Continuing his studies at Oxford, Gunter was awarded an M.A. in 1606 but he remained at Oxford until 1615 when he received the divinity degree of B.D. on 23 November. Gunter was ordained and in 1615 became Rector of St George's Church in Southwark and of St Mary Magdalen, Oxford. He held these positions in the Church until his death.
Gunter became a friend of Briggs, and would spend much time with him at Gresham College discussing mathematical topics. Briggs was the first Professor of Mathematics at Gresham College holding the position from the founding of the College. Gresham College had been established in 1597 and there was also a Professor of Astronomy from the founding of the College. The first Professor of Astronomy was Edward Brerewood (about 1565-1613), educated at Brasenose College, Oxford, who had a broad range of expertise from mathematics, to antiquities, to logic and to languages. He died on 4 November 1613 and Briggs supported Gunter to be appointed as the second Professor of Astronomy at Gresham College. However, Gunter failed to gain the appointment which went to Thomas Williams. He had been educated at Christ Church, Oxford, and was appointed as Professor of Astronomy on 11 November, just a week after Brerewood died. Thomas Williams had entered Christ Church in 1599, only one year before Gunter, so could not have had much greater experience. However, by 1618 Gunter must have spent so much time at Gresham College that William Oughtred believed that he was already a professor there. Oughtred wrote:-
In the Spring 1618 I being at London went to see my honoured friend Master Henry Briggs at Gresham College: who then brought me acquainted with Master Gunter lately chosen Astronomical lecturer there, and was at that time in Doctor Brooks his chamber. With whom falling into speech about his quadrant, I showed him my Horizontal Instrument. He viewed it very heedfully: and questioned about the projecture and use thereof, often saying these words, it is a very good one. And not long after he delivered to Master Briggs to be sent to me mine own Instrument printed off from one cut in brass: which afterwards I understood he presented to the right Honourable the Earl of Bridgewater, and in his book of the sector printed six years after, among other projections he setteth down this.The second Gresham College professor of astronomy Thomas Williams resigned in a letter of 4 March 1619. It reads:-
I Thomas Williams of the University of Oxford, Master of Arts, Reader of the astronomy lecture at Gresham house London, do fully and absolutely resign all the right and interest which I have to the place and office of astronomy lecturer in the same house ...We do not know why he resigned, but it may have been so that he could marry. Nothing further seems to be known about Williams. Gunter was appointed to fill the vacancy on 6 March 1619, two days after Williams resigned, largely on the recommendation of Briggs. Certainly Gresham College made remarkably rapid appointments.
An episode concerning the appointment of the first professor of geometry at Oxford occurred in 1619 which is reported by John Aubrey. This chair of geometry at Oxford was founded by Sir Henry Savile who was keen to improve the state of mathematical studies in England which, at that time, were considered extremely poor. Savile required the professor to be "of good character", at least 26 years old and from a Christian country, preferably England. The professor had to be thoroughly acquainted with Aristotle and Plato but also have studied the latest developments in science. The first person Savile interviewed for the professorship was Gunter. Aubrey writes that Seth Ward told him that Gunter came to the interview:-
... and brought with him his sector and quadrant, and fell to resolving triangles and doing a great many fine things. Said the grave knight [Savile], "Do you call this reading of geometry? This is showing of tricks, man!", and so dismissed him with scorn, and sent for Henry Briggs.We must point out that if this is true, then certainly Seth Ward learnt this story from others since he was only two years old when Savile interviewed Gunter.
Jon V Pepper writes in :-
Gunter's works, written in English, reflected the practical nature of his teaching and linked the more scholarly work of his time with everyday needs; the tools he provided were of immense value long afterward.Gunter published seven figure tables of logarithms of sines and tangents in 1620 in Canon Triangulorum sive Tabulae Sinuum et Tangentium Artificialum Ⓣ (see  for details). This was a Latin text but an English translation with title Canon of Triangles: or Tables of Artificial Sines and Tangents was published in the same year. Although the words sine and tangent were already in use, Gunter invented the words cosine and cotangent. This was the first ever publication of logarithms of trigonometric functions and Gunter deserves much credit for this innovation.
He made a mechanical device, Gunter's rule, to multiply numbers based on the logs using a single scale and a pair of dividers. It was called the 'gunter' by seamen and was an important step in the development of the slide rule. Gunter published his description in 1623 in Description and Use of the Sector, the Crosse-staffe and other Instruments. It is worth noting that in this work Gunter uses the contractions sin for sine and tan for tangent in his drawing of his scale although not in the text of the book. Charles Cotter writes in :-
This book must be reckoned, by every standard, to be the most important work on the science of navigation to be published in the seventeenth century. It opened the whole subject of mathematical application to navigation and nautical astronomy to every mariner who was sufficiently interested in devoting time to the perfecting of his art.A second edition of this book was published in 1636 and purchased by Isaac Newton for 5 shillings in 1667. Newton's copy of this second edition is in the library of Trinity College, Cambridge.
C J Sangwin writes :-
A sector is a mathematical instrument which consists of two hinged rulers on which there are engraved scales. The scales allow various questions in trigonometry to be resolved by using the property that two similar (equiangular) triangles have sides in a constant ratio. The issue of who first invented by the sector is not without controversy. ... What singles out Gunter's sector is that it is the first mathematical instrument to be inscribed with a logarithmic scale to facilitate the resolution of numerical problems. This is not a slide rule in any sense of the term; the single logarithmic scale is used in conjunction with a pair of compasses. Such a rule is frequently referred to as a Gunter line. A two foot long boxwood ruler inscribed with a variety of scales was a standard navigator's tool up until the end of the nineteenth century.Gunter was the inventor of instruments but he needed an instrument maker to produce them. Elias Allen (1592-1653) who had his workshop beside St Clement Danes Church, the Strand, London, was the instrument maker whom Gunter used. Hester Higton writes :-
Allen appears to have had a close working relationship with Gunter. He was the most highly regarded instrument maker of his day. Having been an apprentice of the famous maker Charles Whitwell, he set himself up in business shortly before Whitwell's death in 1611, first [near] Fleetbridge, then beside St Clement Danes church in the Strand, probably in his former master's workshop.Gunter gained from having Allen make his instruments and the illustration of the sector in Description and Use of the Sector, the Crosse-staffe and other Instruments is from plates engraved by Allen. But the relationship between the two worked the other way round too, for Allen made use of this engraving of Gunter's sector to advertise his own business.
Gunter also invented 'Gunter's chain' which was 22 yards long with 100 links. It was used for surveying and the unit of area called an acre is ten square chains. Gunter also studied magnetic declination and was the first to observe the secular variation. Higton writes in :-
In 1622 Gunter's investigations at Limehouse, Deptford, of the magnetic variation of the compass needle produced results differing from William Borough's, obtained more than forty years earlier. He assumed an error in Borough's measurements, but this was in fact the first observation of temporal change in magnetic variation, a contribution acknowledged by his successor, Henry Gellibrand, who discovered the phenomenon.His fascination with mathematical instruments went right back to his days at school and his main mathematical contributions are rightly seen to be in this area. In fact he made a new sundial with many plates containing a considerable amount of astronomical data and erected it in the gardens of Whitehall, a royal residence, in 1622. Prince Charles, who became King Charles I in 1625, persuaded Gunter to write and publish a description of the sundial which he did as The Description and Use of his Majesties Dials in Whitehall Garden (1624).
His contributions are summed up in  as follows:-
Gunter was a firm advocate of the use of instruments in mathematics for easing the work of various mathematical practitioners, notably surveyors and navigators. His instruments were designed with these aims in mind. In particular his work on logarithms, their applications to trigonometry, and their inclusion on instruments greatly simplified the processes of mathematical calculation. His books were popular for many years after his death: an edition of all his works was produced by Samuel Foster in 1636 and this had three more editions, the last in 1680 ...Gunter died at Gresham College and was buried one day later in the churchyard of St Peter-le-Poer, Old Broad Street. St Peter-le-Poer was rebuilt on the site of its churchyard in 1788 and the church no longer exists having been demolished in 1896.
Article by: J J O'Connor and E F Robertson