# 1911 Encyclopædia Britannica/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* (Στοιχεῖα) (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* (Δεδομένα) 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, &c.—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* (Εἰσαγωγὴ ἁρμονική), and the
*Section of the Scale* (Κατατομὴ κανόνος), 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* (Φαινόμενα) 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* (Ὀπτικά, Κατοπτρικά) 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 τὸ περὶ διαιρέσεων βιβλίον. 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* (Περὶ τῶν πορισμάτων) 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 Τόπων πρὸς ἐπιφανείᾳ,
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* (βιβλία τέσσαρα Κωνικῶν), 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* (Περὶ ψευδαρίων) 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-Dīn Al-Tūsī
(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*
*MathematicaruF disciplinarū Janitoris: Habent in hoc volumine*
*quicūqF ad mathematicā substantiā aspirāt: elemētorum libros xiii*
*cū expositione Theonis insignis mathematici ... Quibus ... adjuncta.*
*Deputatum scilicet Euclidi volumē xiiii cū expositiōe Hypsi.*
*Alex. ItidēqF Phaeno. Specu. Perspe. cum expositione Theonis ac*
*mirandus ille liber Datorum cum expostiōe Pappi Mechanici una*
*cū Marini dialectici protheoria. Bar. Zāber. Vene. Interpte.*

The first printed Greek text was published at Basel, in 1533, with the title Εὐκλείδου Στοιχεῖων βιβλ. ιέ ἐκ τῶν Θέωνος συνουσιῶν. 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, Εὐκλείδου τὰ σωζόμενα. *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 Œuvres*
*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
Εὐκλείδου Στοιχεῖα. *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 Husaīn Rifkī, 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.)