SUPPLEMENT. BOOK I. PROP. V. and VI, &c. THE demonstrations of the 5th and 6th propositions require the method of exhaustions, that is to say, they prove a certain property to belong to the circle, because it belongs to the rectilineal figures inscribed in it, or described about it according to a certain law, in the case when those figures approach to the circles so nearly as not to fall short of it or to exeeed it, by any assignable difference. This principle is general, 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 the 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 most necessary to observe, that 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 that 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, should for a moment have supposed to be true. The proposition 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: Bab, C: D::c:d, E: Fef, then the sum of A, C and E will be to the sum of B, D and F, as the sum of a, c and e, to the sum of b, d and f, or A+C+E: B+D +F::a+c+e: b+d+f. Now, this proposition, which Torelli supposes to be perfectly general, is not true, except in two cases, viz. either first, when A: C: a: c, and A: Eae; and consequently, B: D:: bd, and B: Fb:f; or, secondly, when all the ratios of A to B, C to D, E to F, &c. are equal to one another. To demonstrate this, let us suppose that there are four magnitudes, and four others, thus A B a: b, and CD::c: d, then we cannot have A+C: B+D::a+c: b+d, unless either A: C:: a: c, and B : D::6: d; or A: C:: b: d, and consequently a : b:: c : d. K, A, B, L, Take a magnitude K, such that a:c:: A: K, and another L, such that bd: BL; and suppose it true, that A+C: B+D:: a+c: b+d. Then, because by inversion; K: A::c: a, and, by hypothesis, A: B::a: b, and also B: L::b:d, ex æquo, K: L::c:d; and consequently, K: L:: C: D. Again, because A: K:: a: c, by addition, c, a, b, d. A+K: K::a+c: c; and for the same reason, L:B+L::d:b+d. And, since it has been shewn, that K: L::c:d; therefore, ex æquo, A+K, K, L, B+L, a+c, c, d, b+d. A+K: B+La+c: b+d; but by hypothesis, A+C: B+D::a+c: b+d, therefore A+K: A+C:: B+L: B+D. Now, first, let K and C be supposed equal, then it is evident that L and D are also equal; and therefore, since by construction a: c :: A : K, we have also ac:: A: C; and, for the same reason, b: d:: B : D, and these analogies from the first of the two conditions, of which one is affirmed above to be always essential to the truth of Torelli's proposition. Next, if K be greater than C, then, since A+K: A+C:: B+L: B+D, by division, K-C:K:: L-D: L, therefore, ex æquo, A+K:K::B+L: L, and lastly, by division, A: K:: B: L, or A: B :: K: L, that is, A: BC: D. Wherefore, in this case the ratio of A to B is equal to that of C to D, and consequently, the ratio of a to b equal to that of c to d. The same may be shewn, if K is less than C; therefore in every case there are conditions necessary to the truth of Torelli's proposition, which he does not take into account, and which, as is easily shewn, do not belong to the magnitudes to which he applies it. In consequence of this, the conclusion which he meant to establish respecting the circle, falls entirely to the ground, and with it the general inference aimed against the modern analysis. It will not, I hope, be imagined, that I have taken notice of these circumstances with any design to lessen the reputation of the learned Italian, who has in so many respects deserved well of the mathematical sciences, or to detract from the value of a posthumous work, which by its elegance and correctness, does so much honour to the English editors. But I would warn the student against that narrow spirit which seeks to insinuate itself even into the abstractions of geometry, and would persuade us, that elegance, nay, truth itself, are possessed exclusively by the ancient methods. of demonstration. The high tone in which Torelli censures the modern mathematics is imposing, as it is assumed by one who had studied the writings of Archimedes with uncommon diligence. His errors are on that account the more dangerous, and require to be the more carefully pointed out. PROP. IX. This enunciation is the same with that of the third 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 of the circumference. 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 of 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 to the simplest cases, the ratio which one of them bears to another. 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 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. ABCD, having some 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 arcs of great circles making angles with one another, on opposite sides alternately, the plane angles at the centre of the sphere that stand on these arcs may evidently exceed four right angles, and that too, by multiplying and extending the arcs 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. B C 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. Whether Euclid meant entirely to exclude the consideration of figures of the latter kind, in all that he has said of solids, 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. I cannot conclude this note without remarking, with the historian of the Academy, that it is extremely singular, that not one of all those who had read or explained Euclid before M. le Sage, appears to have been sensible of this mistake. (Memoires de l'Acad. des Sciences, 1756, Hist. p. 77.) A circumstance that renders this still more singular is, that another mistake of Euclid on the same subject, and perhaps of all other geometers, escaped M. le Sage also, and was first discovered by Dr. Simson, as will presently appear. 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 greatly simpler than that of Euclid. It has even an advantage that does not belong to Legendre's, that of requiring no particular construction or determination of any one of the lines, but reasoning from properties common to every part of them. The 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 of before that of Legendre. The solution given here is more simple than his, or than any other that I have yet met with it also leads more easily, if it be required, to a trigonometrical computation. |