the antecedent of that which immediately follows, the first of the antecedents has to the last of the consequents a ratio which evidently depends on the intermediate ratios, because if they are determined, it is determined also; and this dependance of one ratio on all the other ratios, is expressed by saying that it is compounded of them. Thus, be any series of ratios, such as described above, the if ABCD ratio For of A to E is said to be compounded of the ratios A. The ratio is evidently determined the ratios by E B'C' A B B' &c. A B &c. because if each of the latter is fixed and invariable, the former cannot change. The exact nature of this dependence, and how the one thing is determined by the other, it is not the business of the definition to explain, but merely to give a name to a relation which it may be of importance to consider more attentively. BOOK VI. DEFINITION II. This definition is changed from that of reciprocal figures, which was of no use, to one that corresponds to the language used in the 14th and 15th propositions, and in other parts of geometry. PROP. XXVII, XXVIII, XXIX. As considerable liberty has been taken with these propositions, it is necessary that the reasons for doing so should be explained. In the first place, when the enunciations are translated literally from the Greek, they sound very harshly, and are, in fact, extremely obscure. The phrase of applying to a straight line, a parallelogram deficient, or exceeding by another parallelogram, is so eliptical, and so little analogous to ordinary language, that there could be no doubt of the proprie. ty of at least changing the enunciations. It next occurred, that the Problems themselves in the 28th and 29th propositions are proposed in a more general form than is necessary in an elementary work, and that, therefore, to take those cases of them that are the most useful, as they happen to be the most simple, must be the best way of accommodating them to the capacity of a learner. The problem which Euclid proposes in the 28th is, "To a given "straight line to apply a parallelogram equal to a given rectilineal figure, and deficient by a parallelogram similar to a given parallelo"gram;" which may be more intelligibly enunciated thus: "To cut " a given line, so that the parallelogram which has in it a given angle, " and is contained under one of the segments of the given line, and a 1 " straight line which has a given ratio to the other segment, may be " equal to a given space;" instead of which problem I have substituted this other : "To divide a given straight line so that the rectangle " under its segments may be equal to a given space." In the actual solution of problems, the greater generality of the former proposition is an advantage more apparent than real, and is fully compensated by the simplicity of the latter, to which it is always easily reducible. The same may be said of the 29th, which Euclid enunciates thus : "To a given straight line to apply a parallelogram equal to a given rec"tilineal figure exceeding by a parallelogram similar to a given paral " lelogram." This might be proposed otherwise: "To produce a "given line, so that the parallelogram having in it a given angle, and "contained by the whole line produced, and a straight line that has a " given ratio to the part produced, may be equal to a given rectilineal "figure." Instead of this, is given the following problem, more simple, and, as was observed in the former instance very little less general. "To produce a given straight line, so that the rectangle contain"ed by the segments, between the extremities of the given line, and "the point to which it is produced, may be equal to a given space." PROP. A, B, C, &c. Nine propositions are added to this Book on account of their utility and their connection with this part of the Elements. The first four of them are in Dr. Simson's edition, and among these Prop. A is given immediately after the third, being, in fact, a second case of that proposition, and capable of being included with it, in one enunciation. Prop. D. is remarkable for being a theorem of Ptolemy the Astronomer, in his Μεγαλη Συνταξις, and the foundation of the construction of his trigonemetrical tables. Prop E is the simplest case of the former; it is also useful in trigonometry, and, under another form, was the 97th, or, in some editions, the 94th of Euclid's Data. The propositions F and Gare very useful properties of the circle, and are taken from the Loci Plani of Apollonius. Prop. H is a very remarkable property of the triangle; and K is a proposition which, though it has been hitherto considered as belonging particularly to trigonometry, is so often of use in other parts of the Mathematics, that it may be properly ranked among the elementary theorems of Geometry. SUPPLEMENT. T BOOK I. PROP. V. and VI, &c. HE 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 6gures inscribed iu it, or described about it according to a certain law, in the the case when those figures approach to the circles so nearly as not to fall short of it, or to exceed 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 more 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 shew, 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: B::a: b, C:D::cd, E:F::e: f, 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:E::a:e; and consequently, B:F::b: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 C:D::c:d, then we cannot have A+C:B+D::a+e:b+d, unless either, A:C::a:e, and B: D::b:d; or A: C:: 6: 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 b:d:: B: L; 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: 6, and also B: L ::b: d, ex 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:::d; therefore ex æquo, A+K:B+L::a+c:b+d; but by hypothesis, A+C:B+D::a+c: b+d, therefore A+KA+C::B+L:B+D. Ss 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 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 essential to the truth of Torelli's proposition. Next, if K be greater than C, then since 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] |