Book V. THE ELEMENT S OF EUCLID. A BOOK V. DEFINITIONS. I. LESS magnitude is faid to be a part of a greater magnitude, when the lefs measures the greater, that is, when the lefs is 'contained a certain number of times exactly in the greater.' II. A greater magnitude is faid to be à multiple of a lefs, when the greater is measured by the lefs, that is, when the greater contains the less a certain number of times exactly.' III. "Ratio is a mutual relation of two magnitudes of the fame kind to See N. "one another, in refpect of quantity." IV. Magnitudes are faid to have a ratio to one another, when the lefs can be multiplied fo as to exceed the other. V. The first of four magnitudes is faid to have the fame ratio to the fe cond, which the third has to the fourth, when any equimultiples whatsoever of the firft and third being taken, and any equimultiples whatsoever of the fecond and fourth; if the multiple of the first be less than that of the fecond, the multiple of the third is álfo less than that of the fourth; or, if the multiple of the first be equal to that of the fecond, the multiple of the third is alfo equal to that of the fourth; or, if the multiple of the first be greater than Book V. than that of the fecond, the multiple of the third is also greater See N. than that of the fourth. VI. Magnitudes which have the fame ratio are called proportionals. N. B. When four magnitudes are proportionals, it is ufually expreffed by faying, the firft is to the fecond, as the third to 'the fourth.' VII. When of the equimultiples of four magnitudes (taken as in the 5th Definition) the multiple of the first is greater than that of the fecond, but the multiple of the third is not greater than the multiple of the fourth; then the firft is faid to have to the fecond a greater ratio than the third magnitude has to the fourth; and on the contrary, the third is faid to have to the fourth a lefs ratio than the first has to the second. VIII. "Analogy, or proportion, is the fimilitude of ratios." IX. Proportion confifts in three terms at least. X. When three magnitudes are proportionals, the firft is faid to have to the third the duplicate ratio of that which it has to the second. XI. When four magnitudes are continual proportionals, the first is faid to have to the fourth the Triplicate ratio of that which it has to the fecond, and fo on Quadruplicate, &c. increasing the denomination ftill by unity, in any number of proportionals. Definition A, to wit, of Compound ratio. When there are any number of magnitudes of the fame kind, the first is faid to have to the laft of them the ratio compounded of the ratio which the firft has to the fecond, and of the ratio which the fecond has to the third, and of the ratio which the third has to' the fourth, and fo on unto the last magnitude. For example, If A, B, C, D be four magnitudes of the fame kind, the first A is faid to have to the last D the ratio compounded of the ratio A to B, and of the ratio of B to C, and of the ratio of C to D; or, the ratio of A to D is faid to be compounded of the ratios of A to B, B to C, and C to D. And if A has to B, the fame ratio which E has to F; and B to C, the fame ratio that G has to H; and C to D, the fame that K has has to L; then, by this Definition, A is faid to have to D the ra- Book V. tio compounded of ratios which are the fame with the ratios of E to F, G to H, and K to L. and the fame thing is to be underftood when it is more briefly expreffed by saying A has to D, the ratio compounded of the ratios of E to F, G to H, and K to L. In like manner, the fame things being fuppofed, if M has to N the fame ratio which A has to D, then, for fhortnefs fake, M is faid to have to N, the ratio compounded of the ratios of E to F, G to H, and K to L. XII. In proportionals, the antecedent terms are called homologous to one another, as alfo the confequents to one another. Geometers make use of the following technical words to fignify ' certain ways of changing either the order or magnitude of proportionals, fo as that they continue ftill to be proportionals.' XIII. Permutando, or Alternando, by Permutation, or alternately; this See it. word is used when there are four proportionals, and it is inferred, that the firft has the fame ratio to the third, which the fecond has to the fourth; or that the first is to the third, as the fecond to the fourth. as is fhewn in the 16th Prop. of this 5th Book. XIV. Invertendo, by Inverfion; when there are four proportionals, and it is inferred, that the fecond is to the firft, as the fourth to the third. Prop. B. Book 5th. XV. Componendo, by Compofition; when there are four proportionals; and it is inferred, that the firft together with the fecond, is to the second, as the third together with the fourth, is to the fourth. 18th Prop. Book 5th. XVI. Dividendo, by Divifion; when there are four proportionals, and if is inferred, that the Excefs of the firft above the fecond, is to the fecond, as the Excefs of the third above the fourth, is to the fourth. 17th Prop. Book 5th.. XVII. Convertendo, by Converfion; when there are four proportionals, and it is inferred, that the first is to its Excefs above the fecond, as the third to its Excefs above the fourth. Prop. E. Book 5th. XVIII, Ex H Ex aequali (fc.diftantia), or,ex aequo, from equality of distance; when there is any number of magnitudes more than two, and as many others, so that they are proportionals when taken two and two of each rank, and it is inferred, that the first is to the last of the first rank of magnitudes, as the firft is to the laft of the others. 'of this 'there are the two following kinds, which arife from the diffe' rent order in which the magnitudes are taken two and two.' XIX. Ex aequali, from aequality; this term is used fimply by itself, when the first magnitude is to the fecond of the first rank, as the first to the fecond of the other rank; and as the fecond is to the third of the first rank, fo is the fecond to the third of the other; and fo on in order, and the inference is as mentioned in the preceeding Definition; whence this is called Ordinate Proportion. It is demonftrated in 22d Prop. Book 5th. XX. Ex aequali, in proportione perturbata, feu inordinata, from equality, in perturbate or diforderly proportion *; this term is used when the firft magnitude is to the fecond of the first rank, as the laft but one is to the last of the second rank; and as the second is to the third of the first rank, so is the last but two to the last but one of the fecond rank; and as the third is to the fourth of the firft rank, fo is the third from the laft to the last but two of the fecond rank; and fo on in a crofs order. and the inference is as in the 18th Definition. It is demonftrated in 23d Prop. of Book 5th. E AXIOM S. I. QUIMULTIPLES of the fame, or of equal magnitudes, are equal to one another. Thofe magnitudes of which the fame, or equal magnitudes, are equimultiples, are equal to one another. III. A multiple of a greater magnitude is greater than the fame multiple of a lefs. 4. Prop. Lib. 2. Archimedis de fphaera et cylindro. IV. That IV. That magnitude of which a multiple is greater than the fame multiple of another, is greater than that other magnitude. PROP. I. THEOR. IF F any number of magnitudes be equimultiples of as many, each of each; what multiple foever any one of them is of its part, the fame multiple fhall all the first magnitudes be of all the other. El Let any number of magnitudes AB, CD be equimultiples of as many others E, F, each of each; whatsoever multiple AB is of E, the fame multiple shall AB and CD together be of E and F together. Because AB is the fame multiple of E that CD is of F, as many magnitudes as are in AB equal to E, fo many are there in CD equal to F. Divide AB into magnitudes equal to E, viz. AG, GB; and CD into CH, HD equal each of A them to F. the number therefore of the magnitudes CH, HD fhall be equal to the number of G the others AG, GB. and because AG is equal to È, and CH to F; therefore AG and CH together are equal to E and F together. for the fame reafon, because GB is equal to E, and HD to F; GB and HD together are equal to E and F together. Wherefore as many magnitudes as are in AB equal to E, fo many are there in AB, CD together equal to E and F together. Therefore whatfoever multiple AB is of E, the fame multiple is AB and CD together of E and F toge ther. B H D Therefore if any magnitudes, how many foever, be equimultiples of as many, each of each, whatfoever multiple any one of them is of its part, the fame multiple fhall all the first magnitudes be of all the other. for the fame Demonftration holds in any number of 'magnitudes, which was here applied to two.' Q. E. D. Book V. |