A treatise of Algebra. 75

nuently — — — (by p. 17 J or at : bs :: cg : dh. Hencebf dh

it follows that, if four Quantities are proportional, theirSquares, Cubes, &c. will likewise be proportional.

THEOREM VI.

Iffour Quantities a, b,c, d, or 2,6, 5, 15 are proportional.

c

-c

h

1. inversely, b : a :: d : c, or 6 : 2 :

2. alternately, a : c : : b : d, or 2 : 5 :

3. compoundedly, a\a-\-b.'.c\c-\-d,ox2 : 8 :

4. dividedly, a:b—a:.c:d —r,or 2:4:

^5. mixtly, b-\~ad — awd-fic.d —c,or 8:4:

6. by Multiplication, ra : rb :. c : d , or 2r.6r.

a b 2 6

Is: 5

6 : 15S : 205:1020 : 10

5 : 15

7. by Division, — : — :: c : d, or — : — :: 5 : re

I r r r r

Because the Product of the Means, in each Cafe, is equalto that of the Extremes, and therefore the Quantitiesare proportional, by Theor. III.

THEOREM VII.

If three Numbers a, b, c (or 2, 4, J) be in continued Pro-portion, the Square of the first will be to that of the secondas the firfl Number to the third ; that is a 2 : b 2 :: a : c.

For ac — bb, by Theorem II; therefore aac — abb , byequal Multiplication, and consequently a 2 : b 2 :: a : c,by Theorem III.

In like manner it may be proved that, in four Quan-tities continually proportional, the Cube of the first is tothat of the second, as the first Quantity to the fourth.

THEOREM VIII.

The Differences, b—a, c — b, d — c, e — d, tsfe. of aRank of continued Proportionals, a, b, c, d, e, (stc. arealso a Series of continued Proportionals.

For since, by Theorem II, ac—b 2 , it is plain flatat —» ab, otaxe-m-b is equal to b 2 —ab, or bxb — a,

and