A R I T H 
flotation'was continued to any length. But, when a num¬ 
ber lefs than fixty was to be joined with any of the. fex¬ 
agefimal integers, their proper expredion was annexed 
without the da(h ; thus 4 times 6o and 25 is I V'XXV ; the 
fum of twice 6® and 10 times 3600 and 15 is-,X''lI'XV.. So 
near did the inventor of this method approach to the Ara¬ 
bic notation : indead of the fexagefimal progretlion, he 
had only to fubditute decimal; and to make the figns of 
numbers, from 1 to 9., fnnple characters, and to introduce 
another character, which Ihould fignify nothing by itfelf, 
but ferving only to fill up places. The fexagence integro- 
Tum were loon laid afide, in ordinary calculations, after 
the introduction of the Arabic notation; but the fexage¬ 
fimal fractions continued till the invention of decimals, 
and indeed are dill tiled in the fubdivifions of the degrees 
of circular arcs and angles. Sam. Reyher has invented a 
kind of fexagenal rods, in imitation of Napier’s bones, by 
which the fexagefimal arithmetic is eafily performed. 
Specious Arithmetic, is that which gives the calculus of 
quantities as defigned by the letters of the alphabet; a 
method which was more generally introduced into algebra 
by Vieta ; being the fame as literal arithmetic or algebra. 
Dr. Wallis has joined the numeral with the literal calcu¬ 
lus; by w hich means he has demondrated the rules for 
fractions, proportions, extraction of roots, Sec. of which 
a compendium is given by himfelf, under the title of Ele- 
menta Arithmetics; in the year 1698. 
Tabular Arithmetic, is that in which the operations of 
multiplication, divifion, &c. are performed by means of 
tables calculated for that purpofe: fuch as thofe of Her- 
wart, in 1610; and Hutton’s tables of powers and pro¬ 
ducts, pnblilhed by order of the CommilTioners of Lon* 
gitude, in 1781. 
Tetratic Arithmetic, is that in which only the four cha¬ 
racters o, 1, 2, 3, are ufedi A treatrfe of this kind of arith¬ 
metic is extant by Erhard or Echard Weigel. But both 
this, and binary arithmetic, are little better than curiofi- 
ties, efpecially with regard to practice; as all numbers 
are much more compendioufly and conveniently expretfed 
by the common decuple feale. 
Universal Arithmetic, is the name given by Newton t® the 
fcience of algebra; of which lie left at Cambridge an ex¬ 
cellent treatife, being the text-book drawn up for the ufe 
of his lectures, while he was profeffor of mathematics in 
that univerfity. 
Of NOTATION and NUMERATION. 
The ffrffi elements of arithmetic are acquired during our 
infancy. The idea of one, though the fimpled of any, 
and fuggefted by every fingle object:, is perhaps rather of 
the negative kind, and confids partly in the exclusion of 
plurality, and is not attended to till that of number be ac¬ 
quired. Two is'formed by placing one objeCt near ano¬ 
ther; three, four, and every higher number, by adding one 
continually to the former collection. As we thus advance 
from lower numbers to higher, we foon perceive that there 
is no limit to this increafing operation; and that, whatever 
number of objects be collected together, more may be 
added, at leaft in imagination ; fo that we can never reach 
the highed poflible number, nor approach near it. As we 
are led to underhand and add numbers by collecting ob¬ 
jects, fo we learn to diminifh them by removing the ob¬ 
jects collected; and, if we remove them one by one, 
the number decreafes through all the tteps by which it ad¬ 
vanced, till only one remain, or none at all. When a 
child gathers as many hones together as fuits his fancy, 
and then throws them away, he acquires the firlt elements 
of the two capital operations in arithmetic. The idea of 
numbers, which is firft acquired by the obfervation of fen- 
fible objeCts, is afterwards extended to meafures of fpace 
and time, affeCtions of the mind, and other immaterial 
qualities. Small numbers are mod eafily apprehended : a 
child foon knows what two and what three is ; but has not 
any diitinCt notion of feventeen. Experience removes this 
difficulty in tome degree; as we become accuftomed to 
Vo l. II. No. 64. 
MET! C. t C s 
larger collections, we apprehend clearly the number of a 
dozen or a fcore; but perhaps could hardly advance to an 
hundred without the aid of claffical arrangement, which 
is the art of forming fo many units into a clafs, and fo 
many of thefe clali'es into one of a higher kind, and thus 
advancing through as- many ranks of claffes as occalion 
requires. If a boy arrange an hundred itones in one row, 
he would be tired before he could reckon them ; but} if 
he place them in ten rows of ten (tones each, he will reckon 
an hundred with eafe ; and, if he colleCt ten fuch parcels, 
he will reckon a thou (and. In this cafe, ten is the lowed 
clafs, an hundred is a clafs of the fecond rank, and athoq- 
fand is a clafs of tire third rank. 
It is proper, whatever number of units conditutes a- 
clafs of the lower rank, that the fame number of each 
clafs (hould make one of the next higher; This is obfer- 
ved in our arithmetic, ten being tire 11 niverbal feale : but 
is not regarded in the various kinds of monies, weights, 
and the like, which do not advance by any univerfal mea- 
fure; and much of the difficulty in the practice of arith¬ 
metic arifes from that irregularity. As higher numbers 
are fomewhat difficult to apprehend, we naturally fall on’ 
contrivances to nx them in our minds, and render them 
familiar: but, notwithftanding all the expedients we can 
fall upon, our ideas of high numbers are (till imperfect, 
and generally far Iliort of the reality; and though we can 
perform any computation with exaCtnefs, the anfwer we 
obtain is often incompletely apprehended. Hence it may 
not be amifs to illuftrate, by a few examples, the extent 
of numbers which are frequently named, without being 
attended to. If a perfon employed in telling money reckon 
an hundred pieces in a minute, and continue at work ten 
hours each day, he will take leventeen days to reckon a 
million; a thoufand men would take forty-five years to 
reckon a billion. If we fuppofe the whole earth to be as 
well peopled as Britain, and to have been fo from the cre¬ 
ation, and that the whole race of mankind had confiantly 
fpent their time in telling from a heap confiding of a qua¬ 
drillion of pieces, they would hardly have yet reckoned the 
thoufanth part of that quantity. 
All numbers are represented by the ten following Ara¬ 
bic characters: 1, 2, 3, 4, 5, 6, £, 8, 9, o. The nine fird 
are called frgnijicant figures or digits', and Sometimes repre¬ 
sent units, TometimeS tens, hundreds, or higher claffes. 
When placed fingly, they denote the Simple numbers Sub¬ 
joined to the characters. When feveral are placed toge¬ 
ther, the fil'd or right-hand figure only is to be taken for 
its fimple value: the fecond lignifies fo many tens, the 
third lo many hundreds, and the others So many higher 
claffes, according to the order they dand in. And as it 
may Sometimes be required to expreSs a number confiding 
of tens, hundreds, or higher clalfes r without any units or 
clades of a lower rank annexed ; and, as this can only be 
done by figures danding in the fecond, third, or higher, 
place, while there are none to fill up the lower ones; 
therefore an additional character or cypher (o) is necedary, 
which has no fignification when placed by itfelf, but ferves 
to fupply the vacant places, and bring the figures to their 
proper dation. The following table (hews the names and 
divilions of the clades: 
4 3 7 > 9 82. 5 6 4, 7 38. 9 72, 6 45 
C/5 CO 
c c 
o o 
rG 
0-0 
3 0 
ca c 
H <42 
6 
T* C 
<D <D 
-o H 
C 
£ ~ 
O S 
J O 
M’S 
co co 
C G 
O O 
s s s 
O £ OJ <L> 
£ 
O IT' 
*T3 G 
<U <u 
T 3 H 
c 
D 
5 < *2 "O 
i 3 3 
^ J 0 
§ 
rG -G ^ 
’U C 
CD <D 
*0 H 
s 
3 
X 
Utt 
h 
CO C/5 C/D C/5 CO C/5 
"d "C xi "d c .ti 
C G G D c3 
o?o 3 5 
G rG —- fG t-f - * 
►J 
M-d c 
The 
