3W5 N U M B E R. 
vention of what lie called his v-ov.y.i w, or ficve; by which 
lie excluded from the general feries tbofe that were compa- 
J:le, and confequently the remaining numbers were prime. 
The principle of this method confided, firft, in -writing 
down every odd number from i to any extent propofed, 
and then pointing off every 3d, 5th, 7th, See. numbers, 
each of which would neceffarily be compofite, and thofe 
which remained without points, prime numbers. Let 
there be written, for example, the following feries: 
I 
3 
s 
7 
9 
11 
13 
15 
17 
19 
2 I 
23 
2 5 
2 7 
29 
3 i 
33 
35 
37 
39 
41 
43 
45 
47 
49 
5 i 
53 
55 
57 
59 
Ci 
63 
65 
67 
69 
7 i 
73 
75 
77 
79 
8*1 
83 
8j 
8*7 
89 
91 
93 
95 
97 
99 
We begin with the firft prime number 3, and over every 
third number from 3 a point is placed ; each of thefe num¬ 
bers being divifible by 3, as 9, 15, 21, See. Then, from 5, 
o point is placed over every fifth number; thefe being di¬ 
vifible by. 5, as 15, 25, 35, &c. Again, from 7, every Se¬ 
venth number is pointed in the fame manner, fuch as 21, 
35, 49, &c. And having done this, all that remain with¬ 
out points in the above feries are primes ; for there is no 
prime number between 7 and 3/100; and it is ufelefs try¬ 
ing any prime humber greater than the fquare-root of the 
number propofed, which w-e have fuppofed here to be 100. 
Adding, therefore, to the above the prime number 2, 
which is the only even prime number, we fliail have, 
3 . S. 7 , 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 
61, 67, 71, 73, 79, 83, 89, 97, which are the primenumbers 
under 100. 
Such was the method purfued by Eratofthenes; and the 
fame, aided probably by fome mechanical contrivance, has 
been adopted by modern authors. Of thefe, Vega lias 
publifhed a table of prime numbers to 400,000; and a work 
lately publifiied in Holland, not only contains the prime 
numbers up to 1,000,000, but alfo the faftors of all com¬ 
pofite numbers to the fame extent; a performance which, 
it muft be allowed, difplays the induitry of its author to 
much more advantage than either his genius or his judg¬ 
ment. 
it is obvious, from what is flated above, that this me¬ 
thod cannot be employed for afeertaining whether any 
propofed number be a prime, without going through the 
whole procefs up to the fame extent; the dcfideratum, 
therefore, of modern mathematicians, has been todifeover 
fome rule, whereby any propofed number may be deter¬ 
mined to be a prime or compofite number; in which, how¬ 
ever, they have not fucceeded. Others, again, have en¬ 
deavoured to find fome formula, as .r 2 + r4-4i, .r 2 -{-:r4-i 7, 
&c. which fhall contain prime numbers only, 
whatever value may be given to x; but neither in this 
have they been more fuccefsful. It is in faft demonftra- 
ble, that no fuch formula can be found ; though fome for¬ 
mula of this kind are remarkable forthe numberof primes 
included in them. Thus, the firft of thofe given above, by 
making fucceftively a—o, 1,2, 3, See. will give a feries, the 
firft forty terms of which are prime numbers; the fecond, 
in the fame way, gives feventeen of its firft terms prime; 
and the latter, twenty-nine. Thefe cafes (how the great 
danger of induftive conclulions in mathematical invefti- 
gations ; for it will feldom happen that we have greater 
reafon to draw' a general conclufion, on thofe principles, 
than in the feries above mentioned. It was doubtlefs by 
not being fufficiently guarded on this point, that led 
Fermat, the father of the prefent theory of numbers, to 
affert that 2* + i would always be a prime number, while 
x was taken forany terrain the feries, 1, 2, 2 2 , 2 3 , See. but 
Euler found that it failed in the cafe x—^2, which gives 
a* 2 + 1=641X6700417. 
It may not be amifs to obferve, with refpeft to what i3 
ftated above, viz. that no rule has yet been found for af- 
certaming whether a given number be prime or not; that 
this is only meant with reference to a ready method; for, 
in fa< 51 , it we divide a number fucceftively by all prime 
numbers lets than the fquare-root of itfelf, and no one of 
them will divide it without a remainder, that number is a 
prime, which, though long and tedious, may be called a 
rule for that purpofe; there are alfo other rules depend¬ 
ing upon the quadratic forms of prime numbers, but they 
are all extremely laborious for large numbers. Waring, 
in his Meditationes Algebraicse, gives alfo a rule for this 
purpofe, which he informs us is due to fir John Wilfon, 
and which, confidered in ahjiradio, is certanly as complete 
as can be defired. This is as follows s If n be a prime 
number, then the continued produft (r, 2, 3, 4, 5, &c. 
ji—i)-|-i, will be divifible by n. And, as this property 
belongs exclufively to prime numbers, nothing can be 
more complete ; but unfortunately, the great magnitude 
of the produft renders it totally ufelefs as a practical rule. 
Prime numbers, the number 2 excepted, can never be 
even, nor can any of them terminate in 5, except 5 itfelf; 
hence it follows, that except thofe contained in the firft 
period of ten, they muft neceffarily terminate in 1 or 3, or 
7 or 9. Another curious property of prime numbers is, 
that every fuch number, 2 and 3 excepted, if increafed or 
dirninilhed by unity, is divifible by 6. This may be rea¬ 
dily feen in any numbers taken at pleafure, as 5, 7, 11, 13, 
1 7 > 19, 23, 29, 31, &c. But the inverfe of this is not true ; 
for every number which, when increafed or ftiminifned 
by unity, is divifible by 6, is not neceffarily a prime num¬ 
ber. . Indeed we have already laid, that no ready and eafy 
rule is yet known (as the above would be, if the inverfe of 
it would hold) for the difeovery of prime numbers. 
The number of primes is infinite ; but the quantity of 
them, under any given number N, is very nearly expreffed 
N 
by the formula 
as may be verified by 
hyp. log. N—1-08366 
means of the following Table, which contains the number 
of the primes, under and between certain periods, from 
1 to 400,000. But, for the demonftration of it, we muft 
refer to part iii. of Legendre’s Effai fur la Theorie des 
Nombres, 
IOOOO 
the number of primes 
is 1230 
20000 
- 
- 
2263 
30000 
- 
3246 
40000 
- 
- 
4204 
50000 
- 
5134 
60000 
- 
6058 
70000 
- 
- 
6936 
80000 
* 
7837 
9OOOO 
- 
8713 
100000 
- 
9592 
I50000 
- 
- 
13849 
200000 
- 
17984 
25OOOO 
22045 
300000 
- 
- 
25998 
35OOOO 
-• 
29977 
400000 
** 
33861 
Between 10000 and 20000 number of primes 1033 
20000 
30000 
- 
- 
983 
30000 
40000 
- 
- 
958 
40000 
50000 
- 
- 
930 
50000 
60000 
- 
- 
9 2 4 
60000 
T 70000 
- 
- 
878 
70000 
* 5 £oooo 
- 
- 
901 
80000 
90000 
- 
- 
876 
100000 
150000 
- 
4 2 57 
j 50000 
200000 
- 
- 
4 i 33 
200000 
25OOCO 
- 
- 
4061 
250000 
300000 
- 
3953 
300000 
3 5OOOO 
- 
- 
3979 
350000 
400000 
3884 
Pcrfett Numbers. —If prime numbers are very numerous, 
perfeft numbers, which are deduced from them, are very 
few. A perfeft number is that which is equal to the fum 
1 of 
