disapproved of Euclid's arrangement have vainly attempted to change it without weakening the force of the demonstrations. Their unavailing attempts, he considers to be the strongest proof of the difficulty of substituting for the chain formed by the ancient geometer, any other equally strong and valid.* WOLF also acknowledges how futile it is to attempt to arrange geometrical truths in a natural or absolutely methodical order, without either taking for granted what has not been previously established, or relaxing in a great degree the rigour of demonstration. One of the favourite arrangements of those who object to that of Euclid, has consisted in establishing all the properties of straight lines considered without reference to their length, intersecting obliquely and at right angles, as well as the properties of parallel lines, before the more complex magnitudes called triangles are considered. In attempting this, it is curious to observe the difficulties into which these authors fall, and the expedients to which they are compelled to resort. Some find it necessary to prove that every point on a perpendicular to a given right line is equally distant from two points taken on the given right line, at equal distances from the point where the perpendicular meets it. They imagine,' says Montucla, that they prove this by saying that the perpendicular does not lean more to one side than the other.' Again, to prove that equal chords of a circle subtend equal arcs, they say that the uniformity of the circle produces this effect: that two circles intersect in no more than two points, and that a perpendicular is the shortest distance of a point from a right line, are propositions which they dispose of very summarily, by appealing to the evidence of the senses. They prefer an imperfect demonstration, or no demonstration at all, to any infringement of the order which they have assumed.

'There is a kind of puerility in this affectation of not mentioning a particular modification of magnitude, triangles, for

* Montucla, tom. i. p. 205.

+ Element. Math. tom. v. c. 3. art. 8.

example, until we have first treated of lines and angles; for if any degree of geometrical rigour be required, as many and as long demonstrations are necessary as if we had at once commenced with triangles, which, though more complex modifications of magnitude, are still so simple that the student does not require to be led by degrees to them. Some have even gone so far as to think that this affectation of a natural and absolutely methodical order contracts the mind by habituating it to a process of investigation contrary to that of discovery.'

The mathematicians who have attempted to improve the reasoning of Euclid, have not been more successful than those who have tried to reform his arrangement. Of the various objections which have been brought against Euclid's reasoning, two only are worthy of notice, viz. those respecting the twelfth axiom of the first book, which is sometimes called Euclid's postulate, and those which relate to his doctrine of proportion. On the former, I have enlarged so fully in Appendix II. that little remains to be said here. I have there shown that what is really assumed by Euclid, is, that 'two right lines which diverge from the same point cannot be both parallel to the same right line;' or that more than one parallel cannot be drawn through a given point to a given right line.' The geometers who have attempted to improve this theory, have all either committed illogicisms or assumed theorems less evident than that which has just been expressed, and which seems to me as evident as several of the other axioms. In the Appendix I have stated at length some of the theories of parallels which have been proposed to supersede that of Euclid, and have shown their defects. Numerous have been the attempts to demonstrate the twelfth axiom by the aid of the first twenty-eight propositions. Ptolemy, Proclus, Nasireddin, Clavius, Wallis, Saccheri, and a cloud of editors and commentators of former and later times, have assailed the problem without success.

Montucla, p. 206.

The second source of objection, on the score of reasoning, is the definition of four proportional magnitudes prefixed to the fifth book. By this definition, four magnitudes will be proportional if there be any equimultiples of the first and third which are respectively equal to equimultiples of the second and fourth. This is the common popular notion of proportion. But it is necessary to render the term more general in its geometrical application. Four magnitudes are frequently so related, that no equimultiples of the first and third are equal respectively to other equimultiples of the second and fourth, but yet have all the other properties of proportional quantities, and therefore it is necessary that they should be brought under the same definition. Euclid adapted his definition to embrace these, by declaring four magnitudes to be proportional, when every pair of equimultiples of the first and third were both greater, equal to, or less than equimultiples of the second and fourth. I agree with Playfair, in thinking that no other definition has ever been given from which the properties of proportionals can be deduced by reasonings, which, at the same time that they are perfectly rigorous, are also simple and direct. Were we content with a definition which would only include commensurate magnitudes, no difficulty would remain. But such a definition would be useless; for in almost the first instance in which it should be applied, the reasoning would either be inconclusive, or the result would not be sufficiently general.

In the second and fifth books, in addition to Euclid's demonstrations, I have in most instances given others which are rendered more clear and concise by the use of a few of the symbols of algebra, the signification of which is fully explained, and which the student will find no difficulty in comprehending. The nature of the reasoning, however, is essentially the same, the language alone in which it is expressed being different.

The commentary and deductions are distinguished from the text of the Elements, by being printed in a smaller character;

and those articles in each book which are marked thus *, the student is advised to omit until the second reading.

*

*

No part of Euclid's Elements has obtained the same celebrity, or been so universally studied as the first six books. The seventh, eighth, and ninth books treat of the Theory of Numbers, and the tenth is devoted to the Theory of Incommensurable Quantities. Instead of the eleventh and twelfth books, I have added a Treatise on Solid Geometry more suited to the present state of mathematical knowledge. For much of the materials of this treatise I am indebted to Legendre's Geometry.

Appendix I. contains a short Essay on the Ancient Geometrical Analysis, which may be read with advantage after the sixth book. The Second Appendix contains an account of the Theories of Parallels.

I have directed that the cuts of this work shall be published separately, in a small size, for the convenience of students who are taught in classes where the use of the book itself is not permitted.

33, Perry Street, Bedford Square, May, 1828.

ERRATA.

P. 65, 1. 8 from the bottom, for the product, read half the product.

P. 288, 1. 20 from the bottom, dele cylindrical,

(4)

IV.

(5)

(6)

A right line is that which lies evenly
between its extremities.

V. A surface is that which has length and
breadth only.

VI. The extremities of a surface are lines.

(7) VII. A plane surface is that which lies evenly between its extremities.

*

(8) These definitions require some elucidation. The object of Geometry is the properties of figure, and figure is defined to be the relation which subsists between the boundaries of space. Space or magnitude is of three kinds, line, surface, and solid. It may be here observed, once for all, that the terms used in geometrical science, are not designed to signify any real, material or physical existences. They signify certain abstracted notions or conceptions of the mind, derived, without doubt, originally from material objects by the senses, but subsequently corrected, modified, and, as it were, purified by the operations of the understanding. Thus, it is certain, that nothing exactly conformable to the geometrical notion of a right line ever existed; no edge, which the finest tool of an artist can construct, is so completely free from inequalities as to entitle it to be considered as a mathematical right line. Nevertheless, the first notion of such an edge being obtained by the senses, the process of mind by which we reject the inequalities incident upon the nicest mechanical production, and substitute for them, mentally, that perfect evenness which constitutes the essence of a right line, is by no means difficult. In

*From y, terra; and μśrgov, mensura.

B