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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.

PROPOSITION I.

THEOREM. 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 equimultiples o. 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. Divide AB into magnitudes
equal to E, viz. AG, GB; and CD into CH, HD, equal Ay
each of them to F: therefore the number of the mag- GE
nitudes CH, HD, shall be equal to the number of the

В!
others AG, GB; and because AG is equal to E, and а
CH to F, therefore AG and ch together, are equal
to

H
E and F together: for the same reason, because
GB is equal to E, and HD to F, GB and HD together, D
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,
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 equi

• 2 Ax. 1.

multiples 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 demonstra'tion holds in any number of magnitudes, which was here applied to two. Q. E. D.

PROPOSITION 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 to

D

A gether with the fifth, be the same multiple of c the second, that DH, the third together with

E

BI the sixth, is of f the fourth.

Because AB is the same multiple of C that Gld on DE is of F, there are as many 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 F; therefore as many as there are in the whole AG equal to C, so many are there in D the whole DH equal to F: therefore AG is the Ay E same multiple of C that DH is of F; that is AG, B the first and fifth together, is the same multiple K

G of the second C, that DH, the third and sixth together, is of the fourth F. If, therefore, the first

HC LF magnitude be the same multiple, &c. Q. E, d.

COR. “ From this it is plain, that if any number of magnitudes AB, BG, GH, be multiples of another C; and as many DE, EK, KL, be the same multiples of F, each of each ; then the whole of the first, viz. AH, is the same multiple of C, that the whole of the last, viz. DL, is of F."

PROPOSITION III.

THEOR.-If 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.

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 equimultiples EF, GH be taken: then EF shall be the same 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 there are
in GH equal to C; let EF be divided
into the magnitudes EK, KF, each equal F

H
to A; and GH into GL, LH, each equal
to C; therefore the number of the mag-
nitudes EK, KF, shall be equal to the K

L
number of the others, GL, LH: and be-
cause A is the same multiple of B, that
c is of D, and that EK is equal to A, E A BG
and GL equal to C; therefore EK is
the same 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: there-
fore, because the first 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 sixth
LH is of the fourth D; EF, the first together with the fifth,
is the same multiple * of the second B, which gh, the third
together with the sixth, is of the fourth D. If, therefore, the
first, &c.

Q. E. D.

2. 5.

PROPOSITION IV.

See N,

THEOR.-If the first of four magnitudes has the same ratio

to the second which the third has to the fourth, then any equimultiples whatever of the first and third, shall have the same ratio to any equimultiples of the second and fourth; viz. the equimultiple of the first shall have the same ratio to that of the second, which the equimultiple of the third has to that of the fourth.

3. 5.

Let A the first have to B the second, the same ratio which the third c has to the fourth D; and of A and c, let there be taken any equimultiples whatever E, F; and of B and D, any equimultiples whatever G, H: then E shall have the same ratio to G, which F has to H.

Take of E and F, any equimultiples what. ever K, L, and of G, H, any equimultiples whatever M,N: then, because E is the same

KE A BG M multiple of A, that F is of C, and of E and F

LTC DH N have been taken equimultiples K, L, therefore K is the same multiple of A*, that L is of C: for the saine reason, M is the same multiple of B, that N is of D. And because *, as A is to B, so is C to D, and of A

Нур. and c have been taken certain equimultiples K, L, and of B and D have been taken certain equimultiples M, N, therefore if K be greater than M, L is greater than N; and if equal, equal ; if less, less * : but K, L are * 5 Def. 5. any equimultiples † whatever of E, F, and M, N, any what- + Constr. ever of G, H; therefore as E is to G, so is * F to H. There- • 5 Def. 5. fore, if the first, &c. Q. E. D.

Cor. Likewise, if the first has the same ratio to the second, which the third has to the fourth, then also any equimultiples whatever of the first and third, shall have the same ratio to the second and fourth : and in like manner, the first and the third shall have the same ratio to any equimultiples whatever of the second and fourth.

Let A the first have to B the second, the same ratio which the third c has to the fourth D, and of A and c, let E and F be any equimultiples whatever ; then E shall be to B, as F to D.

Take of E, F, any equimultiples whatever K, L, and of B, D, any equimultiples whatever G, H: then it may be demonstrated, as before, that K is the same multiple of A, that L is of C: and because + A is to B as C is to D, and of A and † Hyp. C, certain equimultiples have been taken, viz. K and L; and of B and D, certain equimultiples G, H, therefore if K be greater than G, L is greater than H; and if equal, equal;

* 5 Def. 5. + Constr.

if less, less * : but K, L are any + equimultiples whatever of E, F, and G, H any whatever of B, D; therefore, as E is to B +, so is F to D. · And in the same way the other case iş demonstrated.

+ 5 Def. 5,

PROPOSITION V.

See N.

THEOR. If one magnitude be the same multiple of another,

which a magnitude taken from the first is of a magnitude taken from the other, the remainder shall be the same multiple of the remainder, that the whole is of the whole.

1. 5.

Let the magnitude AB be the same multiple of CD, that AE taken from the first is of CF taken from the other: the remainder EB shall be the same mul- G tiple of the remainder FD, that the whole AB is of the whole CD.

A Take AG the same multiple of FD, that AE is of CF; therefore AE is * the same multiple of CF, EY that EG is of CD: but AE, by the hypothesis, is the same multiple of CF, that AB is of CD; there- B D fore EG is the same multiple of CD that AB is of CD; wherefore EG is equal * to AB: take from each of them the common magnitude AE, and the remainder AG is equal to the remainder EB. Wherefore, since AE is the same mul. tiple of CF +, that AG is of FD, and that AG is equal to EB, therefore AE is the same multiple of CF, that EB is of FD: but AE is the same multiple of CF + that AB is of CD; therefore EB is the same multiple of FD, that AB is of CD. Therefore, if one magnitude, &c. Q. E. D.

# 1 As. 5.

+ Constr.

+ Нур.

PROPOSITION VI.

See N.

THEOR.-If two magnitudes be equimultiples of two others,

and if equimultiples of these be taken from the first two, the remainders are either equal to these others, or equimultiples of them.

Let the two magnitudes AB, CD be equimultiples of the two E, F, and let AG, CH, taken from the first two, be equimultiples of the same E, F: the remainders GB, HD shall be either equal to E, F, or equimultiples of them.

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