314 N U M 
d u< 5 l,8, is the excefs of 2377 39 580 above the neareit multiple 
of nine. With eleven likewife the remainders are 4. and 3, 
whofe product, 12, gives an excefs of 1 for the remainder 
of the divifion of the refult by eleven. 
In the duodenary J'cale, the correfponding property mull 
belong to the numbers 11 and 13. Thus, calling the 
elevens out of 276 and 2957, the excelles are 4 and 1; but 
their produ< 5 l, y^tpitpb, being divided by eleven, leaves 4. 
Calling 13 out of the fame factors, there remain 1 and 9 ; 
and the product, divided by 13, leaves 9. Again, calling 
out the elevens from 3878 and 194^1, there are left 
4 and 3; which, being multiplied, give 12, or 1 for the 
excefs of the product 67750938 above the neared mul¬ 
tiple of 11 ; but, cading 13 out of the fame numbers, the 
remainders are 6 and 9, which give 54, or 10 for the ex¬ 
cefs of the product. The agreement in thefe examples 
affords a drong prefumption of the correilnefs of the 
operations ; as any difcordance would be an abfolute 
proof of inaccuracy. 
We lhall in this place farther illudrate the properties 
of the number 9, by fome extrafts from Mr. Snart’s inge¬ 
nious pamphlet on the “ Power of Numbers.” Thefe 
properties will appear very extraordinary indeed to thofe 
who have not previoully turned their minds to the fub- 
jeCl; but, to the mathematician, it will be evident that 
they arife naturally from the fydem of our common 
arithmetic. 
Whenever two figures reciprocally change places, it in¬ 
variably is the cafe, upon fubtra&ing thefmaller from the 
greater, that nine, or fome power or multiple thereof, is 
the refult. But any other number of figures will produce 
the fame effe£l, and whether following numbers or not, 
provided that all of them be permuted ; for which there 
appear two realons, one growing out of the other. 
Fird, that there are but nine digits, or units; therefore, 
any digit going into the ten’s place by permutation, be- 
coriies as many tens as that digit indicated units before: 
which is the fecond reafon ; confequently, its worth in- 
creafes by as many nines as it was the fymbol of units be¬ 
fore. Strange as this may appear to perfons unacquainted 
with the nature of numbers, yet a little examination will 
make it fe!f-evident. For, although a digit or number 
increafes ten-fold by one removal toward the left hand, 
yet, as it lofesits unit’s value, it increafes in reality only by 
as many nines as the laid digit reprefented units before : 
thus one, when it becomes ten by having a cipher added, 
■has gained but nine, and two becomes twenty by the 
fame rule ; yet the two has only acquired two nines, or 
eighteen more, which is a multiple of nine of the lowed 
order. The fame law governs any other number, com¬ 
pound as well as fimple. Thus, 98 lofes nine by inverfion 
into 89, and as naturally regains it by re-aflumption of 
its former order; and 234 gains by inverfion 198, or 
twenty-two nines, producing432. But to make the mat¬ 
ter plainer, it may not be amifs to demondrate, that the 
increafe by nine operates, without inverfion, only by put¬ 
ting any figure, or number of figures, one place to the 
left. So 25 becomes 250 by one removal to the left hand, 
or ten times as many as before, but does not gain twenty- 
five tens, but twenty-five nines, or 225, becaufe it relin- 
quilhes its former power in obtaining the latter. 
But, perhaps an example in a tabular form, will eluci¬ 
date this matter better than either aflertion or argument. 
For indance, let the numbers 1234 be arranged in all 
the twentyrfour forms they are fufceptible of, and in fuch 
an order (which Mr.Snart calls his gi-adus) that each row 
lhall make a largerfum than the one immediately above it; 
till at length, in the lad row, we lhall find the figures in 
a totally .reverfed order, and forming the larged fum, 4321. 
If each term be compared with the one immediately pre¬ 
ceding, it, will be found that the condant increment 
is 9, or fome power or multiple of 9; as in the following 
Table;'. , 
, ■ 
BE R. 
Gradfts of the 
Numbers. 
Inere. 
of No. 
Number of Nines, or Multiples 
or Powers thereof. 
1234 
1243 
9 
rr 1 Nine. 
1324 
81 
=2 9 Nines, the fquare of 9, or 9 X 9- 
134a 
18 
2= 2 Nines, the binary of 9, or 9+9. 
1423 
81 
2= 9 Nines, the fquare of 9, or 9X9- 
14 3 ^ 
9 
22: 1 Nine. 
2134 
JOZ 
22:78 Nines, the cube of 9—9X3- 
2143 
9 
=2 1 Nine. 
2314 
171 
= 19 Nines, the fquare of 9X2+9. 
2341 
27 
=2 3 Nines, the fquare of 9-2-3. 
2413 
72 
2= 8 Nines, the fquare of 9—9. 
2431 
18 
222 2 Nines, the binary of 9, or 9+9. 
3124 
693 
—77 Nines, the cube of 9—9X4. 
3x42 
18 
22 2 Nines, the binary of 9, or 9+9. 
3214 
72 
=2 8 Nines, the fquare of 9—9. 
3241 
27 
22: 3 Nines, the fquare of 9-2-3. 
22*9 Nines, the fquare of 9X2+9. 
3412 
171 
3421 
9 
— 1 Nine. 
4123 
702 
2278 Nines, the cube of 9—9X 3- 
4132 
9 
22 1 Nine. 
4213 
81 
— 9 Nines, the fquare of 9, or 9 X 9 
4231 
18 
22 2 Nines, the binary of 9, or 9+9 
4312 
81 
22 9 Nines, the fquare of 9, or 9X 9 
4321 
9 
— 1 Nine. 
Total increal 
e 3087 
22:343 Nines. 3087 is alfo the re 
mainder, after fubtrafting the fird or lead fum from the 
lad or greated, which proves that the operation is correft. 
Or, if leveral of the above fum's w'ere taken at once, the 
nines W'ould dill refult to the fame amount as when taken 
in detail. Thus, if the eleventh fum, 2413, were fubtraiSl- 
ed from the fourteenth, 3142, it would amount to the 
fame as if the intercepted furns had been taken feparately, 
and produce a remainder of 729, the cube of 9, or 81 times 
nine, as they do all together. 
Of Square Numbers, Cube Numbers, &c. 
Square Numbers are the produft of any number multi¬ 
plied by itfelf: thus 4, being the fadlum of 2X2, is a 
fquare number ; and every fquare numbernecedarily ends 
with one of thefe five figures, 1,4, 5, 6, 9; or with an 
even number of ciphers preceded by one of thefe figures. 
This may be eafily proved, and is of great utility in ena¬ 
bling us to difcover when a number is not a fquare ; for, 
though a number may end as above-mentioned, it is not 
always however a perfeft fquare; but, at any rate, when 
it does not end in that manner, we are certain that it'is 
not a fquare, which may prevent ufelefs labour. The 
above-mentioned five concluding figures of a fquare num¬ 
ber, always lie equally didant on each fide of the middle. 
When a number ends in. 1, its fquare root mud end in 
1 or 9 ; when it ends in 4, the root ends in 2 or 8 ; when it 
ends in 5, the root will likewife end in 5 ; when it ends in 
6, the root will end in 4 or 6; and when it ends in 9, the 
root will end in 3 or 7. Unlefs therefore in the cafe of 5, 
there are always two correfponding terminations of the 
root, making together the number 10. 
Every fquare number is divifible by 3, or becomes fo 
when diminiflied by unity. This may be eafily tried on 
any fquare number at pleafure. Thus 4 lefs 1, 16 lefs 1, 
25 lefs 1, 121 lefs 1, &c. are all divifible by 3; and the 
cafe is the fame with other fquare numbers. Every fquare 
number is divifible alfo by 4, or becomes fo when dimi- 
nilhed by unity. This may be proved with the fame eafe 
as the former. Every fquare number is divifible likewife 
by 5, or becomes fo when increafed or elfe diminiflied by 
unity. Thus, for example, 36—1, 49-f-i, 64—1, 81—j, 
&c. are all divifible by 5. Every odd fquare number is a 
multiple of 8 increafed by unity. We have examples of 
this property in the numbers 9, 25, 49, 8x,&c. from which, 
if 1 be deducted, the remainders will be divifible by 8. 
