**Roger Paman** was an English mathematician who took part in a voyage hoping to circumnavigate the world. He made contributions to the development of the calculus.

- Although there is no record of Paman being a student at the University of Cambridge, we know that Mr Frank of St John's College, Cambridge gave him a copy of George Berkeley's The Analyst (1734).
- Paman wrote a paper on this treatise which he called The Harmony of the Ancient and Modern Geometry Asserted.
- On 18 September 1740 Paman set off with George Anson on his journey around the world.
- It returned to England, reaching Spithead on 15 June 1744 but, since Paman was back in England long before this, we know that he was not on board the Centurion.
- Five of the ships, the Gloucester, the Wager, the Tryal, the Anna and the Industry, were destroyed during the journey, three of them while rounding Cape Horn, so Paman was not on board any of these ships.
- We can therefore deduce that Paman must have been on one of the two remaining ships, either the Severn or the Pearl.
- The Severn and Pearl, with Paman on board, left England together with the other ships.
- The two ships, with Paman certainly on one of them, made contact again on 21 May.
- Although we know nothing specific about Paman, he must have suffered along with the other members of the two crews.
- Given these feelings it is clear that Paman and the other the men from the Severn and the Pearl would not have been treated as heroes when they returned to England.
- It is reasonable to ask exactly what Paman's role had been on the voyage.
- Perhaps the best indication of this is given in an advertisement for a book which Paman intended to write.
- It is possible that Paman wrote this book, but no trace of it has been found nor is there any evidence that it was published.
- Before setting out on the voyage, Paman had given his paper, The Harmony of the Ancient and Modern Geometry Asserted, to his friend Dr Hartley, and when he returned, in February 1742, Paman sent it to the Royal Society.
- Paman's paper was published the paper as a book in 1745.
- As this states, Paman had written his work to answer the criticisms of the calculus presented by Berkeley in The Analyst published in 1734.
- The way that Paman attempted to avoid "considerations of velocity, time or motion of indivisibility or infinity" was to introduce certain new concepts.
- A "radical Quantity" is close to what today would be called a "variable", even though Paman implies that an expression can be composed of this variable only by taking powers of it, and by multiplicating by scalars, which means that Paman is thinking of polynomials or power series.
- Paman's explanation looks like our present definition of infimum and supremum, but whereas our infimum and supremum given a set (of real numbers) gives a real number, the Maximinus and Minimajus of a quantity is a quantity of the same kind.
- With these concepts Paman defines fluxions.
- Paman's terminology will of course seem strange at first glance.
- But as all of the terms he introduces cover concepts developed by Paman, we cannot blame him for using new names.
- Paman managed to give foundations to the calculus using these concepts, and he needed all of them.
- But we would not be able to define the derivative using Paman's terms since we consider more complicated functions than the polynomials or power series which Paman considered.
- It must therefore be said that far from introducing concepts for their own sake, Paman introduced interesting new concepts that were useful to him and that would have been useful to mathematics if other mathematicians had noticed them.
- This was after Paman's first version of his paper, but before the publication of Paman's book.
- There seems no way of discovering whether Paman made major changes to his manuscript after reading Maclaurin's treatise.
- There were three authors, Robins, Maclaurin and Paman, who attempted to answer Berkeley in different ways.
- However, it is interesting to see Paman's remarkably modern work - with concepts resembling our neighbourhood concept and lim inf and lim sup.
- It only in 1989, when Breidert and Sageng made detailed studies of Paman's work, that its contents became known.
- Thanks to Bjorn Smestad for his permission to use material from his article on Paman.

Born not known. Died 1748.

View full biography at MacTutor

Origin England

**Oâ€™Connor, John J; Robertson, Edmund F**: MacTutor History of Mathematics Archive