15. A maltster had a round malt-kiln of 16 feet diameter, but is to build a square one that will contain 3 times as Anf 24.5557 feet. 16. The folidity of a sphere is 47016 cubic inches. Requi. red the side of a cube whose content is equal to it. Ans. 36 inches. 46. PROPORTION. When two quantities of the same kind are compared, their relation or ratio is obtained by enquiring how often the first contains the second. Thus, the ratio of 12 to 4 is 3; of 4 to 3 is iź ; and of 3 to 8 is z, or .375. 47. When four quantities, a, b, c, d, are proportional, it is usually expressed by saying, the first is to the second as the third is to the fourth; or, a :b::c:d, and the quantities are said to be in geometrical proportion. 48. The quantity whose ratio is enquired into is called the antecedent, and the quantity, with which it is compared, the consequent. 49. The first and third terms, a and c, are called antecedents. The second and fourth, b and d, are called the consequents. 50. Ifa:b::c:d, the product of the means, is equal to the product of the extremes, Euclid vi. 16. thus ad=be. 51. If the product of two quantities, od, be equal to the product of two others, be, the quantities are proportional, and a: 5 ::c:d; that is, a factor of the first is to a factor of the second second as the remaining factor of the second is to the remaining factor of the first. Euclid, vi. 16. duct of the means divided by either extreme quotes the other extreme, and the product of the extremes, divided by either of the means, quots the other mean. 53. If a:b::c:d, they will remain proportionals under the following varieties.' Euclid, v. Definitions. Thus, a :b::c:d Alternando, a :c:: b;d Invertendo, b:a::c:d Componendo, atb:b::c+d:d In all these varieties, the product of the means is equal to that of the extremes. 54. EQUATIONS. An equation is a proposition asserting the equality of two quantities; it is usually expressed by the sign=. Thus, 2x6=12, or bd=e. 55. AXIOMS. 1. Quantities that are equal to one and the same quantity are equal to each other. 2. If equal quantities be added to equal quantities the sums are equal. 3. If equal quantities be taken from equal quantities, the remainders are equal. 4. Quantities 4. Quantities which are double of the fame quantity are equal; and the contrary. 5. If equal quantities be multiplied by the same quantity, the products are equal. 6. If equal quantities be divided by the same quantity, the quotes are equal. 1 56. The value of an unknown quantity is found by changing the form of the equation till it stand alone on one side, and the known quantities on the other. But it frequently happens that the unknown quantity is variously combined with others, and so its value not easily discovered. We shall therefore lay down a few general rules for the solution of equations, and which depend on the foregoing axioms. RULES. 1 1. A quantity is said to be transposed, if it be taken from one side of the equation to the other with the opposite sign. Thus, 2+4=6, and 4=6—2. Ax. III. 2. If the unknown quantity be multiplied by any other quantity, divide both sides of the equation by that other quantity. b Thus, if ax=b, then x= Ax. VI. 3. If the unknown quantity be divided by any other quantity, multiply both sides of the equation by the divisor. Thus, if a 4. If that member of the equation which involves the un known 1 known quantity be a surd root, make that member stand alone on one side of the equation, remove the radical sign, and raise the other side to the corresponding power. Thus, 4x+b=a; then, 4x=a—b, by the rule 4x=a2—2ab+62. Ax. VII. and VIII. 5. If the same quantity be found on both sides of the equation, with the same lign prefixed, expunge it from both. 6. If a : 3::c:d, an equation is obtained by allerting the product of the means equal to that of the extremes, EXAMPLES. I. A person being asked his age, answers, If to my age you add triple my age, the sum will be 100. Required his age. From London to Edinburgh, by the Carlisle road, is 399 miles. A mesienger is dispatched irom Edinburgh, who travels at the rate of 36 miles per day : and, after fix days, another is dispatched from London to meet the former, who tra vels 25 miles per day. Required their distance from London when they meet, and how many days will the latter take. Suppose they meet in x days, then, 25* 61x=183 A=3 days EXAMPLE III. Suppose a messenger, who travels 331 miles per day, is difpatched, and after 8 days another is sent on horseback to overtake the former, who rides 80 miles a day, how many days does each travel before the first is overtaken? Suppose in x days. 270=46.25% EXAMPLE IV. Suppose the fun to proceed one degree per day in the eccliptic, and the moon 13°, and that the sun is in the beginning of Capricorn, and after three days the moon enters Aries Re. quired the place of their next conjunction. When the moon enters Aries the fun is advanced 273° from Aries. *+-273=13* 22 days 273=124 3 221=x days. 251 That is, the next conjunction will be in 25° 45' of Capricorn. EXAMPLE |