fimilarly defcribed Figure on the fecond; which was to be demonftrated.

Figures that are fimilar to the fame Right-lin'd Figure, are alfo fimilar to one another.

ET each of the Right-lin'd Figures A, B, be fimilar to the Right-lin'd Figure C. I fay, the Right-lin❜d Figure A, is alfo fimilar to the Rightlin'd Figure B.

For because the Right-lin'd Figure A is fimilar to

*

the Right-lin'd Figure C, it fhall be equiangular ⋆ Def. 1. thereto; and the Sides about the equal Angles pro- of this. portional. 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❜d Figures A, B, are equiangular to C, and they have the Sides about the equal Angles proportional. Wherefore the Right-lin'd Figure A is equiangular to the Right-lin'd Figure B; and the Sides a. bout 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 Right-lin❜d Figures fimilar and fimilarly defcribed upon them, fhall be proportional; and if the fimilar Right-lin'd Figures fimilarly deferibed upon the Lines, be proportional, then the Right Lines fhall also be proportional.

L

ET four Right Lines AB, CD, EF, GH, be proportional; and as AB, is to CD, fo let EF be to GH.

*

Now let the fimilar Figures KAB, LCD, be fimilarly defcribed upon AB, CD; and the fimilar * 18 of this. Figures MF, NH, fimilarly defcribed upon the Right

*11 of this.

† 22.5.

Cor. 20.

of this.

* 11. 8.

+12 of this.

$9.5.

Lines E F, GH. I fay, as the Right-lin❜d Figure
KAB is to the Right-lin'd Figure LCD, fo is the
Right-lin❜d Figure MF to the Right-lin'd Figure N H.

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 O; it fhall be f by Equality of Proportion, as A B is to X, fo is E F to O. But A B is to X, as the Right-lin❜d Figure K AB is to the Right-lin❜d Figure LCD; and as EF is to O, fo is the Right-lin❜d Figure MF, to the Rightlin'd Figure NH. Therefore as the Right-lin❜d Figure KAB is to the Right-lin❜d Figure LCD, fo is the Right-lin❜d Figure MF to the Right-lin❜d Figure NH.

And if the Right-lin'd Figure KAB be to the Right-lin❜d Figure LCD, as the Right-lin❜d Figure MF is to the Right-lin'd Figure NH; I fay, as AB is to CD, fo is EF to GH.

For make EF to PR, as AB is to CD, and describe upon PR a Right-lin❜d Fignre SR fimilar, and alike fituate, to either of the Figures MF and NH.

Then because AB is to CD, as EF is to PR, and there are described upon AB, CD, fimilar and alike fituate Right-lin'd Figures K AB, LCD, and upon EF, PR, fimilar and alike fituate Figures MP, SR; it fhall be (by what has been already proved) as the Right-lin'd Figure KAB is to the Right-lin❜d Figure LCD, fo is the Right-lin'd Figure MF to the Rightlin'd Figure RS: But (by the Hyp.) as the Right-lin'd Figure KAB is to the Right-lin'd Figure LCD, fo is the Right-lin❜d Figure MF to the Right-lin'd Figure NH. Therefore as the Right-lin'd Figure M F is to the Right-lin'd Figure NH, fo is the Right-lin'd Figure MF to the Right-lin❜d Figure SR: And fince the Right-lin❜d Figure M F has the fame Proportion to NH, as it hath to SR, the Right-lin'd Figure NH fhall be equal to the Kight-lin'd Figure SR; it is alfo fimilar to it, and alike defcribed; therefore GH is equal to PR. And because A B is to CD, as EF is to PR; and PR is equal to GH, it shall be as A B is to CD, fo is EF to GH. Therefore, if four Right Lines be proportional, the Right-lin'd Figures, fimilar and fimilarly defcribed upon them, fhall be proportional;

and

and if the fimilar Right-lin'd Figures fimilarly defcribed upon the Lines, be proportional, then the Right Lines fhall also be proportional; which was to be demonftra

ted.

LEMM A.

Any three Right Lines A, B, and C,'being given, the Ratio of the firft A to the third C, is equal to the Ratio compounded of the Ratio of the first A to the fecond B, and of the Ratio of the Second B to the third C.

FOR

OR Example, let the Number 3 be the Exponent, or Denominator of the Ratio of Ă 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, 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 univerfally demonftrated thus: The Quantity of the Ratio of A to B, is the Number Aviz. which multiplying the Confequent, produces the Antecedent. So likewife the Quantity of the Ratio of B to C, is B. 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 fo the faid Ratio of the Rectangle under A and B, to the Rectangle under A B C B and C, is that which in the Senfe 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 Rectangle contained under B and C, as A is to C; therefore the Ratio of A to C, is equal to the

AxB

Bx C'

Ratio compounded of the Ratio's 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 Second В 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's compounded of the Ratio's of A to C, and of C to D; and it has been already demonftrated, that the Ratio of A to C is equal to the Ratio compounded of the Ratio's of A to B and B to C. Therefore the Ratio of A to D is equal to the Ratio compounded of the Ratio's of A to B, of B to C, and of C to D. 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 Ratio's ABCD of the first to the second, of the fecond to the third, of the third to the fourth, and fo on to the last.

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 firft 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 Ratio's 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 firft Mag nitude to the laft, is equal to the Ratio compounded of the Ratio's of the first Magnitude to the fecond, of the fecond to the third, and fo on to the laft; which was to be demonftrated.

PRO

PROPOSITION XXIII.

THEOREM.

Equiangular Parallelograms have the Proportion to one another that is compounded of their Sides.

TET AC, CF, be equiangular Parallelograms, having the Angle B CD equal to the Angle ECG. I fay, the Parallelogram AC, to the Parallelogram CF, 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 CG.

Then DC fhall be in a ftrait Line with CE, * 14. 1. and compleat the Parallelogram DG; and then † † 12 of this. as BC is to CG, fo is fome Right Line K to L; and as DC is to CE, 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 CG, and DC to CE; but the Proportion of K to M is compounded of the Proportion of K + Com. to L, and of the Proportion of L to M. Wherefore proceed. alfo K to M hath a Proportion compounded of the

Sides. Then because BC is to CG as the Parallel

*

ogram AC is to the Parallelogram CH: And fince * 1 of this. BC is to CG as K is to L, it fhall be † as K is to L, † 11.5. fo is the Parallelogram A C, to the Parallelogram CH. Again, because DC is to CE 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 CH be to the Paralt lelogram CF; and confequently fince it has been proved that K is to L, as the Parallelogram AC is to the Parallelogram CH, and as L is to M, fo is the Parallelogram CH to the Parallelogram CF; it fhall be by Equality of Proportion, as K is to M, fo is 20. 5. the Parallelogram AC to the Parallelogram CF; but K to M hath a Proportion compounded of the Sides: Therefore also the Parallelogram AC, to the Parallogram C F, hath a Proportion compounded of the Sides. Wherefore equiangular Parallelograms have

the