E. 20. A maltfter hath a kiln, which he finds too large for his bufi nefs, its diameter being 21,2 feet; the diameter of another, which will hold half the quantity, is required?

First 21,2X21,2=450 nearly; then 2)450

225(15 Feet, the answer.

I

25)125

125

By having the bung and head diameters of a cask given, to find the diagonal line,

RULE. Add the square of half the fum of the head and bung dia. meters, to the fquare of half the length; the square root of that fum is the diagonal of the cask.

LIII. EXTRACTION OF THE CUBE ROOT.

T

O extract the cube root, is to find out a number, which being mul tiplied into itself, and then again into the product, produceth the given number.

As the cube root of 512 is 8, confequently 8X8X8=512, the given number; and fo of others, as in the following:

2

TABLE.

4 5

6

7

8

9

Roots I
3
¡Cube I 8 27 64 125 216 343 512

729

RULE. 1. Make a Point over every third figure given, beginning at the units place; feek the greatest cube to the first point on the left-hand (by the table) whofe root place in the quotient; then fubtract its cube from the period, and to the remainder (if any) bring down the three next figures, or your next period, and call it your dividend.

2.

Find a divifor, by calling your quotient figure, with a cypher joined to it, r; then three times the fquare of r will be your divifor, feek how often it is contained in the dividend, and put the answer in the quotient, as in divifion, only with this difference, call the faid quotient figure laft put up e, and multiply your divifor by it, and place the produce underneath the dividend; then multiply the fquare of e, by three times r and place it alfo under the dividend. Laftly, cube the figure you called e, and place it under the dividend; then add the three products together, which gives the fubtrahend, which fubtract from your last dividend, and to the remainder bring down the next period, and proceed as before.

EXPLANATION. The nearest cube to 32, the first period, is 27, which is fet under, and fubtracted therefrom, and 3, the root of the faid cube, is placed in the quotient, and to the remainder 5, the period 768 is

annexed,

annexed, which makes 5768 for a dividend; then a cypher is joined to the quotient figure 3, making 30, which is called, and being fquared, and that square multiplied by 3, produces 2700 for a divifor, which being contained twice in the dividend, 2 is placed in the quotient, and called e, by which the divifor is multiplied, and the product 5400 fet under the dividend. Then 3 times r90, is multiplied by 4, the fquare of e, and the product 360 is placed under 5400; and laftly, 8 the cube of e, is placed under, and added to the other two numbers under the dividend; and the fum 5768 being the fame as the dividend, and no more periods to be brought down, the work is finished, and 32768 is found to be a cube number, and 32 its cube root.

Now 452,08452,08×452,08+4197361088 92398647 Proof. ANOTHER CONCISE METHOD OF EXTRACTING THE CUBE ROOT. RULE 1. Point every third figure of the given number, beginning at the units place; then find the nearest cube to the first point, fubtract, and bring down the three next figures in the next period to the remainder for a refolvend.

2. Square the quotient, and multiply it by 3, for a divifor ; find how often it is contained in the refolvend, rejecting units and tens, and put the answer in the quotient!

3. Square this new figure, and put it on the right hand of the divifor'; but if the new figure fhould be 1, 2, or 3, then put 01, 04, or 09, to the right hand.

4. Multiply the laft figure in the quotient by 30, and multiply it by the former figures; add this product to the divifor, and multiply the fum by the laft figure in the quotient; fubtract that product from the refolvend bring down the next three figures, and proceed as before.

EXPLANATION. The square of 3×327, the divifor; and the fquare of 2 is 4, which (per rule) is 04, this put on the right-hand of the

2 G

divifor

divifor 27, makes 2704; then 2 X 30X3=180, which added to 2704, makes 2884, for a new divifor, which multiplied by 2, the last figure in the quotient, the product is 5768, to be fet under the dividend and fubtracted therefrom, and nothing remains; therefore 32768 is found to be a cube number, and 32 its cube root; the fame as Example 1 in this fection. E. 6. What is the cube root of E. 7. What is the cube root of 618470208? 27407028375?

M. de la Hire has given us a very odd property common to all powers, which M. Carre had obferved with regard to the number 6, which is this that all the natural cubic numbers, 8, 27, 64, 125, whofe root is lefs than 6, being divided by 6, the remainder of the divifion is the root itfelf; and if we go further, 215, the cube of 6, being divided by 6, leaves no remainder, but the divifor 6 is the root itself. Again, 343, the cube of 7, being divided by 6, leaves 1, which added to the divifor 6, makes 7 the root, &c.

The above gentleman, on confidering this property of 6, has found that all numbers, raised to any power whatever, have divifors, which have the fame effect with regard thereto, that 6 hath with regard to cubic numbers.

For finding of these divisors, observe the following:

RULES. 1. If the exponent of the power of a number be even, i. e. if the number be raised to the fecond, fourth, fixth power, &c. it must be divided by 2; the remainder of the divifion, in cafe there be any, added to 2, or to a multiple of 2, gives the root of this number, corresponding to its power, i. e. the fecond, fixth, &c. root.

2. If the exponent of the power be an uneven number, i. e. if the number be raised to the third, fifth, feventh power, &c. the double of that exponent will be the divifor, which has the property mentioned. Thus it is found in 6, double of 3, the exponent of the power of all the thus also 10 is the divifor, of all the numbers raised to the fifth power, &c.

cubes;

To EXTRACT THE CUBE OF A VULGAR FRACTION. RULE. Extract the cube root of the numerator for a new numerator, and the cube root of the denominator for a new denominator; and this new fraction will be the cube root of the given fraction. The fractions must be reduced to their lowest terms; if it be a mixed number, to an improper fraction; and if a furd to a decimal. EXAMPLE 1. What is the cube root of 27? 273; and the 3 3437; then is the root required.

Firft 9

343