be expressed by the number of its units, and the two magnitudes are then said to be commensurable. If the second magnitude do not contain the measuring unit an exact number of times, there may perhaps be a smaller unit which will be contained an exact number of times in each of the magnitudes. But if there is no unit of an assignable value, which is contained an exact number of times in each of the magnitudes, the magnitudes are said to be incommensurable. It is plain, however, that if the unit of measure be repeated as many times as it is contained in the second magnitude, the result will differ from the second magnitude by a quantity less than the unit of measure, since the remainder is always less than the divisor. Now, since the unit of measure may be made as small as we please, it follows, that magnitudes may be represented by numbers to any degree of exactness, or they will differ from their numerical representatives by less than any assignable magnitude. 4. We will illustrate these principles by finding the ratio between the straight lines CD and AB, which we will suppose commensurable. From the greater line AB, cut off a part equal A to the less CD, as many times as possible; for example, twice, with the remainder BE From the line CD, cut off a part, CF, equal to the remainder BE, as many times as possible; once, for example, with the remainder DF. From the first remainder BE, cut off a part equal to the second, DF, as many times as possible; once, for example, with the remainder BG. From the second remainder DF, cut off a part equal to BG, the third remainder, as many times B as possible. E -G -F D Continue this process, till a remainder occurs, which is contained exactly, a certain number of times, in the preceding one. Then, this last remainder will be the common measure of the proposed lines. Regarding this as unity, we shall easily find the values of the preceding remainders; and at last, those of the two proposed lines, and hence, their ratio in numbers. Suppose, for instance, we find GB to be contained exactly twice in FD; BG will be the common measure of the two proposed lines. Put BG=1; we shall then have, FD-2; but EB contains FD once, plus GB; therefore, we have EB-3: CD contains EB once, plus FD; therefore, we have CD-5 and lastly, AB contains CD twice, plus EB; therefore, we have AB=13; hence, the ratio of the lines is that of 5 to 13. If the line CD were taken for unity, the line AB would be 13; if AB were taken for unity, CD would be 13° 5. What has been shown, in respect to the straight lines, CD and AB, is equally true of any two magnitudes, A and B.. For, we may conceive A to be divided into M number of units, each equal to A': then AMXA': let B be divided into N number of equal units, each equal to A'; then B-NXA'; M and N being integer numbers. Now the ratio of A to B, will be the same as the ratio of MXA' to NXA'; that is, the same as the ratio of the numerical quantities M and N, since A' is a common unit. 6. If there be four magnitudes, A, B, C, and D, having such values that then A is said to have the same ratio to B, that C has to of A to B is said to be equal to the When four quantities have this relation D; or, the ratio ratio of C to D. to each other, they are said to be in proportion. To indicate that the ratio of A to B is equal to the ratio of C to D, the quantities are usually written thus, A: B:: C: D, and read, A is to B as C is to D. The quantities which are compared together are called the terms of the propor tion. The first and last terms are called the two extremes, and the second and third terms, the two means. 7. Of four proportional quantities, the last is said to be a fourth proportional to the other three, taken in order. The first and second terms, are called the first couplet of the proportion; and the third and fourth terms, the second couplet: the first and third terms are called the antecedents, and the second and fourth terms, the consequents. 8. Three quantities are in proportion, when the first has the same ratio to the second, that the second has to the third; and then the middle term is said to be a mean proportional between the other two. 9. Magnitudes are in proportion by alternation, or alternately, when antecedent is compared with antecedent, and consequent with consequent. 10. Magnitudes are in proportion by inversion, or inversely, when the consequents are taken as antecedents, and the antecedents as consequents. 11. Magnitudes are in proportion by composition, when the sum of the antecedent and consequent is compared either with antecedent or consequent. 12. Magnitudes are in proportion by division, when the difference of the antecedent and consequent is compared either with antecedent or consequent. 13. Equimultiples of two quantities are the products which arise from multiplying the quantities by the same number: thus, m×A, m×B, are equimultiples of A and B, the common multiplier being m. 14. Two varying quantities, A and B, are said to be reciprocally proportional, or inversely proportional, when their values are so changed that one is increased as many times as the other is diminished. In such case, either of them is always equal to a constant quantity divided by the other, and their product is constant. PROPOSITION I. THEOREM. When four magnitudes are in proportion, the product of the two extremes is equal to the product of the two means. Let A, B, C, D, be any four magnitudes, and M, N, P, Q, their numerical representatives; then, if. we shall have M : N : P: Q M×Q=N×P. For, since the magnitudes are in proportion, we have Cor. If there are three proportional quantities, the product of the extremes will be equal to the square of the mean (D. 8). For, if N=P, we have MXQ=N2 or P2. PROPOSITION II. THEOREM. If the product of two magnitudes be equal to the product of two other magnitudes, two of them may be made the ex tremes and the other two the means of a proportion. If we have MX Q=NXP; then will M: N:: P: Q. For, if P have not to Q, the ratio which M has to N, let P have to Q', (a number greater or less than Q,) the same ratio which M has to N: that is, let M : N :: P: Q: M×Q'=N×P; then (P. 1), NXP hence, Q'= NXP ; but, Q= M M Consequently, Q'=Q, and the supposition that it is either greater or less, is absurd; hence, the four magnitudes M, N, P, Q, are proportional. PROPOSITION III. THEOREM. If four magnitudes are in proportion, they will be in pro. portion when taken alternately. N: Q. Let M, N, P, Q, be four quantities in proportion; so that M: N:: P: Q: then will M P: For, since M: N :: P: Q: we have M×Q=N×P; therefore M and Q may be made the extremes, and. N and P the means of a proportion (P. 2); If there be four proportional magnitudes, and four other proportional magnitudes, having the antecedents the sume in both, the consequents will be proportional. and Let M: N : P : S, giving MXS=R×P, S. M: R :: then will N: Q :: R: For, multiplying the equations crosswise, we have M×QXRXP=M×S×N×P; cancelling MXP in both numbers, we have, Cor. If there Q×R=S×N: hence (P. 2), N : Q : : R: S. be two sets of proportionals, in which the ratio of an antecedent and consequent of the one is equal to the ratio of an antecedent and consequent of the other, the remaining terms will be proportional. For, if we had the two proportions, M : P :: N: Q and R : S :: T: V, we shall also have |