2. A merchant would mix wines at 14s. 15s. 19s. 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. 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. 24, and the same of 30 and 36. Now 14 bushels at 24d. is 336d. and 14 bushels at 34d. 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. 1 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. * Questions are solved in the same way when several of the ingredients are limited to certain quantities, by 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 putting 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 10lb. 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, 7 8-2 And 3:10::2 : 6-lb. copper. } Ans. ARITHMETICK, PART II. SECTION I. POWERS AND ROOTS. Involution. 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 multiplied 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. 3×3= 9 is the second power or square of 3 3x3x=27 is the third power or cube of 3 3x3x0x3=81 is the fourth power or biquadrate of 3 3 =52 53 =34* The small figures, 1, 2, 3, 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. * 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. 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. 3 The square root is denoted by the radical sign, ✓, placed before the power, and other roots by the same sign, with the index of the root placed over it. Thus 9 denotes the square root of 9, 27 the cube root of 27; and ☑ 16 the biquadrate root of 16. 4 Roots are also denoted by fractional indices. Thus 9 denotes the square root of 9; 273, the cube root of 27, and 64 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 consisting of as many figures as are denoted by the index of the root, designating 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. 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-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 number cannot have more than three times the number 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; 1×1×1=1, two less than the number of factors, &c. Again, 10 is the least root consisting of two figures; 10×10= 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 that one figure alone, or standing on the left hand of other full periods, may of itself constitute a period. + The reason of this rule may be shown 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 |