DIO 
compofe n‘. We may, therefore, fuppofe n‘, and confe- 
quently z, to have no divifors, but prime numbers of the 
2 ob +3 13 and, in the fir place, let us fee whether a 
ee 8, a 4 Bs an 
the fun of two igoares ouly one w 
hud f° = (a? + 6")? + 4076", norc one 
two {quares ns other igs io te fourth 
+ (408 
power, we fhall find g* = (a1 — 6 _ 
By), pia pa? + pb’. All thefe two are all ae ways 
that p', that be retoived into two fquares, therefore 
' f th ll anfwer the 
of 
now try the produG& 
prime numbers o of the fame on ae and p'; then, accord- 
ing o what iv been fhewn, we fhall have 
— 60° B + Bt 
eae 
‘two 
alues of p* with t , the ebiquadrati 
ppt will be the el uét of the fums of two {quares fou 
different biel ; but the produ& of two fums of two ee 
of two gare twice; for (M? +N’). ae + Q’) 
IP +N ‘y+ MQ—NP)=(MP — NQ )? 
+(MQ+NP/ 
Thcrelon. 2 combining the two values ef p* with the 
the fourth power of 
primes of the form 4x + I is 
jefs than twelve different ways. 
We fee then, that no prime number of the form 4.x + 1 
ean fatisfy the eee se in the queflion, a num- 
ber which is the fum o fuch primes is the fum of tw 
{quares twelve pane ae one {quare of each fum bine 
even and one 0 herefore, a8 every combination of 
three out of thetwe elve will give one pede of the queition, 
3 admit of 
he fans of two cae no 
excluded as pro eens egative numb 
that one combination of three will only give one integral 
foluti 
The t wo laft primes of the form 4 + 1 are 5 and 13, 
their produ@ 65, is confequently the leaft number 
= 2? + 13, and 
and 
et will ee the queftion. 
if = 34 Whence 654 = 3713" + 20167 
65) = 3047) At 300 
5 145° + 3640 
which are only three of the dsc ways that ee is the 
And taking the three even {quares for 
0, b= 2237536, 
ae is ove folution. 
Tn like manner, if we oe ce three of the twelve ways 
that 65+ is the fum of two a as 
654 == 2047 Hee 
654 = 2145? + 3640 
5¢ = 40957 + 1040? we fhall derive an- 
other fokution ; @ == 740208, 6 = 12914208, ¢c = 335392, 
and d = 35 5481 7. 
DIOPHANT US, in Biography, a celebrated mathema- 
tician and analyft of Alexandria, who flourifhed at a period 
n precifely afcertained. Accordin 
as not bee gt 
pe eee in his * Hik, Dynatt.” he flourithed coat 
a 
: S 
re) 
x 
DIO 
the emperor Julian, — the year 366 of the Chriftian 
era, It is certain that he could not be later than this time, be- 
caufe the ingenious female yea commented on his work ; 
it is well knowa that fhe flourithed towards the commence- 
ment of the one ese 
i reek Anthologia, which furnithes 
of arithmetical prea. the following particulars of 
j i hen he was 3 F 
aes whence i ae eee Diophantus was 84 years of 
e when he died. T’b em amvunts to this: to fin 
€ 
he is the firft of the Greeks who has written on this fubject. 
Although we ffiould not be warranted in afcribing the in« 
vention of algebra to Diophastus, he introduced the ufe 
of various fymbols into this feietce enotes the une 
he cube he called xufos, and 
oe it by x” the bi sotak by 367; the pies aad 
by 2 But the diicoyery which demands ore 
Pp renee attention was the method adopted by Diephantus, 
of applying t 
blems. 
o more than fix 
e prefumed that no more 
, in fhe 7 ee to his Algebra, antes im 
the year 1972, lave that there were but fix of the bo 
rary 
, hed at Bafil by Xylander, in the 
year 1575, Latin verfion, with the Greek {cholia of 
Maximus Planudes upon t o firft books, and cbferva- 
i The 
own. fame books were afterwards pub= 
lifhed in Greek and Latin ct Paris, in 1621, by Bachet, 
who made a new rae ibibo and added learned commen. 
taries. He regard to Xylander’s notes, but 
treated thofe of the “feholiat Planudes with the utmoft con- 
tempt. He feems to intimate, that the fix books of Dio- 
cient analyft, e fame wat a has engaged the attention 
of Ozanam, Preftet, Kerfe de Lagni, Frenicle, Wallis, 
Saunderfon, Euler, Playfair, Ivory, &c.——Montucle, Hitt, 
ath. vol. 
DIOPOLIS, in Ancient Geography, a town of Afi 
Armenia Minor, formerly called Cabira, and ee 
ebafte 
DIOPSIS, i in Entomology, a genus of the dipterous order 
of 
