44-t Mr. Walsh's Account 



surd propositions. The first of these was invented by the 

 ancients ; the others were invented by the moderns, of which 

 the last asserts, — that however small the variable x may be 

 taken, the variable // may be taken still smaller; which is 

 asserting that h may be less than itself. It is true, that how- 

 ever small x may be, h may be taken as small ; but it is ab- 

 surd to say that however small x may be taken, // may be 

 taken smaller, or however small h may be taken, la? may 

 be taken smaller. After this I began to dabble at the binomial 

 theorem, which appeared to me not to have received a satis- 

 factory explanation. I thought this theorem could be de- 

 monstrated by numbers, as it may be applied to numbers : 

 and I succeeded in demonstrating the law of its binomiation 

 when the exponent is a negative whole number. And shortly 

 after, I demonstrated the law of binomiation when the expo- 

 nent is a positive fraction. Encouraged by my success so far, 

 I endeavoured to apply this formula to the doctrine of curves. 

 The most illustrious Des Cartes first represented curve lines 

 by algebraic equations ; and that impressed me with the idea 

 that the development of these equations involved the general 

 theory of curves. With these ideas fixed in my mind, and 

 having demonstrated that all those propositions were absurd 

 which were made the bases of so many theories of calculation, 

 I devoted myself with some ardour to apply the binomial 

 theorem in drawing tangents to curve lines. The equation 

 y l = ax of the common parabola, is that which I made the 

 subject of investigation, as being the most simple. From this 



1 S ot /*(«)*+ *(«>£;- hat) f + &c 



h being any arbitrary increase of x. I saw clearly that the 



term \{ax) — , was the equation of a straight line. But here 



a question arose: What straight line it was of which that 

 term was the equation? I saw clearly it was the tangent 

 straight line. But then it was necessary to demonstrate this. 

 Now I saw clearly that this would be accomplished, if I could 

 prove that y + \{axY, was greater than y. 



In order to prove this I squared those two terms, and this 

 gave me , . 7 » . , , v A* 



which was greater than a(x +h), the square of the entire 

 series, whatever may be h. This broke down the barrier which 

 opposed itself to my advancement, and opened to my view a new 

 field more fertile than any that was before explored, and freed 

 the human mind from the shackles of an absurd logic by which 



it 



