205, 266. SQUARE koor. 109 employed as factors; the cube of 1 is (1X1X13) 1, two figures less than the number employed as factors, and so on. The least root consisting of two figures is 10, whosc square is (10X10_) 100, which has one figure less than the number of figures in the factors, and whose cube is (10x fox 105) 1000, two figures less than the number in the factors; and the same may be shown of the least rools consisting of 3, 4, &c. figures: Again, the greatest root consisting of only one figure, is 9, whose square is (9X9=) 81, which has just the number of figures in the factors, and whose cube is (9X9 X95) 720, just equal to the number of figures in the factors, and the greatest root consisting of two figures, is 99, whose square is (99X995) 9801, &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 figures in those factors; nor full short of the number of figures in the factors by the number of fuctors, wanting one. From this, it is clear that a square 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 ihree times as many, and so on for the higher powers. Therefore, 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 2. How many figures in the square, cube, and biquadrate square and cube root of 68101 root of 348753421 ? 2.1416 ? 348753 4 2 i square root 5. 6810121416 square 5. 34875342 i cube root 3. 681012141600 cube 4: 34875242 i biquadrate 3. 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. ANALYSIB 266. To extract the square root of a given number is to ind a sumber, which, multiplied by itself, will produce the given number, or it u to find the lengih of the side of a square of which we given number expressed 10 the area. 1. If 629 feet of boerus 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 rool of 529 ? From what was shown (264), wc kuow the root must consist of two fig. ures, 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 Ibe square of a root consisting of "two figures is formed. For this pur pose we will take the number 23, and square it. By this operation, it appears that the square of a number consisting of lens and unils is made up of the 9 square of units. square of the units, plus twice the pro60 / iwice the product of duct of the tens, by the units, plus tho 60 / the tens by units. square of the tens. See this exhibited 400 square of the lens, in figure F. As 10x10=100, the square of the tens can never make a part of the 529 square of 23. two right hand figures of the whole square. Hence the square of the tens is always contained in the second peri5 29 (20 od, or in the 5 of the present example. 4 00 The greatest square in 5 is 4, and its root %; hence, we conclude, that the 1 29 tens in the root are 2–20, and 20X20 400. Lut as the square of the tens can never contaiu significant figures below hundreds, we need only write the square of the figure denoting tens under second period. From what precedes it appears that 400 of the 529 see of boards are now disposed of in a square for E measuring 20 feet on each 20 side, and that 1 feet are to be added to this square in such a uner as not to alter its form: uid in order to > this, the additions must bo ade des of the square, E_204 -10 feel. N if 129, the number of feet to 20 bedded, be divided by 40, the length of the additions, or, dropping the cipher and ġ, if 12 be divided by 4, the quotient will be the width of. 400 A. the additions; and as 4 in 12 is had 3 times, we conclude the addition will be 3 feet wide, and 40X3=120 feet, the quantity added upon the two sides. But since these additions are 20 honger than the sides of the square, E, there must be a deficiency at the corner, as exhibited in F', whose sides 20 nt. 3 ft. are equal to the width of the additions, 20X3=10. 19 or 3 feel, and 3x339 feel, required to fill out ihe corner, so as to complete the square. The whole operation may be F arranged as on the next page, where it will be seen, that we first' find the roof of the greatest square in the left hand 20X0-400 upon two 20 ft. 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 ebe rool and place the quotient for the second figure of the runt ; and the square of this 23 A. 'VOZ 120X360. rool. 329 ( 23 figure being necessary to preserve the + forin of the square, by filling the corner, we place it ai the right of the divisor, in 43 ] 129 place of the cipher, which is always un. 129 derstood there, and thien multiply the whole divisor by the last figure of the 23X2355% proof. 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 RULE. 207. Distinguish the given numbers into periods ; find the root 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 times 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 aivisor by the figure in the root last found; subtract the proauct 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 rooh and so on, till all the periods are brought down. ? Ans. f. QUESTIONS FOR PRACTICE. 1. What is the square root 6. What is the square root of 529? of 5 2. What is the square root Ans. .64549+ of 2 ? Ans. 1,4:12+. Reduce it to a decimal and The decimals are found by elle then extract the root (130). he xing pairs of ciphers continually to the remaindler for a new dividend, 7. What is the square root In this way a surd root may be ohsained to any assigned degree of of s*? 3. What is the squarc root 8. What is the square root of of 182.25 ? Ans. 13.5. 194? 4. What is the square root 9. An army of 567009 men of .0003272481 ? are drawn up in a solid body, Ans. .01809. in form of a square; what is Hence the root of a decimal is the number of men in rank greater than its powers. Ans. 753. 5. What is the square mot of 54990257 Ans, 2;345. 10. What is the length of exactness. Ans. and file ? 1 aquare the side of a square, which / ameter of a circle 4 times as shall contain an acre, or 160 large ? Ans. 24. rods ? Ans. 12,619+ rods, Circles are in one another as the 11. The area of a circle is squares of their diameter ; therefore 234.09 rods; what is the length square the given diameters, multiply of the side of a square of or divide it by the given proportion, equal area? as the required diameters iš ļo be greater or less than the given diam. Ans. 15.3 rods, eter, and the square root of the pro. 12. The area of a triangle duci, or groticit, will be the diam. eter required ! is 44944 feet; what is the length of the side of an equal 14. The diameter of a circle Ans. 212 feet. is 121 feet; what is the diam13. The diameteș of a circle i eter of a circle une half as is 12 inches; what is the di- large ? Ans. 85,5+ feet 268. Having two sides of a right angled triangle given to find the other side, RULE. -Square the two giyen sides, and if they are the two sides 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 ; sub, tract the square of the less from that of the greater, and the square root of the remainder will be the length of the side re, quired. (See demonstration, Part I. Art. 68.) QUESTIONS FOR PRACTICE. 1. In the right angled tri- If A B be 45 inches, and angle, A B C, the side A C is A C 36 inches, what is the 36 inches, and the side B C, length of BC? 27 inches ; what is the length A B2=45X45=2025 pf the side A B? A C2=36X36=1296 B B C2# 729 BC=1729=27 in. Ans, If A B345, (=27in,, what is the length of A C? A B2=452, B C2=272, A base c2 and A C=71296.3in A C2=36X3681296 Ans. perpendicular. bypotbeause. A B2= 2025 AB=\/2025=45 in, Ans, $69, 270. CUBE ROOT. 113 2. Suppose a man travel east and one on the other side of 40 miles (from A to C), and the street, 21 feet from the then turn and travel north 30 | ground; what is the width of miles (from C to B); how far the strect? is he from the place (A) where Ans. 56.64 + feet. he started ? Ans. 50 miles. 5. A line 81 feet long, will 3. A ladder 48 feet long exactly reach from the top of will just reach from the oppo- a fort, on the opposite bank of site side of a ditch, known to a river, known to be 69 feet be 35 feet wide, to the top of broad; the height of the wall a fort; what is the height of is required. the fort? Ans. 32.8+ feet. Ans. 42.42 6 feet. 4. A ladder 40 feet long, 6. Two ships sail from the with the foot planted in the same port, one goes due east same place, will just reach a 150 miles, the other due north window on one side of the 252 miles; how far are they street 33 feet from the ground, I asunder? Ans. 293.26 miles. 269. To find a mean proportional between two numbers. RuĻE.—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 2. What is the mean pro tional between 4 and 36 ? portional between 49 and 64 ? 36X4=144 and v144=12 Ans. 56. Ans. 3. What is the mean proThen 4 : 12 : : 12 : 36. portional 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 and 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 cubica! pile; what will be the length of the sides ? or, in other words, what is tie cube root of 12167 1 By distinguishing 12167 into periods, we find the root will consist of iwo 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 in wbie left hand period. The greatest cube in 12 is 8, whose root is %, |