Considering how few, and how simple the Principles are, upon which the whole Art of <small>PERSPECTIVE</small> depends, and withal how useful, nay ho… - Brook Taylor

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.

In order to this, I found it absolutely necessary to consider this Subject entirely anew, as if it had never been treated of before; the Principles of the old Perspective being so narrow, and so confined, that they could be of no use in my Design: And I was forced to invent new Terms of Art, those already in use being so peculiarly adapted to the imperfect Notions that have hitherto been had of this Art, that I could make no use of them in explaining those general Principles I intended to establish.

Hence if <math>y</math> be the Root of any Expression formed of <math>y</math> and known Quantities, and supposed equal to nothing, and <math>z</math> be a part of <math>y</math>, and <math>x</math> be formed of <math>z</math> and the known Quantities, in the same manner as the Expression made equal to nothing is formed of <math>y</math>; and let <math>y</math> be equal to <math>z + v</math>; the difference <math>v</math> will be found by Extracting the Root of this expression <math>x + \frac {\dot{x}v}{1} + \frac {\ddot{x} v^2}{1 \cdot 2} + </math> ... etc. <math>= 0</math>.