VOL. XIV.] PHILOSOPHICAL TBANSACTIONS. 41 



are constant portions of the square of the resolvend and of the diorlstic limit; 

 and if so. Cardan's rules will have their defect supplied. 



If the binomial roots of a series of quadratics be squaredly squared, and those 

 results be constant portions of the cube of the resolvend, and the dioristic 

 limit; it will be certain there may be general surd canons for equations of the 

 4th dimension ; and Monsieur Cavalerius, {now in London) positively asserts, 

 he has a general method to obtain them for all dimensions. 



As Cardan's are surd canons derived from the resolvend and dioristic limit, 

 so it may be worthy consideration, whether other surd canons do not arise out 

 of the limits of those particular cases and equations, and whether the glimpse 

 of a general method might thence be derived, for all other equations, though 

 encumbered with negative quantities? which Mr. Gregory, a little before his 

 death, said he had attained. 



The learned Dr. Pell has often asserted, that after the limits of an equation 

 are once obtained, then it is easy to find all the roots to any resolvend 

 proposed. 



Suppose I should propound two cubic or biquadratic equations, in both of 

 which all the signs are +• It is propounded out of these two, to denve a 

 third equation, whose root shall be the sum, difference, or rectangle of the 

 roots of the two equations propounded. This Mr. Gregory, a little before his 

 death, wrote that he had obtained, and in the following series for finding the 

 half of an hyperbolic logarithm I suppose made use of. 



From a number proposed subtract an unit, and let that be numerator; also to 

 it add an unit, and let that be denominator: and call that fraction N. 



Then +N + iN^ + iN^ + J-N' + -lN' + V4-N" + ^vN^ &c. is equal 

 to half the hyperbolic logarithm sought. 



Example in the Number 1. 

 N 

 The fraction is -f. i 



3 



5 



The rank N is easily 7 



made by dividing every Q 



preceding number by 9, 1 1 



13 



which is the hyperbolic logarithm of 2 sought. 

 VOL. III. G 



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