NUMBER. 
315 
It has been demonftrated by Legendre, that the jiro- 
duft of two numbers, each of which is the fum of four 
fquares, is itfelf the fum of four fquares ; and hence, by 
a fimple deduction, that every number whatever is either 
a fquare, or divifible into two, or three, or fcyir, fquares. 
Thus, 30 is equal to 25+4-+1, 31=25+4+1 Hr 1 ; 33=16 
+ 16+1> 63 =499+4+ 1 s or —3^+ z 5+ I + I - 
The fquare, and indeed every power, of 5 or of 6, ne- 
ceflarily ends with 5 or with 6. 
If two numbers are of fuch a nature, that their fquares, 
when added together, forma fquare, the product of thefe 
two numbers is divifible by 6. Of this kind, for example, 
are the numbers 3 and 4, the fquares of which, 9 and 16, 
when added, make the fquare number 25 5 their produft, 
12, is divifible by 6. 
From this property, a method may be deduced for find¬ 
ing two numbers, the fquares of which, when added to¬ 
gether, fhall form a fquare number. For this purpofe, 
multiply any ..two numbers together; the double of their 
product will be one of the numbers fought, and the dif¬ 
ference of their fquares will be the other. Thus, if we 
multiply together 2 and 3, the fquares of which are 4 and 9, 
their produft will be 6; if we then take 12, the double of 
this produft, and 5, the difference of their fquares, we fhall 
have two numbers, the fum of whole fquares is equal to 
another fquare number; for thefe fquares are 144and 25, 
which, when added, make 169, the fquare of 13. 
When two numbers are fuch, that the difference of 
their fquares is a fquare number, the fum and difference 
of thefe numbers are themfelves fquare numbers, or the 
double of fquare numbers. Thus, for example, the num¬ 
bers 13 and 12, when fquared, give 169 and 144, the dif¬ 
ference of which, 25, is alfo a fquare number; then 25, the 
fum of thefe numbers, is a fquare number, and alfo their 
difference, 1. In like manner, 6 and 10, when fquared, 
produce 36 and 100, the difference of which, 64, is alfo a 
fquare number; then it will be found, that their fum, 16, is 
a fquare number, as well as their difference, 4. The num¬ 
bers 8 and 10 give for the difference of their fquares 36 ; 
and it may be readily feen, that 18, the fum of thefe num¬ 
bers, is the double of 9, which is a fquare number, and 
that their difference, a, is the double of 1, which is alfo a 
fquare number. 
If two numbers, the difference of which is 2, be mul¬ 
tiplied together, their product, increafed by unity, will be 
the fquare of the intermediate number. Thus, the pro- 
duff of J2 and 14 is 168, which, being increafed by x, 
gives 169, the fquare of 13, the mean number between 
12 and 14. Nothing is eafier than to demonftrate, that 
this muff always be the cafe; and it will be found, in ge¬ 
neral, that the produft of two numbers, increafed by the 
fquare of half their difference, will give the fquare of the 
mean number. 
It may be thought wonderful that the whole population 
of this country could Hand collefted on conliderably lefs 
fpace than one fquare mile. Allowing 6 men to a fquare 
yard, the mile would accommodate 18,585,600 men. 
Cubic Numbers are thole which arife from the produft 
of three equal integral faftors, or the product of a fquare 
number by its foot; thus, 27 is a cubic number, being the 
produft of the fquare number 9 by its root 3. 
Cubic numbers may end with any figure whatever, not 
being fubjeft to the ambiguity we noticed as to fquare 
numbers. If they terminate in ciphers, thefe mull be in 
number either three, or fix, or nine, &c. If a cube fhould, 
however, terminate in one of the fore-mentioned figures, 
x, 4, 5, 6, or 9, its root will end in the fame figure; but, 
if it terminate in any of the remaining digits, 2, 3, 7, or 8, 
the correfponding root will end in 8, 7, 3, or 2, that is, in 
the difference of each from 10. 
All cubic numbers, whofe root is lefs than 6, being di¬ 
vided by 6, the remainder is the root itfelf; thus 27-7-6 
leaves the remainder 3, its root; 216, the cube of 6, being 
divided by 6, leaves no remainder; 343, the cube of 7, 
leaves a remainder 1, which, added to 6, is the cube-root; 
and 512, the cube of 8, divided by 6, leaves a remainder 2, 
which, added to 6, is the cube-root. Hence the remain¬ 
ders of the divifions of the cubes above 216, divided by 6, 
being added to 6, always give the root of the cube fo di¬ 
vided till that remainder be 5, and confequently 11 is the 
cube-root of the number divided. But the cubic num¬ 
bers above this being divided by 6, there remains nothing, 
the cube-root being 12. Thus, the remainders of the 
higher cubes are to be added to 12 and not to 6, till you 
come to 18, when the remainder of the divifion mull be 
added to 18 ; and fo on, ad infinitum. 
The difference between the fquare and the cube of a 
given number, is found by a very eafy procefs : Multiply 
the fquare by the number lefs one, and it will produce 
the difference. What is the difference between the fquare 
and the cube of 53 ?—Anf. 53 s x 52=146068. 
If we write down a feries of the fquares of the natural 
numbers, viz. 1, 4, 9, 16, 25, 36, 49, &c. and take the 
difference between each term and that which follows it, 
and then the differences of thele differences; the latter 
will each be equal to 2, as may be feen in the following 
Example : 
Squares 1 4 9 16 25 36 49. 
ill Diff. 3 5 7 9 11 13 
2d Diff. 2 2 2 2 2 
It hence appears, that the fquare numbers are formed by 
the continual addition of the odd numbers 1, 3, 5, &c, 
which exceed each other by 2. 
In the feries of the cubes of the natural numbers, viz. 
1, 8, 27, &c. the third, inftead of the fecond differences, 
are equal, and are always 6, as may be feen in the follow¬ 
ing Example: 
Cubes 1 8 27 64 125 216 
ill. Diff. 7 19 37 61 91 
2d. Diff. 12 18 24 30 
3d. Diff. 6 6 6 
In regard to the feries of the fourth powers, or biqua¬ 
drates, of the natural numbers, the fourth differences only 
are equal, and are always 24. In the fifth powers, the 
fifth differences only are equal, and are invariably 120. 
Thefe differences, 2, 6, 24, 120, &c. may be found by mul¬ 
tiplying the feries of the numbers 1,2, 3, 4, 5, 6, &c. For 
the fecond power, multiply the two firft; for the third 
power, the three firft ; and fo on. 
The progreflion of the cubes 1, 8, 27, 64, 125, See. of 
the natural numbers, 1, z, 3, 4, 5, 6, &c. poffeffes this re¬ 
markable property, that if any number of its terms what¬ 
ever, from the beginning, be added together, their fum 
will always be a fquare. Thus, 1 and 8 make 9 ; if we add 
to this fum 27, we lhall have 36, which is ftill a fquare num¬ 
ber; and if we add 64, we (hall have 100, and fo on. 
Legendre has demonftrated, that “ Neither the fum nor 
the difference of two biquadratics can be equal to a fquare.” 
Alfo, that “ Neither the fum nor the difference of two 
cubes can be equal to a cube.” The latter of thefe pro¬ 
perties is only a particular cafe of Fermat’s general pro- 
pofition, which may be thus briefly exprefled : “ Neither 
the fum nor the difference of any two equal powers above 
the fecond can be equal to a power of the fame dimenfion; 
or the equation x’ , ±?/*=z’' is always impoflible in integers, 
when n is greater than 2.” 
Compofite Numbers are thofe which are produced by the 
multiplication of two or more integral faftors ; or that 
may be divided in(o two or more equal integral parts, 
each greater than unity. 
Prime Numbers, on the contrary, are fuch as cannot be 
produced by the multiplication of any integral factors, or 
which cannot be divided by any number greater than unity. 
Thefe numbers have formed a fubjeft of inveftigation 
and inquiry from the earlieft traces of arithmetic to the 
prefent time ; either with a view of finding them, or fe- 
lefting them from the common feries of numbers, or for 
inveftigating certain properties which are peculiar to them. 
Eratofthenes is the firft author amongrt the ancients who 
attempted the former problem, which led him to the in¬ 
vention 
