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I.
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 a 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 See N, “ to one another, in respect of quantity.”

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

V.
The first of four magnitudes is said to have the same ratio to the fee

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 same 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

sth Definition, the multiple of the first 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 said to have to the
fecond a greater ratio than the third magnitude has to the
fourth ; and on the contrary, the third is said to have to the
fourth a less fatio 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.
See N. 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. increafing the de
nomination still by unity, in any number of proportionals.

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

the first A is faid to have to the last D the ratio compoun ied
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 fame ratio which E has to F; and B to C,

the same ratio that has to H; and C to D, the fame that K

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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
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 magnitude of pro-
portionals, so as that they continue still to be proportionals.'

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

word is used when there are four proportionals, and it is infer- See N.
red, that the first has the same ratio to the third, which the se-
cond has to the fourth ; or that the first is to the third, as the
second to the fourth. as is thewn in the 16th Prop. of this sth
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 sth.

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 Excess above the fourth. Prop. E. Book sth.

H

Book V.

XVIII. wEx acquali (fc. diftantia,) 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 firft 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 kiods, which
• arise from the different order in which the magnitudes are
I taken two and two.'

XIX.
Ex acquali, 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, fo 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 Propor-
tion. It is demonstrated in 22d Prop. Book sth.

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

lity, 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 se-
cond is to the third of the first rank, fo 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, fo 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,

E Е

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.

Book V.

IV.
That magnitude of which a multiple is greater than the fame mul-

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

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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

DI
E, the fame multiple is AB and CD together of
E and F together.

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.

HH

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