Wcjuir&d only to substitute decimal, to make the 
signs of numbers from 1 to 0 simple characters, 
and to introduce the O, a character which sig- 
nifies nothing of itself, but which serves to fill 
tip places. Though the sexagesimal whole num- 
bers were soon laid aside after the introduction 
of the Arabic notation, sexagessimal fractions 
Continued tiii the invention of decimals, and are 
even still used in the subdivisions of circular arcs 
and angles. 
The method of notation, which we now use, 
came into Europe from the Arabians, by the 
way of Spain. Hie Arabs, however, do not 
E retend to be the inventors of the characters, 
at acknowledge that they received them from 
the Indians. Some imagine that the)' - were found 
out by the Greeks, which is not probable ; as 
Maximus Plumules, who lived towards the close 
pf the 13th century, is the first Greek who 
makes use of them; and Dr. Wallis is of opinion 
that these characters must have been used in 
tngland at least as long ago as the year 1050, if 
not in ordinary affairs, at least in mathematical 
Ones, and in astronomical tables. 
The oldest treatises extant upon the theory 
of arithmetic, are the seventh, eighth, and ninth 
books of Euclid’s Elements, which treat of pro- 
portion and of prime and composite numbers. 
Nicomachus the Pythagorean, wrote a treatise 
on the theory of arithmetic, consisting chiefly 
of the distinctions and divisions of numbers into 
classes, as plain, solid, triangular, quadrangular, 
and the rest of the figurate numbers as they are 
called, numbers odd and even, &c. with some 
of the more general properties of the several 
kinds. His arithmetic was published at Paris in 
1538. The next remarkable writer on this sub- 
ject is Boethius, who is supposed to have copied 
most of his work from Nicomachus. 
Fro hi this time no remarkable writer on arith- 
metic appeared till about the year 1200, when 
Jordanus of Namur wrote a treatise on this sub- 
ject, which was published and demonstrated by 
Joannes Faber Stapulensis in the 15th century: 
and, as learning advanced in Europe, the num- 
ber of writers on arithmetic increased. About 
the year 1464, Regiomontanus, in his triangular 
tables, divided the radius into 10,000 parts in- 
stead of 60,000; and thus tacitly expelled the 
\ sexagesimal arithmetic ; which, however, still 
remains in the division of time. Ramus, in his 
arithmetic, written about the year 1550, and 
published by Lazarus Schonerus in 1586, uses 
decimal periods in carrying on the square and 
Cube roots to fractions. The same had been 
thine before by our countrymen Buckley and 
Record; but the first who published an express 
treatise on decimals was Simon Stevinius, about 
the year 1582. Dr. Wallis is the first who took 
much notice of circulating decimals, and the 
honour of inventing logarithms is unquestion- 
ably due to lord Napier, baron of Merchiston 
in Scotland, about the end of the 16th or be- 
ginning of the 17th century. Arithmetic has 
thus advanced to a degree of perfection which 
the ancients could never have imagined possible, 
much less hoped to attain ; and it may now be 
reckoned one of those few sciences which is in 
its nature capable of little further improvement. 
The following marks are used as abbrevia- 
tions in arithmetic. 
zz is the sign of equality. 
-}- signifies Addition thus ; 2 -j- 3 — 5, is 2 add- 
ed to 3 equal to 5. 
— signifies Subtraction thus ; 5 — 2 — 3, is 5 
less 2 equal to 3. 
X signifies Multiplication ; 9 X 5 — 45, is 9 
multiplied by 5 is equal to 45. 
.signifies Division ; 54 9, is 54 divided by 
9 is equal to 6. 
•signifies the Square Root, \/ 9 is the square ! 
root of 9, which is equal to 3, 
ARITHMETIC. 
Notation and Numeration. 
The first elements of arithmetic are acquired 
during our infancy : small numbers are most 
easily apprehended : a child soon understands 
what is meant by two, or three, or four, but has 
no distinct notion of twenty or thirty. Experi- 
ence removes this difficulty, and enables us 
to form many units into a class, and several of 
these smaller classes into one of a higher kind, 
and thus to advance through as many ranks of 
classes as occasion requires. If a boy arrange an 
hundred stones in one row, he would be tired 
before he could reckon them ; but if he place 
them in ten rows of ten stones each, he will 
reckon an hundred with ease ; and if he collect 
ten such parcels, he will reckop a thousand. 
There does not seem to be any number na- 
turally adapted for constituting a class of the 
lowest, or any higher rank to the exclusion of 
others. However, as ten has been generally 
used for this purpose by most nations who 
have cultivated this science, it is probably the 
most convenient for general use. Other scales 
may be assumed : thus, if eight were the 
scale, 6 times 3 would be two classes and two 
units, and the number of 18 would then be re- 
presented by 22. If 12 were the scale, 5 times 
9 would be three classes and nine units, and 45 
would be represented by 39, &c. By not ob- 
serving the same scale in the various kinds of 
monies, weights, and the like, much of the diffi- 
culty in the practice of arithmetic arises. 
All numbers are represented by the ten fol- 
lowing characters. 
1 2 3 4 5 6 7 8 
One, two, three, four, five, six, seven, eight, 
9 0 
nine, cypher. 
The nine first are called significant figures or 
digits. When placed singly, they denote the 
simple numbers subjoined to the characters. 
When several are placed together, the first or 
right-hand figure only is to be taken for its sim- 
ple value : the second signifies so many tens, the 
third so many hundreds, and the others so many 
higher classes, according to the order in which 
they stand. And the cypher in any place de- 
notes the want of a number in that place: thus, 
20 denotes two tens and no units or simple 
number. 
The following table shows the names and di- 
visions of the classes. 
8. 4 3 7, 9 8 2. 5 6 4, 7 3 8. 9 7 2, 6 4 5 
a a c a a 
c o o o o 
: 42 42 
1 ^ 'TJ 
i n y 
=3 =3 
42 IS 
-a a 
; <V <u 
1 •- ’—i 
! -a r-< 
i a 
' 3 
;5J 
. 2 . 2 . 2 . 2 .2 
il'STl 
•£ ‘a ”3 "3 S3 
. O O j- oj <2 
i'H’S 
2 g 3 o § 
*4 p S >-r< 
J o O U ^ 
w"S-5 
T5 a 
c a T3 ’■d 
ctf 03 rt P-i 
c .3 
- 3 
D 
3 § 
H3 
K 
The first six figures from the right hand are 
called the unit period, the next six the million 
period, after which - the trillion, quadrillion, 
quintillion, sextillion, septillion, octillion, and 
nonillion periods, follow in their order. 
The whole art of arithmetic is comprehended 
in two operations, Addition and Subtraction. 
But as methods have been invented for facili- 
tating these operations, and distinguished by 
the names of Multiplication and Division ; these 
four rules are the foundation of all arithmetical 
operations. 
Addition. 
Addition is that operation by which several 
numbers of the same denomination are collected 
into one total. 
T 2 
767345 
234672 
142131 
223164 
3898751 
ur 
£ \ ample. 
Rule. Write the numbers distinctly, 468632 
units under units, tens under tens, and 
so on. Then reckon the amount of the 
right-hand column. If it be under ten, 
mark it down. If it exceed ten, mark 
the units only, and carry the tens to the 
next place. In like manner, carry the 796543 
tens of each column to the next, and 
mark down the full sum of the left- 3022366 
hand column. 
The best method of proving the truth of 
sums in Addition, is to add up the lines in a 
contrary direction : thus, if I begin at the bot- 
tom of the lines, when the sum is done, I add 
the figures again together, beginning each line 
from the top ; and if the total in both cases cor- 
respond, it may be supposed right. 
Compound Addition. 
Compound Addition teacheth to collect several 
numbers of different denominations into one 
total. 
Rule. (1) Place the numbers so that those 
of the same denomination may stand direct*- 
ly under each other, and draw a line below 
them. (2) Add up the figures in the lowest de- 
nomination, and find how many ones of the next 
higher denomination are contained in their sum, 
(3) Write down the remainder, and carry the 
ones to the next denomination ; with which pro- 
ceed as before ; and so on, through all the de- 
nominations to the highest, whose sum must be 
all written down ; and this sum, together with, 
the several remainders, is the total sum required. 
The method of proof is the same as in simple 
addition. 
£■ s. d. 
34 15 2f 
27 12 
30 9 Ilf 
79 15 4i 
172‘ 13 Si 
£■ s. J. 
51 18 9i 
15 9 ll| 
76 4 9 
59 19 7i 
203 13 U 
The reason of this rule is evident : for, in ad- 
dition of money, as 1 in the pence is equal to 4 
in the farthings; 1 in the shillings to 12 in the 
pence ; and in the pounds to 20 in the shillings ; 
carrying as directed is nothing more than pro- 
viding a method of digesting the money arising’ 
from each column properly in the scale of de- 
nominations : and, this reasoning will hold good 
in the addition of numbers of any denomination 
whatsoever. Thus to take an example in Troy ' 
20 pennyweights one 
ounce, 
and 
12 ounc 
pound. 
lb. 
oz. 
dxvts. gr. 
ib. 
OZ . 
dxvts. 
s r - 
45 
11 
19 22 
54 
9 
17 
15 
53 
9 
17 15 
97 
8 
15 
7 
24 
10 
18 23 
41 
3 
19 
23 
99 
9 
10 8 
88 
n 
7 
16 
224 
6 
6 20 
282 
10 
0 
13 
Hence it is evident, that for a person to be 
expert in compound addition requires- only a 
knowledge of the several tables of weights and 
measures : if, for instance, I have to pay for the 
carriage of four packages, marked A, B, C, and 
D ; A weighing 4 tons, J 6 cwt. 3 qrs. ; B 1 ton, 
14cwt. 2 qrs. 24 lb. ; C 12 cwt. 3 qrs. 25 lb. ; 
and D 3 tons, 17 cwt. 0 qr. 26 lb. ; to be able to 
ascertain the weight of the whole, it is necessary 
that I should know the Avoirdupoise Table, of 
which a part is, that 28 lb. make 1 quarter of a 
hundred weight, 4 quarters 1 hundred weight, 
and 20 hundred 1 ton ; then I proceed as fol- 
lows : 
Tons. 
exit. 
qr. 
lb. 
4 
16 
3 
0 
1 
14 
2 
24 
0 
12 
3 
25 
3 
17 
0 
26 
11 
1 
2 
•IS 
