Elements of geometry and trigonometry . ^.At last the value of tt, developed to a certain order of decimals,wasfound by other calculators to be :)897932. &r ; BOOK V. 123 and some have had patience enough to continue these decimalsto the hundred and twenty-seventh, or even to the hundredand fortieth place. Such an approximation is evidently equi-valent to pirlcct correctness : the root of an imperfect poweris in no case mure accurately known. Tiie followinsr problem will exhibit one of the simplest ele-mentary mcthuds of obtainuig those approximations. PROrOSITION XIII. PROBLEM. Fh
Elements of geometry and trigonometry . ^.At last the value of tt, developed to a certain order of decimals,wasfound by other calculators to be :)897932. &r ; BOOK V. 123 and some have had patience enough to continue these decimalsto the hundred and twenty-seventh, or even to the hundredand fortieth place. Such an approximation is evidently equi-valent to pirlcct correctness : the root of an imperfect poweris in no case mure accurately known. Tiie followinsr problem will exhibit one of the simplest ele-mentary mcthuds of obtainuig those approximations. PROrOSITION XIII. PROBLEM. Fhe surface of a rc^ukir inscribed polygon^ and that of a simiiar poll/gun , being given ; to find the surfaces ofthe regular inscribed and circumscribed polygons havingdouble the number of Let AB be a side of the giveninscribed polygon ; EF, parallel toAB, a side of the circumscribedj)olvgon ; C the centre of the cir-cle. If the chord AM and thetangents AP, BQ, be drawn, AMwill be a side of the inscribedpolvgon, having twice the num-ber of sides ; and AP 4- PM =:2PMor PQ, will be a side of the simi-lar circumscribed polygon ( Cor. 3.). Now, as the sameconstmction will take place at each of the angles equal toACM, it will be suflicient to consider ACM by itself, the tri-angles connected with it being evidently to each other as thewhole polygons of which they form j)art. I^et A, then, bethe surface of the inscribed polygon wliose side is AB, B thatof the similar circumscribcnl polygon ; A the surface of thepolygon whose side is AM, B that of the sunilar circumscribedj)olygon : A and B aw given ; we have to find A and B. First. The triangles AC I), ACM, having the common ver-tex A, are to each other as their bases CD, CM ; they are like-wise to each other as the po
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