NQTATION. 
Exam.—Given the breadth and area of a rectangle, 
ee to 24 feet 9 inches, and 971 feet 10 inches, to find 
its 
24 . 9 in. = 20°9, and g71 ft. 10 in. =. 68y"D 
20°9) 68y"¢ (33°323 
623 
68) 
623 
670 
Proof by 11. 623 
Therefore its length is 39 ft. 3 in. 2! 3!", 
‘And the fame principles are equally applicable to the ex- 
pape of the fquare root, as is evident by the following 
ie e. 
L£xam.—Having given the area of a fquare equal to 
17 ft. 4 a 6‘, required the length of its fide. 
15°46(4°2029 
14 
82 146 
2 144 
- Proof. 
8402 20000 
2 14804 NS 
PR 
8404 737800 3 
©4404 
47378 
Therefore the fide is : ft. 2 in. of 2! roll, 
And thus may any other oe operation be per- 
formed with as much Facilit on arithmetic. 
Let us pi ew t preceding prin- 
ciples t three curious = ems propofed by 
Euler, in ae “ Brava oaitorum 
Prop. VII. 
Every number lefs than cae is compounded of fome 
number of terms in the feri 
2, 2%, a. 24, 25, &e. 2". 
This is made evident by oa the given number 
< a"t' into = peal {cale, which 
rom what has been 
obferved at Cor. Prop. 1, will affume the form 
N=a.2"+4. . 2 epee + Q.2+ mW, 
where a, b, c, &c. are each lefs than 2, = eg etl 
either o or 1; and as every number lefs than 27+! ma 
panies into this form; therefore, with ae above oe 
very number whatever within the afligned limits may be 
compounded of fome number of thofe terms. 
hat is faid in the above demonftration not cigs 
Peek oe truth of the theorem, but alfo points out the 
J; 
method by which it is to be effeGed; and at the fame time 
it is evident, that there is only one way in which the felec- 
can be made. 
Cor. 2.—In the above theorem the greateft power of 
2 is 2"; and, confequently, the greateit number that can be 
formed is 2"+ 
> 
Having a feries of weights of tlb. alb. "ab. "SIb. 
16lb. &e. -» it is required to afcertain which of them mutt 
be feleGted to weigh 1719 lb. 
Firft, 1719, in the binary feale, is a oleae by 
IIOLOIIOIII; the weights, therefore, to be eal are 
i + 2lb, + 27lb. + 2tlb. + 2°Ib. + a'lb. +2 
"Tb. 
Prop. VIII. 
E number whatever may be furmed hy the fums 
and diference of the terms of the geometrical feries 1, 3, 
For transforming “the given number into the ter- 
nary {cale of notation, it will affume the for 
N = a 3” + 63"! + 3777. ae +93+4, 
where each of the co-efficients a, 4,¢, &c. are lefs than n 3s 
and, confequently, they muft be either 2 or 1 or o 
in order to prove the truth of the theorem, it a be 
better to felect a partial example, the reafoning on which 
will be evidently applicable to every other cafe. Firft, then, 
itis obvious, that if no one of thefe co-efficients be greater 
than 1, the queftion is refolved agreeably to the conditions of 
the propofition. We need, therefore, only confider the cafe, 
which fome one or more of the co-efficients are equal to 2. 
then 
N— gtd a, tt Ogt tg Bg Fh A O.3 FH 
o 
Andfince 3. 3° = 3° 5, and 3.373 == 3"~*, 3.3"! = 3” 
The above expreflion is the fame as . 
2243" 43" es) Ss og = 
(3"~" a Ts 3 pe ts 3 3”) 
6 
agreeably to the ine of the propofition. 
Remark rt of the above demonftration is 
only for a iene ‘cake, a it is evident that the fame 
realoning will apply to any cafe, o 
ut it w 
ti 
the fame time it aie the truth of the theorem, and, like 
0, ae there is only one way in which the 
iy ita can be effe 
—It ae pain mo theorem, that with a el of 
3b. 37 lb. * Ib. &e., an mber of 
a 
thofe weights in one fcale, and fom 
eam, 1 Required in what manner the weights muft 
be 
