3d. Reduce the vulgar fraction to a decimal, and extract its root. 4th. Mixed numbers may either be reduced to improper fractions, and extracted by the first or second rule, or the vulgar fraction may be reduced to a decimal, then joined to the integer, and the root of the whole extracted. 31)81 Therefore, the root of the given fraction. 81 By Rule 2. 16x1681 26896, and 26896-164. Then, 1881-184-409756+ 1681)16(-0095181439+. And √.0095181439=·09756+. 2d. What is the square root of 2283? 3d. What is the square root of 421 ? Ans. Ans. 6. Note. In extracting the square or cube root of any surd number, there is always a remainder or fraction left, when the root is found. To find the value of which, the common method is, to annex pairs of cyphers to the resolvend, for the square, and ternaries of cyphers to that of the cube, which makes it tedious to discover the value of the remainder, especially in the cube, whereas this trouble might be saved if the true denominator could be discovered. As in division the divisor is always the denominator to its own fraction, so likewise it is in the square and cube, each of their divisors being the denominators to their own particular fractions or numerators. In the square the quotient is always doubled for a new divisor; therefore, when the work is completed, the root doubled is the true divisor or denominator to its own fraction; as, if the root be 12, the denominator will be 24, to be placed under the remainder, which vulgar fraction, or its equivalent decimal, must be annexed to the quotient or root, to complete it.* If to the remainder, either of the square or cube, cyphers be annexed, and divided by their respective denominators, the quotient will produce the decimals belonging to the root. Although these denominators give a small matter too much in the square root, and too little in the cube, yet they will be sufficient in common use, and are much more expeditions than the operation with cyphers. A a APPLICATION AND USE OF THE SQUARE ROOT. PROB. I. To find a mean proportional between two numbers. RULE. Multiply the given numbers together, and extract the square root of the product; which root will be the mean proportional sought. Note. When the first is to the second as the second is to the third, the second is called a mean proportional between the other two. Thus, 4 is a mean proportional between 2 and 8, for 2 : 4 :: 4×4 4 :: 2 Το =8, or 4 is as much greater than 2, as 8 is greater than 4. By Theorem I. of Geometrical Proportion 2×8=4×4=43. find a mean proportional between 2 and 8, take the square root of their product. The same must be true in every case, and is the rule. EXAMPLE. What is the mean proportional between 24 and 96 ? 96X24=49. Answer. PROB. 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.* EXAMPLES. 1st. If the area of a circle be 184-125, What is the side of a square equal in area thereto ? 184 125=13-569+ Answer. 2d. If the area of a triangle be 160, what is the side of a square equal in area thereto ? 16012-649+ Answer. PROB. 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 Answer.t PROB. IV. Let 10952 men be so formed, as that the number in rank may be double the file. 74 in file, and 148 in rank. PROB V. If it be required to place 2016 men so as that there may be 56 in rank and 36 in file, and to stand 4 feet distance in rank and as much in file, How much ground do they stand on? To answer this, or any of the kind, use the following proportion: As unity: the distance :: so is the number in rank less by one to a fourth number; next, do the same by the file, and mul 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 Prob. 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 4, 3, 4, &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, tiply the two numbers together, found by the above proportion, and the product will be the answer.* And, as 1 : 4 :: 36-1: 140. Then. As 1 4 56--1: 220. :: 220×140=30800 square feet, the Answer. PROE. VI. Suppose I would set out an orchard of 600 trees, so that the length shall be to the breadth as 3 to 2, and the distance of each tree, one from the other, 7 yards: How many trees must it be in length, and how many in breadth? and, How many square yards of ground do they stand on? To resolve any question of this nature, say, as the ratio in length is to the ratio in breadth :: so is the number of trees to a fourth number, whose square root is the number in breadth. And as the ratio in breadth is to the ratio in length :: so is the number of trees to a fourth, whose root is the number in length. : As 3 2: 600; 400, : And √400=20=number in breadth, As 2 3 600 900. And 900=30=number in length. As 17: 30-1: 203. And as 17 :: 20-1 to 133. And 203×133=26999 square yards, the Answer. PROB. VII. Admit a leaden pipe inch diameter will fill a cistern in 3 hours; I demand the diameter of another pipe which I will fill the same cistern in 1 hour. RULE. As the given time is to the square of the given diameter, so is the required time to the square of the required diameter † 375 and 75X75=5625. Then, as 3h.: ·5625 :: : 1h. 1-6875 inversely, and √1·6875-13, inches nearly, Ans. PROB VIII. If a pipe whose diameter is 1 5 inches, fill-a cistern in 5 hours, in what time will a pipe whose diameter is 3-5 inches fill the same? 1.5X1.5=2.25; and 3.5 × 3·5=12.25. Then, as 2.25; 5: 12:25 : 918+hour, inversely, 55 min. 5 sec. Answer. PROB. IX. 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 four inches bore, be in discharging double the quantity? 6×6=36. 4x4=16, and 16×3=48. Then, as 36: 4h. :: 48 ; 3h. inversely, and as 1w.: 3h. :: 2w. : 6h. Answer. PROB X. Given the diameter of a circle, to make another circle which shall be 2, 3, 4, &c. times greater or less than the given circle. RULE. Square the given diameter, and if the required circle be greater, multiply the square of the diameter by the given proportion, and the square root of the product will be the required diameter. But if the required circle be less, divide the square of the diameter by the given proportion, and the root of the quotient will be the diameter required. The above rule will be found useful in planting trees, having the distance of ground between each given. + For more water will run as the area of the pipe is greater, and the areas of circular pipes vary as the square of their diameters. There is a circle whose diameter is 4 inches; I demand the diameter of a circle 3 times as large? 4x4=16; and 16×3=48; and √48=6•928+ inches, Answer. PROB. XI. To find the diameter of a circle equal in area, to an ellipsis, (or oval) whose transverse and conjugate diameters are given.* KULE. Multiply the two diameters of the ellipsis together, and the square root of that product will be the diameter of a circle equal to the ellipsis. Let the transverse diameter of an ellipsis be 48, and the conjugate 36 What is the diameter of an equal circle? 48×36=1728, and 1728=41.569+the Answer. Note. The square of the hypothenuse, or the longest side of a right angled triangle, is equal to the sum of the squares of the other two sides; and consequently the difference of the squares of the hypothenuse and either of the other sides is the square of the remaining side. PROB. XII. 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? 36×36=1296; and 24×24=576. Then, 1296-576=720, and √720=20•33+yards, the Answer. PROB. XIII. 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 teet. What is the breadth of the stream, provided the land on each side of the water be level? 256x256 65536; and 75 75-5625: Then, 65536-5625= 59911 and 59911=244·76+ and 244·76—4=222·76 feet, Ans. PROB. XIV. 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? 102.64 feet the Answer. PROB. XV. Two ships sail from the same port; one goes due north 45 leagues, and the other due west 76 leagues: How far are they asunder ? 88-32 leagues, Answer. PROB. XVI. Given the sum of two numbers, and the difference of their squares, to find those numbers RULE. Divide the difference of their squares by the sum of the numbers, and the quotient will be their difference. The two num * The transverse and conjugate are the longest and shortest diameters of an ellipsis; they pass through the centre, and cross each other at right angles, and the diameter of the equal circle is the square root of the product of the diameters of the ellipsis. The square root may in the same manner be applied to navigation; and, when deprived of other means of solving problems of that nature, the following proportion will serve to find the course. As the sum of the hypothenuse (or distance) and half the greater leg (whether difference of latitude or departure) is to the less leg; so is 36, to the angle opposite the less leg. bers may then be found, from their sum and difference, by Prob. 4, page 57. Ex. The sum of two numbers is 32, and the difference of their squares is 256, what are the numbers ? Ans. The greater is 20. The less 12. PROB. XVII. Given the difference of two numbers, and the dif ference 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. Then pro Ex. The difference of two numbers is 20, and the difference of their squares is 2000; what are the numbers? Ans. 60 the greater. 40 the less. Examples for the two preceding problems. 1. A and B played at marbles, having 14 apiece at first; B having lost some, would play no longer, and the difference of the squares of the numbers which each then had, was 336; pray how many did B lose? Ans. B lost 6. 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. Ans. Harry's share 12. Jack's share EXTRACTION OF THE CUBE ROOT. 6. A cube is any number multiplied by its square. To extract the cube root, is to find a number which, being multiplied into its square, shall produce the given number. FIRST METHOD. RULE. *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 divi · dend. 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. * The reason of pointing the given number, as directed in the rule, is obvious from Coroll. 2, to the Lemma made use of in demonstrating the square root. 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 |