A DEFINITIONS. J. Lefs 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 a multiple of a lefs, when the greater is measured by the less, 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 See N. kind to one another, in respect 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 fecond, which the third has to the fourth, when any equimultiples whatsoever of the firft and third being taken, and any equimultiples whatfoever of the fecond and fourth; if the multiple of the first be less than that of the fecond, the multiple of the third is also lefs than that of the fourth; or, if the multiple of the firft be equal to that of the second, the multiple of the third is alfo equal to that of the fourth; H 4 or, Book V. See Ne or, if the multiple of the firft be greater than that of the fecond, the multiple of the third is alfo greater 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 saying, the firft is to the second, as the third to the fourth.' VII. When of the equimultiples of four magnitudes (taken as in the fifth definition) the multiple of the firft is greater than that of the fecond, but the multiple of the third is not greater than the multiple of the fourth; then the first 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 less ratio than the first has to the fecond. VIII. "Analogy, or proportion, is the fimilitude of ratios.” IX. Proportion confifts in three terms at least. X. When three magnitudes are proportionals, the first is faid to have to the third the duplicate ratio of that which it has to the fecond. XI. When four magnitudes are continual proportionals, the firft 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 propor tionals. Definition A, to wit, of compound ratio. When there are any number of magnitudes of the fame kind, the firft 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 laft D the ratio compounded of the ratio of 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 And if A has to B, the fame ratio which E has to F; and B Book V. to C, the fame ratio that G has to H; and C to D, the fame that K has to L; then, by this definition, A is faid to have to D. the ratio 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 understood when it is more briefly expreffed, by faying 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 ufe of the following technical words to fignify certain ways of changing either the order or magni'tude of proportionals, fo as that they continue ftill to be • proportionals.' XIII. Permutando, or alternando, by permutation, or alternately; See N. this word is used when there are four proportionals, and it is inferred, that the first has the fame ratio to the third, which the fecond has to the fourth; or that the first is to the third, as the second to the fourth: As is hown in the 16th prop. of this 5th book. XIV. Invertendo, by inversion: When there are four proportionals, and it is inferred, that the second is to the firft, as the fourth to the third. Prop. B. book. 5. XV.. Componendo, by compofition; when there are four proportionals, and it is inferred, that the firft, together with the fecond, is to the fecond, as the third, together with the fourth, is to the fourth. 18th prop. book 5. XVI. Dividendo, by divifion; when there are four proportionals, and it is inferred, that the excefs of the first above the fecond, is to the fecond, as the excefs of the third above the fourth, is to the fourth. 17th prop. book 5. XVII. Convertendo, by converfion; when there are four proportionals, and it is inferred, that the firft is to its excefs above the fecond Book V. fecond, as the third to its excefs above the fourth: Prop. E book 5. XVIII. Ex aequali (fc. diftantia), or ex aequo, from equality of diftance; 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 firft is to the laft of the firft rank of magnitudes, as the first is to the laft of the others: Of this there are the two following kinds, which arife from the different order in which the magnitudes are taken two and two.' XIX. Ex aequali, from equality; this term is ufed 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 firft rank, fo is the second to the third of the other; and fo on in order, and the inference is as mentioned in the preceding definition; whence this is called ordinate proportion. It is demonftrated in 22d prop. book 5. XX. Ex aequali, in proportione perturbata, feu inordinata; from equality, in perturbate or diforderly proportion*; this term is ufed when the firft magnitude is to the fecond of the first rank, as the laft but one is to the laft of the fecond 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 first rank, fo is the third from the laft to the laft but two of the second rank; and fo on in a cross order: And the inference is as in the 18th definition. It is demonftrated in the 23d prop. of book 5. A XI O I. Etudes, are equal to one another of equal magni. 4. Prop. lib. 2. Archimedis de fphaera et cylindre. II. Those II. 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. IV. That magnitude of which a multiple is greater than the fame multiple of another, is greater than that other magnitude. PROP. I. THEOR. 1 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 shall all the first magnitudes be of all the other. 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 fhall 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 A CH, HD equal each of them to F: The number therefore of the magnitudes CH, HD shall be equal to the number of the others AG, GB: And because AG is equal to E, and CH G C to F, therefore AG and CH together are B El Therefore, if any magnitudes, how many foever, be equimultiples of as many, each of each, whatsoever 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 demonstration • holds & Ax. 2. 1. |