The Method of Fluxions

This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1736 Excerpt: ...as with the former; and so on in infinitum. The Quo tient therefore Is j----+---, &c. (or j into i--I + ji--ji, &c.) So that by this Operation the Number or Quantity jq, (or1 x-f-x_I) is reduced from that Scale in. Arithmetick whose Root is b + x, to an equivalent Number, the Root of whose Scale, (or whose converging quantity) is. And this Number, or infinite Series thus found, will converge so much the faster to the truth, as b is greater than x. To apply this, by way of illustration, to an instance or two in common Numbers. Suppose we had the Fraction f, and would reduce it from the septenary Scale, in which it now appears, to an equivalent Series, that shall converge by the Powers of 6. Then we shall have j =---and therefore in the foregoing general Fraction j, make a= r, £ = 6, and x=i, and the Series will become f---f. 1--±t Sec. which will be equivalent to.7. Or if we would reduce it to a Series converging by the Powers of 8, because f =-f, make a =5 1, b = 8, and.v =--1, then then f = r +--t-p-+-g tec. which Series will converge faster than the former. Or if we would reduce it to the common Denary.(or Decimal.) Scale, because i =-T, make taijeio, and X--3 » t"en T == T7 "4" Too "+" TooJ H Too o o' "+" To oTTTJT C« s= o, 1428, &c. as may be easily collected. And hence we may observe, that this or any other Fraction may be reduced a great variety of ways to infinite Series; but that Series will converge fastest to the truth, in which b shall be greatest in respect os x. But that Series will be most easily reduced to the common Arithmetick, which converges by the Powers of 10, or its Multiples. If we should here resolve 7 into the parts 3-4-4, or 2-+-5, or 1-f-,6, &c. instead of conv...