Ahead of the festive season, the library has been treated to a brand-new book for the Rare Book collection: an interesting edition of Briggs’ *Trigonometria Britannica*.

## The Trigonometria Britannica

The *Trigonometria *alone is interesting for its place in mathematical history. Other principles of arithmetic progressions existed prior to it; Beery [1] mentions Zhu Shijie’s (1249-1314) and Yang Hui’s (ca. 1238-1298) writings on binomial coefficients – also known as Pascal’s Triangle – al-Bannā’s (1256-1321) continued fractions, and Levi ben Gerson’s (1288-1344) simple combinatorial identities. The idea of a mathematical table was far from new as well – we know of Ptolemy’s table of chords (2^{nd} century AD), and Āryabhata’s (476-550) sine table, for example.

However, the publication and use of logarithmic tables truly started in 1614, with John Napier’s *Mirificis Logarithmorum Canonis Descriptio* – detailing the [2] “Napierian”, or natural logarithm. Logarithmic tables could be used by mathematicians, astronomers, and navigators, to simplify problems – a sort of calculator before its time. Napier’s tables used “natural” (or “base-e”) values for calculations, and this – coupled with an error Napier made in his own tables – led to discrepancies and inaccurate results. This is where Brigg’s common base-10 logarithms, and the *Trigonometria*, come into play.

## Briggs, Gellibrand, Bainbridge

Mathematician and astronomer Henry Briggs (1561 – 1630) is primarily known for these base-10 logarithmic tables. He first published them in his *Logarithmorum Chilias prima* (1617), which he worked on with Napier himself, and then in the *Arithmetica Logarithmica* (1624). Briggs then started working on extending these into trigonometric functions into the tables we now see in the *Trigonometria Britannica*, finishing it with instructions on how to use these tables to solve trigonometrical problems [3].

Unfortunately, he passed away before completing his work, and his friend Henry Gellibrand (1597 – 1636) would eventually complete the book in his stead. At the time, Gellibrand was Professor of Astronomy at Gresham College (London), which would see the first Royal Society meeting in November 1660 [4][5]. It is possible that astronomer John Bainbridge (1582 – 1643) frequented the same circles – the correspondence of Sir Christopher Heydon indicating that both scientists were frequently invited to his estate. The relationship between Gellibrand and Bainbridge was sufficiently important for the latter to receive a *Trigonometria Britannica* fresh off the press – hence the signature we find on our copy’s first flyleaf.

## A Sharp mathematician

If its status as an important mathematical work was not enough, our copy of the *Trigonometria* has the peculiarity of having being heavily annotated, likely by past owner Abraham Sharp (1653-1742).

Sharp was not destined to be a mathematician. Born the son of a wealthy merchant, he was sent to apprentice at a mercer aged 16, but apparently left to teaching arithmetic sometime after his father’s death three years later. Little is known of his studies, whereabouts or profession between 1672 and 1684, year in which he became the assistant of John Flamsteed, astronomer royal. [6]

Sharp stayed at the Royal Observatory, at Greenwich, between 1684 and 1685, and again between 1688 and 1690, at the same time as Flamsteed worked on his Mural Arc. He then left London in early 1691, and returned to his family’s property in 1693 after a short period in Portsmouth. It seems that over the following decade, he continued to write about mathematics, collect scientific instruments, and exchange correspondence with other scientists of his time [7]. He resumed communication with Flamsteed, who seemed to appreciate his mathematical talent more than his – apparently poor – astronomer skills [8].

Flamsteed re-hired Sharp in 1705, this time as a ‘Calculator’ [9]. It is possible that Sharp used the tables in the *Trigonometria* to help with his mathematical work. Sharp expanded, and sometimes corrected the logarithmic tables, on top of annotating some of the text explaining how to use the tables themselves. It seems that he may also have altered some one or two of the figures published in the first half of the *Trigonometria*, although the neatness of his hand makes it difficult to recognise.

Another page seems to have been corrected in another hand than Sharp’s – maybe that of the next owner, an Abram Ogden, who we have not been able to identify more clearly. The inscription left by Ogden seems to indicate that he was aware of the prestigious history of this *Trigonometria*.

After him, we have little information on what happened to this book between 1758 and our days. However, we can assure that it has found its place on ours shelves, and will be well-cared for as long as this library stands!

*Blog post researched, written and published by Raphaëlle Goyeau, Library Assistant.*

### References

[1] Janet L. Beery. Formulating figurate numbers. *Journal of the British Society for the History of *Mathematics, 24:2, 78-91. https://doi.org/10.1080/17498430902820879 https://www.tandfonline.com/doi/full/10.1080/17498430902820879

[2] Ernest William Hobson. *John Napier and the invention of logarithms, 1614; a lecture*. Cambridge: Cambridge University Press. 1914.

[3] Ian Bruce. Henry Briggs: The Trigonometria Britannica. *The Mathematical Gazette*, 88:513, 457-474. http://www.jstor.org/stable/3620722

[4] The Royal Society. *About us: History of the Royal Society. *Retrieved from https://royalsociety.org/about-us/history/

[5] Francis R. Johnson. Gresham College: Precursor of the Royal Society. *Journal of the History of Ideas*, 1:4 (Oct. 1940), 413-438. https://doi.org/10.2307/2707123

[6] Frances Willmoth. Sharp, Abraham. *Oxford Dictionary of National Biography*. 2011. https://doi.org/10.1093/ref:odnb/25206

[7] Elizabeth Connor. Abraham Sharp, 1653-1742. *Publications of the Astronomical Society of the Pacific*, 54:321, 237-243. https://doi.org/10.1086/125455

[8] J.J. O’Connor and E.F. Robertson. Abraham Sharp. *MacTutor History of Mathematics Archive*. Retrieved from https://mathshistory.st-andrews.ac.uk/Biographies/Sharp/

[9] Graham Dolan. People: Abraham Sharp. *The Royal Observatory, Greenwich*. Retrieved from http://www.royalobservatorygreenwich.org/articles.php?article=1148