word for word identical in Athelhard and Campanus; and this is the case not only where both have the same text as R but where they diverge from it. Hence it would seem that Campanus used Athelhard's translation and only developed the proofs by means of another redaction of the Arabian Euclid. It is true that the difference between the proofs of the propositions in the two translations is considerable; Athelhard's are short and compressed, [p. 96] Campanus' clearer and more complete, following the Greek text more closely, though still at some distance. Further, the arrangement in the two is different; in Athelhard the proofs regularly precede the enunciations, Campanus follows the usual order. It is a question how far the differences in the proofs, and certain additions in each, are due to the two translators themselves or go back to Arabic originals. The latter supposition seems to Curtze and Cantor the more probable one. Curtze's general view of the relation of Campanus to Athelhard is to the effect that Athelhard's translation was gradually altered, from the form in which it appears in the two Erfurt MSS. described by Weissenborn, by successive copyists and commentators who had Arabic originals before them, until it took the form which Campanus gave it and in which it was published. In support of this view Curtze refers to Regiomontanus' copy of the Athelhard-Campanus translation. In Regiomontanus' own preface the title is given, and this attributes the translation to Athelhard; but, while this copy agrees almost exactly with Athelhard in Book I., yet, in places where Campanus is more lengthy, it has similar additions, and in the later Books, especially from Book III. onwards, agrees absolutely with Campanus; Regiomontanus, too, himself implies that, though the translation was Athelhard's, Campanus had revised it; for he has notes in the margin such as the following, “Campani est hec,� “dubito an demonstret hic Campanus� etc.

We come now to the printed editions of the whole or of portions of the Elements. This is not the place for a complete bibliography, such as Riccardi has attempted in his valuable memoir issued in five parts between 1887 and 1893, which makes a large book in itself20 . I shall confine myself to saying something of the most noteworthy translations and editions. It will be convenient to give first the Latin translations which preceded the publication of the editio princeps of the Greek text in 1533, next the most important editions of the Greek text itself, and after them the most important translations arranged according to date of first appearance and languages, first the Latin translations after 1533, then the Italian, German, French and English translations in order.

It may be added here that the first allusion, in the West, to the Greek text as still extant is found in Boccaccio's commentary on the Divina Commedia of Dante21 . Next Johannes Regiomontanus, who intended to publish the Elements after the version of Campanus, but with the latter's mistakes corrected, saw in Italy (doubtless when staying with his friend Bessarion) some Greek MSS. and noticed how far they differed from the Latin version (see a letter of his written in the year 1471 to Christian Roder of Hamburg)22 .

[p. 97]

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' translation23 . 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. Kästner24 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 letters25 . 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 latitudinibus26 . 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 [p. 98] 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 title27 , 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,� or “to that most barbarous translator� who filled a volume purporting to be Euclid's “with extraordinary scarecrows, nightmares and phantasies,� one object of Zamberti's translation is clear. His animus against Campanus appears also in a number of notes, e.g. when he condemns the terms “helmuain� and “helmuariphe� used by Campanus as barbarous, un-Latin etc., and when he is roused to wrath by Campanus' unfortunate mistranslation of v. Def. 5. He does not seem to have had the penetration to see that Campanus was translating from an Arabic, and not from a Greek, text. Zamberti tells us that he spent seven years over his translation of the thirteen Books of the Elements. As he seems to have been born in 1473, and the Elements were printed as early as 1500, though the complete work (including the Phaenomena, Optica, Catoptrica, Data etc.) has the date 1505 at the end, he must have translated Euclid before the age of 30. Heiberg has not been able to identify the MS. of the Elements which Zamberti used; but it is clear that it belonged to the worse class of MSS., since it contains most of the interpolations of the Theonine variety. Zamberti, as his title shows, attributed the proofs to Theon.

1509. As a counterblast to Zamberti, Luca Paciuolo brought out an edition of Euclid, apparently at the expense of Ratdolt, at Venice (per Paganinum de Paganinis), in which he set himself to vindicate Campanus. The title-page of this now very rare edition28 begins thus: “The works of Euclid of Megara, a most acute philosopher and without [p. 99] question the chief of all mathematicians, translated by Campanus their most faithful interpreter.� It proceeds to say that the translation had been, through the fault of copyists, so spoiled and deformed that it could scarcely be recognised as Euclid. Luca Paciuolo accordingly has polished and emended it with the most critical judgment, has corrected 129 figures wrongly drawn and added others, besides supplying short explanations of difficult passages. It is added that Scipio Vegius of Milan, distinguished for his knowledge “of both languages� (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) smoke29 .� 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. 530 . 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 proofs31 . The date is not on the title-page nor at the [p. 100] 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

II. Editions of the Greek text.

1533 is the date of the editio princeps, the title-page of which reads as follows: EUKLEIDOU STOICHEIÔN BIBL<*> IE<*><*> EK TÔN THEÔNOS SUNOUSIÔN. Eis tou autou to prôton, exêgêmatôn Proklou bibl. d_. Adiecta praefatiuncula in qua de disciplinis Mathematicis nonnihil. BASILEAE APVD IOAN. HERVAGIVM ANNO M.D.XXXIII. MENSE SEPTEMBRI.

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 arithmetic32 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� (Jean Ruel, a French doctor and a Greek scholar), while the commentaries of Proclus were put at [p. 101] 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,� 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.

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 quotes33 , 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 Euclid34 �

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 words35 .

1557 (also 1558). Stephanus Gracilis published another edition (repeated 1573, 1578, 1598) of the enunciations (alone) of Books I.--XV. [p. 102] 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.�

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 Euclid36 . 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 [p. 103] 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 French37 . 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 Berlin38 . 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.

[p. 104]

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 Paris39 . 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� 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� on the title-page. Commandinus not only followed the original Greek more closely than his predecessors but added to his translation some ancient scholia as well as good notes of his own. The title of his work is

Euclidis elementorum libri XV, una cum scholiis antiquis. A Federico Commandino Urbinate nuper in latinum conversi, commentariisque quibusdam illustrati (Pisauri, apud Camillum Francischinum).

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 [p. 105] 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.

1574. The first edition of the Latin version by Clavius40 (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.

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). [p. 106]

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.

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.

1802. Euclidis elementorum libri priores XII ex Commandini et Gregorii versionibus latinis. In usum juventutis Academicae...by Samuel Horsley, Bishop of Rochester. (Oxford, Clarendon Press.)

IV. Italian versions or commentaries.

1543. Tartaglia's version, a second edition of which was published in 156541 , 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).

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). [p. 107]

1731. Elementi geometrici piani e solidi di Euclide by Guido Grandi. No translation, but an abbreviated version, of which new editions followed one another up to 1806.

1749. Italian translation of Dechales with Ozanam's corrections and additions, re-issued 1785, 1797.

1752. Leonardo Ximenes (the first six Books). Fifth edition, 1819.

1818. Vincenzo Flauti's Corso di geometria elementare e sublime (4 vols.) contains (Vol. I.) the first six Books, with additions and a dissertation on Postulate 5, and (Vol. II.) Books XI., XII. Flauti also published the first six Books in 1827 and the Elements of geometry of Euclid in 1843 and 1854.

V. German.

1558. The arithmetical Books VII.-IX. by Scheubel42 (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)43 . 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,� 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 [p. 108] 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,� 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.

VI. French.

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.

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.

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.

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.� [p. 109]

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.

VIII. English.

1570 saw the first and the most important translation, that of Sir Henry Billingsley. The title-page is as follows: THE ELEMENTS OF GEOMETRIE of the most auncient Philosopher EVCLIDE of Megara

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.

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....�

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 [p. 110] 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.

Billingsley was admitted Lady Margaret Scholar of St John's College, Cambridge, in 1551, and he is also said to have studied at Oxford, but he did not take a degree at either University. He was afterwards apprenticed to a London haberdasher and rapidly became a wealthy merchant. Sheriff of London in 1584, he was elected Lord Mayor on 31st December, 1596, on the death, during his year of office, of Sir Thomas Skinner. From 1589 he was one of the Queen's four “customers,� or farmers of customs, of the port of London. In 1591 he founded three scholarships at St John's College for poor students, and gave to the College for their maintenance two messuages and tenements in Tower Street and in Mark Lane, Allhallows, Barking. He died in 1606.

1651. Elements of Geometry. The first VI Boocks: In a compendious form contracted and demonstrated by Captain Thomas Rudd, with the mathematicall preface of John Dee (London).

1660. The first English edition of Barrow's Euclid (published in Latin in 1655), appeared in London. It contained “the whole fifteen books compendiously demonstrated�; several editions followed, in 1705, 1722, 1732, 1751.

1661. Euclid's Elements of Geometry, with a supplement of divers Propositions and Corollaries. To which is added a Treatise of regular Solids by Campane and Flussat; likewise Euclid's Data and Marinus his Preface. Also a Treatise of the Divisions of Superficies, ascribed to Machomet Bagdedine, but published by Commandine at the request of J. Dee of London. Published by care and industry of John Leeke and Geo. Serle, students in the Math. (London). According to Potts this was a second edition of Billingsley's translation.

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.

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� [p. 111] 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.

1714. W. Whiston's English version (abridged) of The Elements of Euclid with select theorems out of Archimedes by the learned Andr. Tacquet.

1756. Simson's first English edition appeared in the same year as his Latin version under the title:

The Elements of Euclid, viz. the first six Books together with the eleventh and twelfth. In this Edition the Errors by which Theon or others have long ago vitiated these Books are corrected and some of Euclid's Demonstrations are restored. By Robert Simson (Glasgow).

As above stated, the Latin edition, by its title, purports to be “ex versione latina Federici Commandini,� but to the Latin edition, as well as to the English editions, are appended

Notes Critical and Geometrical; containing an Account of those things in which this Edition differs from the Greek text; and the Reasons of the Alterations which have been made. As also Observations on some of the Propositions.

Simson says in the Preface to some editions (e.g. the tenth, of 1799) that “the translation is much amended by the friendly assistance of a learned gentleman.�

Simson's version and his notes are so well known as not to need any further description. The book went through some thirty successive editions. The first five appear to have been dated 1756, 1762, 1767, 1772 and 1775 respectively; the tenth 1799, the thirteenth 1806, the twenty-third 1830, the twenty-fourth 1834, the twenty-sixth 1844 The Data “in like manner corrected� was added for the first time in the edition of 1762 (the first octavo edition).

1781, 1788. In these years respectively appeared the two volumes containing the complete translation of the whole thirteen Books by James Williamson, the last English translation which reproduced Euclid word for word. The title is

The Elements of Euclid, with Disserlations intended to assist and encourage a critical examination of these Elements, as the most effectual means of establishing a juster taste upon mathematical subjects than that which at present prevails. By James Williamson.

In the first volume (Oxford, 1781) he is described as “M.A. Fellow of Hertford College,� and in the second (London, printed by T. Spilsbury, 1788) as “B.D.� simply. Books V., VI. with the Conclusion in the first volume are paged separately from the rest.

1781. An examination of the first six Books of Euclid's Elements, by William Austin (London).

1795. John Playfair's first edition, containing “the first six Books of Euclid with two Books on the Geometry of Solids.� The book [p. 112] reached a fifth edition in 1819, an eighth in 1831, a ninth in 1836, and a tenth in 1846.

1826. Riccardi notes under this date Euclid's Elements of Geometry containing the whole twelve Books translated into English, from the edition of Peyrard, by George Phillips. The editor, who was President of Queens' College, Cambridge, 1857-1892, was born in 1804 and matriculated at Queens' in 1826, so that he must have published the book as an undergraduate.

1828. A very valuable edition of the first six Books is that of Dionysius Lardner, with commentary and geometrical exercises, to which he added, in place of Books XI., XII., a Treatise on Solid Geometry mostly based on Legendre. Lardner compresses the propositions by combining the enunciation and the setting-out, and he gives a vast number of riders and additional propositions in smaller print. The book had reached a ninth edition by 1846, and an eleventh by 1855. Among other things, Lardner gives an Appendix “on the theory of parallel lines,� in which he gives a short history of the attempts to get over the difficulty of the parallel-postulate, down to that of Legendre.

1833. T. Perronet Thompson's Geometry without axioms, or the first Book of Euclid's Elements with alterations and notes; and an intercalary book in which the straight line and plane are derived from properties of the sphere, with an appendix containing notices of methods proposed for getting over the difficulty in the twelfth axiom of Euclid.

Thompson (1783-1869) was 7th wrangler 1802, midshipman 1803, Fellow of Queens' College, Cambridge, 1804, and afterwards general and politician. The book went through several editions, but, having been well translated into French by Van Tenac, is said to have received more recognition in France than at home.

1845. Robert Potts' first edition (and one of the best) entitled:

Euclid's Elements of Geometry chiefly from the text of Dr Simson with explanatory notes...to which is prefixed an introduction containing a brief outline of the History of Geometry. Designed for the use of the higher forms in Public Schools and students in the Universities (Cambridge University Press, and London, John W. Parker), to which was added (1847) An Appendix to the larger edition of Euclid's Elements of Geometry, containing additional notes on the Elements, a short tract on transversals, and hints for the solution of the problems etc


1862. Todhunter's edition.

The later English editions I will not attempt to enumerate; their name is legion and their object mostly that of adapting Euclid for school use, with all possible gradations of departure from his text and order.

IX. Spanish.

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.

[p. 113]

X. Russian.

1739. Ivan Astaroff (translation from Latin).

1789. Pr. Suvoroff and Yos. Nikitin (translation from Greek).

1880. Vachtchenko-Zakhartchenko.

(1817. A translation into Polish by Jo. Czecha.)

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.

1836. H. Falk, the first six Books.

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.

XII. Danish.

1745. Ernest Gottlieb Ziegenbalg.

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).

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).

1 De oratore III. 132.

2 Tusc. 1. 5.

3 Gromatici veteres, I. 97 sq. (ed. F. Blume, K. Lachmann and A. Rudorff. Berlin, 1848, 1852).

4 Censorinus, ed. Hultsch, pp. 60-3.

5 Martianus Capella, VI. 724.

6 ibid. VI. 708 sq.

7 Cf. Cantor, I_{3}, p. 565.

8 The fragment was deciphered by W. Studemund, who communicated his results to Cantor.

9 Cassiodorus, Variae, I. 45, p. 40, 12 ed. Mommsen.

10 See 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.

11 Cantor, Gesch. d. Math. I3, p. 906.

12 Boncompagni, Della vita e delle opere di Gherardo Cremonese, Rome, 1851, p. 5.

13 Described in an appendix to Studien über Menelaos' Sphärik (Abhandlungen zur Gesckichse der mathematischen Wissenschaften, XIV., 1902).

14 4 See Bibliotheca Mathematica, VI3, 1905-6, PP. 242-8.

15 Anaritii in decem libros priores Elementorum Euclidis Commentarii ex interpretatione Gherardi Cremonensis in codice Cracoviensi 569 servata edidit Maximilianus Curtze, Leipzig (Teubner), 1899.

16 Cantor, II_{1}, p. 88.

17 Tiraboschi, Storia della letteratura italiana, IV. 145mdash;160.

18 H. Weissenborn in Zeitschrift für Math. u. Physik, XXV., Supplement, pp. 143mdash;166, and in his monograph, Die Ãœbersetzungen 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.

19 Quoted by Halliwell in Rara Mathematica (p. 56 note) from MS. Bib. Reg. Mus. Brit. 17 A. 1. f. 2^{4}mdash;3.

20 Saggio di una Bibliografia Euclidea, memoria del Prof. Pietro Riccardi (Bologna, 1887, 1888, 1890, 1893).

21 I. p. 404.

22 Published in C. T. de Murr's Memorabilia Bibliothecarum Norimbergensium, Part I. p. 190 sqq.

23 Curtze (An-NairÄ«zÄ«, p. xiii) reproduces the heading of the first page of the text as follows (there is no title-page): PreclariffimÅ© opus elemento<*> Euclidis megarÄ“fis [vmacr]na cÅ« cÅ�mentis Campani pfpicaciffimi in artÄ“ geometriÄ� 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 elementorÅ« euclidis megarenfis in geometriÄ� artÄ“ Jnid quoq<*> Campani pfpicaciffimi CÅ�mentationes finiÅ©t. Erhardus ratdolt Augustensis impreffor folertiffimus. venetijs impreffit . Anno falutis . M.cccc.lxxxij . Octauis . Caleñ . Juñ . Lector . Vale.

24 Kästner, Geschichte der Mathematik, I. p. 289 sqq. See also Weissenborn, Die Ãœbersetzungen des Euklid durch Campano und Zamberti, pp. 1-7.

25 â€œMea industria non sine maximo labore effeci vt qua facilitate litterarum elementa imprimuntur ea etiam geometrice figure conficerentur.â€�

26 Curtze in Zeitschrift für Math. u. Physik, XX., hist.-litt. Abth. p. 58.

27 The 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.â€� For a description of the book see Weissenborn, p. 12 sqq.

28 See Weissenborn, p. 30 sqq.

29 â€œAtque utinam et alii cognoscere vellent non ostentare aut ea quae nesciunt veluti fumum venditare non conarentur.â€�

30 Campanus' 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 minuuntâ€� (!), 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.â€�

31 â€œEuclidis 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.â€� 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.â€� For description see Weissenborn, p. 56 sqq.

32 De arté supputandi libri quatuor.

33 Kästner, I. p. 260.

34 Heiberg, vol. V. p. cvii.

35 Kästner, I. p. 359.

36 EUKLEIDOU TA SÔZOMENA. Euclidis quae supersunt omnia. Ex recensione Davidis Gregorii M.D. Astronomiae Professoris Saviliani et R.S.S. Oxoniae, e Theatro Sheldoniano, An. Dom. MDCCIII.

37 Euclidis quae supersunt. Les Å’uvres 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).

38 Euclidis 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.

39 Described by Boncompagni, Bullettino, II. p. 389.

40 Euclidis elemcntorum librit XV. Accessit XVI. de solidorum regularium comparatione. Omnes perspicuis demonstrationibus, accuratisque scholiis illustrati. Auctore Christophoro Clavio (Romae, apud Vincentium Accoltum), 2 vols.

41 The 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.

42 Das 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....

43 .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.