Book V. pofition] AB is greater than CD; therefore G B is greater than H D. But fince A G is equal to E, and C H to F; AG and F will be equal to c H and E. But if equal magnitudes be added to unequal ones, the whole will be unequal: Wherefore fince G B, HD are unequal, and G B is the greater, if AG and F be added to GB, and C H and E to HD; A B and F will be greater than CD and E. If therefore four magnitudes be proportional, the greateft and leaft of them taken together will be greater than the other two taken togerher. Which was to be demonftrated. b If the antecedent of one of the ratios be a maximum, the confequent of the other will be a minimum. And contrariwise, if the antecedent of one of the ratios be a minimum, the confequent of the other will be a maximum. But all the four magnitudes must be of the fame kind, viz. lines, fuperficies, or folids, otherwise the propofition will not hold good: For the greatest and leaft put together cannot make one magnitude, no more than the other two can. It is eafy to demonftrate this theorem in right lines, by means of p prop. 7. and 35. lib. iii. and prop. 16. lib. vi. N. B. Some of the Commentators have added several other propofitions to this fifth book, but I think there is no occafion for them. But the following useful propofition fhould not be omitted, viz. If four magnitudes be proportionals, the fum of the firft and fecond will be to their difference, as the fum of the second and third is to their difference. The proof eafily follows [by prop. 11. 5.] EUCLID's EUCLID's ELEMENTS. BOOK VI. I. DEFINITION S. Imilar right lined figures are those which have their feveral angles equal each to each, and the fides about the equal angles proportional. 2. Figures are reciprocal, when in each figure, the antecedents and confequents are the terms of ratios2. a In the trian D gles A B C, C D F or the parallelograms A B, D F, if the fide BC of the one be to the B fide C F of the o- former. Where E B A the firft antecedent в c and the laft confequent a c of the equal ratios B C to CF, as CD to AC, are both in one of the figures, and also the second antecedent CD and the first confequent c F are also both in the fame figure, thofe triangles and parallelograms are faid to be reciprocal figures. Two figures are therefore said to be reciprocal, or their fides are faid to be reciprocally proportional when their fides are fo proportional, that there is one antecedent of one of two equal ratios, and the confequent of the other equal ratio, in each figure. And as the fides about two equal angles of equal triangles 3. A 3. A right line is faid to be cut or divided into mean and extreme ratio, when that line fhall be to the greater fegment, as the greater fegment is to the leffer. 4. The altitude of any figure is the perpendicular drawn from the vertex [or top] to the base b. 5. A ratio is faid to be compounded of ratios when the quantities of the ratios multiplied between themselves pròduce that ratio c. are always reciprocally proportional, fo will the fides about the equal angles of equiangular triangles, be always directly proportional. Thefe diftinctions of reciprocal and direct proportion are very useful, and ought therefore to be particularly regarded. But the definition of reciprocal figures feems to be useless. b The altitude or height of a figure is the diftance or shorteft length from the bottom of it to the top; and the perpendicular rather measures that distance, than is the distance itself; and when figures have the fame altitude, the perpendiculars drawn from their vertexes or tops upon their bafes are equal. As let there be any number of quantities generally reprefented by the letters A, B, C, D, E, the ratio of the extremes and E is compounded of the ratios of A to B, of в to C, of c to D, and of D to E; that is, if A divided by в be the mea A fure or quantity of the ratio of A to B ; then will a fraction, be the quantity of the ratio of A to B quantity of the ratio of s to c; the quantity of the ra D tio of c to D, and the quantity of the ratio of D to E. And E the product of the multiplication of all these quantities, according to the rules of multiplication of fractions, that is, multiplying all the numerators A, B, C, D, together, and dividing them by all the denominators B, C, D, E, multiplied together (or A into в into c into D divided by в into c, into D into E) will be the compound ratio of a into ɛ. For the product в into c into D being both in the numerator, and the denominator of that fraction divided by E; that is the ratio of a to E. Many have complained of this definition, and taxed Euclid (if he was the author) with not only explaining one unknown thing by another, for he has not told us what the quantity of a ratio à ratio is, nor what is to be understood by multiplying the quan tities of the ratios into one another. But alfo for putting the definition as general to all quantities, when nevertheless it is only particular, viz. relates to the ratios of numbers and commenfurable quantities. But altho' Euclid (or the author of this Definition whoever he was) cannot perhaps be intirely freed from blame in this matter, yet the difficulty of the subject, I mean the nature of ratios, requiring more to be faid to be quite right in the matter, than he thought neceffary to mention, and what he has mentioned, imperfect as it is, is fo far intelligible and neceffary, that his reader understands him to mean by compound ratio the multiplication of the quantities of fimple ratios, and not their addition; and that he has faid enough to make the following part of his work intelligible. I fay, all this confidered, will fufficiently excufe him. He might think it was difficult to give a better definition, and fome one or other was to be given. The fixth book could not be understood without it; therefore he gave the beft he could, confiftent with brevity, and the intelligence of the following part of his work. Scarborough, in feveral paffages of his obfervations upon this fifth definition, fays, amongst other things, 1. That the tenth definition of the fifth book is a fufficient definition of compound ratio. And fecondly, that this fifth definition is never used in thefe Elements of Euclid, or in the Conics of Apollonius, or elfewhere in Archimedes. But in all these and other geometrical writers, compofition of proportion is ever taken in the fenfe and notion of def. 10 lib. v. and to make this affertion appear in fome measure good, Scarborough, in his demonftration of the twenty-third propofition of this fixth book, makes Euclid, (or the author of the demonstration of this propofition whoever he was) to fay (fee the figure) But the ratio of « to м is compounded of the ratio of K to L, and of L to M, according to def. 10. of the fifth book; when at the fame time Euclid himself really fays it is fo by the fifth definition of the fixth book. Wherefore Euclid himself plainly uses this definition in the twenty-third propofition of the fixth book. And as to the tenth definition of the fifth book, it is only a particular cafe of compound ratio, viz. where the ratios are the fame, and so is useless where the ratios are unequal; whereas this fifth definition extends to any unequal ratios whatsoever. PROPOSITION I. THEOREM. Triangles and parallelograms which have the fame altitude, have the fame ratio to one another, as their bafes &. Let the triangles ABC, ACD, and the parallelograms EC, CF have the fame altitude, viz. the perpendicular drawn from the point A to B D: I say as the base B C is to the bafe CD, fo is the triangle ABC to the triangle ACD, and the parallelogram EC to the parallelogram C F. For continue out BD both ways to the points H, L, and take any number of right lines BG, GH each equal to the base BC, and any number DK, KL each equal to the base And join AG, AH, AK, AL. CD. Then becaufe CB, BG, GH, are equal to one another; [by 38. 1.] the triangles AGH, AGB, ABC will be equal to one another: Therefore the base HC is the fame multiple of the bafe BC, that the triangle AHC is of the trian I EA F HG BC D K gle ABC: By the fame reafon the bafe LC is the fame multiple of the bafe CD, that the triangle ALC is of the triangle ACD and [by 38. 1.] if the base HC be equal to the base LCL, the triangle AN c is equal to the triangle ALC If the base HC exceeds the bafe CL, the triangle AHC will alfo exceed the triangle ALC; and if the base HC be less than the base CL; the triangle AHC will likewife be less than the triangle ALC. There are therefore four magnitudes, viz. the two bases BC, CD, and the two triangles ABC, ACD, and there are equimultiples taken of the base BC, and the triangle ABC, viz. HC of the base, and the triangle AHC as alfo other equimultiples of the base CD, and triangle A C D, viz. c L of the bafe and the triangle ALC. And it has been proved if the base HC exceeds the base C L, the triangle AHC exceeds the triangle ALC; and if the base H c be equal to the base C L, the triangle AHC is equal to the triangle ALC: If the one be less, the other will be lefs too. Therefore [by 5. def. 5.] as the bafe |
