And if A has to B the same ratio which E has to F; and B Book V. to C, the same ratio that Ghas to H; and C to D, the same that K has to L; then, by this definition, A is said to have to D the ratio compounded of ratios which are the same with the ratios of E to F, G to H, and K to L: And the same thing is to be understood when it is more briefly expressed, by saying, A has to D the ratio compounded of the ratios of E to F, G to H, and K to L.

In like manner, the same things being supposed, if M has to N the same ratio which A has to D; then, for shortness sake, M is said to have to N the ratio compounded of the ratios of E to F, G to H, and K to L.

XII.

In proportionals, the antecedent terms are called homologous to one another, as also the consequents to one another. Geometers make use of the following technical words to ' signify certain ways of changing either the order or mag' nitude of proportionals, so as that they continue still to be ' proportionals.'

XIII.

Permutando, or alternando, by permutation, or alternately. See N. This word is used when there are four proportionals, and it is inferred that the first has the same ratio to the third, which the second has to the fourth; or that the first is to the third, as the second to the fourth: As is shown in the 16th Prop. of this 5th book.

XIV.

Invertendo, by inversion; when there are four proportionals, and it is inferred, that the second is to the first, as the fourth to the third. Prop. B. Book 5.

XV.

Componendo, by composition; when there are four proportionals, and it is inferred, that the first, together with the second, is to the second, as the third, together with the fourth, is to the fourth. 18th Prop. Book 5.

XVI.

Dividendo, by division; when there are four proportionals, and it is inferred, that the excess of the first above the second, is to the second, as the excess of the third above the fourth, is to the fourth. 17th Prop. Book 5.

XVII.

Convertendo, by conversion; when there are four proportionals, and it is inferred, that the first is to its excess above the

Book V.

second, as the third to its excess above the fourth. Prop. E. Book 5.

XVIII.

Ex æquali (sc. distantia,) or ex æquo, from equality of distance, when there is any number of magnitudes more than two; and as many others, so 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 first rank of magnitudes, as the first is to the last of the others: ' Of this there are the two fol'lowing kinds, which arise from the different order in which ' the magnitudes are taken two and two.'

ΧΙΧ.

Ex æquali, from equality. This term is used simply by itself, when the first 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 first rank, so is the second to the third of the other; and so on in order, and the inference is as mentioned in the preceding definition; whence this is called ordinate proportion. It is demonstrated in 22d Prop. Book 5.

xx.

Ex æquali, in proportione perturbata, seu inordinata, from equality, in perturbate or disorderly proportion.* This term is used when the first magnitude is to the second of the first rank, as the last but one is to the last of the second rank; and as the second is to the third of the first rank; so is the last but two to the last but one of the second rank, and as the third is to the fourth of the first rank, so is the third from the last to the last but two of the second rank; and so on in a cross order: And the inference is as in the 18th definition. It is demonstrated in the 23d Prop. of Book 5.

AXIOMS.

I.

EQUIMULTIPLES of the same, or of equal magnitudes, are equal to one another.

II.

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

III.

A multiple of a greater magnitude, is greater than the same

multiple of a less.

IV.

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

PROP. I. THEOR.

If any number of magnitudes be equimultiples of as many others, each of each; what multiple soever any one of the former magnitudes is of its part, the same multiple will all the former magnitudes be of all the latter.

Let any number of magnitudes AB, CD be equimultiples of as many others, E, F, each of each; whatsoever multiple AB is of E, the same multiple will AB and CD together, be of E and F together.

Book V.

Because AB is the same multiple of E that CD is of F, as many magnitudes as are in AB equal to E, so many are there in CD equal to F. Divide AB into magnitudes equal to E, viz. AG, GB; and CD into CH, A HD, equal each of them to F: The number therefore of magnitudes CH, HD, in the one, will be G equal to the number of magnitudes, AG, GB, in the other: And because AG is equal to E, and CH to F, therefore AG and CH together are equal B toa E and F together: For the same reason, because GB is equal to E, and HD to F; GB and C HD together are equal to E and F together. Wherefore, as many magnitudes as there are in AB equal to E, so many are there in AB and CD H together, equal to E and F together. Therefore, whatsoever multiple AB is of E, the same multiple D is AB and CD together of E and F together.

E

a Ax.2.5.

F

Therefore, "If any magnitudes, how many soever, be equimultiples of as many, each of each, whatsoever multiple any one of the former magnitudes is of its part, the same multiple will all the former magnitudes be of all the latter." For the same de

H

Book V.

monstration holds in any number of magnitudes, which was here applied to two.' Q. E. D.

PROP. II. THEOR.

If the first magnitude be the same multiple of the second that the third is of the fourth, and the fifth the same multiple of the second that the sixth is of the fourth; then shall the first together with the fifth be the same multiple of the second, that the third together with the sixth is of the fourth.

Let AB the first, be the same multiple of C the second, that DE the third is of F the fourth; and BG the fifth, the same multiple of C the second, that EH the sixth is of F the fourth: Then is AG the first, together with the fifth, the same multiple of C the second, that DH the third, together with the sixth, is of F the fourth.

A

D

E

B

Because AB is the same multiple of
C, that DE is of F; there are as many
magnitudes in AB equal to C, as there
are in DE equal to F: In like manner, G

C d

H F

as many magnitudes as there are in

BG equal to C, so many are there in EH equal to F: As

many magnitudes, then, as are in the whole AG equal to C, so many are there in the whole DH equal to F: therefore AG is the same multiple of C, that DH is of F; that is, AG the first and fifth together, is the same multiple of the second

C, that DH the third and sixth together

D

is of the fourth F. "If, therefore, the

"first be the same multiple," &c. Q. E. D. A

E

K

Cor. From this it is plain, that if B 'any number of magnitudes AB, BG, GH, be multiples of another magnitude C; and as many others DE, EK, KL G 'be the same multiples of another magnitude F, each of each; the whole of the 'former, viz. AH, will be the same multiple of C, that the whole of the latter, HCLF viz. DL is of F.'

PROP. III. THEOR.

If the first of four magnitudes be the same multiple of the second, as the third is of the fourth; and if of the first and third there be taken equimultiples, these will also be equimultiples, the one of the second, and the other of the fourth.

Let A the first, be the same multiple of B the second, that C the third is of D the fourth; and of A, C let the equimultiples EF, GH be taken; then will EF be the same multiple of B, that GH is of D.

Because EF is the same multiple of A, that GH is of C, there will be as many magnitudes in EF equal to A, as are in GH equal to C: Let EF be

divided into the magnitudes F

EK, KF, each equal to A,

and GH into the magnitudes
GL, LH, each equal to C:
Then will the number of mag-
nitudes EK, KF, in the one, K
be equal to the number of
magnitudes GL, LH in the
other: And because A is the
same multiple of B, that C is
of D, and EK is equal to A,
and GL to C; therefore, EK E
is the same multiple of B, that

H

L

Book V.

AB

GCD

GL is of D: For the same reason, KF is the same multiple of B, that LH is of D; and so, if there be more points in EF, GH equal to A, C: Because, therefore, the first EK is the same multiple of the second B, as the third GL is of the fourth D, and the fifth KF is the same multiple of the second B, as the sixth LH is of the fourth D: EF the first, together with the fifth, is the same multiplea of the second B, that GH the third, together with the sixth, is of the fourth D. " If, therefore, the first," &c. Q. E. D.

a 2. 5.

H2