On Taylor's Theorem. 189 



been objected to; and I strongly suspect that it must finally 

 be classed with the disputed axiom of Euclid, and other points 

 in which the mathematics touch upon metaphysics. The 

 proofs which have been given maybe briefly classed as fol- 

 lows: 



1. Those which are derived from the binomial theorem, to 

 the proofs of which the above-mentioned objections equally 

 exist. 



2. Those which are derived from combined considerations 

 of expansions and limits ; which are liable to the objections 

 urged against both methods. 



3. That of Lagrange, which assumes expansions only, and 

 to which the objections of Woodhouse and others are well 

 known. 



But I am not aware that any proof is commonly circulated 

 which depends upon limits only, though the various considera- 

 tions upon which I have put together the following are not 

 unfrequently mentioned. The fundamental theorems from 

 which it is deduced are marked I., II., and III., and the form 

 in which the theorem is deduced is that of Lagrange, after 

 the method of M. Cauchy in his Lemons sur le Calciil Differ- 

 entiel. But the last-mentioned writer has not established it 

 entirely on the theory of limits. 



I now proceed to the point in question, and enunciate the 

 several steps of the process. 



Axiom, commonly assumed in what is called the law of 

 continuity. If $ x be a function which continually decreases 

 or increases from x = a to x = b, without becoming infinite 

 for any value of x contained in that interval, then if $ a = A 

 and <p b = B, any value lying between A and B may be given 

 to the function <px, by giving a value to x lying between a 

 and b. 



Theorem 1. If the second of the following series of quan- 

 tities be all positive 



a b c .... 

 a /3 y . . . . 

 then the fraction 



a + b + c + . . . . 

 « + /3 + y + . • • • 



must lie between the greatest and least (algebraically speak- 

 ing) of the following fractions, 



a b c „ 

 Theorem 2. If S and X be given quantities, and if a, a\ a l \ 



