A R I 
A R t 
191 
and from t!ie quotient Tub trait one-twelfth of the fquare 
of the a (Turned root; the fquare root ot t lie remainder 
added to half the a (fumed root will give the true root re¬ 
quired. When the given refolvend is a furd number, and 
great exaftnefs is required, the operation mult be repeated 
once or ot'tener, always taking the laft-lound root for the 
alTumed root, and by each operation you will double the 
number of decimals in the root preceding. 
Ex. What is the cube root of 79895922443 ? 
Affume the root 4000. Then 4000X3 =i2000 > three 
times the alTumed root. 
From 79895922443-7-1 2000 22: 6657993 the quotient 
Take 4000 J -i-i2 - — 1 333333 
Remainder - - 53 2 466o 
Then f 5324660 
To which add 4000-ft- 2 
The fum is - •*■ 
— 2307 
— 2000 
4307 the root required. 
Proof 4307 X 4307 X 4307=79 S 959 22 443 - 
To EXTRACT the ROOTS of all POWERS. 
Prepare the given number for extra £1 ion, by pointing off 
from the units place as the index of the given root directs. 
Find the fil'd figure of the root by trial, and fubtrael its 
power from the firft period of the given number. To the 
remainder bring down the firft figure of the next period, 
and call it the dividend. Involve the root to the next in¬ 
ferior power to that which is given, and multiply it by the 
Humber denoting the given power for a divifor. Find 
how many times the divifor may be had in the dividend, 
and the quotient will be another figure of the root. In¬ 
volve the whole root to the given power, and fubtraft it 
from the two firft periods of the given number as before. 
Bringdown the firft figure of the next period to the remain¬ 
der, to which find a new divifor as before, and fo on till the 
whole is completed. 
Ex. What is the cube root of 82312875? 
82312875(435 root 
64 222 cube of 4 
4 ! X3=4 -S)iS 3 dividend 
82312 — two firft periods 
Subt. 79507 zee cube of 43 
43'x 3=5547)_28o58_new dividend 
82312875 22: three periods 
Subt. 82312875 2= cube of 435. 
The extraction of roots is greatly expedited by obferv- 
ing what integer numbers multiplied together produce the 
index of the required root, and making fuch extractions 
as are denominated by thefe numbers. Thus, inftead of 
the biquadrate root, extraCt the fquare root twice, i. e. 
extraft the fquare root, and then the fquare root of that 
root; inftead of the fixth root extract the firft fquare root, 
and then the cube root of that, and fo on. 
To ex trad the Cube Root of a Vulgar Fraction. Reduce 
the fraction to its loweft terms, then extraCt the cube root 
of both terms for the terms of the required root; but, if 
the terms w ill not extraCt exactly, multiply the numera¬ 
tor by the fquare of tire denominator, and the cube root 
of the produCt divided by the denominator will give the 
root required. Or, multiply the denominator by the fquare 
of the numerator, then divide the given numerator by the 
cube root of the produCt for the root required. That is, 
3 , 2 
3 f -22: * ■■-— - . Or, univerfally, thus, let n 
y y f j yx‘ 
denote the index of the required root; then 
xy n x . 
- =2-- 2g=fr , the root required, 
y\ y 
Or, reduce the vulgar fraction to a decimal, and then 
extraCt the root 
Ex. a. What is the cube root of 
then the cube root of 216-is 6, and ot 343 is 7 ; therefore 
y is the cube root required. 
Ex. 2. Required the cube root of By the theorem, 
27 a x 8=5832, whole cube root is 18, then is the 
root fought. Or, 8XSX 2 7 = i7 28, whole cube root is 
12 ; therefore, the root as before. 
Arithmet ic, is inconologically deferibed by a very 
beautiful but penlive woman fitting, and having the nu¬ 
meration table before Iter, her garment of divers colours 
and ftrewed with nuilTcal notes, on the (kirts of it the 
words par & impar , (even and odd.) Her beauty denotes 
that the beauty of all things refult from her, for. God 
made all things by number, weight, and mcafure ; her per¬ 
fect age (hews the perfection of this art; and the various 
colours, that (he gives the principles of all parts of the 
mathematics. 
ARITILME'TICAL, adj. According to the rules or 
method of arithmetic.—The principles of bodies may be 
infinitely finall, not only beyond all naked or affifted fenfe, 
but beyond all arithmetical operation or conception. Grew. 
ARITHMET ICALLY, adv. In an arithmetical man¬ 
ner; according to the principles of arithmetic.—Though 
the fifth part of a xeftes being a fimple fraction, and arith¬ 
metically regular, it is yet no proper part of that meafure. 
At buthnot. 
AklTHMETI'CIAN, f A matter of the art of num¬ 
bers.— A man had need be a gbod arithmetician, to under¬ 
hand this author’s works. His delcription runs on like a 
multiplication table. Addifon. 
ARIT'ZAR, a town of European Turkey, in the pro¬ 
vince of Bulgaria, ten miles fouth ot Viddin. 
A'RIUS, a divine of the fourth century, the head and 
founder of the Arians, a feCt which denied the eternal 
divinity and fubftantiality of the Word. He was born in 
Lybia, near Egypt. Eufebius, bifhop of Nicomedia, a 
great favourite of Conftantia, lifter of the emperor Con- 
ftantine, became a zealous promoter of Arianifm. He 
took Arius under his protection, and introduced him to 
Conftantia; fo that the feCt increafed, and fieveral bifhops 
embraced it openly. There arofe, however, fuch difputes 
in the cities, that the emperor w\as obliged to atTemble the 
council of Nice, where, in the year 325, the doClrine of 
Arius was condemned. Arius was baniftied, his books 
were ordered to be burnt, and capital punifhment w as de¬ 
nounced againft whoever dared to keep them. After five 
years banilhment, he was recalled to Conftantinople, where 
he prefented the emperor with a confeflion of his faith, 
draw n up fo artfully, that it fully fatisfied him. Notwith- 
ftanding which, Athanafius, bilhop of Alexandria, refuted 
to admit him and his followers to communion. This fo 
enraged them, that, by their intereft at court, they pro¬ 
cured that prelate to be depofed and baniftied. But the 
church of Alexandria ftill refuting to admit Arius into 
their communion, the emperor fent for him to Conftanti¬ 
nople ; where, upon delivering in a freih confellion of 
his faith in terms lets oftenfive, the empei'or commanded 
Alexander, the bilhop of that church, to receive him the 
next day into his communion ; but that very evening Arius 
died. The planner of his death was very extraordinary': 
as his friends were conducting him in triumph to the great 
church of Conftantinople, Arius, prefted by a natural ne- 
ceftity, ftepping afide to eafe himfclf, expired on the (pot, 
his bowels gulhing out. But the herefy did not die with 
the herefiarch ; Ins party continued ftill in great credit at 
court. In (hort, this feCt continued with great luftre above 
300 years: it was the reigning religion of S. ain for above 
two centuries; it prevailed in Italy, France, Pannonia, and 
Africa; and was not extirpated till about the end of the 
eighth century. It was again fet on foot in the weft by 
Servetus, who, in 1531, wrote a treatife againft the myf- 
tery of the Trinity. After his death Arianifm got footing 
its 
