Page:Philosophical Transactions - Volume 014.djvu/184

Rh varying the same, both the other roots (as in the Postscript): for every number or magnitude capable of a cube root, is capable of two more, see Section the 11th. following.

10 If the roots in the former Section, be assumed in Arithmetical progression, and the equation with its several Resolvends be depressed, there will come out a regular Series of Quadratick Equations, whence an easie method will rise of writing down such ranks as multiplied by an Arithmetical progression, shall always beget the same cubick equation, the Resolvend only varying.

11 Let the roots of this series of quadraticks be found as usual in binomials, let these binomials be cubed, and then let it be observed, whether the results are constant portions of the square of the Resolvend and of the dioristick limit: and if so, Cardans Rules will have their defect supplyed.

12 In breaking biquadratick, 'tis asserted that by leaving the Resolvend at liberty, it may be infinitely and rationally done, without the Aid of the separating cubick Equation.

13 But Supposing such separating cubick in store, of which Bartholimus in his dioristick hath given us great furniture in Spicies, why may not several roots of that equation be assumed rational, and thence the biquadratick broken into as many pairs of quadratick equations?

14 May not from hence a method arise of writing down 2 Series of quadraticks that multiplied together shall always beget the same biquadratick Nomes, the Resolvend only varying? and hence the Locus of the equation is easily described.

15 Here again (as in the 11) if the binomial roots of these quadraticks be squaredly squared, and those results are constant portions of the cube of the Resolvend, and the dioristick limit; it will be certain there may be general surd Canons for equations of the 4th. dimension, and Monsieur Cluverius (now at London) positively asserts he hath a general method to obtain them for all Dimensions.

16 As Cardans are surd canons deriv'd from the Resolvend, and dioristick limit, so it were worthy disquisition, whether other furd Canons (of which many are fitted to particular cases by your self, Leibnitz and others) do not arise out of the limits of those particular cases and equations, and whether the glimpse of a general Method might thence be deriv'd for all other equations, though encumbred with negative quantities? which Mr. Gregory, a little before his death, said he had attained.

17 The Learned Dr. Pell hath often asserted that after the Limits of an equation are once obtain'd, then it is