N U M 
of all its divifors. or aliquot parts; thus 6=14-2+3 ; or 
£=■!+.■§+!; therefore 6 is a perfect number. The num¬ 
ber 28 pofreiTes the fame properry; for its aliquot parts 
are r, 2, 4, 7, 14, the firm of which is 28. 
To find all the perfect numbers of a numerical pro- 
greffion, take the double progrefiion 2, 4, 8, 16, 32, 64, 
128, 256, 512, 1024, 2048, 4096, 8192, &c. and examine 
thofe terms of it which, when diminished by unity, are 
orime numbers. Thofe to which this property belongs, 
wiil be found to be 4, 8, 32, 128, 8192 ; for thefe terms, 
when diminished by unity, are 3, 7, 31, 127, 8191. Mul¬ 
tiply therefore each of thefe numbers by that number in 
the geometrical progrefiion which preceded the one from 
which it is deduced; for example 3 by 2, 7 by 4, 31 by 16, 
127 by 64, 8191 by 4096, Sec. and the refult will be 6, 28, 
496, 8128, 33550336, which are perfect numbers. The 
following are all the perfect numbers at prefent known : 
6 33550336 
28 8589869056 
496 137438691328 
8128 2305843008139952128 
The difficulty of finding perfeCt numbers arifes out of that 
offinding prime numbers of a high value, which is a very 
laborious operation. Euler afcertained that 214748347 is 
a prime number, and this is the greatell at prefent known 
to be fuch ; and confequently, the lait of the above perfeft 
- numbers, which depends upon it, is the greateft perfeCt 
number at prefent known, and, in all probability,the greateft 
that ever will be known : for, being merely matter of cu- 
riofity, it is not likely that any one will ever attempt to 
find one beyond it. “ The rarity of perfect numbers,” 
fays a certain author, “is a fymbol of that perfection.” 
All the perfect numbers terminate with 6 or 28, but not 
alternately or converfely. 
Jmptrfi ft Numbers are thofe whofe aliquot parts, added 
together, make either more or lefs than the whole. And 
tiiefe are distinguished into abundant and defective; an in- 
ftance in the former cafe is 12, whofe aliquot parts, 6, 4, 3, 
2, 1, make 16; and, in the latter cafe, 16, whofe aliquot 
parts, 8, 4, 2, and 1, make but 15. The number 120 has 
the property of being equal to half the fum of its aliquot 
parts, or divifors, viz 1, 2, 3, 4, 5, 6, 8, 10,12, 15, 20, 24, 
30, 40, 60, which, together, make 240. The number, 672, 
is alfo equal to half the fum of its aliquot parts, 1344. 
Several other numbers of the like kind may be found ; and 
fotne even which would form only a third or fourth of 
the fum of their aliquot parts, or which would be the 
double, triple, or quadruple, of that fum ; but what has 
been here faid will be fufficient to exercife thofe who are 
fond of fuch refearches. 
Amicable Numbers, are fo called on account of a certain 
property which gives them a kind of affinity or recipro¬ 
city, and which confifts in their being mutually equal to 
the fum of each other’s aliquot parts. Of this kind are 
the numbers 220 and 284; for 220 is equal to the aliquot 
parrs of 284, viz. 1, 2, 4, 71, 142 ; and, reciprocally, 284 
is equal to the aliquot parts, 1, 2, 4, 5, 10, : 1, 20, 22, 44, 
55, 11«, of 220. 
Amicable numbers may be found by the following me¬ 
thod. Writedown, as in the fubjcined Example, the terms 
of a double geometrical progrefiion, or having the ratio 2, 
and beginning with 2 ; then triple each of thefe terms, 
and place thefe triple numbers each under that from which 
it has been formed ; thefe numbers, diminiffied by unity, 
5, 11, 23, &c. if placed each over its correfponding num¬ 
ber in the geometrical progrefiion, will form a third feries 
above the latter. In the laft place, to obtain the numbers 
of the lowedferies 71, 2S7, See. multiply each of the terms 
01 the feries, 6, 12, 24, See. by the one preceding it, and 
i'*btra£t unity from the produCt. 
5 
1 I 
23 
47 
95 
i 9 i 
383 
2 
4 ' 
8 
16 
32 
64 
128 
6 
12 
24 
48 
96 
192 
384 
7 i 
287 
1151 
4607 
18431 
73727 
Vql. XVII. No. 1179. 
ft E ft. 317 
Take any number of the loweft feries, for example 71, of 
which its correfponding number in the firfi feries, viz. 11, 
and the one preceding the latter, viz. 5, as well as 71, are 
prime numbers; multiply 5 by n,and the product, 55, by 4, 
the correfponding term of the geometrical feries; and the 
lad product, 220, will be one of the numbers required. 
The fecond will be found by multiplying the number 71 
by the fame number, 4, which will give 284. In like man¬ 
ner, 1151, with 47 and 23, which are prime numbers, we 
may find two other amicable numbers, 17296 and 18416; 
but 4607 will not produce any amicable numbers, becaufe, 
of the two other correfponding numbers, 47 and 95, the 
latter is not a prime number. The cafe is the fame with 
the number 18431, becaufe 95 is among its correfponding 
numbers; but the following number, 73727, with 383 and 
191, will give two more amicable numbers, 9363584 and 
9437056. By thefe examples it may be feen, that, if per¬ 
fect numbers are rare, amicable numbers are much more 
fo, the reafon of which may be eafily conceived. 
Of Combinations and Permutations. 
In order to facilitate the kind of calculations of which 
we are going to treat, we ffiail avail ourfei ves of a kind of 
Table called the Arithmetical Triangle, which is con- 
ftruCted in the following manner: Form a row, A B, of 
ten equal fquares; and below it another, C D, of the like 
kind, but Shorter by one fquare on the left, fo t hat it (hall 
contain only nine fquares ; and continue in this manner 
1 
1 
1 
1 
I 
I 
I 
I 
I 
I 
c 
1 
2 
3 
4 
5 
6 
7 
8 
9 
I 
3 
6 
IO 
!5 
2 I 
28 
36 
X 
4 
10 
20 
35 
56 
84 
' 
5 
15 
35 
7 0 
126 
I 
6 
21 
56 
126 
I 
7 
28 
84. 
I 
3 
36 
I 
9 
1 
E 
always making each fucceffive row a fquare Shorter. We 
Shall thus have a feries of fquares diSpoled in vertical and 
horizontal rows, and terminating at each end in a fingle 
fquare, fo as to form a triangle. 
The numbers with which the Table is to be filled up, 
mufi- be difpofed in the following manner: In each of 
the fquares of the firfi: row, AB, inferibe unity, as well ns 
in each of thofe on the diagonal AE. Then add the num¬ 
ber in the firll fquare of the row, CD, which is unity, to 
that in the fquare immediately above it, and write down 
the fum, 2, in the following Iquare. Add this number, in 
the like manner, to that in the Square above it, which 
will give 3, and write it down in the next Iquare. By 
thefe means, we (hall have the feries of the natural num¬ 
bers, 1, 2, 3, 4, 5, &c. The fame method mufi be followed 
to fill up the other horizontal rows ; that is to fay, each 
fquare ought always to contain the fum of the number in 
the preceding fquare of the fame row, and that which is 
immediately above it. Thus, the number 15, which oc¬ 
cupies the fifth fquare of the third row, is equal to the 
fum of 10, which Stands in the preceding fquare, and of 5, 
which is in the fquare above it. The cale is the fame with 
21, which is the fum of 15 and 6 ; with 35, in the fourth 
row, which is the fum of 15 and 20 ; See. 
The firfi property of this Table is, that it contains, in 
4 M its 
