Then because the Polygon ABCDE is similar to the Polygon FGHKL, the Angle BAE is equal to the Angle GFL; and BA is to AE as GF is to FL. Now since ABE, FGL, are two Triangles, having one Angle of the one equal to one Angle of the other, and the Sides about the equal Angles proportional; the Triangle A BE will be * equiangular * 6 of this. to the Triangle FĞL; and also funilar to it. Therefore the Angle ABE, is equal to the Angle FGL; but the whole Angle ABC is t equal to the whole t Def. 1. Angle FGH, because of the Similarity of the Poly-of this gons. Therefore the remaining Angle EBC is equal to the remaining Angle LGH: And since (by the Similarity of the Triangles . ABE, FGL) as EB is to B A, fo is LG to GF: And since also by the Similarity of the Polygons) A B is to BC, as F G is to GH; it shall be by Equality of Proportion, as † 29. 5. EB is to BC, fo is LĠ to GH, that is, the Sides about the equal' Angles EBC, LGH are proportional. Wherefore the Triangle EBC is equiangular to the Triangle LGH; and consequently also similar to it. For the same Reason, the Triangle ECD, is likewise similar to the Triangle LHK; therefore the fimilar Polygons ABCDE, FGHKL, are divided into equal Numbers of similar Triangles. I say, they are also homclogous to the Wholes, that is, that the Triangles are proportional; and the Antecedents are ABE, EBC, ECD, and their Confequents FGL, LGH, LHK. And the Polygon ABCDE; to the Polygon FGHKL, is in the duplicate Proportion of an homologous Side of the one, to an homologous Side of the other, that is, AB to FC. For because the Triangle ABE is similar to the Triangle F GL, the Triangle ABE, shall be * to the * 19 of this. Triangle F GL, in the duplicate Proportion of BE to GŁ: For the fame Reason, the Triangle BEC, to the Triangle GLH, is * in a duplicate Proportion of BE to GL: Therefore the Triangle ABE is † to † 11. 5. the Triangle F GL, as the Triangle BEC is to the Triangle GLH. Again, because the Triangle EBC is fimilar to the Triangle LGH; the Triangle EBC to the Triangle LGH, shall be in the duplicate Proportion of the Right Line CE to the Right Line HL; and so likewise the Triangle E CD to the Tri angle M4 angle LHK, shall be in the duplicate Proportion of CE to HL. Therefore the Triangle BEC is to the Triangle LGH, as the Triangle CED is to the Triangle LHK. But it has been proved, that the Triangle EBC is to the Triangle LGH, as the Triangle ABE is to the Triangle F GL: Therefore as the Triangle ABE is to the Triangle F GL, fo is the Triangle BEC to the Triangle GHL; and fo is the Triangle ECD to the Triangle LHK. But as one of the Antecedents is to one of the Conse| 12.5. quents, fo are | all the Antecedents to all the Conse quents. Wherefore as the Triangle ABE is to the be demonstrated after the fame manner that fimilar quadrilateral Figures are to each other in the duplicate Proportion of their homologous Sides; and this has been already proved in Triangles. A F Coroll. 1. Therefore universally similar Right-lin'd Fi gures, are to one another in the duplicate Proportion of their homologous Sides; and if X be taken a third Proportional to AB and FG, then AB will have to X a duplicate Proportion of that which AB has to F G ; and a Polygon to a Polygon, and a quadrilateral*Figure to a quadrilateral Figure, will be in the duplicate Proportion of that which one homoBGX logous fide has to the other; that is, AB to FG; but this has been proved in Triangles. 2. Therefore universally it is manifest, if three Right Lines be proportional, as the first is to the third, so is a Figure described upon the first, to a similar and fimilarly fimilarly described Figure on the second ; which was to be demonstrated. PROPOSITION XXI. THEOREM. Figure, are also similar to one another. milar to the Right-lin'd Figure C. I say, the Right-lin'd Figure A, is also similar to the Right-lin'd Figure B. For because the Right-lined Figure A is similar to Def of the Right-lin'd Figure C, it shall be * equiangular ebis. thereto; and the Sides about the equal Angles proportional. Again, because the Right-lin'd Figure B is fimilar to the Right-lin'd Figure C, it shall * be equiangular thereto ; and the Sides about the equal Angles will be proportional. Therefore each of the Right-lin's Figures A, B, are equiangular to C, and they have the Sides about the equal Angles proportional. Wherefore the Right-lind Figure A is equiangular to the Right-lin'd Figure B; and the Sides about the equal Angles are proportional; wherefore A is fis milar to B; which was to be demonstrated. PROPOSITION XXII. THEORE M. If four Right Lines be proportional, the Right-lin'd Figures similar and similarly described upon them, proportional; and as AB, is to CD, fo let EF, be to GH. Now let the similar Figures KAB, LCD, be fi milarly described * upon Å B, CD; and the fimilar 18 of this. Figures + 22. 5. of tbis. * 11. 5: Figures MF, NH, similarly described upon the Right Right-lip'd Figure MF to the Right-lined Figure NH. in 11 of this. For take * X a third Proportional to AB, CD, and O a third Proportional to EF, GH. Then because AB is to CD, as EF is to GH, and as CD is to X, fo is GH to 0; it shall be + by Equality of Proportion, as AB is to X, fo is EF to Ó. Cor. 20. But AB is to X, as the Right-lined Figure KAB is I to the Right-lined Figure LCD; and as E F is to 0, lo is the Right-lined Figure MF, to the Right-lined Figure NH. Therefore as the Right-lined Figure KĀB is to the Right-lined Figure LCD, fo is *the Right lined Figure MF to the Right-lined Figure N H. And if the Right-lined Figure KAB 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 EF to GH. + 12 of this. For make + EF to PR, as AB is to CD, and de fcribe upon PR a Right-lined Figure SR similar, and alike fituate, to either of the Figures MF and NH. Then because AB is to CD, as E F is to PR, and there are described upon AB, CD, fimilar and alike fitúate Right-lined Figures KAB, LCD, and upon EF, PR, fimilar and alike situate Figures MF, SR, it shall be (by what has been already proved) as the Right-lined Figure KAB is to the Right-lined Figure LCD, so is the Right-lined Figure MF to the Rightlined Figure RS: But (by the Hyp.) as the Right-lined Figure KAB is to the Right-lined Figure LCD, so is the Right-lined Figure MF to the Right-lined Figure NH. Therefore as the Right-lined 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 MF has the fame Proportion to NH, as it hath to SR, the Right-lined Figure NH shall be f equal to the Right-lined Figure SR; it is also similar to it, and alike described ; therefore GH is equal to PR. And because AB is to CD, as EF is to PR ; and PR is equal to GH, it shall be AB is to CD, so is EF to GH. Therefore, if four Right Lines be proportional, the Right-lined Figures, similar and similarly described upon them, shall be proportional; and as and if the similar 'Right-lined Figures fimilarly defcribed upon the Lines, be proportional, then the Right "Zines shall also be proportional; which was to be de monstrated. L E M M A.. Any three Right Lines A, B, and C, being given, ihe. 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 of the sea cond B to the third C. 4 B 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 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, 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, produces the Antecedent. So likewise the Quantity of the Ratio of B to C, is T: And these two. Quantities multiplied by each other, produce the Number 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 Ratio of the A B C 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 Ratio's of A to B, and B to C; but (by 1. 6.) the Rectangle contained under A and B, is to the Rettangle contained under B and C, as A is to C; therefore the Ratio of A to C, is equal to the Ratio compounded of the Ratio's of A to B, and of B to C. AXB |