Therefore it is not less than any assignable angle,' for it may be assigned. Hence the demonstration is self-contradictory: if the angle ACa be less than any assignable angle, any person except this profound and inventive geometer' would see at once that no multiple of it can be finite.

Mr. Leslie subjoins an Appendix to his Elements, divided into two parts. In the first, which is avowedly taken principally from a scarce tract by Schooten, problems are solved by means of the ruler only; in the second part, drawn from Mascheroni's Geometry of the Compass, problems are solved by means of compasses only. Here, a very useful problem is omitted, viz. To describe a circle to pass through three given points.' This is the more to be regretted, because the construction by the help of the compasses alone is more simple than that usually given in the Elements, and because it follows at once as a corollary to our author's 19th proposition.

6

It is now time to direct our attention to the Treatise on Geometrical Analysis, which the Professor says, is the part that has cost him the greatest pains.' It is divided into three books, of which the first consists of a series of the choicest problems rising above each other in gradual succession.' With the demonstrations of these we cannot profess ourselves to be always satisfied. Thus, prop. 10, From two given points in the circumference of a given circle, to inflect, to another point in the circumference, straight lines that shall have a given ratio.' The analysis and composition here given are accurate; but it would have been advantageous to the student, had Mr. Leslie shewn that this problem is essentially the same, as having the base of a plain triangle given, together with the vertical angle and the ratio of the legs, to find the triangle.' A construction like the following might then have been given. Let Q be the given angle, or in other words, the angle in the segment. Set off upon the lines whose inclination constitutes the angle, from the point Q, the lines QM, QN, in the assigned ratio, and join MN: then upon the given base AB make a triangle ABC similar to MNQ, by VI. 31, of this work, and the thing is effected. The demonstration follows at once from VI. 13.

Prop. 17. Two straight lines being given, to draw, through a given point, another straight line, cutting off segments which are together equal to a given straight line.' The demonstration of this is very tedious; and after all the author has not shewn between what limits the problem is possible.

The Second Book on Geometrical Analysis comprises restitutions of the attempts of Apollonius and his illustrious contemporaries.' The first four propositions relate to what is technically

termed the 'section of ratio; the general problem being, 'Through a given point to draw a straight line intercepting seg ments on two straight lines, which are given in position, from given points and in a given ratio.' Propositions 5 to 10 inclusive relate to the section of space:' here the general problem is, "Through a given point, to draw a straight line cutting off segments from given points on straight lines given in position, and which shall contain a rectangle equal to a given space.' Nine of the 10 propositions which are devoted to these two general problems deserve commendation: we are sorry we cannot say as much of the remainder of the disquisitions on analysis.

The problem concerning 'inclinations,' or that in which it is proposed to draw a right line through a given point, so that the part thereof intercepted between two given lines may be of an assigned length,' is considered by Mr. Leslie in props. 20, 26, of Book II.-On these we have to remark, that in prop. 21, no limitation is traced; in prop. 22, there is no subdivision of cases; in prop. 23 the cases and limitations are not specified; in props. 24, 25, no limitations stated; though these are absolutely necessary to the complete resolution of the problem. Two important propositions relating to this inquiry are also omitted, namely, those which make the 4th and 6th problems in Burrow's Restitution.

The remainder of the 2d book on analysis is employed about the problem of Tangencies,' which is thus enunciated in Halley's Translation of Pappus :-E punctis rectis et circulis, quibuscunque tribus positione datis, circulum ducere per singula data puncta, qui, si fieri possit, contingat etiam datas lineas.' This naturally falls into ten distinct propositions, which, if a point be represented by (.), a line by (1), and a circle by (0), may be stated very briefly, according to the several data, in the following order:-(.. 1), (. 11), (1 10), (. 10), ( 100 ), (.. 0) (.00), (000), (...), (111). Mr. Leslie has solved all but the 5th, 7th, and 8th of our enumeration. Prop. 27, Book II. agrees with our (..1). The author properly enough divides it into two cases, but does not notice that the second case (viz. when the given line is not parallel to the line joining the given points) is susceptible of two answers. As to the remaining problems we are told that they are easily reduced to the cases already solved:' yet Mr. Leslie seems to think otherwise in his notes; where he gives the prop. marked by our (.00). Even, in this supplementary proposition we trace his usual deficiencies, for he has only presented one case out of three into which the problem manifestly divides. With regard to the other problems omitted, viz. (100), and (000), we have only to observe that each comprises four cases, the construction of which ought by all

means to have been given, in order to render this part of the work an adequate representation of the problem of tangencies.

Book the Third commences with seventeen propositions relative to Plane Loci.' Here, of course, our author has availed himself of the labours of Dr. Simson, so that this part of the work may be expected to partake of the excellencies and defects of that great geometer's disquisition on this subject. The general problem discussed by Apollonius, according to the enunciation of Pappus, is this:

Any number of right lines being given in position, if from any point there be drawn to right lines given in position, other right lines under given angles; and if the rectangle under one of the right lines so drawn, and a right line given in magnitude, augmented by the rectangle under another of those right lines and another right line given, is equal to the rectangle made by a third of those right lines and another right line given in magnitude, and so on: that point is in a right line given in position.'

Such is the general problem as originally proposed: which is widely different from that given by Mr. Leslie, he having, rightly enough, included circles among the Loci. Still his deduction of propositions is excessively defective, as he has neither considered the case where three parallel lines are given in position, nor those in which the two, three, or more given lines diverge from different points: besides this, in what he has done his method has the inconvenience of proceeding successively from a given number of right lines, to the number next greater by unity; while the mutual connection between the several cases is never shewn perspicuously. Thus it happens that the most curious properties respecting plane loci are not exhibited at all, and especially the three following:

1st. Any number of right lines being given in position on a plane, and any other number of lines also being given in position on the same plane: a right line is the locus (if it be determinate) of points from each of which demitting perpendiculars upon all those right lines; the sum of the rectangles of the perpendiculars let fall upon the first set of right lines, into right lines given in magnitude; is to the sum of the rectangles of the perpendiculars demitted upon the remaining right lines into so many other right lines given in magnitude, in a given ratio.

2d. Any number of right lines being given in magnitude and position upon a plane, a right line is the locus of the vertices of triangles having those right lines for bases, and of which the aggregate is given. This is a generalization of Euclid I. 37.

3d. Any number of points being given in position on a plane, and any number of right lines given in position on the same plane:

a circumference of a circle is the locus of points from each of which drawing right lines to the points given, and perpendiculars to the right lines given in position, the sum of spaces which have to the squares of the first right lines given ratios, is to the sum of the rectangles of the perpendiculars into right lines given in magnitude, in a given ratio. This is a generalization of propositions 5 and 7, lib. ii. of Apollonius de Locis Planis.

We may farther remark, respecting Professor Leslie's propositions on Plane Loci, that some of the demonstrations are merely algebraical, exhibited in a very disdavantageous form.

The remainder of Book III. on Geometrical Analysis relates to porisms, and isoperimetrical figures. With respect to the seven propositions devoted to the latter, we have a few remarks to offer.

Prop. 26. In a straight line given in position, to find a point, whose distances from two given points on the same side, shall together be the least possible.' In the demonstration of this propo

sition the synthesis is omitted

Prop. 27. Straight lines drawn from two given points to the circumference of a given circle are the least possible, when they make equal angles with a tangent applied at the point of inflexion.' This proposition is demonstrated, in our author's estimation, by referring to 1. 19, of his elements: but the proposition referred to is not applicable to the point in hand, at least without additional steps; the proposition, therefore, is not demonstrated.

Prop. 28. To find a point, whose distances from three given points are [together] the least possible.' The demonstration depends upon that of the preceding proposition, and is, of course, inadmissible.

Prop. 30. To find a triangle with a given perimeter, and standing on a given base, which shall contain the greatest area.' The demonstration of this is made to rest upon two assertions, one of which is without proof, the other a disputable proposition which is ás often false as true. First it is affirmed, that with a given base, and the area 'a maximum, the corresponding altitude must evidently be the greatest possible.' Why is it evident? Mr. Leslie has no where proved, that when the bases of triangles, are given, the surfaces are as the altitudes. 2dly. Lines inflected from the points A and C, to any point in the parallel DE, must be together greater than those drawn to any other parallel.' This is only true under certain conditions, and when true, requires proof; so that, notwithstanding the Professor's absolute assertion, this proposition must be considered as undemonstrated.

Prop. 31. If a polygon have all its sides given, except one,

it

will contain the greatest area, when it can be inscribed in a semicircle of which that indeterminate side is the diameter.' In this proposition also, the demonstration is made to stand upon the merely repeated affirmation that maximum triangles with given bases have the greatest possible altitudes. But Mr. Leslie ought to know that repeating an assertion and establishing its truth, are not perfectly similar operations.

Prop. 32. A circle contains, within a given perimeter, the greatest possible area.' Here again Professor Leslie's reasoning is unsatisfactory. He proves that an isoperimetrical figure has its area always increased by doubling the number of its sides. Continuing this duplication, therefore, the regular polygons which arise in succession will have their capacity perpetually enlarged. Whence the circle, as it forms the limit, or extreme boundary of all those polygons, must with a given circumference, contain the greatest possible space.' This reasoning, as far as it relates to polygons, is correct; but when it is transferred to circles in the loose manner of Mr Leslie, it is highly ungeometrical. However much the number of sides of the polygons be augmented, they still remain polygons, and what is predicated of them cannot, by any process of fair deduction, be transferred to circles. The dif ficulty is not overcome by talking of limits, with our author, as 'extreme boundaries,' the process is as absurd as it would be to infer the nature of a fence from a chemical analysis of the soil which it encloses. T. Simpson and Legendre managed this matter much better; but they had not learnt that it was 6 more philosophical to invert the usual procedure.'

Indeed the theorem, so far as relates to rectilinear figures, may be demonstrated very easily from a few propositions, each of which may be established by a perfectly elementary process. Thus

(a). Of all figures of the same number of sides, and of the same perimeter, the greatest is regular or equilateral.

(b). If a circle and a figure that may be circumscribed about another circle are isoperimeters, the surface of the circle is a geometrical mean between that figure, and a similar figure circumscribed about the first circle.

(c). If a circle and a figure circumscribable about another circle are equal; the perimeter of that figure is a geometrical mean between the periphery of the circle, and the perimeter of a similar figure circumscribed about that circle.

Now let C represent a circle, F a figure isoperimeter to that circle, and circumscribable about some circle, and F' a figure similar to F and circumscribed about C. Then...

Prop. (b.) F: C: C: F'; where CF.. F

C.