It seems that those, who have hitherto treated of this Subject, have been more conversant in the Practice of Designing, than in the Principles of Geo… - Brook Taylor

In doing this he makes use of a Table of Products (...he calls Speculum Analyticum,) by which he computes the Coefficients in the new Equation for finding the Difference mentioned. This Table, I observed, was formed in the same Manner from the Equation propos'd, as the s are, taking the Root sought for the only flowing Quantity, its Fluxion for Unity, and after every Operation dividing the Product successively by the Numbers 1, 2, 3, 4, etc.

Dr. Halley..., has publish'd a... compendious and useful Method of extracting the Roots of affected Equations of the common Form, in Numbers. This Method proceeds by assuming the Root desired nearly true... (...by a Geometrical Construction, or by some other convenient way) and correcting the Assumption by comparing the Difference between the true Root and the assumed, by means of a new Equation whose Root is that Difference, and which he shews how to form from the Equation proposed, by Substitution of the Value of the Root sought, partly in known and partly in unknown Terms.

<math>z</math> and <math>x</math> being two flowing Quantities (whose Relation... may be exprest by any Equation...) by [the aforesaid] Corollary, while <math>z</math> by flowing uniformly becomes <math>z+v</math>, <math>x</math> will become<math>x + \frac {\dot{x}}{1 \cdot \dot{z}}v + \frac {\ddot{x}}{1 \cdot 2 \cdot \dot{z}^2}v^2 +</math>... etc. or