() of ALEXANDRIA. The
length of this article will not be blamed by any one who considers that, the sacred writers
excepted, no Greek has been so much read or so variously translated as Euclid. To this it may
be added, that there is hardly any book in our language in which the young scholar or the
young mathematician can find all the information about this name which its celebrity would
make him desire to have.
Euclid has almost given his own name to the science of geometry, in every country in which
his writings are studied; and yet all we know of his private history amounts to very little.
He lived, according to Proclus (Comm. in Eucl. 2.4), in the time of the
first Ptolemy,
Euclid, says Proclus, was younger than Plato, and older than Eratosthenes and Archimedes,
the latter of whom mentions him. He was of the Platonic sect, and well read in its doctrines.
He collected the Elements, put into order much of what Eudoxus had done, completed many things
of Theaetetus, and was the first who reduced to unobjectionable demonstration the imperfect
attempts of his predecessors. It was his answer to Ptolemy, who asked if geometry could not be
made easier, that there was no royal road (Ep. 91, cited by August) of Alexander.
Pappus (lib. vii. in praef.) states that Euclid was distinguished by
the fairness and kindness of his disposition, particularly towards those who could do anything
to advance the mathematical sciences: but as he is here evidently making a contrast to
Apollonius, of whom he more than insinuates a directly contrary character, and as he lived
more than four centuries after both, it is difficult to give credence to his means of knowing
so much about either. At the same time we are to remember that he had access to many records
which are now lost. On the same principle, perhaps, the account of Nasir-eddin and other
Easterns is not to be entirely rejected, who state that Euclid was sprung of Greek parents,
settled at Tyre; that he lived, at one time, at Damascus; that his father's name was
Naucrates, and grandfather's Zenarchus. (August, who cites Gartz, De Interpr.
Eucl. Arab.) It is against this account that Eutocius of Ascalon never hints at it.
At one time Euclid was universally confounded with Euclid of Megara, who lived near a
century before him, and heard Socrates. Valerius Maximus has a story (8.12) that those who
came to Plato about the construction of the celebrated Delian altar were referred by him to
Euclid the geometer. This story, which must needs be false, since Euclid of Megara, the
contemporary of Plato, was not a geometer, is probably the crigin of the confusion. Harless
thinks that Eudoxus should be read for Euclid in the
passage of Valerius.
In the frontispiece to Whiston's translation of Tacquet's Euclid there is a bust, which is
said to be taken from a brass coin in the possession of Christina of Sweden; but no such coin
appears in the published collection of those in the cabinet of the queen of Sweden. Sidonius
Apollinaris says (Epist. 11.9) that it was the custom to paint Euclid
with the fingers extended (laxatis,) as if in the act of measurement.
The history of geometry before the time of Euclid is given by Proclus, in a manner which
shews that he is merely making a summary of well known or at least generally received facts.
He begins with the absurd stories so often repeated, that the Aegyptians were obliged to
invent geometry in order to recover the landmarks which the Nile destroyed year by year, and
that the Phoenicians were equally obliged to invent arithmetic for the wants of their
commerce. Thales, he goes on to say, brought this knowledge into Greece, and added many
things, attempting some in a general manner (physical discovery in geometry, by measurement of instances. Next is mentioned
Ameristus, the brother of Stesichorus the poet. Then Pythagoras changed it into the form of a
liberal science (
Barocius, whose Latin edition of Proclus has been generally followed, singularly enough
translates quae non exlpicari
possunt, and Taylor follows him with " such things as cannot be explained." It is
strange that two really learned editors of Euclid's commentator should have been ignorant of
one of Euclid's technical terms. Then come Anaxagoras of Clazomenae, and a little after him
Oenopides of Chios; then Hippocrates of Chios, who squared the lunule, and then Theodorus of
Cyrene. Hippocrates is the first writer of elements who is recorded. Plato then did much for
geometry by the mathematical character of his writings; then Leodamos of Thasus, Archytas of
Tarentum, and Theaetetus of Athens, gave a more scientific basis (EUDOXUS], increased the number of general theorems, added three proportions to the
three already existing, and in the things which concern the section (of the cone, no doubt)
which was started by Plato himself, much increased their number, aud employed analyses upon
them. Amyclas Heracleotes, the companion of Plato, Menaechmus, the disciple of Eudoxus and of
Plato, and his brother Deinostratus, made geometry more perfect. Theudius of Magnesia
This history of Proclus has been much kept in the background, we should almost say
discredited, by editors, who seem to wish it should be thought that a finished and
unassailable system sprung at once from the brain of Euclid; an armed Minerva from the head of
a Jupiter. But Proclus, as much a worshipper as any of them, must have had the same bias, and
is therefore particularly worthy of confidence when he cites written history as to what was
not done by Euclid. Make the most we can of his preliminaries, still the
thirteen books of the Elements must have been a tremendous advance, probably even greater than
that contained in the Principia of Newton. But still, to bring the state of our opinion of
this progress down to something short of painful wonder, we are told that demonstration had
been given, that something had been written on proportion, something on incommensurables,
something on loci, something on solids; that analysis had been applied, that the conic
sections had been thought of, that the Elements had been distinguished from the rest and
written on. From what Hippocrates had done, we know that the important property of the
right-angled triangle was known; we rely much more on the lunules than on the story about
Pythagoras. The dispute about the famous Delian problem had arisen, and some conventional
limit to the instruments of geometry must have been adopted; for on keeping within then, the
difficulty of this problem depends.
It will be convenient to speak separately of the
Some of the characteristics of the Elements are brietly as follows:
First. There is a total absence of distinction between the various ways in which we know the meaning of terms: certainty, and nothing more, is the thing sought. The definition of straightness, an idea which it is impossible to put into simpler words, and which is therefore described by a more difficult circumlocution, comes under the same heading as the explanation of the word "parallel." hence disputes about the correctness or incorrectness of many of the definitions.
Secondly. There is no distinction between propositions which require demonstration, and those which a logician would see to be nothing but different modes of starting a preceding proposition. When Euclid has proved that everything which is not A is not B, he does not hold himself entitled to infer that every B is A, though the two propositions are identically the same. Thus, having shewn that every point of a circle which is not the centre is not one from which three equal straight lines can be drawn, he cannot infer that any point from which three equal straight lines are drawn is the centre, but has need of a new demonstration. Thus, long before lie wants to use book i. prop. 6, he has proved it again, and independently.
Thirdly. He has not the smallest notion of admitting any generalized use of a word, or of parting with any ordinary notion attached to it. Setting out with the conception of an angle rather as the sharp corner made by the meeting of two lines than as the magnitude which he afterwards shews how to measure, he never gets rid of that corner, never admits two right angles to make one angle, and still less is able to arrive at the idea of an angle greater than two right angles. And when, in the last proposition of the sixth book, his definition of proportion absolutely requires that he should reason on angles of even more than four rifht angles, he takes no notice of this necessity, and no one cantellwhether it was an overshigt, whether Euclid thought the extension one which the student could make for himself, or whether (which has sometimes struck us as not unlikely) the elements were his last work, and he did not live to revise them.
In one solitary case, Euclid seems to have made an omission implying that he recognized
that natural extension of language by which unity is considered as a
number, and Simson has thought it necessary to supply the omission
(see his book v. prop. A), and has shewn himself more Euclid tian Euclid upon the point of
all others in which Euclid's philosophy is defective.
Fourthly. There is none of that attention to the forms of accuracy with which translators
have endeavoured to invest the Elements, thereby giving them that appearance which has made
many teachers think it meritorious to insist upon their pupils remembering the very words of
Simson. Theorems are found among the definitions : assumptions
Fifthly. Euclid never condescends to hint at the reason why he finds himself obliged to adopt any particular course. Be the difficulty ever so great, he removes it without mention of its existence. Accordingly, in many places, the unassisted student can only see that much trouble is taken, without being able to guess why.
What, then, it may be asked, is the peculiar merit of the Elements which has caused them to retain their ground to this day? the answer is, that the preceding objections refer to matters which can be easily mended, without any alteration of the main parts of the work, and that no one has ever given so easy and natural a chain of geometrical consequences. There is a never erring truth in the results; and, though there may be here and there a self-evident assumption used in demonstration, but not formally noted, there is never any the smallest departure from the limitations of construction which geometers had, from the time of Plato, imposed upon themselves. The strong inclination of editors, already mentioned, to consider Euclid as perfect, and all negligences as the work of unskilful commentators or interpolators, is in itself a proof of the approximate truth of the character they give the work; to which it may be added that editors in general prefer Euclid as he stands to the alterations of other editors.
The Elements consist of thirteen books written by Euclid, and two of which it is supposed that Hypsicles is the author. The first four and the sixth are on plane geometry; the fifth is on the theory of proportion, and applies to magnitude in general; the seventh, eighth, and ninth, are on arithmetic; the tenth is on the arithmetical characteristics of the divisions of a straight line; the eleventh and twelfth are on the elements of solid geometry; the thirteenth (and also the fourteenth and fifteenth) are on the regular solids, which were so much studied among the Platonists as to bear the name of Platonic, and which, according to Proclus, were the objects on which the Elements were really meant to be written.
At the commencement of the first book, under the name of definitions (common notions (axiom (which was
not used till after Euclid) to them all. The intention of Euclid seems to have been, to
disinguish between that which his reader must grant, or seek another system, whatever may be
his opinion as to the propriety of the assumption, and that which there is no question every
one will grant. The modern editor merely distinguishes the assumed problem (or construction) from the assumed theorem. Now there is
no such distinction in Euclid as that of problem and theorem ; the common term proposition, includes
both, and is the only one used. An immense preponderance of manuscripts, the testimony of
Proclus, the Arabic translations, the summary of Boethius, place the assumptions about right
angles and parallels (and most of them, that about two straight lines) among the postulates;
and this seems most reasonable, for it is certain that the first two assumptions can have no
claim to rank among common notions or to be placed in the same list with " the whole is
greater than its part."
Without describing mintutely the contents of the first book of the Elements, we may
observe that there is an arrangement of the propositions, which will enable any teacher to
divide it into sections. Thus propp. 1-3 extend the power of construction to the drawing of
a circle with any centre and any radius; 4-8 are the basis of the theory of equal triangles;
9-12 increase the power of construction; 13-15 are solely on relatiolis of angles; 16-21
examine the relations of parts of one triangle; 22-23 are additional constructios ; 23-26
augment the doctrine of equal triangles; 27-31 contain the theory of parallels; Penni Cyclopaedia, art. "
Parallels,"' for some account of this well-worn subject.
The second book shews those properties of the rectangles contained by the parts of divided straight lines, which are so closely connected with the common arithmetical operations of multiplication and division, that a student or a teacher who is not fully alive to the existence and difficulty of incommensurables is apt to think that common arithmetic would be as rigorous as geometry. Euclid knew better.
The third book is devoted to the consideration of the properties of the circle, and is
much cramped in several places by the imperfect idea already alluded to, which Euclid took
of an angle. There are some places in which he clearly drew upon experimental knowledge of
the form of a circle,
The fourth book treats of regular figures. Euclid's original postulates of construction give him, by this time, the power of drawing them of 3, 4, 5, and 15 sides, or of double, quadruple, &c., any of these numbers, as 6, 12, 24, &c., 8, 16;, &c. &c.
The fifth book is on the theory of proportion. It refers to all kinds of magnitude, and is wholly independent of those which precede. The existence of incommensurable quantities obliges him to introduce a definition of proportion which seems at first not only difficult, but uncouth and inelegant; those who have examined other definitions know that all which are not defective are but various readings of that of Euclid. The reasons for this difficult definition are not alluded to, according to his custom; few students therefore understand the fifth book at first, and many teachers decidedly object to make it a part of the course. A distinction should be drawn between Euclid's definition and his manner of applying it. Every one who understands it must see that it is an application of arithmetic, and that the defective and unwieldy forms of arithmetical expression which never were banished from Greek science, need not be the necessary accompaniments of the modern use of the fifth book. For ourselves, we are satisfied that the only rigorous road to proportion is either through the fifth book, or else through something much more difficult than the fifth book need be.
The sixth book applies the theory of proportion, and adds to the first four books the
propositions which, for want of it, they could not contain. It discusses the theory of
figures of the same form, technically called similar. To give an idea
of the advance which it makes, we may state that the first book has for its highest point of
constructive power the formation of a rectangle upon a given base, equal to a given
rectilinear figure; that the second book enables us to turn this rectangle into a square;
but the sixth book empowers us to make a figure of any given rectilinear shape equal to a
rectilinear figure of given size, or briefly, to construct a figure of the form of one given
figure, and of the size of another. It also supplies the geometrical form of the solution of
a quadratic equation.
The seventh, eighth, and ninth books cannot have their subjects usefully separated. They treat of arithmetic, that is, of the fundamental properties of numbers, on which the rules of arithmetic must be founded. But Euclid goes further than is necessary merely to construct a system of computation, about which the Greeks had little anxiety. He is able to succeed in shewing that numbers which are prime to one another are the least in their ratio, to prove that the number of primes is infinite, and to point out the rule for constructing what are called perfect numbers. When the modern systems began to prevail, these books of Euclid were abandoned to the antiquary: our elementary books of arithmetic, which till lately were all, and now are mostly, systems of mechanical rules, tell us what would have become of geometry if the earlier books had shared the same fate.
The tenth book is the development of all the power of the preceding ones, geometrical and arithmetical. It is one of the most curious of the Greek speculations : the reader will find a synoptical account of it in the
The eleventh and twelfth books contain the elements of solid geosnetry, as to prisms,
pyramids, &c. The duplicate ratio of the diameters is shewn to be that of two circles,
the triplicate ratio that of two spheres. Instances occur of the method of
exhaustions, as it has been called, which in the. hands of Archimedes became an
instrument of discovery, producing results which are now usually referred to the
differential calculus: while in those of Euclid it was only the mode of proving propositions
which must have been seen and believed before they were proved. The method of these books is
clear and elegant, with some striking imperfections, which have caused many to abandon them,
even among those who allow no substitute for the first six books. The thirteenth,
fourteenth, and fifteenth books are on the five regular solids: and even had they all been
written by Euclid (the last two are attributed to Hypsicles), they would but ill bear out
the assertion of Proclus, that the regular solids were the objects with a view to which the
Elements were written : unless indeed we are to suppose that Euclid died before he could
complete his intended structure. Proclus was an enthusiastic Platonist: Euclid was of that
school; and the former accordingly attributes to the latter a particular regard for what
were sometimes called the Platonic bodies. But we think that the author himself of the
Elements could hardly have considered them as a mere introduction to a favourite speculation
: if he were so blind, we have every reason to suppose that his own contemporaries could
have set him right. From various indications, it can be collected that the fame of the
Elements was almost coeval with their publication; and by the time of
The
While on this subject, it may be useful to remind the reader how difficult it is to judge of the character of Euclid's writings, as far as his own merits are concerned, ignorant as we are of the precise purpose with which any one was written. For instance: was he merely shewing his contemporaries that a connected system of demonstration might be made without taking more than a certain number of postulates out of a collection, the necessity of each of which had been advocated by some and denied by others? We then understand why lie placed his six postulates in the prominent position which they occupy, and we can find no fault with his tacit admission of many others, the necessity of which had perhaps never been questioned. But if we are to consider him as meaning to be what his commentators have taken him to be, a model of the most scrupulous formal rigour, we can then deny that he has altogether succeeded, though we may admit that he has made the nearest approach.
The literary history of the writings of Euclid would contain that of the rise and progress of geometry in every Christian and Mohammedan nation: our notice, therefore, must be but slight, and various points of it will be confirmed by the bibliographical account which will follow.
In Greece, including Asia Minor, Alexandria, and the Italian colonies, the Elements soon
became the universal study of geometers. Commentators were not wanting; Proclus mentions
Heron and Pappus, and Aeneas of Hierapolis, who made an epitome of the whole. Theon the
younger (of Alexandria) lived a little before Proclus (who died about edition of Euclid, with
some slight additions and alterations: he tells us so himself, and uses the word Comment. in priora Analyt.
Aristot.) mentions as the fourth of the tenth book that which is the fifth in all
manuscripts. Again, in several manuscripts the whole work is headed as
The two books of Geometry left by BOETHIUS contain nothing but
enunciations and diagrams from the first four books of Euclid. The assertion of Boethius that
Euclid only arranged, and that the discovery and demonstration were the work of others,
probably contributed to the notions about Then presently described. Until the restoration of
the Elements by translation from the Arabic, this work of Boethius was the only European
treatise on geometry, as far as is known.
The Arabic translations of Euclid began to be made under the caliphs Haroun al Raschid and Al Mamun; by their time, the very name of Euclid had almost disappeared from the West. But nearly one hundred and fifty years followed the capture of Egypt by the Mohammeddans before the latter began to profit by the knowledge of the Greeks. After this time, the works of the geometers were sedulously translated, and a great impulse was given by them. Commentaries, and even original writings, followed; but so few of these are known among us, that it is only from the Saracen writings on astronomy (a science which always carries its own history along with it) that we can form a good idea of the very striking progress which the Mohammedans nade under their Greek teachers. Some writers speak slightingly of this progress, the results of which they are too apt to compare with those of our own time: they ought rather to place the Saracens by the side of their own Gothic ancestors, and, making some allowance for the more advantageous circumstances under which the first started, they should view the second systematically dispersing the remains of (Greek civilization, while the first were concentrating the geometry of Alexandria, the arithmetic and algebra of India, and the astronomy of both, to formn a nucleus for the present state of science.
The Elements of Euclid were restored to Europe by translation from the Arabic. In
connection With this restoration four Eastern editors may be
The first European who found Euclid in Arabic, and translated the Elements into Latin, was Athelard or Adelard, of Bath, who was certainly alive in 1130. (See "Adelard," in the
The seed thus brought by Adelard into Europe was sown with good effect. In the next century
Roger Bacon quotes Euclid, and when he cites Boethius, it is not for his geometry. Up to the
time of printing, there was at least as much dispersion of the Elements as of any other book
: after this period, Euclid was, as we shall see, an early and frequent product of the press.
Where science flourished, Euclid was found; and wherever he was found, science flourished
more or less according as more or less attention was paid to his Elements. As to writing
another work on geometry, the middle ages would as soon have thought of composing another New
Testament: not only did Euclid preserve his right to the title of solo introdultore delle scientie mathematice: and algebra was not at that
time considered as entitled to the name of a science by those who had been formed on the
Greek model; "arte maggiore" was its designation. The story about Pascal's discovery of
geometry in his boyhoud (A. D. . 1635) contains the statement that he had got "as far as the
32nd proposition of the first book" before he wits detected, the exaggerators (for much
exaggerated this very circumstance shews the truth must have been) not having the slightest
idea that a new invented system could proceed in any other order than that of Euclid.
The vernacular translations of the Elements date from the middle of the sixteenth
century,from which time the history of mathematical science divides itself into that of the
several countries where it flourished. By slow steps, the continent of Europe has almost
entirely abandoned the ancient Elements, and substituted systems of geometry more in
accordance with the tastes which algebra has introduced : but in England, down to the present
time, Euclid has held his ground. There is not in our country any system of geometry twenty
years old, which has pretensions to anything like currency, but it is either Euclid, or
something so fashioned upon Euclid that the resemblance is as close as that of some of his
professed editors. We cannot here go into the reasons of our opinion; but we have no doubt
that the love of accuracy in mathematical reasoning has declined wherever Euclid has been
abandoned. We are not so much of the old opinion as to say that this must necessarily have
happened; but, feeling quite sure that all the alterations have had their origin in the
desire for more facility than could be obtained by rigorous deduction from postulates both
true and evident, we see what has happened, and why, without being at all inclined to dispute
that a disposition to depart from the letter, carrying off the spirit, would have been
attended with very different results. Of the two best foreign books of geometry which we
know, and which are not Euclidean, one demands a right to "imagine" a thing which the writer
himself knew perfectly well was not true; and the other is content to shew that the theorems
are so nearly true that their error, if any, is imperceptible to the senses. It must be
admitted that both these absurdities are committed to avoid the fifth book, and that English
teachers have, of late years, been much inclined to do something of the same sort, less
openly. But here, at least, writers have left it to teachers to shirk
At the time of the invention of printing, two errors were current with respect to Euclid
personally. The first was that he was Euclid of Megara, a totally different person. This
confusion has been said to take its rise from a passage in Plutarch, but we cannot find the
reference. Boethius perpetuated it. The second was that Theon was the demonstrator of all the
propositions, and that Euclid only left the definitions, postulates, &c., with the Euclid, so called, contained only enunciations; all that
contained demonstrations were said to be Euclid with the commentary of
Theon, Campanus, Zambertus, or some other. Also, when the enunciations were given in Greek
and Latin, and the demonstrations in Latin only, this was said to constitute an edition of
Euclid in the original Greek, which has occasioned a host of bibliographical errors. We have
already seen that Theon did edit Euclid, and that manuscripts have described this editorship
in a manner calculated to lead to the mistake : but Proclus, who not only describes Euclid as
Praelectiones tresdecim in principium elementorum Euclidis; Oxonü habitae
M.DC.XX. Oxoniae, 1621.quidam
stulti et perridiculi, above discussed: the second, that of Peter Ramus, who held the
whole to be absolutely due to Theon, propositions as well as demonstrations, false, quis negat? the third, that of Buteo of Dauphiny, a geometer of merit, who
attributes the whole to Euclid, quae opinio aut vera est, aut veritati certe
proxima. It is not useless to remind the classical student of these things : the middle
ages may be called the "ages of faith " in their views of criticism. Whatever was written was
received without examination ; and the endorsement of an obscure scholiast, which was perhaps
the mere whim of a transcriber, was allowed to rank with the clearest assertions of the
commentators and scholars who had before them more works, now lost, written by the
contemporaries of the author in question, than there were letters in the stupid sentence
which was allowed to overbalance their testimony. From such practices we are now, it may well
be hoped, finally delivered: but the time is not yet come when refntation of " the scholiast
" may be safely abandoned.
All the works that have been attributed to Euclid are as follows:
1.
2.
3. Treatise on Music ; and 4. the Division of the Scale
: one of these works, most likely the former, must be rejected. Proclus says that
Euclid wrote
5.
6. on Optics ; and 7.
on Catoptrics.
Proclus mentions both.
The preceding works are in existence; the following are either lost, or do not remain in the original Greek.
8. On
Divisions. Proclus (l.c.) There is a translation from the Arabic,
with the name of Mohammed of Bagdad attached, which has been suspected of being a translation
of the book of Euclid : of this we shall see more.
9. Four books on
Conic Sections. Pappus (lib. vii. praef.) affirms that Euclid
wrote four books on conics, which Apollonius enlarged, adding four others. Archimedes refers
to the elements of conic sections in a manner which shews that he could
not be mentioning the new work of his contemporary Apollonius (which it is most likely he
never saw). Euclid may possibly have written on conic sections; but it is impossible that the
first four books of APOLLONIUS (see his life) can have been those
of Euclid.
10. Three
books of Porisms. These are mentioned by Proclus and by Pappus (l.c.), the latter of whom gives a description which is so corrupt as to be
unintelligible.
11. Two books on Plane Loci. Pappus mentions these, but not Eutocius, as Fabricius
affirms. (Comment. in Apoll. lib. i. lemm.)
12. Loci ad Superfuiem, were, neither Pappus nor Eutocius inform us; the latter
says they derive their name from their own
13.
We now proceed to some bibliographical account of the writings of Euclid. In every case in which we do not mention the source of information, it is to be presumed that we take it from the edition itself.
preface to the 14th book, by which it is made
almost certain that Euclid did not write it (for Euclid's books have no prefaces) is
omitted. Its Arabic origin is visible in the words helmuaym and helmuariphe, which are used for a rhombus and a trapezium. This edition is
not very scarce in England; we have seen at least four copies for sale in the last ten
years.
Megarensis.
Specularia and Perspectiva), and the Data with the preface of Marinus, being the editio
princeps of all but the Elements, has the title Euclidis Megarensis philosophici Platonici,
mathematicarum disciplinarū janitoris : habent in hoc votumine quicūque ad
mathematicā substantiā aspirāt : elemētorum libros, (&c. &c. Zamberto Veneto Interprete. At the end is Impressum
Venetis, &c. in edibus Joannis Tacuini, &c., M. D. V. VIII. Klendas Novēbris
-- that is, 1505, often read 1508 by an obvious mistake.exposition of Theon or Hypsicles, by which he probably means that Theon or Hypsicles
gave the demonstrations. The preceding editors, whatever their opinions may have been, do
not expressly state Theon or any other to have been the author of the demonstrations : but
by 1505 the Greek manuscripts which bear the name of Theon had probably come to light. For
Zambertus Fabricius cites Goetz mem. bibl. Dresd. ii. p. 213: his edition is beautifully
printed, and is rare. He exposes the translations from the Arabic with unceasing severity.
Fabricius mentions (from Scheibel) two small works, the four books of the Elements by Ambr.
Jocher, 1506, and something called "Geometria Euclidis," which accompanies an edition of
Sacrobosco, Paris, H. Stephens, 1507. Of these we know nothing.
Euclidis Megarensis philosophi aculissimi mathematicorumque omnium sine
controversia principis opera, &c. At the end, Venetüs
impressum per ... Payaninum de Payaninis ... anno...MDVIIII... Paciolus adopts the Latin of Adelard, and occasionally quotes the comment
of Campanus, introducing his own additional comments with the head " Castigator." He opens
the fifth book with the account of a lecture which he gave on that book in a church at
Venice, August 11, 1508, giving the names of those present, and some subsequent laudatory
correspondence. This edition is less loaded with comment than either of those which precede.
It is extremely scarce, and is beautifully printed : the letter is a curious intermediate
step between the old thick black letter and that of the Roman type, and makes the derivation
of the latter from the former very clear.
With these editions the ancient series, as we may call it, terminates, meaning the complete Latin editions which preceded the publication of the Greek text. Thus we see five folio editions of the Elements produced in thirty-four years.
so far as it
remains; all Greek, without Latin. On Grynoeus and his reverend i. e. at
Oxford), he visited and studied in most of the libraries, searched after rare books of the
Greek tongue, particularly after some of the books of commentaries of Proclus Diadoch.
Lycius, and having found several, and the owners to be careless of them, he took some
away, and conveyed them with him beyond the seas, as in an epistle by him written to John
the son of Thos. More, he confesseth." Wood.Athen. Oxon. in verb.) The Oxford editor is studiously
silent about this Basle edition, which, though not obtained from many manuscripts, is even
now of some value, and was for a century and three-quarters the only printed Greek text of
all the books.
With regard to Greek texts, the student must be on his guard against bibliographers. For
instance, Harless Italian, being " I quindici libri degli element di Euclide,
di Greco tradotti in lingua Thoscana." Here is another instance in
which the editor believed he had given the whole of Euclid in giving the enunciations. From
this edition another Greek text, Florence, 1545, was invented by another mistake. All the
Greek and Latin editions which Fabricius, Murhard, &c., attribute to Dasypodius (Conrad
Rauchfuss), only give the enunciations in Greek. The same may be said of Euclid, Gr. Lat. thirteen books in the title-page, but contains only six in all copies that
we have seen : it is attributed to the celebrated mathematician Briggs.
any single known manuscript.
This document was part of the plunder sent from Rome to Paris by Napoleon, and had belonged
to the Vatican Library. When restitution was enforced by the allied armies in 1815, a
special permission was given to Peyrard to retain this manuscript till he had finished the
edition on which he was then engaged, and of which one volume had already appeared. Peyrard
was a worshipper of this manuscript, No. 190, and had a contempt for all previous editions
of Euclid. He gives at the end of each volume a comparison of the Paris edition with the
Oxford, specifying what has been derived from the Vatican manuscript, and making a selection
from the various readings of the other 22 manuscripts which were before him. This edition is
therefore very valuable; but it is very incorrectly printed: and the editor's strictures
upon his predecessors seem to us to require the support of better scholarship than he could
bring to bear upon the subject. (See the
The Elements of Euclid with
disseritatios, by James Williamson, B.D. 2 vols. 4to., Oxford, 1781, and London,
1788.
Camerer and Hauber published the first six books in Greek and Latin, with good notes, Berlin, 8vo. 1824.
We believe we have mentioned all the Greek texts of the Elements; the liberal supply with which the bibliographers have furnished the world, and which Fabricius and others have perpetuated, is, as we have no doubt, a series of mistakes arising for the most part out of the belief about Euclid the enunciator and Theon the demonstrator, which we have described.
Of Latin editions, which must have a slight notice, we have the six books by
We have not thought it necessary to swell this article with the various reprints of these and the old Latin editions, nor with editions which, though called Elements of Euclid, have the demonstrations given in the editor's own manner, as those of Maurolycus, Barrow, Cotes, &c., &c., nor with the editions contained in ancient courses of mathematics, such as those of Herigonius, Dechâles, Schott, &c., &c., which generally gave a tolerably complete edition of the Elements. Commandine and Clavius are the progenitors of a large school of editors, among whom Robert Simson stands conspicuous.
We now proceed to English translations.
We find in Tanner (Bibl. Brit. Hib. p. 149) the following short
statement : " Candish, Richardus, patria Suffolciensis, in linguam patriam transtulit
Euclidis geometriam, lib. xv. Claruit A. D. MDLVI. Bal. par. post. p. 111." Richard Candish is mentioned
elsewhere as a translator, but we are confident that his translation was never published.
Pathway to Knowledge, 1551, containing enunciations only of the first four
books, not in Euclid's order. Recorde considers demonstration to be the work of
Theon.Ath. Oxon. in verb.) we find it stated
(and also how the information was obtained) that he studied three years at Oxford before he
was apprenticed to a haberdasher, and there made acquaintance with all " eminent
mathematician" called Whytehead, an Augustine friar. When the friar was " put to his shifts"
by the dissolution of the monasteries, Billingsley received and maintained him, and learnt
mathematics from him. " When Whytehead died, he gave his scholar all his mathematical
observations that lie had made and collected, together with his notes on Euclid's Elements."
This was the foundation of the translation, on which we have only to say that it was
certainly made from the Greek, and not from any of the Arabico-Latin versions, and is, for
the time, a very good one.
Elements of Geometry contains the
first six books of Euclid; but the solid geometry is supplied from other sources. The first
edition is of Edinburgh, 1795, octavo.
Euclid's Elements of Plane Geometry, by John Walker, London,
1827
The Diagrams of Euclid's Elements by the Rev. W.
Taylor, York, 1828, 8vo. size (part i. containing the first book; we do not know of any
more), is a collection of lettered diagrams stamped in relief, for the use of the blind.
The remaining writings of Euclid are of small interest compared with the Elements, and a shorter account of them will be sufficient.
Euclidis Dalorum Liber by Horsley, Oxford, 1803,
8vo.
The
Of the two works on music, the
The book on
Proclus Pappus; August ed cit.; Fabric. Bibl.
Graec. vol. iv. p. 44, &c.; Gregory, Pracf. edit. cit.;
Murhard, Bibl. Math. ; Zamberti, ed. cit.; Savile,
Prelect. in Eucl.; Heilbronner, Hist. Mathes. Unit.
; Schweiger, Handb. der Classisch. Bibl. ; Peyrard, ed. cit., &c. &c.: tall editions to which a reference is not added
having been actually consulted.