Thus the four numbers (5, 54, 6, 45) form a proportion in the four following ways:— For the product of the two extremes remains always equal to the product of the means. Similarly, in every proportion, we may invert-Ist. The two means only. 2nd. The two extremes only. 3rd. The two means as well as the two extremes. When the two means are equal, the fourth term is called a third proportional, and the mean is termed the geometrical mean of the two extremes; thus (in the proportion 4: 6:: 6 : 9) 6 is called the geometrical mean between 4 and 9. (f). If two proportions have two terms common to both, the two other terms of the first form a proportion with the two other terms of the second. (g). When two ratios are equal, the ratio of the sum, or the difference of the numerators to the sum, or the difference of the denominators, equal to either of the equal ratios. Let us again take the proportion 56 48 since 56 contains 8 times seven, and 48 8 times 6, the sum 56 + 48, or 104, will contain 8 times the sum 7 + 6, or 13; we may then write and because 104 13 = 48 we may conclude that, by taking the difference of the numerators and that of the denominators, a ratio is obtained equal to 56 7° N 2 And generally, if we have a series of equal ratios, for example, the sum of a certain number of numerators divided by the sum of the corresponding denominators gives a ratio equal to each of the given ratios. The magnitudes which form the proportion need not all be of the same kind; two may be of one kind and two of another. Again, the units measuring the magnitudes are not necessarily the same; for instance 54 yards: 6 yards :: 45s. : 5s. 54 days 6 days :: 45 years: 5 years. We must, however, always have one extreme and one mean of the same kind of magnitude and measured by the same unit, the other two terms being alike in kind and also measured by the same unit. Proportion of Straight Lines.-When each of two straight lines can be exactly measured by a certain unit of length, the ratio between them is the ratio of the two numbers which express how many times this unit of length is contained in each of them. If two lines are respectively 3 yards and 2 yards in length, the ratio of the two lengths is 3: 2 or 3. If, instead of measuring the two lines by the yard, we measure them by some other unit―the foot, for instance—the length of the first ruler would be 9 feet, that of the second 6 feet, and the ratio of the two 2 rulers would have been 9 to 6, or still 2. The ratio of two lengths is then quite independent of the unit which is used to measure them, and therefore in treating of the ratio no mention need be made of this unit. The ratio of two lines, AB, CD, will be represented simply by AB where AB is a number of times a certain unit, and CD a CD number of times the same unit; but we need not trouble ourselves to inquire what this unit is. When two magnitudes have no common measure, by taking a very small length as the unit we may express by means of it the lengths of the lines very nearly, and we can always make the remainders as small as we please. Hence the properties of proportion which apply to all commensurable lines may be considered as applicable also to incommensurable lines; but in order that nothing in our course which admits of proof shall be passed by without it, we will first establish the properties of proportion in connection with commensurable magnitudes and then extend them to those which are incommensurable. Extension of Preceding Illustration.-The properties of proportion illustrated by the particular numbers 56, 7, 48 and 6, may be shown to belong to all numbers, and therefore to all magnitudes having a common measure. Let two commensurable magnitudes contain their G. C. M. A and B A times respectively, then their ratio is expressed by A: B or B Thus, if A proportion is the equation formed by two equal ratios. the two ratios A: B and C : D are equal, the equation A : B = C: D is a proportion, and it may also be written (a). The product of the means of a proportion is equal to the product of its extremes. Proof. If the fractions of a proportion This proposition is called the test of proportion. It follows that when four commensurable lines are proportionals, the rectangle contained by the extremes is equal to the rectangle contained by the means. (b). If four quantities are such that the product of the first and last of them is equal to the product of the second and third, these four quantities form a proportion. Proof.-Let A, B, C, D, be such that which, reduced to lower terms and written in the form of ratios, is A:BC: D. Hence, if two rectangles are equal, the sides of one form the extremes of a proportion of which the sides of the other form the means. (c). The terms of a proportion may be transposed in any way, provided the product of the means is retained equal to that of the extremes, and the proportion will not be destroyed. Thus, the preceding proportion gives, by transposition, If both the means of the proportion are of the same magnitude, this mean is called the mean proportional between the extremes. Thus, if A: BB: D, B is a mean proportional between A and D. (d). The mean proportional between two quantities is the square root of their product. Proof.—The application of the test to the preceding proportion gives B2 = A x D, A succession of several equal ratios is called a continued proportion. Thus, A: BC:D=E:F= &c. is a continued proportion. (e). The sum of any number of antecedents of a continued proportion is to the sum of the corresponding consequents as one antecedent is to its consequent. Proof.—Denote the common value of the ratios in the above continued proportion by M, we have (f). The sum of the antecedents of a proportion is to the sum of its consequents as either antecedent is to its consequent; and the difference of the antecedents is to the difference of the consequents in the same ratio. The sum of the antecedents of a proportion is to their difference as the sum of the consequents is to their difference. Proof. The proportion A:BC: D gives, by the preceding proposition, A+C:B+D=A-C:B-D; whence, by transposing the means, 7. A+ CA-C=B+D: B-D. (g). The sum of the first two terms of a proportion is to the sum of last two as the first term is to the third, or as the second is to the fourth; and the difference of the first two terms is to the difference of the last two in the same ratio; also the sum of the first two terms is to their difference as the sum of the last two is to their difference. from which we obtain, by the preceding propositions, A+B:C+D=A−B: C-DA:C=B: D may evidently be multiplied together term by term, and the result, A XE:BX F=Cx G = D x H, is a new proportion. Likewise, a proportion may be multiplied by itself any number of times in succession, and the squares, cubes, fourth powers. &c., of the terms form a new proportion. Thus, the proportion |