. An elementary course of infinitesimal calculus . Fig. 3. polygon of perimeter ITj (say), and a corresponding circum-scribed polygon of perimeter Ila; and it is evident thatn2> III, whilst n2< n/. The process can be continued toany extent, and if we imagine it to be followed out accordingto some definite law, we obtain an ascending sequence ofmagnitudes 111, -l-ls) iiji ••• J and a descending sequence n/, n/. n;,.... Also, every member of the former sequence is less than everymember of the second. Hence the former sequence will havean upper limit 11, and the latter a lower limit 11, and
. An elementary course of infinitesimal calculus . Fig. 3. polygon of perimeter ITj (say), and a corresponding circum-scribed polygon of perimeter Ila; and it is evident thatn2> III, whilst n2< n/. The process can be continued toany extent, and if we imagine it to be followed out accordingto some definite law, we obtain an ascending sequence ofmagnitudes 111, -l-ls) iiji ••• J and a descending sequence n/, n/. n;,.... Also, every member of the former sequence is less than everymember of the second. Hence the former sequence will havean upper limit 11, and the latter a lower limit 11, and wecan assert that 11 :^ 13. We can further shew that if the law of construction ofthe successive polygons be such that the angle subtended atthe centre by any two successive^ points on the circumference 4] CONTINUITY. is ultimately less than any assignable magnitude, the limitsn and n are identical. For, let PQ be a side of the. Fig. 4. inscribed polygon, PT and QT the tangents at P and PQ meets OT in ]Sf, we have PQ _P]Sr_ONTP + TQ~PT OP Now if 2 be a symbol of summation extending round thepolygons, we have n,=2(PQ), n,=S(rp+TQ). Hence, by a known theorem, the ratio n, tjPQ) U/ X{TP + TQ) will be intermediate in value between the greatest and leastof the ratios OF OP But in the limit, when the angles POQ are indefinitelydiminished, each of the ratios ON/OP becomes equal tounity. Hence the limit of Hg is identical with that of n/, or n = H. Finally, whatever be the law of construction of thesuccessive polygons, subject to the above-mentioned condition. 8 INFINITESIMAL CALCULUS. [CH. I the limit obtained is the same. For suppose, for a moment,that in one way we obtain the limit 11, and in another thelimit P. Regarding 11 as the limit of an inscribed, and Pas that of a circumscribed polygon, it is plain that n :^ like manner, regarding P as the limit of an inscribed, andn as that of a circumscribed po
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