242
ARITHMETIC.
tional to their positions, nor the errors to the difference of thetrue number and each position ; yet, in all suc h cases, it is a verygood approximation, and in exponential equations, as well as manyother things, succeeds better than perhaps any other method.
Those questions, in which the results are proportional to theirsuppositions, belong to single position ; such are those which re-quire the multiplication or division of the number sought bv amnumber, or in which it is to be increased or diminished bv itselfany number of times, or by any part or parts of it. Hut those inwhich the results are not proportional to their positions, belongto the double rule ; such arc those, in which the number sought,is increased or diminished by some given number, which is noknown part of the number re quired.
Rule. l ake any number, and perform the same operationswith it, as, in the question, are described to be performed withthe number sought ; then, if the result he the same with that inthe question, the suppose d number is the number sought ; hut ifit be not, say, as the result of the operation is to the position, sois the result in the question to the number required.
ExAMI'LK.
A person after spending J and (. of his money, has yet remain-ing 60/. : what had he at first r
1. Suppose he had at first ICO/. Proof.
Now I of 120 is
40
•4 of 144 is
48
^ of it is
30
^ of it is
36
their sum is
70
their sum
84
which taken from
120
taken from
144
--
—
leaves
50
leaves
60
as per question.
Then, 50: 120 :: 60 : 144.
Double Position.
(76.) Pule. Having taken any two convenient numbers, forthe positions, proceed with each, according to the conditions ofthe question, as if it were the true number sought; and iindhow much the results are different from the result in the question."Next multiply each of these errors or differences by the other’sposition; then if the errors be of the same affection, that is, if theresults be both either too great or too little, divide the differenceof the products by the difference of the errors, and the quotientwill be the answer ; but if the errors be of different affections,that is, if one result be too great, and the other too little, dividethe sum of the products by the sum of the errors, and the quo-tient will he Ihe answer.
Or, having found the errors, say, As the sum or difference of(he errors, according as they are of a different, or the same kind,is to the difference of the suppositions, so is the least error to the■correction of the supposition belonging to this error; which mustbe added to, or subtracted from it, according to the followingconditions, viz. If the errors be of the same kind, add, or sub-tract the correction, to or from this supposition, according as it isgreater or less than the other supposition ; but if the errors be ofdifferent kinds, add, or subtract, according as the supposition isthe less or the greater of the two; and the stun, or difference willbe the number sought.
Example.
What number is that, which being multiplied bv 6, the productincreased by 18, and the sum divided bv 9, the quotient willbe 20 ?
First, suppose 30 to be the number sought.
30 x 6 + 18
Then- - • — = 22 but ought to be 20 ; therefore the error
9
is 2 in excess.
Again, suppose 18 to be the number sought.
18 X 6 ■+ 18
Then-= 14; but ought to be 20; therefore the
9
error is 6 in defect.
And the errors are of different kinds or affections.
30 x 6 -f 18 x 2
Whence, by the fust rule,---= 27, the num-
2+0
ber sought.
2 X 12
And, bv the second rule, 2 +■ 6 : 30 — 18:: 2?: -= 3,
8
the correction ; (hen 30 — 3 = 27, the number sought.
Involution.
(77.) The first power of any number is the number itself; thesecond poweii or square is the product of the number into itself;the third power or cube, is the product of the first and second ;the fourth power is the product of the first and tniid, or of thesecond and second ; the fifth power is the product of the first andfourth, or i f the second and third, &c. that is the number whichdenominates the power, and which is called the index, is expressedby the number of multiplications added to 1.
Thus: 3 is the 1st power of 3.
3 x 3 — 9 is the 2nd power of 3.
3 X 3 x 3 = 27 is the 3rd power of 3.
3 X 3 x 3 x 3'= 81 is the 4lh power of 3.
(78.) Involution is the finding of powers; to do which we havethis Rule. Multiply the given number or first power continuallyby itself, (ill the number of multiplications he 1 less than the indexof the power required, which power will he then found.
Powers are sometimes denoted by affixing the index, thus, the5th power of S is 8'.
Or if two powers of the same number be multiplied togetherthe product will be the power denoted by the sum of the indicesof the two powers.
Examp. 1. What is the cube or third power of 12..’.
12.M = 12.5x 12.5 x 12.5 = 1953.125 Ans.
2. What is the 5th power of 1 -Jor
■3 »
— 3 y 3. y 3 y 3 y 3 — U y .9 —
^ 2 . ^ J A j A £ 1 " — ^ ^ “
w
7.59373.
(79.) The root of any number proposed is a number whic h mul-tiplied into itself, a certain number of times, will produce theproposed number or power, and this root is denominated the 1st,2nd, 3rd, 4th, Ac. respectively; as the multiplications arc 0, 1,2, 3, Ac. that is, the index of the root is more by 1, than the mul-tiplications of that root requisite to produce the power or proposednumber.
(80.) Evolution is the finding of the roots of numbers, eitheraccurately or by decimals, to any degree of exactness.
Roots are .sometimes denoted, as in algebra, by writing the cha-racter t/ before the power, with the index of the root against it:thus, the third root of 70 is expressed \/70, Ihe second root ofit is 2 ^/70 or rather \/70, and thus the 5 root is \/70.
If the power be expressed by several numbers, with the sign +or — between them, a line is drawn from the top of the signover all the parts of it; thus, the third root of 28— 13 is
V28 - 13.
Sometimes roots are designated like powers, with fractional in-dices ; finis, the square root of 5 is 5’-, the third root of 19 is 19 1 ',
and the fourth root of 40 — 12 is 40— 12* * 4 ", or (40 — 12)’ 4 ', Ac.
To prepare a power for extraction, divide it from the place ofunits, to the left in integers, and to the right in decimal fractions,into periods ; each containing as many figures as the index con-tains units; if there be more than complete periods to the left, theremaining figures are accounted the lirst period ; if there be a de-fective period to the right, it is completed bv cyphers, and wholeperiods of cyphers may be annexed to continue the extractionwith: this division may be made by writing a dot over the unitsplace, and then over the last figure of each period on bothsides.
(81.) To Extract the Square Root.
•
Rule I. Having pointed the number into periods of twofigures, find a square either equal to, or next less than the firstperiod, which subtract from it, and place the root, as a quotientfigure in division, for the lirst figure of the required root.
II. To