JE 



1818.] Lemmas and Propositions. 37 



tangent greater than the arc. (Dr. Hutton's Course of Mathe- 

 matics, vol. iii. p. 39.) 



Otherwise. 



By putting the radius C A = r, the arc AB = «, and tan- 

 gent A D = t ; we have from the nature of series, t = a -f 



fl 3 2a> 17 a> „ 



It is also apparent from the tables of natural tangents and 

 circular arcs to radius unity. 



Proposition I. 



Any arc of continued curvature is less than its corresponding 

 tangent. 



Let A B be an arc of continued 

 curvature, concave towards the or- 

 dinate A E, whose vertex is B, and 

 let the tangent at A meet the ab- 

 scissa E B in D. /jrl^-ta?^ 



Conceive the curve A B to be 

 divided into indefinitely small por- 

 tions, either equal or unequal, such 

 as F G; through the points F G 

 draw H I, K L, parallel to ED; 

 and let F N be a tangent to the curve at F : also draw F P 

 parallel to A D, and F M parallel to A E. 



Now since F G is an indefinitely small part of the curve A FB, 

 it will evidently coincide with its osculating circle ; therefore F N 

 will be a tangent of the circular arc which coincides with F G, 

 and F N greater than F G. (Lemma II.) By construction FP 

 is parallel to AD; therefore the angle D A E is equal to the 

 angle P F M ; but D A E is greater than N F M. (Lemma I.) 

 Hence the angle P F M is greater than the angle NFM, and 

 consequently F P greater than FN. It has been shown that 

 F N is greater than F G ; much more, then, is F P, or its equal 

 I L, greater than F G. 



In like manner it may be proved that each of the increments 

 of A D is greater than its corresponding arc; therefore the sum 

 of all the increments of A D is greater than the sum of all the 

 increments of A F B ; that is, the tangent A D is greater than 

 the arc A F B. 



Corollary. 



To the curve A B continued, draw the tangent D R, and join 

 A R. We then have, according to the proposition, the tangent 

 R D greater than the arc R B. Hence it follows that the sum of 

 the tangents A D and R D is greater than the curve A B R. 



