ing to which these materials are put together. The volume now laid before the public, as Professor Leslie informs us, is the first of a projected course of Mathematics.' The subjects treated of, are Geometry, including Geometrical Analysis, and Trigonometry: The Elements of Geometry are comprehended in six books. Of these the first two books relate principally to parallel lines, triangles, and quadrilaterals ; the third and fourth to circles, lines, and figures, drawn in and about them, their dependent angles, &c.; the fifth to the doctrine of ratios and proportions ; and the sixth to similar figures, their division by parallel lines, a summary of the chief propositions that depend upon proportionality, and one or two that relate to the rectification of the circle. Such are the constituents of Professor Les. lie's Elements; by which it will be seen that the geome. try. of solids is omitted altogether. er. The author, instead of this, has given an Appendix, in two parts; in which several problems in plane geometry are constructed, some by means of the ruler only, others solely by means of the compasses. In the first portion of this Appendix, Professor Leslie acknowledges himself indebted to a scarce tract of Schooten; in the second to Mascheroni's “Geometrie du Compas”, an ingenious work, well known to most of our mathematical readers. The treatise on Geometrical Analysis is comprized in three books. The first of these is somewhat miscellaneous. In the second and third books, Mr. Leslie professes to have given “ all that relates , to the ancient analysis in its most improved state, as extended by the labours of Apollonius and his illustrious contemporaries." Of course, these books develope the general principles, constructions, and operations, known to geometers under the terms Data, Section of a Ratio, Section of a Space, Determinate Section, Inclinations, Tangencies, Loci, Porisnis, and Isoperimeters. When discussing these particulars, the author is necessarily indebted to Euclid, Apollonius, and Pappus, among the ancients; as well as to Ghetaldus, Alexander Anderson, Halley, Dr. Simson, and Professor Playfair, among the moderns. The Elements of plane Trigonometry are included in 21 propositions, occu. pying about 50 pages. Spherical Trigonometry is, of necessity, omitted ; since the propositions relative to it could not be demonstrated independently of Solid Geometry. The work is preceded by two tables, one, of correspondence between these Books of Geometry and the Elements of Eu, clid;' the other, of correspondence of the Elements of EMclid with these Books of Geometry. From either of these tables it will be seen, that Mr. Leslie has departed greatly from the logical order of the Alexandrian Geometer. In our opinion, his deviations are often extremely wanton and ill judged. They give the work, however, an air of noveity, which it would not otherwise possess; and in this quality, indeed, it is by no means deficient. The matter, of course, is in the main very well known; but the manner is frequently original : and, if the artificer may be judged of from his workn anship, Professor Leslie is a most extraordinary and non-descript character. There is great variety in this gentleman's demonstrations. They are sometimes good, sometimes indifferent, sometimes bad; sometimes strict, sometimes loose. The good and legitimate demonstrations are the scarcest. We select one which we really think the best in the book. It relates to a very simple proposition, demonstrated about 30 ytars ago by Reuben Burrow in his Diary; and since then adu itted into some of our elementary books. But to demonstrate even a very simple theorem more simply than any other person, is a species of merit which ought not to be withholden' from Mr. Leslie on the present occasion. The proposition and demonstration are extracted below : the diagram will be readily supplied by our scientific readers. • The difference between two sides of a triangle is less than the third side. Let the side AC be greater than AB, and from it cut off a part AE equal to AB; the remainder EC is less than the third side BC. For the two sides A B. and BC are together greater than AC (1. 16.); take away the equal lines AB and AE, and there remains BC greater than EC; or EC is less than BC.' We now proceed to the second part of our task, which is, to select a few particular excellences. Upon these we shall not be able to descant so largely as Professor Leslie might expect; but there are other publications in which there can be little doubt of his receiving all the consolation that friendship can bestow. Tat we may deliver our remarks in some sort of order, we shall follow that adopted by the author himself: beginning with the Principles.' Here we have the following definition of a straight line, or, rather, of the idea of a straight line: The uniform description of a line which through its whole extent stretches in the same direction gives the idea of a straight line.' What is here meant by stretching, and what by direction? We bave some notion of stretching a cord; but none, certainly, of stretching a geometrical line, And as to direction, we believe Mr. Leslie would find some difficulty in defining it, without mentioning a right line in his definition, Direction ne cessarily implies rectilineality, and therefore canpot with any sort of propriety be included in the definition of a right line. 'Two points, we are informed, ascertain the position of a straight line.'' But to determine the position of a plane, it requires three points. Our mathematical Professor should have added, that these three points must not be in one and the same right line. This is an essential condition: neither three nor twenty times three points would determine the position of a plane if they were all in one right line. We are not told, either in the Principles' or in the 'De-. finitions', what a point, or what an angle is. It is merely stated that we derive the idea of divergence or angular magnitude, from revolving motion. Presently after, we are told, that the straight lines which contain an angle are termed its sides, and their point of origin or intersection, its vertex.' All this is very confused. Geometers often speak of the sides of a triangle; but none that we are aware of, before Mr. Leslie, ever talked of the sides of, an angle. We may expect to hear next of the dimensions of a geometrical point. But Mr. Leslie proceeds to tell us, that'a right angle is the fourth part of an entire circuit or revolution ;' meaning, perhaps, to be more simple, accurate, and satisfactory, than all his predecessors. Yet we doubt whether even Mr. Leslie would venture to say, that when a planet had described a right angle in its elliptical orbit, referring the angle to the focus, it had passed through the • fourth part of an entire circuit or revolution. After all this, Mr. Leslie takes care, in his corresponding note (p. 455), to affirm that Euclid's definition of an angle is obscure and altogether defective ;' and that it is curious to observe the shifts to which the author of the Elements is hence obliged to have recourse.' This is a discovery which neither Simson, Play fair, nor any other of our modern geometers can boast of. But to Mr. Leslie it was urgently necessary; for when he comes to prove,-(Prop. 26. of his 2nd Book,) that the angle in a semicircle is a right angle, he actually speaks of the angle made by two segments of a right line at a point in that line! Well' may he censure that definition, according to which " A plane rectilineal angle is the inclination of two straight lines 10 one another, which meet together, but are not in the same straight line :" and well may he remark that the conception of an angle is one of the most difficult in the whole compass of Geometry. But we must proceed to another definition connected with this difficult subject of angles. The retro-fected divergence of the two sides, or the defect of the angle from four right angles, is named a reverse angle.' Let not the unsophisticated reader complain of this as unintelligible ; but pause and bend to what follows: - In the definition of reverse angle, I find that I have been anticipated by Stevin of Bruges. It is satisfactory to have the countenance of such respectable authority. It is : even in support of 'affectation carried to the extreme of absurdity. Richard Brothers had the countenance of the respectable authority' of Mr. Halhed. Between the 10th definition relating to a reverse angle, and the 11th definition, Mr. Leslie, who manfully spurns at the trammels of order with which poor Euclid was hampered, presses into the opening formed by the regression of AB through the points D and E,' and there demonstrates a theorem, namely the 15th of Euclid's 1st book. Happy for those, whose comprehension of Euclid's theorem does not depend upon their understanding Mr. Leslie's phraseology! Def. 12. “Straight lines which have no inclination are parallel. This is incomplete : straight lines may have some inclination, viz. to a third line, and yet be parallel to one another. Every mathematician will be aware that this is not hyper-criticism. Def. 24. Of quadrilateral figures, a square has one right angle, and all its sides equal.' Mr. Leslie gives this, because he thinks the common definition, which describes a square “as having all its angles right”, errs by excess. He adds, “The original Greek, and even the Latin version, by employing the general terms odoyávrov, and rectangulum, dexterously avoided that objection. Mr. L. might have avoided it with equal dexterity, by simply calling,' a square a quadrilateral, equilateral, rectangular figure. One grand object of a definition is not accomplished, unless what is intended by it is put out of all doubt : this is not effected by Mr. L.; for when he says in so pointed a way that a square has one right angle, and all its sides equal,' a novice might hesitate till he could inquire whether it had only one right angle ; – and thus the justly boasted precision and certainty of geometry would be sacrificed. We have only to remark farther, with respect to the definitions in the first book, that Mr. Leslie, contrary to the usage of all preceding geometers, makes a trapezium a less general term than a trapezoid; and errs in confining the term diagonal to quadrilaterals. The first proposition is a problem, viz.' To construct a triangle, of which three sides are given.' The proof of the truth of the construction, is defective and unsatisfactory : for it is not shewn that the circles employed must neces. sarily intersect: nor indeed could it be shewn, independently, of other propositions. A similar observation applies with equal force to the second proposition, which affirms that Two triangles are equal, which have all the sides of the one equal to the corresponding sides of the other.' Euclid would have added “ each to each:” but this old fashioned geometer, as Mr. L. remarks,'had recourse to’ sad shifts for the sake of perspicuity and accuracy. Our Professor bas no such scruples : but, very adroitly failing in the demonstration of his first two propositions, by necessary consequence leaves all that follows undemonstrated. Such is the way by which the mathematician of the north strengthens the loose and defective' structure of Euclid. And we may add, too, that he at the same time ' enlarges the basis,' by taking away the foundations altogether! The science of Geometry, he tells us, owes its perfection to the extreme simplicity of its basis, and derives no visible advantage from the artificial mode of its construction. The axioms are now rejected as totally useless, and rather apt to produce obscurity! In our opinion, to take away the axioms, is to remove the basis' itself, a measure, of which the extreme simplicity is by no a sufficient recommendation: and as to the reason alledged, that this foundation is totally useless,' we be ieve he will find it no easy task to prove his assertion, without admitting either that nothing, or that every thing, in geometry, is self evident. In Prop. 4. Mr. L. constitutes a series of isosceles triangles having all their vertices at one common point: he adds, co It is evident that this addition is without limit, and that the angle so produced may continue to swell, and its expanding side make repeated revolutions.” We have heard of the swellings of vanity, the swelling of the sea, and the swellings of a wounded limb; but never before of the swelling of an angle! Surely the extravagances of affectation are without limit.' Prop. 9. The demonstration is defective. It ought also to have been supposed that AB exceeds CB, and the reductio ad absurdum employed. Prop. 10. is demonstrated by means of a proposition of which we have already spoken, as included in the definitions. Farther, the enunciation of the proposition includes a definition. This, we think, is not very consistent with what Mr. L. terms the Scholastic arrangements.' But what man of genius can endure the shackles of good sense and antiquity? He furnishes similar specimens at pages 27, and 216, means |