DIOPHANTINE, 
Lemma.—36. There are many numbers which cannot be 
divided into two rational {quares. 
it, Every even {quare aumner is divifible by 4: for its 
root ae >? and 4 4° the numbe 
2d, Every a fquare number — 1 13 divifible by 4: for 
its root ond 1 and Aw’ -b 4% +1 the number. 
d. Every number = the ium of two even f{quares is di- 
vifiole by 4. Per rf 
th, Every number = the fum of two odd fquares — 2 
is divifible by 4. Per 2d 
5th. Every number = the fum of an even and an odd 
{quare — 1 is divifible Per rftand 2. Hence, 
6th. If any number ae div:ded by 4 leave a remainder 
ual 3, it cannot be compof<d of two integral {quares. 
Nor of two {quare fra&tions. 
Pros. XXI. 
37. To divide any nun.ber compofed of two known 
gia as a? + + 2 into two other fquares, 
tr be greater than s, and not asa: 4, norasa + é: 
as 
Q Na 
_ Aff oe a’, and oar ae = J, or a® + 
rx? + s*x? — 2arx — 2bsx), or 
2ar+ 2 
= r +s a's r 
ar pe tina es 
Sues uers 2bs, and x 
2ar+2bsx—ar—as’ 
—a= 2 1 52 
cee ioe —bs? 
ret s? 
= fide of lefs fquare. 
= fide of greater{quare, andsx—)= 
2ars+bst—b2? 
r + S. 
Let da=rn-—st,e rs, and f = a a rs then 
ad 
arr—as'+2brs=ad+b 
a 
gars—brit+b t= ae—ba 
Ob.1. Ifb=0,this prob. and its folution, becomes the 
fame as Prob. VIII. viz. 7 and 7 are the fides of the re- 
a = = fide of greater fq. 
seeaiekas == fide of lefs {q. 
quired {quares. 
Ob 2. Ifsx+4\? had been afflumed = 2%, the sai 
b 
would have been a = fide of greater, and = a 
= fide of the leffer fquare. 
The four fractions exhibiting both folutions are, —= 
bd be 
— 
E) 
? 7 
Hence, if d, ¢, and 7 be the fides of a right-angled 
triangle, and a and 4 the roots of the original fquares, and 
the two fides of the ares = multiplied cp the roots of 
ac bd 
PPP 
then the fum of the extremes and difference of the 
the faid {quares, and fet down in order, thus, 
be. 
means will be the fides of the two {quares required. 
pad ges 
38. Re a to divide 2 = 1? 
eel fae . Let the Gale be 3, 4 and 53 
the fraftions are - 
3 
" 
_ 
I? into two un 
then 
7 . s and ? and : the fides of 
tet ale 
2 rai the triangle be eee » 103 then the arabhlons3 are 
8 
Tul 
“, and =, as above. 
eS 
To’ 10’ 10’ 10° To 
If the triangle be 5, 12, 13, then 5, —s 5, = Z, 
a 13, 13 «33 
and 1 are the fides of 2 4 42 — 35° 2 
3- Lat 1 be divided ino two fare. Here 6 = o, and 
ad 
f 
Br 7 are the iides of the {quares required. 
If the triangle be 3, 4, 5, we have san : 
If the triangle be 5, 12, 13, we have a : the fides 
25 ds 
169 
of - a and 
Pros. XXII. 
39. To find four pe angled triangles, in whole numbers, 
all! as the far ie thenufe. 
oluti ‘Take 
any two right-angled triangles, not 
7 € I 
| bs d, 
1 2 
fimilar, as {ob 4s 5s } and ve ne 
Multiply the three fides of the firft ee and we have af, 
b fre f= 39,52.65. Mult.2™ by c, wehave de, ec, cf = 
25, 60, 65, per 37, ad + be, ae mbd, and oe 635 
16, 65; or,a¢ + bd,ad mbe, andcf= 56, 33, 65 
Pros, XXIII. 
. To find four numbers, which, being feverally — 
o, or fubtraéted from, the fquare of their fum, fhall m 
the feveral fums, or nanandee. all {quare num mbes 
Find (per 39.) four right-angled triangles, having the 
ae ae a which fuppofe = 4. 
Let p, p’, p’. pl’, denote the double product of their fides. 
Then A+ p,P tp + pp P+ 2 are all {quares 
23.)3 ee esi B? x? le eM ap es Ox +p 
an Bx? + p” x, are all {qua 
. But the condition | i aha a x thall be the {quare of 
te fam of = + px x pl” x 
Le? tp bp +p ae 
the fquare of ex*, or bx = ex, and x = -, or x? 
i, _ ob 
Then A? x? is to be = 
a 
Therefore, p x*, p' x*, p" x, p'" x’, are = 
p” 4}? 
—? 3? 
which are refpectively the numbers meer 
Demonftration—For the fum of thefe four numbers is 
t ph? + p" B+ pra? eh? 
7 
ph + == 5 and the _ of 
& 
2 fh? a 
att g ie 
their fum is — . But — + B+ 
p= a {quare ea 235) ue with the ei 
Example. The four triangles in 39 give 4° oo, 
p = 4056, p’ = 3000, p” = 2016, ee ae ee 
17136600 12675000 
27888049, and the four numbers are 3788049? 27888049" 
8517600 15615600 ih __ 53944800, 
27888049. BoE 27888049” ane ume 27888049 
Pros. XXIV. 
41. To find three numbers a, 4,c, fuch that, ¢ being 
any a number, 4 b + e,ac + ¢,and dc + e, may be all 
{quar 
Firk, 
