rUirniMfiTic
2 44
(84.) General Rvi.es FOR EXTRACTIN’'-, ANY root out of aVulgar Fraction, or Mixed Number.
1. If the given fraction have a finite root of the kind required,it is best to extract the root out of the numerator and denomi-nator, for the terms of the root required.
But if the fraction be not a complete power, it may be throwninto a decimal, and then extracted. Or,
3. Take either of the Uirnts of the given fraction for the c r-responding term of the root; and for the other term of the rootextract the required root of the product, arising from the multi-plication of such a power of the first assigned term of the rootwhose index is less by l than that of the‘given power, by tire otherterm of the given number. This rule will do when the root iseither finite or infinite : That is, for any root n in general.
n n n . i nn a </a x / a b _ —
= ' = 0 ~~ \/l>a u ' 1
on
Jb
4. Mixt numbers mav be reduced either to improper fractions
or decimals, and then extracted. ’
In treating this subject we have proceeded chiefly after the planof the celebrated Dr. Hutton’s “ Complete Treatise on PracticalArithmetic,” which we are persuaded is the best extant.
For further information see Multiplication, Decimals ,Logarithms, Algebra , & c.
INDEX.
Definition of Arithmetic, Art. 1.
History of Arithmetic, 2—6.
Roman Notation, 4.
Sexagesimal Arithmetic, account of, 5.
Explanation of the signs, 7.
Notation and Numeration, 8—10.
To express in figures any number proposed in words, 9.
To read any number expressed in figures, 10.
Simple Addition, 11.
Simple Subtraction, 12.
Simple Multiplication , 13, 14.
Simple Division, IS, 16.
Reduction, 17, 18.
Tables, 18.
Compound Addition, 19.
Compound Subtraction, 20.
; Compound Multiplication, 21.*
Compound Division, 22.
. The Rule of Three, 23, 24.i The Rule of Five, 23, 26.
Rules of Practice, 2/.
Tare and Tret, 28.
Vulgar Fractions, 29—42.
Definitions, 29.
| Reduction of Vulgar Fractions, 30—38>
; Addition of Vulgar Fraction', 39.j Subtraction of Vulgar Fractions, 40.
| Multiplication of 4'ulgar Fractions, 41.
Division of Vulgar Fractions, 42.
S Decimal Fractions, 43—51.
Addition and Subtraction of Decimals, 44,
, Multiplication of Decimals, 43.
. Interest Definitions, 32—34.
: Simple Interest, S3.
: Discount, 55! liquation of Payments, 56.
Single Fellowship, 57.‘
Double Fellowship, 58.j Barter , 59.
| Loss and Gain, 60.
' Exchange, 61—68.j General Rules, 68.
| Arbitration of Exchange, G9, 71.
' Simple Arbitration, 70.j Compound Arbitration, 71.
| Alligation, 72—74.
■ Alligation Medial, 73.
j Alligation Alternate, 74.
Single Position, 75-
■ Double Position, 76.
: Involution, 77—78.
j Evolution, 79—84.
! To Extract the Square Root, 81.j To Extract the Cube Root, 82.l To Extract any Root whatever, 83.
I General Rules for extracting any Root out of a Vulgar Fraction,' or Mixed Number, S4.
ARITHMETIC, Binary, or Dyadic , is that wherein only itwo figures, unity, or 1, and 0, are used. See Binary.
Arithmetic, Decadal, or Arithmetic, Decadary, Arith- jmetic performed by a series of ten characters, such as we now juse.
Arithmetic, Decimal, the arithmetic of decimal fractions.See Arithmetic.
Arithmetic, Instrumental, a species of arithmetic, whereinthe common rules are performed by means of instruments con-trived for expedition.
Arithmetic, Literal or Specious, arithmetic performed byletters instead of figures. See Algebra .
Arithmetic, Logarithmical. See Logarithms .
Arithmetic of Infinites, a method invented bv Dr. Wallis,of summing up a series of numbers, consisting of infinite terms,or of finding the ratios thefeof; every thing pertaining to it ismore easily performed by Fluxions.
Arithmetic, Political, is the application of arithmetic topolitical subjects; as, the strength and revenues of princes,number of inhabitants, births, burials, &c.
Arithmetic, Sexagenary, or, Arithmetic, Sexagesimal.See Arithmetic.
Arithmetic, Tetractical, is that wherein only the figures1, 2, 3, and 0, are used. We have a treatise of this arithmeticby Erhard Weigel ; but both binary and tetractic are little betterthan curiosities, especially with regard to practice.
Arithmetical Complement, the remainder when the figureon the right hand of any number is subtracted from 10, and eachof the other digits lrom 9, thus 236 is the arithmetical comple-ment of 764. J his is otlen used to change subtraction into addi-tion, thus, if 764 is to be subtracted from 962, the remainder willbe found by adding the arithmetical complement arul rejectingone in the last figure of the sum, that is, 9(52-(- 230 = 1198,which leaving out the one, gives 198, the true remainder. It isot frequent use in trigonometrical calculations, when radius orlO’OOOOOOO is not the first term, to save the labour‘of subtraction.
Arithmetical Medium or Mean , the middle term of threequantities in arithmetical progression.
Arithmetical Progression. See Progression.
Arithmetical Proportion. See Proportion.
Arithmetical Ratio. See Ratio.
ARIUS, founder of the sect of the Arians , was horn, it is gene-rally thought, in Libya . He became popular at Alexandria, andwas orthodox till be was disappointed of church-preferment, whenhe broached his opinion against the divinity of the Word , whichoccasioned such disputes, that the emperor called a council atNice, in 325, to put an end to them, in this council the heresyof Arius was condemned, and the celebrated confession of faith,known by the name of the Nicene Creed, drawn up. Arius wasthen banished by the emperor; but two years after ‘he was recalledto Constantinople , and made a confession of his faith, which wasreceived as orthodox. In 331, he went to Alexandria, where* Athanasius