CHAMBERS'S INFORMATION FOR THE PEOPLE. 



If there are four magnitudes, A, B, C, D, which 

 are proportionals, it is thus expressed : 



A : B : : C : D, 



and is read A to B as C to D. 



Euclid's test of proportion is the following : 

 The first of four magnitudes is said to have the 

 same ratio to the second which the third has to 

 the fourth, when any equimultiples whatever of 

 the first and third being taken, and any equi- 

 multiples whatever of the second and fourth ; if 

 the multiple of the first be equal to that of the 

 second, then the multiple of the third is equal to 

 that of the fourth ; if greater, greater ; and if less, 

 less. 



PROP. XXIV. Straight lines have to one an- 

 other the same ratio as rectangles of equal altitude 

 described upon them. 



Let AK, BP be the two lines, MK and NP two 

 rectangles described on them of the same altitude. 

 Then the ratio of AK to BP is the same as the 

 ratio of MK to NP. Let AK be divided into any 

 number whatever of equal parts AC, CD, &c., 



then MK will be divided into the same number of 

 equal rectangles. If also from BP parts BE, EF, 

 &c., each equal to AC, be laid off till there is no 

 remainder, or a remainder less than AC, and lines 

 be drawn through E and F parallel to BN, then 

 NP will contain the same integral number of 

 parts each equal to MC as BP does of parts equal 

 to AC; 



/. AK has the same ratio to BP that MK 



has to NP. 



Cor. In a similar manner it may be shewn that 

 straight lines have to one another the same ratio 

 as parallelograms or triangles of equal altitude 

 described upon them. 



PROP. XXV. If four straight lines are propor- 

 tional^ the rectangle contained by the extremes is 

 equal to the rectangle contained by the means. 



Let the four straight lines A, B, C, D be pro- 

 portional, then A D = B C On A and B con- 



struct rectangles, of which C shall be the altitude 

 of each. On C and D construct rectangles, of 

 which A shall be the altitude of each. 



Then A : B : : rect AC : rect BC. 

 But A : B : : C : D ; 



.'. rect AC : rect. BC : : C : D. 

 But C : D : : rect. AC : rect. AD ; 

 .'. rect. AC : rect. BC : : rect. AC : AD ; 



.'. rect. BC = rect AD. 

 Hence, if A : B : : C : D, 



AD = BC 



622 



Conversely, if A, B, C, D be four straight lines 

 such that AD =-= BC. 



Then A : B : : C : D. 



The same construction being made. 



Since AD = BC ; 



.-. rect AC : rect BC : : rect AC : rect AI 



But rect AC : rect BC : : A : B, 

 and rect AC : rect. AD : : C : D ; 



.'. A : B : : C : D. 



Cor. Hence if three straight lines be propor- 

 tional, the rectangle contained by the first and 

 third is equal to the square of the mean, and 

 conversely. 



PROP. XXVI. If A : B : : C : D. 



It follows (i.) That A : C : : B : D ; 

 for AD = BC. 



then A : C : : B : D. 



(2.) Similarly, that B : A : : D : C 



PROP. XXVII. If A : B : : C : D, 

 A + B:B::C + D:D, 

 and A B:B::C D:D. 



For AD = BC, 



A C 



. A + B C+D 



B D ; 



.'. A + B : B : : C + D : D. 

 Similarly, A B:B::C D:D; 

 .'. A + B : A B : : C + D : C D. 



PROP. XXVIII. If A:B::C:D::E:F: 

 G: H, 



r A C E G 



then 



For let 



...A+C + E + G 



. A+C + E + G 

 "B+D+F+H 



. A C 

 K- g = s =&c. 



PROP. XXIX. If A :B 

 then A : C 



A = B 



B/"* 

 \^ 



. A B 



B 



A 2 



Q 



B a . 



For 



or 



B 2 



B 2 



PROP. XXX. If a straight line be drawn 

 parallel to one of the sides of a triangle, it divides 



