A R I T H M E T I C, 
Spain' by the 'A.rSbs or'Saracens; whither the learned men 
from all parts of Europe repaired, to learn the arts and 
iciences of them. This, Dr. Wallis proves, began about 
the year 1000 ; particularly that Gilbert, a monk, after¬ 
wards pope Sylvefter II. who died in the year 1003, brought 
this art from Spain into France, long before the date of 
Jiis death : and that it was known in Britain before the 
■ year 1150, where it was'brought into common ufc before 
*250, as .appears by the treatife of arithmetic of Johannes 
de'Sacro Bofco, or Halifax, who died about 1256. Since 
that time, the principal writers on this art have been, Bar¬ 
gain, Lucas de Burgo, Tondall, Aventinus, Purbach, 
Cardan, Schcubelius, Tartalia, Faber, Stifelius, Recorde, 
:.j?aimts, Maurolychus, Hemifchius, Peletarius, Stevinus, 
Xylander, Kerfey, Snellius, Tacquet, Clavius, Metius, 
.-Gemma Frilius, Buteo, Urlinus, Romanus, Napier, Ceu- 
ien, Wingate, Kepler, Briggs, Ulacq, Oughtred, Cruger, 
Van Schooten, Wallis, Dee,. Newton, Morland, Moore, 
jeake, Ward, Hatton, Malcolm, &c. &c. The feveral 
ipecies.or ikinds of arithmetic are included under thefe 
heads, viz. theoretical, practical, injlrumental, logarithmical, 
numerous, fpecious , univerfal, common or decadal, frattional, 
radical or.-cf furds, decimal, duodecimal, fexagejimal, dynami¬ 
cal or binary, tetraclycal, political, &c. 
Theoretical Arithmetic, is the fcience of the properties, 
relations, &c. of numbers, abftraffly confidered; with the 
■reafons and jdemonfixations of the feveral rules. Such is 
that contained in the 7th, 8th, and 9th, books of Euclid’s 
Elements; the Logidics of Barlaam the monk, publifhed 
in Latin by j. Chambers, in 1600; the Surnma Arithme- 
tica of Lucas de Burgo, printed 1494, who gives the fe¬ 
veral divilions of numbers from Nichomachus, and their 
properties from Euclid, with the algorithm, both in in¬ 
tegers, fractions, extraction of roots, &c. Malcolm’s New 
Syftem of Arithmetic, theoretical and praffical, in 1730, 
in which the fubjeff is very completely treated, in all its 
branches, &c. 
PraBical Arithmetic, is the art or praffice of numbering 
or computing; that is, from certain numbers given, to find 
others which (hall have any propofed relation to the for¬ 
mer. As, having the numbers 4 and 6 given; to find 
their fum, which is 10; or their difference, which is 2 ; 
.or their product, 24; or their quotient, ; or a third,pro¬ 
portional to them, which is 9, &c. Lucas de Burgo’s 
works contain the whole practice of arithmetic, then ufed, 
as well as the theory. Tundall gave a neat practical treatife 
of Arithmetic in 1526; as did Stifelius, in 1544, both on 
the practical and.other parts. Tartalea gave an entire bo¬ 
dy of practical arithmetic, which was printed at Venice in 
1556, confiding of two parts; the former, the application 
of arithmetic to civil ufes; the latter, the grounds of al¬ 
gebra. 
Binary Arithmetic, is that in which two figures or cha¬ 
raffers, viz. 1 and o, only are ufed; the cypher multi¬ 
plying every thing by 2, as in the common arithmetic by 
10; thus 1 is x, 10 is 2, 11 is 3, 100 is 4, 101 is 5, 110 is 
6, hi is 7, iqoo is 8, 1001 is 9, 1010 is 10; being found¬ 
ed on the fame principles as common arithmetic. This 
fort of arithmetic was invented by Leibnitz, who pretended 
that it is better adapted than the common arithmetic, for 
difcovering certain properties of numbers, and for con- 
ftrlifting tables; but he does not recommend it for ordinary 
uie, on account of the great number of places of figures 
req.uifite to exprefs all numbers, even very fmall ones. Jof. 
Pelican of Prague has more largely explained the princi¬ 
ples and praffice of the binary arithmetic, in a book en- 
titied Arithmeticus PerfeBus, qui tria numerare ncfcit, 1712. 
And De Lagni propofed a new fyftem of logarithms, on 
the plan of the binary arithmetic ; which he finds fhorter, 
and more eafy and natural, than the common ones. 
Common or Vulgar Arithmetic, is that which is concerning 
integers and vulgar fraffions. 
Decimal or Decadal Arithmetic, is that which is performed 
by a f'eries of ten charaffers or figures, the progrefiion be¬ 
ing fen-fold, or from 1 to 10’s, xoo’s, &c. which includes 
* 
both integers and decimal fraffions, in the common fcal^ 
of numbers; and the. charaffers. ufed are the ten Arabic 
or Indian figures o, 1, 2, 3, 4, 5, 6, 7, 8, 9. This method 
of arithmetic was not known to the Greeks and Romans; 
but was borrowed from the Arabs while they poffeffed a 
great part of Spain. It is probable that this method took 
its origin from the ten fingers of the hands, which were 
ufed in computation before arithmetic was brought into an 
art. The eadern midionaries affure us, that to this day 
the Indians are very expert at computing on their fingers,' 
without any ufe of pen and ink. And it is afferted, that 
the Peruvians, who perform all computations by the dif¬ 
ferent arrangements of grains of maize, outdo any Euro¬ 
pean, both for certainty and difpatch, with all his rules. 
Duodecimal Arithmetic, is that which proceeds from 12 
to 12, or by a continual fubdivifion according to 12. This 
is greatly ufed by mod artificers, in calculating the quan¬ 
tity of their work; as bricklayers, carpenters, painters, 
tilers, &c. 
FraBional Arithmetic, or of fra&ions, is that which ti'eats 
of fraffions, botli vulgar and decimal. 
Harmonical Arithmetic, is fo much of the theory and doc¬ 
trine of numbers, as relates to making the comparilons and 
reduffions of mufical intervals, which are exprefled by 
numbers, for finding their mutual relations, compofitions, 
and refolutions. 
Arithmetic of Infinites, is the method of fumming up a 
feries of numbers, of which the number of terms is infi¬ 
nite. This method was firft invented by Dr. Wallis, as 
appears by his treatife on that fubjeff; where he fliews its 
ufes in geometry, in finding the areas of fuperfices, the 
contents of folids, &c. But the method of fluxions, which 
is a kind of univerfal arithmetic of infinites, performs all 
thefe more eafily ; as well as a great many other things, 
which the former will not reach. 
Injlrumental Arithmetic, is that in which the common rules 
are performed by infiruments, or fome fort of tangible or 
palpable fubftanre. Such are the methods of computing 
by the ten fingers and the grains of maize, by the Eaft- 
Indians and Peruvians, above-mentioned ; by the abacus or 
fhwanpan of the Chinefe; the feveral forts of feales and 
Aiding rules; Napier’s bones or rods; the arithmetical 
machine of Pafcal, and others ; Sir Samuel Morland’s in- 
ftrument; that of Leibnitz, deferibed in the Mifcel. Berol. 
that of Polenus, publifhed in the Venetian Mifcellany, 
1709; and that of Dr. Saunderfon, of Cambridge, de¬ 
feribed in the introduffion to his algebra. 
Integral Arithmetic, or of integers , is that which refpeffs 
integers, or whole numbers. 
Literal or Algebra Arithmetic, is that which is performed 
by letters, which reprefent any numbers indefinitely. 
Logarithmical Arithmetic, is performed by the tables of 
logarithms. Thefe were invented by baron Napier; and 
the bed treatife on the fubjeff is, Briggs’s Arithmetics 
Logarithmica, 1624. 
Numerous or Numeral Arithmetic, is that which teaches 
the calculus of numbers, or of abftraff quantities ; and is 
performed by the common numeral or Arabic charaffers. 
Political Arithmetic, is the application of arithmetic to 
political fubjects; fuch as, the (Length and revenues of 
nations, the number of people, births, burials, See. See 
Political Arithmetic. To this head may alfo be 
referred the doffrine of chances, gaming, Sec. 
Sexagejimal or Sexagenary Arithmetic, is that which pro¬ 
ceeds by fixties ; or the doffrine of fexagefimal fraffions: 
a method which, it is fuppofed, was invented by Ptolemy, 
in the 2d century; at lead they were ufed by him. In this 
notation, the integral numbers from 1 (059 were expreffed 
in the common way, by the alphabetical letters: then fix- 
ty was called Tfexagejima prima, and marked with a dafh 
to the charaffer, thus I'; twice fixty, or 120, thus IP; 
and fo on to 59 times 60, or 3540, which is LIX'. Again, 
60 times 60, or 3600, was called fexagefima fecunda, and 
marked with two daffies, thus I"; twice 3600, thus 11"; 
and ten times 3600, thus X", &c. And in this way the 
notation 
