Bangma's Method of solving Equations. S69 



the arc of any curve. Binomiating from this term, taking 

 dx negative, and, in the binomiation, for dx substituting x, 

 the sinomial or arc is equal to the sum of the remaining terms 

 of the series. If, then, for x we substitute any constant in- 

 volved in the equation of the curve, the sinomial will be de- 

 termined in terms of this constant. Applying this to the cir- 

 cle, we get for the length of the fourth part of the circum- 

 ference the well known series 



• «= < 



, , J , 1-3 , 1 -3.5 



From which it follows, that the circumferences of circles are 

 to one another as their diameters. 



The preceding proposition does not appear to have been 

 demonstrated before the invention of the binomial calculus. 

 Euclid does not demonstrate it. His reasoning is founded on 

 a lemma (the base of the method of exhaustions, of fluxions, 

 of the differential calculus, &c), which asserts the absurdity 

 that a magnitude may be less than itself. And his attempt 

 to prove this absurdity involves the assertion of the opposite 

 absurdity, that a magnitude may be greater than itself. The 

 second proposition ot his twelfth book was deduced by analogy 

 from the property of similar polygons ; and he was obliged to 

 heap absurdity on absurdity, to give his postulate the colour 

 of demonstration. This is the only blemish in the most im- 

 portant work on science that was ever composed, or that can 

 hereafter be composed. Every other work on science, either 

 physical or mathematical, falls into insignificance when com- 

 pared with the stupendous work of this immortal geometer. 



With respect to Mr. Ivory's paper. I require of Mr. Ivory 

 to construct the triangle, of which the base is not any thing. 

 He is not to prove his construction by a simple appeal to 

 " every body." Such a mode of reasoning does not belong to 

 geometry. Neither is he to introduce the " ghosts of departed 

 quantities." For it is demonstrated that such things are ab- 

 surd. He is to prove his construction by reasoning referred 

 to our intuitive knowledge. His paper is a very awkward 

 surrender of the point he would maintain, 



Cork, May 15, 1824. J. Walsh. 



LXL On a Method of folding the Limits of the Roots of the higher 

 Powers of Numerical Equations. By Mr. J. Rowbotham. 



To /he Editors of the Philosophical Magazine and Journal. 



Walworth, May 17, 1824. 

 wHOULI) you think the following method of solving, or 

 rather of finding the limits of the roots of the higher 

 Vol. 6X No, 913. May 1824. '{ A powers 



