DIGPHANTINE, 
Examples, 
x. Find the proper faGtors of the diff. x, when x?+ x 
— 1and «* — 1 are to be fquares. 
a & 
Let z and — = 4 diff, = —--— —, or 
z 
the factors; the 
2% 3 
x x 4 Zz? 2g 
——-— +e SH wR 1. 
4x 2 4 
2 
. .- , 
tional value, i if —,= x*,or42?= 1, andz=d 
4% 
We fee that x may have a ra- 
20 oe eae 
a ake roper fators for making two fquares, m 
and. n? ne ailferee ce is d, fuch that the leaft, 2°, may ex- 
ceed an ny given number ¢ 
2 aig)? 
d z d 
FaGors x and— ; é difference =— — —, or— 
% 2 2% 22 
—2 a a? 
t =a, is to be greater than ¢, or 2+ — 
2dzxz? + a greater than 42’¢, orz?— 42°¢ —2d2? + d? 
greater t 
it a —2ds° greater thano, or 27—4¢ 
2 ge than 0; and 2” = or greater than 4¢ + 2 de 
Ife =ard, then ater than d.4n + 2. 
Hence, if we take d= fa 
angies as we pleafe may be found, all having the fame 
ae 
=a, Or are 
lare, as many right-angled tri- 
bafe 
— 
Pros. XVI. 
o find two numbers fuch, that the fquare of each, 
bene added to the fum of beth, may make two fqaure 
-bumbers. 
1. Let » == one of the numbers; its fquare is x, and 
x? + 2x +1 is a fquare. Let 
both ana then x +1 e greater, its ecu is 
wt 2 1, and if we add a fom of both, x 
is to be “nade a {quare, fuppofe = « — 2)? = x* — 2x% 4 28, 
2 
—, where x may be 
2B+4 
ru number greater than 14, if z = 2,« = 
» KC. 
rt 
a & 
or, 22+ 4x = x) — sand x = — 
4, and*x« +1= 
Pros. XVII. 
2. To find two fuch numbers, that their produét + their 
fum may be f{quares. 
y. Let « = one of the numbers, and a*x + 8?x =the other: 
the: a?x? + ox? = their produ&t: but a?x* + 24 
+ 2x? = ax + bxl' =a fquare. We hav e only to make 
2abx? = ax + bx + « = the fum of the two numbers: 
or, x= eth == » where a and may be any numbers 
at pleafure: ifa = 1,5= 2, then x =3, and ~ = = greater: 
36 Y 9 
for their produ@ = and fum = 2; and —, or=, are 
or el Pp ba ri 4° 4 
{quares. 
Fros. XVIII: 
To find three numbers fuch, that their fum and the 
fum of every two of them may be {quare 
xt -+toax+1=-x +11 be the peas ieee three: 
Let/ 7 be ie fum of the art and fecond: x-fi 
third number: Let «? — 2x +1 be the fam of ae fecond 
and third numbers; then *?— 4x is the fecond number, 
which taken from «* leaves 4 oi the fit number .°. the: 
thre numbers required are 4, x? —4 x, and 2x +1 
» But the fum of the firft ae thied = ae + Lis alfo to 
2 
aa 
then x = : where a? may 
be a fquare, fuppofe = q?: 
be any seas saan greater than 25; becaufe the fecond 
numbe or x — 4 mutt be greater than o .:. x 
greater re 4, ie ‘greater than 24, and bx +1 greater 
than 25, 
i] 
Examples. 
1. Suppofe at = 49 > thenx = 8,46” = 32,07 —4x= 
= “<0 80 
ae and2x-+1i1= 
41, &e. 
Pros. XIX. 
34. To find a number which being divided into any two 
arts, the {quare of either part + 10% times the other part, 
fhall make a fquare nur ae 
Let y =the ess Pia soe x one iol and y — h 
then v? +1 o be a {quare, eee 
2)? x 
other: — 100 xis 
a IoOo x= ZP— 22 Ke 
Now as the aon isto fiud y without limiting #; kt22=— 
100, or == 50: then 100y = 2500, or y= 25: there- 
fore x? + 100. 25 —x will bea {quare, let «be what it will, 
GENERALLY. — , 
To determine y when s?x? +2. y9—a isa {qnare. 
Let Px? + ry —remsx —2) al hal 
y 
let r= 25%, then 252 y= 2, andy = = but z= <“s 
7 s 
2% -+1 be the fum of y= = where r may be = any fquare number, and s = 1. 
OB. AX. 
35. To find two jibes in the ratio of a: 4, fuch that 
either of them, added to the {quare of the other, may make 
a fquare. 
Firft, let 2 — = the Iefs of the two numbers fought, 
thenz —«P +4anx—=2+ x! one 
and 4.2 = greater: 
condition fatisfied. Second, but a:b::n—x*%:4n: or 
4anu“ =>bn—br; Serer 
Therefore, pee et. ee 
4an+b6 ~~ gan-b 
lefs number, and 424 = as = greater. the {quare 
of which + the lefs i - alfo to be a ie 
Or, 16 bn 
"36 an? + Sabn Rt ae _ 
16 6? n+ + 16da + 4abn? . _ : 
16a2+ 8abn+e ° ia ed 
is to be made a fquare: fuppole = = eal - eu 
aneree than a4) =42'nt—4bhez n> zn, n + 
x — ab 
- a orto 
mie 163? 24+ 16 a? n? +4abnr'a ae apace it 
4br—a2n? (x? being greater than 444). Then 
1a -+8hban3 = x90 — gabn’; ora = 
4620 = 2° n? —abn®: or n= 
es Sha 
Ife = 10, a= 2, b=3, thenn = 7 and = and 2 the 
numbers required. 
: Lemma. 
