1837] 



Circular bodies and their Squares. 



301 



VI. — On the relative proportion between Circular bodies and their 

 Squares, — By Captain Cortlandt Taylor, of the Madras Artillery. 



Being on a mission to the Malabar Coast connected with limber for 

 the Ordnance Department at Madras, I found the native dealers in 

 that article at Calicut disposed of timber cut to the square, varying 

 about thirty per cent, from their, measurement in the round log. I 

 was induced in consequence to fall back upon school recollections, to 

 ascertain how far this method was correct or borne out by theory, and 

 which the subjoined mathematical process enabled me to do.* As the 

 results, for comparative data, may be of practical utility, perhaps its 

 dissemination will not be unattended with benefit. 



Then, as the circumferences of all circles are to their diameters, 

 as 3.1416 is to unit; so 3.1416 is to 1, as the circumference of any given 

 circle ABCD, is to its diameter, DB: but DB the diameter is the double 

 of EB the radius, and EB is one leg or side of the right angle triangle 

 AEB, of which AE is the other leg, and AB, is the third side, or hypo- 

 thenuse. 



Then, as in right angle triangles, the square of the hypothenuse is 

 equal to the sum of the squares of the other two sides; in the right 

 angle triangle AEB,— BE * -f AE _*. AB^ : but as BE, and AE are 

 equal, their squares are equal also, and 2, EB JL AB 2 or V2, EBjl AB, 

 the one-fourth of the pevimeter of the required square. 



Thus knowing the circumference of any circle, the pevimeter, or 

 measurement of the inscribed square, can be easily ascertained; and 



C 



Let ABCD be a circle, and the 

 diameters AC, and BD, perpendi- 

 cular to each other ; then are the 

 triangles AEB, BEC, CED, and 

 DEA (formed by the radii of the 

 circle and the points ABC and D 

 being joined), right angle triangles 

 and similar; having the common 

 centre E a right angle in each ; 

 the legs AE, BE, CE, and DE (or 

 circle's radii), equal ; and the lines 

 or hypothenuses AB, BC, CD, and 

 AD, also equal, and together form- 

 ing an inscribed square to the cir- 

 cle ABCD. 



* The sale of the slabs pays the sawing expenses of squaring. 



