202 Method of Progressions. 



but so long as m is anv imaginable number these lines will not 

 meet. It is therefore clear that wo cannot express the area of 



a curvilinear space accurately by if it depends upon the 



sum of the progression being accurately given by the same ex- 

 pression. But it is obvious that in proportion as we increase the 

 value of w, the progression approaches nearer to the true area of 



the space, because — becomes less ; and if we suppose this ap- 

 proach to accuracy to be measured by a deflexion of the linn AC 

 towards B b"\ the two lines must ultimately meet. And as we 

 (an always suppose the value of m to be such that they would 



meet ; — must truly express the area. 



The truth of the result is a consequence of the causes of error 

 neutralising one another, and which necessarily flows from the 

 method of investigation. The result is essentially the same as is 

 obtained by the method of fluxions ; indeed it is one of the most 

 important of the rules of that celebrated method investigated by 

 common processes. I much suspect that neither Sir Isaac New- 

 ton nor Leibnitz ever understood the real nature of the method 

 of fluxions ; if this suspicion be well founded (and the oI)scnre 

 or erroneous reasoning of its authors renders such a suspicion 

 justifiable), there will be little difficulty in accounting for Sir 

 Isaac's tardiness in publishing his discoveries. Uiider any other 

 point of view his conduct appears inconsistent. 



I shall most likely be accused of presumption in making the 

 preceding remarks; but i would rather believe that the method 

 of fluxions was the result of repeated trials, than that its author 

 was indiiferen; about the progress of science, or that he wished 

 to reserve to himself, as a miser does his gold, that which was a 

 thousand times more valuable to his fellow men. 



I am, sir, yours &c. 

 No. 2, GroTC Terrace, Maixli IJ, l;:'l'l. TlIOMAS TrecGOLD. 



XXX I II, On 



