278 DME0NSTRAT10N OF THE COTESIAN THEOREM. 



IV. 



Demonstration of the Coiesian Theorem. By Mr. P. 

 Barlow. 



To Mr. NICHOLSON. 



* SIR, Oct, 1 3th, 1809. 



Cotesian thco- .1.1 F the following demonstration of the celebrated theorem 

 rem> of Cotes appear to you to possess a sufficient degree of ori- 



ginality, to entitle it to a place in your Journal, its inser- 

 tion will oblige 



Yours, &c. 

 Royal Military Academy, P. BARLOW . 



Woohvkh. 



j a r^n^e's I was led to the consideration of this theorem from an 



conjectu e of observation made by La Grange, in his Theorie des Fonc- 

 toit snecestetl tlons Analytiques, where he hazards a conjecture as to the 

 the den.onstra- probable circumstances that led Cotes to the discovery of 

 tionhea given. this elegant property of the circle; and by pursuing the 

 hints there given I arrived at the following demonstration, 

 which appears to me to be, at least, as satisfactory as any 

 one that I have at present seen : and on this I rest my apo- 

 logy for intruding into the pages of the Philosophical Jour- 

 nal a subject, that has so long, and so often, engaged the 

 attention of many very celebrated mathematicians, without 

 any of them having arrived at what may be considered an 

 unobjectionable demonstration. 



It will be proper, however, for the information of some 

 of your readers* before we enter upon the demonstration 

 to sta'e the theorem itself, which is as follows. 



C'jtts's Theorem. 



Theorem. Let ABC, &C, PI. VIII, fig. 31, 32, be any circle, 



divided into any even number of equal parts, 2 m, as A B, 

 DC, CD, &c.j also let P be any point in the diameter, 



e/itlier 



