2. Therefore, universally, it is manifest, if three Right Lines be proportional, as the first is to the third, so THEOR EM. Figure, are also similar to one another. similar to the Right-lined Figure C. I say, the Right-lined Figure A is also similar to the Right-lined Figure B. For, because the Right-lined Figure A is similar to • Defit. ofthe Right-lined Figure C, it shall be * equiangular thereto, and the sides about the equal Angles proportional. Again, because the Right-lined Figure B is similar to the Right-lined Figure C, it shall * be equiangular thereto; and the sides about the equal Angles will be proportional. Therefore each of the Right-lined Figures A, B, are equiangular to C, and they have the Sides about the equal Angles proportional. Wherefore the Right-lined Figure A is equiangular to the Right-lined Figure B; and the sides about the equal Angles are proportional. Wherefore, A is similar to B ; which was to be demonstrated. PROPOSITION XXII, THEOREM. lined Figure, similar and similarly described up- proportional : and a6 A B is to CD, so let E F be to G H. Now, Now, let the fimilar Figures KAB, LCD, be fimilarly described || upon A B, CD; and the similar || 18 of bico Figures MF, NH, similarly described upon the Right Lines E F, G H I say, as the Right-lined Figure K A B is to the Right-lined Figure LCD, so is the Right-lined Figure MF to the Right-lined Figure NH. For, take * X a third Proportional to A B, CD;. 11 of ubis. and O a third Proportional to E F, GH. Then, because AB is to CD, as E F is to GH; and as Ć D is to X, so is G H to 0; it shall be t, by + 21. 3. Equality of Proportion, as A B is to X, fo is E F to O. But A B is to X, as the Right-lined Figure KAB is $ to the Right-lined Figure LCD; and as I Cor. 26 EF is to 0, fo I is the Right-lined Figure MF, to of tbis. the Right-lined Figure N H. Therefore, as the Right lined Figure K A B is to the Right-lined Figure LCD, fo is * the Right-lined Figure M F to the Right-lined 11. s. Figure N H. And, if the Right-lined Figure K A B be to the Right-lined Figure LCD, as the Right-lined Figure MF is to the Right-lined Figure NH; I say, as AB is to CD, fo is E F to GH. For, make + E F to PR, as A B is to CD, and de- 12 of this scribe upon P Ra Right-lined Figure SR fimilar, and alike situate, to either of the Figures MF, NH. Then, because AB is to CD, as E F is to PR; and there are described upon A B, CD, similar and alike fituate Right-lined Figures K AB, LCD; and upon EF, PR, fimilar and alike fituate Figures MF, SR; it Thall be (by what has been already proved), as the Right-lined Figure K A B is to the Right-lined Figure L CD, so is the Right-lined Figure M F to the Right-lined Figure SR: But (by the Hyp.) as the Right-lined Figure K A B is to the Right-lined Figure LCD, so is the Right-lined Figure MF to the Right-lined Eigure N H. Therefore, as the Righlined. Figure M F is to the Right-lined Figure NH, fo is the Right-lined Figure M F to the Right-lined Figure SR: And since the Right-lined Figure M F has the fame Proportion to NH, as it hath to SR, the Right-lined Figure N H shall be f equal to the 19.50 Right-lined Figure SR: it is also fimilar to it, and alike ascribed, therefore GH is equal to P R. And because A B is to CD, as EF is to PR; and PR 1 is equal to GH; it shall be as A B is to CD, fo is E F to G H. Therefore, if four Right Lines be proportional, the Right-lined Figures, similar and similarly described upon them, fall be proportional ; and if similar Right lined Figures, fimilarly described upon Lines, be proportional, then the Right Lines shall also be proportional; which was to be demonstrated. LEM M A. Any three Right Lines A, B, and C. being given, ibe Ratio of the first A, to the third C, is equal to the Ratio compounded of the Ratio of the first A to the second B, and of the Ratio to the second B to the third C. Fo OR Example, Let the Number 3 be the Exponent or Denominator of the Ratio of A to B that is, let A be three Times B, and let the Number 4 be the Exponent of the Ratio of B to C; then the Number 12, produced by the Multiplication of 4 and 3, is the compaunded Exponent of the Ratio of A to C: For fince A contains B thrice, and B contains C four Times, A will contain C thrice four Times, that is, 12 Times. This is also true of other Multiples, or Submultiples; but this Theorem may be universally demonstrated thus : The Quantity of the Ratio of A to B, is the Number : viz. which multiplying the Consequent, produced the Antecedent: So likewise the Quantity of the Ratio of B to C, is B And these two Quantities, multiplied by each other, o AXB produce the Number B XC which is the Quantity of the Ratio that the Rectangle, comprehended under the Right Lines A and B, has to the Rectangle comprehended under the Right Lines B and C ; and so the said Rario of the Rectangle under A and B, to the Rectangle under B, and C, is that which, in the Sense of Def. 5. of this Book, is compounded of the Ratios of A to B, and B to C; but (by 1.6.) the Rectangle contained under A and B, is to the Rectangle contained under B and C, as A is to C ; therefore the Ratio of A to C, is equal to tbc B.to C. the Ratio compounded of the Ratios of A to B, and of If any four Right Lines A, B, C, and D, be proposed, the Ratio of the first A to the fourth D, is equal to the Ratio compounded of the Ratio of the first A to the second B, and of the Ratio of the second B to the third C, and of the Ratio of the third C to the fourth D. For, in three Right Lines A, C, and D, the Ratio of A to D is equal to the Ratio compounded of the Ratios of A to C, and of C to D; and it has been already demonArated, that the Ratio of A to C is equal to the Ratio compounded of the Ratios of A to B, and of B to C. Therefore the Ratio of A to D is equal to the Ratio compounded of the Ratios of A to B, of B to C, and of C to D. After the same manner we demonstrate, in any Number of Right Lines, that the Ratio of the firft to the last is equal to the Ratio compounded of the Ratios of the first to the second, of the second to the third, of the third À BCD to the fourth, and jo on to the laft. This is true of any other Quantities befides Right Lines, which will be manifeft, if the same Number of Right Lines A, B, C, &c. as there are Magnitudes, be assumed in the same Ratio, viz. fo that the Right Line A is to the Right Line B, as the first Magnitude is to the second, and the Right Line B to the Right Line C, as the second Magnitude is to the third, and so on. It is manifest (by 22. 5.), by Equality of Proportion, that the first Right Line A is to the last Right Line, as the first Magnitude is to the last; but the Ratio of the Right Line A to the last Right Line is equal to the Ratio compounded of the Ratios of A to B, B to C, and so on to the last Right Line : But (by the Hyp.) the Ratio of any one of the Right Lines to that nearest to it, is the fame as the Ratio of a Magnitude of the same Order to that nearest And therefore the Ratio of the first Magnitude to the laft, is equal to the Ratio compounded of the Ratios of the first Magnitude to the second; of the second to the third, and so on to the last; which was to be demonArated. 14. I. PROPOSITION XXIII. THEOREM. having the Angle B C D equal to the Angle ECG. I say, the Parallelogram A C to the Parallelogram CF, is in the Proportion compounded of their Sides, viz. compounded of the Proportion of B C to CG, and of D C to CE. For, let B C be placed in the same Right Line with C G Then D C shall be * in a strait Line with CE, and + izof ibis, compleat the Parallelogram DG; and then t, as B C is to CG, so is some Right Line K to L; and as D C is to CE, fo let L be to M. Then the Proportions of K to L, and of L to M, are the same as the Proportions of the Sides, viz. of BC to CG, and DC to CE; but the Proportion Lemma of K to MI is compounded of the Proportion of K preced. to L, and of the Proportion of L to M. Therefore, also K to M hath a Proportion compounded of the Sides. Then, because B C is to C G as the Paralle• suf ibis, logram A C is * to the Parallelogram CH: And since tu.. B C is to CG, as K is to L; it shall be t, as K is to L, so is the Parallelogram A C to the Parallelogram CH. Again, because D C is to CE, as the Parallelogram C H is to the Parallelogram CF; and since as D'C is to CE, lo is L to M; therefore as L is to M, fo Thall the Parallelogram C H be to the Parallelogram CF: And, consequently, since it has been proved that K is tu L, as the Parallelogram A C is to the Parallelogram CH; and as L is to M, so is the Parallelogram CH to the Parallelogram CF; it shall be I by Equality of Proportion, as K is to M, so is the Parallelogram A C to the Parallelogram CF; but K to M hath a Proportion compounded of the Sides : Therefore, also, the Parallelogram AC, to the Parallelogram CF, hath a Proportion compounded of the Sides. Wherefore, equiangular Parallelograms bave tbe |