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III. “ Ratio is a mutual relation of two magnitudes of the fame kind See N, “ to one another, in respect of quantity.”
cond, which the third has to the fourth, when any equimultiples
Book v. than that of the second, the multiple of the third is also greater
expressed by saying, the first is to the second, as the third to
sth Definition, the multiple of the first is greater than that of
to have to the fourth the Triplicate ratio of that which it has
Definition A, to wit, of Compound ratio.
first is said to have to the last of them the ratio compounded of
third has to the fourth, and so on unto the last magnitude.
the first A is faid to have to the last D the ratio compoun ied
of the ratios of A to B, B to C, and C to D.
the same ratio that has to H; and C to D, the fame that K
has to L ; then, by this Definition, A is said to have to D the Book V. ratio compounded of ratios which are the same with the ratios of E to F, G to H, and K to L. and the fame thing is to be understood when it is more briefly expressed, by faying A has to D
the ratio compounded of the ratios of Eto F, G to H,and K to L. In like maoner, the same things being supposed, if M has to N
the same ratio which A has to D, then, for shortness fake, M
one another, as also the consequents to one another.
word is used when there are four proportionals, and it is infer- See N.
it is inferred, that the second is to the first, as the fourth to the
and it is inferred, that the first together with the second, is to
is inferred, that the Excess of the first above the second, is to
and it is inferred that the first is to its Excess above the fecond,
XVIII. wEx acquali (fc. diftantia,) or, ex aequo, from equality of distance ;
when there is any number of magnitudes more than two, and
the first magnitude is to the second of the first rank, as the first
lity, in perturbate or disorderly proportion * ; this term is used
A X Ι Ο M S.
1. QUIMULTIPLES of the same, 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 maltiple of a less.
• 4. Prop. Lib, m Archimedis de sphaera et cylindro.
tiple of another, is greater than that other magnitude.
PROP. I. THEOR.
IF any number of magnitudes
be equimultiples of as many, each of each; what multiple foever any one of them is of its part, the same multiple shall all the first magnitudes be of all the other.
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 shall AB and CD together be of E and F together.
Because AB is the same multiple of E that CD is of F, as many magnitudes as are in AB equal to E, fo many are there in CD equal to F. Divide AB into magnitudes equal to E, viz. AG, GB; and CD into CH, HD e Ap qual each of them to F. the number therefore of the magnitudes CH, HD shall be equal to the
El number of the others AG, GB. and because AG is equal to E, and CH to F; therefore AG and
B CH together are equal to · E and F together.
a. A2. 3. 1, for the fame reason, because GB is equal to E, C and HD to F; GB and HD together are equal to E and F together. Wherefore as many mag F nitudes as are in AB equal to E, fo many are there in AB, CD together equal to E and F together. Therefore whatsoever multiple AB is of
Therefore if any magnitudes, how many foever, be equimaltiples of as many, each of each, whatsoever multiple any one of them is of its part, the fame multiple shall all the first magnitudes be of all the other. for the same Demonstration holds in any number
of magnitudes, which was here applied to two. Q. E. D.