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Pup. I understand the reason of obtaining the first figure of the root as we do, and of subtracting its cube from the left hand period; but I do not understand why the divisor is obtained as it is.
Tut. In order to give you a distinct understanding of the whole work, we will prepare blocks in the following manner, and build them up into a cubic form, as directed by the rule.
First make a cubic block of any given size, and mark it with the letter A. Then make three other blocks of a square form, of an indefinite thickness, but all equal to each other, each of which will just cover one side of the block A. and mark them, B. C. and D. Place these blocks on three adjoining sides of the block A. when there will be deficiencies at the three points where the blocks B. C. and D. meet. These deficiencies must be filled with three other blocks, each of which must be just equal in length to one side of the block A. and mark these blocks with E. F. and G. When the blocks E. F. G. are put in their places, there will be a deficiency at the place where the ends of these blocks meet This deficiency must be filled with another block, which mark H. To illustrate the rule, I will take the number before used, and extract the cube root of it, explaining every part.
In the first place I seek the greatest cube in the left hand period, and place its root, 2, in the quotient. The cube of 2 is 8, which I place under the left hand period, and subtract it therefrom, which leaves a remainder of 2. Now as there are two periods in the given number, there must be 2 figures in the root, consequently, 2, in the quo. tient, does not express 2, merely, but 20; and the cube of 20 is 8000, which 8, under the period 10, represents ; thus 8000 of the parts of 10648, are disposed of into a cubic body, the length of each side of which is equal to 20 of those parts, and to render the explanation more plain, we will consider these parts as cubic feet, so that each side of this body is 20 feet square, and this body we will have represented by the block A. Now as each side of this
block is 20 feet square, there are 400 feet on each side of it. Now 8000 feet are disposed of in this block, consequently there are 2648 cubic feet to be added to the block A. in such a manner, that its cubic form will be preserved. To do this, the additions must be made to three sides of the block, and these additions are represented by the blocks, B. C. and D. each of which containing 400 feet, the sum of the whole is 1200. Thus it is evident, that if there were 1200 feet more, there would be just enough to cover three sides of the block A.; and it is to find the contents of these three sides, that the rule directs to " multiply the square of the quotient by 300.” The square of the quotient shows the superficial contents of one side of the block A. viz. 400, for 2 in the quotient is in reality 20, and 20 x 20 = 400, and it is because the cipher is not annexed to the quotient figure, that we are directed to multiply the square of the quotient by 300 instead of 3, as in the following work.
The rule directs to 6 multiply the quotient by 30." This is to obtain the contents of the blocks E. F. and G. and the sum of these is taken for a divisor, because the number of times the dividend contains the divisor, will be equal to the number of feet, the additions to the block, A. are in thickness.
The rule next directs to 6 multiply the triple square by the last quotient figure.” Now the triple square represents the superficial contents of the three blocks B. C. and D. and the last quotient figure shows the thickness of those blocks, consequently, multiplying the triple square by the last quotient figure gives the cubic contents of
those blocks. Then the rule directs to “ multiply the square of the last quotient figure by the triple quotient." Now the triple quotient is the length of the blocks E. F. and G. and the quotient figure shows their breadth, and their thickness; hence multiplying the square of the last quotient figure by the triple quotient, gives the cubic contents of these blocks. Next we are directed to cube the last quotient figure. This cube shows the cubic contents of the block H. and the sum of these is equal to the cubic contents of the blocks B. C. and D.; and E. F. and G.
Pup. This appears much plainer than I expected it would, before I heard your explanation. I can see no difficulty in finding the cube root of any number, if the explanation is well attended to.
Tut. It is not so difficult to extract the cube root as it appears to be to those unacquainted with the reasons of the rule. If you understand the nature of the rule you will never be troubled in performing any operations in it.
What is the cube root of 1092727 ?
What is the small figure, which denotes the order of the power, called ?
How do you find the power of any number?
Why is the given number pointed into periods of two figures each, for extracting the square root ?
Why do you subtract the square of the root from the period in which it was taken ?
Why do you double the root for a divisor ? In dividing, why do you except the right hand figure of the dividend ?
Why do you place the root in the quotient, and at the right of the divisor ?
How is a number pointed, when accompanied by decimals ?
How is the root of a vulgar fraction found?
What is it to extract the cube root?
found the first figure of the root, why do you subtract its cube from the period in which it was taken ?
Why do you multiply the square of the quotient by 300 ?
Why should the sum of the triple square and triple quotient be taken for a divisor?
Why do you multiply the triple square by the last quotient figure? the square of the last quotient figure by the triple quotient ? and cube the last quotient figure, and take their sum for a subtrahend ?
Examples for Practice.
1. What is the square root of 30138696025?
Ans. 173605. 2. What is the square root of 234,09 ?
Ans. 15,3. 3. What is the square root of ,045369 ? Ans. ,213. 4. What is the square root of 964,5192360241 ?
Ans. 31,05671. 5. A general has an army of 4096 men; how many must he place in rank and file to form them into a square ?
Ans. 64. 6. If the area of a triangle be 160, what is the side of a square equal in area thereto?
Ans. 12,6491. 7. There is a circle whose diameter is 4 inches ; what is the diameter of a circle 5 times as large ?
Ans. 8,94 inches.
To solve questions of this kind, square the given diameter; then multiply this square by the given proportion, and extract the root of the product, which will be the
When the circle of the required diameter is to be less than the circle of the diameter, the square of the given diameter must be divided by the proportion.
8. If the diameter of a circle be 20 feet, what is the diameter of a circle one third as large ? Ans. 11,54 ft.
9. The eaves of a house are 18 feet high; there is a wharfing in front of the house 12 feet wide; what is the length of a ladder that will just reach from the edge of the wharfing to the eaves of the house? Ans. 21,63 ft.
In every right angled triangle, like that formed by the house, the wharfing, and the ladder, the square of the bypotenuse, or slanting side, is equal to the sum of the squares of the other two sides.
10. The height of a tree growing in the centre of a circular island, 44 feet in diameter, is 75 feet, and a line stretched from the top of it over to the hither edge of the water, is 256 feet; what is the breadth of the stream, provided the land on each side of the water be level ?
Ans. 222,76 ft. 11. Two ships sail from the same port; one goes due north 45 leagues, and the other due west 76 leagues ; how far are they asunder?
Ans. 88,32 leagues. What is the cube root of 16194277 ? Ans. 253. 13. What is the cube root of 27054036008 ?
Ans. 3002. 14. What is the cube root of ,0001357 ?
Ans. ,05138 &c. *15. What is the cube root of ? Ans. ,873 &c.
16. If a cube of silver, whose side is 4 inches, be worth $150 ; I demand the side of a cube of the like silver, which shall contain 3 times as much ? Ans. 6,349 in.
17. If a globe of lead, 8. inches in diameter, weigh 3 cwt. ; what is the weight of another globe 4 times as large?
Ans. 24 cwt.
BARTER AND ALLIGATION.
Tut. What is bapter ?
Pup. Barter is exchanging one commodity for another in such a manner that neither party shall sustain loss.