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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.O. 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.

(1) The Oxford edition in folio, published in 1703 by David Gregory, under the title Eŭkleídou owŚóueva. “As an edition 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 Elementaire 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 devéloppement des Méthodes en Géo. métrie...

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 equi. lateral 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 it has been given, it is conceivable that the same angle may be greater than one right angle, and less than another right angle, that is, obtuse and acute at the same time.

The definition of a square assumes more than is necessary. For if a four-sided figure have all its sides equal and one angle a right angle, it may be shewn that all its angles are right angles ; or if a four-sided figure have all its angles equal, it may be shewn that they are all right angles.

Postulates. The postulates state what processes we assume that we can effect, namely, that we can draw a straight line between two given points, that we can produce a straight line to any length, and that we can describe a circle from a given centre with a given distance as radius. It is sometimes stated that the postulates amount to requiring the use of a ruler and compasses. It must however be observed that the ruler is not supposed to be a graduated ruler, so that we cannot use it to measure off assigned lengths. And we do not require the compasses for any other process than describing a circle from a given point with a given distance as radius ; in other words, the compasses may be supposed to close of themselves, as soon as one of their points is removed from the paper.

Axioms. The axioms are called in the original Common Notions. It is supposed by some writers that Euclid intended his postulates to include all demapds which are peculiarly geometrical, and his common notions to include only such notions as are applicable to all kinds of magnitude as well as to space magnitudes. Accordingly, these writers remove the last three axioms from their place and put them among the postulates ; and this transposition is supported by some manuscripts and some versions of the Elements.

The fourth axiom is sometimes referred to in editions of Euclid when in reality more is required than this axiom expresses. Euclid says, that if A and B be unequal, and. C and D equal, the sum of A and C is unequal to the sum of B and D. What Euclid often requires is something more, namely, that if A be greater than B, and C and D-be equal, the sum of A and C is greater than the sum of B and D. Such an axiom as this is required, for example, in I. 17. A similar remark applies ta the fifth axiom.

In the eighth axiom the words “that is, which exactly fill the same space,” have been introduced without the authority of

the original Greek. They are objectionable, because lines and angles are magnitudes to which the axiom may be applied, but they cannot be said to fill space.

On the method of superposition we may refer to papers by Professor Kelland in the Transactions of the Royal Society of Edinburgh, Vols. XXI. and XXIII.

The eleventh axiom is not required before I. 14, and the twelfth axiom is not required before I. 29; we shall not consider these axioms until we arrive at the propositions in which they are respectively required for the first time.

The first book is chiefly devoted to the properties of triangles and parallelograms.

We may observe that Euclid himself does not distinguish between problems and theorems except by using at the end of the investigation phrases which correspond to Q.E. F. and Q.E.D. respectively.

I. 2. This problem admits of eight cases in its figure. For it will be found that the given point may be joined with either end of the given straight line, then the equilateral triangle may be described on either side of the straight line which is drawn, and the sides of the equilateral triangle which are produced may be produced through either extremity. These various cases may be left for the exercise of the student, as they present no difficulty.

There will not however always be eight different straight lines obtained which solve the problem. For example, if the point A falls on BC produced, some of the solutions obtained coincide; this depends on the fact which follows from I. 32, that the angles of all equilateral triangles are equal.

I. 5. “Join FC.Custom seems to allow this singular expression as an abbreviation for “ draw the straight line PC," or for "join F to C by the straight line PC.

In comparing the triangles BFC, CGB, the words “and the base BC is common to the two triangles BFC, CGBare usually inserted, with the authority of the original. As however these words are of no use, and tend to perplex a beginner, we have followed the example of some editors and omitted them.

A corollary to a proposition is an inference which may be deduced immediately from that proposition. Many of the corol. laries in the Elements are not in the original text, but introduced by the editors.

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