106 Mr, Adamses Demonstraiion of a Proposition. [Feb. 



Proposition. — ^To demonstrate that two sides of a triangle 

 that is inclosed within another, may together be greater than any 

 two sides of the triangle that includes it, in any ratio which is 

 less than that of tico to one. 



Demonstration.—- Let ABC 

 be a scalene triangle whose 

 shortest side is A B, in the 

 side B C ; take C D equal to 

 A B, and join A D ; make 

 D F also equal to A B, and 

 divide A F into indefinitely 

 small equal parts, such as 

 Am, inv, &c. Join77«C, FC; 

 then. (20 . \ e ,) will A rn 



fit 



F + w C > A C, and 



m C > A C, much more will 

 5 D + m C > A B + A C. 



Now suppose A C and the angle B A C to remain constant 

 while A B, B C, vary ; then if A B decrease, C B will increase, 

 for A B + B C will always be greater than A C ; and when A B 

 and its equals, C D, D F, become indefinitely small, the points 

 D and F will approach to the point C, and the point B to the 

 point A as their limits, but to which they never can arrive as 

 long as the triangles BAG and D /« C have any magnitude : 

 hence the variable lines B D, B C, m D, approach to the fixed 

 line A C as their limit; so that the difference between them 

 may at length become leas than any assignable line. If the 

 points D and F be conceived actually to coincide with the point 

 C, and the points B and ;w* with the point A, the triangles 

 BAG and i) 7n G will cease to exist, for their equal bases A B 

 and G D will vanish together ; then m G +9^1) would become 

 AC4-AG = 2AG, andAG + AB = AG + 0=AG; 

 therefore, the ratio of m C + m D to A B + AC, may be any 

 ratio, less than two to one. 



Corollary. — Neither an isosceles triangle standing on its 

 shortest side, nor an equilateral triangle, will answer the condi- 

 tions of the proposition, because the straight line A D, drawn 

 within the triangle ABC, will be less than either A B, or its 

 equal A C. 



It IS stated at page 301, before quoted, that Pappus Alexan- 

 drinus has demonstrated this proposition in book the third of his 

 mathematical collections, which 1 have never seen, neither do I 

 know that a demonstration of the property has been pubhshed 

 elsewhere. 



• In tlie former part of the deinonstration, A F is supposed to be divided into inde- 

 finitely small equal parts, and D F is supposed to be diminished continually ; therefore, 

 A F would, in consequence, increase, and become any length less than A C; but since 

 any given quantity divided by an itidefinitely great quantity, will produce an indefinitely 

 Minall quantity, A m may be considered indefinitely small. 



