LIMITS AND FLUXIONS 



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before the end of that time, approach nearer to each 

 other than by any given difference, become ulti- 

 mately equal. 



" If you deny it, let them be ultimately unequal ; 

 and let their ultimate difference be D. Therefore, 

 they cannot approach nearer to equality than by 

 that given difference D. Which is against the 

 supposition." 



Principia, Book /, Sectiott /, Lemma II 

 Translation by Motte :^ 



9. ''If in any figure AacK, terminated by the 

 right lines Aa, A E, and the curve acY., there be 

 inscribed any number of paral- 

 lelograms Ab, B<:, Cd, etc, 

 comprehended under equal bases 

 AB, BC, CD, etc, and the sides 

 ^b. Ce, Dd, etc, parallel to one 

 side Aa of the figure ; and the 

 parallelograms aKb/, bhejn, 

 eMdn, etc, are completed. 

 Then if the breadth of those 

 parallelograms be supposed to 

 be diminished, and their number be augmented in 

 infinitum ; 1 say, that the ultimate ratios which 

 the inscribed figure AK<^L<:M<^D, the circumscribed 

 figure Aalbmcndo^., and curvilinear figure AabedK, 

 will bave to one another, are ratios of equality. 



^ T/ie Mathematical Principks of Naturai Philosophy, by Sir Isaac 

 Newton; transiated into English ^j/ Andrew Motte, London, 1729. 

 (Two volumes.) 



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