1824.] connected with the Trisection of an Arc. 357 



though they failed of arriving at the ultimate object which they 



Eursued. And I believe that geometrical science would not 

 ave remained as stationary, as it has been for many years, if the 

 problem had continued to share the attention of modern geome- 

 ters. 



But nothing so much paralyzes exertion as despondency of 

 success ; and I am aware that some of the most profound mathe- 

 maticians of the present day do not hesitate to pronounce, that 

 the problem alluded to is impossible. Their assertion is, I think, 

 unwarrantable and rash. If I may hazard an opinion, in which 

 many will think me over sanguine, I avow my expectation that 

 some of the principles, which I offer in this Memoir, will yet lead 

 to the solution of that problem ; though I am now obliged to 

 dismiss the subject from my mind, on account of other avoca- 

 tions and declining health. 



Prop. I. Problem. (Plate XXVII.) Fig. 1. 



In a given segment of a circle to inscribe a triangle, whose 

 sides shall be in arithmetical progression. 



Let the given segment be A b a. From B, the vertex of the 

 opposite segment, draw the lines B T, B t, trisecting A a in T 

 and t, and meeting the periphery in E and e. From A inscribe 

 the chord A x equal to E e ; and draw ax. I say that A x a is 

 the triangle required. 



Dem. — Let y be the point in which a x intersects E e. Draw 

 T y : and parallel to this draw x z, meeting A a in z, and inter- 

 secting E e in s. Lastly, draw E z and sf. I shall now prove 

 that a x is equal to a z, and that E z is perpendicular to A a ; 



whence it will follow that a z, or a x = —£■ + — , and is there- 

 fore an arithmetical mean between A a and E e, i. e. between 

 A a and A x. 



For the triangles B T t, a t F are similar [having the angles at 

 t equal, and als'o the angles at B and a, as standing upon equal 

 arcs]. .-. B t : T t : : (a I) T t : t F. .'. rectangle B«F = 

 T t 1 ; and 2 B * F = 2 T t* = rectangle A t a = B t e. :.te 

 is bisected in F. .-. the similar triangles at F,y e F are mutually 

 equilateral, and y e = a t = T t . .'. e t is parallel to yT, 

 and /. to xz. .'.ax = a z. But also E y is bisected in s [for 

 the triangles* E y x, a y e, and x y s, ¥ ye, being similar, E y : 

 (y x) y s :: a y : (y ej y F :: 2 . 1]. .•. E s — z T, and E z is 

 parallel to s T," and perpendicular to E e. /. a z (or a D + D z) 



i. e. a x = ^- + ?— = A a * A * . .'. a x is an arithmetical mean 



between A a and A x. Q. E. D. 



Cor. 1. (Fig. 2.)— The lines A b, a b trisect the line Ee; for 

 since the line t e bisects the angle A e a, A e : (a e) A E :: A t : 



• The linca E jr, t a, are not drawn in the figure, but may be easily supposed. 



