in point of morals and argument. The philosophers, vexed to be foiled by musty pieces of divinity,' brought serious accusations against the whole body; the divines modestly refuted them. The philosophers re-advanced their charges; the divines remon strated, but abstained from recrimination. The matter was laid before the General Assembly, where it was eagerly contested, and where the point was finally decided in his favour by a majority of twelve. Yet genuine philosophy' had here no cause for exulta. tion. The best talents (it was ruefully observed) that Scotland could furnish from the church, the peerage, the academy, and the bar, were powerfully displayed (yet with so slender a triumph!) in defending maxims of received and demonstrated philosophy against the incoherent rhapsodies, the boisterous defamation, and the ignorant jargon of men, among whom the truth seemed disregarded, or utterly unknown: that is, the demonstrated philosophy' of men who affirmed that what uniformly preceded another in the order of occurrence' was the cause of it, against the ignorant jargon' of those who affirmed that though the ebbing of the tide uniformly preceded' the flowing of the tide, and morning uniformly preceded noon in the order of occurrence,' yet was not the ebb the cause of the flow, nor morning the cause of noon. So high, indeed, was the estimation in which Mr. Leslie's talents were held, and so loud the outcry against the persecution which he experienced, that nearly one fourth of a celebrated journal was occupied in detailing his merits, and calling down public execration upon the vindictive malice' of his enemies; malice' which pursued a man distinguished above his competitors; long known to his countrymen as a profound and inventive geometer: a man who honoured his country by his genius, and by a life devoted to the labours of science.' The consequence of all this was that. Mr. Leslie at length obtained the professorship; the jarring tones of controversy gave way to the song of victory; and the philosophers of Europe were invited to join in celebrating this triumph of intellect and metaphysics over ignorance and fanaticism.

We feel no inclination to re-judge the question: we must, however, be permitted to exercise our faculties in examining the pretensions of Professor Leslie to so much notoriety, and in guessing at the state of philosophical and mathematical talent in a country, where acquirements of so moderate a cast as this gentleman really possesses could be so highly extolled.

That our readers may judge how far we are justified in adopting this language, we shall proceed to give them an account of the book now on our table: first allowing Professor Leslie to speak for himself, to describe his own plan, and to appreciate his own merits-by making a copious extract from his preface.

We should form a wrong estimate, did we consider the Elements of Euclid, with all its merits, as a finished production. That admirable work was composed at a period when geometry was making its most rapid advances, and new prospects were opening on every side. No wonder that its structure should now seem loose and defective. In adapting it to the actual state of the science, I have, therefore, endeavoured carefully to retain the spirit of the original, but have sought to enlarge the basis, and to dispose the accumulated materials into a regular and more compact system. By simplifying the order of arrangement, I hope to have considerably smoothed the toil of the student. The numerous additions which are incorporated in the text, so far from retarding, will rather facilitate his progress, by rendering more continuous the chain of demonstration. To multiply the steps of ascent, is in general the most expeditious mode of gaining a summit.

The view which I have given of the nature of proportion, in the fifth book, will, I flatter myself, be found to remove the chief difficulties attending that important subject. The sixth book, which exhibits the application of the doctrine of ratios, contains a copious selection of propositions, not only beautiful in themselves, but that pave the way to the higher branches of geometry, or lead immediately to valuable practical results. Yet the appendix, without claiming the same degree of utility, will not be deemed the least interesting portion of the volume, since the ingenious resources which it discovers are calculated to afford a very pleasing and instructive exercise.

The part which has cost me the greatest pains, is that devoted to geometrical analysis. The first book consists of a series of the choicest problems, rising above each other in gradual succession. The second and third books are almost wholly occupied with the researches of the ancient analysis. In framing them, I have consulted a great variety of authors, some of whom are of difficult access. The labour of condensing the scattered materials will be duly estimated by those, who, taking delight in such fine speculations, are thus admitted at once to a rich and varied repast. The analytical investigations of the Greek geometers are indeed models of simplicity, clearness, and unrivalled elegance; and though miserably defaced by the riot of time and barbarism, they will yet be regarded by every person capable of appreciating their beauties as some of the noblest monuments of human genius. It is matter of deep regret, that algebra, or the modern analysis, from the mechanical facility of its operations, has contributed, especially on the Continent, to vitiate the taste and destroy the proper relish for the strictness and purity so conspicuous in the ancient method of demonstration. The study of geometrical analysis appears admirably fitted to improve the intellect, by training it to habits of precision, arrangement, and close application. If the taste thus acquired be not allowed to obtain undue ascendancy, it may be transferred with eminent utility to algebra, which, having shot up prematurely, wants reform in almost every depart

ment.

The elements of trigonometry are as ample as my plan would allow. I have explained fully the properties of the lines about the circle, and

the calculation of the trigonometrical tables; nor have I omitted any proposition which has a distinct reference to practice. The last problem is of essential consequence in marine surveying.

Having already exceeded the ordinary limits, it perhaps unfortu nately became requisite to curtail the notes and illustrations, with which the volume concludes: yet the more advanced student may peruse the few historical and critical remarks with CONSIDERABLE advantage. Some of the disquisitions, and the solutions of certain more difficult problems relative to trigonometry and geodesiacal operations, in which the modern analysis is sparingly introduced, are of a nature sufficiently interesting, I would presume, to claim the notice of proficients in science.

Abstract pursuits will be found nowise unfriendly to the cultivation of elegant literature, or incompatible with the most vigorous play of imagination. When the connexion and mutual dependence of the several branches of knowledge are clearly understood, it may be expected that our academical institutions, happily released from the trammels of antiquated forms, will hasten to show their liberality, in extending to the mathematical studies the same protection which has hitherto been almost exclusively confined to the scholastic arrangements.

'It is the nature of mathematical science to advance in continual progression. Each step carries it to others still higher. As its domain swells on the sight, new relations are descried, and the more distant objects seem gradually to approximate. But, while science thus enlarges its bounds, it likewise tends uniformly to simplicity and concentration. The discoveries of one age are, perhaps in the next, melted down into the mass of elementary truths. What are deemed at first merely objects of enlightened curiosity, become, in due time, subservient to the most important interests. Theory soon descends to guide and assist the operations of practice. To the geometrical speculations of the Greeks, we may distinctly trace whatever progress the moderns have been enabled to achieve in mechanics, navigation, and the various complicated arts of life. A refined analysis has disclosed the harmony of the celestial motions, and conducted the philosopher, through a maze of intricate phenomena, to the great laws appointed for the government of the universe.'-Preface, page vi to xii.

In the preceding quotation, we know not which most to admire, the modesty and correctness of the author's statements, the elegance of his language, or the matchless beauty and congruity of his imagery. On each of these we could descant with pleasure, were we not anxious to proceed to the body of the work.

The part devoted to the Elements of Geometry is comprised in six books. Book I. contains principles, definitions, and forty propositions, in which are developed the chief truths given in the first 46 propositions of Euclid's first book. Mr. Leslie, who is determined to be original, has omitted the axioms altogether; because, as he says, they are now rejected as totally useless, and rather apt to produce obscurity.' This is a point which we shall not stop to contest with the Professor: we presume that in the oral

delivery of his lectures, he takes care to explain whether every thing or nothing in geometry be self-evident, and thus to establish the perfect inutility of axioms. But this is not the only particular in which the Professor deems it more philosophical to invert 'the usual procedure;' for he gives no definition of a point, line, or angle; which indeed can scarcely be necessary in geometry, because, as Mr. Leslie remarks, with his usual propriety of metaphor, that science is supereminently distinguished by the luminous evidence, which constantly attends every step of its march.' He gives, however, instead of definitions, what he calls ideas of a straight line, an angle, and the position of a plane; which are any thing but perspicuous and correct. Considering the diversity of particulars which lie before us, we cannot stop to specify all our objections; but shall merely remark that Mr. Leslie defines a right angle' as the fourth part of an entire circuit or revolution,' and 'a reverse angle' as the retroflected divergence of the two sides, or the defect of the angle from four right angles; leaving our readers to wonder at the hardihood of the man who, after this ridiculous display, could complain of Euclid's definition of an angle as obscure and altogether defective,' and add it is curious to observe the shifts to which the author of the Elements is obliged to have recourse.'

We have also to complain of several of the enunciations and demonstrations in this book. Thus, the enunciations of propositions 10 and 34 include'definitions; a thing which is not very consistent with the finest specimens of logical deduction,' any more than the professor's introduction of a demonstration among his definitions, def. 10, page 7. The demonstrations to props. 1, 2, 9, 11, 12, 20, 23, and 25, of this book, are loose, defective, and unsatisfactory. In the first and second, it is not shewn that the circles used in the constructions must necessarily intersect: hence neither the problem nor the theorem is demonstrated; and of course all the dependent propositions, that is, all the rest of the work, is unsupported. Such is the mode in which Mr. Leslie smooths the toil of the student,' and 'renders more continuous the chain of demonstration.' But farther, the demonstrations of propositions 9, 11, 20, are not sufficiently general; that of proposition 12 contains a petitio principii; and that of proposition 25 is all but nonsense. The professor, however, is very proud of it; for, after accusing the unhappy Euclid (against whom he seems to entertain a mortal antipathy) of evading the difficulty' respecting parallel lines, he adds, the investigation now given, seems the best adapted to the natural progress of discovery. It is almost ridiculous to scruple about the idea of motion, which I have employed for the sake of clearness.' Be it so; we will not, then, object to the demonstration on that account; our objection is, that the author has una

wares fallen into the fallacia suppositionis,' and that therefore his vaunted demonstration is of no more use, than tables of compound interest would be to determine the nature of thunder.

We proceed to Book II. where we have five definitions, and thirty five propositions. The latter contain most of those in Euclid's second book, with others collected from West, Stone, &c. Among the demonstrations, we complain of the 3d, as defective and unsatisfactory; of the 4th, as unnecessarily operose; and of the 19th, as incomplete, for want of a diagram. The proposition is, In any triangle, the square described on the base is equivalent to the rectangles contained by the two sides and their segments intercepted from the base by perpendiculars let fall upon them from its opposite extremities. To the demonstration the author has only attached the figure belonging to the case when the vertical angle of the triangle is obtuse; but he says in a note, at page 461, The figure representing the other case of this elegant proposition, where the vertical angle is acute, was inadvertently omitted in the text, and has since been accidentally mislaid.' But could he not supply the deficiency in a note, as in other cases? could not his invention, animated by the zeal, and supported by the active perseverance,' of which he boasts in his preface, enable him to overcome the difficulty' of sketching a diagram, which almost any boy in the lowest Edinburgh class might draw

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in five minutes?

The author ought to have told his readers, with respect to prop. 18, 'In any triangle, the rhomboids described on two sides are together equivalent to a rhomboid described on the base, and limited by these and by parallels to the line which joins the vertex with their point of concourse', (a theorem of which the celebrated 47th of Euclid's 1st book is only a particular case,) that it is given in Pappus' Mathematical Collections, and that it has survived what he calls, with peculiar elegance, 'the riot of time and barbarism.' In the notes to prop. 15, 28 of this book, the Professor has exhibited three methods of finding right-angled triangles whose sides shall be rational. Two of these are given analytically, as below. Let n denote any odd number; then, according to Pythagoras, n, n2-1, and n2+1, or, according to Plato, 2n, n2—l, and n3+1, will represent the perpendicular, base, and hypotenuse respectively of a right-angled triangle. A more general rule, however, should have been inserted: for example, ma+n3 = hypotenuse, m3-n-base, 2 m n = perpendicular; where m and n are taken at pleasure, provided m be greater than n. There are also two curious series for the same purpose, which should not

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