series, the difference or ratio of each consequent and the antecedent of the next following couplet be the same as the common difference or ratio of the couplets, the proportion is said to be continued, and the numbers themselves a series of continued arithmetical or geometrical proportionals. So 2, 4, 6, 8, form an arithmetical progression; for 4—2=6—4=8— 6=2; and 2, 4, 8, 16, a geometrical progression; for -16=2. 8 But if the difference or ratio of the consequent of one couplet and the antecedent of the next couplet be not the same as the common difference or ratio of the couplets, the proportion is said to be discontinued. So 4, 2, 8, 6, are in discontinued arithmetical proportion; for 4—2=8—6=2, but 8—2=6; also 4, 2, 16, 8, are in discontinued geometrical proportion; for 162, but 16-8. Four numbers are directly proportional, when the ratio of the first to the second is the same, as that of the third to the fourth. As 2: 4:3: 6. Four numbers are said to be reciprocally or inversely proportional, when the first is to the second, as the fourth is to the third, and vice versa. Thus, 2, 6, 9, and 3, are reciprocal proportionals; 2: 6:: 3 : 9. Three or four numbers are said to be in harmonical proportion, when, in the former case, the difference of the first and second is to the difference of the second and third, as the first is to the third; and, in the latter, when the difference of the first and second is to the difference of the third and fourth, as the first is to the fourth. Thus, 2, 3, and 6; and 3, 4, 6, and 9, are harmonical proportionals; for 3—2 16- -3=3:: 2:6; and 4—3=1 : 9—6=3: :3 : 9. Of four arithmetical proportionals the sum of the extremes is equal to the sum of the means.* Thus of 2-4 :: 6·8 * DEMONSTRATION. Let the four arithmetical proportionals be A, B, C, D, viz. A" B:: C. D; then, A—B—C—D, and B+D being added to both sides of the equation, A—B+B+D =CD+B+D; that is, A+D the sum of the extremes =C+B the sum of the means. And three A, B, C, may be thus expressed, A. B:: B. C; therefore A+C=B+B=2B, Q. E. D. the sum of the extremes (2+8)= the sum of the means (4+6) 10. Therefore, of three arithmetical proportionals, the sum of the extremes is double the mean. Of four geometrical proportionals, the product of the extremes is equal to the product of the means.* Thus, of 2:4:: 8: 16, the product of the extremes (3x16) is equal to the product of the means (4x8)=32. Therefore of three geometrical proportionals, the product of the extremes is equal to the square of the mean. Hence it is easily seen, that either extreme of four geometrical proportiouals is equal to the product of the means divided by the other extreme; and that either mean is equal to the product of the extremes divided by the other mean. SIMPLE PROPORTION, Or RULE OF THREE. The Rule of Three is that, by which a number is found, having to a given number the same ratio, which is between two other given numbers. For this reason it is sometimes named the Rule of Proportion. It is called the Rule of Three, because in each of its questions there are given three numbers at least. And because of its excellent and extensive use, it is often named the Golden Rule. RULE.f 1. Write the number, which is of the same kind with the answer or number required. * Demonstration. Let the proportion be A: B:: C: D, and let =r; then A=Br, and C=Dr; multiply the for A с B D mer of these equations by D, and the latter by B; then AD= BrD, and CB=DrB, and consequently AD the product of the extremes is equal to BC the product of the means.- -And three may be thus expressed, A: B:: B: C, therefore AC=Bx B=B*. Q. E. D. t Demonstration. The following observations, taken col N 2. Consider whether the answer ought to be greater or less than this number; if greater, write the greater of the lectively, form a demonstration of the rule, and of the reductions mentioned in the notes subsequent to it. 1. There can be comparison or ratio between two numbers, only when they are considered either abstractly, or as appiied to things of the same kind, so thai one can, in a proper sense, be contained in the other. Thus there can be no comparison be- . tween 2 men and 4 days; but there may be between 2 and 4, and between 2 days and 4 days, or 2 men and 4 men. Therefore, the 2 of the 3 given numbers, that are of the same kind, that is, the first and the third, when they are stated according to the rule, are to be compared together, and their ratio is equal to that, required between the remaining or second number and the fourth or answer. 2. Though numbers of the same kind, being either of the same or of different denominations, have a real ratio, yet this ratio is the same as that of the two numbers taken abstractly, only when they are of the same denomination. Thus the ratio of 11. to 21. is the same as that of 1 to 2 =1; Is. has a real ratio to 21. but it is not the ratio of 1 to 2; it is the ratio of Is. to 40s. that is, of 1 to 40. Therefore, as the first and third numbers have the ratio, that is required between the second and answer, they must, if not of the same denomination, be reduced to it; and then their ratio is that of the abstract numbers. 3. The product of the extremes of four geometrical proportionals is equal to the product of the means; hence, if the product of two numbers be equal to the product of two other numbers, the four numbers are proportionals; and if the product of two numbers be divided by a third, the quotient will be a fourth proportional to those three numbers. Now as the question is resolvable into this, viz. to find a number of the same kind as the second in the statement, and having the same rati^ to it, that the greater of the other two has to the less, or the less has to the greater; and as these two, being of the same denomination, may be considered as abstract numbers; it plainly follows, that the fourth number or answer is truly found by multiplying the second by one of the other two, and dividing the product by that> which remains. two remaining numbers on the right of it for the third, and the other on the left for the first number or term; but if less 4. It is very evident, that, if the answer must be greater than the second number, the greater of the other two numbers nWst be tne multiplier, and may occupy the third place; but, if less, the less number must be the multiplier. 5. The reduction of the second number is only performed for convenience in the subsequent multiplication and division, and not to produce an abstract number. The reason of the reduc-. tion of the quotient, of the remainder after division, and of the product of the second and third terms, when it cannot be divided by the first, is obvious. 6. If the second and third numbers be multiplied together, and the product be divided by the first; it is evident, that the answer remains the same, whether the number compared with the first be in the second or third place. Thus is the proposed demonstration completed. There are four other methods of operation beside the general one given above, any of which, when applicable, performs the work much more concisely. They are these: 1. Divide the second term by the first, multiply the quotient by the third, and the product will be the answer. 2. Divide the third term by the first, multiply the quotient by the second, and the product will be the answer. 3. Divide the first term by the second, divide the third by the quotient, and the last quotient will be the answer. 4. Divide the first term by the third, divide the second by the quotient, and the last quotient will be the answer. The general rule above given is equivalent to those, which are usually given in the direct and inverse rules of three, and which are here subjoined. The Rule or Three Direct teaches, by having three numbers given, to find a fourth, that shall have ths same proportion to the third, as the second has to the first. RULE. I. State the question; that is, place the numbers so, that the first and third may be the terms of supposition and demand, and th» second of the same kind with the answer required. write the less of the two remaining numbers in the third place, and the other in the first. 2. Bring the first and third numbers into the same denomination, and the second into the lowest name mentioned. 3. Multiply the second and third numbers together, and divide the product by the first, and the quotient will be the answer to the question, in the same denomination you left the second number in; which may be brought into any other denomination required. EXAMPLE. If 241b. of raisins cost 6s. 6d. what will 18 frails cost, each weighing net 3qrs. 18lb.? The rule is founded on this obvious principle, that the magnitude or quantity of any effect varies constantly in proportion to the varying part of the cause: thus the quantity of goods bought is in proportion to the money laid out; the space gone over by an uniform motion is in proportion to the time, &c. The truth of the rule, as applied to ordinary inquiries, may be made very evident by attending only to the principles of compound multi |