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when any equimultiples whatsoever of the first and third being taken, and any equimultiples 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 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.'

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VII.
When of the equimultiples of four magnitudes (taken

as in the fifth 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 second 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 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.

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. 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

same kind, the first A is said 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 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 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 bomo

logous 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 proportionals, so that they continue
still to be proportionals.”

XIII.
Permutando, or alternando, by permutation or alter- See N.

nately. This word is used when there are four pro-
portionals, 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 shewn in the 16th Prop. of this fifth Book.

XIV.

Invertendo, by inversion; when there are four propor

tionals, 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 propor

tionals, 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 pro

portionals, and it is inferred, that the first is to its excess above the second, as the third to its excess above the fourth. Prop. E. Book 5.

XVIII.

Ex æquali (sc. distantiâ), or ex æquo, from equality of

distance ; when there is any number of magnitudes more than two, and as many others, such 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 different order in which the magnitudes are taken, two and two.'

XIX.

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 the 22d Prop. Book 5.

XX. Ex æquali in proportione perturbatâ seu inordinatâ,

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,

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.
Let any number of magnitudes AB, CD be equimul-

4 Prop. lib. 2. Archimedis de Sphæra et Cylindro.

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tiples 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 there are in AB equal to
E, so many are there in CD equal to F. Di-
vide AB into magnitudes equal to E, viz. AG,
GB; and CD into CH, ÂD equal each of GE
them to F: therefore the number of the mag-

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nitudes CH, HD, shall be equal to the num-
ber of the others AG, GB: and because AG

HT is equal to E, and CH to F, therefore AG and • 2 Ax. 1. CH'together are equal to * E and F together: D

for the same reason, because GB is equal to E,
and HD to F, GB and 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, CD
together equal to E and F together: therefore, whatso-
ever multiple AB is of E, the same multiple 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 others : For • the same demonstration 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 shall AG, the first together with the fifth,

E

BI
be the same multiple of C the second, that
DH, the third together with the sixth, is G
of F the fourth.

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

D

A

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