Prop. 11. Any two angles of a triangle are together less than two right angles.' The demonstration of this simple proposition is incomplete. And in the demonstration of Prop. 12, Every triangle has two acute angles,' there is a petitio principii. Prop. 18. The shortest line that can be drawn between two given points, is a straight line.' This simple proposition, which might without any hesitation have been included in the axioms, had not Mr. L. thought them 'rather apt to produce obscurity,' is here demonstrated very circuitously by the consideration of limits. Yet Mr. L. himself says in the notes (p. 454.) that a straight line has two radical properties, which are distinctly marked in different languages. It holds the same undeviating course, and it traces the shortest distance between its extreme points.' Why, then, does this author attempt to demonstrate a property, which, according to his own account is essentially included in its definition, and even in its name? Prop. 20. The demonstration is defective. It ought to be shewn that the point C falls below AC. Other loose, defective, or unsatisfactory demonstrations, in this book, are those of Prop. 23, 24, and 25. Book. II. Def. 2. The altitude of a triangle is a perpendicular let fall from its vertex upon the extension of its base. According to this definition, an acute angled triangle has no altitude. Exact geometrician! Def. 4. The complements of rhomboids about the diagonal of a rhomboid, annexed to either of them, forms what is termed a gnomon.' Accurate grammarian! 6 These complements,' as we should also observe, are no where defined. Consummate logician! Prop. 3. The demonstration is incomplete. The indirect reasoning ought to include the case when BE falls above BD. Prop. 4. The deduction in the corollary might be demonstrated clearly in a fourth part of the compass. In this book, the valuable property of a triangle,demonstrated by Simson at p. 128 of his Select Exercises, ought certainly to have been included. " Book III. Def. 4. A straight line is said to be inflected in a circle, when it terminates at the circumference.' According to this definition, a chord of a circle is a straight line bent: for this, we submit, is the meaning of the word inflected.' The same intrepid defiance of custom and etymology, occurs also at pages 192, and 207, where the Professor speaks of straight lines inflected; but we have sought in vain for an instance of the corresponding phrase, 'curved lines straight ened,' Prop. 7. The proof is not sufficiently general: for DE may be drawn from some point in AB, when the demonstration will not hold-at least, without an additional diagram. Prop. 8. The third figure destroys the generality of the corollary. Prop. 13. In the diagram, FC should be less than CG. Prop. 23. In the enunciation of this theorem, Professor Leslie has tacitly admitted Euclid's definition of an angle, which notwithstanding, at p. 455, he calls obscure and defective.' Prop. 26 we have noticed above. Prop. 28. The perpendicular at the extremity of a diameter is a tangent to the circle.' In the demonstration of this... theorem, the Professor has brought a certain line of the name of HBG into a sad predicament; it seems to have been confined in a sort of lock-up-house, denominated a circle, and we are told it would again meet the circumference before it effected its escape ' The Professor appears to us to have been placed pretty much in the same predicament; having no chance, in the mathematical world at least, of effecting his escape from obscurity except by meeting derision. Book IV. Prop. 12. In the construction of this problem, angles are said to be "adjacent," which are removed at the greatest possible distance from each other. But this, we suppose, is conformable to "the Scholastic arrangements." Prop. 13. might be demonstrated more simply. In Prop. 18. we read of accrescent triangles,' of which we cannot be supposed to know any thing, having never been introduced to them before. Perhaps it is these triangles that have the swelling angles. re The construction of Prop. 20, is described with the Professor's usual felicity of phrase; the student is told to peat the equal triangles about the vertex O.' This repetition of triangles is not at all necessary. Book. V. in Mr. Leslie's Elements, like Book V in Euclid's, is devoted to the subject of Ratios and Proportions. But the Alexandrian is prodigiously excelled by the Scotsman, in point of accuracy and perspicuity. His account of proportion may serve for an example: Quantities viewed in pairs, may be considered as having a similar composition, if the corresponding terms of each pair contain its measure equally. Two pairs of quantities of a similar composition, being thus formed by the same distinct aggregations of their elementary parts, constitute a proportion.' It is actually of this sort of explication, that our author says, the view which I have given of the nature of proportion, in the fifth book, will, I flatter myself, be found to re move the chief difficulties attending that important subject! Props. 1 and 2 are, The product of a number into the sum or difference of two numbers, is equal to the sum or difference of its products into those numbers' and, The product which arises from the continued multiplication of any numbers, is the same, in whatever order that operation be performed.' In demonstrating these, our geometer' views' the units contained in' A, B, C, &c. as known: but, suppose A = √2, B = 3/3, C=/5, &c. how will this kind of proof hold? Certainly, not at all. Thus, then, the basis' (to adopt Mr. Leslie's most favourite term) giving way, the 6th proposition, and all the dependent part of the structure,' fall into ruins. So much for our author's method of 'removing the chief difficulties,' and chasing away the 'obscurity that confessedly pervades the fifth book. of Euclid!' We must now proceed, as Mr. Leslie says, to survey the contours of the distant amphitheatre' in the sixth book. Here the 1st proposition, that parallels cut diverging lines proportionally,' is not strictly demonstrated; for the Professor affirms that incommensurables may be expressed numerically to any required degree of precision;' an assertion, which we need not be at any pains to refute. Prop. 11. A straight line which bisects, either internally or externally, the vertical angle of a triangle, will divide its base into segments, internal or external, that are proportional to the adjacent sides of the triangle.' This proposition, though true enough when Anglicised, is, we believe, perfectly unintelligible as enunciated by this desperate adventurer after originality. In the demonstration, we are told expressly that equal angles are straight lines! The assertion, that the constant difference AC between' certain distances must always bear a sensible relation to them,' is not true. Prop. 17. Cor. 1. It should be added, that AC: CB:: AB: BD. For this is a very useful property, flowing naturally from the theorem. Prop. 20 'To divide a straight line, whether internally or externally, &c.' Had not our author been possessed with a sort of nervous antipathy to established phrases, he would have said, To divide a straight line or its continuation. Prop. 35. The arcs of a circle are proportional to the angles which they subtend at the centre." In demonstrating this theorem, Mr. Leslie supposes one of the angles ACB divided by continual bisections till an angle ACa is obtained 'less than any assignable angle:' he then applies this infini tesimal of an angle ACa, or one equal to it BCb, repeatedly, till by its multiplication it fills up the other angle BCD Dearer than by any possible difference,' and thus infers the equality of the ratios of the arcs AB and BD, and 'the angles ACB, BCD. Now, we have to remark respecting this strange kind of demonstration, that if the angle ACa (which we will call I) is less than any assignable angle, no multiple of it can be equal to a finite angle BCD or C: for suppose m times I to be equal to the known angle C, then is I equal to C m , a known quantity, and not less than any assignable angle. This demonstration is therefore contradictory and self-destructive; and, consequently, all the propositions that depend upon it are undemonstrated. Prop. 39. The circumference of a circle is proportional to the diameter, and its arcs to the square of that diameter.' The truth of this proposition is inferred from the inscription of polygons of 6, 12, 24, &c. sides in the circle. Proceeding thus,' says the Professor, by repeated duplications, -the perimeters of the series of polygons which emerge in succession, will continually approximate to the curvilineal boundary which forms their ultimate limit. Wherefore this extreme term, or the circumference,' &c. All this is exces. sively loose and ungeometrical. Does approximation characterize identity? If the writer of this article were to travel from London to Edinburgh he would continually approximate to the author of this book; but the Reviewer would not therefore become the Professor, nor could the qualities of the latter be with any fairness ascribed to, the former. If the reasoning of Archimedes in his celebrated treatise, Kuxio Magnos, had not been far more strict and logical, it would scarcely have survived its author. Prop. 38. Here Mr. Leslie gives a concise approximation to the quadrature of the circle, which he says was first published, at Padua, in the year 1668, by my illustrious predecessor James Gregory.' It should be observed, however, that the corollary to the proposition from whence this quadrature is made to flow, is not James Gregory's; and farther, that it is inadequately demonstrated, being effected in the loose manner adopted in Prop. 35. In the three books on Geometrical Analysis our Professor scrupulously preserves consistency of character, being as 'loose and defective' as in other parts of his work but as we have not room to augment our selections under this head, we must only say, generally, that in many instances he omits the synthesis, which necessarily renders his solutions incom plete; and that in many others what he presents as de monstrations, are, in fact, no demonstrations at all. Let us now pass to the Elements of Plane Trigonometry, a science, it seems, which depends upon that universal standard derived from the partition of a circuit! We leave our ingenious readers to decypher this riddle; and proceed to observe, that, out of five definitions, the 1st and 4th are expressed in defective language Farther, our author says, p. 406, an arc may, by a simple extension of analogy, be conceived to comprehend innumerable other arcs.' simple extension of analogy we certainly do not understand but we think we understand that the learned author writes ungrammatically, when he adds, in the same page, 'the sine or tangent of an arc a ARE the same with the sine or tangent of any arc n. 360° + a.” This Of the five first propositions, the demonstrations are every one incomplete and unsatisfactory. Thus, in the first proposition, which affirms that the rectangle under the radius and the sine of the sum or difference of two arcs, is equal to the sum or difference of the rectangles under their alternate sines and cosines,' it is not enough to demonstrate its truth when the sum of the arcs A and B is less than a quadrant; it is, likewise, necessary to establish it, not only when A+B, but when either A or B, or both, exceeds a quadrant. M. Legendre, a mathematician to whom Mr. Leslie refers, has demonstrated this theorem in its utmost generality at p. 343, of his "Elemens de Geometrie et Trigonometrie," 5th edition. The demonstrations of the succeeding four propositions in Mr. Leslie's book are defective for like reasons. Prop. 10 has a corollary, the object of which is not specified, and can hardly be guessed. The table of solutions at Prop. 10, does not contain the simplest rule in the case where the three sides of a plane triangle are given to find an angle: and, in the well known ambiguous case, Professor LesTie does not point out the limits between which the ambiguity exists. We have thus endeavoured to display the merits of this author, as a mathematician, in a proper light. We might have added greatly to the preceding selection of the Professor's beauties; but, as our patience began to tire, we could not but sympathize with our readers, who will doubtless be more than satisfied with the materials of this second course of our rich and varied repast. We now proceed, lastly, to establish incontrovertibly, by proofs drawn from the volume before us, the truth of a |