Tag Archives: Brook Taylor

Kirby Live Again!

I am giving a talk at MathFest in Hartford on August 1.  Intended for a wide audience, this talk will show how Kirby’s networks of subscribers evolved over the publication of his series of books between 1748 and 1754 to trace the patronage circles that ended with Kirby’s appointment as tutor in perspective to the then Prince of Wales.

Here’s the abstract:

How Brook Taylor Got Joshua Kirby a Position

In 1748, Joshua Kirby was a provincial coach-painter in Ipswich, Suffolk. By 1755 he was tutor in perspective to the Prince of Wales (the future George III). In between, he published Dr. Brook Taylor’s Method of Perspective Made Easy, a book that aimed to explain Brook Taylor’s notoriously difficult Linear Perspective. Using the subscription lists of the three works he published during this period, we trace how Kirby’s expanding social networks brought him to the notice of those in power.

Brook Taylor against Horizontals

We have seen how Brook Taylor’s Euclidean conception of space without a coordinate system and without a privileged reference system led him to condemn the use in perspective theory of horizon lines. In the expanded preface to the second edition of Linear Perspective, he had said,

The Term of Horizontal Line, for instance, it apt to confine the Notions of a Learner to the Plane of the Horizon, and to make him imagine, that the Plane enjoys some particular Privileges, which make the Figures in it more easy and more convenient to be described, by the means of that Horizontal Line, than the Figures in any other Plane; as if all other Planes might not as conveniently be handled, by finding other Lines of the same nature belonging to them.

In his own work, of course, Taylor is having none of this:

But in this Book I make no difference between the Plane of the Horizon, 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.

Taylor had not felt a need to burden the reader with such lengthy explanations in the first edition, but he very subtly got his point across right at the beginning of his text. He opens Section One of the book, before he even gives his first formal definitions with an explanation of what exactly perspective is:

Perspective is the Art of drawing on a Plane the Appearances of any Figures, by the Rules of Geometry.

The eighteenth century contained a long-rumbling debate about the extent to which artists should be bound by the rules of geometry in laying out their paintings and how much geometric rules could be trumped by other optical considerations. Taylor the mathematician comes down strictly on the side of geometry. (Kirby, with less confidence in his mathematical ability, equivocates.)

Taylor continues:

In order to understand the Principles of this Art, we must consider, That a Picture painted in its utmost degree of Perfection, ought so to affect the Eye of the Beholder, that he should not be able to judge whether what he sees be only a few Colours laid artificially on a Cloth, or the very Objects there represented, seen thro’ the Frame of the Picture, as thro’ a Window.

This conception of painting as representing a view through a window goes back at least to Alberti. Taylor explains further that this effect is not solely due to following the rules of geometry, that the geometrical rules of perspective are necessary, but not sufficient:

To produce this Effect, it is plain the Light ought to come from the Picture to the Spectator’s Eye, in the very same manner, as it would do from the Objects themselves, if they really were where they seem to be; that is every Ray of Light ought to come from any Point of the Picture to the Spectator’s Eye with the same Colour, the same strength of Light and Shadow, and in the same Direction, as it would do from the corresponding Point of the real Object, if it were placed where it is imagined to be.

As will be his usual practice, Taylor here gives a general description of what is to be done, and then gives a specific simple example:

So that (Fig. 1.) if EF be a Picture, and abcd be the Representation of any Object on it, and ABCD be the real Object placed where it should seem to be to a Spectator’s Eye in O; then ought the Figure abcd to seem exactly to cover the Figure ABCD, and the Rays AO, BO, CO, &c. that go from any Points A, B, C< &c. of the original Objects to the Spectator’s Eye O, ought to cut the Picture in the corresponding Points a, b, c, &c. of the Representation.

Now this is an admirably clear, simple and straightforward description of how the method of perspective is supposed to work. The kicker is in Figure 1, the first figure in the book a reader would encounter, and one that clearly establishes Taylor’s priorities.

Not a Horizon Line to be seen.

See also:

Brook Taylor on Education

Brook Taylor’s Second Preface

Brook Taylor’s First Preface

Brook Taylor’s Linear Perspective

More on Brook Taylor’s Linear Perspective

Who was Brook Taylor?

Taylor on Education

Brook Taylor’s complaints about his readers and predecessor writers on perspective reflected a general philosophy of education that he explained in his preface to the second edition of Linear Perspective. His objection was to students who learned by rote, and to teachers who taught students how to follow simple recipes without attaining deeper understanding of the underlying principles. He saw his role as laying out the general theory and sketching in the directions a student should follow to gain mastery of particular cases, without burdening the beginner with too much explication. He emphasized that his work was not for the complete beginner, that is, it assumed a familiarity with Euclidean geometry:

The Reader, who understands nothing of the Elements of Geometry, can hardly hope to be much the better for this Book, if he reads it without the Assistance of a Master; but I have endeavour’d to make everything so plain, that a very little Skill in Geometry may be sufficient to enable one to read the 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.

At this point in his Preface, Taylor is responding, in tones that appear somewhat hurt and largely uncomprehending, so critics of the earlier, purer, first edition of his book:

I find that many People object to the first Edition that I gave of these Principles, in the little Book entituled, Linear Perspective, &c because they see no Examples in it, no curious Descriptions of Figures, which other Books of Perspective are commonly so full of; and seeing nothing in it but simple Geometrical Schemes, they apprehend it to be dry and unentertaining, and so are loth to give themselves the trouble to read it. To satisfy these nice Persons in some measure, II have made the Schemes in this Book something more ornamental, that they may have some visible Proofs of the vast Advantages these Principles have over the common Rules of Perspective, by seeing what simple Constructions, and how few Lines are necessary to describe several Subjects, which in the common Method would require an infinite Labour, and a vast Confusion of Lines. It would have been easy to have multiplied Examples, and to have enlarged upon several things that I have only given Hints of, which may easily be pursued by those who have made themselves Masters of these Principles.

Taylor can be as prolix as any eighteenth-century author when he puts his mind to it. But now we come to the nub of the matter.

Perhaps some People would have been better pleased with my Book, if I had done this: but I must take the freedom to tell them, that tho’ it might have amused their Fancy something more by this means, it would not have been more instructive to them: for the true and best way of learning any Art, is not to see a great many Examples done by another Person; nut 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.

Theory first, then practice. Taylor expands upon this advice at length, with special reference to art education for which he suggest wholesale reform along his preferred lines.

See also:

Brook Taylor’s Second Preface

Brook Taylor’s First Preface

Brook Taylor’s Linear Perspective

More on Brook Taylor’s Linear Perspective

Who was Brook Taylor?

Brook Taylor’s Second Preface

Brook Taylor’s elegant but austere treatment of perspective in the first edition of Linear Perspective proved difficult for all but the most determined scholars to absorb. Treating his topic in the most general fashion possible, and largely eschewing examples or particular cases, gave little help for his readers to hold onto. In an attempt to meet his audience part way, Taylor rewrote the book, extending and expanding his explanation and introducing more examples in a largely unsuccessful attempt to make the work more palatable. It is clear from his comments that he was as baffled by the inability of ordinary mortals to read his perfectly clear work, as they were defeated by his uncompromising generality. In the much-expanded preface to the second edition, Taylor laid out his reasoning and articulated his underlying philosophy of education.

The changes began with the title. The first edition had been called, Linear Perspective, and only further down did it mention its novelty, `a new method of representing justly all manner of objects as they appear to the eye’. In the second edition, the novelty was emphasized right from the beginning, the New Principles of Linear Perspective, with the further explanation that the book expounded: `the art of designing on a plane the representations of all sorts of objects, in a more general and simple method than has been done before.’

The preface to the first edition spanned two paragraphs; that of the revised second edition ran twelve pages. The new preface is also much more conventional in beginning with a justification of the new work by declaring all previous treatments to be lacking, while again emphasizing the simplicity of his own approach:

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 sorts of Designing; I have often wonder’d, that is 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.

Reflecting his own emphasis on the mathematical qualities of generality and elegance, his criticisms of previous authors center on their excessive attention to details:

Some of those Books indeed are very voluminous: but then they are made so, only by long and tedious Discourses, explaining of common things; or by a great number of Examples, which indeed do make some of these Book valuable, by the great Variety of curious Cuts that are in them; but do not at all instruct the Reader, by any Improvements made in the Art it self.

Speaking as a high-level mathematician, Taylor’s specific charge against his predecessors is that while they might have had a lot of experience, they were insufficiently skilled mathematicians.

For 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; and therefore when, in their Practice, the Occasions that have offer’d, have put them upon inventing particular Expedients, they have thought them to be worth communicating to the Public, as Improvements in this Art; but they have not been able to produce any real Improvements in it, for want of a sufficient Fund of Geometry, that might have enabled them to render the Principals of it more universal, and more convenient for Practice.

Taylor now explains what is different about his own text and how in contrasts with earlier treatments of perspective.

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 the preface to the first edition Taylor said that he had found it necessary to ‘lay aside’ such terms as ‘Horizontal Line’ from the study of perspective, but had not troubled the reader with an explanation of why this should be necessary. In the second edition preface, he now uses this very term as an example with an explanation of why he was forced to begin anew. First he reiterates his general point:

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 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.

Following the general criticism, Taylor gives a specific example:

The Term of Horizontal Line, for instance, is apt to confine the Notions of a Learner to the Plane of the Horizon, and to make him imagine, that that Plane enjoys some particular Privileges, which make the Figures in it more easy and more convenient to be described, by the means of that Horizontal Line, than the Figures in any other Plane; as if all other Planes might not as conveniently be handled, by finding other Lines of the same nature belonging to them.

Thus speaks the mathematician.

See also:

Brook Taylor’s First Preface

Brook Taylor’s Linear Perspective

More on Brook Taylor’s Linear Perspective

Who was Brook Taylor?

Brook Taylor’s First Preface

The first edition of Brook Taylor’s Linear Perspective was a short work and contained an equally brief preface, called here, ‘To the Reader’ and spanning only two paragraphs. Here is the preface in its entirety.

In this Treatise I have endeavour’d to render the Art of Perspective more general, and more easy, than has yet been done. In order to this, I find it necessary to lay aside the common Terms of Art, which have hitherto been used, such as Horizontal Line, Points of Distance, &c, and to use new ones of my own; such as seem to be more significant of the Things they express, and more agreeable to the general Notion I have formed to my self of this Subject.

Thus much I thought necessary to say by way of Preface; because it always needs an Apology to change Terms of Art, or any way to go out of the common Road. But I shall add no more, because the shortness of the Treatise it self makes it needless to trouble the Reader with a more particular Account of it.

The first edition of Linear Perspective presents the material in the way that Taylor thought best. The second edition, by contrast, represents his attempts to meet the common reader part way, having found that his purist conception of the topic was largely indigestible to the wider public. Here, in his preface, Taylor admits that he needs to give fair warning to his prospective readers, that he has completely changed the terminology of perspective, the `Terms of Art’, but does not give any reason other than the fact that he personally found his own terminology `more significant’ by which he means that they better signify the concepts to which they refer, and `more agreeable to the general Notion I have formed my self of the Subject’. Thus, the reader is warned to expect an idiosyncratic terminology, but not given much of an explanation as to why.

Mathematicians are well-used to the introduction of new definitions and terminology when treating of new topics, or of old topics in a new manner. Other discourses are less freighted with the new. Taylor thus situates himself within a mathematical audience, rather than an artistic one. I want to explore this issue of discourse, presentation and style in further posts.

See Also:

Brook Taylor’s Linear Perspective

More on Brook Taylor’s Linear Perspective


Kirby Live!

I am giving a talk at the American Mathematical Society meeting at Boston College, the weekend of April 5-6.

My talk is on Perspective, Painting, Publishing, and Patronage: Joshua Kirby and Brook Taylor. It will be aimed at historians of mathematics and focus on a comparison of the content and discourse between Brook Taylor’s Linear Perspective and Joshua Kirby’s Method of Perspective.


Related Posts

Method of Perspective

Who was Brook Taylor?

Brook Taylor’s Linear Perspective

More on Brook Taylor’s “Linear Perspective”

More on Brook Taylor’s “Linear Perspective”

Brook Taylor’s book, Linear Perspective (1715) is short, only 40 pages, but dense. For later comparison with Kirby’s exposition, I want to bring out a few more features of Taylor’s text.

Taylor divides his book into five sections:

Section I: Containing an Explanation of those things that are necessary to be understood in order to the Practice of Perspective.

    This section contains his 10 definitions, along with several ‘Corollaries’, and the first four propositions.

Section II: Propositions relating to the General Practice of Perspective.

    The longest section, this one includes Propositions 5-18, with a set of examples between Propositions 16 and 17.

Section III: Of finding the Shadows of given Figures

    A short section with three propositions on shadows.

Section IV: Of finding the Representations of the Reflections of Figures on polish’d Planes.

    Taylor dispenses with reflections in three quick propositions.

Section V: Of the Inverse Practice of Perspective and of the manner of Examining Pictures already drawn.

    The mathematician comes to the fore with the study of inverse problems. The first proposition is, “Having given the Representation of a Line divided into two Parts in a given Proportion; to find its Vanishing Point.” He then tackles similar problems for triangles, parallelograms, trapezia, and “right-angled Parallelepipedons”.


For a flavor of Taylor’s style and terminology, here are the first few definitions, along with the figure they reference.

DEF. I. The Center of the Picture is that Point where a Line from the Spectator’s Eye cuts it (or its Plane continued beyond the Frame, if need be) at Right Angles.

If the Plane CD be the Picture, and O the Spectator’s Eye, then a Perpendicular let fall on the Picture from O, will cut it in its Center P.

DEF. II. The Distance of the Picture, or principal Distance, is the Distance between the Center of the Picture and the Spectator’s Eye

    In the same Figure PO is the Distance of the Picture.

DEF. III. The Intersection of an Original Line is that Point where it cuts the Picture.

    If IK be an Original Line cutting the Picture in C, then is C the Intersection of the Line IK.

DEF. IV. The intersection of an Original Plane, is that Line wherein it cuts the Picture.

    AB is an Original Plane cutting the Picture in the Line CQ, which therefore is its Intersection.

DEF. V. The Vanishing Point of an Original Line, is that Point where a Line passing thro’ the Spectator’s Eye, parallel to the Original Line, cuts the Picture.

    Such is the Point V, the Line OV being parallel to the Original Line IK.

COROL. I. Hence it is plain, that Original Lines, which are parallel to each other, have the same Vanishing Point. For one Line passing thro’ the Spectator’s Eye, parallel to them all, produces the Vanishing Point of ’em all, by this Definition.

COROL. 2. Those Lines that are parallel to the Picture have no Vanishing Points. Because the Lines which should produce the Vanishing Points, are in this Case also parallel to the Picture, and therefore can never cut it.

COROL. 3. The Lines that generate the Vanishing Points of two Original Lines, make the same Angle at the Spectator’s Eye, as the Original Lines do with each other.

DEF. VI. The Vanishing Line of an Original Plane, is that Line wherein the Picture is cut by a Plane passing thro’ the Spectator’s Eye parallel to the Original Plane.

    Such is the Line VS, the Plane EF, being parallel to the Original Plane AB.

COROL. I. Hence Original Planes, that are parallel, have the same Vanishing Line. For one Plane passing thro’ the Spectator’s Eye, parallel to them all, produces that Vanishing Line.

COROL. 2. All the Vanishing Points of Lines in parallel Planes, are in the Vanishing Line of those Planes. For the Lines that produce those Vanishing Points, (by Def. 5.) are all in the Plane that produces that Vanishing Line, (by this Def.)

COROL. 3. The Planes which produce the Vanishing Lines of two Original Planes, being parallel to the Original Planes, and passing both thro’ the Spectator’s Eye, (by this Def.) have their common Intersection passing thro’ the Spectator’s Eye, parallel to the Intersection of the Original Planes, and are inclined to each other in the same Angle as the Original Planes are. And hence,

COROL. 4. The Vanishing Point of the common Intersection of two Planes, is the Intersection of the Vanishing Lines of those Planes.

COROL. 5. The Vanishing Line, and Intersection of the same Original Plane, are parallel to each other. Because they are generated by parallel Planes. (By this Def. and Def. 4.)

I think this is all admirably clear, and shows how he treats lines and planes in a similar fashion. It also shows how his concepts of vanishing point and vanishing line are central to the way he is going to proceed.

Related Posts

Brook Taylor’s Linear Perspective

Who was Brook Taylor?

Method of Perspective

Related Works

Andersen, K., 1992, Brook Taylor’s Work on Linear Perspective: A Study of Taylor’s Role in the History of Perspective Geometry. Including Facsimiles of Taylor’s Two Books on Perspective. New York: Springer.

Andersen, K., 2006. The Geometry of an Art. The History of the Mathematical Theory of Perspective from Alberti to Monge. New York: Springer.