Page:Philosophical Transactions - Volume 014.djvu/185

Rh easy to find all the roots to any Resolvend offer'd.

Now for instance (according to Huddens method) in a biquadratick equation you must multiply all the terms beginning with the highest, and so in order by 4, 3, 2, 1 and the last term or Resolvend by O. whereby it is destroyed, and you come to a cubick Equation, the same as Harriot uses to take away the penultimate Term of the biquadratick, the roots whereof being found, and as roots having Resolvends raised thereto in the biquadratick Equation, are the dioristick Limits thereof.

18 And if this easy method were known, we may come down the Ladder to the bottom, and fall into irrational quantities, and ascend again. Against which assymetry, an Equation might be assumed low, as rational quadratick, and thence a cubick Equation formed, whole limits should be found by aid of the quadratic Equation, and out of that cubick a Biquadratick Equation, whose limits should be found by the aid of that cubick Equation, &c.

19 Equations may be so continued of two Nomes, that both the dioristick and base limits, should be rational, then supposing such Equation incomplete, the increasing or diminishing the roots, fills up all the vacant places.

Q. Whether or in what place one or both sorts of Limits shall loose their rationality? And what is the nature of the roots thus drawn? in this I think you have already determined in divers of your surd Canons.

20 What Dr. Pells method mention'd in Section 17 should be I cannot guess, unless it be either.

To make surd Canons. Or good approaches.

Or that raising Resolvends out of assumed roots, those should make a store from whence to derive the roots of the Resolvend offered.

Or making quadratick Equations out of the dioristick and base limits, those might be interpoled, by aid of a Table of figurate numbers, or otherwise thereby, as in quadratick Equations to attain two roots of a biquadratick at once. which if performed the greatest difficulties are overcome, and why should not this seem probable, in regard the Curve or Locus, be the Equation what it will, makes indented porches.

21 Suppose I should propound two cubick or biquadratick Equations, in both whereof all the signs are +. It is propoundded out of these two, to derive a third Equation, whose root shall be the Summm, Difference, or Rectangle of the Roots of the two Equations propounded. This Mr Gregory a little before his death writ word he had obtained and in the following Series for finding the Moity of a Hyperbolick Logarithm I suppose made use of. From