If this process be continued, and the triangles be supposed to be taken away, there will at length remain segments of circles which are together less than the excess of the circle EFGH above the space S, by the preceding Lemma.

Let then the segments EK, KF, FL, LG, GM, MH, HN, NE be those which remain, and which are together less than the excess of the circle above S;

therefore the rest of the circle, namely the polygon EKFLGMHN, is greater than the space S.

In the circle ABCD describe the polygon AXBOCPDR similar to the polygon EKFLGMÍN.

Then the polygon AXBOCPDR is to the polygon EKFLGMHN as the square on BD is to the square on

FH,

[XII. 1. that is, as the circle ABCD is to the space S. [Hyp., V. 11. But the polygon AXBOCPDR is less than the circle ABCD in which it is inscribed,

therefore the polygon EKFLGMHN is less than the space S;

but it is also greater, as has been shewn; which is impossible.

[V. 14.

Therefore the square on BD is not to the square on FH as the circle ABCD is to any space less than the circle EFGH.

In the same way it may be shewn that the square on FH is not to the square on BD as the circle EFGH is to any space less than the circle ABCD.

Nor is the square on BD to the square on FH as the circle ABCD is to any space greater than the circle EFGH.

For, if possible, let it be as the circle ABCD is to a space T greater than the circle EFGH.

Then, inversely, the square on FH is to the square on BD as the space 7' is to the circle ABCD.

But as the space T is to the circle ABCD so is the circle EFGH to some space, which must be less than the circle

ABCD, because, by hypothesis, the space T is greater than the circle EFGH

[V. 14.

Therefore the square on FH is to the square on BD as the circle EFGH is to some space less than the circle ABCD;

which has been shewn to be impossible.

Therefore the square on BD is not to the square on FH as the circle ABCD is to any space greater than the circle EFGH.

And it has been shewn that the square on BD is not to the square on FH as the circle ABCD is to any space less than the circle EFGH.

Therefore the square on BD is to the square on FH as the circle ABCD is to the circle EFGH.

Wherefore, circles &c. Q.E.D.

NOTES ON EUCLID'S ELEMENTS.

THE article Eucleides in Dr Smith's Dictionary of Greek and Roman Biography was written by Professor De Morgan; it contains an account of the works of Euclid, and of the various editions of them which have been published. To that article we refer the student who desires full information on these subjects. Perhaps the only work of importance relating to Euclid which has been published since the date of that article is a work on the Porisms of Euclid by Chasles; Paris, 1860.

Euclid appears to have lived in the time of the first Ptolemy, B.C. 323-283, and to have been the founder of the Alexandrian mathematical school. The work on Geometry known as The Elements of Euclid consists of thirteen books; two other books have sometimes been added, of which it is supposed that Hypsicles was the author. Besides the Elements, Euclid was the author of other works, some of which have been preserved and some lost.

We will now mention the three editions which are the most valuable for those who wish to read the Elements of Euclid in the original Greek.

1703 by David "As an edition

(1) The Oxford edition in folio, published in Gregory, under the title Εὐκλείδου τὰ σωζόμενα. of the whole of Euclid's works, this stands alone, there being no other in Greek." De Morgan.

(2) Euclidis Elementorum Libri sex priores...edidit Joannes Gulielmus Camerer. This edition was published at Berlin in two volumes octavo, the first volume in 1824 and the second in 1825. It contains the first six books of the Elements in Greek with a Latin Translation, and very good notes which form a mathematical commentary on the subject.

(3) Euclidis Elementa ex optimis libris in usum tironum Græce edita ab Ernesto Ferdinando August. This edition was published at Berlin in two volumes octavo, the first volume in 1826 and the second in 1829. It contains the thirteen books of the Elements in Greek, with a collection of various readings.

A third volume, which was to have contained the remaining works of Euclid, never appeared. "To the scholar who wants one edition of the Elements we should decidedly recommend this, as bringing together all that has been done for the text of Euclid's greatest work." De Morgan.

An edition of the whole of Euclid's works in the original has long been promised by Teubner the well-known German publisher, as one of his series of compact editions of Greek and Latin authors; but we believe there is no hope of its early appearance.

Robert Simson's edition of the Elements of Euclid, which we have in substance adopted in the present work, differs considerably from the original. The English reader may ascertain the contents of the original by consulting the work entitled The Elements of Euclid with dissertations...by James Williamson. This work consists of two volumes quarto; the first volume was published at Oxford in 1781, and the second at London in 1788. Williamson gives a close translation of the thirteen books of the Elements into English, and he indicates by the use of Italics the words which are not in the original but which are required by our language.

Among the numerous works which contain notes on the Elements of Euclid we will mention four by which we have been aided in drawing up the selection given in this volume.

An Examination of the first six Books of Euclid's Elements by William Austin...Oxford, 1781.

Euclid's Elements of Plane Geometry with copious notes...by John Walker. London, 1827.

The first six books of the Elements of Euclid with a Commentary...by Dionysius Lardner, fourth edition. London, 1834.

Short supplementary remarks on the first six Books of Euclid's Elements, by Professor De Morgan, in the Companion to the Almanac for 1849.

We may also notice the following works:

Geometry, Plane, Solid, and Spherical,...London 1830; this forms part of the Library of Useful Knowledge.

Théorèmes et Problèmes de Géométrie Eleméntaire par Eugène Catalan...Troisième édition. Paris, 1858.

For the History of Geometry the student is referred to Montucla's Histoire des Mathématiques, and to Chasles's Aperçu historique sur l'origine et le developpement des Méthodes en Géométrie...

THE FIRST BOOK.

Definitions. The first seven definitions have given rise to considerable discussion, on which however we do not propose to enter. Such a discussion would consist mainly of two subjects, both of which are unsuitable to an elementary work, namely, an examination of the origin and nature of some of our elementary ideas, and a comparison of the original text of Euclid with the substitutions for it proposed by Simson and other editors. For the former subject the student may hereafter consult Whewell's History of Scientific Ideas and Mill's Logic, and for the latter the notes in Camerer's edition of the Elements of Euclid.

We will only observe that the ideas which correspond to the words point, line, and surface, do not admit of such definitions as will really supply the ideas to a person who is destitute of them. The so-called definitions may be regarded as cautions or restrictions. Thus a point is not to be supposed to have any size, but only position; a line is not to be supposed to have any breadth or thickness, but only length; a surface is not to be supposed to have any thickness, but only length and breadth.

The eighth definition seems intended to include the cases in which an angle is formed by the meeting of two curved lines, or of a straight line and a curved line; this definition however is of no importance, as the only angles ever considered are such as are formed by straight lines. The definition of a plane rectilineal angle is important; the beginner must carefully observe that no change is made in an angle by prolonging the lines which form it, away from the angular point.

Some writers object to such definitions as those of an equilateral triangle, or of a square, in which the existence of the object defined is assumed when it ought to be demonstrated. They would present them in such a form as the following: if there be a triangle having three equal sides, let it be called an equilateral triangle.

Moreover, some of the definitions are introduced prematurely. Thus, for example, take the definitions of a right-angled triangle and an obtuse-angled triangle; it is not shewn until I. 17, that a triangle cannot have both a right angle and an obtuse angle, and so cannot be at the same time right-angled and obtuseangled. And before Axiom 11 has been given, it is conceivable