2. A merchant would mix wines at 114s. 15s. 195. and 22s. per gallon, so that the mixture may be worth 18s. per gallon; how much must he take of each sort ? 4 at 14s. 5 at 14s. 1 at 15s. I at 15s. 1 Ans. 2 Ans. 3 at 19s. 7 at 19s. 4 at 22s. 4 at 22s. &c. { 1.25_100 Rule 2. When the quantity of the whole composition is limited to a certain sum ; find the differences by linking as before; then say, as the sum of the quantities or differences is to the given quantity, so is each of the differences to the required quantity of each rate. Examples. 1. How much water at Octs. 3. A grocer would mix teas at per gallon, must be mixed with 3s. 4s. and 4s. 6d. per pound, and brandy at $1.25 per gallon, so as would have 30 lb. of the mixture to fill a vessel of 80 gallons, and I worth 3s. 6d. per lb. how much that a gallon of the mixture inay 1 of each must he take? be worth $1? lb. 25 SO100 18 at 3s. Aps. 6 at 4s. 6 at 4s. 6d. s 25 : 16 water. 4. How many gallons of water worth Os. per gallon, must be 80 given mixed with wine worth 3s. per quantity gallon, so as to fill a cask of i00 2. How much silver of 15, of gallons, and that a gallon of the 17, of 18, and 22 carats fine, must inixture may be afforded at 2s. be melted together to form a 16d. ? composition of 40 oz. 20 carats gall. fine ? Ans. 16 . 5 of 15 831 5 of 17 car. fine. 24, and the same of 30 and 36. Now 14 bushels at 24d. is 336d, and 14 bushels at 34ů. the mean rate, is 476d. and 476-336=140d. so that there is here a loss of 140d. And again, 10 bushels of rye.at 48d. is 480d. and 10 bushels at 34d. is 340d. and 480—340=140d. ; here there is a gain of 140d. precisely the sum that was lost by the other, so that the balance is preserved, and the same is true of the 30 and 36, or of any two numbers connected in this way, a greater with a less than the mean. Questions in this rule will admit of as many answers as there are different ways of linking the rates of the ingredients together ; and after as many answers are found by linking the rates as can be, more answers may be formed by multiplying or dividing these by 2, 3, 4, &c. Rule 3. When one of the ingredients is limited to a certain quantity ; find the differences as before; then as the difference standing against the given quantity is to the given quantity, so are the other differences severally, to the several quantities required. Examples. 1. A grocer would inix teas at 2. How much wine at 5s. at 12s. 10s. and 6s. with 20 lb. at 4s. 5s. 6d. and 6s. per gallon, must per Ib. ; how much of each sort | be mixed with 3 gallons at 4s. must he take to make the compo- per gallon, so that the mixture sition worth 8s. per pound ? may be worth 5s. 4d. per gallon ? 4- 4 against the given gal. 3 at 5s. Ans. 6 at 58. 6d. per gall. 12 4 6 at 6s. Ib. 2 : 10 at 6s. 4: 20 :: 2 : 10 at 10s. Ans. 4: 20 at 12s. 8 QUESTIONS 1. What is Alligation ? 7. When the quantity of the whole 2. Of how many kinds is it? composition is limited, what is 3. What is Alligation Medial? the rule ? 4. What is the rule ? 8. What is the rule when one of 5. What is Alligation Alternate ? the quantities is limited ? 6. What is the rule for reckoning 9. How do you prove Alligation and linking quantities? Alternate? * Questions are solved in the same way when several of the ingredients are limited to certain quantities, hy finding first of one limit, and then of another. The second rule in Alligation Alternate may be employed for finding the specific gravities of bodies. A curious instance of the application of this rule to the detection of fraud, is recorded of the celebrated Archimedes. Hiero, king of Syracuse, suspecting his crown, which he had ordered to be made entirely of pure gold, to be alloyed with some baser metal, employed Archimedes to ascertain the fact. The philosopher procured two other masses, the one of pure gold, and the other of silver or copper, and each of the same weight of the crown, to be examined, and by pntting each of these separately into a vessel of water, he found the quantity of water expelled by each, and thus determined their specific gravity, and by that means the amount of gold, and also of alloy, in the crown. Thus, if we suppose the weight of each of the masses to be 101b. and the water expelled by the copper or silver to be 8, that expelled by the gold to be 5, and that expelled by the crown, 7, so the rates of the simples will be 8 and 5, and that of the compound, 7. Then, 582 3 : 10 :: 2 : blb. copper. Ans. 3 : 10 :: 1 : 3jlb. gold, SECTION I. POWERS AND ROOTS Xnvolution. INVOLUTION is the raising of powers. A power is a number produced by multiplying any given number continually by itself a certain number of times. Any number is itself called the first power; if it be muitiplied by itself, the product is called the second power, or square; if this be multiplied by the first power again, the product. is called the third power, or cube, and so on. 3= 3 is the first power of 3 3 3x3= 9 is the second power or square of 3 =32 3X3 X j=27 is the third power or cube of 3 =93 3X3 XuX=81 is the fourth power or biquadrate of 3 The small figures, 4, placed over the 3, and used to designate the power, are called the indices, or exponents. The index of the first power is always omitted. Examples 1. What is the 5th power of 6 ? 2. What is the second power 6 of 45 ? Ans. 2025. 6 =34* 1 2 3 9 * The index of the power is always one more than the number of multiplications performed ; thus 3 multiplied 3 times by itself continually, is raised to the fourth power. + A vulgar fraction is involved by raising both its terms to the power required. The involution of fractions diminishes their value. QUESTIONS 1. What is Involution ? by the first, what is the product 2. What is a power ? called ? 3. What is any number itself called ? 6. How are powers desigpated ? 4. What is the product called, if a 7. Of which of the powers is the in number be multiplied by itself? dex always omitted ? 5. If the second power be multiplied 2. Evolution. Evolution is the method of extracting roots. The root of any number, or power, is a number, which being multiplied by itself a certain number of times, will produce that power. Roots are denominated from the powers of which they are the root, and are called square, cube, biquadrate, or 2d, 3d, 4th root, &c. Thus 3 is the square root of 9, because 9 is the 2d power, or square of 3; 3, also, is the cube root of 27, because 27 is the 3d. power, or cube of 3. Again; 2 is the 4th, or biquadrate root of 16, because 16 is the 4th power of 2, &c. The following table exhibits the 2d, 3d, 4th, 5th, and 6th powers of the 9 digits, considered as roots or first powers. TABLE. Rotarl:t powers 2 3 4 61 9 4 16 25 36 49 64 81 8! 27 64! 125 216 343 512 729 161 811 2561 6251 1296) 2401 4096) 6561 1391243:024 3125 7776 16807 32768) 5! 049 1/641729 4096 :5625/46656'117649'262144531441 3 4 The square root is denoted by the radical sign, v, placed before the power, and other roots by the same sign, with the indes of the root placed over it. Thus vi denotes the square root of 9, 727 the cube root of 27 ; and 716 the biquadrate root of 16. Roots are also denoted by fractional indices. Thus 93 denotes the square root of 9; 27), the cube root of 27, an:1 65 the biquadrate root of 16. The latter method of designating roots is most rational, and at present generally practised. Although every number has a root, yet the complete root of the greatest part of numbers cannot be ascertained. The roots of all can, however, by the help of decimals, be obtained to a sufficient degree of accuracy for practical purposes. A power is complete, when its root of the same name can be accurately extracted. A power is imperfect, when its root cannot be accurately found, and the root of such a power is called a surd, or irrational quantity. To prepare any number, or power, for extracting its root. Rule.*_Beginning at the right hand, distinguish the given number into periods, each consistiug of as many figures as are denoted by the index of the root, desiguating the periods by points placed over the first figures in each ; by the number of periods will be shown the number of figures of which the root is to consist. Examples 1. Prepare 348753421 for ex 2. Prepare 681012.1416 for ex'tracting the square, cube, and tracting its square and cube roots. biquadrate roots. Square. 681012.1416 For the square root. 318753421 Cube. 681012.141600 For the cube root. 348753421 In preparing decimals, proceed For the biquadrate root. 348753421 hand, and if the last period happen to be incomplete, complete it by an nexing ciphers. 1. TO EXTRACT THE SQUARE ROOT. To extract the square root is to find the number which, multiplied into itself, will produce the given number. A square is a figure bounded by 4 equal straight lines, having 4 right angles, and its root is the length of one of their sides. Rule.t-1. Having distinguished the given number into periods, find the root of the greatest square number in the left hand period, * The reason of this rule will appear by considering that the product of any two numbers can have at most but just as many places of figures as there are in both the factors, and at least but one less, of which any one can satisfy himself by trial. From this fact, it is clear that a square number can have at most but twice as many places of figures as there are figures in the root, and at least but one less; and that a cube nuruber cannot have more than three times the numher of figures that there are figures in the root, and at least but two less, and so on. Example.-1 is the least possible root of a square number ; 1>1 = 1, which is one less than the number of factors ; 1x1x)=1, two less than the number of factors, &c. Again, 10 is the least root consisting of two figures ; 10x10== 100, one less than the number of places in the factors, and 10*10*10=1000, two less, &c. ; and the same may be shown of the least roots consisting of 3, 4, &c. figures. Again, the greatest root consisting of one figure only, is 9; its square, or 2d power, is 9*9=81, consisting of just twice as many places as there are in 9, the root, and the cube of 9 is 9*9*9==729, consisting of three times the number of places in the root. The same may in like manner be shown of 99, the greatest root consisting of two places, 999, the greatest consisting of three places, &c. These observations must make the reason for pointing off the number into periods, obvious, and must also make it evident, that each period will give one figure in the required root, and no more. They also show ihat one figure alone, or standing on the left hand of other full periods, may of itself constitute a period. t. The reason of this rule may be own from the first example. Now if we suppose 529 to be so many square feet of boards, which we wish to lay down in the form of an exact square, it is evident that the square root of 529 will be the |