nitudes are said to be commensurable, 6. The terms of the ratio of two magwhen there is some magnitude which is nitudes, are the numbers which denote contained in each of them a certain how often a common measure of the number of times exactly. two is contained in each of them. They Magnitudes which have no common are distinguished by the names of antemeasure, are said to be incommen- cedent and consequent, according to the surable. corresponding magnitudes. In the foreMagnitudes A, B, which have one going example 5 and 6 are the terms of common measure M, have also many the ratio of A to B, 5, the antecedent, others, indeed, an unlimited number and 6, the consequent. of common measures, for (as will be The terms of the same ratio of A to shown in Prop. 1.) every magnitude B, will be different according to the which is contained an exact number of common measure by which they are detimes in M, is contained also an exact termined ; the lowest terms being in all number of times in A and B; and cases those which are determined by the whether M be divided into two, or three, greatest common measure. It must be or any other number of equal parts, one observed, however, that no other terms of these parts will be contained an exact can express the same ratio, but such as number of times in M. are either the lowest, or equimultiples of Among the common measures of the the lowest terms; for the magnitudes same two magnitudes, there is, however, compared, can have no common meaalways one which is greater than any of sure, which is not either the greatest, or the others, and which (as will be shown contained a certain number of times in in Prop. 6.) is measured by every other. the greatest common measure. On This greatest common measure is al- the other hand, any terms whatever ways to be understood when “ the com- which are equimultiples of the lowest mon measure" is spoken of without terms will express the same ratio: thus, further specification. if A contain ths of B, it will contain 5. The numerical ratio of one mag- also ths of B, işths, and, generally, nitude to another with which it is com 3Xп mensurable, is a certain number, whole -ths of B, where any number whator fractional, which expresses how ever may be substituted for n. many, and what parts of the second are The ratio of B to A has the same contained in the first: for example, if terms with the ratio of A to B, but in an the common measure of A and B be inverse order: thus, if 5 : 6 be the nucontained in Å five times, and in B six merical ratio of A to B, 6 : 5 will be times, or, which is the same thing, if A that of B to A. contain ths of B, then A is said to have The ratio of B to A is accordingly to B the numerical ratio “5 to 6" said to be the inverse or reciprocal of the which is thus written 5; 6, or, in the ratio of A to B. fractional form, . If the terms of the ratio of A to In fact, the particular ratio of two B be equal, it is evident that the maggiven magnitudes, whether commensur- nitudes A, B, must likewise be equal. able or otherwise, can be conceive In this case the ratio is said to be a only by means of the numbers which ratio of equality. denote how often the same magnitude is Def. (7.) If there be two magnitudes of contained, or nearly contained, in each: the same kind, and other two, and if without these, no idea can be formed of the first contain a measure of the second, their relative magnitude; they constitute as often as the third contains a like meaits measure, true or approximate.* sure of the fourth; or, which is the same To these numbers, therefore, when thing, if the ratios of the first to the speaking of commensurable magnitudes, second, and of the third to the fourth be the term “ ratio” alone, i. e. without the expressed by the same terms; the first addition of “ numerical,” will be found is said to have to the second the same commonly applied in what follows. ratio which the third has to the fourth; and the four magnitudes are called * The numerical_ratio is accordingly designated, proportionals. by some writers, " The measure of the ratio" of one magnitude to another. This term has, however, been applied in a different sense, to which deference is more particularly due, as it has given rise to the word " logarithm," of which it is the literal interpretation, A 5 Xn e. BE For example, let A B C D, and of the fourth : also, when this is the case, EFGH be two rectangles, having the third is said to have to the fourth equal altitudes, and let their bases A B, a less ratio than the first has to the EF, contain the same straight line M second. 5 and 6 times respectively : divide A B, This definition can, in no case, apply E F, into the parts A b, &c. Ef, &c. each to the same magnitudes which come equal to M, and through the points of under def. [7], i. e. one magnitude cannot division draw right lines b c, &c. fg, &c. have to another the same ratio as a parallel to AD, and EH, thereby divid- third to a fourth by def. [?], and at the ing the rectangles ABCD, EFGH, same time a greater or a less ratio than into 5 and 6 smaller rectangles respec- the third has to the fourth, by this detively, Abc D, &c. Efg H, &c. all equal finition. (See Prop. 9. Cor. 1 and 2.) to one another (I. 25.). Then, 5 : 6 is the Much less can one magnitude have ratio of the rectangle A B C D to the to another a greater ratio than a third rectangle E FGH, because A B C D and has to a fourth, and at the same time a EFGH contain the same rectangle 5 less ratio than the third has to the fourth, and 6 times respectively; and the same by this definition. 5: 6 is also the ratio of the base AB 9. When four magnitudes A, B, C, D, to the base EF, because A B and EF are proportionals, they are said to concontain the same straight line 5 and 6 stitute a proportion,which is thus written, times respectively. Therefore, the ratios A:B::C:D. of the first to the second, and of the i. e. “A is to B as C is to D.” third to the fourth, are expressed by Of a proportion, the first and last the same terms, and the two rectangles, terms are called the extremes, and the and their two bases, are proportionals. second and third the means ; thus A, D In the preceding, and in every other are the extremes, and B, C the means, of instance of commensurable proportion- the proportion A:B::C:D. The terms als, the first two, and the second two, A and C are said to be homologous, have a common numerical ratio : and, as also B and D; the former being anin every case, if two magnitudes, and tecedents, and the latter consequents, in other two, have a common numerical the proportion. ratio, the four magnitudes are, accord- It is indifferent in the statement of a ing to this definition, proportionals. proportion which of the ratios precedes It is evident from the observations the other, for it is evident, that if A on def. 6, that if four magnitudes be has to B the same ratio as C has to proportionals, any other terms express- D, C has to D the same ratio as A to B. ing the ratio of the first to the second, Hence, A:B::C:D, and C:D::A:B, must likewise express the ratio of the signify the same proportion—the only third to the fourth. For the terms which difference being, that the extremes of one determine the proportion, are either expression are the means of the other. the lowest terms, or equimultiples of Since magnitudes cannot be comthem (see Prop. 6. Cor. 1.): and in pared, except they be of the same kind, either case, the lowest terms which it is manifest that the first and second express the ratio of the first to the terms of a proportion must be of the same second, and of the third to the fourth, kind, as also the third and fourth; yet the must be the same; therefore, because first and third may be of different kinds : any other terms expressing the ratio of e. g. 10lbs : 6lbs::15 ft. : 9ft. is a true the first to the second must be equi- proportion; for the ratio of the first to multiples of the lowest terms, that is, the second, as well as of the third to the of the lowest terms of the ratio of the fourth, is 5 : 3, and yet 10lbs. and 15ft. third to the fourth, such terms express are not magnitudes of the same kind. also the ratio of the third to the fourth 10. Three magnitudes of the same kind (see observations on Def. 6.) are said to be proportionals or in conThe same will be demonstrated more tinued proportion, when the first has to at large in Prop. [9]. the second the same ratio which the Def. [8]. If there be two magnitudes second has to the third. Magnitudes of the same kind, and other two, the first A, B, C, which are in continued prois said to have to the second a greater portion, may be written thus, A:B:C, ratio than the third has to the fourth, i.e. “A is to B, as B to C.” In this case, when the first contains some measure of B is called a mean proportional or geothe second a greater number of times metrical mean between A and C, and than the third contains a like measure C a third proportional to A and B. D 11. Any number of magnitudes of the same may be said, if, instead of three and same kind are said to be in continued four, any other numbers whatever be proportion, when the first is to the taken, i. e. if A be any multiple of B, second, as the second to the third, as the and B any multiple of 0. Third to the fourth, and so on. Magni- Therefore, &c. tudes A, B, C, D, &c. which are in con- Cor. 1. If one magnitude measure tinued proportion, may be thus written, another, it will measure any multiple of A:B:0:D:&c. that other. The magnitudes of such a series are Cor. 2. Hence, the ratio of A to B said to be in geometrical progression being expressed by certain given terms, and B, C, are called two geometrical as 5:6, the same ratio may be expressed means between A and B ; again, B, C, D, by any terms which are equimultiples of three geometrical means between A the given terms, as 10 x 5:10 x6. For, and E; and so on. if M be the common measure which is Also, in this case, the first A is said contained in A five times and in B six to have to the third C, the duplicate times, any measure of M, as the tenth ratio of that which it has to the second part, will also measure A and B (Cor. 1.) B-to the fourth D, the triplicate ratio and will be contained in A 10x 5 times, of that which it has to B, and so on : and in B 10 X 6 times. and reciprocally, A is said to have to B Cor. 3. Hence, also, reversely, the the subduplicate ratio of that which it ratio of A to B being expressed by any has to C, the subtriplicate ratio of that terms, as 10 x 5 : 10 x6, which have a which it has to D, and so on. common factor, the same ratio may be 12. If there be any number of magni- expressed by any terms, as 5 : 6, which tudes of the same kind, A, B, C, D, the are like parts of the given terms. first A is said to have to the last D a For, if M be the common measure ratio which is compounded of the ratios which is contained in A 10x5 times, of A to B, B to C, and C to D. and in B 10 x 6 times, it is evident that Also, if K and L, M and N, P and Q, 10 M will be contained in A 5 times, be any other magnitudes, and if the ra- and in B 6 times. tios of K to L, of M to N, and of P to Q, be the same respectively with the PROP. 2. (Euc. V. 3.) ratios of A to B, of B to C, and of C to If two magnitudes be equimultiples of D, A is said to have to D a ratio which two others, and if these be likewise equiis compounded of the ratios of K to L, multiples of two third magnitudes, the M to N, and P to Q. two first shall be equimultiples of the two Axioms. (Euc. v, Ax, 1, 2, 3, 4.) third. Let A and A' contain B and B' respec1. Equimultiples of the same or of equal magnitudes are equal to one ano tively three times, and let B and B'contain ther, C and C respectively four times; then, 2. Those magnitudes of which the as in the demonstration of the preceding same, or equal magnitudes are equi- proposition, A and A' being equal to three multiples, are equal to one another. times B and three times B' respectively, 3. A multiple of a greater magnitude contain C and C' respectively 3 X 4 times, is greater than the same multiple of a i.e. the same number of times exactly ; less, therefore A and A' are equimultiples of C and C'. 4. That magnitude of which a multiple is greater than the same multiple The same may be said, if, instead of of another, is greater than that other 3 and 4, any other numbers be taken, i. e, magnitude. if A, A' be any equimultiples of B, B', and B, B' any equimultiples of C, C'. Therefore, &c. If one magnitude be a multiple of an- Cor. If two magnitudes A, A' be equiother, and if this be likewise a multiple multiples of two others B, B', and like, of a third magnitude, the first shall be a wise of two third magnitudes C, C', and multiple of the third. if one of the second, B, be a multiple of Let A contain B three times, and let the corresponding one, C, of the third, B contain C four times; then because B the other second, B', shall be the same contains C four times, three times B or A multiple of the other third, C'. must contain C thrice as often, or twelve times, i.e. a certain number of times ex PROP. 3. actly; therefore A is a multiple of C. The A common measure M of any two a PROP. 1. any other 3 magnitudes A and B, measures also their shown that every one of the latter measum A + B, and their difference A-B. sures both A and B; therefore the For, if M be contained in A any num- greatest among them is the greatest ber of times, as 7, and in B any num- common measure of A and B. ber of times, as 4; it will evidently be contained in the sum of A and B, 7 + 4 or PROP. 5, 11 times, and in their differ By repeating the process indicated in ence 7 - 4 or 3 times, and the last proposition, with the remainder therefore will measure their and the lesser magnitude, and again sum and difference. with the new remainder (if there be one) The same may be said, M and the preceding, and so on, the greatest if, instead of 7 and 4, common measure of two given commennumbers be surable magnitudes A, B may be found. taken.* Let B, for instance, be contained in Therefore, &c. A twice (as in the last proposition), with a remainder R; let R be contained in PROP. 4. B three times, with a second remainder If there be two magnitudes A, B, and R, ; let R, be contained in R four times, if one of them be contained in the other with a third remainder Ry, and let R, be à certain number of times with a re contained in R, five times exactly. Then, mainder; any common measure of the because (by 4. Cor.) the greatest comtwo magnitudes shall measure the re- mon measure of A and B is the greatest mainder, and any common measure of common measure of B and R, that is the remainder and the lesser magnitude, (by the same Cor.) of R and Rg, that is, shall measure the greater also. again, of R, and Rz, and because Rg, Let B be contained in A twice being contained in itself once and a cerwith a remainder R, and let M be A tain number of times in R,, is the greatany common measure of A and est common measure of R, and Rg, it is B; then, since M is contained likewise the greatest common measure a certain number of times in B, of A and B. it is also contained a certain The same may be said, if instead of number of times in twice B (1.)* 2,3,4,5, any other numbers, supposed to and measures twice B: but it arise from a similar examination of any also measures A ; therefore (3.) two given commensurable magnitudes, be it measures the difference of A and taken. At every step of the process, the twice B, that is R. remainder, as R, is diminished by the folNext, let N be any common measure lowing remainder R, or by as many of R and B: then, as before, N mea- times R, as are contained in it, that is, sures twice B; therefore (3.) it measures in either case, by a magnitude greater the sum of R and twice B, that is, A. than the supposed greatest common And the reasoning, in either case, is measure, to procure the new remainder independent of the particular numbers Rg. In all cases therefore, after a numassumed. ber of steps, which is less than the numTherefore, &c, ber of times the lesser magnitude conCor. The greatest common measure tains the supposed greatest common of the remainder and lesser magnitude measure, a remainder will be found which is also the greatest common measure is equal to the greatest common measure. of the two magnitudes. For, since Therefore, &c. every common measure of A and B is Cor. 1. If the process admit of being also a common measure of B and R; continued through an unlimited number the greatest common measure of A and of steps, without arriving at a remainder B will be found among the common which measures the next preceding, the measures of B and R; and it has been magnitudes which are subjected to it, have no common measure, i. e. they are * The example of Euclid has been followed in incommensurable. annexing straight lines to illustrate this and many subsequent propositions, which are, however, not Cor. 2. By proceeding in a similar the less to be understood as applicable to and de- manner, the greatest common measure monstrated of magnitudes generally, as is evident from the language of the enunciation and demonstra- found; for if M'be taken, the greatest of three magnitudes A, B and C may be tion. + The reference is here to the first proposition of common measure of A and B, and M, the present Book; and generally, in such references the greatest common measure of M and as have no Roman numeral to indicate the Book, the current Book is always to be understood, C, then because the greatest common D 2 3 measure of A, B and C is to be found common factor must still remain, viz. among the common measures of M and the number which denotes how often the C, and because every one of the latter, first is contained in the greatest common being a measure of M, measures A and B factor. (1.), the greatest among them, that is, Thus, the greatest common factor of M,, is the greatest common measure of 204 and 240 is 12, as found by the Rule; A, B and C. therefore those numbers have no comIt is obvious, that in the same manner mon factor which is not a factor of 12; the Rule may be extended to any num- and if they be divided by any factor of ber of magnitudes. 12, as 6, the quotients 34 and 40 have Cor. 3. By help of this proposition still a common factor 2, which is the the lowest terms of the ratio of two given number of times the factor 6 is concommensurable magnitudes may be de. tained in 12. termined : for the lowest terms of their It is impossible to have a clear and ratio are the numbers which denote how correct apprehension of the subject beoften their greatest common measure is fore us, without a reference, not merely contained in each (see def. 6.). to numbers, but also to the properties Scholium. just mentioned (see Arithmetic, art. 54, It may be observed that the foregoing 55, 56, 57, 58, 63.). process includes the arithmetical rule for PROP. 6. finding the greatest common factor of two numbers: which is to divide the If a magnitude measure each of two greater number by the lesser, and find others, it shall either be the greatest the remainder; the lesser by the remain- common measure of the two, or it shall der, and find the second remainder, if be contained an exact number of times there be one; the preceding remainder in the greatest common measure. by this, and find the third remainder ; For, in the process of Prop. 5, it was and so on, until a remainder be found seen that every common measure of the which is contained an exact number of two magnitudes A and B measures also times in the next preceding ; this last the successive remainders, the last of remainder will be the greatest common which is the greatest common measure factor required. of A and B. Thus, if the numbers be 628 and 272, Therefore, &c. the successive remainders will be 84, 20, Cor. 1. The lowest terms of the ratio and 4, of which 4 is contained in 20 an of two magnitudes being determined by exact number of times : therefore, 4 is the greatest common measure of the the greatest common factor of the num- two, and any other common measure bers 628 and 272. being contained an exact number of If there be found no remainder which times in the greatest, any other terms is exactly contained in the preceding, expressing the same ratio must be equiuntil the course of the Rule produces a multiples of the lowest terms. remainder 1, the numbers have no com- This corollary has been cited by anmon factor but 1, and are said to be ticipation in the observations upon prime to one another. def. 6. The greatest common factor of two Cor. 2. The numerical ratio of two magnumbers being thus found, if the num- pitudes being given, if not already in its bers be divided by it, the quotients will, lowest terms, may be reduced to them by manifestly, be prime to one another. dividing the terms by their greatest comWith regard to other common factors of mon factor: for the lowest terms, being the same two numbers, every other com- determined by the greatest common mon factor must be contained an exact measure of the two magnitudes, must be number of times in the greatest: for it prime to one another, and there is no is contained an exact number of times other common factor but the greatest, in each of the remainders of the Rule, by which if two numbers be divided, the the last of which is the greatest common quotients will be prime to one another. factor. (See PROP. 5. Scholium.) Hence it follows, that, any two num- For example, the terms of the ratio bers being given, there is no other com- 628 : 272 have 4 for their greatest common factor but the greatest, by which if mon factor : therefore, dividing them by the numbers be divided, the quotients 4, the quotients 157 and 68 are the will be prime to one another; for, after lowest terms in which the ratio can be division by any other common factor, a expressed. |