and place it on the right hand of the given number, in the manner of the quotient in division, and it will be the first figure of the root required. 2. Subtract the square of the root already found, from the left hani period, and to the remainder bring down the next period for a dividend. 3. Double the root already found, for a divisor; seek how often the divisor is contained in the dividend, (excepting the right hand figure) and place the answer for the second figure of the root, and also on the right hand 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. 4. Find a divisor as before, by doubling the figures in the root, and proceed as before to find the third figure in the root, and so on through all the periods. PROOF.--Multiply the root by itself; add the remainder, if any, and if it be right, the sum will equal the given number. Examples. 1. What is the square root of 3. What is the square root of 529? 2 ? Ans. 1.41427. 529(23 root. The decimals are found by annex4 ing pairs of ciphers continually to the remainder for a new dividend. 43)129 In this way, a surd root may be ob129 tained to any assigned degree of exactness. length of one side of this square, for that will be the number, which, multiplied into itself, will produce the given number of feet. Now by distinguishing 529 into periods, s 29 we find the root or length of one side of the square will be expressed by two figures. Now the greatest square number in 5, the left hand period, is 529(20 4, and its root 2; putting 2 in the quotient, 1 subtract 4 from the left hand period, and to 1, the remainder, bring down the next period, making the sum 129. Here it is plain that 2 is in 129 the place of tens, because the root is to consist of two figures : its true value is therefore 20, and its square 400. Thence it 529(23 appears that 400 feet of the boards are disposed of in a square 4 form, measuring 20 feet on each side, and that there are 129 feet remaining to be added to the square, and in order that the 129 form should continue equare, it is necessary that the additions 129 should be made upon two sides. Now the length of the two sides, to which the additions are to be made, is found by doubling 2040, and dividing 129, the number of feet to be added, by 40, the length to which the addition is made, evidently gives the breadth of the addition. But if the length of the additions be only equal to the length of the sides to which they are made, there will be a deficiency at the corner of a small square, one side of which will be just equal to the width of the additions ; the length upon which the addition is made, should therefore be increased by the breadth of the addition, and this is done by placing 3 on the right hand of the divisor. By this means, additions are made to the two sides 3 feet wide, and the corner filled up by a little square, measuring 3 feet on each side, which disposes of all the boards, and leaves them in the form of a complete square, 23 feet on each side. 3. What is the square root of 6. What is the square root of 185.25 ? Ans. 13.5. Ans. .64549. Reduce to a decimal, and 4. What is the square root of then extraci the root. .000327 2481 ? Ans. .01809. 7. What is the square root of Ans. ** 5. What is the square root of 8. What is the square root of 5499025 ? Ans. 2345. 144 Ans. 11. Application. 1. An arıny of 567009 men are 5. The diameter of a circle is odrawn up in a solid body in the 12 inches; what is the diameter form of a square ; what is the of a circle 4 times as large ? number of mer in rank and file ? Aos. 24. Aus. 753. Circles are to one another as the 2. What is the length of the squares of their diameters; therefore, side of a square which shall con square the given diameter, multiply tain an acre, or 160 rods? or divide it by the given proportion, Ans. 12.649 +roils. as the required diameter is to be 3. The area of a circle is 234.09 greater or less than the given diamrous ; what is the length of the eter, and the square root of the proside of a square of equal area ? duct or quotient will be the diameAns. 15.3 rods. ter required. 4. The area of a triangle is 6. The diameter of a circle is 44944 feet ; what is the length of | 121 feet ; what is the diameter of the side of an equal square ? a circle one half as large ? Ans. 212 feet. Ans. 85.5-4 feet. Having two sides of a right angled triangle given to find the other side. Rulet--Square the two given 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, subtract the square of the less from that of the greater, and the square root of the remainder will be the length of the side required. *When the terms of the fraction are complete powers, extract the root of the numerator for the numerator of the root, and the root of the denominator for the denominator of the root. +A right angle is an angle that is formed hy a line falling perpendicularly upon another line, as the angle C in the triangle A B C, and a right angled triangle is a triangle, which has one such angle. The rule is founded on the celebrated proposition of Pythagoras, which is the 47th proposition in the 1st Book of Euclid, riz: that the square formed on the line subtending, or opposite to the right angle, in a right angled triangle, is equal to the sum of the squares formed in both the other sides ; that is, the square formed in the line A B is equal to the sum of the squares formed on the sides A C and C B, which may be demonstrated to be true in all cases. B Examples. 1. In the right angled triangle 3. A ladder 48 feet long wil A B C, the side A C is 36 inches, just reach from the opposite side and the side B C 27 inches; what of a ditch, known to be 35 feet is the length of the side A B? wide, to the top of a fort; what 36 27 is the height of the fort ? Ans. 32.8+ feét. 216 189 4. A ladder 40 feet long, with 108 54 the foot planted in the same place, will just reach a window 1296 729 on one side of the street 33 f feet from the ground, and one on the 1296 square of A C=36. other side of the street 21 feet 729 square of B C=27. from the ground; what is the width of the street ? 2025 sum. Ans. 56.66 + feet. 2025/45 in. Ans. 16 5. A line 81 feet long, will exactly reach from the top of a 85)425 fort, on the opposite bank of a 425 river, known to be 69 feet broad; the height of the wall is required? Ans. 42.425 feet. 2. Suppose a man travel east 40 miles, (from A to C) and then 6. Two ships sail from the turn and travel north 30 miles ; ! same port, one goes due east 150 (from C to B) how far is he from | miles; the other due due north the place (A) where he started ? 252 miles; how far are they asun Ans. 50 miles. der ? Ans. 293.25 miles. ? 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. Examples. 1. What is the mean propor 2. What is the mean proportional between 4 and 36? tional between 49 and 64 ? Ans. 56. 36 144(12 Ans. 1 3. What is the mean propor144 22)44 tional between 16 and 64? 44 Ans. 32. Then 4 : 12 : : 12 : 36 QUESTIONS. 1. What is Evolution ? 13. What is shown by the number 2. What is meant by the root of any of periods ? power? 14. How are decimals prepared for 3. How are roots denominated ? extracting their root? 4. How is the square root denoted ? 15. What is the first step in the rule 5. How are other roots denoted ? for extracting the square root ? 6. Is there any other way of denot- 16. What is the second ? ing roots ? 17. What the third ? 7. Has every number a root? 18. What the fourth ? 8. Can the complete root of all num 19. What is the method of proof? bers be ascertained ? 20. How do you extract the root of 9. When is a power complete, and a Vulgar Fraction? when incomplete ? 21. What is a square? 10. What is the root of an incom- 22. What proportion have circles to plete power called ? another? 11. How do you prepare any num 23. When two sides of a right angled ber or power for extracting its triangle are given, what is the root? rule for finding the other side ? 12. How do you designate the peri-24. How do you find a mean proporods ? tional between two numbers ? NOTES. 1. Why do you subtract the square figure of the dividend excepted ? from the period in which it is taken? 4. Why do you place the quotient figure 2. Why do you double the root for a in the divisor as well as in the root ? divisor 5. What is the 47th proposition in Eu3. In dividing, why is the right hand clid, which is referred to ? TO EXTRACT THE CUBE ROOT. The cube root of a number is a number which multiplied into its square, will produce that number. A cube is a solid body comprehended under six equal sides, each of which is an exact square, and its root is the length of one of the sides. Rule.*--1. Having distinguished the given number into periods of three figures each, find the greatest cube in the left hand period, and place its root in the quotient. 2. Subtract the cube from the left hand period, and bring down the next period for a dividend. 3. 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. * The reason of the rule will appear by a consideration of the first example. Having distinguished the given number into periods, we find that the root will consist of two figures. Now if we suppose the given number 10648 to be so many solid feet of wood, which are to be piled into a cubical heap, the two figures of which the root is to consist will express the length of one side of that heap. By trial we find 8, whose root is 2, the greatest cube in the left hand period; we therefore place 2 for the first figure of the root, and subtract 8 from the left hand period. But as 2 is in the place of tens, its value is 20, and its 4. Seek how often the divisor may be had in the dividend, and place the result in the quotient. 5. Multiply the triple square by the last quotient figure, and write the product under the dividend ; multiply the triple quotient by the square of the last quotient figure, and place this product under the last ; under these write the cube of the last quotient figure, and call their sum the subtrahend.. cube 8 is 8000 ; therefore 8000 of the given number of feet are piled into a cubical beap, whose side is 20 feet, and there 10648(20 are 2648 feet to be added to the pile in such 8 manner that it shall still retain its cubical form. In order to do this, it is evident that 2648 the additions must be made to 3 sides of the cube already formed. Here the rule directs 2x2x300=1200 10648(22 to multiply the square of the last quotient figure by 300. This gives the superfices of 2x30= 60 8 the 3 sides to which the additions are to be made, as may be thus shown : 20 has been 1260 ) 2648 found to be the length of the several sides of the cube, 20 * 20=400, the superfices of one 1200x2=2400 side; this multiplied by 3 gives (400 X3=) 60x2x2= 240 1200 for the superfices of the 3 sides, the 2x2x2= 8 same as by the rule for squaring 2. (2x2=4). and multiplying it by 300 (4 X 300=1200) is. 2648 the same as squaring 20, and multiplying it by 3. Again, the rule directs to multiply the quotient figure by 30. Now it is evident that there will be three deficiencies between the additions which are made upon the 3 sides, of the length of those additions; that is, 3 deficiencies, each 20 feet long; or in the whole, (20*3=) 60 feet; but because the cipher is omitted in the quotient by the rule, and the 2 only used, we must annex the cipher to 3, the number of deficiencies, and multiply the 2 by 30 for the length of the deficiencies. These two, 1200 and 60=1260, show the points upon the cube to which the additions are to be made. The 2648 feet being divided by this, shows the thickness of the additions to be made, which is 2 feet, therefore 2 is the other figure of the root. Now to gee what timber is used in making these additions, we are directed first to multiply the triple square (1200, which is the superfices of the 3 sides to wbich the addi. tions are made) by the last quotient figure. This gives (1200x2=) 2400 feet for the additions upon the sides. Then to find how much it takes to fill up the deficiencies between the additions upon the sides, we are directed to multiply. the triple quotient (60, the length of the deficiencies) by the square of the last quotient figure. This gives (60>4=) 240 feet, employed in filling the deficiencies between the other additions. The reason for multiplying the triple quotient by the square of the last quotient figure, is that two of the dimensions of this addition are just equal to the thickness of the additions upon the sides. But after these additions there is still evidently a deficiency at the corner, between the ends of the last additions, the 3 dimensions of which are just equal to the thickness of the other additions, and to fill this, we are therefore directed to cube the last quotient figure, (2*2*2=8.) Then the quantities employed in these additions are 2400 feet, 240 feet, and 8 feet, which, added together, give 2648 feet, a sum just equal to the dividend, which shows that the cube is complete, measuring 22 feet on each side, and that all the 10648 feet of timber is used. The steps in this rule may be very clearly illustrated by the help of a cubical block, with other small blocks in the for of several add |