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. RULE. Multiply the given numbers together, and extract the square root of the product; which root will be the mean proportional fought. What is the mean proportional between 24 and 96? 96X24-48 Auf. 1 PROBLEM II. To find the fide of a Square equal in Area to any given Superficies whatever. RULE. : 1 Find the Area, and the square root is the side of the square fought. EXAMPLES. If the area of a circle be 184,125, what is the dide of a square equal in area theretos છ?? છે 2 r ✓ 184,125-13,569+ Ans. 2. If the area of a triangle be 160; what is the fide of a square equal in area thereto? √160=12,649+ Ans. PROB 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 3 of 15625-75 Ans.* PROB. IV. Let 10952 men be so formed, as that the number in rank may be double the file. 10952=74 in file, and 74×2=148 in rank. 2 PROB. V. If it be required to place 2016 men fo 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 anfwer this, or any of the kind, use the following proportion-As unity to the distance :: fo is the num ber in rank lefs by one to a fourth number; next do the same by the file, and multiply the two numbers together, found by the above proportion, and the product will be the answer.t Ast:4::56-1: 220. And as 1 : 4 :: 36-1: 140. Then 220×140-30800 square feet, the Ans. PROB. VI. Suppose I would fet 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 93f you would have the number of men be double, triple, or quadruple, &c. as many in rank as in file; extract the square foot of&c. of the given number of men, and that will be the number of men in file, which double, triple, quadruple, s&c, and the product will be the number in rank. + The above rule will be found useful in planting trees, Chaving the distance of ground between each given. To refolve any question of this nature; fay, as the ratio in length: is to the ratio in breadth :: so is the num ber 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. : And ✓ 400=20=number And ✓ 900=30=numbe As 1 : 7:30-1: 203. And as 1:7:: 20-1 to 133. And 203X133=26999 square yards the Anfwer. PROB. VII. Admit a leaden pipe inch diameter will fill a ciftern in 3 hours; I demand the diameter of nother pipe, which will fill the same ciftern in 1 hour. RULE. As the given time is to the square of the give en diameter, so is the required time to the square of the required diameter. =,75; and,75,75=5625: Then, as 3h.:,5625 :: the: 1,6875 inversely, and 1,6875=1,3 inch nearly, Answer. PROB. VIII. If a pipe, whose diameter is 1,5 inch, fill a cistern in 5 hours; in what time will a pipe, whose diameter is 3,5 inches fill the fame ? 1 1,5X1,5=2,25; and 3,5×3,5=12,25: Then, a 2,25:5:: 12,25 : ,91 hour inversely, 54 minutes 39 feconds. Anf. 1 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 4 inches bore, be in difcharging double the quantity? 6×6=36, 4×4=16, and 16×3=48. Then, as 36:4h.:: 48: 3h. inversely, and as I w. : 3h. :: 2w.: 6h. Answer. alozs 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 rea quired circle be greater, multiply the square of the diam. eter by the given proportion, and the 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. There is a circle, whose diameter is 4 inches ; I demand the diameter of a circle 3 times as large? 4×4=16; and 16×3=48; and inches, Answer. 48=6,928+ PROB. XI. To find the diameter of a circle equal in area to an ellipfis (or oval) whose transverse and conjugute diameters are given.* RULE. Multiply the two diameters of the ellipfis together; and the Square Root of that product will be the diameter of a circle equal to the ellipfis. Let the transverse diameter of an ellipfis be 48, and the conjugate 36; What is the diameter of an equal circle? 48×36-1728, and ✓ 1728=41,569+the Anf. PROB. XII. Two ships fail from the fame port; one goes due north 45 leagues, and the other due west 76 leagues: How far are they asunder ? 45×45=2025. 76×76=5776. Then 5776X2025 =780, and, 7801=88,32 leagues, the Anf. EXTRACTION * The transverse and conjugate are the longest and shortefe diameters of an ellipsis; they pass through the centre, and cross each other at right angles, EXTRACTION of the CUBE ROOT. 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 faid period, and to the remainder bring down the next period, and call this the dividend. 4. Multiply the square of the quotient by 300, calling it the triple fsquare, and the quotient by 30, calling it the triple quotient, and the sum of these call the divifor. 5. Seek how often the divisor may be had in the dividend, and place the refult in the quotient. 6. Multiply the triple square by the last quotient figure, and write the product under this dividend; multiply the square of the last quotient figure by the triple quotient, and place this product under the last; under all, fet the cube of the last quotient figure, and call their fum the fubtrahend. 7. Subtract the fubtrahend from the dividend, and to the remainder bring down the next period for a new dividend, with which proceed as before, and fo on till the whole be finished. Note. The fame rule must be observed for continuing the operation and pointing for decimals, as in the square root. EXAMPLES. |