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"For, there is little doubt, that the student's curiosity and attention will be more excited and sustained, when he finds history blended with science, and the demonstration of formulae accompanied with the object and causes of their invention, than by a mere analytical exposition of the principles of the subject." Robert Woodhouse

Robert Woodhouse (1773-1827) was the eighth occupant of the Lucasian Professorship of Mathematics at Cambridge University. A mathematician by profession, he was considered to be a reformer. He was one of the first to introduce to England and Cambridge the methods of calculus developed on the Continent and the notation devised by Leibniz. He was senior wrangler at Caius College, Cambridge when he was awarded his BA in 1795 and was the winner of the Smith's Prize.¹ His M.A. was awarded in 1798 and he was elected a fellow that same year.² He was elected to the Royal Society in 1802.

At Cambridge, he was elected to the Lucasian professorship in 1820, but held the chair for only two years. He took the Plumian Chair of Astronomy and Experimental Philosophy at Cambridge, in 1822, which he held for five years until his death in 1827. Woodhouse was the first superintendent of the Cambridge astronomical library.³ He was a moderator for the Senate House mathematics examinations for the years 1803, 1804, 1807 and 1808. He used the standard Cambridge notation in devising calculus problems for the examinations despite his personal preferences for the continental style.⁴It appears that Cambridge did not hold a grudge against him for his earlier publication indicating his choice of Leibniz's notation.

Woodhouse was primarily interested in what was then called the "metaphysics of mathematics."⁵ This area was concerned with questions about the theoretical foundations of calculus, the role of geometric and algebraic methods, the importance of notation, and the nature of imaginary numbers. His major work was in the writing of texts on pure mathematics that helped to inspire the Analytic Society (1813) by influencing students like Charles Babbage and George Peacock. ⁶ He has been overlooked as an innovator and contributor to the mathematical revolution that took place in England in the nineteenth century. William Whewell, a respected mathematician and historian of mathematics, wrote in the 1830s about the lack of standards and expectations in mathematics at Cambridge. ⁷ He felt the teaching methods were just as poor, partly because tutors did not understand the basic principles themselves. The notable exception to these criticisms was Woodhouse.⁸

Woodhouse's publication of Principles of Analytical Calculation in 1803 was the first publication in England to use the differential notation that had become common on the European continent.⁹In this work Woodhouse defended analytic methods, the differential notation, and a theory of calculus based, like that of Lagrange, on series expansions. He reviewed the methods of infinitesimals, limits, and expansions, and severely criticized the principles adopted by Lagrange in his theory of functions, regarding them as logically insufficient. By thus exposing the lack of soundness of some continental methods, he rendered his general support of the system far more weighty than if he had appeared to embrace it as a blind partisan.¹⁰

British mathematicians refused to acknowledge that alternatives to Newton's dot notation existed. Although Woodhouse had pioneered Leibniz's notation in England, he was largely ignored. He gave clear, logical reasons for the superiority of the Leibniz notation, and even added a little to it. Cambridge did not switch over to it, nor was there any reference to it in professional writings.

It is somewhat curious that although Woodhouse attacked a fundamental belief of English mathematicians, no one seemed upset enough to condemn him or even argue about it with him in public. He urged everyone to read George Berkeley's book, The Analyst, published in 1734, which was an attack on Newton's calculus.¹¹ When it was first published there was a great deal of argument in published attacks and defenses of Newton. The unfortunate and illogical result was the further entrenchment of Newton's second-rate notation. Woodhouse had hoped that the book would demonstrate that the resistance to the notation was simply prejudice and not mathematics.¹²

Woodhouse's publication of Elements of Trigonometry in 1809 caused the renowned mathematician George Peacock to say that this book "... more than any other contributed to revolutionize the mathematical studies of this country."¹³ The key was that this book was aimed at the students, where his previous work was aimed at the teachers.

This table is taken from Woodhouse's 1810 publication The Calculus of Variations. It presents a clear picture of the two forms of calculus notation demonstrating the difficulties of the Newton form (English), designated by E. and the Leibniz form (Foreign), designated by F. The Leibniz form is the one found in today's textbooks on calculus. It is obvious that the dots are more easily obscured when in complicated equations containing these terms.

Woodhouse was one of the earlier proponents of the history of science as a method for teaching. The quote at the beginning of this section is taken from the preface of his history of calculus, The Calculus of Variations, written almost one hundred years after its invention.

Footnotes

1. Crosbie Smith, " `Mechanical Philosophy' and the Emergence of Physics in Britain: 1800-1850," Annals of Science 3 (1976): 21.
2. Charles Gillispie, ed., Dictionary of Scientific Biography (New York: Charles Scribner's Sons, 1980), 500.
3. Gillispie, 500.
4. Dubbey, Charles Babbage, 37.
5. Gillispie, 500.
6. H.W. Becher, "Woodhouse, Babbage, Peacock and Modern Algebra," Historia Mathematica 7 (1980): 389-400.
7. William Whewell, Thoughts on the Study of Mathematics as Part of a Liberal Education (Cambridge: Cambridge University Press, 1836), 43.
8. Dubbey, Charles Babbage, 19.
9. J.M. Dubbey, "The Introduction of the Differential Notation to Great Britain," Annals of Science 19 (1963): 35-48.
10. Dictionary of National Biography.
11. Robert Woodhouse, The Principles of Analytical Calculations (Cambridge: Cambridge University Press, 1803), xxvii.
12. Dubbey, Charles Babbage, 13.
13. Dictionary of National Biography.