DIOPHANTINE, 
x, es a = a found. ae ab +e =n, then 
ab = find a, 
6, and n, without trials, 
take an ae ace r ae s, rthe greater, fo that the 
{quare of either may be Breater than e, ~ let a=ent+yr, 
~s, then ab =n _ sn? —e(1), or 
rs— r—e ome 
a= 3 At Trmeas sna—-sa b= ° 
~— r—s 
From a & + ¢, already made a fquare, take a ¢ + e 
the difference isab —acma.b—c the faGors (25.) The 
até—e we -ab+e . 
a. the 2 Z difference is —-—-——. 4. Again, 
4 fum is 
ony an 
af. can ae 
a 
at+b+-e 
omaha pte he 4 othe ae isab—be=}. 
a+o0—e 
a“ —c, the £ fum is , and £ 
Therefore if fucha value of ¢ can be found, that 
/ 
may be = a6 +- e = n”, the problem is folved. 
=+ 2, then c= a +6+ 22, either 
of which aie a ¢ with a and 4 above are the numbers re-« 
quired, 
Se Buiamaterandb=n—s.t.atbaor—st+ 
an,anda-+b—wn=r—s= lefs value of ¢, &e. 
oroilary. Hence if r—s= 1, then 2, a, 2, &c. 
will be integers. 
. Example. Let ¢ = 3, r = 4, and 
= 6, anes 1, or 37, fothat ¢ + ¢, b+ e, 
If the prob. had been to find a4 — e,'we 
—e,and dc —e all fquares, the only oa a be to 
e 
ne | 
s == 3, thena = £3, 
e,and a + ¢, are 
change the fign of e, or find a= a b= and 
s — 
; 45. 
the lefs value of c= + — 5, or he greater s = ‘the excefs of 
above r — s as befor 
Pros. KV. 
o find four numbers, fuch that, if 1 or any fquare 
number be added a = 
sd fums may be fq 
Take a, 4, a ¢, in cael progreffion, their com- 
A B Cc 
Then if x, ax +24, B?x + 26, and 
the four a fought, every condi- 
= fquare. 
Rpabtadb 
mon dilforence . 
L 2K n 
Gee is tae except BD - 
2, b= 3, c= 4, then the four numbers fought are x, 
B Cc 
ax+ 4, 9x + 6, aad aa BD+1= 64%? + 
Let Sx |? = 64x07 — 96% + 565 192% 
96% + 33 
= 3, and x = rm 
3. Let tx — 12)? = 64 x? — 192 « + 1443 288 x = IIT, 
and x = wise = 3 Let 8» — gl)? = 64.07 — 1446 + 
288 ~ gO 
eee = 
Sr; 740 x = 46, and x = oa Therefore, = 
a, cig 112, or, a ia 39 and 58 are the numbers res 
jo Io 10 
quired, when 1 is the a number. But if we fuppofe 
it fome other {quare number, as 100, and multiply each of 
thefe numbers by its aoe we fhall have 2, 48, 78, 112, fot 
the numbers required. 
Vou. XI. 
rodu& of every two of them, all: 
Pros. XXVI. 
43. To find x and y, when mee xy, and x? mA are 
all {quares. Affume x —y=a’, + y = ma’, or x* 
—y" = m° at= a fquare. . 
ra+ a 
Now, 2 x m a + a’, or x = —-——, and y = 
ie a 
2 2 2 6 
m a—a a’.3mt+ I i 
——. Bux? v= ie eslaa ahd =, or 3 mi + rmuft 
2 
be made a fquare ; 3; mmuftbe = 2. Hence this 
R y = any {quare es and « -+ y= 
4 times that number; and find x and y required, 
LExampls, Let x—y = 4,and « + y= a then « = 10, 
aud y = 6, and the three expreflions 4, 64, and 784. 
Pros. XXVIII. 
44. To find three numbers in geometrical ce ch 
that a given number being added to each may m 
all ae 
and x? denote the three numbers fought, and 
the rit condition i ts fatiafied. 
Find two {quares whofe difference is s (24), ak let the 
lefs = = a , and the fecond condition is fatisfied. 
and x? -- s, are alfo to be {quares. Their d: erence’ is x*— ax. 
a 
Fa@tors « and x—a, the 4 difference is = — =axtss 
4 
orax= a s = 2d number, and the third is found by 
proportion. 
Ob. 
Daas 
a’ : ; 
ax=a — —s mut be affirmative .*. a® greater than 
is a two [quares whofe difference is s, and the 
efs, of fier ew than 4.5, (30. 2.) call this lefs fquare a? 
the mean term is $ a’—s, ate the other extreme is fund by 
propertion. - 
Lxamples. 
1. Find three numbers in geometrical i ha fuck, 
that 21 =-+s —_ io each, may make the fums fquares. 
The twa 
84 .°. 
Factors 21 and — = and 3 diff, = 103 — 3 = 10. 
fquares are 121 and 100 = a@° greaterthan 4s = 100% 
4 and 4 are the numbers required, 
— 
@ greater than 4¢ 
Fa&ors 
2. Lets = 19, then 100 and 81 
25 
1296 
: andi dif. =gi—4=9. 
Pros. XXVIII. 
48. To find three fquare numbers in arithmetical pro» 
are the numbers. 
z= 76and S81, 3 and 
1gand 
. 1+4d,and1 + 8 d= denote the i numbers, 
then 1 en 4 dand 1 + § dare to be sc 
Difference is 4d, faStors 2, and 2 d, 
@+2d TES Ge diad da oor d= 6, 
chat 1, 25, and 49 arethe numbers. et x7, 2? y? > oe 
2 x* x? denote the three fquare mumbers, then r, y 
*—1 are {quares; and y= 5 «%. Ty 25, 49 nui iplied 
by x°, will give as many aatwer as we. pieafe. 
4 | 
nN 
Peor, 
