NUMBERS. 
Thus for example, 99 = 17 + 2.7%, can only be di- 
vided Aha the following ce which are all of the fame 
form; viz 
and the fame for all “other numbers a: the above form. 
a VII. 
ing 
prime to each other, can only a ve ial afr canbe 
aoe are of the fame form. T ample, 98 = 107 
. 1°, have the four llawing divifors, bee of which 
is ee the fame form, viz. 
2=2%—2. 17 
3°— 2.1 
yA yP-—2.Vr 
49>=9—2.47 
and the fame for all other numbers of this form. 
For the divifors of the powers of numbers, fee Powrrs. 
Of the Produ@ and Transformation of certain numerical 
Formule. 
Prop. VIII. 
The produ& of the fum, and difference of two integers, 
is equal co the difference of their fquares. For 
(ety) (w@-y)Hwr—y. 
Prop. IX. 
The product of the fum of ad See by double a 
{quare, 1s alfo the fum of two fquar For 
(+9) X 227 = (x + y) X z+ (x —y)* 
Hence, if a number be a {um of two f{quares, its sabre 
is alfo the fum of two {quares. Alfo, if a number N be the 
fum of two f{quares, ce 2” N will be alfo the fum of two 
{quares. 
Thus for i : 
Sa P+ 5 xX 2S 10 = 374 1°3 
IO x era te X 2= 40= 6? 4 2°53 &e, 
Prop. X. 
The produ& arifing from the fum of two = by the 
fum of two {quares, is alfo the fum of two fquares. For 
ge ai cs + yy’)? + ey! — 2" 9)" OF 
CEE SV EE IN Vat yy + (By! + ay)? 
Hence alfo it appears, i: the product may be divided 
into two fquares two different ways. And if this produ& 
be again multiplied by another, which is the fum of two 
{quares, the refulting produ€t may be divided into two 
{quares f.ur different ways; and generally, if a number N 
be the produét of a faCtors, each of which is the fum of 
two fquares; then will N be the fum of two fquares, and 
may be refolved into two fquares 2” different ways. 
For example : 
g? + r, or 7 + 4 
The produ& 65 = 
Again : 1y = 
41 
The produ& {1195 = 3 + of = 33+ 4° 
oS 31> + 127 = 247 + 2 23° 
nd this refolution of the propofed produ into a 
parts, is readily effeted by the foregoing formule. For 
8° - be 8+ 1)? + (8.1—4. 1)? 
(8° + 1) (47 +1) 4.8—1)> 4 (8.1 + 4.1)? 
2 At (4 = i. 7+ 1.4)? + 44-71) 
PET N= yas G at Tay 
and in the fame manner may any other produc arifing from 
factors of this form be refolved into its {quare parts. 
Prop. XI. 
The produé& of the fum of yt la es the fum of 
two f{quares, is the fum of four {qu 
(al te bx") (2® + ') = 
(x x! cae ae wy! _ yay i at? + y? gt 
which is evidently the fum of ee {quares. 
For example: 14 = 37+ 274 1? 
5 = 27+ 1” 
70 = (3-242.1)? +(2.2—3.1)? 
The produé& { $2 4a Pra as 7 
Prop. XII. 
The produé& of the fum of four {quares by the fum of 
two {quares, is alfo the fum of four fquares. For 
(a? ox? + y + at) (x? + y?) = 
(w* + 2") (we + + GF A a") (a + y") 
each of which produéts is the fum of two fquares, by the 
10th propofition above, _— sy confeuenty the whole produ& 
is the fum of four {quar 
Proe. XIII. 
The produc of the fum of four Tenia by ag fum of 
four {quares, is alfo the fum ef four {quar 
(w* + x + 9? 4 2) (w? Saye ea) 
(w aw! + xxl + yy! + 2!)* + (wal— vw + yx! 
= —ylz) + (wy! — x2! —yw'+ zx!) 4+ (wet 
+ ay — 90 — 8 _ 
as will appear immediately from the developement of the 
above formule ; coniequently, the produ in queftion 
is the fum of four {quares 
a in this produ@ there are only complete {quares enter, 
may change at pleafure the figns of the fimple quanti- 
ties; and, confequently, there will refult feveral different 
formule equal to the fame product, and each equal to the 
fum of four fquares; and in fo many ways may numbers 
ae from factors of the above form be refolved into the 
our fquares. La Grange has rendered this propofi- 
tion more a by the following enunciation. 
duét of the two formul 
(w? — ie — cy + bex*) (w™ — ba? — cy”? + bee”) 
(w w! + bx x, teyy' + be z')* — 
b(wel + we +eye' + cy's)? — 
c(wy! — bee! ee + bx x’)? 4 
be (xy! — wa! + zw! + ye’)? 
which equality will appear from the developement of the 
formulz. 
PRopP- 
