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Now the first figure of the root shows the number of pieces there are on a side of Fig. 1, viz. 10. In order to preserve the square form, the additions must be made on two adjoining sides of the square, as in Fig. 2. Now it is evident that if there were just 20 pieces left, after disposing of 100, there would be just enough to make a row on two sides of Fig. 1, and if there were 40 pieces left, they would make two rows, on two sides, as represented by the rows a, e, and o, n, Fig. 2. Hence the reason of placing the double of the root on the left of the dividend for a divisor. In making the additions a, e, and
o, n, you will observe there is a deficiency, A. which is not filled.— To fill this deficiency, the rule directs to " except the right hand figure," and likewise to "place the quotient figure on the right hand of the divisor." Now the deficiency A. must be limited by the additions a, e, and o, n, consequently the figure, expressing the width of these additions, expresses the root of this deficiency, which, multiplied into itself, gives the superficial contents of the deficiency. Thus Fig. 2 shows the disposition of 144 pieces into a square form.
2. Required the square root of 575.5 ?
Rules for extracting the Square Root of Vulgar Fractions and Mixed Numbers.
1. Reduce the fraction to its lowest terms for this and all other roots.
2. Extract the root of the numerator for a new numerator, and the root of the denominator for a new denominator.
3. Or reduce the vulgar fraction to a decimal, and extract its root.
4. Mixed numbers may be reduced to improper fractions, and the root of the numerator and denominator extracted, or the vulgar fraction may be reduced to a decimal, and annexed to the whole number, and the root of the whole extracted.
Or, 1681)16 (0095181439+. AndV-00951314390975€.
291. What is the rule for extracting the Square Root of Vulgar Fractions and mixed numbers?
APPLICATION AND USE OF THE SQUARE ROOT.
PROBLEM I. To find a mean proportional between two numbers. RULE.-Multiply one of the given numbers by the other, and extract the square root of the product, and the root will be the mean proportional required.
NOTE. When the first number is as many times greater than the second, as the second is times greater than the third, the second number is called a mean proportional between the other two.
1. What is the mean proportional between 36 and 144 ?
PROBLEM II. To find the side of a square equal in area to any given superficies whatever.
RULE. Find the area, and the square root is the side of the square sought.*
1. If the area of a triangle be 160, what is the side of a square equal in area thereto ?
160 12.649+ Ans.
PROBLEM III. A certain general has an army of 5625 men: pray, how many must he place in rank and file, to form them into a square ? 5625 75 Ans.†
PROBLEM IV. If a pipe 6 inches bore, will be 4 hours in running off a certain quantity of water, in what time will 3 pipes, each 4 inches bore, be in discharging double the quantity?
6×6=36. 4×4=16, and 16×3=48. inversely, and as 1w.: 3h.:: 2w.: 6h.
Then, as 36 : 4h. :: 48: 3h.
* A square is a figure of four equal sides, each pair meeting perpendicularly, or, a figure whose length and breadth are equal. As the area, or number of square feet, inches, &c. in a square, is equal to the product of two sides which are equal, the second power is called the square. Hence the rule of PROBLEM II. is evident.
If you would have the number of men be double, triple, or quadruple, &c. as many in rank as in file, extract the square root of ... &c. of the given number of men, and that will be the number of men in file, which double, triple, quadruple, &c. and the product will be the number in rank.
292. How do you find a mean proportional between two numbers?
PROBLEM V. A line 36 yards long will exactly reach from the top of a fort to the opposite bank of a river, known to be 24 yards broad. The height of the wall is required?
36X36=1296 ; and 24×24576. 720 26.83+yards, the Answer.
Then, 1296-576-720, and
PROBLEM VI. 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?
256X256=65536 and 75X75-5625: Then, 65536-5625-59911 and/59911 244-76+and 244-76-44-222-76 feet; Answer.
PROBLEM VII. Suppose a ladder 60 feet long be so planted as to reach a window 37 feet from the ground, on one side of the street, and without moving it at the foot, will reach a window 23 feet high on the other side; I demand the breadth of the street?
Ans. 102-64 feet.
PROBLEM VIII. Given the difference of two numbers, and the difference of their squares, to find the numbers.
RULE. Divide the difference of the squares by the difference of the numbers, and the quotient will be their sum. ceed by Prob. 4, p. 57.
1. The difference of two numbers is 20, and the difference of their squares is 2000; what are the numbers ?
2. Said Harry to Charles, my father gave me 12 apples more than he gave brother Jack, and the difference of the squares of our separate parcels was 288: Now, tell me how many he gave us, and you shall have half of mine.
EXTRACTION OF THE CUBE ROOT.
A CUBE is any number muliplied by its square. To extract the cube root, is to find a number which being multiplied into its square, shall produce the given number.
1. Separate the given number into periods of three figures each, by putting a point over the unit figure and every third figure beyond the place of units.
2. Find the greatest cube in the left hand period, and put its root in the quotient.
3. Subtract the cube thus found, from the said period, and to the remainder bring down the next period, and call this the dividend.
4. Multiply the square of the quotient by 300, calling it the triple square, and the quotient by 30, calling it the triple quotient, and the sum of these call the divisor.
5. Seek how often the divisor may be had in the dividend, and place the result in the quotient.
6. Multiply the triple square by the last quotient figure, and write the product under this dividend; multiply the square of the last quotient figure by the triple quotient, and place this product under the last; under all, set the cube of the last quotient figure and call their sum the subtrahend.
7. Subtract the subtrahend from the dividend, and to the remainder bring down the next period for a new dividend, with which proceed as before, and so on till the whole be finished.*
*The process for extracting the cube root may be illustrated in the same manner as that for the square root. Take the same number 37, and multiply as before, collecting the twice 21 into one sum, as they belong to the same place, and the operation will be simplified, 373-50653.
49 72 42.2X3X7 9.32
37 the multiplier.
293. What is a Cube?294. What is the method of extracting the cube root of a given number ?—295. I wish you to illustrate the process under this rule, by one of the examples given.