PROPOSITION G. THEOREM. If several ratios be the same to several ratios, each to each; the ratio which is compounded of ratios which are the same to the first ratios, each to each, shall be the same to the ratio compounded of ratios which are the same to the other ratios, each to each. Let be to B, as E to F; and C to D, as G to H: and let A be to B, as K to L; and C to D, as L to M. Then the ratio of K to M, by the definition of compound ratio, is compounded of the ratios of K to L, and L to M, which are the same with the ratios of A to B, and C to D. Again, as E to F, so let N be to 0; and as G to H, so let O be to P. Then the ratio of N to P is compounded of the ratios of N to 0, and 0 to P, which are the same with the ratios of E to F, and G to H: and it is to be shewn that the ratio of K to M, is the same with the ratio of N to P; or that K is to M, as N to P. A.B.C.D. K.L.M. Because K is to L, as (A to B, that is, as E to F, that is, as) N to 0: and as L to M, so is (C to D, and so is G to H, and so is) O to P: ex æquali K is to M, as N to P. (v. 22.) Q. E. D. If a ratio which is compounded of several ratios be the same to a ratio which is compounded of several other ratios; and if one of the first ratios, or the ratio which is compounded of several of them, be the same to one of the last ratios, or to the ratio which is compounded of several of them; then the remaining ratio of the first, or, if there be more than one, the ratio compounded of the remaining ratios, shall be the same to the remaining ratio of the lust, or, if there be more than one, to the ratio compounded of these remaining ratios. Let the first ratios be those of A to B, B to C, C to D, D to E, and E to F; and let the other ratios be those of G to H, H to K, K to L, and L to M: also, let the ratio of A to F, which is compounded of the first ratios, be the same with the ratio of G to M, which is compounded of the other ratios; and besides, let the ratio of A to D, which is compounded of the ratios of A to B, B to C, C to D, be the same with the ratio of G to K, which is compounded of the ratios of G to H, and H to K. Then the ratio compounded of the remaining first ratios, to wit, of the ratios of D to E, and E to F, which compounded ratio is the ratio of D to F, shall be the same with the ratio of K to M, which is compounded of the remaining ratios of K to L, and L to M of the other ratios. A.B.C.D.E.F. Because, by the hypothesis, A is to D, as G to K, and as A is to F, so is G to M; (hyp.) If, therefore, a ratio which is, &c. Q.E.D. PROPOSITION K. THEOREM. If there be any number of ratios, and any number of other ratios such, that the ratio which is compounded of ratios which are the same to the first ratios, each to each, is the same to the ratio which is compounded of ratios which are the same, each to each, to the last ratios ; and if one of the first ratios, or the ratio which is compounded of ratios which are the same to several of the first ratios, each to each, be the same to one of the last ratios, or to the ratio which is compounded of ratios which are the same, each to each, to several of the last ratios ; then the remaining ratio of the first, or, if there be more than one, the ratio which is compounded of ratios which are the same each to each to the remaining ratios of the first, shall be the same to the remaining ratio of the last, or, if there be more than one, to the ratio which is compounded of ratios which are the same each to each to these remaining ratios. Let the ratios of A to B, C to D, E to F, be the first ratios : and the ratios of G to H, K to L, M N, O to P, Q to R, be the other ratios : and let A be to B, as S to T; and C to D, as T to V; and E to F, as V to X: therefore, by the definition of compound ratio, the ratio of S to X is compounded of the ratios of S to T, T to V, and V to X, which are the same to the ratios of A to B, C to D, E to F: each to each. Also, as G to H, so let Y be to Z; and K to L, as Z to a; M to N, as a to b; O to P, as b to c; and Q to R, as c to d: therefore, by the same definition, the ratio of Y to d is compounded of the ratios of Y to Z, Z to a, a to b, 6 to c, and c to d, which are the same, each to each, to the ratios of G to H, K to L, M to N, O to P, and Q to R: therefore, by the hypothesis, S is to X, as Y to d. Also, let the ratio of A to B, that is, the ratio of S to T, which is one of the first ratios, be the same to the ratio of e to g, which is compounded of the ratios of e to f, and f to g, which, by the hypothesis, are the same to the ratios of G to H, and K to L, two of the other ratios; and let the ratio of h to l be that which is compounded of the ratios of h to k, and k to l, which are the same to the remaining first ratios, viz. of C to D, and E to F; also, let the ratio of m to p, be that which is compounded of the ratios of m to n, n to o, and o to p, which are the same, each to each, to the remaining other ratios, viz. of M to N, O to P, and Q to R. Then the ratio of h to l shall be the same to the ratio of m to p; or h shall be to I, as m to p. h, k, l. m, n, o, p. Because e is to f, as (G to H, that is, as) Y to Z; and f is to g, as (K to L, that is, as) Z to a; wherefore S is to T, as Y to a; (v. 11.) but S is to X, as Y to D; (hyp.) therefore, ex æquali, T is to X, as a to d: also, because h is to k as (C to D, that is, as) T to V; (hyp.) and k is to l as (E to F, that is, as) V to X; therefore, ex æquali, h is to l, as T to x : in like manner, it may be demonstrated, that m is to p, as a to d; and it has been shewn, that T is to X, as a to d; therefore h is to l, as m to p. (v. 11.) Q.E.D. The propositions G and K are usually, for the sake of brevity, expressed in the same terms with propositions F and H: and therefore it was proper to shew the true meaning of them when they are so expressed; especially since they are very frequently made use of by geometers. NOTES TO BOOK V. In the first four books of the Elements are considered, only the absolute equality and inequality of Geometrical magnitudes. The fifth book contains an exposition of the principles whereby a more definite comparison may be instituted of the relation of magnitudes, besides their simple equality or inequality. Def. 1, 11. In the first four books the word part is used in the same sense as we find it in the ninth axiom, “The whole is greater than its part:” where the word part means any portion whatever of any whole magnitude : but in the fifth book, the word part is restricted to mean that portion of magnitude which is contained an exact number of times in the whole. For instance, if any straight line be taken two, three, four, or any number of times another straight line, by Euc. I. 3; the less line is called a part, or rather a submultiple of the greater line; and the greater, a multiple of the less line. The multiple is composed of a repetition of the same magnitude, and these definitions suppose that the multiple may be divided into its parts, any one of which is a measure of the multiple. And it is also obvious that when there are two magnitudes, one of which is a multiple of the other, the two magnitudes must be of the same kind, that is, they must be two lines, two angles, two surfaces, or two solids. Euclid does not seem to consider one triangle a part or multiple of another, except when both are between the same parallels ; and a triangle is doubled, trebled, &c., by doubling, trebling, &c. the base. The same may be said of parallelograms. Also the arcs of two circles are not considered as part and multiple, unless the circles have equal radii. Angles, arcs, and sectors of equal circles may be doubled, trebled, or any multiples found by Prop. XXVI—XXXix, Book 111. Two magnitudes are said to be commensurable when a third magnitude of the same kind can be found which will measure both of them; and this third magnitude is called their common measure: and when it is the greatest magnitude which will measure both of them, it is called the greatest common measure of the two magnitudes : also when two magnitudes of the same kind have no common measure, they are said to be incommensurable. The same terms are also applied to numbers. Unity has no magnitude, properly so called, but may represent that portion of every kind of magnitude which is assumed as the measure of all magnitudes of the same kind. The composition of unities cannot produce Geometrical magnitude ; three units are more in number than one unit, but still as much different from magnitude as unity itself. Numbers may be considered as quantities, for we consider every thing that can be exactly measured, as a quantity. Unity is a common measure of all rational numbers, and all numerical reasonings proceed upon the hypothesis that the unit is the same throughout the whole of any particular process. Euclid has not fixed the magnitude of any unit of length, nor made reference to any unit of measure of angles, surfaces, or volumes. Hence arises an essential difference between number and magnitude; unity, being invariable, measures all rational numbers; but though any quantity be assumed as the unit of magnitude, it is impossible to assert that this assumed unit will measure all other magnitudes of the same kind. All whole numbers therefore are commensurable ; for unity is their common measure : also all rational fractions proper or improper, are commensurable ; for any such frac tions may be reduced to other equivalent fractions having one common denominator, and that fraction whose denominator is the common denominator, and whose numerator is unity, will measure any one of the fractions. Two magnitudes having a common measure can be represented by two numbers which express the number of times the common measure is contained in both the magnitudes. But two incommensurable magnitudes cannot be exactly represented by any two whole numbers or fractions whatever ; as, for instance, the side of a square is incommensurable to the diagonal of the square. For, it may be shewn numerically, that if the side of the square contain one unity of length, the diagonal contains more than one, but less than 2 units of length. If the side be divided into 10 units, the diagonal contains more than 14, but less than 15 such units. Also if the side contain 100 units, the diagonal contains more than 141, but less than 142 of such units. It is also obvious, that as the side is successively divided into a greater number of equal parts, the error in the magnitude of the diagonal will be diminished continually, but never can be entirely exhausted ; and therefore into whatever number of equal parts the side of a square be divided, the diagonal will never contain an exact number of such parts. Thus the diagonal and side of a square having no common measure, cannot be exactly represented by any two numbers. The term equimultiple in Geometry is to be understood of magnitudes of the same kind, or of different kinds, taken an equal number of times, and implies only a division of the magnitudes into the same number of equal parts. Thus, if two given lines are trebled, the trebles of the lines are equimultiples of the two lines : and if a given line and a given angle or triangle be trebled, the trebles of the line and angle or triangle are equimultiples of the line and angle or triangle : as (fig. vi. 1.) the straight line HC and the triangle AHC are equimultiples of the line BC and the triangle ABC: and (fig. vi. 33.) the arc EN and the angle EHN are equimultiples of the arc EF and the angle EHF. Def. ΙΙι. Λόγος έστι δύο μεγεθών ομογενών η κατα πηλικότητα προς άλληλα ποια σχέσις. . By this definition of ratio is to be understood the conception of the mutual relation of two magnitudes of the same kind, as two straight lines, two angles, two surfaces, or two solids. To prevent any misconception, Def. iv. lays down the criterion, whereby it may be known what kinds of magnitudes can have a ratio to one another; namely, Λόγον έχειν προς άλληλα μεγέθη λέγεται, α δύναται πολλαπλασιαζόμενα αλλήλων υπερέχειν. . Magnitudes are said to have a ratio to one another, which, when they are multiplied, can exceed one another ;” in other words, the magnitudes which are capable of mutual comparison must be of the same kind. The former of the two terms is called the antecedent ; and the latter, the consequent of the ratio. If the antecedent and consequent are equal, the ratio is called a ratio of equality; but if the antecedent be greater or less than the consequent, the ratio is called a ratio of greater or less inequality. Care must be taken not to confound the expressions " ratio of quality” and “equality of ratio :" the former is applied to the terms of a ratio when they, the antecedent and consequent, are equal to one another, but the latter, to the ratios, when they are equal. Arithmetical ratio has been defined to be the relation which one number bears to another with respect to quotity; the comparison being made by considering what multiple, part or parts, one number is of the other. An arithmetical ratio, therefore, is represented by the quotient which arises from dividing the antecedent by the consequent of the ratio ; or by the fraction which has the antecedent for its numerator and the consequent for its denominator. Hence it will at once be obvious that the properties of arithmetical ratios will be made to depend on the properties of fractions. |