DIOPHANTINE, 
. The fquare of the hypothenufe -+ the double produ& 
of ile fides == a {quare, but os double produd of the fides 
<= 4 times the area ,*. the fquare of the hypothenufe = 4 
times the area = a fquare fie tr 
Find, (per 49.) Lae right-angled triangles having im 
the fame bie ne ex tr hypothenufes be a, 4, and ¢, le 
m= 4 times the common area, and sm a+é+c 
a + m, 3° = m, aa - ++ m, are all fquare numbers. 
. Affume any indeterminate a as x°; then 
2 tx? mat, a nd are fquar 
ae to find fuch a ailie - a nat mx* may be = ax + 
bx +e¢x =the fum of the haan canoes let mx nx*be = = 
atb+x 
axtbx+ex: thanx = . Therefore the 
2. 
then 
as bs 
three numbers ax, bx, cx, are —»—, and —, re{pec- 
m m 
tively. 
Example. 4. The three triangles having the fame area in 
495 are ee 42, 58 — 24, 70, 74 — and 15, 112, 113 
.4= 59, b= 74, and c= 113, s= 245, m= B40. 43300, 
ae _ 49 _ a Se 518 791 
and — aoe a a oe i aaers: a: =~, are the 
three numbers required: for their fum is oe 
6 9216 
and 42°) = TSC an 54836416460 _ a9 329479 196 
99 921 9216 “9216 "9216 
574 14 
the {quares of 56 —}, or 56 &c. 
Pros. XXXIV, 
. To find three fuch numbers that the fum or difference 
of any two we them m val be a {quare. 
1, Affum and 1 +- 4.x, for the three num- 
bers fought: 
and each of the others, are fquares : 
the fum and difference of the two laft fquares 
to make 5x7+ — 3, {quares; their pr 
5 15 = 15 .xt— 1 muft bea fquare. If «= 2, it 
is 15. 15. 15 evidently a fquare, and the three 
= 8, 8, ond pe the firft and fecond the fame 
; . Putxw=z—2,then15 xt — 15 = 225 — 4802 + 3602? 
—12023+ 152%+= a {quare, fuppofe=15 — ax+bx\*= 
30 
225 — 504% + 7 
gi—2abz3+ 5x4, where it is 
evident that the co efficients of the correfponding powers » 
of z mult a bien elfe all the terms brought to one fide 
could not . From 30 4= 480,we have a= 16; from 
306 + a? = 360, 306 = 1043 and b= sa and from 15 2+ 
— 20283 = bh 2t—24ab23, 152 —-10=he—24a)b; 
I3z2—-P2= 120 — 2ad,and z= A= — 
342 698 
X= Z—-2= ee eee . Which, fubftituted in 
671 671 O71 
the affumed expreffions, gives 4 . eek ——, 4 eel ,and 1+ 
671 671 
6yo)? _ 2792 2288168 2399057 
—, and 2224 ejecti 
4: 671! 671. “oqil 671) ee 
the common denominator, we have the numbers fought. 
1873433 
2399957 a 
2285168 
Sum 
ce hee 25 Oo of 2165. 
Difference 
“gnesee = 0 of 333. 
—e 
2288168 
1873432 
a ny 
41616c0 
ee a eng 
414736 
Sum O of 2040. 
Difference = O of 644. 
2399057 
1973432 
Sum = == 0) of 3067. 
4272459 
Difference 525625 oO of 725, 
ae 
Pros. XXXV. 
To find the leaft biquadratic number, that can be divided 
into four eli, ae fuch that the fum of every two of 
them may be a fquare 
Let x‘ denote the biquaoeie number fought, and a, b, 
¢, and d, its four integral parts; then the quettion requires, 
2 a+b 
that a =pepy ate =a re es 
ce ae cre =? Hence we getat = a+) -+ 
d= ~= 92 yj? 172 yl"? 
ef 
We e. Gear that the biquadratic number fought 
muft be refolvable -into three fquares, three different ways, 
a this condition being fulfilled, we derive 
oe we oo. eis E(u ae ) 
c= EC 2p 9"? — 1), d=ni— 2 (py? + pl? bh yf), 
from eck formule it may be inferred, chat the three 
{quares p?, »’”, and »*, muft either be all even, or two of 
them muft be odd and only one even, in order that no frac- 
tions may arife from the divifions by 2, and ret al po 
the fum of every two of them muft be greater t 
third ; ; and, at the fame time, the fum - all three iefe ree 
2 2, in order to avoid negative number 
It is then neceflary to find a biquadeatic number se 
may be divided into two fquares three differer 
we to Fermat the curious difcovery, that ev - prime num- 
ber, ay an a eet ae ef 4, by unity, or comprehended 
in the form 3 the fum of two fquares; and this 
property, anche is cet for prime numbers of the form 
4x + I, belongs exclufively to that form, fo that no iba 
number, which is not - the form 4x + 1, can poffibly b 
ie fum of two [quar his curious propofition was Yirtt 
demonftrated by the blessed Euler, and to effe& the de- 
monftration, it was neceflary to sah this other area: 
Ww 
a=9lu 
m of two fquares. 
gather, that the biquadratic number, 7‘, which 
ng, muft have prime divifors of the form 4.x + 1; for if it it 
and as the leaft value of n* is required, we m 
other divifors ; for 1f «+ had any other divifors, thefe could 
only enter as common meafures, into every two {quares that 
452 compole 
