TOL. XLVIII.] PHILOSOPHICAL TRANSACTIONS. 431' 



given a, to find b such, that t/) + 1 = {ac ^=:.") aa -\- ab \ then is bb — ab ■=. aa 

 — 1 : and therefore b ■=. \a -\- \ ^ baa — 4. Hence, to make b a rational in- 

 teger number, baa — 4 must be a square, which it will be, if a =: l ; and then 

 b will also be 1, and c will be 1: and having continued the series, every number 

 will have the properties mentioned. 



The 2d thing which Alljert Girard mentions, is a way of exhibiting a series of 

 rational fractions, that converge to the square root of any number proposed, and 

 that very fast. He tells nothing about the way of forming it, and only gives the 

 two following examples, viz. He says, ^"2 is equal nearly to ^i^: or, if you 

 would have it nearer, to ^-^-S-^ . 



His other example is of a/ 10, which, he says, is nearly equal to 3^y/,»^, 

 i. e. to '-jS^jV-s' . And these are the fractions your lordship has turned, at first 

 sight into continued fractions of the same value.* 



The way of making a series of rational fractions, which converge to the square 

 root of any number proposed, in such a manner, that the square of the nume- 

 rator of any of them being lessened by a unit, or in some cases increased by a 

 unit, the remainder or sum, divided by the square of the denominator, shall be 

 exactly equal to the number proposed, depends on the following propositions. 



Prop. 1. Let a be any number proposed, and - be such a fraction, that — ^^— 

 ■= a, i. e. bh = ace + 1 ; then if two other fractions be taken, one of which is 



b • * c 



-, the first divided by the proposed number a, and the other is y the reciprocal 



of the first fraction; then the fraction — —. — , whose numerator is the sum of 

 the products of the numerators, and of the denominators of the fi-actions 

 - and — ; and its denominator the sum of the products of the numerators, and 



c ac 



b c 



of the denominators of the fractions and -, shall have the same property 



with the fraction -i.e. ^^ — 7T b )'' ~ ~ "* because bb = ace + 1, therefore 



bb — ace := 1 , and squaring 



b* — lab-c'' + a''c* = 1 . And adding 4a^'c^ gives 



b* + 2a6^c^ + aV = 4ab\'' + 1. Hence £^l±ggl=-L = a. 



Prop. 1. If - be such a fraction, that — — = a, i.e. bb -{■ 1 := ace, all 

 • N. B. That the continued fraction here alluded to, for expressing the square root of 10, waa 



i 



— TT 



__ f 



TT 



— -^, ice. ad infinitiun. — Orig. 



