**Thomas Harriot** was an English mathematician who did outstanding work on the solution of equations, recognising negative roots and complex roots in a way that makes his solutions look almost like a present day solution.

- We know very little of Harriot's youth.
- As an undergraduate at Oxford, Harriot was a student at St Mary's Hall.
- He became friends with Richard Hakluyt and Thomas Allen, both lecturers at the university, but not at St Mary's Hall.
- Harriot graduated in 1580 and went to London.
- Harriot wrote a text called Arcticon which was never published and unfortunately no copies have ever been found.
- This work was essentially his lecture course given at Durham House, Raleigh's lodgings in The Strand in London, where Harriot lived at this time.
- It was not only as a navigational instructor that Raleigh employed Harriot.
- Harriot was certainly on a voyage to Virginia organised by Raleigh in 1585-86.
- Harriot made many notes during his time in the New World, being particularly interested in the language and customs (particularly the eating habits) of the inhabitants.
- Although Drake met up with the colonists, in June 1586 there were severe storms and there was a hurried return to England by Harriot and most of the party.
- Harriot, together with Drake's ships, landed at Portsmouth in July 1586 and he went immediately to Raleigh to report on the expedition.
- By the time Harriot had returned, Raleigh had turned his attention to Ireland.
- Harriot carried out surveys of the Lismore estate, which was owned by Raleigh, beginning in 1589.
- Harriot felt that this was a reference to him and he discussed the allegations with John Dee (who also felt that the charges might relate to him).
- There is no reason to believe that Harriot (or Raleigh) were atheists but certainly they were free thinkers and Harriot's scientific approach to the world was, to say the least, viewed with great suspicion by the church.
- As well as problems caused by allegations, Dee and Harriot discussed scientific and mathematical matters in the 1590s.
- Harriot had now moved from working for Raleigh to working for Henry Percy, Duke of Northumberland.
- Raleigh's life became so chaotic that Harriot had sought the support of a patron who could provide more stability for his scientific pursuits.
- In 1595 the Duke made property in Durham over to Harriot and he moved up the social ladder becoming a member of the "landed gentry".
- Harriot also later held estates in Cornwall and Norfolk.
- Not long after the Durham transaction, the Duke gave Harriot the use of one of the houses on the estate at Syon (near Kew outside London) which Harriot used both as a residence and as a scientific laboratory.
- We certainly know from manuscripts which survive that Harriot was engaged in deep studies of optics at Syon by 1597.
- As with all his other mathematical discoveries, however, Harriot did not publish his findings.
- Snell's discovery was in 1621, about 20 years after Harriot's discovery, but the result was not published until Descartes put it in print in 1637.
- One of the optical problems which Harriot did study in the 1590s was Alhazen's problem.
- Optics was not the only topic to occupy Harriot during this period.
- Harriot resolved the forces acting on the projectile into horizontal and vertical components.
- Other topics which Harriot began to work on before 1600 were problems of chemistry.
- Perhaps it is less clear that Harriot, by this time not so closely associated with Raleigh, would find problems too.
- He then sought Harriot's help in obtaining evidence on his behalf.
- Poor Harriot was singled out in the judgement as being an atheist and an evil influence.
- His attempts to help Raleigh had been based on Christian principles (to which undoubtedly he adhered) but this had rather damaged Raleigh as Harriot was seen an atheist using Christian principles for convenience.
- Harriot was devastated and for about a year undertook no new scientific work as he tried to come to terms with what was happening.
- Four others, including Thomas Percy, the grandson of Henry Percy, were also arrested as the main conspirators.
- Harriot was held on suspicion of being involved and imprisoned in the Gatehouse.
- On 27 November Henry Percy, Harriot's patron, was also put in the Tower where he remained until 1621 when he was released.
- No evidence seems to have been found against Harriot and, although he remained in the Gatehouse for some while writing several letters requesting his release, he was a free man probably by the end of 1605.
- As soon as he was released, Harriot returned to his work on optics.
- He began to develop a theory for the rainbow and, by 1606, Kepler had heard of the remarkable results on optics achieved by Harriot.
- Kepler wrote to Harriot, but the correspondence never really achieved any significant exchange of ideas.
- Perhaps Harriot was too wary of the difficulties that his work had nearly brought on him, or perhaps he did (as he claimed to Kepler) still intend to publish his results if his health permitted.
- The appearance of a comet attracted Harriot's attention and turned his scientific mind towards astronomy.
- Kepler had discovered the comet six days earlier but it would be the observations of Harriot and his friend (and student) William Lower which eventually were used by Bessel to compute its orbit.
- His astronomy brought back to the fore, Harriot went on to make the earliest telescopic observations in England.
- As with all his scientific discoveries, Harriot did not publish his results.
- Of the few pieces of work done by Harriot after 1614, one was his observation of another comet in 1618 (there were three visible comets that year and Harriot observed the third) from Syon House.
- Raleigh was executed on 29 October 1618 in a public execution, with Harriot present to witness the event.
- However, by this time Harriot was already suffering from the cancer of the nose which eventually led to his death.
- The cancer seems to have started around 1613, about the time when Harriot lost interest in pushing forward his mathematical and scientific research.
- Harriot would suffer this "evil" for a further three years before the cancer took his life.
- There are a few other major mathematical achievements due to Harriot which we should mention.
- The loxodromes are the straight lines on the Mercator map, which Harriot computed with great precision.
- In order to achieve this degree of precision, Harriot introduced finite-difference interpolation.
- There is an interesting history to a problem which has only recently been solved, yet originated with Harriot.
- Raleigh asked Harriot to solve certain problems regarding the stacking of cannonballs.
- On a manuscript dated 12 December 1591 (Sunday), Harriot set out a table to answer Raleigh's questions.
- Raleigh posed a second question, which Harriot also answered, namely given the pyramid of cannonballs, compute the number in the pile.
- Harriot was too much the mathematician to stop there, however.
- Later, in his correspondence with Kepler about atomic theory, Harriot mentioned the packing problem.
- This seems intuitively obvious, but resisted proof until 1998 when Thomas Hales of the University of Michigan (with the help of hours of computer generated data) finally proved the conjecture.
- The one part of Harriot's work which we have not yet described is the mathematical work for which, in some ways, he is best known, namely his work on algebra.
- We have only made one slight change to Harriot's notation.
- Harriot invented certain symbols which are used today.
- However, the symbols < for "less than" and > for "greater than" were not due to Harriot (as is often claimed), but were introduced by the editor of Artis Analyticae Praxis ad Aequationes Algebraicas Resolvendas Ⓣ(The analytical art by which algebraic equations can be resolved) - Harriot himself used different symbols.
- There is still scholarly debate on how much Harriot was influenced by Viète, or whether notation and ideas introduced by Viète were learnt by him from Harriot.
- As we have seen from the example above, Harriot did outstanding work on the solution of equations, recognising negative roots and complex roots in a way that makes his solutions look like a present day solution.
- This is a major step forward in understanding which Harriot then carried forward to equations of higher degree.

Born 1560, Oxford, England. Died 2 July 1621, London, England.

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Algebra, Astronomy, Origin England, Physics

**O’Connor, John J; Robertson, Edmund F**: MacTutor History of Mathematics Archive