J52 TO CONVERT ROUND MEASURE INTO SQUARE, 



To find a right Line, the Square of which Jhall be equal to a 

 given Circle. 



Problem. Draw (wo diameters (IF and C T) bife&ing each other at 



ftraft"on b the° n " nght a " gleS '" lhe Ce " ter (°) ° f the g lVe " C ' rcle 5 bife61 ° n6 

 fide of a fquare of the radii fo formed (O T) in W, and from the extremity (I) 

 equal to a circle. f t j ie next adjoining radius, through W draw the right line 

 (IB) to the circumference: This line (IB) is the line re- 

 quired. 



Proof. 



Prosf by tr'»r. Let a fquare be formed equal to the given circle by the third 

 corol. prop, fifth, of Archimedes; then take a fquare formed 

 by the line I B, and place it on this other fquare, fo that one 

 angle and the fide adjacent to it of one, fliall fall on one angle 

 and the fide adjacent to that angle of the other ; then will it 

 be feen that all the other angles and fides of each will coin- 

 cide, and the whole of one be equal to the whole of the 

 other. 



This kind of proef is nearly the fame as that of the fourth 

 prop, firft book of Euclid, on which fo many other propofi- 

 tions depend ; and having often tried this method in the above 

 manner, I could never perceive any difference between the 

 two fquares : In thofe trials I ufed circles of card paper for the 

 more exafl meafurement of the circumference in the mechani- 

 cal procefs directed in the method of Archimedes.* 



To the above I have to add the farther proof which follows 

 of the exa&nefs of my method, which may make it appear (till 

 more certain. 



Sometime after I difcovered the above, looking into a work 

 of the learned Kircher for a ready method of defcribing a pa- 



* In order to fliewhow near Mr. B.'s confhu&ion approaches to 

 the truth, we may obferve, that when the diameter is = 1, the 

 area is n 0.7854, and the fide of the equal fquare = 0.8862. 

 But in the figure, I W is found by adding the fquare of the radius 

 to that of the half radius, and extracting the fquare root ; and then 

 by the property of fimilar triangles, as the radius is to I W, fo is 

 the diameter to I B, which will be 0.8944 when the diameter is 

 — 1. But this line exceeds 0.8862, or the true fide of the fquare, 

 by 0,0082, or nearly one hundredth part. N. 



4 rabola 



