the other: And each of them is contained by four plane Book VII. angles, which are equal to one another, each to each, or are the felf fame, as has been proved: And indeed, there may be innumerable folid angles all unequal to one another, which are each of them contained by plane angles that are equal to one another, each to each: It is likewife manifeft, that the before mentioned folids are not fimilar, fince their solid angles are not all equal.' PRO P. XXXIII. The principal departure from the method of Euclid, introduced into this Book, refpects the pyramid; in treating of which, the method of exhaustions is first employed, as in this propofition. The demonftration ufed here has the advantage of applying very generally to all the folids which require that method, and which can be treated of in the Elements; and it has also a very close and evident affinity to the methods used in the higher geometry. BOOK VIII. THE object of this Book being to compare with one ano-Book VIII. ther, and with the parallelepiped, the folids that depend on the circle, it is neceffary to begin with the quadrature of that figure, or the comparison of the space contained within it with rectilineal figures. The firft eight propofitions relate entirely to this fubject, and contain the rules both for finding the length of the circumference nearly, and also the space contained within it, and in demonftrating them, I have followed partly the method of Thomas Simpfon, in his Elements, and partly that of Archimedes, in his Dimenfio circuli. As the foundation of the propofition for finding the D d length Book VIII. length of the circumference, an Axiom is prefixed to the be ginning of this Book, the fame in effect with that which Ar chimedes employs as the bafis of his investigation, though fomewhat more concifely expreffed. I have already take notice of this Axiom, in the notes on parallel lines, (p. 372.) as a propofition which it has been found impoffible to de monftrate, though it certainly might be expected to follow neceffarily from the ideas of a straight, and a curve line. Another thing remarkable concerning this Axiom is, that in the fimpleft cafe of it, when the figures are rectilineal, it admits of demonstration; for it then coincides with the 20th and 21ft of the first of Euclid, in the former of which it is fhewn, that any two fides of a triangle are greater than the third, and in the latter, that of two triangles which have the fame base, that which is within the other, has the fum of its fides the least. But when, instead of two fides of a triangle, we have a curve AEB, and a straight line AB terminated in the fame points A and B, it then ceases to be poffible to prove, that AEB is greater than AB; as alfo, that any curve line which includes AEB, and is ter- A minated in A and B, is greater than E AEB; and the truth of both these propofitions must be taken for granted. At least, after Archimedes has confidered them as Axioms, we may reasonably despair of ever seeing them demonftrated. Here, therefore, we find ourselves under the neceffity of taking for granted a general propofition, after having demonftrated the fimpleft cafe of it. For all this, the trouble taken, to demonstrate that cafe, ought not to be confidered as fuperfluous, for we are thereby prepared to admit the general Axiom, and have the fatisfaction of obferving its perfect agreement with truths, previously demonftrated. It is probably from the circumftance of a curve line having no definition, but one which is merely negative, viz. that it is a line of which no part is a ftraight line, that the impoffibility of demonstrating this propofition takes its rise. PROP Book VIII. PROP. IV. V. &c. The demonstrations of the 5th and 6th propofitions require the method of exhauftions, that is to fay, they prove a certain property to belong to the circle, because it belongs to the rectilineal figures infcribed in it, or described about it according to a certain law, in the case when those figures approach to the circle fo nearly as not to fall fhort of it, or to exceed it by any affignable difference. This principle is general, and is the only one by which we can poflibly compare curvelineal, 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 injuftice to the latter methods to reprefent them as standing on a foundation less fecure than the former; they stand in reality on the fame, and the only difference is, that the application of the principle, common to them both, is more general and expeditious in the one cafe than in the other. This identity of principle, and affinity of the methods ufed in the elementary and the higher mathematics, it seems the more neceffary to obferve, that fome learned mathematicians have appeared not to be fufficiently aware of it, and have even endeavoured to demonstrate the contrary. An inftance 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, whofe labours have done fo much to elucidate the writings of the Greek geometer, but who is fo unwilling to acknowledge the merit of the modern analyfis, undertakes to prove, that it is impoffible from the relation which the rectilineal figures infcribed in, and circumfcribed about, a given curve, have to one another, to conclude any thing concerning the properties of the curvelineal space itself, except in certain circumftances which he has not precifely defcribed. With this view he attempts to fhew, that if we are to reason from Dd 2 爨 the Book VIII. the relation which certain rectilineal figures belonging to the 'circle have to one another, notwithstanding that thofe figures may approach fo near to the circular spaces within which they are infcribed, as not to differ from them by any affignable magnitude, we fhall be led into error, and shall seem to prove, that the circle is to the fquare of its diameter exactly as 3 to 4. Now, as this is a conclufion which the difcoveries of Archimedes himfelf, prove fo clearly to be falfe, Torelli argues, that the principle from which it is deduced must be falfe also; and in this he would no doubt be right, if his former conclufion had been fairly drawn. But the truth is, that a very grofs paralogifm is to be found in that part of his reasoning, where he makes a tranfition from the ratios of the fmall rectangles, infcribed in the circular fpaces, to the ratios of the fums of thofe rectangles, or of the whole rectilineal figures. In doing this, he takes for granted a propofition which, it is wonderful, that one who had studied geometry in the school of Archimedes, should for a moment have supposed to be true. The propofition taken in the fimpleft view of it is this: If A, B, C, D, E, F, be any number of magnitudes, and a, b, c, d, e, f, as many others; and if A: B:: a: b C:D:: :4, E: F ef, then the fum of A, C and E A: Eae; and confequently, B: Fb:f; or, fecondly, when all the ratios of A to B, C to D, E to F, &c. are equal to one another. To demonftrate this, let us fuppofe that there are four magnitudes, and four others, thus, A: B: a: b, and C: D::c:d, then we cannot have A+C: B+D:: a+c : b+d, unless either, A: C: :a: c, and B: Db:d; or A: C:: b: d, and confequently a: b:: c: d. 2 Take Take a magnitude K, fuch that a:c:: A: K, and another Book VIII L, fuch that b:d:: B:L; and fuppofe it true, that A+C : B+D : :a+c:b+d. Then, becaufe by inverfion, K: A :: ca, and, by hypothefis, A: Bab, and alfo B:L::b:d, ex æquo, K : L :: cd; and confequently, K:L:: C; D, Again, because A: K:; a: c, by addition, K, A, B, L. c, a, b, d. A+K: K :: a+c: c; and, for the same reason, L: B+L::d: b+d. And, fince it has been fhewn, that KL::c:d; therefore, ex æquo, A+K, K, L, B+L. a+c, c, d, b+d. A+K: B+L:: a+c: b+d; but by hypothefis, Now, firft, let K and C be fuppofed equal, then, it is evident, that L and D are alfo equal; and therefore, fince by construction a:c:: A:K, we have also a:c::A:C; and, for the same reason, b: d :: B : D, and these analogies form the first of the two conditions, of which one is affirmed above to be always effential to the truth of Torelli's proposition. Next, if K be greater than C, then, fince A+K : A+C :: B+L: B+D, by divifion, A+K: K-C:: B+L: L-D. But, as was fhewn, Wherefore, in this cafe the ratio of A to B is equal to that of C to D, and confequently, the ratio of a to b equal to that of |