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"Whiston was dismissed for having too much religion, and Saunderson preferred for having none." Edmund Halley

Nicolas Saunderson (1682-1739) replaced William Whiston as the Lucasian Professor of Mathematics in 1711. He was respected at the time; moreover, it was hoped that he would deflect attention from the previous problems of Whiston and his heretical views.

He was blinded at about the age of one by small pox. Nevertheless, he attended school where he learned Latin, Greek and mathematics, demonstrating considerable abilities. He learned Euclid in the original Greek and learned to speak French. He did not attend college because of the obstacles encountered by one who was blind, although he had more than enough ability and education. He was fortunate that his father made sure that he learned mathematics along with his other subjects. He was also fortunate in that not only did he find friends who would read to him, but further helped him by providing support through other friends. He had access to books that were difficult even for students with sight. So liked and respected was he, when at Cambridge, the sitting Lucasian Professor, William Whiston, consented to allow him to lecture in mathematics, Whiston's own field.

Saunderson was brought to Cambridge by a number of his friends when he was twenty-five years old. He came as a master in 1707. He began by giving lectures in mathematics that were purported to be so crowded that he could not attend to all his students. In spite of being blind, he lectured in optics. The usual explanation was that because optics is really geometry, there was nothing unusual about his mastery. He also studied Newton's Principia, making it more accessible to students. After a time he met with Newton to receive help in understanding the details of the Principia. By 1710, when Whiston was removed, Saunderson had built up a reputation of high merit. This combined with his expertise in Newton's theories made him an obvious choice for the Lucasian Chair. Because he had not received a university degree, the heads of the colleges, with help from several influential people, convinced Queen Anne to decree him Master of Arts. Shortly thereafter, he was chosen Lucasian Professor. He delivered his inauguration speech in Latin, as expected. He spent the rest of his days at Christ's College lecturing and helping his students.

His work focused on algebra and fluxions, which we know today as calculus. It should be noted here that calculus was still new during his day. All of his written work was collected and published after his death by his son, John. Saunderson spent his energy teaching, not writing for publication. His book, The Elements of Algebra , basically his lectures and notes, consists of almost eight hundred pages covering all aspects of algebra. It is divided into an introduction, ten books and some appendices. The approach is definitions, lemmas, problems and explanations of the problems, a masterful textbook. Saunderson's friends strongly encouraged him to write down his lectures in 1733 when he had suffered a fever. They realized he could have died without having written anything. This was an intolerable situation. Once he recovered he worked long hours on his book to finish it. However, he passed away before it was actually published.

The introduction is preceded by a short postulata which covers the basic knowledge required before one begins to study the main part of the book. Saunderson goes through multiplication, division, square roots and the Rule of Three . If the reader is not proficient in these basics, Saunderson suggested he stop, returning when he is able to handle them. The introduction follows with thirty pages of a very thorough explanation of fractions, organized into twenty-four articles or topics.¹

Having set the stage for the study of algebra with these introductory materials, Saunderson began his ten books on algebra by carefully laying the groundwork for algebra itself. The first three books are comprised of one hundred thirty-three articles that start with the necessary definitions and operations, go through problem formulation, and equations, simple to quadratic. These three books are approximately two hundred pages in length.

The fourth book begins the work in earnest with the presentation of problems. It deals with general problems and general theorems showing the methods that are synthetic in their application. In addition to the thirty-three problems of his own, he presents two more from Abraham de Moivre. The fifth book contains eighteen propositions, thirteen lemmas, followed by seventeen problems, including the classical "Magic Square" construction. There are a few numbered articles that have lemmas and definitions that are not numbered, with the result that the total count is incorrect. The topic of the sixth book is Diophantus and the set of problems known as Diaphantine problems. The problems are concerned mainly with squares and triangles, for example, article 228 states "In every right-triangle, if the double product of the legs be either added or subtracted from the square of the hypotenuse, both the sum and the remainder will be square numbers." To show this, imagine a right triangle of sides 3, 4 and 5 in length. Multiplying 3 times 4 and doubling it gives us 24. The square of 5 is 25. The difference of 25 and 24 is 1, which is a square. The sum of 24 and 25 is 49, which is the square of 7. The examples continue with items such as finding three square numbers that, when added together, give another square. These problems are not quite the application of algebra to geometry, but the relationship is clear.

Book seven covers the topics of proportion and ratios, with a discussion of the fifth book of Euclid's Elements of Geometry. Saunderson's seventh book is almost all definitions and examples. Saunderson is continuing to move towards geometry by building a foundation. The eighth book is divided into two parts: the first part is titled "The application of Algebra to plain Geometry." Here he returns to an earlier format, where the majority of articles are problems followed by a large number of lemmas. The second part presents solid geometry, prisms, cylinders, spheres, etc. Articles 342 and 343, describe the circumference of a circle as somewhat larger than three diameters. He gives the multiplier value as somewhat less than 3 10/70 and somewhat more than 3 10/71, the value worked out by Archimedes in the third century. This places the decimal value of the number we know as pi between 3.140845 and 3.142857. Saunderson refers to van Ceulen's ² calculation of pi in article 345. Van Ceulen spent most of his life calculating pi to thirty-five digits. The accepted value today is 3.141592.... His life's work can now be done by a three year old in less than a second with a hand-held calculator costing less than ten dollars.

For some perspective, it was not until 1770 that Lambert ³ proved pi was irrational and it was only in 1882 that Lindemann ⁴ proved that pi was a transcendental number.⁵ The proof that pi was transcendental showed that it is impossible to solve the ancient Greek problem of squaring the circle, which Saunderson discusses in article 346. Although mathematics had achieved much by Saunderson's time, there was still much to be discovered. In fact, the Greek letter pi was not used as a symbol to represent this number until 1706, by William Jones.⁶

The ninth book is divided in four parts. The first part presents powers and indices, with an explanation of Newton's theorem for binomials. The second book explains both common and natural logarithms. Common logarithms are also called Briggsian logarithms after Briggs.⁷ These use ten as a base. Natural logarithms are also called Napierian logarithms after Napier.⁸ These use the transcendental number e as a base. The two men were contemporaries and collaborated on the development of logarithms and the publications of tables of logarithms.⁹ The use of logarithms was of great importance at the time because all calculations were done manually, which could consume a man's lifetime. When compared to the computers of today, it is staggering to think that today calculations can be performed in a few seconds that would have taken a brilliant professional of Saunderson's time his whole life to accomplish. Furthermore, there will be no computational error from the machine, whereas a calculation by hand would almost certainly contain many errors, with no certain means to discover them. Saunderson presents a lucid explanation of each method with appropriate precautions and caveats for their use by his students.

The topic of the third part of his ninth book is the invention of divisors. The ideas of both compound divisors and multi-dimensional divisors are covered. The last part explains surds, which is more familiar to us as irrational numbers, for example, the the square root 3 and 5 + square root 2 This part is mainly an explanation of methodology.

The tenth and last book is divided into two parts, dealing with general equations and their roots. The first part covers the generation of equations and the second covers cubic and biquadratic equations. The appendix contains a letter from Saunderson to de Moivre and an explanation of de Moivre's method for finding the roots of a cubic equation.

Saunderson invented his own form of a calculator to compensate for his lack of sight. It is an ingenious representation of a number using a board with holes and pins placed in the holes. There is a central hole where a large pin is placed, then a second pin is placed in holes surrounding it. There are eight positions around the central pin, forming a square, so that a pin may be placed at the corners of the square and at the median of each face of the square. This is similar to a compass that shows, north, south, east, west, northeast, northwest, southeast and southwest. Because there are only eight positions, Saunderson used a large pin by itself to represent zero and the small pin standing alone in the center to represent one. Two was the large pin in the center with the small pin directly above it. The small pin was moved clockwise one spot for each higher number, until nine was reached. By placing an arbitrary number of these squares in a horizontal row, he could represent an arbitrary number of digits. By placing an arbitrary number of these horizontal rows in a vertical fashion, he could use the board to do calculations, just as others would write numbers to be added or subtracted.

He had a second board made that was similar in that there were holes drilled in it for the insertion of pins. The holes were one half inch from each other. He would insert pins, after wrapping twine around these pins, he then stretched the twine to other pins to create geometrical shapes. The lines were more true than one could draw by hand. For solid geometry, he had the shapes made in three dimensions for spheres, cones, pyramids, etc.

The Royal Society elected Saunderson to membership in 1719. He was also a member of the Spitalfields Mathematical Society.¹⁰ In 1728, King George II visited Cambridge, requesting an audience with Saunderson. The King was so impressed with Saunderson, that he conferred the degree of Doctor of Laws upon him. It was unfortunate that Saunderson did not live past his fifty-seventh year. He had planned a book on fluxions (calculus), which surely would have been a welcome contribution.

Footnotes

1. Nicholas Saunderson, Elements of Algebra (Cambridge: Cambridge University Press, 1740), 1.
2. Ludolph van Ceulen (Holland, 1540-1610).
3. Johann Lambert (Germany, 1728-1777).
4. Carl Lindemann (Germany, 1852-1939).
5. Robert James, Mathematics Dictionary, 5th ed. (New York: Van Nostrand Reinhold, 1992), 316.
6. Alexander Hellemans, The Timetables of Science (New York: Simon & Schuster, 1988), 173.
7. Henry Briggs (England, 1561-1630).
8. John Napier (Scotland, 1550-1617).
9. James, 255.
10. Dictionary of National Biography.