JOURNAL 



NATURAL PHILOSOPHY, CHEMISTRY, 



AND 



THE ARTS. 



SEPTEMBER, 1808. 



ARTICLE I. 



A Mathematical Prohleyn : hy John Gough, Eaq. 



To Mr. NICHOLSON. 

 SIR, Middleshaiv, August 5, 1808. 



T. 



HE insertion of the following problem, with the inves- 

 tigation of it, in your valuable Journal, will oblige 



Yours, &c. 



JOHN GOUGH. 



Problem. To a given arc of a circle a, let it be required The problem 

 to add another s, making the sum of the two arcs equal to P''°P®*^°« 

 the tangent of the latter, t, viz. a-{-z7z:t, 



(A) We may show in the following manner, that the ContainsTio 

 problem contains nothing absurd in it ; but that, on the ^ *"^ ^ * 

 contrary, there is a value of z to each value of a, which 



would fulfil the conditions of the question, were we but able 

 to rectify the circle. 



(B) Let A B C, PI. I, fig. 6, be the given arc, consist- This assertion 

 ing of n quadrants, n being any positive number, whole or °®"^o"strated, 

 fractional ; to this add the quadrant C P D, in which take 



Vol. XXI. No. 91— Sept. 1808. B the 



