## Phillip S Jones on Brook Taylor

 An International Congress of Mathematicians was held in Cambridge, Massachusetts, USA from 30 August to 6 September 1950. Several lectures given in the "History" part of Section VII "History and Education". Phillip S Jones gave a lecture in "History" on Brook Taylor and the Mathematical Theory of Linear Perspective, his Contributions and Influence. Here are some notes which indicate the content of the lecture:-

### BROOK TAYLOR AND THE MATHEMATICAL THEORY OF LINEAR PERSPECTIVE, HIS CONTRIBUTIONS AND INFLUENCE

Phillip S Jones

The last of four periods in the development of the theory of linear perspective began in the eighteenth century when Willem 'sGravesande, Brook Taylor, and J H Lambert constructed a general and abstract mathematical theory.

The only original English work prior to Taylor's Linear Perspective (London, 1715) was that of Humphrey Ditton published in 1712. The only mathematically significant later English writer was John Hamilton who admittedly based his work on Taylor's but added much to it.

Taylor's book is a concise, generalized, mathematical treatment formulated in terms of axioms and theorems. The concepts of horizon line and principal point are generalized to vanishing lines and vanishing points for all planes and lines. The object, eye, and picture are considered to be free to assume all possible relative positions. Use is made of the invariance of a ratio which is essentially the cross ratio of four points so projected that one of them goes to infinity. Taylor's basic theorem is equivalent to Desargues' theorem although phrased in his own perspective terminology with no mention of Desargues.

In his first edition Taylor gave an interesting construction for the perspective of a circle. He explained it by translating standard Euclidean geometric constructions into perspective rather than by constructing the original circle and then drawing its perspective. The procedure actually amounts to the construction of a conic given a point of the curve and a point and line which are pole and polar with respect to it. Taylor gave "perspective" proofs, terming lines "parallel" if they met on the vanishing line of their common plane. He also gave the first constructions for a line from a point to the inaccessible intersection of two given lines. One procedure is equivalent to using harmonic sets and complete quadrangles.

A final unique feature of Taylor's work is his treatment of inverse problems. Although the first mention of such problems was in Monte's Perspectivae Libri Sex (1600), little had been done prior to Taylor. Later Lambert further expanded the treatment and so earned credit for originating the modern science of photogrammetry. Lambert also carried Taylor's concept of doing geometric constructions and proofs directly in perspective much farther. Max Steck lists Jacquier's French translation of Taylor's work among the books in Lambert's library.

Taylor's work appeared in four English editions and three translations. Nine other authors published books appearing in 22 editions from 1738 to 1888 which claimed explicitly to be based on Taylor's work and to give his methods. These facts alone would establish Taylor's wide influence and justify having made a special study of his work.

University of Michigan,
Ann Arbor Mich., U.S.A.

JOC/EFR March 2006