36 Lemmas and Propositions. [Jan. 



For the mediate terms + a* x + 6» = a* b*, which doubled 

 s2« 9 b*, and — a b squared = -+■ a* b 1 , which added =: -f 

 3d« b*, exactly measuring the mediate quantity. 



Sir, I remain your obedient servant, 



Joseph Reade, M.D. 



Article VII. 



Lemmas and Propositions. By Mr. J. Adams. 

 (To Dr. Thomson.) 



SIR, Slonehouse, Oct. 9, 1817. 



Should the following lemmas and propositions merit a place 

 in your Annals of Philosophi/, your inserting them therein will 

 much oblige 



Your humble servant, 



James Adams. 



Lemma I. 



The angles made by ordinates and tangents at different points 

 of a curve are unequal, those being less that are nearest the- 

 vertex. 



Let A F B be a curve of continued cur- 

 vature ; A H, F G, ordinates to the ab- 

 scissa B H ; and let the tangents A D, 

 FC, to any two points, A, F, on the same 

 side of the vertex B, intersect each other 

 in the point E, and the tangent B I in the 

 points D and C. 



From this construction it will appear that the external angle, 

 A D I, is greater than the internal angle, ICE; that is, the 

 angle DAH greater than the angle CFG. Q. E. D. 



Lemma II. 



The arc of a circle is less than its corresponding tangent. 



With the radius C A describe the arc 

 A B, draw the tangent A D, and join C D. 



Per mensuration \ C A x A D = the 

 area of the triangle CAD, and iCA x 

 A B = area of the sector CAB; there- 

 fore, aren of triangle CAD: area of 

 sector CAB :: AD : A B. But the triangle is greater than 

 the sector ; therefore A D is greater than A B ; that is, the 



