CASE III. mA: BmC: D.

D. 1 H. & Sch. 3, V.. A: B = C: D, .. mA: BmC: D;
2 Sim.
In like manner, An BC:n D.

•.• A : B = C : D, & A,B are equims. of A, B by m;

m

m

D. 1 H.

Dividendo. If magnitudes taken jointly, be proportionals, they shall also be proportionals when taken separately; that is, if two magnitudes together have to one of them, the same ratio which two others have to one of these, the remaining one of the first two shall have to the other the same ratio which the remaining one of the last two has to the other of these.

N.B. The General Enunciation of this 17th Proposition is variously given;

"If magnitudes be proportional, they will also be proportional by division."--EUCLID.

1

"If four magnitudes, A, B, C, D, be proportionals, they shall also be proportionals, when taken dividedly: that is, the difference of the first and second shall be to the second as the difference of the third and fourth to the fourth; or dividendo, A~B: B = C ~ D: D."---DE MORGAN.

"If four magnitudes be proportional, the first being greater than the second and the third greater than the fourth; then the excess of the first above the second shall be to the second, as the excess of the third above the fourth is to the fourth."---HOSE.

CON. Pst. 1, V. DEM. 1, V.---If any number of Ms be equims. of as many, &c., 2, V. If the 1st M be the same m of the 2nd, &c. DEF. 5, V. Criterion of the equality of two ratios. Ax. 4, I. If equals be added to unequals the wholes are unequal. Ax. 5, I. If equals be taken from unequals the remainders are unequal.

E1 Hyp. 1

" 2

3 Conc.

Let A B, BE, CD, D F, be Ms taken jointly and proportionals,

i. c., AB: BE CD: DF;-AB > BE and CD > DF;

and let A E be the excess of A B above B E, and CF the excess of C D above DF;

C.1 Pst. 1, V. ¡Of A E, E B, CF, FD take equims. G H, HK,

LM, MN;

and of E B, F D any equims. X K, NP.
GH, HK are equims. of A E, BE;
.. G H same m of A E that G K is of A B;
But GH same m of AE that L M is of CF;
.. GK

LM

of CF

LM

C F.

AB
Again. L M same m of C F that MN is of FD;
LN CD;
but L M same m of C F that G K is of AB,
GK, L N are equims of A B, C D.

Next,
and

HK, M N are equims of EB, FD; KX, NP equims, of E B, FD, .. HK+KX & MN + NPequims. of EB, FD; i. e. IIX and MP are equims of EB, FD;

Otherwise. If A + BB = C + D :D, then by division A, B

C.1 Pst. 1, V.

D. Sup. 1.

Take mA, nB, mults. of A, B.

Let m A, >nB.

Q. E. D.

C: D;

2 Add. Ax. 4, I, To both sides add mB; .. mA + m B > mBnB,

or (m + n) B,

but. A+B:B=C+D:D,

.. ifm (A+B) >(m +n)B, m (C +D) > (m + n)D; thus mC+m D >mD + nD;

from both sides take m D, .. mC > nD;

i. e., if m A > n B, mC >n D.
If m A

=

n B, then m C = n D,
and if m A <n B, then m C <nD.
.. A:B= C: D, by division Def. 16, V.
Wherefore, if magnitudes, &c.

2:

Alg. and Arith. Hyp. Take a, 8 : b, : c, 12: d, 3.

COR. 1. Convertendo.-If four Ms, A,B,C,D be proportionals A B C D, they shall also be proportionals by conversion, i.e., the 1st 1st 2nd 3rd : 3rd~4th,—or A: A~B = C: C~ D.

~

For, invertendo, Pr. B,V. B: A=D:C; dividendo, 17, V.

B~A : A=D~C : C;

invertendo, B, V. A: AB = C: C~D; or convertendo.

COR. 2. If four Ms of the same kind, A,B,C,D, be proportionals, A: BC: D, then the greatest the least are together greater than the other two.

D. 1 Sup. 1.

2 14, V.

Let A one of the extremes be the greatest;

as 12: 8 = 3:2.

:: B<A, .. D<C; and C < A, .. D <B .. D is the least.

3 H. & 17, V. .. A : B = C : C, .. A−B: B = C—D : D. and B>D, .. A-B>C-D.

4 D. 2, 5 Add.

6 Sup. 2.

7 Pr. B, V.
8 Sim.

9: Rec.

To each add B + D; .. A+D>C+B;

i. e. the greatest 12+ the least 2 > the other two 8+ 3.

Let B one of the means be greatest;

as 4:16 = 3:12;

invert. B: A

D: C.

So, as before, B+C>A+D,

i. e. the greatest B 16+ the least C 3

> the other two 4 + 12.

.. the sum of the greatest and least > the sum

of the other two.

N. B. This Corollary is identical with Prop. 25, Bk. V.

COR. 3. In three Proportionals, A: B: C, as 2: 4: 8, or 9 : 6, 4, the sum of the extremes, A + C is greater than twice the mean, 2 B; and therefore half the sum, A+ C of the extremes is greater than the

mean, B.

2

For, if the mean B>C, then B<A; and if B<C, then B>A. .. the extremes A & C are the greatest M, and the least. and as before, in Cor. 2, A+ C> B+ B, or 2 B;

USE & APP. When half the sum of two Ms,

A+ C
2

is as much greater than the one, as it is less than the other, that half sum is an arithmetical mean

2+8
2

between the two; thus, in = 5,8—5=3, & 5—2—3; and 25: 8 are in arithmetical progression. Except when the magnitudes are equal, the arithmetical mean between two magnitudes, A and C, is therefore greater than > B the geom. mean;

the geometrical mean, i.

e,

the arith. mean,

A+ C
2

Geom. Plane, Sol. & Sph. pp 41 & 42.

PROP. XVIII.-THEOR.

Componendo. If magnitudes taken separately, be proportionals, they shall also be proportionals when taken jointly, by composition; that is, if the first be to the second, as the third is to the fourth, the first and second together shall be to the second, as the third and fourth together to the fourth.

"If magnitudes be proportional, they will also be proportional by composition."-EUCLID'

66

The terms of an analogy are proportional by composition."-BELL.

CON. Pst. 1, V. DEM. Def. 5, V.Criterion of the equality of ratios.
Ax. 3, V. A m of a gr. M is gr. than the same m of a less.

5, V. If one M be the same m of another which a M taken from the 1st
is of a M taken from the other; the rem. is the same m of the rem.
that the whole is of the whole.

6, V. If two Ms be equims. of two others, and if equims. of these be
taken from the first two; the rems. either these others, or are equims.
Cor. 4, V. If 1st 2nd =
:
3rd 4th, then any equims. of 1st and 3rd
the same ratio to 2nd and 4th; and 1st: 3rd = equim. 2nd equim
4th.

:

Pr. A, V. If 1st 2nd 3rd 4th; then if 1st 2nd, 3rd > 4th, and if=,=; if <, <.

Ax. 2, V. Those Ms of which the same or equal Ms are equims. are equal.

Ax. 4, V. 'That M of which a m is greater than the same m of another is gr. than that other M.