Monthly Archives: March 2013

Brook Taylor’s Linear Perspective

Joshua Kirby claimed in his Method of Perspective that he was making Brook Taylor’s work easier to understand for gentlemen and practitioners. Brook Taylor’s Linear Perspective was published in 1715, with a revised edition in 1719. His work is austere, rigorous and mathematically challenging. Perhaps as a result, it went largely unread, and, even after it was popularized by Kirby, Highmore, and others, it seems to have been more appreciated than read. In recent scholarship, Kirsti Andersen has studied Taylor’s work most deeply.

When Brook Taylor wrote Linear Perspective, few in Britain had gone before him, and certainly not attempting a coherent theoretical approach; in this respect Britain was far behind the Continent. The inadequacy of his predecessors caused Taylor to sweep away the old vocabulary and replace it with his own set of terms and concepts, including linear perspective. As he wrote in his Preface,

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 do 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.

He succeeded in his aim of rendering the theory of perspective `more general’, but perhaps not `more easy’ to mere mortals. Like many mathematicians, his definition of `easy’ did not match that of the general populace. His constructions are simple, they use few ideas and few construction lines, but they require a great deal of mathematical maturity and do not lend themselves easily to actual practice.

The first clue to his generality lies in his getting rid of the horizon line. Taylor inhabited a Euclidean geometrical world, not a Cartesian one. To him,

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

Definition I. The Center of the Picture is that Point where a Line from the Spectator’s Eye cuts it …at Right Angles.

Taylor has three ingredients: a spectator, a picture plane, and the original plane of the objects to be represented. He sees no reason why the picture plane should be perpendicular to the ground plane, and hence, he has no need of horizons and horizontals. Here’s his illustration, showing the ‘leaning plane’.

I just love his approach, but then I don’t have to use it to paint. He introduces the terms vanishing point and vanishing line, and treats the one and two-dimensional cases as on equal footing in a manner which is wonderful to behold. Many of Taylor’s propositions take various pieces as given (in the Euclidean sense) and require finding the remaining points. The proofs unfold in a Euclidean manner, and there are practically no examples. He also, as you would expect, leans heavily on ratio theory. Here is Theorem 2:

Any Line in the Representation of a Figure parallel to the Picture, is to its Original Line, as the Principal Distance is to the Distance between the Spectator’s Eye, and the Plane of the Original Figure.

And here is Proposition 10:

Having given the Center and Distance of the Picture, and the Vanishing Line of a Plane, and the Vanishing Point of the Intersection of that Plane, with another Plane perpendicular to it; to find the Vanishing Line of that other Plane.

He covers the whole theory of perspective in 40 pages.

Related Posts

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.

Who was Brook Taylor?

Kirby’s Method of Perspective was called Dr. Brook Taylor’s Method of Perspective Made Easy. So who was Brook Taylor, and why did his method of perspective need to be made easy? In this post I will answer the first question.

Taylor’s name is known to generations of calculus students through Taylor series. A Taylor series represents a function as an infinite series with coefficients calculated from the derivatives of the function at a particular point. If the series converges nicely, it allows you to approximate the value of a function by a simple finite sum. Back in the early 18th century (Taylor’s theorem is from 1715), using infinite series as a way to deal with intractable functions was a popular topic, although the properties of infinite series were not completely understood. However, Brook Taylor (1685—1731) did a lot more than prove that one theorem. He wrote his first paper while still an undergraduate at Cambridge, and belonged to a circle of mathematicians who corresponded, and sometimes challenged each other with problems, but did not see a need always to publish their results, and certainly not in a timely manner.

Taylor was elected to the Royal Society in 1712, and became its secretary in 1714. He then published a stream of papers, mostly in the Philosophical Transactions of the Royal Society, on a wide variety of subjects, not all of which would be considered mathematics today. In 1715 he published two books, his major works. The first was Methodus Incrementorum Directa et Inversa, the first book on the calculus of finite differences, which included Taylor’s Theorem (actually, he proved two versions of this result). The other was Linear Perspective, about which we shall have more to say later.

Taylor’s personal life was marred with tragedy and ill-health. In 1721, he married a Miss Brydges. His father did not approve of the match and broke off relations with his son. The unfortunate Mrs. Taylor died in 1723 in childbirth with their first-born, who also did not survive. Her death led to a rapprochement with his father, and in 1725 he married again, to Sabetta Sawbridge, with his father’s approval. In 1729, he inherited an estate from his father, but in 1730 his wife died, again in childbirth. This time, the baby, Elizabeth, survived, but Taylor’s own fragile health gave out and he died at Somerset House in 1731. His poor orphan grew up and married Sir William Young, Bart. Kirby dedicated a plate to her.

Related Posts

Brook Taylor’s Linear Perspective

Method of Perspective

Method of Perspective

Joshua Kirby’s main claim to fame rests on his book, Method of Perspective, or, to give its full title in the 18th century way, `Dr. Brook Taylor’s Method of Perspective Made Easy, Both in Theory and Practice. In Two Books. Being an Attempt to make the Art of Perspective easy and familiar; To Adapt it intirely to the Arts of Design; And To make it an entertaining Study to any Gentleman who shall chuse so polite an Amusement’.

A textbook on perspective may not seem an obvious sequel to his previous antiquarian volume, `An Historical Account of the Twelve Prints of Monasteries, Castles, antient churches, and Monuments, in the County of Suffolk’. While the earlier work had been successful, it was targeted at a Suffolk audience of clergymen, gentry, and local politicians. The Method of Perspective drew in an quite different subscriber list, as we shall see, despite Kirby’s remaining attention to ‘Gentlemen’ and their Amusements. By this time, 1754, Kirby was good friends with both the young Thomas Gainsborough, and the much older William Hogarth. Hogarth supported the endeavor. In his own book, Analysis of Beauty, published in 1753, the only mention of perspective is to give a reference to Kirby’s forthcoming work. More famously, Hogarth supplied the eccentric, and wonderful, frontispiece to Kirby’s book, the Satire on False Perspective, now reproduced in almost every book on perspective.

Related Posts

Who was Brook Taylor?

Brook Taylor’s Linear Perspective


Swift’s Exploding Mountain

Newspapers of the 18th century sometimes carried the most extraordinary reports, with absolutely no commentary, presumably leaving their readers to determine whether the reports were serious or humorous. Here is a fine example of the absurd genre from 1733 that appeared in several newspapers, including the Ipswich Journal, featuring the inimitable Dean Swift.

Dublin May 19th. Last Saturday the 12th of this Instant, the Right Hon. the Earl of Orrery, the Rev. Dr. Swift, Dean of St. Patrick’s, and the Rev. Dr. Sherridan, rid from Dublin to Tallow Hill, to take a Prospect of the adjacent Country. As they were mounting a Rock, they observed a Stream running thro’ the Middle of it, which fell into a natural Bason, and was thence conveyed thro’ some subterraneous Cavities, but they could not any where discover by what secret Passage it was conveyed out again; so that they concluded the Waters were still in some Reservoir within the Bowels of the Hill, which must infallibly come to burst forth in time, and fall directly upon the City. The Doctor sent for a Milking-Pail, to compute what Quantity ran out, which held about two Gallons, and it was filled in the Space of a Minute, so that it runs in 24 Hours 2,880 Gallons; this multiplied by 365, produces 1,051,200, and shews the Quantity that runs from the Rock in a Year; so that in three Years, about the 13th Day of November, he computes that it must burst the Belly of the Mountain, and emit an Inundation which will run to all the Points of the Boyne, and greatly endanger the City of Dublin.

I wonder, did they worry in Dublin?

William Wollaston

William Wollaston (1693—1757) subscribed to Kirby’s Historical Account. At one time a fabulously wealthy family, the Wollastons made their money in the wool trade and bought Finborough Hall in Suffolk for £10000 in the 1650s, although this was not their primary residence. Wollaston’s father, also William (1660—1724), however was a schoolteacher and philosopher, who tried to suppress his own writings. His most popular work, Religion of Nature Delineated, was only published shortly before his death, but quickly sold ten thousand copies and went through many editions. By living a quiet life, he drew the attention of his cousin William Wollaston, who had inherited the bulk of the estates, had no surviving sons, and was much irritated by importunate relatives. He left pretty much everything to the retired schoolteacher when he died in 1688. Leslie Stephen has a lovely article on the father William Wollaston in the old DNB.

Our William Wollaston lived at Finborough Hall and became MP for Ipswich in 1733 running unopposed in a by-election to replace the deceased former MP. Returned in the 1734 election, he served until 1741, being then replaced by Edward Vernon. In 1730, William Hogarth painted a conversation piece of the Wollaston family.

William Wollaston married Elizabeth Fauquier, whose father was governor of the Bank of England, and together they had eight children. In 1739, he had four of his children inoculated against smallpox, with the Ipswich Journal reporting that they were ‘in a fair way of Recovery’.  His eldest surviving son, William (1731-1797) was himself MP for Ipswich from 1768 to 1784. An amateur musician, he also gave Thomas Gainsborough two important commissions shortly before Gainsborough moved to Bath. One is this portrait:

The other is Gainsborough’s first (surviving) full-length.

Rosenthal (1999) suggests that the two portraits were intended to hang in Gainsborough’s new picture room in Bath to show how successfully he could catch a likeness, the two paintings being recognizably of the same person.

For more on Suffolk MPs, see A Clique of Politicians.

Oddly enough, a Wollaston is currently a member of parliament.

Francis Folkard

Rev. Francis Folkard subscribed to Kirby’s Historical Account. He also subscribed to Richard Canning’s Ipswich Gifts and Legacies. Francis Folkard (1688—1753) was rector of Clopton and Hasketon. The Folkards are another of those extensive, intertwined Suffolk families, and Francis came from a branch that for several generations had lived in a house in Parham, near the church. He went to Cambridge, was ordained deacon in 1712 and priest in 1714, becoming rector of Clopton in 1722, and of Hasketon in 1737, holding both livings until his death. He married Deborah Chaplin (1698—1779), daughter of the Rev. Peter Chaplin, rector of Higham. Deborah inherited estates from her father spread across several counties, which added to the substantial holdings of her husband. They had five children, three of whom died young. Their daughter Elizabeth married Rev. Montagu North. When Francis Folkard was drawing up his will, they were yet to marry but close enough that he had already made out a marriage settlement giving her various estates. The remaining daughter Deborah, still under age, received the most concern. If she were to die without issue the property she inherited was to pass to his step-brother Thomas. Deborah in fact married, but had no children, and so that is what happened.

A careful and detailed history of the Folkard family, complied by some members of the family is available here.

Charles Scrivenor

Charles Scrivenor subscribed to Kirby’s Historical Account. He was High Sheriff of Suffolk in 1745. The Scrivenors (or Scriveners) were lords of the manor at Sibton in Suffolk. John Scrivener acquired Sibton abbey and built a house adjoining the ruins in 1655. His son Thomas married in 1740, had his lands sequestered, and spent time in prison during the Interrregnum, but retired to the new house and died there in 16667. The estate then passed to his son Charles and then down to our Charles. Charles Scrivenor married Margaret Bedingfield, but they had no children, and he died in 1751. The estate went to his sister Ann, and then to her son John who took the name Scrivenor, and married Dorothea Howman, whose portrait was painted by the young Thomas Gainsborough.

Charles Scrivener has a memorial at Sibton church, which praises him as having had a “Mind adorned with every Virtue to which were added by the Assistance of an Excellent Understanding all the Accomplishments becoming the Son, the Husband, and the Friend”.