1^4 ARITHMETIC. 



ancl second is to the difference of the second and third, a5 

 the first is to the third ; and, in the latter, when the dif- 

 ference of the first and second is to the difference of the 

 third and fourth as the first is to the fourth. Thus, 2, 3 

 and 6 ; and 3, 4, 6 and 9, are harmonical proportionals j 

 for 3 — 2== I 16^-3 = 3 : : 2 : 6; and 4—^3 = 1 : 9 — 6=3 



• • 3 • 9- 



Of four arithmetical proportionals the sum of the ex- 

 tremes is equal to the sum of the means.* Thus of 

 2 •• 4 : : 6 .. 8 the sum of the extremes (2-j-8)zz the sum' 

 of the means (4+6)zzio. Therefore, of three arithmet- 

 ical proportionals, the sum of the extrenfes io double the 

 mean. 



Of four geometrical proportionals the product of the 

 extremes is equal to the product of the"^ means.f Thus, 

 of 2 : 4 : : 8 : 16, the product of the extremes (2X16) is 

 equal to the product of the means (4X8)1:332. Therefore 

 of three geometrical proportionals, the product of the ex- 

 tremes is equal to the square of the mean. 



Hence it is easily seen, that either extreme of four geo- 

 metrical proportionals is equal to the product of the means 



divided 



* Demonstration. Let the four aiithmetical proportionals 

 be ^, B, C, D, viz. A- B '.\ C- D \ then, J—B=C—D and 

 B-\-D being added to both sid^s of the equation, A — B-\-B-\-D 

 zzC — Z)-|-^-f-Z) ; that is, A-\-D the sum of the extremes 

 z=.C-\-B the sum of the means. — And three A, B, C, may he 

 thus expressed, A .. B : : B - C ', therefore yl-^C:=B-^B—2B. 



Q^ E. 1>. 



-}■ Demonstratioj^. Lettlie proportion he A : B : : C : Dy 



A C 

 and let^=2>~^ 5 ^^^^" Az=:Br, and Cz=.Dr ; multiply the for- 

 mer of these equations by /), and the latter by B ; then ADzz 

 BrD, and CBzzzDrB, aftd consequently AD the product of the 

 extremes is equal to BC the product of die means. — And three 

 may be thus expressed, A : B : : B : C, therefore A C=zBx 

 Bz=B\ Q, E. D. 



