ZL A Treatise of Algebra.

viding the Whole , by any Quantity, is equal to the Quo-tients of all the Parts, by the fame Quantity.) After theo c od “ * '• be

fame Manner will — — — be transformed to- ;

b a bd

2 xy ix c ixybd -f i$a*dx — *a*bc

and - — — — to -—-. Also

5 a 7 - b d sa*bd

2 dx 2 dx ,b idx-\-ab

-1- b, or —- -f- — will be transformed to--

a a l a

±xy , b % 4 cxy —f" lab * — 2 aed

and — H- d to -. Moreover

2 a c lac

tab lab a

- -f-a, or -H-, by Reduction, become*

a — b a — b I

lab-^-aa — ab ab~\~a x a* — lab

--- or —-—: And —

a — b a — b a~\~b a — b

becomes a l± * x TZI ~ b * ^~J ab X or

a-{~b X a — b

2a3 —2 a 1 b — ab x — bl

a 1 — b 1

a 1 a*

So, likewise, by Reduction,- 1 - ; - zee

_ a — x a — t~ x

... , a'xa-f-a—f-a*xa— x — zaxa ~i~ xxa — x

will become --—--— —— --

a —at xn + *

2axt . S</xi . i°* — 5* ,

or -;, and t - V ■ H-r»=r—- will become

a 1 — x a V xy -f- a ’

5*v - 4 - 5<Jv / A i y -+* 10a* — <;ax ,

11 1 - -—— 7 i and the like m other

a\? xy -)- a

Cafes.

The Reason of the two Kinds of Reduction hithertoexplained is grounded on this obvious Principle, that theEquimultiples, or like Parts of Quantities, are in the fameRatio to each other as the Quantities themselves, or that,the Quotient which arises by dividing one Quantity byanother is the fame as arises by dividing any Part orMultiple of the former by the like Part or Multiple of