DIO 
DIONYSOPOLIS, in —— Geography, a town of the 
Hither India, cae to Ptolemy, who a . ’ at it was 
called Na Arrtan a es it betwee e Indus 
d the Cophenes. It was the WVy/a of Aes a the 
aie Nughz, or Nagaz.— Ifo, a town of Lower Meelia, 
fea. Pliny fays, that it was 
~ called Crunos ; ae Pomponius Mela fays, that a 
as the name of the port of Dionyfopolis. It is 
ie had this name ne Po a ftatue of Bacchus, whieh was 
carried by fea to this place.x—Alfo, an epifcopal town of 
Alia, in the fecond Pacatian Phrygia; founded by Eume- 
neg and Attalus, who found in this place a ftatue of Bac- 
ehus. Steph. Byz 
| ‘7 , or Dions, a town of the Peloponnefus, in 
Arc 
DIOPHANTINE Fakeluas fo called from the in- 
ventor, Diophantus, a fpecies of indeterminate pro oblems 
relating to {quare and cube nutabers; and oftea admitting a 
aaa of anfwers. The folution of them depends, 
On affuming fuitable pofitions to denote the number 
fou; ht. 
fair os — the propofed expreflion a fquare 
3. making two propofed expreffions cae &ec, 
called fe ive: ofa duplicate equality. 
Thefe problems {eldom involve apr of a high order; 
neverthelefs, it is extremely difficult ive any general rules 
by which their folution can, with eek ee and in all cafes, 
be effected. 
Yet it will be found, that there are two or three very 
general principles, on a due application of which the folu- 
tion of almoft all of them, in great meafure, depends: 
slic e are, 
1. To refolve a Jingle equality, or to — an repens 
a lets porn is fully illuftrated in Prob. oO. 
the fides vf whatever pair casles triangles 
an ired: thisis founded on Euc. 47.1, and is fully 
iluttrated i in Prob. VII. and the fils wius ones, ae foes a 
To refolvea ane las or is make 
two ian expreffions laity fee . XI. &c. and 
rt. 30, i vol. ii. p. re hig is of 
very estenfiv application, ae is wake foundation of the 
th Principle. To refolve two given fquares into as 
ollow 
he a’ a 5 be the hia abi and al eee the re- 
quired ones, Then a* + x? + 9, —ah ~ 4%, 
SI ee ree iy ee es oO 
b+y xX mb—my. Now, ifx+a= 
Mx —- Ma = 
aay b+y 
mb — my, thenmx —ma=b POL MSA ae as 
and x = TYEE, whence —T 2+ m4 _ mb —~ 
my — a, of by + ma wh — my — ma, or wy 
mb —2ma—b 
m? -- I 
» and x 
+ y= mb — 2ma—b, and y= 
ma + amb—a 
= 
m+ 
Cor. 1. If 5 =0, or one es a be to be — os 
two others, we have x = oe andy = — or 
—nt_, which ig Prob. VIIL 
m+ 1 
DIO 
Prosiem I. 
make a Propofed Expreffion a Square. 
e.—Affume a fuitable fide for it, and make its fecond 
power equal to the propofed expreffion. 
LExampees. 
1. Required fuch a value of n, that n? --+ 1 may be a 
fquare. 
Afflume 7? +1 = apezVP =m t+anz + ema fquare ; 
ze 1 
then 27% = 2?—1, and 4 = —-—, and x may be any 
2% 
number greater than 1. 
25 
he = 25 n=, and fretye 2 = 25a fquare 
IO 
8 O4 i 1990 
JP ea 3, a aaa gar ice 7 =: a fquare, 
&c. &c. Hence we ma as many right-angled triama 
gles as we pleafe, all oonie 1 for their bai 
2. Required « when x? + x is a fquare. 
Affume x? Femu—se 2 + 2x% +2’, then 2x2 
+x= 
= x 
2, and w= 
z22+1 
7 1 3_4 
Ifg=i,x=-,andx*+xn=-4+2=2= a fauare. 
3 9°39 9 
f z= 2,xis = 4, and x fxm — peed a 
2 25 25 5 
f{quare, &c. 
3. Find an si value of », fuch that 16x? +*— 1 
may be a fqua 
iets bate —«}?= 16x? — 8ax+ a’, 
: 7 _@tt 2 5 021 
ax-+-xmai+tx, and x ‘Soak F ear ers 
26 37 50 65 
—sI —~a ——3 — = I, 
149 57 95 
Anf. Va=16"= — almoft 2. 
4. Make 2x? — 2x +1, a fquare. 
Afflume 2x29 —2xen +124 
+ I, then 2x — 
x—iIP= 4a o*°— gan 
222 4a’x— 44, Orda’ xn—2x=4a 
Se Xe. $ fac ox is 
= 4, and the given eeciton a &e. : 
ts Find a value of y, fuch as may make SP + l2x+-9, 
Aime 33 Se =95'—18y + 95 
Idd. 
then 3 y Qy—18.6y= 30, andyas, 
= fquare cata ¢ 
6. Given nVja +t = m, to find ma whole number. 
Since 72% + 1 = m*, m is lefs than 32. Affume 
7Fet+i=32%2—aPage — Santa’, 22°—6ax- 
—~P +135 e— Zaz ead Lita we — gan poe 
4 
2 +2 ., 
= it, and Pe cle AD . lf a= 1,2>= 3, 
and m = 8. 
7. Required a rational value of v* — 2. 
Affume vt — 2 = vi — Q’ ‘= vt— 2a? vu? + at, 2 ay? 
4 4 
a+ 2 24a — 
eat+2; c= = cag Here 2a*+ 4 maf 
4R2 be 
