Book I. ral, and is the only one by which we can possibly compare curvilineal with rectilineal spaces, or the length of curve lines with the length of straight lines, whether we follow the methods of the ancient or of the modern geometers. It is, therefore, a great injustice to the latter methods to represent them as standing on a foundation less secure than the former; they stand in reality on the same, and the only difference is, that the application of the principle, common to them both, is more general and expeditious in one case than in the other. This identity of principle, and affinity of the methods used in the elementary and the higher mathematics, it seems the more necessary to observe, because some learned mathematicians have appeared not to be sufficiently aware of it, and have even endeavoured to demonstrate the contrary. An instance of this is to be met with in the preface of the valuable edition of the works of Archimedes, lately printed at Oxford. In that preface Torelli, the learned commentator, whose labours have done so much to elucidate the writings of the Greek geometer, but who is so unwilling to acknowledge the merit of the modern analysis, undertakes to prove that it is impossible, from the relation which the rectilineal figures inscribed in, and circumscribed about, a given curve, have to one another, to conclude any thing concerning the properties of the curvilineal space itself, except in certain circumstances which he has not precisely described With this view he attempts to show that, if we are to reason from the relation which certain rectilineal figures belonging to the circle have to one another, notwithstanding those figures may approach so near to the circular spaces within which they are inscribed as not to differ from them by any assignable magnitude, we shall be led into error, and shall seem to prove that the circle is to the square of its diameter exactly as 3 to 4. Now, as this is a conclusion which the discoveries of Archimedes himself prove so clearly to be false, Torelli argues that the principle from which it is deduced must be false also; and in this he would no doubt be right, if his former conclusion had been fairly drawn. But the truth is, that a very gross paralogism is to be found in that part of his reasoning, where he makes a transition from the ratios of the small rectangles, inscribed in the circular spaces, to the ratios of the sums of those rectangles, or of the whole rectilineal figures. In doing this, he takes for granted a proposition, which, it is wonderful, that

one who had studied geometry in the school of Archimedes, Book I. should for a moment have supposed to be true.

PROP. IX.

This enunciation is the same as that of the third proposition of the Dimensio Circuli of Archimedes; but the demonstration is different, though it proceeds, like that of the Greek geometer, by the continual bisection of the 6th part

the circumference.

of

The limits of the circumference are thus assigned; and the method of bringing it about, notwithstanding many quantities are neglected in the arithmetical operations, that the errors shall in one case be all on the side of defect, and in another all on the side of excess, (in which I have followed Archimedes), deserves particularly to be observed, as affording a good introduction to the general methods of approximation.

BOOK II.

DEF. VIII. and PROP. XX.

SOLID angles, which are defined here in the same manner as in Euclid, are magnitudes of a very peculiar kind, and are particularly to be remarked for not admitting that accurate comparison one with another, which is common in the other subjects of geometrical investigation. It cannot, for example, be said of one solid angle, that it is the half, or the double of another solid angle, nor did any geometer ever think of proposing the problem of bisecting a given solid angle. In a word, no multiple or sub-multiple of such an angle can be taken, and we have no way of expounding, even in the simplest cases, the ratio which one of them bears to another.

Book II.

Suppl.

In this respect, therefore, a solid angle differs from every other magnitude that is the subject of mathematical reasoning, all of which have this common property, that multiples and sub-multiples of them may be found. It is not our business here to inquire into the reason of this anomaly, but it is plain, that on account of it, our knowledge of the nature and the properties of such angles can never be very far extended, and that our reasonings concerning them must be chiefly confined to the relations of the plane angles, by which they are contained. One of the most remarkable of those relations is that which is demonstrated in the 21st of this book, and which is, that all the plane angles which contain any solid angle must together be less than four right angles. This proposition is in the 21st of the 11th of Euclid.

A

D

This proposition, however, is subject to a restriction in certain cases, which, I believe, was first observed by M. le Sage of Geneva, in a communication to the Academy of Sciences of Paris, in 1756. When the section of the pyramid formed by the planes that contain the solid angle is a figure that has none of its angles exterior, such as a triangle, a parallelogram, &c. the truth of the proposition just enunciated cannot be questioned. But, when the aforesaid section is a figure like that which is annexed, viz. ABDC, having some of its angles, such as BDC, exterior, or, as they are sometimes called, re-entering angles, the proposition is not necessarily true; and it is plain that in such cases the demonstration which we have given, and which is the same with Euclid's, will no longer apply. Indeed, it were easy to show, that on bases of this kind, by multiplying the number of sides, solid angles may be formed, such that the plane angles which contain them shall exceed four right angles by any quantity assigned. An illustration of this from the properties of the sphere is perhaps the simplest of all others. Suppose that on the surface of a hemisphere there is described a figure, bounded by any number of arches of great circles making angles with one another, on opposite sides alternately; the plane angles at the centre of the sphere, which stand on these arches, may

B

1

8

evidently exceed four right angles, and that, too, by multi- Book II. plying and extending the arches, in any assigned ratio. Now, these plane angles contain a solid angle at the centre of the sphere, according to the definition of a solid angle.

We are to understand the proposition in the text, therefore, to be true only of those solid angles in which the inclination of the plane angles are all the same way, or all directed toward the interior of the figure. To distinguish this class of solid angles from that to which the proposition does not apply, it is perhaps best to make use of this criterion, that they are such, that when any two points whatsoever are taken in the planes that contain the solid angle, the straight line joining those points falls wholly within the solid angle; or thus, they are such, that a straight line cannot meet the planes which contain them in more than two points. It is thus, too, that I would distinguish a plane figure that has none of its angles exterior, by saying, that it is a rectilineal figure, such that a straight line cannot meet the boundary of it in more than two points.

We, therefore, distinguish solid angles into two species, one in which the bounding planes can be intersected by a straight line only in two points, and another where the bounding planes may be intersected by a straight line in more than two points. To the first of these the proposition in the text applies, to the second it does not.

SO

Whether Euclid meant entirely to exclude the consideration of figures of the latter kind, in all that he has said of lids, and of solid angles, it is not now easy to determine. It is certain that his definitions involve no such exclusion; and as the introduction of any limitation would considerably embarrass these definitions, and render them difficult to be understood by a beginner, I have left it out, reserving to this place a fuller explanation of the difficulty,

PROP. IV.

This very elegant demonstration is from Legendre, and is much easier than that of Euclid,

The demonstration given here of the 6th is also much simpler than that of Euclid. It has even an advantage which does not belong to Legendre's, that of requiring no particular construction or determination of any one of the lines, but

Suppl. reasoning from properties common to every part of them. This simplification, when it can be introduced, which, however, does not appear to be always possible, is perhaps the greatest improvement that can be made on an elementary demonstration.

PROP. XIX.

The problem contained in this proposition, of drawing a straight line perpendicular to two straight lines not in the same plane, is certainly to be accounted elementary, although not given in any book of elementary geometry that I know before that of Legendre. The solution given here is more simple than his or any other that I have yet met with: it also leads more easily, if it be required, to a trigonometrical computation.

BOOK III.

Book III.

DEF. II. & PROP. I.

THESE relate to similar and equal solids, a subject on which mistakes have prevailed not unlike to that which has just been mentioned. The equality of solids, it is natural to expect, must be proved like the equality of plane figures, by showing that they may be made to coincide, or to occupy the same space. But, though it be true that all solids which can be shown to coincide are equal and similar, yet it does not hold conversely, that all solids which are equal and similar can be made to coincide. Though this assertion may appear somewhat paradoxical, yet the proof of it is extremely simple.

Let ABC be an isosceles triangle, of which the equal sides are AB and AC; from A draw AE perpendicular to the