PROBLEM V.-To find the area of a piece of land in the form of a triangle. RULE-Add together the three sides; from their half sum subtract each side and note the remainder; then multiply the half sum into all the remainders; and the square root of the last product will be the area. EXAMPLES. What is the area of a piece of land in the form of the adjacent figure, the length of each side being there given. 36+-40+30=106÷÷2 for What is the area of the inner field in the last figure, the length of each side being there given? A. 74,83 nearly. Note. Any irregular four sided piece of land may be measured, by dividing it into two triangles, by a diagonal line, and in the same manner can be measured pieces of land of more than 4 sides by dividing these into triangles. What is the difference between 5 miles square and 5 square miles? A. Five miles square means five miles in length, and 5 in breadth. 5 square miles means 5 miles in length and only one in breadth. five square miles. five miles square containing 25 square miles EXTRACTION OF THE CUBE ROOT. Questions explaining the nature of the extraction of the Cube Root. What is the 3d power of any number called? A. A Cube. What is it to extract the cube root? A. It is only to find what number used as a factor three times, will make the given sum or power; or it is that number which multiplied into its square, will produce the given number. What is the proof then? A. Multiply the answer or root into itself till it is taken as a factor 3 times. What is the cube root of 8? A. 2. Why, or proof? A. Because 2×2×2 is equal to 8. Why? A. Because 3×3×3—27. What is the cube root of 64? A. 4. What is the cube root of 1000? A. 10. Why, or proof? What is the cube root of 8000 ? A. 20. What is the proof? A. 20X20×20=8000. The solid contents of a square pile of wood are 216 feet; I demand the length and breadth of said pile? A. 6 feet. What is the proof? In Involution, you recollect that a boy got by his knowledge of powers a square box filled with marbles, containing 1000; now how many will reach across the bottom in a straight row, and how many from the top to the bottom in a straight line? What is the length of one side of a vessel, which contains 1,000,000 of solid feet? A. 100 feet. Why? A. Because 100X100X100=1,000,000. What is the difference between the cube of 27 and the cube root of 27? A. 19680. What is the RULE for extracting the cube root on the slate? A. Point off every third figure of the given cube, from the unit's place, into periods of three figures each, find the greatest cube in the left hand period, subtract it therefrom, and put the root in the quotient, and bring down the next period, to the right hand of the remainder for a dividend. Multiply the square of the root by 300 for a divisor. Find how many times this divisor is contained in the dividend, and put the answer in the quotient: then multiply the divisor by this quotient figure, writing the product under the dividend. Multiply all the figures in the quotient, except the last, by 30, and that product by the square of the last, and place the whole product under the preceding: then find the cube of the last figure in the quotient (or root) placing it under the last. Add these three last products together: and subtract the sum from the dividend. Bring down the next period to the remainder, and proceed as before, till all the periods are brought down. Square of 2. Exercises for the Slate. 12812904(234 Divisor 8cube of 2 X X300=*1200)4812 Dividend. 1200X3=3600 2X30X3X3= 540 27 4167 Subtrahend, 23×23X300=158700)645904 2d. dividend. 634800 23X30X4X4= 11040 4X4X4= 64 645904 Subtrahend taken from the 2d. Dividend leaves 0. *Note. The reason why 1200 should not be taken 4 times is because it would produce a result greater than the dividend, and this is what is to determine how many times any divisor is contained in the dividend. A. 83. A. 99. A. 126. A. 156. What is the cube root of 17576? A. 26. A. 908. The Application. A. 9009. What is the length of one side of a vessel, which contains 46656 solid inches? A. 36. In a cubical building, that measures 21952 feet, what is the length of a side? A. 28. The side of a cube being given, to find the side of that cube which shall be double, or treble, &c. in quantity to the given cube; what is the rule? A. Cube your given side, and multiply by 2, 3, &c. the cube root of the product is the side sought. To extract the biquadrate root, what is the rule? A. Find the square root of the giver sum, and then extract the square root of that root. Exercises for the Slate. What is the biquadrate root of 236421376 ? A. 124. What is the biquadrate root of 44000935696? A. 458. ALLIGATION MEDIAL. When do you employ this rule? A. When the quantities, and prices of several things, are given to find the mean price of the mixture compounded of those things. What is the RULE? A. As the sum of the quantities, or whole composition is to their whole value: so is any part of the composition to its mean price. Exercises for the Slate. If 37 gallons of rum at $2, be mixed with 24 gallons at $14 and 39 gallons at $23; what is the value of one gallon of the composition? A grocer would mix 10 cwt. of sugar, at $10 per cwt. with 4 cwt. at $4 per cwt. and 8 cwt. at $73 per cwt.; what will 5 cwt. of this mixture be worth? A. $40. A composition being made of 5 lb. of tea at $14 per lb. 9 lb. at $1,80 per lb. and 17 lb. at $1 per lb.; now what is a pound of it worth? A. $1,54,6,7%. If 20 bushels of wheat at $1,35 per bushel, be mixed with 15 bushels of rye at 85 cents per bushel; what will a bushel of this mixture be worth? A. $1,13 cts. 5-7 m. If 4 lb. of gold of 23 carats fine be melted with 2 lb. 17 carats fine; what will be the fineness of this mixture? A. 21 carats. ALLIGATION ALTERNATE. When do you employ this rule ? A. To find what quantity of any number of simples, whose rates are given, will compose a mixture of a given rate, so that it is the reverse of Alligation Medial, and may be proved by it. What is the RULE? A. 1st. Write the rates of the simples in a column under each other. 28. Connect or link with a continued line each price which is less than the compound, to one or more prices, greater than the compound: and each price that is greater, with one or more that is less. 3d. Write the difference between the prices of the compound and each of the simples, opposite the prices with which they are connected. 4th. Then if only one difference stand against any rate : it will be the quantity belonging to that rate: but if there be several their sum will be their quantity. Examples. Suppose you wish to make a mixture worth 15s. per gal. of 3 kinds of wine, one kind being worth 9s. another 12s. and another 20s. how much must you take of each kind? A. 5 gals. at 9s. i. e. 20—15—5. 5 gals. at 12s. i. e. 20-15=5. 66 9 gals. at 20s. i. e. 15--9-6 and Compound 9. 12 66 15 20 15-12-3 and 6=9. How much wine at 5s. per gallon, and 3s. per gal. must be mixed together, that the compound may be worth 4s. per gallon? A. An equal quantity of each sort. How much corn at 42 cts. 60 cts. 67 cts. and 78 cts. per |