ployed as factors; the cube of 1 is (l><l><l=) 1. two figares less than the number employed as factors, and so on. The least root consisting of two figures is 10, whose square is (10× 10-) 100, which has one figure less than the number of figures in the factors, and whose cube is (10 10 10) 1000, two figures less than the number in the factors; and the same may be shown of the least roots consisting of 3, 4, &c. figures. Again, the greatest root consisting of only one figure is 9, whose square is (99 =) 81, which has just the number of figures in the factors, and whosecube is (9×9×9) 729 just equal to the number of figures in the factors; and the greatest root consisting of two figures is 99, whose square is (9999=) 9301, &c. and the same may be shown of the greatest roots consisting of 3. 4, &c. figures. Hence it appears that the number of figures in the continued product of any number of factors cannot exceed the number of figres in those factors; nor fall short of the number of figures in the factors by the number of factors, wanting one. From this it is clear that a quare number, or the second power, can have but twice as many figures as its root, and only one less than twice as many; and that the third power can have only three times as many figures as its root, and only two less than three times as many, and so on for the higher powers; T'herefore, 265. To discover the number of figures of which any root will consist: RULE. Beginning at the right hand, distinguish the given number into portions, or periods, by dots, each portion consisting of as many figures as are denoted by the index of the root: by the number of dots will be shown the number of figures of which the root will consist. EXAMPLES. 1. How many figures in the square, cube, and biquadrate root of 348753421? 2. How many figures in the square and cube root of 68101 2.1416? 348758421 square root 5. 681012.141 6 square 5 348753421 cube root 3. 348752421 biquadrate 3. 681012.141600 cube 4 In distinguishing decimals begin at the separatrix and proceed towards the right hand, and if the last period is incomplete, complete it by annexing the requisite number of ciphers. EXTRACTION OF THE SQUARE ROOT. ANALYSIS. 266. To extract the square root of a given number is to find a num ber, which multiplied by itself, will produce the given number, or it is to find the length of the side of a square of which the given pumber expresses the area. 1. If 529 feet of boards be laid down in a square form, what will be the length of the sides of the square? or, in other words, what is the square root of 529 ? From what was shown [264] we know the root must consist of two figures, in as much as 529 consists of two periods. Now to understand the method of ascertaining these two figures, it may be well to consider how the square of a root consisting of two figures is formed. For this pur 23 23 9 square of units 529 square of 23. 529 (20 129 pose we will take the number 23 and square it. By this operation it appears that the square of a number consisting of tens and units is made up of the square of the units, plus twice the product of the tens, by the units plus the square of the tens. See this exhibited in figure F. As 10 10-100, the square of the tens can never make a part of the two right hand figures of the whole square. Hence the square of the tens is always contained in the second peri od, or in the 5 of the present example. The greatest square in 5 is 4, and its root 2; hence we conclude that the tens in the root are 2 20. and 20 20 400. But as the square of the tens can never contain significant figures below hundreds we need only write the square of the figure denoting tens under the second period. From what precedes it appears that 400 of the 529 feet of boards are now disposed of 20 ft. E 20 20 400 ft. 20 ft. in a square form, E. measuring 20 feet on each side, and that 129 feet are to be added to this square in such manner as not to alter its form; and in order to do this the additions must be made upon two sides of the square, E=204 20-40 feet. Now if 129, the number of feet to be added, be divided by 40, the length of the additions, or dropping the cipher and 9, 12 be divided by 4 the quotient will be the width of the additions; and as 4 in 12 is had 3 times. we conclude the addition will be 3 feet wide, and 403-120 feet, the quantity added upon the two sides. But since these additions are no longer than the sides of the square, A, there must be a deficiency at the corner, as exhibited in F, whose sides are equal to the width of the additions, or 3 feet, and 339 feet, required to co fill out the corner, so as to complete the square. The whole operation may be arranged as on the next page, where x will be seen that we first find the root of the greatest square in the left hand period, place it in the form of a quotient, subtract the square from the period and to the remainder bring down the next period, which we divide, omitting the right hand figure, by double the root, and place the quotient for the second figure of the root; and the square of this 23 ft. 20 ft. 204360 F 20 20-400 23 ft. 3 ft. 9 20 x 360. 20 ft. 529 (23 43 ] 129 129 figure being necessary to preserve the form of the square, by filling the corner, we place it at the right of the divisor, in place of the cipher which is always understood there, and then multiply the whole divisor by the last figure of the root. As we may conceive every root to be made up of tens and units, the above reasoning may be applied to any number whatever, and may be given in the following general 2323529 proof. RULE. 267. Distinguish the given numbers into periods; find the boot of the greatest square number in the left hand period, and place the root in the manner of a quotient in division, and this will be the highest figure in the root required. Subtract the square of the root already found from the left hand period, and to the remainder bring down the next period for a dividend. Double the root already found for a divisor; seek how many tines the divisor is contained in the dividend, (excepting the right hand figure) and place the result for the next figure in the root, and also on the right of the divisor. Multiply the divisor by the figure in the root last found; subtract the product from the dividend, and to the remainder bring down the next period for a new dividend. Double the root now found for a divisor, and proceed as before to find the next figure of the root, and so on, till all the periods are brought down. QUESTIONS FOR PRACTICE. 1. What is the square root of 529 ? 2. What is the square root of 2? Ans. 1.4142+. The decimals are found by an nexing pairs of ciphers continually to the remainder for a new dividend. In this way a surd root may be obtained to any assigned degree of exactness. 3. What is the square root of 182.25? Ans. 13.5. 4. What is the square root of .0003272481? Ans. .01809. Hence the root of a decimal is greater than its powers. 5. What is the square root of 5499025? Ans. 2345. 6. What is the square root of? Ans. .64549. Reduce to a decimal and then extract the root, (190). 7. What is the square root of †? Ans.. 8. What is the square root of $? Ans. 9. An army of 567009 men are drawn up in a solid body, in form of a square; what is the number of men in rank and file? Ans. 753. 10. What is the length of the side of a square, which shall contain an acre, or 160 rods? Ans. 12.6494-rods. 11. The area of a circle is 234.09 rods; what is the length of the side of a square of equal area? Ans. 15.3 rods. 12. The area of a triangle is 45244 feet; what is the length of the side of an equal square? Ans. 212 feet. 13. The diameter of a circle is 12 inches; what is the di ameter of a circle 4 times as Circles are to one another as the squares of their diameters; therefore square the given diameter, multiply or divide it by the given proportion, as the required diameter is to be greater or less than the given diameter, and the square root of the product, or quotient, will be the diameter required. 14. The diameter of a circle is 121 feet; what is the diameter of a circle one half as large? Ans. 85.5+feet. 268. Having two sides of a right angled triangle given to find the other side: RULE-Square the two given sides, and if they are the two aides which include the right angle, that is, the two shortest sides, add them together, and the square root of the sum will be the length of the longest side; if not, the two shortest; subtract the square of the less from that of the greater, and the aquare root of the remainder will be the length of the side required. (See demonstration, Part I. Art. 68.) 2. Suppose a man travel east 40 miles, (from A to C) and then turn and travel north 30 miles; (from C to B) how far is he from the place (A) where he started? Ans. 50 miles. 3. A ladder 48 feet long will just reach from the opposite side of a ditch, known to be 35 feet wide, to the top of a fort; what is the height of the fort? Ans. 32.8+ feet. 4. A ladder 40 feet long, with the foot planted in the same place, will just reach a window on one side of the street 33 feet from the ground, and one on the other side of the street 21 feet from the ground; what is the width of the street? Ans. 56.64+ feet. 5. A line 81 feet long, will exactly reach from the top of a fort, on the opposite bank of a river, known to be 69 feet broad; the height of the wall is required. Ans. 42.425 feet. 6. Two ships sail from the same port, one goes due east 150 miles, the other due north 252 miles; how far are they asunder? Ans. 293.26 miles. 269. To find a mean proportional between two numbers. RULE.--Multiply the two given numbers together, and the square root of the product will be the mean proportional sought. QUESTIONS FOR PRACTICE. 1. What is the mean propor- | portional between 49 and 64 ? tional between 4 and 36? 36-4=144 and v144-12 Ans Ans. 56. 3. What is the mean proportional between 16 and 64? Ans. 32. EXTRACTION OF THE CUBE ROOT. ANALYSIS. 270. To extract the cube root of a given number, is to find a number which multiplied by its square will produce the given number, or it is to find the length of the side of a cube of which the given number expresses the content. 1. I have 12167 solid feet of stone, which I wish to lay up in a cubical pile; what will be the length of the sides? or, in other words, what is the. cube root of 12167? By distinguishing 12167 into periods we find the root wil consist of two figures. (265) Since the cube of tens (264) can contain no significant figures less than thousands, the cube of the tens in the root must be found n the left hand period. The greatest cube in 12 is 8, whose root is 2, |