BOOK II. RATIO AND PROPORTION. cision. On the Relation of Magnitudes to Numbers. The ratios of magnitudes may be expressed by numbers either exactly or approximately; and in the latter case, the approximation can be carried to any required degree of pre Thus, let it be proposed to find the numerical ratio of two straight lines, AB and CD. From the greater line AB, cut A E G B off a part equal to the less, CD, as many times as possible; for example, twice, with a remain- Ç F D der ÊB. From CD, cut off a part equal to the remainder EB as often as possible ; for ex ample, once, with a remainder FD. From the first remainder, BE, cut off a part equal to FD as often as possible ; for example, once, with a remainder GB. From the second remainder, FD, cut off a part equal to the third, GB, as many times as possible. Continue this process until a remainder is found which is contained an exact number of times in the preceding one. This last remainder will be the common measure of the proposed lines; and regarding it as the measuring unit, we may easily find the values of the preceding remainders, and at length those of the proposed lines; whence we obtain their ratio in numbers. For example, if we find GB is contained exactly twice in FD, GB will be the common measure of the two proposed lines. Let GB be called unity, then FD will be equal to 2. But EB contains FD once, plus GB; therefore, EB=3. CD contains EB once, plus FD; therefore, CD=5. AB contains CD twice, plus EB; therefore, AB=13. Consequently, the ratio of the two lines AB, CD is that of 13 to 5. However far the operation is continued, it is possible that we may never find a remainder which is contained an exact number of times in the preceding one. In such cases, the er. act ratio can not be expressed in numbers; but, by taking the measuring unit sufficiently small, a ratio may always be found, which shall approach as near as we please to the true ratio. So, also, in comparing two sur- Unit A B faces, we seek some unit of measure which is contained an exact number of times in each of them. Lot A and B represent two surfaces, and let a square inch be the unit of measure. Now, if this measuring unit is contained 15 times in A and 24 times in B, then the ratio of A to B is that of 15 to 24. And although it may be difficult to find this measuring unit, we may still conceive it to exist; or, if there is no unit which is contained an exact number of times in both surfaces, yet, since the unit may be made as small as we please, we may represent their ratio in numbers to any degree of accuracy required. Again, if we wish to find the ratio of two solids, A and B, we seek some unit of measure which is contained an exact number of times in each of them. If we take a cubic inch as the unit of measure, and we find it to be contained 9 times in A, and 13 times in B, then the ratio of A to B is the same as that of 9 to 13. And even if there is no unit which is contained an exact number of times in both solids, still, by taking the unit sufficiently small, we may represent their ratio in numbers to any required degree of precision. Hence the ratio of two magnitudes in geometry, is the same as the ratio of two numbers, and thus each magnitude has its numerical representative. We therefore conclude that ratio in geometry is essentially the same as in arithmetic, and we might refer to our treatise on algebra for such properties of ratios as we have occasion to employ. However, in order to render the present treatise complete in itself, we will here demonstrate the most useful properties. Definitions. Def. 1. Ratio is the relation which one magnitude bears to another with respect to quantity. Thus, the ratio of a line two inches in length, to another six inches in length is denoted by 2 divided by 6, i. e., į or į , the number 2 being the third part of 6. So, also, the ratio of 3 feet to 6 feet is expressed by i or. A ratio is most conveniently written as a fraction; thus, 01 А the ratio of A to B is written The two magnitudes com B pared together are called the terms of the ratio ; the first is called the antecedent, and the second the consequent. Def. 2. Proportion is an equality of ratios. Thus, if A has to B the same ratio that C has to D, these för quantities form a proportion, and we write it А с BD' A:B:: C:D. The first and last terms of a proportion are called the two extremes, and the second and third terms the two means. Of four proportional quantities, the last is called a fourth proportional to the other three, taken in order. Since А С BD' it is obvious that if A is greater than B, C must be greater than D; if equal, equal; and if less, less; that is, if one antecedent is greater than its consequent, the other antecedent must be greater than its consequent; if equal, equal; and if less, less. Def. 3. Three quantities are said to be proportional, when the ratio of the first to the second is equal to the ratio of the second to the third ; thus, if A, B, and C are in proportion, then A:B::B:C. In this case the middle term is said to be a mean proportional between the other two. Def. 4. Two magnitudes are said to be equimultiples of two others, when they contain those others the same number of times exactly. Thus, 7A, 7B are equimultiples of A and B; so, also, are mA and mB. Def. 5. The ratio of B to A is said to be the reciprocal of the ratio of A to B. Def. 6. Inversion is when the antecedent is made the consequent, and the consequent the antecedent. Thus, if A:B:C:D; then, inversely, B:A.:D:C. Def. 7. Alternation is when antecedent is compared with antecedent, and consequent with consequent. Thus, if A:B::C:D; then, by alternation, A:C::B:D. Def. 8. Composition is when the sum of antecedent and consequent is compared either with the antecedent or con secuent, Thus, if A:B::C:D; then, by composition, A+B: À ::C+D:C, and A+B:B:: C+D:D. Def. 9. Division is when the difference of antecedent and consequent is compared either with the antecedent or con sequent. Thus, if A:B::C:D: then, by division, A-B:A::C-D:C, and A-B:B::C-D :D. Acioms. 1. Equimultiples of the same, or equal magnitudes, are equal to each other. 2. Those magnitudes of which the same or equal magnitudes are equimultiples, are equal to each other. If four quantities are proportional, the product of the two extremes is equal to the product of the two means. It has been shown that the ratio of two magnitudes, whether they are lines, surfaces, or solids, is the same as that of two numbers, which we call their numerical representatives. Let, then, A, B, C, D be the numerical representatives of four proportional quantities, so that A:B::C:D; then will AXD=BXC. For, since the four quantities are proportional, А С BD Multiplying each of these equal quantities by B (Axiom 1). we obtain Вхс D AxD=BXC. A:B::B:C; then, by the proposition, À XC=BxB, which is equal to B'. PROPOSITION II. THEOREM (Converse of Prop. I.). If the product of two quantities is equal to the product of two other quantities, the first two may be made the extremes, and the other two the means of a proportion. Thus, suppose we have A XD=BxC; then will A:B::C:D. Вхс D А с. B-D that is, the ratio of A to B is equal to that of C to D, or, A:B::C:D. If four quantities are proportional, they are also proportional when taken alternately. Let A, B, C, D be the numerical representatives of foui proportional quantities, so that A:B::C:D; then will A:C::B: D. For, since A:B::C:D by Prop. I., AXD=BXC. And, since AxD=BXC, by Prop. II., A:C::B:D. Ratios that are equal to the same ratio, are equal to each other. А с BD we have |