[ 37 ° ] 
and therefore b = ~ lC “- 4 . Whence, to make 
2 
£ a rational integer number, yzw — 4 mud be a fquare; 
which it will be, if a= 1 ; and then b will alfo be 1, 
and c will be 2 : and having continued the feries, 
every number will have the properties mentioned. 
The fecond thing which Albert Girard mentions, 
is a way of exhibiting a feries of rational fractions, 
that converge to the lquare root of any number pro- 
poled, and that very fad. He tells nothing about 
the way of forming it, and only gives the two fol- 
lowing examples ; viz. 
He fays, y" 2 is equal nearly to : or, if you 
would have it nearer, to - 3 g . 
His other example is of y' 10, which, he fays, is 
nearly equal to 3 3 - ^ \ fa ; i. e. to And thefe 
are the fractions your lorddiip has turned, at fird light, 
into continued fradtidns of the fame value *. 
The way of making a feries of rational fractions, 
which converge to the fquare root of any number 
propofed, in l'uch a manner, that the fquare of the 
numerator of any of them being leflened by an unit, 
or, in lbme cafes, increafed by an unit, the remainder, 
or fum divided by the fquare of the denominator, fhall 
be exactly equal to the number propofed, depends 
upon the following propolitions : 
Prop. 
* N. B. That the continued fra&ion here alluded to for exprefling the 
fquare root of io was £ x 19— jV 
— 3 f 
1 
—"IT 
— 3 ^, i5' c. ad inpnitum. 
