A 1 Treatise of Algebra. 37the Definition of Powers, that it’s Cube or third Powerwill be expressed bv 2a* xN xaf x 3 into 2 a^xb>xinto 2a* x P Xn-j-r) 1 ; which, (by Lemma 2,p. 13.) is equal to 2 x a* xb 3 x a -(- * I 5 X2xa i xi* x

s -j-H x 2 x a? xi 3 xa 4- *1 (multiply’d continuallytogether) which f Lemma 1, in the Jame Page) js

1 1 1 * j. -, -

equal to 2X2x2 xa 2 x a 2 x a 2 X b 3 xb 3 xb> xa -(- x 1 <

X ct -f" X a -)- x 1 5 ; and this again (by Lem. 2 ) is equal

to 8 into a 7 x a* X a 1 into b 3 X b> x b> into a *

Xa -)- A 5 xa + rl* j buta^xa^xa 1 expresses the Cubeof a 1 , which (by Notation) is represented by a ; like-

II I

wise b 3 xb 3 x b 3 expresses the Cube of i', which is de-noted by b 3 or b , and in the fame manner a -s- *1* xa 4* x' s x a 4- x* >, expressing the Cube of a 4- *' 5 , is

denoted by a 4- ■*' 5 : Therefore 8 into a 1 xa'xa* into

1 -- .1 -1

b x b 3 X b 3 into a 4- x a 4- x * 1 X a 4- *1 *, or the

Cube of 2 a 1 xb 3 x a 4~1?} V. is trqly represented by 8

into a into^into a-f-j4 5 or by 8 a'xbxa 4- *V ; as wasto be proved ; and the fame Method of Reasoning will,jt is manifest, hold in any other Cafe.

4 0 . A Fraction is involved by raising both the Numer-ator and Denominator to the Power proposed.

a a*

Thus — involved to the second Power gives —; andb b*

2ab . 8 a 3 b 3 3 aH’k

— involved to the third Power gives -; also ~ 2 j

3 c 2-Jcl 2C a

27 a b bt 2 ax .

involved to the third Power gives —-— ; and , j m-. b Sc 6 d 3 )by*

1 4

volv’d to the second Power gives —- — or - —— ; and,

49 by 4 sb z y

D 3 lastly,