NOTATION. 
is of the form m (r — 1) + 13 ana ea m(r— 
whatever integer value is give Q: “and eee it 
follows, that 
ar = _ ) + a4 
bre? = ba! (r—1) +8 
er’? = em! (r—1) +e 
&e. c. 
pr —_ pail (r — 1) + p 
qr = gm" (r—1) + ¢ 
Ww = w 
and confequently 
N = wm (r—1) + (@+ 46+ ¢+..-pigqtw) 
and, therefore, when divided by r — 1, it will evidently leave 
the fame remainder as the fum of its digits (a + 4 + cand 
W i. 
— 
Cor. 1.—Hence, if the fum of the sin in any fyftem 
of notation. be divifible by r — 1, the number is divifible 
b 3 therefore, inthe common feale, if the digits of 
a aoiher be divifible by g, the number itfelf is divifible by 
g; and if the fum of the digits be even, and divifible by 9, 
then will the number ‘itfelf be divifible by 18; becaufe if 
an even number be divifible by an odd number, it is divifible 
by double that number. And fince 3 is a fator of g, the 
fame property that has been fhewn to belong to the number 9, 
belongs alfo to 3; namely, if the fum of the digits of a num- 
ber be divifible by 3, the number itfelf is divifible by 3 ; and 
if the fum be even aifo, then will the number be divifible by 6. 
—It is upon this obvious principle that our rule for 
r— 
mains of this laf ought te be equal a alee remains of the 
produ& of the two former remainders divided by 9, if the 
work be ri 
For let ¢ and 4 reprefent any two faGtors, and make 
a=gnz-+a' 
6=g9m+4+ JI 
then ab = 9 ee nm-+ma' +n6b) + a'b’; and, there- 
~ ab 9, leaves the fame remainder as a BI divided by 
for any other fyftem of notation, by taking the = next 
lefs than the radix for the divifor. Thus, for example, we 
have feen that 215855 = $4yy in the Suodenary xa rae 
215855 ~~ 11, leavea a 2,but¢ + 4 
= Io +10¢ 11+ 11= 46, which ‘vided a - 
ves alfo a remainder 2. a ofe it was required to mul- 
tiply ¢4oyy by $04, the operation and proof would ftand 
thus : 
Operation. 
G40yy rem. 2 
Coq rem. 2 Proof by t1. 
4 
357798 - A 
88112 Sf 4 
88s 172 
95088918 rem. 4 
it is unneceffary to obferve, that in this operatian, as in 
all others in which the radix is r, we muft, in suultiplying» 
dividing, &c. divide by the radix, that is, by 12 in the 
above example, and fet down the overplus, inftead of ae 
ing by ro, and fetting down the overplus, as is done in the 
common feale. 
Prop. V. 
In any fcale of notation whofe radix is r, the differe 
ed by 
dig divided alfo by r + os is ae to the remainder 
the whole number divided by 
Let N= ar” + br? bee se ee had 
then will fie remaihaee of (w + p + 4, &c. ) >+r+iI 
minus the remainder of (et + ¢ +4, hia +r+tt be 
equal to the remainder of N +r + 1 
For make r + 1 = 7', orr =r! — 1, then it is evident 
(f= 1)" 
that . will leave a remainder + 1, or — I, ac- 
r 
cording as # is even or odd; for all the terms in the ex- 
panded binomial (7! — 1)” are divifible by r', except . 
laft, which is + 1 or 1, according as 2 is even or odd, in- 
dependently of any other value of 2; and, therefore, 
r 
r+ 
wiil alfo leave the fame remainder in the fame cafes ; that is, 
every odd power of ris of the form m (r + 1), and ever 
even power of r is of the forma (r+ 1). Therefore, in 
the above expreffion, we have 
Ww _ + w 
qr = qm{r+is) —@g 
pr* = pn (r +1) +6 
er*-? = cm (r+1) —e 
brs = bo! (r +1) +4 
ar” am'(r+1) —a 
&e. &e. 
And confequently, 
N=m"(r+31)+w—qtp-—ce+b—a; 
and, therefore, when polka 2; r + Aa it will leave the 
fame remainder as a) divided 
bye Ligorat(w pbb 8) = (r4 1)—(g+e 
+ 4, Kc.) > 1). 
Cor.—Hence, in the common ar if the fum of the 
digits in the odd places is equal to 
even place; or, if one exceed the 
tiple of 11, the whole number may ie divided 
Cor. 2.—The above propofition furnifhes us a shoe: 
rule (oF proving the truth of the operation in multiplication, 
divifion &c. which, in sa common fcale of notation, the 
radix being 10, is as follow 
From the fum of the digits i the ft, 3d, 5th, &c. places, 
fubtraG& thofe in the 2d, 4th, 6th, &c. places in both fa&o ors, 
and in the produét ; Ae referve the three remainders, when 
each of thofe differences is divided by 1 15 multiply the two 
former together, and caft out the 11’s, which remainder 
ought to be equal to the remainder of the produ, if the 
work be right. Note, If the fum of the 2d, qth, &c. 
digits be greater than the fum of the 1ft, 3d, &c. 11 muft 
be added to the latter. 
Thus, 
