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PROOF. This work may now be proved by adding together all the square rods contained in the several parts of the figure, thus →→

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Point off the given number into periods of two figures each, by putting a dot over the units, another over the hundreds, and so on; and, if there are decimals, point them in the same manner, from units towards the right hand. These dots show the number of figures of which the root will consist.

Find the greatest square number in the left-hand period, and write its root as a quotient in division; subtract the square number from the left-hand period, and to the remainder bring down the next right-hand period for a dividend.

Double the root (quotient figure) already found, and place it at the left of the dividend for a divisor.

Write such a figure at the right hand of the divisor, also the same figure in the root, as, when multiplied into the divisor thrus increased, the product shall be equal to, or next less than the dividend. This quotient figure will be the second figure in the root.

Note.-The figure last described, at the right of the divisor, in the second operation, is the 6 rods, the width, which we add to 60, making 6C: or, omit ting the 0 in 60, and annexing 6, then multiplying 06 by 6, we wrete the 6 in the quotient, at the right of 3, making 36.

Multiply the whole increased divisor by the last quotient Egure, and write the product under the dividend.

Subtract this product from the dividend, and to the remainder bring down the next period, for a new dividend. Double the quotient figures, that is, the root already found, and continue the operation as before, till all the periods are brought down.

More Exercises for the Slate.

16. What is the square root of 65536?

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18. What is the square root of 470596? A. 686.
19. What is the square root of 1048576? A. 1024.
20. What is the square root of 2125764? A. 1458.
21. What is the square root of 6718454? A. 2592.
22. What is the square root of 23059204? A. 4802.
23. What is the square root of 4294967296? A: 65536.
24. What is the square root of 40?

In this example, we have a remainder, after obtaining one figure in the root In such cases, we may continue the operation to decimals, by annexing twe eiphers for a new period, and thus continue the operation to any assignable degree of exactness. But since the last figure, in every dividend thus formed, will always be a cipher, and as there is no figure under 10 whose square number ends in a cipher, there will, of course, be a remainder; consequently, the pupil need not expect, should he continue the operation to any extent, ever to obtain an exact root. This, however, is by no means necessary; for annexing only one or two periods of ciphers will obtain a root sufficiently exact for almost any purpose. A. 6,3245+.

25. What is the square root of 30? A. 5,4772.

96. What is the square root of

?

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Or, we may reduce the given fraction to its lowest terms before the root is extracted.

Thus,

·=√†= }}, Ans., as before.

£7. What is the square root of 450? . }}.

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98. What is the square root of 3? 4.4.

#9. What is the square root of 122 A TỐT.

If the fraction be a sund, the easiest method of proceeding will be to reduce to a decimal first, and extract its root afterwards.

30. What is the square root of ? A.,9128 +

81 What is the square root of? A.,95744

9

2. What is the square root of? A.,83205.

53. What is the square root of 420}?

In this example, it will be best to reduce the mixed number to an improper fraction, before extracting its root, after which it may be converted into a mixed number again. A. 20.

34. What is the square root of 912? A. 301.

35. A general has an army of 5625 men; how many must he place in rank" and file, to form them into a square? 5625=75, Ans.

36. A square pavement contains 24336 square stones of equal size; how many are contained in one of its sides? A. 156.

37 In a circle, whose area, or superficial contents, is 4096 feet, I demand what will be the length of one side of a square containing the same number of feet: A. 64 feet.

38. A gentleman has two valuable building spots, one containing 40 square rods, and the other 60, for which his neighbor offers him a square field, containing 4 times as many square rods as the building spots; how many rods in length must each side of this field measure? 40+ 60 × 4 = 20, Ans.

39. How many trees in each row of a square orchard, containing 14400 trees? A. 120 trees.

40. A certain square garden spot measures 4 rods on each side; what will be the length of one side of a garden containing 4 times as many square rods? A. 8 rods.

41. If one side of a square piece of land measure 5 rods, what will the side of one measure, which is four times as large? 16 times as large? 36 times as large A. 10. 20. 30.

42. A man is desirous of forming a tract of land, containing 140 acres, foods and 20 rods, into a square; what will be the length of each side?

43. The distance from Providence to Norwich, Conn., is computed to be 45 A. 150 rods. miles; now, allowing the road to be 4 rods wide, what will be the length of one side of a eruare lot of land, the square rods of which shall be equal to the equare rous contained in said road? A. 240 rods

EXTRACTION OF THE CUBE ROOT.

¶ LXXXVII. Q. Involution, (¶ LXXXIV.,) you doubtless recol leet, is the raising of powers; car. you tell me what is the 3d power of 3, and what the power is called?

A. 27, called a cube.

Q. Evolution (T LXXXVII.) was defined to be the extracting the 1st power roots of higher powers; can you tell me, then, what is the cube root of 27? A. 3.

Q. Why?

1. Because 3×3× 3, or, expressed thus, 33=27.

Q. What, then, is it to extract the cube root of any number?

A. It is only to find that number, which, being multiplied into itself three times, will produce the given number.

Q. We have seen, (¶ LXXX.,) that, to find the contents of solid bodies, such as wood, for instance, we multiply the length, breadth and depth to gether. These dimensions are called cubic, because, by being thus meitiplied, they do in ict contain so many solid feet, inches, &c., as are expressed by their product; but what do you suppose the shape of a solid body is, which is an exact cube?

A. It must have six equal sides, and each side must be án exact square. See block A, which accompanies this

work.

Q. Now, since the length, breadth and thickness of any regular cube are Exactly alike, us, for instance, a cubical block, which contains 27 cubic feet, can you inform me what is the length of one side of this block, and what the length may be called?

A. Each side is 3 feet, and may be called the cube roo of 27.

Q. Why? A. Because 33=27.

Q. What is the length of each side of a cubical block containing 64 cubin Laches? A. 4 inches.

Q. Why? A. Because 4 X 4 X 4, or 43
. What is the cube root of 64, then? A.
Why? A. Because 43 = 64.

4.

64 cubic inches.

What is the length of each side of a cubical block containing 1000 cuba fool? A. 10.

Q. Why? A. Because 103 = 1000.

1. In a square box which will contain 1000 marbles, how many will it take reach across the bottom of the box, in a straight row? A. 10.

2. What is the difference between the cube root of 27 and the cube of 3?

8. What is the difference between 8 and 23? A. 6.

4. What is the difference between 31 and 13? 4.0.

A. 24

5. What is the difference between the cube root of 27 and the square roc 19? A. 0.

What is the difference between 3/8 and 4? A 0.

Operation by Slate Illustrated.

7 A man, having a cubical block containing 13824 cubic feet, wishes to know the length of each side, without measuring it; what is the length of each side of said block?

Should we attempt to illustrate the season of the rule for extracting the cube root, by exhibiting the picture of the cube and its various parts on paper, it would tend rather to confuse than illustrate the subject. The best method of Coing it is, by making several small blocks, which may be supposed to contain a certain proportional number of feet, inches, &c., corresponding with the opBration of the rule. They may be made in a few minutes, from a small strip of a pine board, with a common penknife, at the longest, in less time than the teacher can make the pupil comprehend the reason, from merely seeing the picture on paper. In demonstrating the rule in this way, it will be an amusing and instructive exercise, to both teacher and pupil, and may be comprehended by any pupil, however young, who is so fortunate as to have progressed as far as this rule. It will give him distinct ideas respecting the different dimensions of square and cubic measures, and indelibly fix on his mind the reason of the rule, consequently the rule itself. But, for the convenience of teachers, blocks, lustrative of the operation of the foregoing example, will accompany this work.

The following are the supposed proportional dimensions of the several block osed in the demonstration of the above example, which, when put together, enght to make an exact cube, containing 13824 cubic feet:

One block, 20 feet long, 20 feet wide, and 20t thick; this we will call A. Three small blocks, each 20 feet long, 20 feet wide, and 4 feet thick; each of these we will call B.

Three smaller blocks, each 20 feet long, 4 feet wide, d 4 feet thick, each of these we will call C.

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One block, and the smallest, 4 feet long, 4 feet wide, and 4 feet thick; thi we will call D.

We are now prepared to solve the preceding example.

In this example, you recollect, we were to find the length of one side of the sube, containing 13824 cubic feet.

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Or,

64|

5824 deducted.
0000

In this example, we know that one side cannot be 30 feet, for 30327000 solid feet, being more than 13824, the given sum; therefore, we will take 20 for the length of one side of the cube.

Then, 20 X 20 X 20=8000 solid feet, which we must, of course, deduct from 13824, leaving 5824. (See Operation 1st.) These 8000 solid feet, the pupil will perceive, are the solid contents of the cubical block marked A. This cor

The same operation, by neglecting the ciphers, responds with the operation; may be performed thus:

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X2 X 300=1200) 5824, dividend

4800

2 x 30 x 4 x4 =

960

X4=

64

5824, subtrahend.
0000

for we write 20 feet, the length of the cube A, at the right of 13524, in the form of a quounder 13824; from which subtient; and its square, 8000, tracting 8000, leaves 5824, as before.

As we have 5824 cubic feet remaining, we find the sides of the cube A are not so long as they ought to be; consequently we must enlarge A; but in doing this, we must enlarge the three sides of A, in order that we may preserve the eubical form of the block. We will now place the three blocks, each of which is marked B, on the three sides of A. Each of these blocks, in order to fit, must be as long and as wide as A; and, by examining them, you will see that this is the case; that is, 20 feet long and 20 feet wide; then 20×20=400, the square contents in one B; and 3 × 400 = 1200, square contents in 3 Bs; then it is plain that 5824 solid contents, divided by 1200, the squire contents, will give the thickness of each block. But an easier method is, to square the 2, (tens,) in the root 20 making 4, and multiply the product, by 300, making 1200, a divisor, the same as before.

We do the same in the operation (which see); that is, we multiply the square of the quotient figure, 2, by 300, thus, 2 x 2 = 4 × 300 = 1200; then, the divisor, 1200 (the square contents) is contained in 5824 (solid contents) 4 times; that is, 4 feet is the thickness of each block marked B. This quotient figure, 4, we place at the right of 5824, and then, 1200 square feet X 4 feet, the thickness, 4800 solid feet.

If we now examine the block, thus increased by the addition of the 3 Bs, we shall see that there are yet three corners not filled up: these are represented by the three blocks, each marked C, and each of which, you will perceive, is as long as either of the Bs, that is, 20 feet, being the length of A, which is the 20 in the quotient. Their thickness and breadth are the same as the thickness of the Bs, which we found, by dividing, to be 4 feet, the list quo tient figure. Now, to get the solid contents of each of these Cs, we multiply their thickness (4 feet) by their breadth (4 feet), = 16 square feet; that is, the square of the last quotient figure, 4, 16; these 16 square contents must be

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