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Book VIII.

PROP. IV. V. &c.

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The demonftrations 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 cafe when thofe 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 poffibly 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 injustice to the latter methods to reprefent them as standing on a foundation lefs 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 demonftrate 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 analysis, 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 precisely defcribed. With this view he attempts to fhew, that if we are to reafon from

Dd 2

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 fpaces within which they are infcribed, as not to differ from them by any affignable magnitude, we shall be led into error, and shall feem 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 himself, prove fo clearly to be false, Torelli argues, that the principle from which it is deduced must be falfe alfo; 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 reafoning, where he makes a tranfition from the ratios of the small 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 ftudied geometry in the fchool 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::c:d,

E: Fef, then the fum of A, C and E will be to the fum of B, D and F, as the fum of a, c and e, to the fum of b, d and f; or A+C+E :B+D+F::a+c+e: b+d+f. Now, this propofition, which Torelli fupposes to be perfectly general, is not true, except in two cafes, viz. either firft, when A: Cac, and

A: Eae; and confequently,

B: D :: b; d, and

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

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C: D: cd, 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.

Take

Take a magnitude K, such 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, because by inverfion, K: A :: c: a, and, by hypothefis, A : B :: a:b, and alfo B:L::b: d, ex æquo, K: L:: c:d; and consequently, 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 fame reason,
B+L: L:: b+d: d, or, by inverfion,

L: B+L::d: b+d. And, fince it has been

shewn, that K: L:: cd; 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,
A+C:B+D ::`a+c: b+d, therefore
A+K: A+C:: B+L: B+D.

Now, first, let K and C be fuppofed equal, then, it is evident, that L and D are alfo equal; and therefore, fince by conftruction a:c:: A:K, we have also a: c:: A: C; and, for the fame reafon, b: d :: B: D, and thefe analogies form the first of the two conditions, of which one is affirmed above to be always effential to the truth of Torelli's propofition.

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 shewn,
KL:: CD, by converfion and alternation,
K-C: KL-D: L, therefore, ex æquo,
A+K: K:: B+L: L, and, laftly, by divifion,
AKB L, or A: B :: K : L, that is,
A: B:: C: D.

Wherefore, in this case 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

Book VIII of c to d. The fame may be fhewn, if K is lefs than G; there fore, in every cafe there are conditions neceffary to the truth of Torelli's propofition, which he does not take into account, and which, as is eafily fhewn, do not belong to the magnitudes to which he applies it..

In consequence of this, the conclufion that is meant to be eftablished respecting the circle falls entirely to the ground, and with it the more general inference that was aimed against the modern analyfis.

It will not, I hope, be imagined, that I have taken notice of thefe circumstances with any defign to leffen the reputation of the learned Italian, who has in fo many refpects deferved well of the mathematical fciences, or to detract from the value of a posthumous work, which, by its elegance and correctnefs, does so much honour to the English editors. But I would warn the student against that spirit of party, which feeks to introduce itself even into the investigations of geometry, and to perfuade us, that elegance, and even truth, are qualities poffeffed exclufively by the ancient methods of demonftration. The high tone in which Torelli cenfures the modern mathematics, is the more calculated to produce an opinion of this kind, that it is affumed by one who had ftudied the writings of Archimedes with fo much diligence; and therefore, the obfervations, that have been made above, may be useful, by teaching us to listen with caution to his decifions.

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This propofition is the foundation of the application of arithmetic to geometry, and m may be faid to be the truth that connects together thefe two branches of the mathematics, though, in elementary treatifes, I think, it has for the most part been omitted. In no cafe do we compute, in numbers, the area of any figure, or even calculate the length of a fide of a triangle, from the other parts which determine it being given, without having recourfe to this Theorem. If, for inftance, from having two fides of a right angled triangle as ABC expreffed in numbers, we would compute the length

of

of the remaining fide BC, we must make ufe of this propo- Book VIIL

fition, as well as of the 47th of the firft Book. If, for inftance, AB=6, and AC=8, and, if it be required to find BC; then, fince. AB is to AC as 6 to 8, the fquare of AB will be to the fquare of AC, by this propofition, as 6x6 to 8x8, or as 36 to 64, and therefore, alfo the fum of the fquares on AB and AC will be to the fquare of AB as 100 to

B

36. But the fquares of AB and AC are equal to the fquare of BC; therefore, the fquare of BC is to the fquare of AB as 100 to 36, and therefore, by the fecond corollary, BC is to BA as 10 to 6. It is the fame in other cafes; and, though the steps may not all be taken fo regularly as is done here, they are nevertheless as certainly implied.

PROP. VIII.

This enunciation is the fame with that of the third of the Dimenfio Circuli of Archimedes; but the demonstration is different, though it proceeds, like that of the Greek geometer, by the continual bifection of the 6th part of the circumference. In this propofition a particular notation is used, by putting the fign+after a number, to denote that something is to be added to it, and the fign to denote the contrary.

Thus, when it is faid, that AQ is to AD as 866.0254+ to 1000, this fignifies, that AQ is to AD as a number greater than 866.0254 is to 1000, and it is the fame thing with faying, that AQ has to AD a greater ratio than 866.c254 to 1000. In the fame manner, when it is demonftrated in the second part of the propofition, that LN is to CD as 32.71927- to 1000, it is meant, that LN is to CD as a number less than 32.71927 to 1000, or that LN has to CD a lefs ratio than that of 32.71927 to 1000.

The advantage that refults from this mode of expreffion is, that when we are to reafon about the fums or differences of two lines, of which the ratio is thus expreffed, it can be done

by

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