tudes. Two or more magnitudes, such that a third magnitude can exist,—whether we are able to find it or not-which can accurately measure them all; or which, by a finite number of repetitions, can be made to equal, first, the least magnitude, then, by continuing the repetitions, the next least, and so on; such magnitudes, when brought into comparison, are, in reference to their having this common measure, called commensurable magnitudes, and are by this name distinguished from an equally extensive class of magnitudes which are often compared one with another, but which do not admit of being accurately measured by any standard whatever; and which, on this account, are called, with respect to each other, incommensurable magnitudes. The side and diagonal of a square are magnitudes of this kind. Not only are we unable to find a line so small that it shall be contained in the side some exact number of times without remainder, and also in the diagonal some other number of times without remainder; but we can prove that no such common measure can possibly exist. We may find an unlimited variety of measures for the side, but no one of these will measure the diagonal; and in like manner, though there are an equal variety of measures for the diagonal, yet no one of them will measure the side. We shall give the proof of this at the end of the Sixth Book. IV. Equimultiples, or like multiples, are those which contain the respective magnitudes of which they are multiples the same number of times. V. And like submultiples are those which are contained in their respective multiples the same number of times. VI. When two magnitudes are compared together, in reference to the enquiry whether the first is contained in the second, or in a multiple of the second, the former is called an antecedent, and the latter its consequent. Such comparison, it is obvious, can be made only with homogeneous magnitudes, or those of the same kind; as, lines with lines, sur faces with surfaces, and solids with solids. A line and surface can never, therefore, stand in the relation of antecedent and consequent. VII. Magnitudes are proportional when, being compared two and two, an antecedent cannot be contained in any multiple of its consequent oftener than any of the other antecedents is contained in a like multiple of its consequent. VIII. If the magnitudes so related be but four, they are denominated simply a proportion.* The first and last of them are called the extremes, and the two intermediate ones the means. The student will readily perceive that this definition of propor tion, as being that relation among four magnitudes which renders it impossible for one antecedent to be contained in any multiple of its consequent oftener than the other antecedent is contained in a like multiple of its consequent-is comprehensive of the common notion of proportion; which notion, however, does not take in incommensurable quantities. One antecedent may be contained in its consequent a certain number of times, and a third, a fourth, a tenth, &c.; and if the other antecedent be contained in its consequent the same number of times, and a third, a fourth, a tenth, &c., the four are said to be in proportion, according to the ordinary notion. Now if, in the cases here supposed, we take the consequents three times, four times, and ten times respectively, then each antecedent will be contained in the multiple of its consequent just as many times as the other antecedent is contained in the like multiple of its consequent; this is what the definition asserts must always have place, be the equimultiples whatever they may; and the same thing is implied in the common notion of each consequent containing its antecedent the same number of times and parts of a time; the sameness, as to number of times, continuing, however we (equally) multiply the consequents. The definition, however, comprehends cases not included in the common notion-cases in which an antecedent is not contained in its consequent any times and parts of a time capable of a numerical expression; cases, in fact, in which antecedent and con. sequent are incommensurable. IX. The magnitudes themselves are called the terms of the proportion; and those are called homologous which have the same name; the antecedents form one pair of homologous terms; the consequents another pair. X. A series of magnitudes is said to form a continued proportion when every consequent is taken for the antecedent of the term next following. XI. If there be but three continued proportionals, the middle term is called the mean, and the others the extremes. Explanation of the SIGNS employed in the succeeding 1. To denote that four magnitudes A, B, C, D, are proportional, they are thus arranged; A: B :: C: D, which is read, "as A is to B so is C to D," understanding, of course, that relation among them which constitutes their proportionality (def. 7). 2. In like manner, to denote a continued proportion, the terms are thus arranged, A: B:: B: C::C: D, &c., which is read, "as A is to B so is B to C and C to D," &c. 3. The symbol + denotes the addition of the quantities between which it is placed. Thus, A + B signifies that B is added to A. To denote subtraction the sign is employed; so that A B means B taken from A. The sign serves, when placed between two quantities, to imply their difference; and is used instead of when it is not stated which of the two magnitudes or quantities is the greater. And lastly, the double sign is employed to express "the sum or difference" of the quantities between which it is placed. 4. The symbol > placed between two magnitudes, implies that the former is greater than the latter; on the contrary, <indicates that the former is less than the latter, while : implies their equality. Thus, by A > B is to be understood that "A is greater than B;" by A < B, that "A is less than B," and by A B, that "A is equal to B." NOTE. When we speak of these relations among A, B, &c., we of course have reference to the magnitudes which these are employed to represent; they may be either lines, surfaces, or solids. AXIOMS. Equimultiples of the same or of equal magnitudes are equal to one another; so also are their like submultiples. II. A multiple of a greater magnitude exceeds a like multiple of a less; and a submultiple of a greater exceeds a like submultiple of a less. III. That magnitude of which a multiple is greater than a like multiple of another, is greater than that other. IV. Of any two magnitudes of the same kind, a multiple of one may be taken so great as to exceed the other. PROPOSITION I. THEOREM. If any number of magnitudes be equimultiples of as many others, each of each, what multiple soever any one of the first is of its part, the same multiple is the sum of all the former to the sum of all the latter. First let there be two magnitudes P, Q, equimultiples of two others A, B; then will P+Q be a like multiple of A+ B P Q B For suppose P to be divided into parts, each equal to A; and Q to be divided into parts, each equal to B; then, by hypothesis, the number of the parts of P is the same as the number of the parts of Q. Let one of the parts of P be added to one of the parts of Q, the sum will be equal to AB. Let another part of P be added to another part of Q; the sum will, in like manner, be equal to A + B. Continue these additions as long as there are parts of P or of Q left; and there will result as many sums, each equal to A+B, as there are parts in P or in Q. But all these sums, taken together, make up P+Q. Hence there are in PQ as many parts equal to A + B, as there are parts in P equal to A, or in Q equal to B. Next let there be three magnitudes P, Q, R, equimultiples of three others, A, B, C; then will P+Q + R be a like multiple of A+B+C. It is proved above that whatever multiple P is of A, or that Q is of B, the same multiple is P + Q of A + B; but R is the same multiple C that P is of A, or Q of B; hence the two quantities (P+Q) R are equimultiples of (A + B) C. Therefore, by the first case, P+Q+ R is the same multiple of A+B+C; and, as the case of three magnitudes was inferred from that of two, so, in like manner, may the case of four magnitudes be inferred from that of three. Consequently, if any number of magnitudes, &c. Q. E. D. |