this restriction has been improperly omitted; a similar omission has been made also in Corollary 1, of the first Proposition, in the sixth book of the same work, where it is inferred that "straight lines which cut diverging lines proportionally are parallel," although it is obvious that these straight lines may cross each other. Such inadvertencies in elements of geometry are of consequence, however trifling they may appear to some, and certainly stand in need of

correction.

In the twelfth and thirteenth propositions, and their corollaries, are comprehended some useful truths relative to the contact and intersections of circles, upon which several of the problems in the fourth book depend.

These two propositions and corollaries include the 11th, 12th, 13th, and 14th propositions of Legendre's second book, and also the converse of those propositions.

On Propositions XIV. XV. XVI. and XVII.

way

The in which Euclid has enunciated and demonstrated the first of these propositions, has rendered it necessary that he should, in the succeeding proposition, demonstrate that "The angles in the same segment of a circle are equal to one another," a proposition which presented two cases requiring separate consideration. It was absolutely necessary that this truth should be established; but it could not be inferred from the preceding proposition, without considering re-entrant angles, which Euclid has made no mention of throughout his Elements. The demonstration given in the text equally avoids the introduction of re-entrant angles, while it dispenses with the proposition which Euclid found it necessary afterwards to demonstrate, as this immediately follows as an obvious inference from the proposition itself.

The fifteenth proposition may appear, perhaps, rather less simple than the corresponding proposition in Euclid, as an additional line has been introduced into the diagram. But I deemed it preferable to establish this proposition before proposition XVII. instead of adopting Euclid's course, and this preference has been given on the ground of simplicity; for this seventeenth proposition, by aid of the foregoing property, becomes susceptible of a much easier demonstration than that which Euclid gives, as will appear from a com

parison of Euclid's demonstration with that in these elements. With regard to the sixteenth proposition, it seems necessary merely to remark, that I have endeavoured to combine, in a single train of reasoning, all the various cases that the proposition presents, and I am in hopes that this reasoning will be found conclusive.

BOOK IV.

The propositions in this book are all problems, in which every practical operation that in the course of the preceding books was admitted to be possible, is actually performed. In Euclid's Elements, the problems are interspersed among the theorems, in order that in every demonstration no operation may be supposed possible, that has not been previously effected, with a view no doubt, as Mr. Playfair observes, "to guard against the introduction of impossible hypotheses, or the taking for granted that a thing may exist, which, in fact, implies a contradiction." There are some advantages, however, connected with a different arrangement; for, by thus keeping the theorems and problems distinct, a continuity is preserved in the chain of reasoning, and the mind proceeds from one truth to another without being interrupted by any thing of a mechanical nature; and, moreover, the problems themselves become, by this separation, sus ceptible of simpler and easier constructions, because we are enabled to avail ourselves of a greater number of previously established principles.

The restriction which Euclid has put upon himself, in this respect, appears to be unnecessary. The learned writer just quoted remarks, that "this rule is not essential to geometrical demonstration, where, for the purpose of discovering the properties of figures, we are certainly at liberty to suppose any figure to be constructed, or any line drawn, the existence of which does not involve an impossibility."

In the construction of problems, however, the case is widely different; for we cannot admit any preliminary construction till it has been actually effected. Every geometrical problem must remain unsolved, while it involves in it the trisection of an angle, since this operation cannot be actually performed; but if the supposition of such trisection, in the course of any demonstration, were necessary to

the establishment of a theorem, the conclusion would be as true and as satisfactory as if the above problem presented no difficulty; all this must be quite obvious, for the truths of geometry necessarily exist independently of any practical operations, and may, therefore, be reached without their aid.

The twentieth problem in this book was taken from the Ladies' Diary, where it was proposed and solved by Mr. Dotchen: the construction here given is somewhat simpler than his.

ever.

BOOK V.

This book is occupied by the doctrine of proportion, a subject of the highest importance, not only in geometry, but in every part of the mathematics. The importance of the subject is not, however, greater than the difficulties that have hitherto attended it; difficulties, the removal of which has resisted the attempts of geometers for a period of more than two thousand years-from the time of Euclid down to the present. Euclid establishes the doctrine of proportion in his fifth book, by reasonings of the most rigorous character, and in a manner so general and comprehensive, that magnitudes of all kinds are included without any restrictions or arbitrary conditions whatThese reasonings, however, are so exceedingly subtle, and it must be confessed, in some instances, so obscure, arising from the metaphysical considerations which they involve, that many, having been unable fully to enter into the spirit of it, have mistrusted his conclusions, and have ventured rashly to question their legitimacy. These circumstances have naturally drawn the attention of succeeding geometers to the formation of a treatise of proportion, of the same extent and universality as that of Euclid, in which the intricacies of his method might be avoided. But all attempts to accomplish this object have either entirely failed, or only very partially succeeded; so that, at the present day, there exists no rigorous and universal treatise on geometrical proportion except the fifth book of Euclid's Elements.

Many and important have been the errors into which geometers have fallen in their deviations from Euclid, on the subject of pro

portion. This assertion will be supported by a reference to some of the most reputable productions of the present day, where Euclid's conclusions have been reached by a shorter path, but by unwarrantable steps in the reasoning, which have consequently rendered those conclusions, though true, illegitimate.

Take, for instance, proposition XVI. of the fifth book of Bonnycastle's Geometry, viz.:

If four magnitudes be proportional, the sum of the first and second will be to the first or second as the sum of the third and fourth is to the third or fourth."

Let AE be to EB as CF is to FD; then will AB be to BE, or AE, as CD is to DF, or CF.

For since AE is to EB as CF is to FD; therefore, alternately, AE will be to CF as EB to FD. And, since the antecedent is to its consequent as all the antecedents are to all the consequents, AE will be to CF as AB is to CD.

But ratios, which are the same to the same ratio, are the same to each other; whence AB will be to CD as EB is to FD; and, alternately, AB to EB as CD to DF.

Again, since AE has been shown to be to CF as AB is to CD; therefore, by alternation, AE will be to AB as CF is to CD. But quantities, which are directly proportional, are also proportional when taken inversely; whence AB will be to AE as CD is to CF.

Now this conclusion is not legitimate, for the above reasoning, if meant to be general, is altogether inadmissible; as will appear obvious from observing, that in every step except the last, the alternation of the proportionals is required, which alternation is not possible except when the magnitudes are all of the same kind; this demonstration, therefore, applies only to a particular case of the proposed theorem. The next proposition in the same work, viz., proposition XVII. is, in like manner, conclusive only in the particular case when the magnitudes are all of the same kind. Mr. Bonnycastle, however, so far from being aware of this circumstance, attributes to these conclusions the same generality that belongs to Euclid's; nay, indeed, he asserts in his notes that they are even more general than those of the Greek

geometer; for he says, "It has been properly observed by Mr. Simpson that the manner in which the composition and division of ratios is treated of by Euclid is defective, as not being sufficiently general. It is also commonly found very abstruse and embarrassing to beginners, on account of the complicated terms in which it is enunciated, and the number of cases to be separately demonstrated. For these reasons it was deemed necessary to give the propositions a more simple and general form, and to render the demonstrations of them as concise and perspicuous as possible."

Having copied this note, it is incumbent on me to add, that Euclid's method (as restored by Dr. Simson) of treating the composition and division of ratios, so far from being defective and not sufficiently general, is undoubtedly complete and universal; and in justice to Mr. Thomas Simpson, as well as to Euclid, I must observe that the remark which Mr. Bonnycastle attributes to the former was in reality never made. It is remarkable that the two propositions just noticed should have been allowed to pass as genuine for so long a period (about 30 years), in a book of such popularity as Bonnycastle's Geometry; and that the foregoing unjust remarks upon the accurate reasonings of Euclid, with which the name of Mr. Simpson is so unwarrantably coupled, should have hitherto escaped the censure which they deserve.

As another example of this inconclusive reasoning, we may refer to Professor Leslie's manner of treating this subject in his Elements of Geometry, where the propositions on proportion are demonstrated to be true only when the magnitudes are both commensurable and homogeneous; that these demonstrations do not extend to incommensurable magnitudes the learned professor seems well aware, but it does not appear that he is also aware of their being restricted to homogeneous magnitudes. That such is the case, however, may be readily shown:-Take, for example, proposition XV. of his fifth book.

If two analogies have the same antecedents, another analogy may be formed having the consequents of the one for its antecedents, and the consequents of the other for its consequents.

Let A: B:: C: D, and A: E:: C: F; then B: E:: D: F.

For alternating the first analogy, A: C:: B: D*, and alter

According to Mr. Leslie's own definition of proportion, this proportion will be mpossible, unless the terms of the proportion A: B:: C: D are homogeneous.