# Euclid

**EUCLID**, Greek mathematician of the 3rd century B.C.; we are ignorant not only of the dates of his birth and death, but also of his
parentage, his teachers, and the residence of his early years. In some of the editions of his works he is called *Megarensis*, as if he had been born at
Megara in Greece, a mistake which arose from confounding him with another Euclid, a disciple of Socrates. Proclus (A.D. 412-485), the authority for most of our
information regarding Euclid, states in his commentary on the first book of the *Elements* that Euclid lived in the time of Ptolemy I., king of Egypt, who
reigned from 323 to 285 B.C., that he was younger than the associates of Plato, but older than Eratosthenes (276-196 B.C.) and Archimedes (287-212 B.C.). Euclid
is said to have founded the mathematical school of Alexandria, which was at that time becoming a centre, not only of commerce, but of learning and research, and
for this service to the cause of exact science he would have deserved commemoration, even if his writings had not secured him a worthier title to fame. Proclus
preserves a reply made by Euclid to King Ptolemy, who asked whether he could not learn geometry more easily than by studying the
*Elements* - "There is no royal road to geometry." Pappus of Alexandria, in his *Mathematical Collection*, says that Euclid was a man
of mild and inoffensive temperament, unpretending, and kind to all genuine students of mathematics. This being all that is known of the life and character of
Euclid, it only remains therefore to speak of his works.

Among those which have come down to us the most remarkable is the *Elements* (see Geometry). They consist of thirteen books; two more are frequently added, but
there is reason to believe that they are the work of a later mathematician, Hypsicles of Alexandria.

The question has often been mooted, to what extent Euclid, in his *Elements*, is a discoverer or a compiler. To this question no entirely satisfactory
answer can be given, for scarcely any of the writings of earlier geometers have come down to our times. We are mainly dependent on Pappus and Proclus for the
scanty notices we have of Euclid's predecessors, and of the problems which engaged their attention; for the solution of problems, and not the discovery of
theorems, would seem to have been their principal object. From these authors we learn that the property of the right-angled triangle had been found out, the
principles of geometrical analysis laid down, the restriction of constructions in plane geometry to the straight line and the circle agreed upon, the doctrine
of proportion, for both commensurables and incommensurables, as well as loci, plane and solid, and some of the properties of the conic sections investigated,
the five regular solids (often called the Platonic bodies) and the relation between the volume of a cone or pyramid and that of its circumscribed cylinder or
prism discovered. Elementary works had been written, and the famous problem of the duplication of the cube reduced to the determination of two mean
proportionals between two given straight lines. Notwithstanding this amount of discovery, and all that it implied, Euclid must have made a great advance beyond
his predecessors (we are told that "he arranged the discoveries of Eudoxus, perfected those of Theaetetus, and reduced to invincible demonstration many
things that had previously been more loosely proved"), for his *Elements* supplanted all similar treatises, and, as Apollonius received the title of
"the great geometer," so Euclid has come down to later ages as "the elementator."

For the past twenty centuries parts of the *Elements*, notably the first six books, have been used as an introduction to geometry. Though they are now
to some extent superseded in most countries, their long retention is a proof that they were, at any rate, not unsuitable for such a purpose. They are, speaking
generally, not too difficult for novices in the science; the demonstrations are rigorous, ingenious and often elegant; the mixture of problems and theorems
gives perhaps some variety, and makes their study less monotonous; and, if regard be had merely to the metrical properties of space as distinguished from the
graphical, hardly any cardinal geometrical truths are omitted. With these excellences are combined a good many defects, some of them inevitable to a system
based on a very few axioms and postulates. Thus the arrangement of the propositions seems arbitrary; associated theorems and problems are not grouped together;
the classification, in short, is imperfect. Other objections, not to mention minor blemishes, are the prolixity of the style, arising partly from a defective
nomenclature, the treatment of parallels depending on an axiom which is not axiomatic, and the sparing use of superposition as a method of proof.

Of the thirty-three ancient books subservient to geometrical analysis, Pappus enumerates first the *Data* of Euclid. He says it contained 90 propositions, the scope of which he describes; it
now consists of 95. It is not easy to explain this discrepancy, unless we suppose that some of the propositions, as they existed in the time of Pappus, have
since been split into two, or that what were once scholia have since been erected into propositions. The object of the *Data* is to show that when certain
things - lines, angles, spaces, ratios, etc. - are given by hypothesis, certain other things are given, that is, are determinable. The book, as we
are expressly told, and as we may gather from its contents, was intended for the investigation of problems; and it has been conjectured that Euclid must have
extended the method of the *Data* to the investigation of theorems. What prompts this conjecture is the similarity between the analysis of a theorem and
the method, common enough in the *Elements*, of *reductio ad absurdum* - the one setting out from the supposition that the theorem is true, the
other from the supposition that it is false, thence in both cases deducing a chain of consequences which ends in a conclusion previously known to be true or
false.

The *Introduction to Harmony* , and the *Section of the Scale* , treat of music. There is good reason for
believing that one at any rate, and probably both, of these books are not by Euclid. No mention is made of them by any writer previous to Ptolemy (A.D. 140), or
by Ptolemy himself, and in no ancient codex are they ascribed to Euclid.

The *Phaenomena* contains an exposition of the
appearances produced by the motion attributed to the celestial Sphere. Pappus, in the few remarks prefatory to his sixth book, complains of the faults, both of
omission and commission, of writers on astronomy, and cites as an example of the former the second theorem of Euclid's *Phaenomena*, whence, and from the
interpolation of other proofs, David Gregory infers that this treatise is corrupt.

The *Optics* and *Catoptrics* are ascribed to Euclid by Proclus, and by Marinus in his preface to the *Data*, but no
mention is made of them by Pappus. This latter circumstance, taken in connexion with the fact that two of the propositions in the sixth book of the
*Mathematical Collection* prove the
same things as three in the *Optics*, is one of the reasons given by Gregory for deeming that work spurious. Several other reasons will be found in
Gregory's preface to his edition of Euclid's works.

In some editions of Euclid's works there is given a book on the *Divisions of Superficies*, which consists of a few propositions, showing how a straight
line may be drawn to divide in a given ratio triangles, quadrilaterals and pentagons. This was supposed by John Dee of London, who transcribed or translated it,
and entrusted it for publication to his friend Federico Commandino of Urbino, to be the treatise of Euclid referred to by Proclus as . Dee mentions that, in the copy from which he wrote, the book was ascribed to Machomet of Bagdad, and adduces
two or three reasons for thinking it to be Euclid's. This opinion, however, he does not seem to have held very strongly, nor does it appear that it was adopted
by Commandino. The book does not exist in Greek.

The fragment, in Latin, *De levi et ponderoso*, which is of no value, and was printed at the end of Gregory's edition only in order that nothing might
be left out, is mentioned neither by Pappus nor Proclus, and occurs first in Bartholomew Zamberti's edition of 1537. There is no reason for supposing it to be
genuine.

The following works attributed to Euclid are not now extant: -

1. Three books on *Porisms* are mentioned both by Pappus and Proclus, and the former gives an abstract of them, with
the lemmas assumed. (See Porism.)

2. Two books are mentioned, named , which is rendered *Locorum ad superficiem* by Commandino and subsequent geometers. These
books were subservient to the analysis of loci, but the four lemmas which refer to them and which occur at the end of the seventh book of the *Mathematical
Collection*, throw very little light on their contents. R. Simson's opinion was that they treated of curves of double curvature, and he intended at one time
to write a treatise on the subject. (See Trail's *Life of Dr Simson*).

3. Pappus says that Euclid wrote four books on the *Conic Sections* , which Apollonius
amplified, and to which he added four more. It is known that, in the time of Euclid, the parabola was considered as the section of a right-angled cone, the
ellipse that of an acute-angled cone, the hyperbola that of an obtuse-angled cone, and that Apollonius was the first who showed that the three sections could be
obtained from any cone. There is good ground therefore for supposing that the first four books of Apollonius's *Conics*, which are still extant, resemble
Euclid's *Conics* even less than Euclid's *Elements* do those of Eudoxus and Theaetetus.

4. A book on *Fallacies* is mentioned by Proclus, who says that Euclid wrote it for the purpose of exercising beginners in
the detection of errors in reasoning.

This notice of Euclid would be incomplete without some account of the earliest and the most important editions of his works. Passing over the commentators of
the Alexandrian school, the first European translator of any part of Euclid is Boëtius (500), author of the *De consolatione philosophiae*. His *Euclidis
Megarensis geometriae libri duo* contain nearly all the definitions of the first three books of the *Elements*, the postulates, and most of the axioms.
The enunciations, with diagrams but no proofs, are given of most of the propositions in the first, second and fourth books, and a few from the third. Some
centuries afterwards, Euclid was translated into Arabic, but the only printed version in that language is the one made of the thirteen books of the
*Elements* by Nasir Al-Din Al-Tusi (13th century), which appeared at Rome in 1594.

The first printed edition of Euclid was a translation of the fifteen books of the *Elements* from the Arabic, made, it is supposed, by Adelard of Bath
(12th century), with the comments of Campanus of Novara. It appeared at Venice in 1482, printed by Erhardus Ratdolt, and dedicated to the doge Giovanni
Mocenigo. This edition represents Euclid very inadequately; the comments are often foolish, propositions are sometimes omitted, sometimes joined together,
useless cases are interpolated, and now and then Euclid's order changed.

The first printed translation from the Greek is that of Bartholomew Zamberti, which appeared at Venice in 1505. Its contents will be seen from the title:
*Euclidis megarēsis philosophi platonici Mathematicaru
disciplinaru Janitoris: Habent in hoc volumine quicuq
ad mathematica substantia aspirat: elemētorum libros xiii cu expositione Theonis insignis mathematici ... Quibus ... adjuncta.
Deputatum scilicet Euclidi volumē xiiii cu expositiōe Hypsi. Alex. Itidēq Phaeno. Specu. Perspe. cum expositione Theonis ac mirandus ille liber Datorum cum expostiōe Pappi Mechanici una cu
Marini dialectici protheoria. Bar. Zaber. Vene. Interpte.*

The first printed Greek text was published at Basel, in 1533, with the title . It was edited by Simon Grynaeus from two MSS. sent to him, the one from Venice by Lazarus Bayfius, and the other from Paris by John Ruellius. The four books of Proclus's commentary are given at the end from an Oxford MS. supplied by John Claymundus.

The English edition, the only one which contains all the extant works attributed to Euclid, is that of Dr David Gregory, published at Oxford in 1703, with
the title, . *Euclidis quae supersunt omnia*. The text is that of the Basel edition, corrected from the MSS.
bequeathed by Sir Henry Savile, and from Savile's annotations on his own copy. The Latin translation, which accompanies the Greek on the same page, is for the
most part that of Commandino. The French edition has the title, *Les OEuvres d'Euclide, traduites en Latin et en Français, d'après un manuscrit
très-ancien qui était resté inconnu jusqu'à nos jours. Par F. Peyrard, Traducteur des œuvres d'Archimède*. It was published at Paris in three volumes,
the first of which appeared in 1814, the second in 1816 and the third in 1818. It contains the *Elements* and the *Data*, which are, says the editor,
certainly the only works which remain to us of this ever-celebrated geometer. The texts of the Basel and Oxford editions were collated with 23 MSS., one of
which belonged to the library of the Vatican, but had been sent to Paris by the comte de Peluse (Monge). The Vatican MS. was supposed to date from the 9th
century; and to its readings Peyrard gave the greatest weight. What may be called the German edition has the title . *Euclidis Elementa ex optimis
libris in usum Tironum Graece edita ab Ernesto Ferdinando August*. It was published at Berlin in two parts, the first of which appeared in 1826 and the
second in 1829. The above mentioned texts were collated with three other MSS. Modern standard editions are by Dr Heiberg of Copenhagen, *Euclidis Elementa,
edidit et Latine interpretatus est J.L. Heiberg*. vols. i.-v. (Lipsiae, 1883-1888), and by T.L. Heath, *The Thirteen Books of Euclid's Elements*, vols.
i.-iii. (Cambridge, 1908).

Of translations of the *Elements* into modern languages the number is very large. The first English translation, published at London in 1570, has the
title, *The Elements of Geometrie of the most auncient Philosopher Euclide of Megara. Faithfully* (*now first*) *translated into the Englishe toung,
by H. Billingsley, Citizen of London. Whereunto are annexed certaine Scholies, Annotations and Inventions, of the best Mathematiciens, both of time past and in
this our age*. The first French translation of the whole of the *Elements* has the title, *Les Quinze Livres des Elements d'Euclide. Traduicts de
Latin en François. Par D. Henrion, Mathematicien*. The first edition of it was published at Paris in 1615, and a second, corrected and augmented, in 1623.
Pierre Forcadel de Beziés had published at Paris in 1564 a translation of the first six books of the *Elements*, and in 1565 of the seventh, eighth and
ninth books. An Italian translation, with the title, *Euclide Megarense acutissimo philosopho solo introduttore delle Scientie Mathematice. Diligentemente
rassettato, et alla integrità ridotto, per il degno professore di tal Scientie Nicolò Tartalea Brisciano*, was published at Venice in 1569, and Federico
Commandino's translation appeared at Urbino in 1575; a Spanish version, *Los Seis Libros primeros de la geometria de Euclides. Traduzidos en lēgua
Española por Rodrigo Camorano, Astrologo y Mathematico*, at Seville in 1576; and a Turkish one, translated from the edition of J. Bonnycastle by Husain
Rifki, at Bulak in 1825. Dr Robert Simson's editions of the first six and the eleventh and twelfth books of the *Elements*, and of the
*Data*.

*Authorities*. - The authors and editions above referred to; Fabricius, *Bibliotheca Graeca*, vol. iv.; Murhard's *Litteratur der
mathematischen Wissenschaften*; Heilbronner's *Historia matheseos universae*; De Morgan's article "Eucleides" in Smith's *Dictionary of
Biography and Mythology*; Moritz Cantor's *Geschichte der Mathematik*, vol. i.

(J. S. M.)

*Note - this article incorporates content from Encyclopaedia Britannica, Eleventh Edition, (1910-1911)*