**[p. 91]**

#### CHAPTER VIII.

#### PRINCIPAL TRANSLATIONS AND EDITIONS OF THE ELEMENTS.

Cicero
is the first Latin
author to mention Euclid^{1} ; 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 place^{2} , 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
mensuris*^{3} , 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 â€œhow to
construct an equilateral triangle on a given straight lineâ€� among a
company of philosophers, who, recognising the first proposition of the
*Elements*, straightway break out into encomiums on Euclid^{5} . But the *Elements* were then (*c*.
470 A.D.) doubtless read in Greek;
for what Martianus Capella
gives^{6} was drawn from a Greek
source, as is shown by the occurrence of Greek
words and by the wrong translation of I. def. 1 (â€œpunctum vero
est cuius pars *nihil*
estâ€�). 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 palimpsest^{7} has part of the â€œMoral reflections on the Book
of Jobâ€� 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 order^{8} . 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, â€œof whom Euclid
was given us translated into the Latin
language by the same great man Boethiusâ€�; also in his collection of
letters^{9} is a letter from Theodoric to Boethius
containing the words, â€œfor in your translations...Nicomachus
the arithmetician, and Euclid
the geometer, are heard in the Ausonian tongue.â€� 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 genuine^{10} ,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-NairÄ©zÄ©'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 â€œcapitolo nonoâ€�
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 â€œque
primo secundo et tertio quarto quinto
et septimo,â€� T becomes â€œtricentissimoâ€� and so on. The Greek
MS. which he used was evidently written in uncials, for DEZTh becomes in
one place â€œquod autem septimo nono,â€�
showing that he mistook DE for the particle de, and kai ho STU is rendered â€œsicut tricentissimo et
quadringentissimo,â€� showing that the letters must have been written
KAIOCTU.

The date of the Englishman
Athelhard (Ã†thelhard) 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.), NÃ¼rnberg (Johannes
Regiomontanus' copy) and Erfurt.

Among the very numerous works of Gherard of Cremona
(1114-- 1187) are mentioned translations of â€œ15 Books of Euclidâ€�
and of the *Data*^{12} . Till recently this translation of the
*Elements* was supposed to be lost; but Axel
Anthon
BjÃ¶rnbo 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 BjÃ¶rnbo 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 â€œsuperficies
equidistantium laterum et rectorum angulorum,â€� found also in Gherard's
translation of an-NairÄ©zÄ©, where Athelhard says â€œparallelogrammum
rectangulum.â€� The translation is much clearer than Athelhard's: it is
neither abbreviated nor â€œeditedâ€� 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 ThÃ£bit's version. It contains several notes quoted
from ThÃ£bit himself (*Thebit dixit*), e.g. about alternative proofs
etc. which ThÃ£bit found â€œin another Greek
MS.,â€� and is therefore a further testimony to ThÃ£bit'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 ThÃ£bit version before him with
other versions as carefully as ThÃ£bit collated the Greek
MSS. Lastly, the new editor speaks of â€œThebit qui transtulit hunc librum
in arabicam linguamâ€� and of â€œtranslatio Thebit,â€� which may tend to confirm
the statement of al-QiftÄ© who credited ThÃ£bit with an independent
translation, and not (as the *Fihrist* does) with a mere improvement
of the version of Is[hnull]Ã£q b. Hunain.

Gherard's translation of the Arabic commentary of an-NairÄ©zÄ© 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 Euclid^{15} : 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 time^{16} , 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 1281^{17} . 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 references^{18} and some particulars. It appears that there
is a MS. at Munich
(Cod. lat. Mon. 13021) written by Sigboto in the 12th c. at PrÃ¼fning 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 â€œelinuam,â€� the Arabic term for â€œrhombus,â€� 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
â€œelinuamâ€� 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 â€œelinuam,â€� 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 â€œBoethius.â€� 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
verses^{19} : â€œThe 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,â€� 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, â€œCampani est hec,â€� â€œdubito an demonstret hic
Campanusâ€� 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 itself^{20} . 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 Dante^{21} . 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' translation^{23} . 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. KÃ¤stner^{24} 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 letters^{25} . 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 KÃ¤stner'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
WindischgrÃ¤tz employed woodcut mathematical figures in printing Oresme's
*De latitudinibus*^{26} . 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 title^{27} , as well as from his prefaces to the
*Catoptrica* and *Data*, with their allusions to previous
translators â€œwho take some things out of authors, omit some, and change
some,â€� or â€œto that most barbarous translatorâ€� who filled a volume
purporting to be Euclid's â€œwith extraordinary scarecrows, nightmares and
phantasies,â€� 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 â€œhelmuainâ€� and â€œhelmuaripheâ€� 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 edition^{28} begins thus: â€œThe 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.â€�* 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 â€œ*of both languages*â€� (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 â€œthat
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^{29} .â€� 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. 5^{30} . 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 proofs^{31} . The date is not on the title-page nor at
the **[p. 100]** end, but the letter of dedication to FranÃ§ois
Briconnet by Jacques
LefÃ¨vre 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 â€œa Greek
copyâ€� (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 STOICHEIÃ”N BIBL<*> IE<*><*>
EK TÃ”N THEÃ”NOS SUNOUSIÃ”N. Eis tou autou to prÃ´ton, exÃªgÃªmatÃ´n 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 TÃ¼bingen, taught last of all at Basel,
where theology was his main subject. His â€œpraefatiunculaâ€� 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 arithmetic^{32} as â€œa farewell to the sciences,â€� 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 â€œLazarus Bayfiusâ€� (Lazare
de BaÃ¯f, then the ambassador of the King of France at Venice),
the other at Paris
by â€œIoann. Rvelliusâ€� (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 â€œIoann. Claymundusâ€� 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 â€œthe other copy,â€� this â€œother copyâ€� is as a rule the Paris
MS., though sometimes the reading of the Paris
MS. is taken into the text and the â€œother copyâ€� 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 â€œZampertusâ€� or to â€œlatinum exemplarâ€�
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*) â€œdemonstrations on the first six books of
Euclid's elements of geometry,â€� â€œin which the Greek
text of Euclid
himself is inserted in its proper places, with the Latin
translation of Barth.
Zamberti of Venice,â€�
which seems to imply that only the enunciations were given in Greek.
The preface, from which KÃ¤stner quotes^{33} , 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 â€œpurged them of six hundred things which did not
seem to savour of the almost divine genius and the perspicuity of Euclid^{34} â€�

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
â€œtogether with true and appropriate proofs of the propositions, without
the use of lettersâ€� (i.e. letters denoting points in the figures), the
various straight lines and angles being described in words^{35} .

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
MondorÃ© (Petrus
Montaureus), published separately at Paris
in 1551. Gracilis also added a few â€œscholia.â€�

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 â€œvocabula quaedam geometricaâ€� 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, â€œcorrected in many placesâ€� (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^{36} . 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 French^{37} . 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 Berlin^{38} . 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 RamÃ©e, 1515-1572) is credited with a translation of Euclid
which appeared in 1545 and again in 1549 at Paris^{39} . 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 â€œGreek
text of Euclidâ€�;
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 â€œangleâ€� of
contact.

1559. Johannes
Buteo, or Borrel (1492-1572), published in an appendix to his book *De
quadratura circuli* some notes â€œon the errors of Campanus, Zambertus,
Orontius, Peletarius, Pena, interpreters of Euclid.â€�
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 (FranÃ§ois de Foix, Comte de Candale, 1502-1594) â€œrestoredâ€� 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 â€œchiefly after the text of Simson.â€� Simson's first (Latin) edition (1756) has â€œex versione Latina Federici Commandiniâ€� 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 â€œscholia antiquaâ€� from a
MS. of the class of Vat. 192 containing the scholia distinguished by Heiberg
as â€œSchol. Vat.â€� 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 Clavius^{40} (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 â€œcommentaries of Theon,â€�
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, â€œa geometer not
of the common sort, who has lately restored Euclid,
in a Latin
translation, to his original brilliancy.â€� 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. AndrÃ© 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
FranÃ§ois 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
1565^{41} , 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 â€œthe first
translation by Campano,â€� â€œthe second made by Bartolomeo Zamberto Veneto
who is still alive,â€� â€œthe editions of Paris
or Germany
in which they have included both the aforesaid translations,â€� and â€œour own
translation into the vulgar (tongue).â€�

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

1613. The first six Books â€œreduced to practiceâ€� 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 Scheubel^{42} (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), â€œfor generals, engineers etc.â€� â€œproved in a new
and quite easy manner.â€� Other editions 1699, 1744.

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

1714. *Euclidis* XV *BÃ¼cher teutsch*, â€œtreated in a special
and brief manner, yet completely,â€� 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 fÃ¼nfzehn BÃ¼cher* 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
â€œsetting-outâ€� into one.

1807. Books I.--VI., XI., XII. â€œnewly translated from the Greek,â€� by J. K. F. Hauff.

1828. The same Books by Joh. Jos. Ign. Hoffmann â€œas guide to instruction in elementary geometry,â€� 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 BÃ¼cher.*

#### VI. French.

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

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 â€œdemonstrated by symbols, by a method very brief and intelligible,â€� by Pierre HÃ©rigone, mentioned by Barrow as the only editor who, before him, had used symbols for the exposition of Euclid.

1672. Eight Books â€œrendus plus facilesâ€� by Claude
FranÃ‡is Milliet Dechales, who also brought out *Les Ã©lÃ©mens d'Euclide
expliquÃ©s d'une maniÃ¨re nouvelle
et trÃ¨s 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. KÃ¤stner, 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, â€œThe Propositions of the Books of Euclid's Elementsâ€�; the fifteen Books in this version â€œenlargedâ€� by Jakob van Leest in 1662.

1695. C. J. Vooght, fifteen Books complete, with Candalla's â€œ16th.â€�
**[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

Imprinted at London by

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 byM. I. Dee,specifying the chiefe Mathematicall SciÄ“ces,what they are,and whereunto commodious: where,also,are disclosed certaine new Secrets Mathematicall and Mechanicall,vntill these our daies,greatly missed.

*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. â€œWherefore 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 toÅ©ge, 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 MathematiciÄ“s, both of old time, and in our age: as by diligent reading it in course, ye shall well perceaue....â€�

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 â€œsixteenthâ€� 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 â€œcustomers,â€� 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 â€œthe whole fifteen books compendiously demonstratedâ€�; 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' â€œElements of Euclid explained in a new but most easy methodâ€� (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 â€œalgebraicalâ€� **[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 â€œex versione latina Federici Commandini,â€� 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 â€œthe translation is much amended by the friendly assistance of a learned gentleman.â€�

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* â€œin like manner correctedâ€� 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 â€œM.A. Fellow of Hertford College,â€� and in the second (London, printed by T. Spilsbury, 1788) as â€œB.D.â€� 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 â€œthe first six Books of
Euclid
with two Books on the Geometry of Solids.â€� 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 â€œon the theory of parallel lines,â€� 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 Ã‡amorano.

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. MÃ¥rten StrÃ¶mer, 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. BrÃ¥kenhjelm, 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
Ã¼ber Menelaos'
SphÃ¤rik* (*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 fÃ¼r 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 Ã¼ber die
Fortschritte der classischen Alterthumswissenschaft*, XL. (188_{4},
III.) pp. 19mdash;22; Heiberg
in *Zeitschrift fÃ¼r 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 . CaleÃ±
. JuÃ± . Lector . Vale.

^{24} KÃ¤stner, *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 fÃ¼r Math. u.
Physik*, XX., hist.-litt. Abth. p. 58.

^{27} The title begins thus: â€œEuclidis
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: â€œQuantitates quae dicuntur continuam habere
proportionalitatem, sunt, quarum equÃ¨ multiplicia aut equa sunt aut equÃ¨
sibi
sine interruptione addunt aut minuuntâ€� (!), to which Campanus adds the
note: â€œContinuÃ¨ proportionalia sunt quorum omnia multiplicia equalia sunt
continuÃ¨ 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 â€œex
Campani...deinde Theonis...et Hypsiclis...*traditionibus*.â€� For
description see Weissenborn,
p. 56 sqq.

^{32} *De artÃ© supputandi libri quatuor*.

^{33} KÃ¤stner, I. p. 260.

^{34} Heiberg,
vol. V. p. cvii.

^{35} KÃ¤stner, I. p. 359.

^{36} EUKLEIDOU TA SÃ”ZOMENA. 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 FranÃ§ais d'aprÃ¨s un manuscrit trÃ¨s-ancien, qui Ã©tait
restÃ© inconnu jusqu'Ã nos jours* Par F. Peyrard. Ouvra<*>e
approuvÃ© 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
integritÃ 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 facilitÃ serÃ capace a poterlo intendere*. In Venetia,
Appresso Curtio Troiano, 1565.

^{42} *Das sibend acht und neunt buch
des hochberÃ¼mbten Mathematici Euclidis Megarensis...* *durch
Magistrum Fohann Scheybl*, *der lÃ¶blichen universitet zu
TÃ¼bingen*, *des Euclidis und Arithmetic Ordinarien*, *auss dem
latein ins teutsch gebracht*....

^{43} .*Die sechs erste b'Ã¼cher Euclidis vom
anfang oder
grund der Geometrj*...*Auss Griechischer sprach in die TeÃ¼tsch
gebracht aigentlich erklÃ¤rt*...*Demassen vormals in TeÃ¼tscher sprach
nie gesehen worden*...*Durch
Wilhelm
Holtzman genant Xylander von Augspurg*. *Getruckht zu Basel.*