ber will give a quot 125 times greater than the dividend; that is, the quot will be equal to the product of the dividend multiplied by 125.
Prob. II. From a given divisor to find a multiplier that gives a product equal to the quot.
Rule. Divide an unit with ciphers annexed by the given divisor, and the quot will be the multiplier sought.
Examp. What multiplier will give a product equal to the quot arising from the same number divided by .008?
Given divisor .008)1.000(125 multiplier sought.
Now, if any number be multiplied by 125, and the same number be divided by .008, the product and quot will be equal; as appears in the example following.
785 .008)785.000(98125 quot.
Rule II. If a finite divisor divide a repeating dividend, work as in integers; but in continuing the division, instead of annexing ciphers to the remainder, annex the repeating figure of the dividend.
Ex. 1. Ex. 2.
5).33(.08 4).518(1.2918
Rule III. If a finite divisor divide a circulating dividend, work as in integers; but in continuing the division, instead of annexing ciphers to the remainder, annex the circulating figures of the dividend.
Rule IV. If the divisor be indeterminate, reduce it to a vulgar fraction, as taught in reduction of decimals, Prob. V.; then multiply the given dividend by the denominator, and divide the product by the numerator.
Here there are six cases; for the divisor may either repeat or circulate, and may divide a finite, a repeating, or circulating dividend.
Case I. When a repeating divisor divides a finite dividend.
Examp. Divide 23.5 by .4 = $\frac{5}{8}$
Case II. When a repeating divisor divides a repeating dividend.
Examp. Divide 43.26 by .3 = $\frac{1}{3}$
Or rather thus:
3)389.40(129.8
Vol. I. No. 18. Case III. When a repeating divisor divides a circulate.
**Example.** Divide .962, by .18, = $\frac{53}{99}$
| 96.296 | |---------| | 96.2 |
Or rather thus:
| 925.185 | |---------| | 92.518 |
| 832.666(208.18) | |-----------------| | 8 |
| 32 | |------------------| | 32 |
| 6 | |------------------| | 4 |
| 26 | |------------------| | *2 |
Case IV. When a circulate divides a finite dividend.
**Example.** Divide 9 by .45 = $\frac{45}{99}$
| 900 | |-----| | 9 |
In order to multiply the dividend 9 by 99, first multiply it by 100, which is done by annexing two ciphers; and from this product subtract the dividend.
| 45 | 891(19.8) | |----|-----------| | 45 | |
| 441 | |-----| | 405 |
| 360 | |-----| | 360 |
Case V. When a circulate divides a repeating dividend.
**Example.** Divide 5.83 by .72, = $\frac{33}{99}$
| 583.83 | |--------| | 5.83 |
| 72 | 577.50(8.02083) | |----|-----------------| | 576 |
In order to multiply the dividend by 99, move the decimal point two places to the right, and then subtract the given dividend.
| 150 | |-----| | 144 |
| 600 | |-----| | 576 |
| 240 | |-----| | 216 |
| *24 |
Case VI. When a circulate divides a circulate.
When the circle of the quot is likely to run on too many places, you may stop the operation, and complete the quot by a vulgar fraction; as in the following example.
**Example.** Divide 34.56097, by 3.592, = $\frac{23463}{9999}$
| 3589 | 34560.97560 | |------|-------------| | 3 | 34.56097 |
| 3589 | 34526.41463(9.6200653) | |------|------------------------| | 32301 |
| 22254 | |-------| | 21534 |
| 7201 | |------| | 7178 |
The quot would run on to 49 figures of a finite part, and then a circle of 65 places; but limit it at seven places of decimals, and then complete it by a vulgar fraction; as follows, viz.
Complete the partial remainder 2724 by annexing to it the circle of the dividend, and placing both, by way of numerator, over the divisor 3589.
The numerator of this complex fraction being a mixt number, reduce it to an improper fraction, by multiplying 2724 by the denominator 99999, and adding the numerator 46341 to the product; as in the margin: and then, instead of the mixt number, the numerator of the complex fraction will be $\frac{272443617}{99999}$. Or rather work thus: Esteem 2724.46341, a circulate; and then you find the numerator of the vulgar fraction by subtracting the finite part. Next divide this fractional numerator by the denominator; which is done by multiplying 3589 by 99999, as in the margin; and now the simple vulgar fraction to be annexed to the partial quot is
\[ \frac{3589}{35896411} = \frac{72444617}{788988417}. \]
If the quot thus completed be multiplied by the divisor, it will produce the dividend.
VII. Decimal Practice.
The price of goods or merchandise may be cast up decimally by any of the methods following.
Method I. Find the decimal of the rate, viz. the value of one yard, one pound, one piece, &c.; and this decimal of the rate multiplied into the number or quantity of the goods gives the price.
Ex. 1. At 3 s. 4 d. what cost 346?
\[ \begin{array}{ccc} 346 & \text{The decimal of the} \\ 15 & \text{rate is } .16 = \frac{1}{6} \\ 1730 & \\ 346 & \\ \end{array} \]
L. s. d.
\[ \begin{array}{ccc} 45 & \\ 69 & \\ 63 & \\ 60 & \\ 54 & \\ \end{array} \]
Table of Rates and Divisors.
| Rates | 0 Farth. | 1 Farth. | 2 Farth. | 3 Farth. | |-------|----------|----------|----------|----------| | d. | Divis. | Divis. | Divis. | Divis. | | 0 | | | | | | 1 | 6,40 | 8,6,4 | 4,30 | 8,40 | | 2 | 4,30 | 4,30+8 | 4,30+4 | 8,0+12 | | 3 | 80 | 80+12 | 80+6 | 80+4 | | 4 | 60 | 60+4 of 4| 60+8 | 60+8+2 of 8 | | 5 | 6,8 | 40-8 | 40-12 | 40-4 of 6 | | 6 | 40 | 40+4 of 6| 40+12 | 40+8 | | 7 | 40+6 | 40+4+4 of 6| 40+4 | 40+4+6 of 4 | | 8 | 30 | 30+4 of 8| 30+2 of 8| 80×3-12 of 80 | | 9 | 80×3 | 80×3+12 of 80| 80×3+6 of 80| 80×3+4 of 80 | | 10 | 8,3 | 30+4+8 of 4| 20-8 | 40+2+2+6 | | 11 | 20-12 | 40×2-8 of 40| 40×2-12 of 40| 8,3+5-8 of .5 |
In the above table the pence stand in the left-hand column, and the farthings on the head, and the divisors in the angle of meeting; which are to be understood and read as follows.
3,8,40. Divide the given number of goods by 3, divide the quot by 8, and again divide this last quot by 40.
4,30,-8. Divide the number of goods by 4, divide the quot by 30, and from this last quot subtract one 8th of itself.
80+12. To an 80th add a 12th of that 80th.
4,30,+4. To a 30th of a 4th add a 4th of that 30th.
Ex. 2. At 6 s. 8 d. what cost 439?
The decimal of the rate is .3
\[ 3)439(146.3 = 146 \quad 6 \quad 8 \]
Meth. II. When the rate consists of pence and farthings, find how often it is contained in one pound Sterling, divide the given number of goods by this number, or by its component parts, or work by aliquot parts, and the result will be the price sought.
We confine this method to such rates as consist of pence and farthings, because when the rate consists of shillings, pence, and farthings, or of pounds, shillings, pence, &c., it is shorter and easier to work by Method I.
To make the practice ready and easy, it will be proper to have at hand a table of rates and divisors, such as the following one. Divide by 60, divide again the quot by 4, and to the first quot add a 4th of the second quot.
80×3=240 of 80. Divide by 80, multiply the quot by 3, and from the product subtract a 12th of the first quot.
Ex. 1. At 1 l. per yard, what cost 432 yards?
One 3d = 144 One 8th of that = 18 One 40th of that = .45 = 9s.
Ex. 2. At 3 f. what cost 728½?
One 8th = 91.0625 L. s. d. One 40th of that = 2.2765625 = 2 5 6¾
Meth. III. The third method is by decimal tables of rates suited to the nine digits; such as those composed and published by the Rev. Mr George Brown in 1718, under the title of *Arithmetica Infinita*, and recommended by Dr John Keill professor of astronomy in the university of Oxford.
These tables are still extant, and extend from 1 farthing to 20s.; a short specimen of which, with their construction, and the manner of using them, we shall here subjoin.
**Decimal Table of Rates, 1 l. the integer.**
| Rate. | Rate. | Rate. | Rate. | |-------|-------|-------|-------| | s. d. | s. d. | s. d. | s. d. | | N. | 11 5 | 11 5¼ | 11 5½ | 11 5¾ | | 1 | 0.57083 | 0.571875 | 0.572916 | 0.5739583 | | 2 | 1.1418 | 1.14375 | 1.14583 | 1.147916 | | 3 | 1.7125 | 1.715625 | 1.71875 | 1.721875 | | 4 | 2.283 | 2.2875 | 2.2916 | 2.29583 | | 5 | 2.85416 | 2.859375 | 2.864583 | 2.8697916 | | 6 | 3.425 | 3.43125 | 3.4375 | 3.44375 | | 7 | 3.99583 | 4.003125 | 4.010416 | 4.0177083 | | 8 | 4.56 | 4.575 | 4.583 | 4.5916 | | 9 | 5.1375 | 5.146875 | 5.15625 | 5.165625 |
In the left-hand column stand the nine digits; and on the right of 1 are the decimals of the respective rates on the head. Thus, .57083 is the decimal of 11 s. 5 d., one pound being the integer; and .571875 is the decimal of 11 s. 5¼ d., &c. Those decimals opposite to 1 being multiplied through the nine digits, make up or compose the rest of the table.
The superior excellency of tables thus constructed is, that we multiply or divide by 10, 100, 1000, &c. by moving the decimal point so many places to the right or left as there are ciphers in the multiplier or divisor.
Hence the price or value of any number of yards, or other things, denoted by a single digit, or by any of its decuples, may be readily found. Thus, the price of 7, 70, 700, 7000, 70000 yards, at 11 s. 5 d. per yard, is found as follows.
Yards. L. s. d. 7 = 3.99583 = 3 19 11 70 = 39.9583 = 39 19 2 700 = 399.583 = 399 19 8 7000 = 3995.83 = 3995 19 8 70000 = 39958.3 = 39958 6 8
Now, every number may be resolved into decuples of the several digits of which it is composed; find therefore the price of each decuple by itself, as already taught, and their sum will be the price of the whole.
**Examp. 1.** Required the price of 7956 yards, at 11 s. 5¼ d. per yard.
Yards. L. 7000 = 4017.708333 900 = 516.5625 50 = 28.697916 6 = 3.44375
L. s. d. 4566.412500 = 4566 8 3
**Examp. 2.** How much money will one spend in a year, or 365 days, at the rate of 11 s. 5¼ d. per day?
Days. L. 300 = 171.875 60 = 34.375 5 = 2.864583
L. s. d. 209.114583 = 209 2 3¾
Tables of this sort may be framed for a great variety of useful purposes, and are easily constructed.
Thus, suppose a table wanted for showing the daily income of any annuity, or yearly pension; in this case, divide 1 by 365, and the quot is the income of 1l. annuity for one day; and by multiplying this quot through the nine digits, the table is constructed as follows.
**Table.**
| 1 | 0.02730726 | | 2 | 0.05479452 | | 3 | 0.08219178 | | 4 | 0.10958904 | | 5 | 0.13698630 | | 6 | 0.16438356 | | 7 | 0.19178082 | | 8 | 0.21917808 | | 9 | 0.24657534 |
The use of the table will best appear by examples; which take as follows.
**Example 1.** If one has a yearly pension of 375l. what is his daily income?
L. 300 = .8219 70 = .1917 5 = .0136
L. s. d. 1.0272 = 1 0 6¾
**Example 2.** The yearly rent of a gentleman's estate is 968l. 10s. what can he afford to spend per day?
L. 900 If the income for any number of days be required, find the income for one day as above; and multiply the decimal answer by the given number of days. Or, multiply the yearly pension by the given number of days, and use the product as the yearly pension. Thus, in Ex. 2, if the gentleman's income for 64 days be demanded, you may either multiply 2.6532 by 64; or multiply 968.5 by 64; and then work for the product as follows.
\[ \begin{align*} 968.5 & \quad 60000 = 164.3835 \\ 64 & \quad 1000 = 2.7397 \\ & \quad 900 = 2.4657 \\ 38740 & \quad 80 = .2191 \\ 58110 & \quad 4 = .0109 \\ \end{align*} \]
The decimals in the table being circles of eight figures, we have used them as approximates, by confining the operations to four decimal places; which, in affairs of this kind, is sufficiently accurate.
If the annual interest of any principal sum be considered as the yearly pension, the interest of the same principal for any number of days may be found by the table as taught above.
The interest of any principal sum for a year is easily found, as being always the hundredth part of the product of the principal multiplied by the rate per cent.
**Example.** Required the interest for 26 days of £685.1. principal, at 5 per cent.
\[ \begin{align*} L. & \quad 685 \\ & \quad 5 \\ & \quad 34.25 \text{ annual interest} \\ 26 & \quad 800 = 2.19178 \\ 20550 & \quad 90 = .24657 \\ 6850 & \quad .5 = .00136 \\ \end{align*} \]
**Duodecimals.**
Decimal practice may be used with great advantage in the multiplication and division of duodecimals, where the integer is divided into twelve equal parts, called primes, and each prime into twelve seconds, each second into twelve thirds, &c.
For the ready conversion of primes, seconds, thirds, &c. into decimals of the integer, the following table is constructed.
| Primes | Seconds | Thirds | Fourths | |--------|---------|--------|---------| | 1 | .083 | .0069 | .000578 | | 2 | .16 | .0138 | .001157 | | 3 | .25 | .0208 | .001736 | | 4 | .3 | .027 | .002314 | | 5 | .416 | .03472 | .002893 | | 6 | .5 | .0416 | .003472 | | 7 | .583 | .0486 | .004050 | | 8 | .6 | .054 | .004629 | | 9 | .75 | .0625 | .005208 | | 10 | .83 | .0694 | .005787 | | 11 | .916 | .0763 | .006365 |
In the column of fourths, the decimals run on to eight places of a finite part, and nine figures of a circle; but the finite part by itself, which alone is inserted in the table, will be found sufficient; and in the column of thirds too, the circle of three figures may in most cases be neglected.
**I. Multiplication.**
**Example 1.**
What is the product of 247 by 18?
\[ \begin{align*} 24 & \quad 7 = 24.583 \\ 18 & \quad 5 = 18.416 \\ \end{align*} \]
Or thus:
\[ \begin{align*} 24.583 & \quad 24.58333 \\ 16575 & \quad 16575 \\ 122916 & \quad 122916 \\ 1720833 & \quad 1720833 \\ 12291666 & \quad 12291666 \\ 14750000 & \quad 14750000 \\ 245833333 & \quad 245833333 \\ \end{align*} \]
\[ \begin{align*} 407468.750 & \quad 407468.750 \\ 452.74309 & \quad 452.74309 \\ 12 & \quad 12 \\ 8.91666 & \quad 8.91666 \\ 12 & \quad 12 \\ 11.000 & \quad 11.000 \\ \end{align*} \]
**Ans.** 452 8 11 In working by the inverted method, for the repeating 6 in the multiplier, take \( \frac{1}{2} \) of the multiplicand. The result wants very little of the true answer.
**Examp. 2.**
Multiply 18 6 by 2 4, and 2 3 continually.
\[ \begin{align*} 18 & \quad 6 = 18.5 \\ 2 & \quad 4 = 2.8 \\ 2 & \quad 3 = 2.25 \\ & \quad 118 \\ & \quad 1866 \\ & \quad 2333 \\ & \quad 43.16 \\ & \quad 2.25 \\ & \quad 21583 \\ & \quad 86333 \\ & \quad 86333 \\ & \quad 97.1250 \\ & \quad 12 \\ & \quad \text{Ans. 97 1 6} \end{align*} \]
II. Division.
**Examp. 1.**
Divide 452 8 11 = 452.74309
by 18 5 = 18.416 = 0.6575
\[ \begin{align*} 18.416 & \quad 4527-43095 \\ 1841 & \quad 452.74309 \\ 16575 & \quad 497468.750(24.583 \\ 33150 & \quad 12 \\ & \quad 75968 \\ & \quad 7.000 \\ & \quad 66300 \\ & \quad 96687 \\ & \quad 82875 \\ & \quad \text{Ans. 24 7} \end{align*} \]
**Examp. 2.**
Divide 97 1 6 = 97.125
by 2 3 = 2.25 and the quot
by 18 6 = 18.5
\[ \begin{align*} 2.25)97.125(43.16(2.8 \\ 900 & \quad 370 \\ & \quad 12 \\ & \quad 712 \\ & \quad 616 \\ & \quad 4.0 \\ & \quad 675 \\ & \quad 555 \\ & \quad 375 \\ & \quad *61 \\ & \quad 225 \\ & \quad 1500 \\ & \quad 1350 \\ & \quad \text{Ans. 2 4} \end{align*} \]
*150
**Sexagesimals.**
Decimal practice might likewise be used to good purpose in the arithmetic of sexagesimals, as it would shorten and facilitate the operations.
Sexagesimals, strictly speaking, are degrees, minutes, seconds, thirds, etc., where each degree is divided into 60 minutes, and each minute into 60 seconds, etc.; but under this title is also usually comprehended the division of a sign into 30 degrees. They are commonly marked as under.
**Signs. deg. min. sec. thirds.**
7 24 36 54 48 etc.
Sexagesimals properly belong to astronomy, being used in computations of motion and time, where the degree of motion, and hour of time, are equally divided into 60 minutes. The preference of the decimal method to that of the sexagesimal will appear from the following example of addition done both ways.
**Sexagesimally.**
| Signs | S. | |-------|----| | 10 20 47 17 | 10.69293,518, | | 7 18 50 40 | 7.62814,814, | | 9 25 30 28 | 9.85018,518, | | 11 10 40 50 | 11.35601,811, |
3 15 49 7 = 3.52728,703,
From the above example it is obvious, that even in addition the decimal operation is more simple and easy than the sexagesimal, especially if care be taken to use no more decimal places than what are absolutely necessary.
But in multiplication and division the advantage of the decimal method is still greater; for in the sexagesimal way the operation is extremely tedious; whereas, by working decimally, it is performed in the same manner, and with the same ease, as in duodecimals.
**Vulgar Fractions.**
Decimal practice may sometimes be profitably used in the the arithmetic of vulgar fractions, the operation being shorter and easier in the decimal than in the vulgar way. This we shall illustrate by a few examples.
I. Addition.
Ex. 1. What is the sum of \( \frac{1}{4} + \frac{3}{5} \)?
\[ \begin{align*} \frac{1}{4} &= .25 \\ \frac{3}{5} &= .666 \\ &= s. d. \\ .918 &= 18 \quad 4 \end{align*} \]
Ex. 2. What is the sum of \( 14\frac{7}{8} + 18\frac{3}{4} + \frac{1}{8} \) of \( \frac{5}{8} \) C.?
\[ \begin{align*} G. & \quad 14\frac{7}{8} = 14.875 \\ & \quad 18\frac{3}{4} = 18.6666 \\ \frac{3}{4} \text{ of } \frac{5}{8} &= \frac{1}{8} = .625 \\ & \quad C. Q. lb. \\ 34.1666 &= 34 \quad 0 \quad 18\frac{3}{4} \end{align*} \]
II. Subtraction.
Ex. 1. From \( \frac{1}{4} \) subtract \( \frac{1}{3} \).
\[ \begin{align*} L. & \quad \frac{1}{4} = .75 \\ & \quad \frac{1}{3} = .333 \\ & \quad s. d. \\ .418 &= 8 \quad 4 \end{align*} \]
Ex. 2. From \( \frac{3}{4} \) of \( \frac{7}{8} \) subtract \( \frac{1}{2} \) of \( \frac{1}{4} \) lb. Troy.
\[ \begin{align*} \frac{3}{4} \text{ of } \frac{7}{8} &= \frac{7}{8} = .875 \\ \frac{1}{2} \text{ of } \frac{1}{4} &= \frac{1}{8} = .125 \\ & \quad 02. dw. \\ .4583 &= 5 \quad 10 \end{align*} \]
III. Multiplication.
Ex. Multiply \( 19\frac{7}{8} \) by \( 22\frac{1}{4} \) feet.
\[ \begin{align*} F. & \quad 19\frac{7}{8} - - - = 19.875 \\ 22\frac{1}{4} = 22.25 &= 22.25 \\ & \quad 201 \\ & \quad 19583 \\ & \quad 3916666 \\ 9)3936.250 \\ & \quad 437.364 = 437 \quad 52. \end{align*} \]
II. Division.
Ex. Divide \( 6\frac{7}{8} \) by \( \frac{4}{5} \).
\[ \begin{align*} \frac{4}{5} &= .36, \quad \text{and } 6\frac{7}{8} = 6.38 \\ .36)638.83( \\ & \quad 6.38 \\ & \quad L. s. d. \\ 36)632.50(17.5694 = 17 \quad 11 \quad 4\frac{3}{5} \end{align*} \]
Rule of Three Direct.
Decimal practice is frequently the shortest and easiest method of operation in the rule of three.
Examp. I. If C. 3 : 1 : 14 of raisins cost L. 10 : 2 : 6, what will 6. C. 3 Q. cost at that rate?
\[ \begin{align*} C. Q. lb. & \quad L. s. d. C. Q. \\ Vulgar state & \quad 3 \quad 1 \quad 14 : 10 \quad 2 \quad 6 :: 6 \quad 3 \\ Decimal state & \quad 3.375 : 10.125 :: 6.75 \\ & \quad 6.75 \\ & \quad 50625 \\ & \quad 70875 \\ & \quad 60750 \\ & \quad L. s. d. \\ 3.375)68.34375(20.25 = 20 \quad 5 \\ & \quad 6750 \\ & \quad 8437 \\ & \quad 6750 \\ & \quad 16875 \\ & \quad 16875 \end{align*} \]
Examp. II. If a wedge of gold, weighing 14 lb., 3 oz., 8 dw. cost L. 514, 4 s., what is that per ounce?
Vulgar:
CHAP. XII. EXTRACTION OF ROOTS.
If unity be multiplied continually by any given number, the products thence arising are called powers of that number; and the given number is called the root, or first power.
Thus, if 2 be the given number, then $1 \times 2 = 2$ is the root or first power; and $2 \times 2 = 4$ is the square or second power; and $4 \times 2 = 8$ is the cube or third power; and $8 \times 2 = 16$ is the biquadrate or fourth power; and $16 \times 2 = 32$ is the surfold or fifth power; and $32 \times 2 = 64$ is the sixth power, or cube squared, &c.
The natural numbers, 1, 2, 3, &c., are sometimes placed over these powers, denoting the number of multiplications used in producing them, or showing what powers they are; and are called indices or exponents, as in the following scheme.
Indices, 0, 1, 2, 3, 4, 5, 6, 7, &c. Powers, 1, 2, 4, 8, 16, 32, 64, 128, &c.
The raising any root or number given to any power required, is called involution; and is performed by multiplying the given root into unity continually, as taught above. But the finding the root of a given power is called evolution, or extraction of roots.
If the root of any power not exceeding the seventh power, be a single digit, it may be obtained by inspection, from the following table of powers.
| Power of Root | 1st Power | 2nd Power | 3rd Power | 4th Power | 5th Power | 6th Power | 7th Power | |---------------|-----------|-----------|-----------|-----------|-----------|-----------|-----------| | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | | 2 | 4 | 8 | 16 | 32 | 64 | 128 | | | 3 | 9 | 27 | 81 | 243 | 729 | 2187 | | | 4 | 16 | 64 | 256 | 1024 | 4096 | 16384 | | | 5 | 25 | 125 | 625 | 3125 | 15625 | 78125 | | | 6 | 36 | 216 | 1296 | 7776 | 46656 | 279936 | | | 7 | 49 | 343 | 2401 | 16807 | 117649 | 823543 | | | 8 | 64 | 512 | 4096 | 32768 | 262144 | 2097152 | | | 9 | 81 | 729 | 6561 | 59049 | 531441 | 4729799 | |
I. EXTRACTION OF THE SQUARE ROOT.
RULE I. Divide the given number into periods of two figures, beginning at the right hand in integers, and pointing toward the left. But in decimals, begin at the place of hundreds, and point toward the right. Every period will give one figure in the root.
RULE II. Find by the table of powers, or by trial, the nearest lesser root of the left-hand period, place the figure so found in the quot, subtract its square from the said period, and to the remainder bring down the next period for a dividend or resolvend.
RULE III. Double the quot for the first part of the divisor; inquire how often this first part is contained in the whole resolvend, excluding the units place; and place the figure denoting the answer both in the quot and on the the right of the first part; and you have the divisor complete.
IV. Multiply the divisor thus compleated by the figure put in the quot, subtract the product from the resolvend, and to the remainder bring down the following period for a new resolvend, and then proceed as before.
Note 1. If the first part of the divisor, with unity supposed to be annexed to it, happen to be greater than the resolvend, in this case place o in the quot, and also on the right of the partial divisor; to the resolvend bring down another period; and proceed to divide as before.
Note 2. If the product of the quotient-figure into the divisor happen to be greater than the resolvend, you must go back, and give a lesser figure to the quot.
Note 3. If, after every period of the given number is brought down, there happen at last to be a remainder, you may continue the operation, by annexing periods or pairs of ciphers, till there be no remainder, or till the decimal part of the quot repeat or circulate, or till you think proper to limit it.
Example I. Required the square root of 133225.
| Square number | 133225 | (365 root) | |---------------|--------|------------| | | 9 | 365 | | 1 div. 66 | 432 | resolvend, | | | 396 | product. | | 2 div. 725 | 3625 | resolvend. |
3625 product. 133225 proof.
Example II. Required the square root of 72, to eight decimal places.
72.00000000(8.48528137 root.
64
164) 800
656
1688) 14400
13504
16965) 89600
84845
169702) 477500
339404
169704) 138006
135763
2333
1697
636
509
127
118
---
Example III. Required the square root of .2916.
.2916 (.54 root.
25
104) 416
416
If the square root of a vulgar fraction be required, find the root of the given numerator for a new numerator, and find the root of the given denominator for a new denominator. Thus, the square root of \( \frac{4}{9} \) is \( \frac{2}{3} \), and the root of \( \frac{1}{4} \) is \( \frac{1}{2} \); and thus the root of \( \frac{1}{4} \) (=6\(\frac{1}{4}\)) is \( \frac{1}{2} = 2\frac{1}{2} \).
But if the root of either the numerator or denominator cannot be extracted without a remainder, reduce the vulgar fraction to a decimal, and then extract the root, as in Example III. above.
II. Extraction of the Cube Root.
Rule I. Divide the given number into periods of three figures, beginning at the right hand in integers, and pointing toward the left. But in decimals, begin at the place of thousands, and point toward the right. The number of periods shows the number of figures in the root.
II. Find by the table of powers, or by trial, the nearest lesser root of the left-hand period; place the figure so found in the quot; subtract its cube from the said period; and to the remainder bring down the next period for a dividend or resolvend.
The divisor consists of three parts which may be found as follows.
III. The first part of the divisor is found thus: Multiply the square of the quot by 3, and to the product annex two ciphers; then inquire how often this first part of the divisor is contained in the resolvend, and place the figure denoting the answer in the quot.
IV. Multiply the former quot by 3, and the product by the figure now put in the quot; to this last product annex a cipher; and you have the second part of the divisor. Again, square the figure now put in the quot for the third part of the divisor; place these three parts under one another, as in addition; and their sum will be the divisor complete.
V. Multiply the divisor, thus completed, by the figure last put in the quot, subtract the product from the resolvend, and to the remainder bring down the following period for a new resolvend, and then proceed as before.
Note 1. If the first part of the divisor happen to be equal to or greater than the resolvend, in this case place o in the quot, annex two ciphers to the said first part of the divisor, to the resolvend bring down another period, and proceed to divide as before.
Note 2. If the product of the quotient-figure into the divisor happen to be greater than the resolvend, you must go back, and give a lesser figure to the quot.
Note 3. If, after every period of the given number is brought down, there happen at last to be a remainder, you may continue the operation by annexing periods of three ciphers till there be no remainder, or till you have as many decimal places in the root as you judge necessary.
**Examp. I.** Required the cube root of 12812904.
Cube number 12812904 (234 root)
| 1st part | 1200 | |----------|------| | 2d part | 180 | | 3d part | 9 |
1 divisor \(1389 \times 3 = 4167\) product.
| 1st part | 158700 | |----------|--------| | 2d part | 2760 | | 3d part | 16 |
2 divisor \(161476 \times 4 = 645904\) product.
**Proof.**
\[ \begin{array}{c} 234 \\ \times 234 \\ \hline 936 \\ + 702 \\ + 468 \\ \hline 54756 \\ \end{array} \]
Square 54756
\[ \begin{array}{c} 234 \\ \times 234 \\ \hline 936 \\ + 702 \\ + 468 \\ \hline 54756 \\ \end{array} \]
Cube 12812904
**Examp. II.** Required the cube root of 28\(\frac{1}{4}\).
\[ \begin{array}{c} 28.750000(3.06 \text{ root}) \\ \times 27 \\ \hline 750000 \\ \end{array} \]
Div. 275436 \(\times 6 = 1652616\) prod.
97384 rem.
**Proof.**
\[ \begin{array}{c} 3.06 \\ \times 3.06 \\ \hline 1836 \\ + 918 \\ \hline 28.652616 \\ \end{array} \]
If the cube root of a vulgar fraction be required, find the cube root of the given numerator for a new numerator, and the cube root of the given denominator for a new denominator. Thus, the cube root of \(\frac{9}{8}\) is \(\frac{3}{2}\), and the cube root of \(\frac{2}{3}\) is \(\frac{1}{2}\); and thus the cube root of \(\frac{1}{2}\) (\(= 15\)) is \(\frac{1}{2} = 2\).
But if the root of either the numerator or denomina-
tor cannot be extracted without a remainder, reduce the vulgar fraction to a decimal, and then extract the root.
**III. Extraction of the Biquadrate Root.**
**Rule.** Extract the square root of the given number; and again extract the square root of the root so found, and the last of these roots is the root sought.
**Examp.** Required the biquadrate root of 5308416.
\[ \begin{array}{c} 5308416 \\ \times 2304(48 \text{ root}) \\ \hline 4 \\ \times 43 \\ \hline 130 \\ \times 88 \\ \hline 704 \\ \end{array} \]
If, in the first extraction, there happen to be a remainder, continue the operation, by annexing pairs of ciphers, till you have twice as many decimal places in the square or first root, as you propose to have in the last root.
**IV. Extraction of the root of the fifth power, or sursolid.**
**Rule I.** Divide the given number into periods of five figures, find the nearest lesser root of the left-hand period, put the figure so found in the quot, subtract its fifth power, and to the remainder bring down the next period for a resolvend.
II. Put \(a\) for the root, and then the sursolid or fifth power will be \(aaaaa + 5aaaa + 10aaaa + 10aaay + 5ayyy + yyyy\). Now, \(aaaaa\) being already subtracted, there remains the other five parts; and to find \(y\), divide by its coefficient, viz. by \(5aaaa + 10aaay + 10aaay + 5ayyy + yyyy\); that is, try how often \(5aaaa\) is contained in the resolvend; and, by the help of the quotient-figure, you make up the other four parts of the divisor.
**Examp.** Required the sursolid root of 33554432
\[ \begin{array}{c} 33554432(32 \text{ root}) \\ \times 243 \\ \hline 9254432 \text{ resolv.} \\ \end{array} \]
Divisor 4627216 \(\times 2 = 9254432\) prod.
**V. Extraction of the root of the sixth power, or cube squared.**
**Rule.** Extract the square root of the given number, and then extract the cube root of that root, the last is the root sought. Or, first extract the cube root, and then extract the square root of that root.
**Examp.** Required the root of 191102976, being the sixth power.
\[ \begin{array}{c} 191102976 \\ \hline 23) \quad 91 \\ \hline 69 \quad 240 \\ \hline 268) \quad 2210 \\ \hline 2144 \quad 1456 \times 4 = 5824 \text{ prod.} \\ \hline 2762) \quad 6629 \\ \hline 5524 \\ \hline 27644) \quad 110576 \\ \hline 110576 \\ \end{array} \]
VI. Extraction of the root of the seventh power.
**Rule.** Put \(a + y\) for the root, and the seventh power will be \(aaaaaaa + 7aaaaay + 21aaaaay + 35aaaaay + 35aaaaay + 21aaayyy + 7ayyyyy + yyyy\), by the aid of which proceed as in extracting the root of the fifth power.
**Examp.** Required the root of 3404825447, being the seventh power.
\[ \begin{array}{c} 3404825447(23 \text{ root.}) \\ \hline 128 \\ \hline 2124825447 \text{ resolv.} \\ \hline 448000000 = 7aaaaaa \\ 201600000 = 21aaaaay \\ 50400000 = 35aaaaay \\ 7560000 = 35aaaaay \\ 680400 = 21aaayyy \\ 34020 = 7ayyyyy \\ 729 = yyyy \\ \end{array} \]
Divis. 708275149 \times 3 = 2124825447 prod.
(o)
VII. Extraction of the root of the eighth power.
**Rule.** Extract the square root of the given number.
continually till you have three roots; the last of these is the root sought.
Thus, let 1785793904896 be the eighth power; by extracting the square root you get the biquadrate or fourth power, viz. 1336336; and by extracting the square root of the biquadrate, you get the square or second power, viz. 1156, whose square root is 34, the root sought.
VIII. Extraction of the root of the ninth power.
**Rule.** Extract the cube root of the given number, and you have the cube or third power, whose cube root is the root sought.
Thus, let 5159780352 by the ninth power; by extracting the cube root you get the cube or third power, viz. 1728, whose cube root 12 is the root sought.
Universally, whatever the given power be, put \(a + y\) for the root, and by involution raise \(a + y\) to the power of the given number; then, with this as your guide or canon, extract the root in the manner prescribed and exemplified in the extraction of the root of the fifth and seventh powers.
But if the index of the given power be a multiple of 2, the work may be rendered easier: For, by extracting the square root of the given number, you obtain a power whose index is one half of the index of the given power. Thus, by extracting the square root of the tenth power, you have the fifth power; and the square root of the twelfth power is the sixth power, &c.
Again, if the index of the given power be a multiple of 3, by extracting the cube root you obtain a power whose index is one third of the index of the power given. Thus the cube root of the ninth power is the cube or third power; and the cube root of the twelfth power is the biquadrate or fourth power, &c.
Involution is directly contrary to extraction or evolution; and therefore, if a square number be squared, it will give the biquadrate or fourth power; and if a biquadrate be squared, it will give the eighth power. Again, if a cube number be cubed, it will give the ninth power; and if the biquadrate be cubed, it will give the twelfth power. See Algebra, Chap. IX. and X.
For the application of Arithmetic to various branches of business, &c. see Alligation, Annuities, Barter, Brokage, Bankruptcy, Exchange, Insurance, Interest, Mensuration, &c., &c.
---
**ARK**
**ARITHMOMANCY,** a species of divination performed by means of numbers.
**ARK,** or Noah's Ark, a floating vessel built by Noah, for the preservation of his family, and the several species of animals, during the deluge. See Plate XXXVIII. fig. 1.
The ark has afforded several points of curious inquiry among the critics and naturalists, relating to its form, capacity, materials, &c.
The wood whereof the ark was built, is called in the Hebrew Gopher-wood, and in the Septuagint square timbers. Some translate the original cedar, others pine, others: others box, &c. Pelletier prefers cedar, on account of its incorruptibility, and the great plenty of it in Asia; whence Herodotus and Theophrastus relate, that the kings of Egypt and Syria built whole fleets thereof, instead of deal.
The learned Mr Fuller in his Miscellanies, has observed, that the wood whereof the ark was built, was nothing else but that which the Greek call κυπαρίστας, or the cypresi-tree; for, taking away the termination, kypar and gopher differ very little in sound. This observation the great Bochart has confirmed, and shewn very plainly that no country abounds so much with this wood as that part of Assyria which lies about Babylon.
In what place Noah built and finished his ark is no less made a matter of disputation. But the most probable opinion is, that it was built in Chaldæa, in the territories of Babylon, where there was so great a quantity of cypresses in the groves and gardens in Alexander's time, that that prince built a whole fleet out of it, for want of timber. And this conjecture is confirmed by the Chaldean tradition, which makes Xithurus (another name for Noah) set sail from that country.
The dimensions of the ark, as given by Moses, are 300 cubits in length, 50 in breadth, and 30 in height, which some have thought too scanty, considering the number of things it was to contain; and hence an argument has been drawn against the authority of the relation. To solve this difficulty many of the ancient fathers, and the modern critics, have been put to very miserable shifts: But Buteo and Kircher have proved geometrically, that, taking the common cubit of a foot and a half, the ark was abundantly sufficient for all the animals supposed to be lodged in it. Snellius computes the ark to have been above half an acre in area, and father Lamy shews, that it was 110 feet longer than the church of St Mary at Paris, and 64 feet narrower; and if so, it must have been longer than St Paul's church in London, from west to east; and broader than that church is high in the inside, and 54 feet of our measure in height; and Dr Arbuthnot computes it to have been 81062 tons.
The things contained in it were, besides eight persons of Noah's family, one pair of every species of unclean animals, and seven pair of every species of clean animals, with provisions for them all during the whole year. The former appears, at first view, almost infinite; but if we come to a calculation, the number of species of animals will be found much less than is generally imagined, not amounting to an hundred species of quadrupeds, nor to two hundred of birds; out of which, in this case, are excepted such animals as can live in the water. Zoologists usually reckon but an hundred and seventy species in all; and bishop Wilkins shews that only seventy-two of the quadruped kind needed a place in the ark.
By the description Moses gives of the ark, it appears to have been divided into three stories, each ten cubits, or fifteen feet high; and it is agreed on, as most probable, that the lowest story was for the beasts, the middle for the food, and the upper for the birds, with Noah and his family; each story being subdivided into different apartments, stalls, &c. Though Josephus, Philo, and other commentators, add a kind of fourth story under all the rest; being, as it were, the hold of the vessel, to contain the ballast, and receive the filth and faeces of so many animals: But F. Calmet thinks, that what is here reckoned a story, was no more than what is called the keel of ships, and served only for a conservatory of fresh water. Drexius makes three hundred apartments. F. Fournier, three hundred and thirty-three; the anonymous author of the Questions on Genesis, four hundred; Buteo, Temporarius, Arias Montanus, Hofius, Wilkins, Lamy, and others, suppose as many partitions as there were different sorts of animals. Pelletier makes only seventy-two, viz. thirty-five for the birds, and as many for the beasts; his reason is, that if we suppose a greater number, as 333, or 400, each of the eight persons in the ark must have had thirty-seven, forty-one, or fifty stalls to attend and cleanse daily, which he thinks impossible to have been done. But it is observed, that there is not much in this; to diminish the number of stalls without a diminution of animals is vain; it being perhaps more difficult to take care of three hundred animals in seventy-two stalls, than in three hundred. As to the number of animals contained in the ark, Buteo computes that it could not be equal to five hundred horses; he even reduces the whole to the dimensions of fifty-six pair of oxen. F. Lamy enlarges it to sixty-four pair of oxen, or an hundred and twenty-eight oxen; so that supposing one ox equal to two horses, if the ark had room for two hundred and fifty-six horses, there must have been room for all the animals. But the same author demonstrates, that one floor of it would suffice for five hundred horses, allowing nine square feet to a horse.
As to the food in the second story, it is observed by Buteo from Columella, that thirty or forty pounds of hay ordinarily suffices for an ox a day, and that a solid cubit of hay, as usually pressed down in our hayricks, weighs about forty pounds; so that a square cubit of hay is more than enough for one ox in one day. Now it appears that the second story contained 150,000 solid cubits, which divided between two hundred and six oxen, will afford each more hay by two thirds, than he can eat in a year. Bishop Wilkins computes all the carnivorous animals, equivalent, as to the bulk of their bodies, and their food, to twenty-seven wolves; and all the rest to two hundred and eighty beesves. For the former he allows 1825 sheep, and for the latter, 109,500 cubits of hay, all which will be easily contained in the two first stories, and a deal of room to spare. As to the third story, no body doubts of its being sufficient for the fowls; with Noah, his sons, and daughters. Upon the whole, the learned bishop remarks, that of the two, it appears much more difficult to assign a number and bulk of necessary things to answer the capacity of the ark, than to find sufficient room for the several species of animals already known to have been there. This he attributes to the imperfection of our list of animals, especially Specially those of the unknown parts of the earth; adding, that the most expert mathematician at this day could not assign the proportion of a vessel better accommodated to the purpose than is here done; and hence finally concludes, that the capacity of the ark, which had been make an objection against scripture, ought to be esteemed a confirmation of its divine authority, since, in those rude ages, men, being less versed in art and philosophy, were more obnoxious to vulgar prejudices than now; so that had it been an human invention, it would have been contrived, according to those wild apprehensions which arise from a confused and general view of things, as much too big, as it had been represented too little.
But it must be observed, that besides the places requisite for the beasts and birds, and their provisions, there was room required for Noah to lock up household utensils, the instruments of husbandry, grains and seeds, to sow the earth with after the deluge; for which purpose it is thought that he might spare room in the third story for six and thirty cabins, besides a kitchen, a hall, four chambers, and a space about eight and forty cubits in length to walk in.
**Ark of the covenant**, a small chest or coffin, three feet nine inches in length, two feet three inches in breadth, and two feet three inches in height, in which were contained the golden pot that had manna, and Aaron's rod, and the tables of the covenant. This coffin was made of shittim-wood, and was covered with the mercy-seat, which was of solid gold; at the two ends whereof were two cherubims, looking toward each other, with expanded wings, which, embracing the whole circumference of the mercy-seat, met on each side in the middle. The whole, according to the rabbins, was made out of the same mass, without joining any of the parts by solder. Here it was that the Schechinah or Divine Presence refted, both in the tabernacle and in the temple, and was visibly seen in the appearance of a cloud over it; and from hence the Divine oracles were given out by an audible voice, as often as God was consulted in the behalf of his people. Plate XXXVIII fig. 2.
**Arklow**, a sea-port town of Ireland, situated in the county of Wicklow, about thirteen miles south of the city of Wicklow, in 6° 20' W. long. and 52° 55' N. lat.
**Arles**, a city of Provence in France, situated on the eastern shore of the river Rhone, in 4° 45' E. long. and 43° 32' N. lat.
**Arleux**, a town of Hainault, in the French Netherlands, situated about six miles south of Douay, in 3° E. long. and 50° 20' N. lat.
**Arlon**, a town of the duchy of Luxemburg, on the Austrian Netherlands, situated in 5° 30' E. long. and 49° 45' N. lat.
**Arm**, in riding, is applied to a horse, when, by pressing down his head, he endeavours to defend himself against the bit, to prevent obeying, or being checked thereby.
**Armada**, a Spanish term, signifying a fleet of men of war, as *armadilla* signifies a squadron.
**Armadabat**, a very large city of Asia, the metropolis of the kingdom of Guzarat.
**Armadillo**, in zoology, a synonyme of the dasypus. See Dasypus.
**Armagh**, once a considerable city of Ireland, but now much reduced, situated about thirty miles south of Londonderry, in 6° 45' W. long. and 54° 30' N. lat. It is still the see of the primate of Ireland, and gives name to the county of Armagh.
**Armagnac**, a district or territory in the north-east part of Gascony in France.
**Arman**, in farriery. See Drench.
**Armed**, in the sea-language. A cross-bar shot, is said to be armed, when some rope-yarn or the like is rolled about the end of the iron bar, which runneth through the shot.
**Armed**, in heraldry, is used when the horns, feet, beak, or talons of any beast or bird of prey, are of a different colour from the rest of their body.
**Armenia**, a large country of Asia, comprehending Turcomania and part of Persia.
**Armeniaca**, in botany. See Prunus.
**Armenians**, in church-history, a sect among the eastern Christians; thus called from Armenia, the country anciently inhabited by them. There are two kinds of Armenians, the one catholic and subject to the pope, having a patriarch in Persia, and another in Poland; the other makes a peculiar sect, having two patriarchs in Natolia. They are generally accused of being manophytes, only allowing of one nature in Jesus Christ. As to the eucharist, they for the most part agree with the Greeks; they abstain rigorously from eating of blood and meats strangled, and are much addicted to fasting.
**Armentiers**, a fortified town in French Flanders, situated about seven miles west of Lille, in 2° 50' E. long. and 50° 42' N. lat.
**Armiers**, a town of Hainault, in the French Netherlands, situated on the river Sambre, about twenty miles south of Mons, in 3° 40' E. long. and 50° 15' N. lat.
**Armiger**, an esquire, or armour-bearer. See Esquire.
**Armillary**, in a general sense, something consisting of rings, or circles.
**Armillary sphere**, an artificial sphere, composed of a number of circles, representing the several circles of the mundane sphere, put together in their natural orders, to ease and assist the imagination, in conceiving the constitution of the heavens, and the motions of the celestial bodies. See Geography.
**Armillustrium**, in Roman antiquity, a feast held among the Romans, in which they sacrificed armed, to the sound of trumpets.
**Armings**, in the sea-language. See Armed.
**Arminians**, in church-history, a sect of Christians which arose in Holland, by a separation from the Calvinists. They are great assertors of free-will. They speak very ambiguously of the presence of God. They look upon the doctrine of the Trinity as a point not necessary to salvation; and many of them hold there is no precept in scripture by which we are enjoined to adore the Holy Ghost; and that Jesus is not equal to God the Father.