Letter received by Banks from Patrick Browne, 10 November1789 (Series 72.010) - No. 0002

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the whole Diagram, and hence,

S as = Triangle, Sa+s, or ye Semilunula.  As + S is also Equall to AsB = Semilunula + B or triangle BS+a.  Take away As (ye Semilunula common to both) and you have left S = B = a and therefore S = s = B = a, & therefore Ss = Sa. 

Yet as there is so little Similarity between S & A I thought proper to give another demonstrative proof of their Equality. Look therefore to ye next Small Square which is divided in ye same manner, & further divided into 4 small triangles xxxx by a Cross Diagonal, which also Divides, A, into L + z and S, into N + o.

Then say L + Z = a - a +0 = x + 3.   }
               N + z + 0 = S + Z + X + 0  } Take away from L + z = a
                                                            [                       ] N + Z =x
or take away Z from both & [                   ] but Let [   ] triangle and

Then Say L + z + o = xx - N, and N + z + o = xx - L, take away z + o from both & remains L = N = xx -zxo then say N + z = x & L + o = x therefore z & o are Equall.  But L+ o is a triangle x therefore N + o = x & L+ o = x is Equall to a triangle.  L + o = L + z = a and N + o = N + z = x = S therefore A = S. Q.E.D.  Now SS ye Segment is Equall to SA ye triangle, therefore ye four Segments Equall 4 triangles, therefore ye Circle Equalls 12 triangles. Q.E.D.  12 triangles is ¾ rs of ye Circumscribed Square therefore ye  area of ye Circle is ¾ rs of ye area of ye Circumscribed Square, & once & a half ye Inscribed Square.

Suppose ye Diamiter of ye Circle 10. 10 x 10 = 100 ¾ of wh is 75.

Suppose ye radius 5 x 5 = 25 x by, 3, for ye ¾ rs = 75.

Suppose one of ye Little Squares = 2 triangles, m, = 12.5 multiplyed by, 6 = 75 & ye Solidity of ye Sphere will be Equall to ½ that of ye Cube, having lost a quarter by ye corners & a quarter (= 1/3 of ye Cylinder) by ye slopings.

Suppose a Circle whose Diamiter is 14.4, by ye old Method ye Area = 162.9216, by ye present method, ye area is Equall only to 155.92.

The truth may be Easily assertained by puting, a Cube and a Sphere of ye Same Diamiter alternately in a Square Even Vessel, whose Diamiter = Diamiter of ye Sphere or Side of ye Cube nearly & Whole Side