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LESS magnitude is said to be a part of a greater magnitude,

when the less measures the greater, that is, ' when the less is
contained a certain number of times exactly in the greater.'

II.
A greater magnitude is said to be à multiple of a less, when the

greater is measured by the less, that is, when the greater con-
• tains the less a certain number of times exactly.'

III.
Ratio is a mutual relation of two magnitudes of the fame kind to See N.
one another, in respect of quantity.”

IV.
Magnitudes are said to have a ratio to one another, when the less can
be multiplied so as to exceed the other.

V.
The first of four magnitudes is faid to have the same ratio to the fe-

cond, which the third has to the fourth, when any equimultiples
whatsoever of the first and third being taken, and any equimul-
tiples whatsoever of the second and fourth; if the multiple of the
first be less than that of the second, the multiple of the third is
also less than that of the fourth; or, if the multiple of the first be
equal to that of the second, the multiple of the third is also equal
to that of the fourth; or, if the multiple of the first be greater

Book v. than that of the second, the multiple of the third is also greater than that of the fourth.

VI.
Magnitudes which have the fame ratio are called proportionals.

N. B. “When four magnitudes are proportionals, it is usually

expressed by saying, the first is to the second, as the third to • the fourth.'

VII.
When of the equimultiples of four magnitudes (taken as in the 5th

Definition, the multiple of the first is greater than that of the se-
cond, but the multiple of the third is not greater than the mul-
tiple of the fourth; then the first is said to have to the second a
greater ratio than thc third magnitude has to the fourth; and on
the contrary, the third is said to have to the fourth a lefs ratio
than the first has to the second.

VIII.
“ Analogy, or proportion, is the similitude of ratios.”

IX.
Proportion consists in three terms at least.

X.
When three magnitudes are proportionals, the first is said to have to
the third the duplicate ratio of that which it has to the second.

XI.
When four magnitudes are continual proportionals, the first is said to

have to the fourth the Triplicate ratio of that which it has to the
second, and so on Quadruplicate, &c. increasing the denomination
still by unity, in any number of proportionals.

Definition A, to wit, of Compound ratio.
When there are any number of magnitudes of the same kind, the first

is said to have to the last of them the ratio compounded of the
ratio which the first 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 so on unto the last 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 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 said to be compounded of the

ratios of A to B, B to C, and C to D. And if A has to B, the same ratio which E has to F; and B to C, the same ratio that G has to H; and C to D, the same that K

has

See Y.

U CLID. has to L; then, by this Definition, A is said to have to D the ra- Book V. tio 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, ånd K to L. În like manner, the same things being supposed, if M has to N the

fame ratio which A has to D, then, for shortness fake, M is said
to have to N, the tario 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 magnitude of pro-
portionals, so as that they continue fill to be proportionals.'

XIII.
Permutando, or Alternando, by Permutation, or alternately; this. See it

word is used when there are four proportionals, and it is infer
red, that the first has the same ratio to the third, which the fe-
cond has to the fourth; or that the first is to the third, as the
fecond to the fourth. as is shewn 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 5th.

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 5th.

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 5th.

XVII.
Convertendo, by Conversion ; when there are four proportionals,

and it is inferred, that the first is to its Excess above the fecond,
as the third to its Exces above the fourti. Prop. E. Book 5th.

}

YVI'I. I,

Book V.

XVIII.
Ex aequali (fc.distantia),or,ex aequo, 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 following kinds, which arise from the diffe* rent order in which the magnitudes are taken two and two.'

XIX.
Èx aequali, from aequality; 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 preceed-
ing Definition; whence this is called Ordinate Proportion. It is
demonstrated in 22d Prop. Book 5th.

XX.
Ex aequali, in proportione perturbata, feu inordinata, from equa-

lity, in perturbate or disorderly proportion *; this term is used
when the firft 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 23d Prop.
of Book 5th.

A X I 0 MS.

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

Í. 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 same multiple of a less. * 4. Prop. Lib. 2. Archimedis de sphaera et cylindro.

IV, That

IV.

Book V. That magnitude of which a multiple is greater than the same mul

tiple of another, is greater than that other magnitude.

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PROP. I. THEOR.
[F any number of magnitudes be equimultiples of as

many, each of each; what multiple soever any one of them is of its part, the same multiple shall all the first magnitudes be of all the other.

IF

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, so many are there in CD equal to F. Divide AB into magnitudes equal to E, viz. AG, GB; and CD into CH, HD equal each of

A. them to F. the number therefore of the magnitudes CH, HD shall be equal to the number of G

El the others AG, GB. and because AG is equal to E, and CH to F; therefore AG and CH together are equal to · E and F together. for the same

a. Ar ats reason, because GB is equal to E, and HD to F; CI GB and HD together are equal to E and F together. Wherefore as many magnitudes as are in

F AB equal to E, so many are there in AB, CD

H Н together equal to E and F together. Therefore whatsoever multiple AB is of E, the same mul

D tiple is AB and CD together of E and F together.

Therefore if any magnitudes, how many soever, be equimultiples of as many, each of each, whatsoever multiple any one of them is of its part, the same 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.

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