DIOPHANTINE. 
24 x7 
2 
+ 
ap s* 49 x7 x? 
Moreover, é 2. = greater number, and = 
24.00 x4 
== lefs; the diff. of their {quares is ———— = 600 x*. 
*he two remaining conditions that 25 0° + 
6oox* are to be fquares. Divide 25x°, then T + 
24 x? are to be fquares; firft make 4 — 24 xa fquare: 
Affume 1 — a x\? = —-2anxen4+ 4x0 = 1 - 
24 x7, OF; . + 240 = 2am; @x+ 24K = 24; 
I I 
and x = ——— (ifa=4 oi and x* = — 
a a 24 ( ) 5 25 
I 
Therefore, + F 19 and Py are the nmbers required. 
° ° 
Pros. V. 
. To find two fquare numbers fuch, that the fum of 
hee may be a fquare, the sali of them a cube, and 
the roots of the fquare and cube e€ 
Since the fum . the two apes is to be a {quare, 
let gx? and 16x* denote the numbers fought, then their 
fum 25 x° isa (ae but their diff. 7 x? is to be a cube, and 
5% aor per the laft condition of the queftion. We 
I 
have 125 x? = 7°, orx = rer 9” = a 3 and 16 x? 
= ee = the numbers required. 
15625 
Pros. VI. 
20. To find three numbers A, B, and Cc, fuch that A? + 
BC, B+ AC, pain , may be e all {quares. 
L x, denote the three numbers required ; 
then the two firft conditions = fats’ fo A? + BC=1 
+ 4x? + 4x= 26 —1),a +AC=x—-1i1"+ 4% 
=x toaxntitdxe='e + a ; fo oe only to make 
c+ AB= 162’ + x—ta {qua 
Affume 4x —a\?= 16x° ee +x—TI; 
Sax+xe=a’t+i,or«= in ae and the three 
er — : Bat’ 
b iit ea rinacommon 
mbers are 1, ——-———— I, and : oF 
= "Sa+1 8a+1’ 
8a+I a—8a 4- a + 
qinator =——, and ——. Or,8a4+1, 
deno — 1 8a+ 1’ 8a+ 
a — 8a, and 4.a°-+ 1, where a may be any oumber greater 
than 8. {f «= 9, the numbers are 73: a and 328. If 
@ == 10, tie wambers are 81, 20, 404, 
Psos. VII. 
a1. To find two fquare integers, whofe fum may be a 
oO; or to find as many right-angled triangles, in whole 
oS as we pleafe. 
T.-ve rand s, any two unequal numbers, r the greater ; 
P— Port 2 res? + stra os SS 4r sy, 
then in > : 
For their ta is rt + 27’ 5? 
the (uate number required. 
+st= Pte = a f{quare. 
Pros. VITI. 
22. To divide a given {quare C* into two other {quares 
@ and 2°: or froma given hypothenufe, to find the fides of 
a right- -angled triang 
1. Weh have C? — a =o O, which put =aa—c}? = 
wa —2nac +03 then c =@?+P=a@ +2 7 
2nac+t @O,or2z2nacm=m a + @ 3 2nemnatanm 
— 
a.n+i,andas a; therefore, i yand n be raticnal, 
| oe 
b will be fo. 
Ife = 5,n=1,thena=ms5,anddb=o0. Ifc=a2, 
n= 3, thena = ‘s 
a= 2 a4 b= 3, 
n= 3 42> 3 b= 4, 
os as ws PO 
a= me = 
Or, thus: 
2. Let C? = the given fquare, a? one of its parts, and 
c? — a’, the other to be made a {quare. Aflame i it = ra—c}" 
> @? 
2g sere 0% Or, bare = + 2, 2reS 
r'a+a,anda= ie as above 
rl ~ > r . 
2 2 
Or, thus: 3. Let 5 alt= 5° a’, and ra—c)° = ra 
zarc+c bethe two spesuatoiy into which ¢? is to be refolved. 
Then? — Samora C3 2arc= + 
— 
F r arse 
Sa; 2re=ra+ts'a, anda= ——..sa=———,, 
res re 5? 
the root of one of the fquares pe and ra—e¢= 
2re c 2rPe— reo sre c— 
- ——; = —"* = root of the 
rs r+s ry 
other. 
2eo tb ste 
ae are the roots of three 
S 
r. Hence ¢, — —— and 
res 
{quares, of which the fum of the two lait fquares is equal to 
the firft, or the three roots are the fides of a right-angled tri- 
ae ae whole hypothenufe is equal c. 
ch of pn ae be eae by 7° + .*, we fhall 
have r7¢+ sc » and rc 2¢ for the ae fides 
of another triangle 5 ; or, oe Soda each by ¢, we have 
xv? +b 7, s, and 7” 
Pros. IX. 
. To find two numbers whofe fum and difference oe 
be ae {quares. If r? + 5° be one of the numbers, and 27 
the other, the problem is folved. But 7* + s? is sada 
the fquare of the hypothenufe of a right-angled pails, 
and 2rs the double produ@ of its fides; therefore, the 
{quare of the hypothenufe +, the double produc of its 
fea is always a {quar 
Pros. X. 
24. Given the difference of two fquares to find the 
{quares. 
Rule.—Refolve the given difference into any two fac- 
tors, and the % fum of them t of the greater 
ee and the 4 difference the root of the leer, For c? — 
a= Cra ae 
—Find two {quares whofe difference may be 12.. 
Fide 2and 6, £fum 4, 3 diff.2. Or, fa€iors 3 and 4, 
§ fam 2, 5 diff. $, and 49 
4 4 
12. + 
Pros XI. 
25. To refolve a duplicate equality, or to make two proe 
pofed expreflions {quar 
GENERAL Rue. 
Refolve 
the difference of the propoled ‘quares into two: 
fuch faGtors, that the {quare of the 3 fum of them m 
ay be 
nade 
