Brook Taylor

"Considering how few, and how simple the Principles are, upon which the whole Art of PERSPECTIVE 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."

"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."

"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."

"I make no difference between the Plane of the , and any other Plane whatsoever; for since Planes, as Planes, are alike in Geometry, it is most proper to consider them as so, and to explain their Properties in general, leaving the Artist himself to apply them in particular Cases, as Occasion requires."

"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."

"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."

"And upon this occasion I would advise all my Readers, who desire to make themselves Masters of this Subject, not to be contented with the Schemes they find here; but upon every Occasion to draw new ones of their own, in all the Variety of Circumstances they can think of. This will take up a little more Time at first; but in a little while they will find the vast Benefit of it, by the extensive Notions it will give them of the Nature of these Principles."

"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..."

"It is generally thought very ridiculous to pretend to write an Heroic Poem, or a fine Discourse upon any Subject, without understanding the Propriety of the Language wrote in; and to me it seems no less ridiculous for one to pretend to make a good Picture without understanding Perspective..."

"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."

"I would recommend it to the Masters of the Art Painting... to establish a better Method for the Education of their Scholars, and to begin their Instructions with the Technical Parts of Painting, before they let them loose to follow the Inventions of their own uncultivated Imaginations."

"[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."

"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."

"Nothing ought to be more familiar to a Painter than Perspective; for it is the only thing that can make the Judgment correct, and will help the Fancy to invent with ten times the ease that it could do without it."

"[H]e should be instructed in the Theory of the Colours; that he should learn... their particular Properties... Relations, and... Effects that are produced by their Mixture; and that he should be made well acquainted with the Nature of the several material Colours... used in Painting."

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

"There may be regular Methods also invented for teaching the Doctrine of Light and Shadow; and other Particulars relating to the Practical Part of Painting, may be improved and digested into proper Methods... But I only hint at these... recommending them to the Masters of the Art to reflect and improve upon."

"The Book it self is so short, that I need not detain the Reader any longer in the Preface..."

"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 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."

"Hence I soon found that this Method might easily and naturally be drawn from Cor 2. Prop. 7. of my Methodus Incrementorum, and that it was capable of a further degree of Generality; it being Applicable, not only to Equations of the common Form, (viz. such as consist of Terms wherein the Powers of the Root sought are positive and integral, without any Radical Sign) but also to all Expressions in general, wherein any thing is proposed as given which by any known Method might be computed; if vice versâ, the Root were consider'd as given: such as are all Radical Expressions of Binomials, Trinomials, or of any other Nomial, which may be computed by the Root given, at least by s, whatever be the Index of the Power of that Nomial; as likewise Expressions of Logarithms, of Arches by the Sines or s, of Areas of Curves by the Abscissa's or any other Fluents, or Roots of Fluxional Equations, etc."

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

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

"[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."

"A general Series for expressing the Root of any Quadratick Equation. Any Quadratick Equation being reduc’d to this Form xx - mqx + my = 0, the Root x will be exprest by this Series of Terms. 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} 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 B = A \times \frac {1}{\frac{mq^2}{y} - 2} 2. ...a, b, c, etc. ...are equal to the whole Divisors of the Fraction in the Terms immediately preceding; thus b = a^2 - 2."

"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 \frac{2}{3} and \frac{4}{3}, the Product \frac{8}{9} is less than Unite, but nearer to it than \frac{2}{3}, 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 \frac{128}{100} and \frac{8}{10}, viz. \frac{2^7}{10^2} and \frac{2^3}{10^1}... one... bigger, and the other less than 1."

"Early in 1717 he returned to London, and composed three treatises, which were presented to the Royal Society, and published in the 30th volume of the Transactions. About this time his intense application had impaired his health to a considerable degree; and he was under the necessity of repairing, for relaxation and relief, to Aix-la-Chapelle. Having likewise a desire of directing his attention to subjects moral and religious speculation, he resigned his office of secretary to the Royal Society in 1718. After this he applied to subjects of a very different kind. Among his papers were found detached parts of a Treatise on the Jewish Sacrifices, and a dissertation of considerable length on the Lawfulness of eating Blood. He did not, however, wholly neglect his former subjects of study, but employed his leisure hours in combining science and art; with this view he revised and improved his treatise on Linear Perspective."

"Drawing continued to be his favourite amusement to his latest hour; and it is not improbable that his valuable life was shortened by the sedentary habits which this amusement, succeeding his severer studies, occasioned."

"The theory of perspective was taught in painting schools from the sixteenth century onward according to principles laid down by the masters... However, their treatises on perspective had on the whole been precept, rule, and ad hoc procedure; they lacked a solid mathematical basis. In the period from 1500 to 1600 artists and subsequently mathematicians put the subject on a satisfactory deductive basis, and it passed from quasi-empirical art to a true science. Definitive works on perspective were written much later by eighteenth-century mathematicians Brook Taylor and J. H. Lambert."

"Brook Taylor... in his Methodus Incrementorum Directa et Inversa (1715), sought to clarify the ideas of the calculus but limited himself to algebraic functions and algebraic differential equations. ...Taylor's exposition, based on what we would call finite differences, failed to obtain many backers because it was arithmetical in nature when the British were trying to tie the calculus to geometry or to the physical notion of velocity."

"The Gregory-Newton interpolation formula was used by Brook Taylor to develop the most powerful single method for expanding a function into an infinite series. In his Methodus Incrementorum Directa et Inversa Taylor derived the theorem... he praises Newton but makes no mention of Leibniz's work of 1673 on finite differences, though Taylor knew this work. Taylor's theorem was known to James Gregory in 1670 and was known... by Leibnez, however these two men did not pubish it. John Bernoulli did publish practically the same result in the Acta Eruditorium of 1694; and though Taylor knew his result he did not refer to it. ...Colin Maclaurin in his Treatise of Fluxions (1742) stated that... [Mclaurin's theorem] was but a special case of Taylor's result."

"I am spared the necessity of closing this biographical sketch with a prolix detail of his character: in the best acceptation of duties relative to each situation of life in which he was engaged, his own writings and the writings of those who best knew him, prove him to have been the finished Christian, gentleman, and scholar."