VOL. XXXII.3 PHILOSOPHICAL TRANSACTIONS. SQl 



remember are Sir Isaac Newton's first two, and from these all his others are 

 easily deduced. And as his irrational forms of the quadratic kind are derived 

 from the rational, so from my general rational form I deduce irrational ones of 

 all kinds. For instance, if - represent any affirmative or negative fraction, 



the fluent of any quantity of this form 



J' 



dzz X e +7^^* ^^ ^^ th\sdzz X J , and so of some others, 



depends upon the measures of ratios and angles. Mr. Leibnitz, in the Leipsic 

 Acts of 1702, p. 218 and 2 1 9, has very rashly undertaken to demonstrate, that 



the fluent of ^ ^ cannot be expressed by measures of ratios and angles ; and 



he swaggers upon the occasion (according to his usual vanity) as having by this 

 demonstration determined a question of the greatest moment. Then he goes 



on thus ; as the fluent of — -— depends upon the measure of a ratio, and the 



fluent of upon the measure of an angle ; so he had more than once 



expressed his wishes, that the progression may be continued, and it be deter- 

 mined to what problem the fluents of ^ ^ , -j-— — 5 , &c. may be referred. 

 His desire is answered in my general solution, which contains an infinite num- 

 ber of such progressions. I can go yet further, and show him how by measures 

 of ratios and angles, without any exception or limitation, the fluent of this 



Gn + - *) — 1 On + - 71 — 1 



general quantity — —7 — ; 5 or even this — —7 — • — , , , , may be had ; 



where 0, as before, represents any integer, and the denominator x of the 

 fraction -, represents any number in this series, 2, 4, 8, 16, 32, &c. any whole 

 number being denoted by its numerator L In truth I am inclined to believe, 

 that Mr. Leibnitz's grand question ought to be determined the contrary way ; 

 and that it will be found at last, that the fluent of any rational fluxion whatever, 

 does depend upon the measures of ratios and angles, excepting those which 

 may be had in finite terms even without introducing measures." 



Dr. Taylor, knowing by this letter what the author had done, was pleased to 

 propose the invention of the fluents of the last two fluxions, as a problem to 

 the mathematicians in foreign parts. Mr. Bernoulli, in the Leipsic Acts of 

 1719, p. 256, showed accordingly how they are reducible to conic areas. The 

 editor has published the author's own solution by measures of ratios and angles; 

 and on this foundation has constructed new tables of logometrical and trigono- 

 metrical theorems, for the fluents of fluxions, reduced to 94 forms, part 



