152 A Treatise of Algebra.

(fit 3 * fi^t

cur Equation will stand thus-s- c-x — —

a->~bt a a z

b z p a z e

and, putting m=c —, z — —, it will be-—

a a a — be

-J- mq 2 e — p ; whence a z e -f- amq z e — mbf-e 1 — ap — bpe tor a z e -f- amq z e — ap — bpe (by rejecting — mbq z e z , as

inconsiderable ;) this, solved, gives e (=; r: -)

a + bp +mq z '

-- : Which Equation, or Theorem more

a + bq + mq 2 '

than quadruples the Number of Places at each Opera-tion. But, if you would have a Theorem still more exact,p 3 b be—ad b z c z — bd

put? = — ,r = — H-,,, t — —+- r .and w

a a ac — bb a z ac — bb

b , ? -f" roo^

;= s -; then will e = —----; which takes in

a 1 + s q +

the four first Terms of the Series and quintuples, atleast, the Number of Places at each Operation, so thatone Operation, in most Cafes, will be sufficient to bring

out the Answer, to less than- Part of the whole

100000

Value: For which Reason it would be needless to carrythis Speculation further; else, by the same Method, otherTheorems might be derived, to take in 5 or 6 Termsof the original Series; but then they would be found moretedious and perplexed in Proportion. I shall therefore ao\uproceed to illustrate what is laid down above by a fewExamples.

Example 1. Let the Equation x z -f- 2ox = 100 beproposed, in order to find the Root thereof. Here x ap-pearing, by Inspection, to be something greater than 4,let 4 -}- e be substituted instead of .v, and we shall have16 -j- 8* -|- e z -f- 8° -)- 20^ = 100, or 282 -f- e z — 4 : ;s j'|;eresQie, in this Cafe a is = z 8, b = r, czo, d= o,

ev.