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 hand 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. 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. 29 we find the root or length of one side of the square will be expressed by two figures. 529(20 4 129 529(23 4 129 129 Now the greatest square number in 5, the left hand period, is 4, and its root 2; putting 2 in the quotient, I 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 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 appears that 400 feet of the boards are disposed of in a square 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 form should continue square, it is necessary that the additions 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 20-40, 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. 1. An army of 567009 men are drawn up in a solid body in the form of a square; what is the number of men in rank and file? Aus. 753. 2. What is the length of the side of a square which shall contain an acre, or 160 rods? Ans. 12.649+rods. 3. 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. 4. The area of a triangle is 44944 feet; what is the length of the side of an equal square? Ans. 212 feet. 5. The diameter of a circle is 12 inches; what is the diameter of a circle 4 times as large ? Ans. 24. 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. 6. The diameter of a circle is 121 feet; what is the diameter of a circle one half as large ? Ans. 85.5+ feet. 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 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 by 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, viz: 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. 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. 1. What is Evolution? QUESTIONS. 2. What is meant by the root of any 3. How are roots denominated? 7. Has every number a root? 9. When is a power complete, and 10. What is the root of an incom- 12. How do you designate the peri ods ? 13. What is shown by the number 14. How are decimals prepared for 17. What the third? 23. 24. NOTES. 1. Why do you subtract the square from the period in which it is taken? 2. Why do you double the root for a divisor? 3. In dividing, why is the right hand When two sides of a right angled triangle are given, what is the rule for finding the other side? How do you find a mean proportional between two numbers? 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 2000; therefore 8000 of the given number of feet are piled into a 10648(20 8 2648 2X2X300=1200 10648(22 8 1260) 2648 1200x2=2400 60×2×2= 240 2×2×2= 8 2648 cubical beap, whose side is 20 feet, and there are 2648 feet to be added to the pile in such manner that it shall still retain its cubical form. In order to do this, it is evident that the additions must be made to 3 sides of the cube already formed. Here the rule directs to multiply the square of the last quotient figure by 300. This gives the superfices of the 3 sides to which the additions are to be made, as may be thus shown: 20 has been found to be the length of the several sides of the cube, 2020-400, the superfices of one side; this multiplied by 3 gives (400X3=) 1200 for the superfices of the 3 sides, the same as by the rule for squaring 2, (2><2=4). and multiplying it by 300 (4 X 300-1200) is 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 see 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 which the additions are made) by the last quotient figure. This gives (1200×2=) 2400 feet for the additions upon the 3 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 form of the several additious. |