Born: 1620 in Eutin, Schleswig-Holstein, Denmark (now Germany)
Died: 14 January 1687 in Paris, France

Nicolaus Mercator was actually called Niklaus Kauffman. He later changed his family name, which was a common thing to do at this time, to Mercator which was the Latin form of 'merchant'. Nicolaus's father was Martin Kauffman who was a school master at Oldenburg in Holstein from 1623 until his death in 1638. Although we have no evidence to prove that Nicolaus attended his father's school, it is impossible to believe that he would attend any other than the one in Oldenburg where his father taught.

He entered the University of Rostock in 1632, received a degree in 1641, then went to Leiden for a short period. After his return to Rostock in 1642 he was appointed to a post in the Faculty of Philosophy at the University. In 1648 Mercator moved to the University of Copenhagen. While he was working there, he published a number of textbooks on spherical trigonometry, geography and astronomy [1]:-

... if not markedly original, they show his firm grasp of essentials.

Three texts were published in 1651, namely Trigonometria sphaericorum logarithmica, Cosmographia, and Astronomica sphaerica. The first tabulates logarithms of the sine, cosine, tangent and cotangent functions at 1° intervals and shows how to solve triangles using logarithmic functions. The second of these texts deals with physical geography, while the third is a text on spherical astronomy. Further publications followed: Rationes mathematicae subductae (1653) sets out to distinguish between rational and irrational numbers pointing out that in music rational ratios lead to harmony while irrational leads to dissonance. Again he gives an astronomical example where rational ratios corresponds to Kepler's structure of the planetary system given in terms of regular polyhedra, while irrational ratios correspond to the observed motions. Probably in the same year Mercator published a text De emendatione annua diatribae duae in which he argues for a new version of the calendar in which there are 12 months with 29, 29, 30, 30, 31, 31, 32, 31, 31, 31, 30, 30 days respectively.

After working at the University of Copenhagen for six years, Mercator had to leave when the university was closed due to the plague. From this time on things were not too good for him. He published nothing for ten years but we do know that he maintained his interests in astronomy. The paper [2] looks at three letters between Boulliau and Mercator in 1659 and early 1660:-

Ismael Boulliau, Nicolaus Mercator, Jeremiah Horrocks : this rare conjunction of post-Keplerian astronomers occurs in a brief exchange of letters in 1659 and early 1660. Three letters exchanged between Boulliau and Mercator, concerning the famous transit observation of Horrocks, symbolically link three distinct phases of post-Keplerian thought. Each astronomer made significant, but distinctive, contributions to the acceptance, modification, and transmission of Keplerian planetary theory. ... Boulliau, although rejecting Kepler's area rule and his vision of a celestial dynamics, became known in the middle decades of the seventeenth century as the leading proponent of elliptical orbits; whereas his younger correspondent, Mercator, author of several mathematical and astronomical texts, argued the superiority of the Keplerian ellipse and area rule over rival elliptical hypotheses in the latter half of the century.

Mercator went to England some time before 1660, almost certainly in an attempt to improve his fortunes. We do know that Cromwell was aware of Mercator proposals for changing the calendar and some historians suggest that Cromwell may have invited Mercator to England. There is no direct evidence for this hypothesis, however. Certainly Mercator would have liked to have received a university appointment but, unable to find such a position, he did private tutoring in London. He was soon recognised in England, however, as a fine mathematician and he began to exchange ideas with Oughtred, Pell and Collins. His first publication in over ten years, and his first in England, was Hypothesis astronomica nova (published in London in 1664) in which he combined Kepler's theory of elliptical orbits with other ideas of his own. It was probably through reading this text that Newton learned about Kepler's claim that the orbits of the planets were ellipses.

One of the big problems of the day was being able to determine the longitude at sea. A known theoretical solution was to devise a clock which would keep accurate time at sea. Mercator invented such a marine chronometer, a pendulum clock, and on the strength of this invention he was elected a Fellow of the Royal Society in November 1666. He also made measurements of air pressure for the Royal Society and got to know Hooke with whom he shared many common interests.

Mercator discovered the well known series, sometimes called Mercator's series,

ln(1 + x) = x - x²/2 + x³/3 - x⁴/4 + ...

He published this in Logarithmotechnia 1668. This series was also investigated by Mengoli.

In the following year of 1669, at the request of Collins and Seth Ward, Mercator translated the Dutch text Algebra ofte Stelkonst by Kinckhuysen into Latin. This brought Mercator into contact with Newton and the two men exchanged letters, discussing among other things the motion of the moon. Mercator published a further two volume work in astronomy Institutiones astronomicae in 1676. These were well written texts covering the latest knowledge on the subject. Newton read these works and in fact Newton's copy containing the notes he made as he read it still exists.

One might have thought that Mercator's reputation was sufficient that he would have been able to acquire a good post but it seems that he could not. In 1676 Hooke proposed Mercator for the post of Mathematical Master at Christ's Hospital, but he was not appointed. Saddened by his inability to get a teaching position, in 1682 he moved to France, this time to fill a specific position, namely to design the waterworks at Versailles. Jean-Baptiste Colbert, founder of the Académie des Sciences in Paris in 1666 and controller general of finance in France, had invited Mercator to undertake the project. Unfortunately Mercator fell out with Colbert so this project was not a success.

1. D T Whiteside, Biography in Dictionary of Scientific Biography (New York 1970-1990). 
   http://www.encyclopedia.com/doc/1G2-2830902917.html

Articles:

1. W Applebaum and R A Hatch, Boulliau, Mercator, and Horrocks's Venus in sole visa: three unpublished letters, J. Hist. Astronom. 14 (3) (1983), 166-179.
2. J Aubrey, Letters ... and Lives of Eminent Men 2 (London, 1813), 450-451, 473.
3. A V Dorofeeva, Nicolaus Mercator (1620-1687) (Russian), Mat. v Shkole (2) (1988), i.
4. J E Hofmann, Nicolaus Mercator (Kauffman), sein Leben und Wirken, vorzugsweise als Mathematiker, Akad. Wiss. Mainz. Abh. Math.-Nat. Kl. 1950 (3) (1950), 45-103.