and K, L are equims, of A & C,

and G, H are equims. of B & D ;

.. K > = or < G, so L > = or < H. But K,L are equims of E, F, & G, H any of B,D; .. E: B = F: D.

In the same way, A: G

C: H

in Dem. 8, Pr. 4, V., if K> or < M, L >=

.. if M > = or < K, N > = or < L. Hence G: E H: F. Therefore, if four magnitudes are proportionals, they will be proportional by inversion.

N.B.-This Cor. is not in its proper place; it correctly forms Prop. B, V.

COR. 3. be taken, as

If A: B = C: D,-and if any like parts of A and C

A
2' 2'

B D

and also any like parts of B and D, as

3' 3

these like parts will also be proportional; i. e.,

Alg. & Arith. Hyp.-Let a 2, b ≈ 5, c = 4, d = 10, m = 2 & n = 3. Alg.-a: b cd, then ma : nb = mc : nd. For, 3, V. equims. of ma and me are equims. of a &c; and equims of nb, nd also equims. of b & d.

But, Def. 5, V., a & c < = or > b & d, ... equims. of a & c <= or > b & d, and also <= or > equims. of b & d. Hence, Def. 5, V. ma : nb = mc : nd.

Now

Arith.. 2:5= 4 10, then 2 X 2 : 3 X 5 = 2 × 4 : 3 x 10. equims. of 4 & 8 are equims. of 2 & 4; and equims of 15 & 30 also equims. of 5 & 10.

But, Def. 5, V. . 2 & 4 < = or > 5 & 5 & 10, and also <=> equims. of 5 & 10.

10.. equims of 2 & 4 <= or > Hence, Def. 5, V. 4: 158:30.

APPL. From Cor. 3, arises the rule in simple proportion in arithmetic, of dividing the 1st and 2nd terms by any common measure, and using the resulting instead of the original numbers.

PROP. V.-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 is the same multiple of the remainder, that the whole is of the whole.

CON.-Pst. 1, V. Dem. 1, V. Ax. 1,V. Equimultiples of the
same or of equal magnitudes are equal to one another.
Ax. 3. 1.-If equals be taken from equals the remainders
are equal.

E. 1, Hyp. 1

2 Conc.

Let A B be the same m of CD
that A E, taken from A B, is
of C F, taken from CD;
then the rem. E B is the same
m of rem. F D, as the whole
A B of the whole CD.

A

18

24

C

Pst. 1, V.

Take A G same m of F D, that
A E is of CF.

D. 1 C.

8

2 1, V.

3 H.

4 1, V.

5 Ax. 1, V.

AG, AE are equims. of 32
FD and CF,

.. A G, A E, i. e., E G, same m of C F, F D,
i.e., CD, that A E is of C F.

but A E same m of C F, that A B is of CD; .. EG same m of C D that A B is of CD, .. EG

AB;

6 Sub & Ax. 3,I. | from each take A E; .. rem. A G = rem. E B.

7 C.

8 D. 6

9 1, V. 10 H.

11 1, V. 12 Conc.

.. A E same m of CF that E B is of FD;
But A E same m of C F that A B is of CD;
.. E B same m of F D that A B is of CD.
Therefore if one magnitude be the same, &c.
Q. E D.

Alg. & Arith. Hyp.-Let A = 32, B8, C (a part of A) = 24, and D (a part of B) = 6, m = 4.

So the rem. is the same m of the rem. as the whole is of the whole.

Or, Let A B =D; to both sides add B,-then AB + D;
.. (1, V.) m Am B+mD;

Subtract m B, and m A-mBmD;

but DA B; ... mA

— mB= m (A—B);

Thus the rem. is the same m of the rem. that m A is of A.

SCH. If from a multiple of a magnitude by any number, a multiple of the same magnitude by a less number be taken away, the remainder will be the same multiple of that magnitude that the difference of the numbers is of unity.

E. 1| Hyp.

2 Conc.

C.

Sum.

D. 1

2

C & 2, V.
Sub.

3 Conc.

Let m A, n A be mults. of A, m being > n;

then m A nA =

Let m

(mn) A.

-nq; then m=n+q.

Here m An A + q A;

from both take n A; then m An A=qA;
.. m A- ―n A = (m —— n) A.

COR. When the difference of the two numbers is equal to unity, or mn= 1, then m An A A; or 2 AA : A.

PROP. VI. 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.

CON. Pst. 2, I.-3, I. From the gr. of two lines to cut off a part equal to the less. Pst. 1, V.

DEM. Ax. 1, V.-Ax. 3, I. Ax. 1, I.—2, V.

Let A B, CD be equims. of E F;
and let A G taken from A B be an equim. of E,
and C H from C D an equim. of F;
then rems. GB, HD either

equims. of them.

= EF, or are

C.

CASE. I. Let G B = E, then HD shall equal F.

Pst. 2 1 & 3 I Make C K = F.

but KC = · F, .. HD = F.

CASE II.-Let G B be a m of E, then HD same m of F.

Of F take C K the same m

that G B is of E.

AG same m of Ethat

K

18

Aly. & Arith. Hyp.-Let m>n express any integers, as 4 and 3; A

B 36, C7, D= 9.

28,

SCH.-The six preceding Propositions are chiefly useful for establishing, by the method of Equimultiples, the Propositions which follow. When this method is not employed some have adopted the Postulate,-Three Magnitudes, A, B, C, being given, let it be granted that there is a 4th magnitude, we may call it x, to which C has the same ratio, as A to B; i. e. A : B = C: x.

PROP. A,-THEOR.

If the first of four magnitudes has the same ratio to the second which the third has to the fourth; then if the first be greater than the second the third is greater than the fourth; and if equal, equal; if less, less.

CON.-Pst. 1, V. DEM.-Def. 5, V.

E.1 Hyp.

2 Conc.

C Pst. 1, V.

D.1 H.

C: D,

and of the 1st and 3rd equims. 2 A, 2 C are
taken,

and of the 2nd and 4th equims. 2 B, 2 D;
.. 2 C is >= or < 2 D, as 2 A is >
2 B.

= or <

But 2 C is >= or < 2 D as C is > = or <
D;

and so, 2 A is >= or < 2 B, as A is >= or
< B;

.. C, is > = or < D, as A is > = or < B. Q. E. D.