have been omitted in a book intended for collegiate education: they are,

17, 211 315 418, 521, 67, &c.

in each of which progressions, if the denominator of any fraction be taken for the base, and the corresponding integer multiplied by the denominator, and the numerator added to the product for the perpendicular, the hypotenuses of the triangles so constructed will be rational.

The Third Book comprises seven definitions and thirty nine propositions, many of which are the same as those in Euclid's Third Book; but like the preceding part of the Professor's work, often thrown into an order widely different from his, and which frequently appears to us to be far from logical and natural. Here the only objectionable definition is the 4th: A straight line is said to be inflected in a circle, when it terminates at the circumference.' We are much inclined to believe, notwithstanding the authority of Professor Leslie, that inflection means bending; and we cannot, therefore, bring ourselves to say, as we ought in conformity with this definition, that a chord is a straight line bent in a circle. Phraseology equally appropriate graces Mr. Leslie's performance at pages 98, 192, 207, 367, 370, &c.

The demonstration of some of the propositions in this book also are unsatisfactory. Thus, in prop. 7, the proof is incomplete, unless a figure be drawn for the case when D E intersects the diameter A B In the demonstration of prop. 9, the words on the same side of the diameter H B' are superfluous. In prop. 23 our author calls the inclination of two straight lines' the angle which they form, thus tacitly admitting the correctness of Euclid's definition of an angle, though at p. 455 he had called it obscure and defective.' But, as Mr. Leslie remarks, the conception of an angle is one of the most difficult in the whole compass of geometry'; which may, perhaps, account for his language in the demonstration of prop. 26, where he speaks of the angle ACD formed by the opposite portions CA, CD, of the diameter,' that is, of the angle formed by two segments of a right line, contrary to the 9th definition of Euclid's first Book, in which it is affirmed that 'a plane rectilineal angle is the inclination of two straight lines to one another, which meet together, but are not in the same straight line.'

Book IV. relates to the construction of polygons, their inscrp tion in, and circumscription about, circles. It contains six definitions, and 24 propositions. In this book we notice no errors, except a verbal one in prop. 12, where angles are called adjacent,' which are at the greatest possible distance from each other. Seve

ral of the problems, however, might be constructed and demonstrated more simply; and the book much improved by the introduction of such of the propositions from L'Huilier's elegant little treatise, Polygonométrie,' as are susceptible of simple geometrical investigations and constructions. Professor Leslie, though often barren in point of information, is always fruitful in the invention of words. thus, at pages 136, and 395, we have accrescent triangles,' a species of geometrical figure which no preceding geometer has characterized.

In a note at p. 461 the Professor says,

A curious and most unexpected discovery was lately made by Mr. Gauss, who has demonstrated, in a work entitled Disquisitiones Arithmetica, published at Brunswic in 1801, that certain very complex polygons can yet be described merely by help of circles. Thus a regular polygon, containing 17,257,65537, &c. sides, is capable of being inscribed, by the application of elementary geometry; and in general, when the number of sides may be noted by 2n+1, and is at the same time a prime number. The investigation of this principle is rather intricate, being founded on the arithmetic of sines and the theory of equations; and the constructions to which it would lead are hence, in every case, unavoidably and most excessively complicated.'

Our author here states a fact, but assigns a wrong reason for it. It by no means follows, that because Gauss's investigations are founded on the arithmetic of sines and the theory of equations,' the constructions should unavoidably be excessively complicated.' The principle of the celebrated Cotesian theorem rests on the arithmetic of sines and the theory of equations; yet the theorem itself, instead of being unavoidably complicated,' is remarkably simple. Indeed, we cannot help conjecturing that Mr. Leslie has never seen Gauss's book, but obtained his information on this subject from Legendre's Geometry. A more complete view of Gauss's method than Legendre has given would be a valuable acquisition to the English student. It is not consistent with our purpose to present it here; we merely state one result. The cosine of the central angle of the polygon of 17 sides, expressed in square roots, is as follows:

Cos.+17+3%√(34—2 √ 17)−√ { (17+3√17) —√(34—2√17)—2√(34+2√17)}:

the cosines of the multiples of that angle have a similar form, and the sines have one radical more. A French translation of M. Gauss's work was published by M. Poullet-Delisle, in 1807.

The Fifth Book, like the Fifth Book of Euclid's Elements, is devoted to the subject of proportion. It abounds in frothy writing,

incorrect reasoning, and contemptuous notices of the Alexandrian geometrician.

"Through the whole contexture of the Elements we may discern the influence of that mysticism which prevailed in the Platonic school.'(p. 463.)

Euclid has contrived rather to evade difficulties than fairly to meet them. He seems not indeed to grasp the subject with a steady and comprehensive hold.' (ibid.)

'Euclid's famous definition leaves us to grope at random after its object, and to seek our escape by having recourse to some auxiliary train of reasoning or induction.' (p. 465.)

How happy for the student, that he is at once relieved from l'action de tater, by the following luminous definition of the Professor!

Two pairs of quantities of a similar composition, being formed by the same distinct aggregations of their elementary parts, constitute a proportion.' (p. 147.)

Mr. Leslie now introduces the concise notation of Algebra into his system of Geometry; though for its introduction at this precise place we can see no substantial reason, unless it be that Mr. John West has done just the same thing in his Elements of Mathematics,' published at Edinburgh, in 1784. Mr. West was assistant to the late Professor Vilant, of St. Andrew's, at the time when Mr. Leslie studied at that famous university. We regret to say that this profound and self-taught geometer (as he is called in the Encyclopædia Britannica) has not acknowledged his obligations to his former preceptor; for that Mr. Leslie has adopted many of his notions from this part of West's book, cannot for a moment be doubted by any person who has seen both performances. His supposed improvements upon West's plan are mere failures; most of the demonstrations are loose in the extreme; the fundamental propositions, on which the succeeding ones rest, being proved only in the case of commensurable quan

tities.

If geometry, however, in the hands of Professor Leslie, be not characterized by the correctness of its processes, and the irresistible conviction of its demonstrations, it is, notwithstanding, favourable, as he remarks, 'to the most vigorous play of imagination:' this we are now about to prove. Speaking of the subject which is exemplified in the 6th book, the author says,

It easily unfolds the primary relations of figures, and the sections of lines and circles; but it also discloses with admirable felicity that vast concatenation of general properties, not less important than remote, which, without such aid, might for ever have escaped the penetration of the geometer. He is thus placed on a commanding emi

nence; from which he views the bearings of the objects below, surveys the contours of the distant amphitheatre, and descries the fading verge of a boundless horizon.' (p. 176)

Now, what would the reader guess is the commanding eminence' from which the fading limit of an unlimited horizon may be descried, and the contours of the distant amphitheatre surveyed? What is this geometrical Pisgah from which the mathematician is to enjoy such delightful prospects? Truly it is none other than Mount Proportion'! Reviewers, it is known, are beings

• Of such vinegar aspect,

That they'll not shew their teeth in way of smile,
Though Nestor swear the jest be laughable.'

But this affected verbiage was so irresistibly ludicrous, that our gravity gave way at once.

Among the demonstrations to the 38 propositions comprehended in this 6th Book, Mr. Leslie takes care to introduce some that are faulty, as if to preserve his character for consistency by the sacrifice of his reputation for accuracy. The first proposition, 'Parallels cut diverging lines proportionally' is proved satisfactorily when the segments of one of the lines are commensurable. But, says the Professor, should the segments AD and AB be incommensurable, they may still be expressed numerically, and that to any required degree of precision.' This we think, at least, questionable; and we shall continue to think so, until Mr. Leslie furnish numerical expressions for the incommensurable

5

10

quantities of 9 and 21, correct only to the fifteenth place of decimals. The principle to which Mr. Leslie appeals, without quoting it, is to be found in the 5th book of West's Elements:If from any magnitude its half be taken away, and from the remainder its half, and so on continually; there shall at length remain a part of that magnitude less than any magnitude proposed.' But in the use of it he does not guard very carefully against the introduction of error; as we shall speedily shew.

Mr. Leslie's 5th proposition in this Book has this general enunciation: To cut off the successive parts of a straight line.' This we do not comprehend. The particular enunciation of the same problem is, Let AB be a straight line from which it is required to cut off successively the half, the third, the fourth, the fifth, &c.' Here, again, the language conveys a wrong idea: for after having taken away the half, and the third, there is not a fourth of the line left. The fact is, that the Professor simply means to shew how to cut off any aliquot part from a given line, beginning at one of its extremities.

In prop. 7 it is affirmed that If a straight line be divided inter

6

nally and externally in the same ratio, half the line is a mean proportional between the distances of the middle from the two points of unequal section.' This enunciation is absurd, and indeed unintelligible without the aid of the demonstration which accompanies it. Who, before Professor Leslie, ever thought of the external parts of a straight line? Besides, in his external division the part AD is greater than the whole AB. If the learned Professor had not asserted that the axioms are totally useless, and rather apt to produce obscurity, we should have ventured to remind him that his mode of dividing is contradictory to Euclid's 9th axiom, in which it is affirmed that The whole is greater than its part.' Mr. Leslie, however, is so fond of this external division, that he introduces it again in his 20th proposition. At prop. 11, too, we have an equally ridiculous phraseology :- 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.' The truth is, that the line which bisects the vertical angle of a triangle, will also bisect it externally, (if the word externally have any meaning,) that is, it will bisect the complement of the angle to four right angles: but such a line will never effect what Mr. Leslie means to designate by the external division of the segments of the base. The enunciation of our author's friend West, though faulty, is preferable to the above: he says, If a straight line bisect the verticle angle of a triangle, or the angle adjacent to it, and meet the base, the segments of the base will be directly proportional to the other two sides of the triangle; and conversely.' Mr. Leslie adds a scholium to this proposition, which contains an assertion relative to a supposed geometrical truth, glaringly er

roneous.

In the demonstration of prop. 35, The arcs of a circle are proportional to the angles which they subtend at the centre,' our author has again recourse to Mr. West's maxim respecting continual bisections which we have already quoted. Of the two angles ACB, BCD, which he compares, he divides the former by continual bisections, until he obtains an angle ACa 'less than any assignable angle.' Then he applies this angle repeatedly about the same centre C until by its multiplication he fills up the angle BCD nearer than by any possible difference.' Now, of all the reasoning that ever entered a system of geometry, this is surely the worst. For, let the multiple of the indefinitely small angle which is nearer the angle BCD than by any possible difference' be Q, then if m represent the number of duplications of the extremely small angle requisite to produce Q, we have for that small angle.

2m