313 
MUM 
By means of the foregoing Examples, a number may 
foe transformed from otic J’cale of notation to another, neither 
of which is the denary, by firft transforming it from the 
o-iven fcale to the common fcale, and then into the parti¬ 
cular one required. 
We have already fhown how the binary, ternary, &c. 
feales, may be transferred into any higher fcale which has 
for its index fome power of the lower. When that is not 
the cafe, we proceed as here directed. One fhort Example 
will fuffice. Let it be required to convert 1534. from the 
fenary to the duodenary fcale. We avail our- g x 3. 
felves of the foregoing Example to affume that I2 ) +1 ?; 
this fum is exprefled by 418 in the common —— 
fcale ; we have therefore only to transfer it to 
the duodenary, which, by the fliort procefs in 
the margin, gives the expreffion ztptp. °l 2 
Curious Properties of certain Numbers. 
Of the Numbers 9, 3, 11, &c.— 'The number 9 poflefles 
this property, that the figures which compofe its multi¬ 
ples, if added together, are alwaysa multiple of 9; fo that, 
by adding them, and reje£ling 9 as often as the fum ex¬ 
ceeds that number, the remainder will always be o. This 
may be eafily proved by trying different multiples of 9, 
finch as 18, 27, 36, See. This obfervation may be of uti¬ 
lity to enable us to difeover whether a given number be 
divifible by 9 ; for in all cafes, when the figures which ex- 
prefs any number, on being'added together, form 9, or 
one of its multiples, we may be aflured that the number 
is divifible by 9, and confequently by 3 alfo. 
But this property does not exclufively belong to the 
number 9 ; for the number 3 has a fimilar property. If 
the figures which exprefs any multiple of 3 be added, we 
fliall find that their fum is always a multiple of 3 ; and, 
when any propofed number is not fuch a multiple, what¬ 
ever the fum of the figures by which it is exprefled exceed 
a multiple of 3, will be the quantity to be dedudted from 
the number, in order that it may be divifible by 3 without 
a remainder. Alfo, if we take any two numbers what¬ 
ever ; then either one of them, or their fum, or their dif¬ 
ference, is neceffarily divifible by 3. Let the numbers 
affumed be 20 and 17; though neither of thefe numbers, 
nor their fum 37, is divifible by 3, yet their difference is, 
for it is 3. It might eafily be demonftrated, that this 
muft neceffarily be the cafe, whatever be the numbers 
affumed. 
Exprefs all the products of 9 by the other figures, in 
the following manner, and they will be found to poffefs 
the following curious properties: 
9 •• 6 + 3—9 
I 8 
T--9 7Z --7+2=9 ' 
9 _ 
i8..i-+82=9 81.. 84-122:9. 
3 
27..2 + 72=9 
4 
36.. 3+6=9 
_ 5 _ 
45 •• 4 + 5=9 
6 
54.. 5 + 42=9 
_ 7 _ 
63.. 
The component figures of the product, made by the mul¬ 
tiplication of every digit into the number 9, when added 
together, make nine. The order of thofe component 
figures is reverfed, after the faid number has been multi¬ 
plied by 5. The component figures of the amount of the 
multipliers (viz. 45), when added together, make nine. 
Vat. XVII. No. 1179. 
B E R. 
The amount of the feveral products, or multiples of 9 
(viz. 405), w'hen divided by 9, gives for a quotient, 45 ; 
that is, 4+ $ — nine. The amount of the firft produft 
(viz. 9), when added to the other produ&s, whofe refpee- 
tive component figures make 9, is 81; which is th efqnarc 
of nine. The faid number 81, when added to the above- 
mentioned amount of the feveral products, or multiples 
of 9 (viz. 405), makes 486 ; which, if divided by 9, gives 
for a quotient 54; that is 5+-4 —nine. 
It is alfo obfervable that the number of changes that 
may be rung on nine bells, is 362880 ; which figures, 
added together, make 27 ; that is, 2+72 -nine. And the 
quotient of 362880, divided by 9, is 40320; that is, 
4+-o-+3+-2+-o==?mW, 
The difference between any number and its reverfe is 
divifible by nine. Thus, the number 2138507 being re¬ 
verfed into 7058312, gives the difference 4919805, divifi¬ 
ble by 9. The reafon is plain ; iince this number and its 
reverfe are exprefled by identical figures, they are both 
multiples of 9 with the fame excels, and confequently 
their difference muft only be fome multiple of 9. 
The difference between a numberand its reverfe is like- 
wife divifible by 11, if it confifts of odd figures : Thus, 
the laft difference, 4919805, isdivifible by 11, for the fums 
of the alternate figures are each 18. But the fum of a 
number and its reverfe is divifible by n when it confifts 
of even figures. Thus, the number 52904682 has 28640925 
for its reverfe, and their fum is 81545607 ; which is evi¬ 
dently divifible by 11, for each fet of alternate figures 
amounts to 18. It is not difficult to perceive the reafon 
of thefe properties of 11. When the number confifts of 
odd figures, they preferve the fame character of abundant 
or defective in its reverfe, and confequently the fubtrac- 
tion of the oppofite numbers will dellroy whatever in¬ 
equality there had before exifted ; but, when the number 
propofed confifts of even figures, the abundant and de¬ 
fective, by reverfing, change mutual places, and hence the 
fum of the number and its reverfe will extinguifh any ori¬ 
ginal inequality between thefe, balancing any furplus of 
the one fet by the equal deficiency of the other. Thus, 
in the firft Example, the number 2138507 exceeds by 8 a 
multiple of 11, and fo does its reverfe 7058312 ; confe¬ 
quently, their difference is a multiple of 11. In the fe- 
cond Example, the number 52904682 is greater than a 
multiple of 11 by 10, and its reverfe 28640925 is greater 
than another multiple of 11 by 1, or is lefs than the next 
lower multiple by 10. Wherefore thefe oppofite numbers, 
added together, will produce a mutual balance, accurately 
divifible by 11. 
The cafting-out of the nines and elevens furnifhes a 
ready, though but a negative, proof of the accuracy of 
arithmetical operations. In addition, for inftance, by 
calling the nines or elevens out of each number, and col¬ 
lecting thofe exceffes, the refult of another ejection fhould 
be the fame as from the fum. This principle applies 
equally to fubtraftion ; but the mode of proceeding in 
multiplication and divifion, will require fome inveftiga- 
tion. By calling the nines out of a number, it is con¬ 
verted into the nonary fcale, of which the excefs would 
occupy the place of units; if another number, therefore, 
were treated in the fame way, and their product repre- 
fented on that fcale, that part of it formed by multiplying 
the exceffes would Hill retain the rank of units, and ex¬ 
hibit the remainder of the divifion by nine. The fame 
principle, it is obvious, will extend to the cafting-out of 
eleven. Thus, the nines, being caft out of the multiplier 
and multiplicand, 472 and 3809, leave 4 and 2, which 
produce 8, the remainder of the divifion of the product 
1797848 by 9. In like manner, on calling 11 out of the 
former numbers, there are left 10 and 3, which, being 
multiplied, yield 30, or an excefs of 8 ; and the produdl, 
1797848, divided by eleven, leaves alfo 8. Again, if nine 
be caft out of the multiplier and multiplicand, 6428 and 
36985, there remain 2 and 4, as formerly ; and their pro- 
4 L dudl. 
