} abis. 2. Therefore, univerfally, it is manifeft, if three Right Lines be proportional, as the firft is to the third, fo is a Figure described upon the firft, to a fimilar and fimilarly defcribed Figure on the second; which was to be demonßrated. PROPOSITION XXI. THEOREM. Figures that are fimilar to the fame Right-lined LET each of the Right-lined Figures A, B, be For, because the Right-lined Figure A is fimilar to Def.1. 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 fimilar 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 fimilar to B; which was to be demonftrated. PROPOSITION XXII. THEOREM. If four Right Lines be proportional, the Rightlined Figure, fimilar and fimilarly defcribed upon them, fhall be proportional; and if fimilar Right lined Figures fimilarly defcribed upon Lines be proportional, then the Right Lines fhall be alfo proportional. LE ET four Right Lines A B, CD, EF, GH, be proportional and as A B is to CD, fo let E F be to GH. Now, 18 of this. Now, let the fimilar Figures K AB, LCD, be fimilarly defcribed || upon A B, CD; and the fimilar Figures MF, NH, fimilarly defcribed upon the Right Lines E F, GH. I fay, as the Right-lined Figure KAB is to the Right-lined Figure LCD, fo is the Right-lined Figure MF to the Right-lined Figure NH. For, take X a third Proportional to A B, C D ;* 11 of this. and O a third Proportional to EF, GH. Then, because A B is to CD, as EF is to GH; and as CD is to X, fo is G H to O; it fhall be t, byt 21. 3. Equality of Proportion, as A B is to X, fo is EF to O. But A B is to X, as the Right-lined Figure KAB is to the Right-lined Figure LCD; and as † Cor. 20 EF is to O, fo is the Right-lined Figure M F, to of this. 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 11.5. Figure N H. And, if the Right-lined Figure K A B be to the Right-lined Figure L C D, as the Right-lined Figure MF is to the Right-lined Figure NH; I fay, as A B is to CD, fo is EF to GH. For, make + E F to PR, as AB is to C D, and de- t 12 of thi fcribe upon P Ra Right-lined Figure S R fimilar, and alike fituate, to either of the Figures M F, NH. Then, because A B is to CD, as EF is to PR; and there are described upon A B, C D, fimilar and alike fituate Right-lined Figures K A B, LCD; and upon EF, PR, fimilar and alike fituate Figures M F, SR; it fhall be (by what has been already proved), 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 Figure SR: But (by the Hyp.) 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 Figure N H. Therefore, as the Rightlined Figure MF is to the Right-lined Figure NH, fo is the Right-lined Figure MF to the Right-lined Figure SR: And fince the Right-lined Figure M F has the fame Proportion to N H, as it hath to S R, the Right-lined Figure NH fhall be equal to the $9.5 Right-lined Figure SR: it is alfo fimilar to it, and alike afcribed; therefore G H is equal to P R. And because A B is to CD, as EF is to PR; and PR is is equal to G H; it fhall be as A B is to CD, fo is EF to GH. Therefore, if four Right Lines be proportional, the Right-lined Figures, fimilar and fimilarly defcribed upon them, fhall be proportional; and if fimilar Right lined Figures, fimilarly defcribed upon Lines, be proportional, then the Right Lines fhall also be proportional; which was to be demonstrated. LEMM A. Any three Right Lines A, B, and C. being given, the Ratio of the first A, to the third C, is equal to the Ratio compounded of the Ratio of the firft A to the fecond B, and of the Ratio to the Jecond B to the third C. FOR 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 compounded Exponent of the Ratio of A to C: For fince A contains B thrice, and B contains C four Times, Awill contain C thrice four Times, that is, 12 Times. This is alfo true of other Multiples, or Submultiples; but this Theorem may be univerfally demonftrated thus: The Quantity of the Ratio of A to B, is the Number; viz. produce the Number AxB A which multiplying the Confequent, produced the Antecedent: So likewife the Quantity of the Ratio of B to C, is B And these two Quantities, multiplied by each other, 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 fo the faid Ratio of the Rectangle under A and B, to the Rectangle under B, and C, is that which, in the Senfe of Def. 5. of this Book, is compounded of the Ratios of Á 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 te C; therefore the Ratio of A to C, is equal to the the Ratio compounded of the Ratios of A to B, and of B to C. If any four Right Lines A, B, C, and D, be propofed, the Ratio of the first A to the fourth D, is equal to the Ratio compounded of the Ratio of the firft A to the fecond B, and of the Ratio of the fecond 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 Ď, 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 demonfrated, 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 After the fame manner we demonftrate, in any Number of Right Lines, that the Ratio of the first to the laft is equal to the Ratio compounded of the Ratios of the firft to the fecond, of the fecond to the third, of the third ABC to the fourth, and jo on to the laft. D. This is true of any other Quantities befides Right Lines, which will be manifeft, if the fame Number of Right Lines A, B, C, &c. as there are Magnitudes, be affumed in the fame Ratio, viz. fo that the Right Line A is to the Right Line B, as the first Magnitude is to the fecond, and the Right Line B to the Right Line C, as the fecond Magnitude is to the third, and so on. It is manifeft (by 22. 5.), by Equality of Proportion, that the firft Right Line A is to the laft Right Line, as the firft Magnitude is to the laft; but the Ratio of the Right Line A to the laft Right Line is equal to the Ratio compounded of the Ratios of A to B, B to C, and fo on to the laft 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 fame Order to that nearest it. And therefore the Ratio of the first Magnitude to the laft, is equal to the Ratio compounded of the Ratios of the firft Magnitude to the fecond, of the fecond to the third, and fo on to the laft; which was to be demonArated. 14. I. t12 of this. PROPOSITION XXIII. THEOREM. Equiangular Parallelograms kave the Proportion to one another that is compounded of their Sides. LE ET AC, C F, be equiangular Parallelograms, having the Angle BCD equal to the Angle ECG. I fay, the Parallelogram A C to the Parallelogram C F, is in the Proportion compounded of their Sides, viz. compounded of the Proportion of BC to CG, and of DC to CE. For, let BC be placed in the fame Right Line with C G. Then D C fhall be in a ftrait Line with C E, and compleat the Parallelogram DG; and then †, as BC is to CG, fo is fome Right Line K to L; and as D C is to C E, fo let L be to M.' Then the Proportions of K to L, and of L to M, are the fame as the Proportions of the Sides, viz. of BC to C G, and DC to CE; but the Proportion of K to Mis compounded of the Proportion of K to L, and of the Proportion of L to M. Therefore, alfo K to M hath a Proportion compounded of the Sides. Then, because B C is to C G as the Parallelogram A C is to the Parallelogram CH: And fince BC is to C G, as Kis to L; it shall be †, as K is to L, fo is the Parallelogram A C to the Parallelogram CH. Again, because D C is to C E, as the Parallelogram CH is to the Parallelogram CF; and fince as DC is to CE, fo is L to M; therefore as L is to M, fo fhall the Parallelogram C H be to the Parallelogram C F And, confequently, fince it has been proved that K is to L, as the Parallelogram A C is to the Parallelogram CH; and as L is to M, fo is the Parallelogram CH to the Parallelogram C F ; it fhall bet by Equality of Proportion, as K is to M, fo is the Parallelogram AC to the Parallelogram CF; but K to M hath a Proportion compounded of the Sides : Therefore, alfo, the Parallelogram A C, to the Parallelogram C F, hath a Proportion compounded of the Sides. Wherefore, equiangular Parallelograms have : the |