Book V.

ས.

See N. If there be four magnitudes, and if any equimultiples whatfoever be taken of the first and third, and any equimultiples whatsoever of the fecond and fourth, and if, according as the multiple of the first is greater than the multiple of the fecond, equal to it, or lefs, the multiple of the third is also greater than the multiple of the fourth, equal to it, or less; then the first of the magnitudes is faid to have to the second the fame ratio that the third has to the fourth.

VI.

Magnitudes are faid to be proportionals, when the first has the fame ratio to the second that the third has to the fourth and the third to the fourth the fame ratio which the fifth has to the fixth, and so on, whatever be their number. When four magnitudes, A, B, C, D are proportionals, it is ufual to say that A is to B as C to D, and to write them thus, A: B:: C: D, or thus, A: B-C: D.

VII.

When of the equimultiples of four magnitudes (taken as in the fifth definition) the multiple of the firft is greater than that of the second, but the multiple of the third is not greater than the multiple of the fourth; then the first is faid to have to the second a greater ratio than the third magnitude has to the fourth; and, on the contrary, the third is faid to have to the fourth a lefs ratio than the first has to the fécond.

VIII.

When there is any number of magnitudes greater than two, of which the firft has to the second the fame ratio that the fecond has to the third, and the second to the third the fame ratio which the third has to the fourth, and fo on, the magnitudes are faid to be continual proportionals.

IX.

When three magnitudes are continual proportionals, the fecond is faid to be a mean proportional between the other

two.

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 first has to the fecond, 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 faid 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 also 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.

A ratio which is compounded of two equal ratios is faid to be duplicate of either of these ratios.

COR. Hence, if the three magnitudes A, B, and C are continual proportionals, the ratio of A to C is duplicate of that of A to B, er of B to C: For by the last definition, the ratio of A to C is compounded of the ratios of A to B, and of B to C; but the ratio of A to B is equal to the ratio of B to C, because A, B, C are continual proportionals; therefore the ratio of A to C, by this definition, is duplicate of the ratio of A to B, or of B to C.

Book V.

XII.

A ratio which is compounded of three equal ratios is faid to be triplicate of any one of these ratios; and a ratio which is compounded of four equal ratios is faid to be quadruplicate of any one of these ratios; and fo on, according to the number of equal ratios. COR. If four magnitudes, A, B, C, D be continual proportionals, the ratio of A to D is triplicate of the ratio of A to B, or of B to C, or of C to D. For the ratio of A to D is compounded of the three ratios of A to B, B to C, C to D; and these three ratios are equal to one another, because A, B, C, D are continual proportionals, therefore the ratio of A to D is triplicate of the ratio of A to B, or of B to C, or of C to D.

XIII.

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

Geometers make use 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 fecond has to the fourth; or that the first is to the third, as the second to the fourth: As is fhewn in the 16th prop. of this 5th book.

XV.

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

XVI.

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

fecond, is to the fecond, fourth, is to the fourth.

as the third, together with the Book V. 18th prop. book 5.

18th

XVII.

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

XVIII.

Convertendo, by conversion: When there are four proportionals, and it is inferred, that the firft is to its excefs 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 distance; 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 first is to the last of the firft rank of magnitudes, as the first is to the last 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 second of the other rank; and as the second is to the third of the firft rank, so is the second 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 demonstrated in the 22d prop. book 5.

Ex aequali, in proportione perturbata, feu inordinata; from equality, in perturbate or diforderly proportion; this term is used when the first magnitude is to the fecond of

K 3

the

Book V.

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

IN the demonftrations of this book there are certain figns or characters which it has been found convenient to employ.

of

1. The letters A, B, C, &c. are ufed to denote magnitudes any kind. The letters m, n, p, q, are used to denote numbers only.

2. The fign+ (plus), written between two letters that denote magnitudes or numbers, fignifies the fum of those magnitudes or numbers. Thus, A+B is the fum of the two magnitudes denoted by the letters A and B ; m + n_is_the fum of the numbers denoted by m and n.

3. The fign(minus), written between two letters fignifies the excess of the magnitude denoted by the first of these letters, which is fuppofed the greateft, above that which is denoted by the other. Thus, A-B fignifies the excess of the magnitude A above the magnitude B.

4. When a number, or a letter denoting a number, is written close to another letter denoting a magnitude of any kind, it fignifies that the magnitude is multiplied by the number. Thus, 3A fignifies three times A; mB m times B, or the multiple of B by m. When the number is intended to multiply two or more magnitudes that follow, it is written thus, m.A+B. which fignifies the sum of A and B taken m times; m.A-B is m times the excefs of A above B.

Alfo, when two letters that denote numbers are written clofe to one another, they denote the product of those numbers, when multiplied into one another. product of m into 1.

2

Thus, mn is the

5. The