This section is from the book "The American House Carpenter", by R. G. Hatfield. Also available from Amazon: The American House Carpenter.

The radius is equal to the sum of the squares of half the chord and of the versed sine, divided by twice the versed sine. This is expressed, algebraically, thus: r = (C/2)2+V2/2V, where r is the radius, c the chord, and v the versed sine (Art. 444).

Example. - In a given arc of a circle a chord of 12 feet has the rise at the middle, or the versed sine, equal to 2 feet, what is the radius?

Half the chord equals 6, the square of 6 is, | 6x6 | = 36 |

The square of the versed sine is, | 2x2 | = 4 |

Their sum equals, | 40 |

Twice the versed sine equals 4, and 40 divided by 4 equals 10. Therefore the radius, in this case, is 10 feet. This result is shown in less space and more neatly by using the above algebraical formula. For the letters substituting their value, the formula r = (C/2)2+V2/2V becomes r = (12/2)2+22/2x2,

2V

and performing the arithmetical operations here indicated equals -

62 + 22 /4 = 36 + 4/4 = 40/40 = 10

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