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Castle Bar
10 Sept. 1789
Dr. Patrick Browne to Sir Joseph Bankes
Sir
Inclosed I send you the Diagram for my method of Squaring ye Circle, wh is compleated without any fractions & every part geometrical; wh I wish to be laid before your Society. The following is ye preparation & Demonstration
preparation. The Circle is Inclosed in a Square whose Sides are Equall to ye Diamiter of ye Circle and within ye circle is another Square whose Sides are ye whole Chords of ye Circle, and by perpendiculars on those, continued, the Cirscribed Square is divided into 16 Equall triangles of a middle Size marked, m. The Internal Square is made up of Eight of these triangles and the Circle, besides contains these Eight triangles and four Segments, SS. And If we prove ye four Segments Equall to 4 triangles, the Circle will be Equall to 12 triangles, m, or to ¾ rs of ye Circumscribed Square, or to Once & a half ye Inscribed Square.
I must premise that ye fundamental part of ye Demonstration is grounded on that proposition of Euclid which proves that ye Square of ye Hyppothenuse [also spelt hypotenuse] is Equall ye Squares of ye two other sides in a right angled triangle, and Consequently that ye Surfaces of regular & simular figures, were as ye Squares of their homologous sides. This posed, raise the half, or oblong square DLME and you will have 2 Squares, each equall to, & divided into two triangles, m, by ye Sides of ye Circumscribed Square.
On the Chord DE (one of ye sides of ye right angled triangle HDE) draw, ye Semicircle DKE, therefore equall to half ye Semicircle on ye Hyppothenuse, or ye quadrant DEC. Now ye Semicircle = triangle 4 + SS & ye quadrant = triangle 3 = triangle 4 + SS, take away both ye triangles & you will have SS = ss. And SA = SB 2 Equall triangles, take away S & S Equall among themselves & you have A = B. And further; ye Semicircle being Equal to ye quadrant, Deduct ye Segment SS wh is common to both, & you have ye Lunula = triangle, 3, = 2 triangles, m. And of Course the half Lunula = one triangle, m, ye same ye Measure the