(No. 14.) by the denominators; then A Bq, C = Dq, and E Fq. Hence by addition, A+C+E

=

and by dividing by B+D+F, we get q =

=

(B+D+F)q; A+C+E

B+D+F

But

or A: B:: A+ C+E: ::A+C+E:

B+D+F. Therefore, of numbers, &c.

PROP. IX. THEOR.

MAGNITUDES have the same ratio to one another that

their like multiples have.

Let A and B be two magnitudes; then, n being a whole number, A: BnA: nB.

Ir four numbers be proportionals; then, (1.) by division, 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: and, (2.) by composition, the sum of the first and second is to the second, as the sum of the third and fourth to the fourth.

If A B C D; then, by division, A-B: B:: C-D:D; and by composition, A + B:B:: C+D: D.

Α C

1. Since (hyp.) fractions from the A-B C-D

B

=

D

2. By adding A+ B C+D

B

=

D

take (No. 11.) the latter former, each from each, and there remains

or A-B: B::C-D:D.

(No. 11.) the same fractions, we obtain

or A+ B: B::C+D: D. If, therefore, &c.

Cor. From the latter analogy we have, alternately, A + B : C + D :: B : D, and from the former A-B: C—D :: B : D. Hence (Sup. 7.) A + B:C+D::A-B: CD, and alternately A+B: A B::C+D: C D: that is, if four numbers be proportional, the sum of the first and second terms is to their difference, as the sum of the third and fourth terms is to their difference. It is evident that if B be greater than A, the analogy will become B +A:B—A::D+Č: D — C.

PROP. XI. THEOR.

Ir four numbers be proportional; then, by conversion, the first is to its excess above the second, as the third to its excess above the fourth.

If A: B:: C: D; then, by conversion, A: A B::C: C- D.

B D

A = C'

A C For, since (hyp. and inver.) and since = ; A take T (No. 11.) the former fractions from the latter, each from each, A-B C-D

and there remains

, or (by inver.) A: A-B::

C: C-D. Therefore, if four numbers, &c.

:

Cor. If A B C D; alternately, A: C:: B: D; and, by conversion, AA-C:: B: BD. Hence, alternately, A: B:: A C: B D; which is the 19th proposition of this book.

PROP. XII. THEOR.

If there be numbers forming two or more analogies which have common consequents, the sum of all the first antecedents is to their common consequent, as the sum of all the other antecedents is to their common consequent.

If A : B : : C: D, and E: B:: F: D; then A + E:B:: C+ F: D.

For (hyp.)

A C
= and
B D'

A+E C+F

11.) A +

B

=

D

, or A+E:B::C+F: D. If, therefore, &c.

PROP. XIII. THEOR.

If there be three or more numbers, and as many others, which, taken two and two in order, have the same ratio; then, ex æquo, the first has to the last of the first rank the same ratio that the first has to the last of the second rank.

If the two ranks of numbers, A, B, C, D, and E, F, G, H, be such that A: B:: E: F, B: CF: G, and C: D:: G: H; then A:D :: E: H.

A E B F
For, since (hyp.) B=F =
Ᏼ F' C G'

C

and

Ꭰ

G
DH'

by multiplying together (Nos. 3. and 12.) the first, third, and fifth fractions, and the second, fourth, and sixth, we obtain ABC EFG

; or, (No. 10.) by dividing the terms of the first of

these fractions by BC, and those of the second by FG, or A: D:: E: H. Therefore, if there be three, &c.

This proposition might also be enunciated thus: If there be numbers forming two or more analogies, such that the consequents in each are the antecedents in the one immediately following it, an analogy will be obtained by taking the antecedents of the first analogy, and the consequents of the last for its antecedents and consequents.

If there be three or more numbers, and as many others, which, taken two and two in a cross order, have the same ratio; then, ex æquo inversely, the first has to the last of the first rank the same ratio which the first has to the last of the second rank.

If the two ranks of numbers, A, B, C, D, and E, F, G, H, be such that A: B:: G: H, B:C::F: G, and C: D:: E: F; then, ex æquo inversely, A: D:: E: H.

A G B F

=

For, since (hyp.) BH' CG'

В

and

C E
D-F

by multiplying together (Nos. 3. and 12.) the fractions as in the

ABC GFE
BCD HGF'

preceding proposition, we get =

whence, by dividing

the terms of the first of these fractions by BC, andthose of the second

A E

by GF, we obtain = or A: D: E: H. If, therefore, &c. D H,

This proposition may also be enunciated thus: If there be numbers forming two or more analogies, such that the means of each are the extremes of the one immediately following it, another analogy may be obtained by taking the extremes of the first analogy, and the means of the last for its extremes and means.

PROP. XV. THEOR.

If there be numbers forming two or more analogies, the products of their corresponding terms are proportionals. If A BC: D; E: F: G: H; and K: L:: M: N; then AEK: BFL :: CGM: DHN.

:

A C EG

:

For (hyp.) D'FH'

=

B

and taking (Nos. 3. and 12.)the products of the corresponding terms

of these fractions, we obtain

AEK CGM

=

BFL DHN'

CGM DHN. Therefore, if there be numbers, &c.

:

Cor. 1. Hence, if there be two analogies consisting of the same terms A, B, C, D, we have A2 : B2 :: C2 : D2; if there be three, we have A3: B3 : : C3 : D3, &c. and it thus appears, that like powers of proportional numbers are themselves proportional.

:

Cor. 2. Like roots of proportional numbers are proportional. Thus, if A: B:: C: D, let √A:√B::√C:√E. Then, by the preceding corollary, A: B:: C: E. But (hyp.) A: B:: C: D; and therefore (Sup. 7.) C: E :: C: D, and (Sup. 6.) E = D, and consequently A: √B:: √C: ✔E, or D.

PROP. XVI. THEOR.

THE sum of the greatest and least of four proportional numbers is greater than the sum of the other two.

If A: B:: C: D, and if A be the greatest, and therefore (Sup. 2. cor. 2.) D the least; A and D are together greater than B and C. For (by conversion) A: AB:: C: C-D, and, alternately, A: C :: A — B: C-D. But (hyp.) A > C, and therefore (Sup. 2. cor. 2.) AB > C-D. To each of these add B; then ABC-D. Add again, D; then, A + D > B + C. Therefore the sum, &c.

Cor. Hence the mean of three proportional numbers is less than half the sum of the extremes.

PROP. XVII. THEOR.

IN numbers which are continual proportionals, the first is to the third as the second power of the first to the second power of the second; the first to the fourth as the third power of the first to the third power of the second; the first to the fifth as the fourth power of the first to the fourth power of the second; and so on.

If A, B, C, D, E, &c., be continual proportionals; A: C : : A2 : B2; A:D :: A5: B5; A: E:: A1: B4, &c.

For, since (V. def. 8.) A: B:: B: C, and since A: B:: A: B, we have (Sup. 15.) A: Bo :: AB: BC, or, dividing the third and fourth terms by B, A2: B2 : : A : C.

2

Again, since A: B:: B: C,

and AB:: C: D,

and also A B:: A: B, we have (Sup. 15.) A3: B3:: ABC: BCD, or dividing the third and fourth terms by BC, A3: B3 :: A : D ; and so on, as far as we please. Therefore, in numbers, &c.

Cor. Hence (V. def. 11. and 12.) the ratio which is duplicate of that of any two numbers, is the same as the ratio of their squares; that which is triplicate of their ratio, the same as the ratio of their cubes, &c.

PROP. XVIII. THEOR.

A RATIO which is compounded of other ratios, is the same as the ratio of the products of their homologous

terms.

Let the ratio of A to D be compounded of the ratios of A to B, B to C, and C to D; the ratio of A to D is the same as that of ABC, the product of the antecedents, to BCD, the product of the consequents.

For, since A: D:: A: D, multiply the terms of the second ratio by BC: then (Sup. 9.) A: D:: ABC: BCD. Therefore, a ratio, &c.

PROP. XIX. THEOR.

IN numbers which are continual proportionals, the difference of the first and second is to the first, as the difference of the first and last is to the sum of all the terms except the last.

If A, B, C, D, E be continued proportionals, A—B: A:: A-E: A+B+C+D.

For, since (hyp.) A: B:: B: C::C:D:D: E, we have (Sup. 8.) A: B:: A+B+C+D: B+C+D+ E. Hence (conv.) A: A-B:: A+B+C+D: A-E; and (inver.) A-BA::A-E: A+B+C+D.

It is evident that if A were the least term, and E the greatest, we should get in a similar manner, B-A: A :: E—A : A +B+C+D. Therefore, in numbers, &c.

Cor. If the series be an infinite decreasing one, the last term will vanish, and, if S be put to denote the sum of the series, the analogy will become A. B: A:: A: S; and this, if rA be put instead of B, and the first and second terms be divided by A, will be changed into 1 -r1:: A: S. The number r is called the common ratio, or common multiplier, of the series, as by multiplying any term by it, the succeeding one is obtained.

This Supplement contains all the principal propositions of the fifth book, and some additional ones of importance. The following list shows the propositions of the fifth book contained in this tract.