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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 kampe 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 bxg 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 fe-
cond has to the fourth; or that the first is to the third, as the
second to the fourth. as is sewn 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 sth.

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

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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 different order in which the magnitudes are s taken two and two.'

XIX.
Ex aequali, 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 5th.

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, 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, 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.

A X I O M S.

I.

E MULTIPLES 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 muitiple of a greater magnitude is greater than the fame multiple

of a less.

4. Prop. Lib. 2. Archimedis de fphaera et cylindro.

Book V.

IV.
That magnitude of which a multiple is greater than the same

multiple of another, is greater than that other magnitude.

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I
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 othet.

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 e. A qual each of them to F. the number therefore of the magnitudes CH, HD shall be equal to the E number of the others AG, GB. and because AG is equal to E, and CH to F; therefore AG and CH together are equal to a E and F together.

B

2. Ax. 2. I. for the same reason, because GB is equal to E, С and HD to F; GB and HD together are equal to E and F together. Wherefore as many magnitudes as are in AB equal to E, fo many are H there in AB, CD together equal to E and F together. Therefore whatsoever 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 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.

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Book V.

PROP. II. THEOR.

I

F the first magnitude be the same. multiple of the

second that the third is of the fourth, and the fifth the fame 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.

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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

D
fame multiple of C the second, that A
DH the third together with the fixth
is of F the fourth.

E
Because AB is the same multiple of
C, that DE is of F; there are as many

B
magnitudes in AB equal to C, as there
are in DE equal to F. in like manner,
as many as there are in BG equal to
C, so many are there in EH equal to

G CH'F
F. as many 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 to-
gether, is the same multiple of the second C, that DH the third
and fixth together is of the fourth F. If

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

A

E
Cor. · From this it is plain, that if any B
number of magnitudes AB, BG, GH be
multiples of another C; and as many De,

K
• EK, KL be the same multiples of F, each G
I of each; the whole of the first, viz. AH
o is the same multiple of C, that the whole
( of the last, viz. DL is of F.'

H CL F

Book v.

PROP. III.

THEOR.

F the first be the same multiple of the second, which

the third is of the fourth; and if of the first and third there be taken equimultiples, these shall be equimultiples the one of the second, and the other of the fourth.

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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 EF is the fame multiple of B, that GH is of D.

Because EF is the same multiple of A, that GH is of C, there are as many magnitudes in EF equal to A, as are in GH equal to C. let EF be divided into the

F magnitudes EK, KF, each e

H qual to A, and GH into GL, LH, each equal to C. the number therefore of the magnitudes EK, KF, shall be

K equal to the number of the

L others GL, LH. and because A is the same multiple of B, that C is of D, and that EK is equal to A, and GL to C; therefore EK is the same E A B G C D multiple of B, that 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 parts in EF, GH equal to A, C. because therefore the firft EK is the same multiple of the second B, which the third GL is of the fourth D, and that the fifth KF is the same multiple of the second B, which the fixth LH is of the fourth D; EF the first together with the fifth is the fame mul. tiple a of the Yecond B, which GH the third together with the a. 2.5 lixth is of the fourth D. If therefore the first, &c. Q. E. D.

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