Give an example-Ans. 1728 is the third power or cube of 12 inches. A father gave his son a piece of land 30 rods long and 15 rods wide ; but told him if he would tell how many rods would make a square, one side of which was to be the length of what he had already given him, he should have the lot, what must his answer be? Ans. 900 rods, A man bought a square piece of land for a building spot, and by measurement one side was found to be 20 feet in length; how many rods did it contain? Ans. 400 feet. A piece of land is 60 rods square, how many rods does it contain? Ans. 3600. A Captain has in each rank and file 20 men, how many are there in the company. In your instructor's school-room, which is exactly square, 10 pupils may be accommodated on each side ; how many may be accommodated in the room. What are the solid contents of a pile of wood 6 feet in length, breadth and depth? Ans. 216 feet. What is the power to which 6 feet was raised, called? Ans. A cube. A boy, having purchased ten marbles, was told, that if he would tell how many it would take to fill a square box, so wide that ten marbles would lie in a row in the bottom, he should have the box filled with marbles; the boy got them, now, let us see if you would have gotten them by your answer, and how many did he get over and above what he had purchased for his mathematical knowledge. A farmer says to his son, our potatoes this year are all of a size, now (as you have cyphered considerably) if you will tell me how many bushels, allowing 200 to the bushel, our little square bin will hold, as you see that 20 lie along on the bottom in a straight line, you may go a fishing one half a day through the summer; the boy gave a correct answer, now I demand how many the bin held, and how many days the boy had to himself. John found a square piece of gold, which measured on each side 2 inches; and was offered 60 dollars for every solid inch it contained, how much was the lump worth at that rate? EVOLUTION Remark--As the design of this work is to embrace merely that portion of arithmetic, which is indispensible to the ordinary transactions of business; therefore the common rules, for the extraction of roots will be omitted. A few examples only will be given, sufficient to initiate the pupil into the principles of their operations. By taking on trial a number, which is supposed will nearly make the given number, when multiplied into itself continually; most sums which are of a practical cast may be generally answered in this way by one or two operations. What is Evolution? Ans. The opposite of Involution. Define it more particularly? Ans. It is finding what number multiplied into itself continually, as in Involution, will make the given sum or power. What is the number so found, called? Ans. The root. What are roots which cannot be perfectly evolved without leaving indefinite fractions, called? Ans. Irrational, or surd roots. What are those which can? Ans. Rational roots. What is a root? Ans. It is that number which by a continual multiplication into itself produces the given power. What is Evolution then? Ans. It is finding a root. What is Involution? Ans. It is finding a power, or a power of a root. EXTRACTION OF THE SQUARE ROOT. What does any number multiplied into itself produce? Ans. A square. What is it then to extract the square root? Ans. It is only to find what number, multiplied into itself will make the given number. What is the square root of 64? Ans. 8. Why? Ans. Because 8 times 8 are 64. What is the proof of the square root then? Ans. Multiply the answer or root into itself. What must the result be like? Ans. The given sum. What is the square root of 144, and the proof? What is the square root of 400 and the proof? What is the square root of 100 and the proof! A man desirous of making his kitchen garden, which is to contain 24 acres, or 400 rods, a complete square; what will be the length of the side? A square lot of land is to contain 22} acres,or 3600 rods of ground, but for the sake of fruit, there is to be a smaller square within the larger, which is to contain 225 rods; what is the length of each square? Ans. 60 rods the outer, 15 rods the inner. One hundred scholars are to be placed in a square roum, how many will that be on each side? Suppose 40 boys should collect together to perform some military evolutions, and should wish to march through the town in a solid phalanx, or square body, how many would the first rank consist of? A General has 400 men, how many must he place in rank and file, to form them into a square? A certain square pavement contaivs 1600 square stones, all of the same size; I demand how many are contained in one of its sides? Ans. 40. EXTRACTION OF THE CUBE ROOT. What is the third power of any number called? Ans. A cube. What is it to extract the cube root? Ans. It is only to find what number used as a factor three times will make the given sun or power; or it is that number which multip.ied into its square, will produce the given number. What is the proof then? Ans. Multiply the answer or root into itself till it is taken as a factor 3 times. What is the cube root of 8? Ans. 2. to 8. What is the cube root of 27? Ans. 3. The solid contents of a square pile of wood are 216 feet ; I demand the length and breadth of said pile? Ans. 6 feet. What is the proof? In Involution you recollect that a boy got by his knowledge of powers a square box filled with mar bles, containing 1000 ; now how many will reach gross the bottom in a straight row, and how many each from the top to the bottom in a straight line. You recollect also the sum in Involution about potatoes, the size, the bin, &c.- the bin held 40 bushels; now how many will reach from one side of row. the bottom to the other in a right line-or straight In Involution John found a piece of gold which brought 80 dollars, that is 10 dollars for each solid inch ; what may be the shape of the piece, and the length of each side. What is the length of one side of a vessel, which contains one million of solid feet? Ans. 100 feet. Why? Ans. Because 100 X 100 X 100=1,000,000. ALLIGATION. What is Alligation? Ans. The mixture of fluids and medicine. How many kinds does it consist of ? Ans. 3. What are they? Ans. Alligation Medial, Alternate and Partial. When do you employ Alligation Medial? Ans. When the quantities and prices of several things are, given to find the mean price of the mixture compound of those materials. In the rule, what make the first term? Ans The whole composition. What make the second? Ans. The whole value. What make the third? Ans. Any part of the composition. What will the fourth term be? Ans. The mean price. Note.--The remaining rules in Alligation as they cannot be abridged, or simplified, may as well be learned from the Arithmetic of which this is an accompaniment. Some useful rules for finding the contents of super ficies and solids. What is a straight line? Ans. The nearest distanen between two points. If two lines are at equal distances from one another in every part, what are they called? Ans. Parallel lines. |