330 DIFFERENTIAL COEFFICIENT OF AN ARC. 



308. Of the axioms on which the preceding demonstra- 

 tion is founded, the first will probably be readily granted ; 

 the second is more difficult, and will not be necessarily true 

 if the arc be net concave towards the chord PQ throughout 

 its extent. It must be understood therefore, in stating it, 

 that the arc PQ must be taken so small that it is always 

 concave towards its chord. 



There is another mode of arriving at the results given in 

 Art 307, which is preferred by some writers : they assert that 

 no precise idea can be formed of the length of an arc, except 

 by regarding it as the limit of the perimeter of a polygon in- 

 scribed in that arc, when the length of each side of the polygon 

 is indefinitely diminished. If we adopt this definition of the 

 length of an arc, we must shew that the limit mentioned 

 does exist, and is the same in whatever manner we suppose 

 the polygon inscribed, provided that each side is ultimately 

 indefinitely diminished. 



Draw two chords dividing the whole arc we are consider- 

 ing into two portions; then in each of these subdivisions 

 place two chords dividing the whole arc into four portions ; 

 in each of the last subdivisions place two chords, and so on. 

 The perimeters of the polygons thus formed constitute a series 

 continually increasing ; and as it is easy to see they cannot 

 increase without limit, we prove the first point, namely, that 

 there is a limit to the perimeter of the inscribed polygon when 

 the length of each side is indefinitely diminished. 



Suppose now two polygons with indefinitely small sides 

 inscribed in the curve, one of them being one of the series just 

 considered, and the other described after any other law. Draw 

 tangents to the curve at the angular points of both polygons, 

 thus forming one polygon circumscribing the arc. Then it is 

 easy to see that any chord of either polygon bears to the cor- 

 responding portion of the circumscribing figure, a ratio which 

 can be made as near to unity as we please by sufficiently 

 diminishing the length of each chord. Hence the perimeter of 

 each inscribed figure bears to that of the circumscribed figure 

 a ratio which is ultimately one of equality, and consequently 

 the ratio of the perimeter of one inscribed figure to that of the 

 other inscribed figure is ultimately one of equality. This 

 proves the second point involved in the definition of the length 



