Book V.

N.

X. When there is any number of magnitudes of the fame kind, the first is faid to have to the last of them the ratio compounded of the ratio which the firft has to the second, and of the ratio which the second has to the third, and of the ratio which the third has to the fourth, and fo on unto the laft magnitude.

For example, if A, B, C, D be four magnitudes of the fame kind, the first A is said to have to the last D, the ratio compounded of the ratio of A to B, and of the ratio of B to C, and of the ratio of C to D; or, the ratio of A to D is faid to be compounded of the ratios of A to B, B to C, and C to D.

And if A: B::E: F; and B: C::G: H, and C: D::K: L, then, fince by this definition A has to D the ratio compounded of the ratios of A to B, B to C, C to D; A may alfo be faid to have to D the ratio compounded of the ratios which are the fame with the ratios of E to F, G to H, and K to L.

In like manner, the fame things being fuppofed, if M has to N the fame ratio which A has to D, then, for fhortness fake, M is faid to have to N a ratio compounded of the fame ratios, which compound the ratio of A to D; that is, a ratio compounded of the ratios of E to F, G to H, and K to L.

XI.

If three magnitudes are continual proportionals, the ratio
of the first to the third is faid to be duplicate of the ratio
of the first to the fecond.
"Thus, if A be to B as B to C, the ratio of A to C is faid
to be duplicate of the ratio of A to B. Hence, fince by
the last definition, the ratio of A to C is compounded of
the ratios of A to B, and B to C, a ratio, which is com-
pounded of two equal ratios, is duplicate of either of
these ratios."

XII.

XII.

If four magnitudes are continual proportionals, the ratio of the first to the fourth is faid to be triplicate of the ratio of the first to the second, or of the ratio of the second to the third, &c.

"So alfo, if there are five continual proportionals; the ratio of the first to the fifth is called quadruplicate of the ratio of the first to the second; and fo on, according to the number of ratios. Hence, a ratio compounded of three equal ratios is triplicate of any one of thofe ratios; a ratio compounded of four equal ratios quadruplicate," &c.

XIII.

In proportionals, the antecedent terms are called homologous to one another, as alfo the confequents to one another.

Geometers make ufe of the following technical words to fignify certain ways of changing either the order or magnitude of proportionals, fo as that they continue still to be proportionals,

XIV.

Permutando, or alternando, by permutation, or alternately; this word is used when there are four proportionals, and it is inferred, that the first has the fame ratio to the third which the second has to the fourth; or that the first is to the third as the fecond to the fourth: See prop. 16th of this book.

XV.

Invertendo, by inverfion: When there are four proportionals, and it is inferred, that the second is to the first, as the fourth to the third. Prop. A. book 5.

XVI.

Componendo, by compofition: When there are four proportionals, and it is inferred, that the firft, together with the

Book V.

Book V.

fecond, is to the fecond, as the third, together with the fourth, is to the fourth. 18th prop. book 5.

XVII.

Dividendo, by divifion: When there are four proportionals, and it is inferred, that the excess of the first above the second, is to the fecond, as the excefs of the third above the fourth, is to the fourth. 17th prop. book 5.

XVIII..

Convertendo, by converfion: When there are four proportionals, and it is inferred, that the first is to its excess above the fecond, as the third to its excefs above the fourth. Prop. D. book 5.

XIX.

Ex aequali (fc. diftantia), or ex aequo, from equality of diftance; when there is any number of magnitudes more than two, and as many others, fo that they are proportionals when taken two and two of each rank, and it is inferred, that the firft is to the laft of the first rank of magnitudes, as the firft is to the laft of the others: Of this there are the two following kinds, which arife from the different order in which the magnitudes are taken two and

two.

XX.

Ex aequali, from equality; this term is ufed fimply by itself, when the firft magnitude is to the second of the first rank, as the first to the fecond of the other rank; and as the fècond is to the third of the first rank, fo is the fecond to the third of the other; and fo on in order, and the inference is as mentioned in the preceding definition; whence this is called ordinate proportion. It is demonftrated in the 22d prop. book 5.

XXI.

Ex aequali, in proportione perturbatą, feu inordinata; from equality, in perturbate, or diforderly proportion; this term is ufed when the first magnitude is to the second of

the

the first rank, as the last but one is to the laft of the fecond Book V. rank; and as the second is to the third of the first rank, fo is the laft but two to the last but one of the second rank; and as the third is to the fourth of the firft rank, fo is the third from the laft, to the laft but two, of the second rank; and fo on in a cross, or inverfe, order; and the inference is as in the 19th definition. It is demonftrated in the 23d prop. of book 5.

AXIOM S.

I.

EQUIMULTIPLES of the fame, or of equal magnitudes, are

equal to one another.

II.

Those magnitudes of which the fame, or equal magnitudes, are equimultiples, are equal to one another.

III.

A multiple of a greater magnitude is greater than the fame multiple of a lefs.

IV.

That magnitude of which a multiple is greater than the fame multiple of another, is greater than that other magnitude.

Book V.

PROP. I.

THEOR.

a Ax. 2. 1.

I'

F any number of magnitudes be equimultiples of as many others, each of each, what multiples foever any one of the firft is of its part, the fame multiple is the fum of all the firft of the fum of all the reft.

Let any number of magnitudes A, B, and C be equimultiples of as many others, D, E, and F, each of each; A+B +C is the fame multiple of D+E+F, that A is of D.

Let A contain D, B contain E, and C contain F, each the fame number of times, as, for inftance, three times. Then, because A contains D three times, A=D+D+D.

For the fame reason, BE+E+E;

And also,

C=F+F+F.

Therefore, adding equals to equals, a A+B+C is equal to D+E+F, taken three times. In the fame manner, if A, B, and C were each any other equimultiple of D, E, and F, it would be fhewn that A+B+C was the fame multiple of D+E+F. Therefore, &c. Q. E. D.

COR. Hence, if m be any number, mD+mE+mF m.D+E+F. For mD, mE, and mF are multiples of D, E, and F by m, therefore their fum is also a multiple of D+E+F by m.

PROP. II. THEOR.

F to a multiple of a magnitude by any number, a multiple of the fame magnitude by any number be added, the fum will be the fame multiple of that magnitude that the fum of the two numbers is of unity.

Let A mC, and BC; A+B=m+n.C.

For, fince AC, A=C+C+C+ &c. C being repeated m times. For the fame reason, B=C+C+ &c. C being repeated