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5

Squaring the circle

(*Journal of the Indian Mathematical Society, v, 1913, 138)*

Let be a circle with centre , of which a diameter is . Bisect at and let be the point of trisection of nearer . Draw perpendicular to and place the chord .

Join , and draw and parallel to . Place a chord , and draw the tangent . Join , and . Cut off . Draw parallel to , meeting at .

Then the square on will be equal to the circle approximately.

For | , |

Therefore | . |

But and are equal to and respectively.

Therefore | , and . |

Hence | , | |

and | . |

But | , | |

and | . |

Therefore | , very nearly. |

*Note.*—If the area of the circle be square miles, then is greater than the true length by about an inch.