Brook Taylor Quotes

[T]he Theory I have endeavour'd to explain in the Appendix, from Sir Isaac Newton, may be of very great use to Learners.

A new Method of computing Logarithms.
This method is founded upon...
1. That the sums of any two Numbers is the Logarithm of the Product of those two Numbers Multiplied together.
2. That the Logarithm of Unite is nothing; and consequently that the nearer any Number is to Unite, the nearer will its Logarithm be to 0.
3rdly. That the Product by Multiplication of two Numbers, whereof one is bigger, and the other less than Unite, is nearer to Unite than that of the two Numbers which is on the same side of Unite with its self; for Example the two Numbers being <math>\frac{2}{3}</math> and <math>\frac{4}{3}</math>, the Product <math>\frac{8}{9}</math> is less than Unite, but nearer to it than <math>\frac{2}{3}</math>, which is also less than Unite. Upon these Considerations, I found the present Approximation... best explain'd by an Example. ...[T]o find the Relation of the Logarithms of 2 and of 10... take two Fractions <math>\frac{128}{100}</math> and <math>\frac{8}{10}</math>, viz. <math>\frac{2^7}{10^2}</math> and <math>\frac{2^3}{10^1}</math>... one... bigger, and the other less than 1.

[I]t may not be amiss to set down here two Approximations I have formerly hit upon. The one is a Series of Terms for expressing the Root of any Quadratick Equation; and the other is a particular Method of Approximating in the invention of Logarithms, which has no occasion for any of the Transcendental Methods, and is expeditious enough for making the Tables without much trouble.

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>.

The Greatest Masters have been the most guilty... The great Occasion of this Fault, is certainly the wrong Method that generally is used in the Education of Persons to this Art: For the Young People are generally put immediately to Drawing, and when they have acquired a Facility in that, they are put to Colouring. And these things they learn by rote, and by Practice only; but are not at all instructed in any Rules of Art. By which means when they come to make any Designs of their own, tho' they... don't know how to govern their Inventions with Judgment, and become guilty of so many gross Mistakes, which prevent themselves, as well as others, from finding that Satisfaction, they otherwise would do in their Performances.

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 Geometry... that might have enabled them to render the Principles of it more universal, and more convenient for Practice. In this Book I have endeavour'd to do this; and have done my utmost to render the Principles of the Art as general, and as universal as may be, and to devise such Constructions, as might be the most simple and useful in Practice.

I have endeavour'd to make every thing so plain, that a very little Skill in Geometry may be sufficient to enable one to read this Book by himself.

A general Series for expressing the Root of any Quadratick Equation.
Any Quadratick Equation being reduc’d to this Form <math>xx - mqx + my = 0</math>, the Root <math>x</math> will be exprest by this Series of Terms. <math> x = \frac {y}{q} + A \times \frac{1}{\frac{mq^2}{y} -2} + B \times \frac {1}{a^2 - 2} + C \times \frac {1}{b^2 - 2} +D \times \frac {1}{c^2 - 2}</math> etc. Which must be thus interpreted.
1. ...A, B, C, etc. stand for the whole terms with their Signs, preceding those wherein they are found, as <math>B = A \times \frac {1}{\frac{mq^2}{y} - 2}</math>
2. ...<math>a, b, c,</math> etc. ...are equal to the whole Divisors of the Fraction in the Terms immediately preceding; thus <math>b = a^2 - 2</math>.

The true and best way of learning any Art, is not to see a great many Examples done by another Person, but to possess ones self first of the Principles of it, and then to make them familiar, by exercising ones self in the Practice. For it is Practice alone, that makes a Man perfect in any thing.

The Art Perspective is necessary to all Arts, where there is any occasion for Designing... but it is more particularly necessary to the Art Painting...

Considering how few, and how simple the Principles are, upon which the whole Art of <small>PERSPECTIVE</small> depends, and withal how useful, nay how absolutely necessary this Art is to all forts of Designing; I have often wonder'd, that it has still been left in so low a degree of Perfection, as it is found to be, in the Books that have been hitherto wrote upon it.

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.

[T]he Method which ought to be follow'd in instructing a Scholar in the Executive Part of Painting; ...first have him learn the most common Effections of Practical Geometry, and the first Elements of Plain Geometry, and common Arithmetic.

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.

When he is sufficiently perfect in these, I would have him learn Perspective. And when he has made some progress in this, so as to have prepared his Judgment with the right Notions of the Alterations that Figures must undergo, when they come to be drawn on a Flat, he may then be put to Drawing by View, and be exercised in this along with Perspective, till he comes to be sufficiently perfect in both.