De sectore et radio. The description and use of the sector in three bookes. The description and use of the Crosse-Staffe in other three bookes. For such as are studious of mathematicall practise. London: Printed by William Jones, and are to be sold by John Tomson at his house in Hosier-lane, 1623. [Bound with:] ibid. Canon Triangulorum or Tables of Artificiall Sines and Tangents. London: Printed by William Jones and are to be sold by Edmund Weaver, 1623. [Offered with:] A brass sector made to Gunter’s design by Elias Allen, the best mathematical instrument maker of his day, c. 1630. Signed ‘Elias Allen fecit’.

London: William Jones, 1623.

First edition, extremely rare complete as here, of Gunter’s book on the sector and other mathematical instruments, “one of the most influential scientific works on navigation” (Waters, p. 359), the work which introduced logarithms into the science of navigation and led to the development of the slide-rule. Gunter’s book is here accompanied by a brass sector made to his design by Elias Allen, c. 1630. The engraved frontispiece carries an advertisement (in Allen's hand) for the maker as provider of the sector: “These instruments are wrought in brasse by Elias Allen dwelling without Tempel barre over against St Clements Church: and in wood by John Thompson dwelling in Hosiar Lane”. “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. The sector described by Gunter consisted basically of two hinged arms (like a carpenter’s ruler) on which were engraved several scales … Gunter’s book was given in two main parts. In the first he concentrated his attention on the sector; and in the second on the cross-staff. In the first part he gave solutions, not only to nautical astronomical problems but also to plane and Mercator sailing. He also provided a novel traverse table, this being the first of its kind and of the type that is now commonly used by navigators. In the second part of his book Gunter described a novel form of cross-staff, the most useful feature of which was the several scales engraved on the staff. These were logarithmic scales by means of which, using a pair of dividers, problems of multiplication and division could be solved easily and quickly” (Cotter, pp. 363-4). Gunter’s sector “allowed calculations involving square and cubic proportions, and carried various trigonometrical scales. Moreover, it had a scale for use with Mercator’s new projection of the sphere, making this projection more manageable for navigators who were only partially mathematically literate. The sector was sold as a navigational instrument throughout the seventeenth century and survived in cases of drawing instruments for nearly three hundred years. The most striking feature of the cross-staff, distancing it from other forms of this instrument, was the inclusion of logarithmic scales. This was the first version of a logarithmic rule, and it was from Gunter’s work that logarithmic slide rules were developed, instruments that remained in use until the late twentieth century” (ODNB). This book is justly renowned as a contribution to navigation, but it seems not to be widely known that it also contains (p. 60 of the second part) the first printed observation of the temporal variation of magnetic declination, the discovery of which is normally ascribed to Henry Gellibrand who published it 12 years later. “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” (ODNB). All of Gunter’s instruments are shown in use on the engraved title page. This particular engraving was used for many of the reprints of Gunter’s work, the central title being changed and various inscriptions being added to the shield at the base (blank in this first edition). The present copy of Gunter’s De sectore is here bound with the second edition of his Canon triangulorum (first, 1620), the first published table of logarithmic sines and tangents. This edition includes the first table of base-ten logarithms, first published by Henry Briggs in 1617 (this is not present in the first edition of the Canon). Our copy of De sectoreis complete with the full text, the engraved and letterpress titles (sometimes omitted), and the volvelle present but not assembled. Only a handful of copies of the first edition of De sectore have appeared at auction since 1957, none complete; the Horblit copy (lacking the letterpress title) was offered by H. P. Kraus in Cat. 168 (ca. 1984) for $4200. ESTC locates only seven institutional copies: British Library, Cambridge (2), St Andrews, UCL, Harvard (2), US Naval Academy Nimitz, and Williams College. The Canon is just as rare, ABPC/RBH listing only two copies (2004 & 1958), and ESTC listing five (three in the UK, two in the US).

Provenance: I. Ownership inscription at head of title page of “John Hope, Tyninghame, 6 October, 1672”. A melancholy provenance: John Hope of Hopetoun (1650-1682) was drowned when HMS Gloucester was wrecked off the coast of Norfolk, carrying the Duke of York (the future James II) to Leith. There are several marginal index notes, apparently in Hope's hand. II. Ownership signature of “I Skene” on blank before title; possibly a descendant of Sir John Skene of Curriehall (1549-1617) and a familial connection, as Skene’s widow married Thomas Hope of Craighall (1573-1646). III. Imposing armorial bookplate of Sir John Hope, fourth earl of Hopetoun (1765-1823), army officer. Hope had a long and distinguished military career; in 1793 he served with the 25th Foot (later the King's Own Scottish Borderers), one of the regiments assigned to make up the numbers of marines on board the Mediterranean and Channel fleets of lords Hood and Howe (the supporters of his bookplate are two figures with anchors). Wellington called him “the ablest man in the Peninsular army” (cited in ODNB).

“Edmund Gunter was born in Hertfordshire in 1581, educated at Westminster School, and then at Christ Church, Oxford. It is probable that while there he was influenced by Sir Henry Savile’s lectures on mathematics. At least it is certain that he soon showed himself to be a mathematician of the first order with a gift for instrumental invention. At the age of twenty-two he gained entrée of the mathematical world in England by a manuscript entitled A New Projection of the Sphere … About the time he became Master of Arts, in 1606, Gunter circulated another manuscript, in Latin, on the application of mathematics to navigation. He called it, probably, De Sectore, for it centred upon an instrument which he had devised for the instrumental solution of navigational problems, and which he called ‘a Sector.’ For reasons best known to himself Gunter did not publish his work for about seventeen years. He did, however, allow many copies to be ‘transcribed and dispersed’. In consequence its contents became known in circles other than strictly mathematical ones. Indeed the fame of his Sector became such that many ‘not understanding the Latine yet were at the charge to buy the Instrument’ which Elias Allen made in brass, and tried to use it – not to pass a pleasant hour in studious practice, but to solve, when they were at sea, the daily problem on the ship’s position. It was not until 1623 that Gunter, partly to satisfy the importunity of those ignorant of Latin but eager to improve their navigation, and partly to relieve himself of the trouble of expounding the contents of his manuscript, was content ‘to give way that it come forth in English.’ The published work appeared under the title of De Sectore & Radio, or, as another impression had it, The Sector and Cross-Staff. Properly speaking the published book represents two distinct works, three ‘Books of the Sector’, each divided into various chapters, and for ‘the most part’ consisting of the Latin manuscript of 1606 or 1607, and three ‘Books of the Cross-Staff’, incorporating Gunter’s subsequent mathematical inventions and his explanation of later mathematical developments. The originality and fundamental nature of much that was contained in the books of the Sector will become apparent if it is remembered that most of the manuscript was written more than sixteen years before the books were published.

“Just as it is reasonable to suppose that Gunter got the inspiration for devising his Sector from Hood’s earlier one [The making and use of the geometricall instrument, called a sector … London, 1598], so it is possible to see that in demonstrating the mathematical solutions of navigational problems he was following the pattern of Hues’s Tractatus de Globis of 1594, and embodied, perhaps unknown to Gunter, in Hariot’s unpublished manuscript of the same year or earlier, The Doctrine of Nautical Triangles Compendious and in Dee’s unpublished manuscript The British Complement of Perfect Navigation, compiled in 1575. Nevertheless, Gunter’s De Sectore & Radio must rank with Eden’s translation of Cortes’ Arte de Navigar and Wright’s Certaine Errors as one of the three most important English books ever published for the improvement of navigation. Eden’s Art of Navigation had introduced to English seamen as a whole the subject for the first time; Wright’s work had improved and publicly explained Mercator’s projection and had made possible the accurate plotting of a ship’s position on a chart; Gunter’s manuscript of 1606 or ’07 opened up to many what was to all save a few an entirely new field, that of arithmetical navigation. It had been glimpsed by Dee, Borough, and Davis, but only as a land that is very far off, it had been brilliantly explored by Hariot but with results kept secret from all save his patrons. Gunter’s treatise brought it under foot to the many. Whether it was original in concept or derived from Dee’s and Hariot’s prior working, Gunter’s exposition of finding a ship’s position by calculation, since it was eventually published to the world, must be classed as one of the most influential scientific works on navigation. Discounting the later reprints of Cortes, Bourne, and Blundeville, all subsequent navigation manuals bear indelibly the stamp of its genius, for they are primarily treatises upon the solution of navigational problems by geometrical and trigonometrical methods. Gunter’s De Sectore was just such a treatise. Its most important features were lucid descriptions of the nature of the trigonometrical functions, sines, chords, tangents, and secants, and their uses, a detailed description of his Sector, including the manner of laying off the lines engraved upon it and their uses, and trigonometrical formulae for solving on the Sector the course, distance, and difference of longitude. The first traverse tables for eliminating tedious calculation in the solution of certain navigational problems, an admirable treatise on the resolution of spherical triangles, and clear descriptions of the drawing of charts to various scales on Mercator’s projection and of the solution of various problems on them are also included. Solutions found by plotting and with the aid of the Sector were compared with the erroneous results of similar workings on a plane chart. There can be little doubt that of Gunter’s contributions to navigation the most outstanding was the enunciation to a wide circle of friends and students of navigation of the various navigational formulae known to Harriot and his circle and first formulated by Dee more than 30 years before, and the explanation of how they could be used to solve problems without tedious calculation, by means of the Sector and of tables.

“How had Gunter made the Sector so practical? He had taken Hood’s Sector and, except for the gnomon, had discarded everything extraneous to the purposes of calculation. He had retained Hood’s Line of Equal Parts and, because it was the measure of the Sector’s radius and thus the base line on it, had renamed it the Line of Lines … The Line of Lines on each leg was divided into tenths and hundredths, and extended from the hinge centre almost to the end of each leg. Thus the Line of Lines was equal to the radius of the Sector and was divided upon the recently introduced decimal system. ‘The Ground of the Sector,’ the principle upon which it was based, was, like Hood’s, that the sides of similar triangles are proportional” (Waters, pp. 358-360).

“It was [Gunter’s] latest inventions and most recent lectures which formed the second half, The Cross-Staff in Three Books, with an appendix on a small portable quadrant, ‘for the more easie finding of the Hour and Azimuth, and other Astronomical and Geometrical Conclusions.’ These books contained his counterpart to Briggs’ Arithmetica logarithmica of 1624; for while Briggs, after the publication of Gunter’s Canon triangulorum in 1620, had busied himself at Oxford with furthering the simplification of calculation by preparing more logarithmic tables, Gunter, with his powerful practical bent, had approached the problem quite differently … Gunter’s solution was an instrument – of the simplest possible sort – a straight ruler … On it he engraved a ‘logarithmic line of numbers.’ The logarithms he took ‘out of the first Chiliad of Mr. Briggs Logarithms,’ and the line he ‘noted with the letter N’. He engraved other straight lines upon the ruler, a ‘Line of Artificial Tangents [i.e., logarithms of tangents] … noted with the letter T, divided unequally into 45 degrees, and numbered both ways, for the Tangent and the Complement’; a ‘Line of Artificial Sines noted with the letter S, divided unequally into 90 degrees, and numbered with 1, 2, 3, 4, unto 90, and a Line of Versed Sines [logarithms of secants] for more easie finding the hour and Azimuth, noted with V …; a Line of Inches … each Inch subdivided into ten parts …; a Line of Several Chords, one answerable to a Circle of twelve Inches semidiameter … another a semidiameter of a Circle of six Inches; and a third of a circle of three inches’; and a Meridian Line ‘of a Sea-chart, according to Mercator’s Projection … known by the letter M …’. The ‘lines of Proportion,’ as he called the logarithmic lines of the trigonometrical functions, he took ‘out of my Canon of Artificial Sines and Tangents.’

“Gunter’s Scale was the logical development of his Sector. This, it will be recalled, was based upon the characteristics of proportional triangles and enabled problems involving proportion to be solved instrumentally. Its basic features were the identical jointed legs that could be opened 180° to form a straight ruler; the Line of Lines divided into equal parts; and the lines of the natural trigonometrical functions and of meridional parts. It was to be used with a pair of compasses. Just as problems involving proportion were solvable on the sector with the aid of compasses, so were they on the Scale … Gunter’s Scale was, in fact, no more than his Sector fully opened and with a logarithmic line of (unequally spaced) numbers substituted for a Line of Lines of equal parts (equally spaced numbers); logarithmic lines of sines and tangents substituted for lines of natural sines and tangents, and a logarithmic line of versed sines substituted for a line of natural secants. Gunter, with his grasp of the characteristics of logarithms and the possibilities of the Sector, must early have appreciated that, as the addition of logarithms amounts to multiplication of the numbers for which they stand and subtraction to division, it should be possible to perform the processes of measuring the logarithms of the quantities concerned on a line graduated in logarithms, and to add or subtract them on this scale with a pair of compasses in order to obtain the products or quotients …

“With the invention of Gunter’s Scale the solution of navigational problems by logarithms was reduced to the simplest of instrumental manipulations. Writing of logarithms in later years William Oughtred {to whom the discovery of the slide-rule is usually ascribed] declared, ‘The honour of the invention next to the Lord of the Merchiston, and our master Briggs belongs … to master Gunter, who exposed these numbers upon a straight line.’ This is indeed true, for Gunter’s Scale was the immediate ancestor of the slide-rule. Except that distances were measured by a pair of compasses and not by another rule, it was a slide-rule. Remove the sliding scale from a modern slide-rule and you have Gunter’s logarithmic Scale.

“Gunter explained his Scale in De Sectore & Radio in the course of describing a cross-staff, of which the staff was a yard long (‘so it may serve for measure’), and was inscribed with four sorts of lines: one served ‘for Measure and Protraction; One for the Observation of Angles; one for the Sea-Chart; and the [four] other for working of Proportions in several kinds. The only lines additional to those already mentioned as being on the Scale in ruler-form were ‘the Lines for observation of Angles’, or the tangent lines on staff and cross ‘inscribed out of the ordinary Table of Tangents.’ These were for use in measuring ‘perpendicular heights and distances and angles’, chiefly in survey work.

“In the ‘first book of the cross-staff’ he described the scales on the cross-staff – which otherwise was like any other cross-staff of the period – and explained their general use for finding heights, distances, and angles, for solving spherical and plane triangles, and for finding proportional numbers. In the second book he described their more particular use ‘in several kinds’, that is to say in measuring places and solids, ‘in gauging of vessels’, i.e., casks; in ‘resolving such Astronomical Propositions as are of ordinary use concerning Longitude, Latitude, Rumb and distance’. The third book dealt with dialling, a science which Gunter, since the death of Wright who had also published a work on the subject, had become the acknowledged expert” (ibid., pp. 416-420).

“[Gunter] records in his Book of the Cross-staffe, in discussing the problem of finding the magnetic variation from an observation of the Sun’s azimuth, that he made magnetic experiments at Limehouse where, in 1580, William Borough had made similar observations. Borough had found the variation to be 11° E., whereas Gunter had found it to be 6° E. Gunter, who must have been puzzled by this observation (for he had sought the exact spot at which Borough had made his observations), seems not to have followed up his discovery; which, ironically, is now attributed to Henry Gellibrand, Gunter’s successor as Gresham Professor of Astronomy” (Cotter, p. 367). The reason Gunter did not follow up his observation was undoubtedly the authority of William Gilbert: one of the conclusions of his De magnete (1600) was that the Earth’s magnetic field could not vary with time, and so Gunter assumed that Borough must have been in error. In 1633, Gellibrand measured the declination in the same location and found it to be 4° E. Because of the care with which Gunter had made his measurements, Gellibrand was confident that the changes were real. In 1635 he published A Discourse Mathematical on the Variation of the Magneticall Needle stating that the declination had changed by more than 7° in 54 years.

The present copy of Gunter’s De Sectore & Radio is bound up with the second edition of his Canon triangulorum, the tables he had used in laying out the logarithmic lines on his Scale. First published in 1620, the Canon was the first published table of ‘artificial sines and tangents,’ i.e., base-ten logarithms of sines and tangents. “What Briggs did for logarithms of numbers, Gunter did for logarithms of trigonometrical functions – no mean contribution to astronomical navigation” (Cotter, p. 363). The logarithms were calculated to seven decimal places for every minute of the quadrant, semi-quadrantically arranged. Gunter does not explain how he obtained the logarithms of the sines and tangents and we can only assume that he used a variant of the radix method later described in detail by Briggs in Arithmetica logarithmica. He must have used the sines from a prior table, perhaps the Opus palatinum of Rheticus or the Thesaurus mathematicus of Pitiscus, and computed their logarithms. Gunter did not give the values of log sin x and log tan x, but values of the logarithms “shifted” by 10 and multiplied by 107. For instance, log10 sin 1° = −1.758144…, 10 + log10 sin 1° = 8.241855… and Gunter’s table contains 82418553.

To this second edition of the Canon was added ‘The first thousand Logarithmes now againe set forth by the Authour Henrie Briggs professor of Geometrie in the Universitie of Oxford, who undertooke this worke at the entreatie [of] John Nepier Baron of Merchiston,’ Briggs’ table of base-ten logarithms (the logarithms of Napier himself were essentially, although not exactly, natural logarithms). Briggs expanded his 1617 tables of numbers in his Arithmetica logarithmica, and Gunter’s tables of logarithms of trigonometrical functions in Trigonometria Britannica (1633), published posthumously and completed by Gellibrand.

“Gunter’s death occurred in 1626, one year after Charles I had succeeded his father James I as King of England. His writings, although rather too mathematical for the navigators of his time, were taken up by teachers of navigation, and Gunter’s novel ideas were soon promulgated among practical seamen. In particular ‘the admirable ruler’, wrote James Wilson in 1786, ‘is so constantly in the practice of our artists that it has got the name The Gunter’. And, indeed, the ‘Gunter’, usually in the form of a two-foot boxwood rule with its several engraved scales, was part of the navigator’s stock-in-trade right up to the end of the nineteenth century” (Cotter, p. 367).

There are several variant states of the first edition of Gunter’s book on the sector. STC notes: “As far as can be told from surviving copies, initial publication was somewhat disorganized, with several booksellers joining in.” The present copy is ‘maximally complete’, having all the parts that any known copy has. It most closely resembles STC 12521.5, although that copy lacks the whole of quire A of the first part (containing the dedication to his patron the Earl of Bridgewater). Other copies commonly lack either the letterpress or engraved title page (the dedication copy held at the Huntington is an example of the former). All issues of the first edition are rare, no complete copy having appeared at auction in the last thirty years. There were two editions of the Canon triangulorum published in 1623, one with English text in quires A and M (as here), STC (2nd ed.), 12517, and one with Latin text in these two quires.

The example of the Gunter Sector offered here is identical to the standard 1623 design, except that it has one additional scale, the purpose of which remains unknown. It consists of two arms pivoted on a flat circular hinge (akin to that of a modem folding carpenter's rule) and engraved with various scales. A short brass strut, pivoted at the end of one arm was used in conjunction with the Line of Lesser Tangents (to draw out hour-lines for sundials and to measure the altitude of the sun) and incidentally provided additional friction when the instrument was open to small angles. All the scales apart from the Line of Tangents, the Line of Secants, the Meridian Line, the Line of Inches (used for simple measurement of lengths) and the Line of Lesser Tangents appear in pairs, with matching lines on each arm, and radiate from the centre of the hinge. The Lines of Tangents and Secants are set along the outer edge of one side of the instrument in such a way that, when the instrument is completely opened out, the Line of Tangents forms a complete scale from the end of one arm to the end of the other, with the Line of Secants running parallel to the second half of the Line of Tangents. The Meridian Line appears in a similar position on the other side of the instrument. The Lines of Inches and Lesser Tangents are set along the outer rims of the arms. For the way in which the Sector was used in practice, see Higton, pp. 114-124 and, for its use in navigation, pp. 134-144.

The present example was constructed by Elias Allen (c. 1588-1653), a mathematical instrument maker who was apprenticed to Charles Whitwell in 1602 and became a Freeman of both the Grocers' and Clockmakers' Company (eventually becoming Master of the latter). Allen was the first English instrument maker to support himself solely through the production of instruments. He was a collaborator of Gunter, William Oughtred and other mathematicians. In the introduction to his book, The New Artificial Gauging Line or Rod (1633), Oughtred speaks of Allen as “a man well knowne and esteemed by all men of his art for his skilfulness in making instruments in metal.” Allen’s workshop was located near St Clement’s Church, near what is now Aldwych, in London.

Higton’s census of examples of Gunter’s sector made by Allen in British museums (pp. 187-197) locates five examples. Of these, only two are signed by Allen (Whipple Museum, Cambridge, and Science Museum, London); the other three are attributed to him on the basis of the style and quality of the engraving.

Macclesfield 959 (lacking five text leaves and the volvelle). Canon: Macclesfield 958; STC (2nd ed.), 12517. Cotter, ‘Edmund Gunter’, Journal of Navigation 34 (1981), pp. 363-7. Higton, ‘Elias Allen and the role of instruments in shaping the mathematical culture of seventeenth-century England.’ Doctoral dissertation, University of Cambridge, 1996. Taylor, Mathematical Practitioners, pp. 196 and 344. Waters, The Art of Navigation in England in Elizabethan and Early Stuart Times, 1958. For an analysis of the Canon, see Roegel, A reconstruction of Gunter’s Canon triangulorum (1620) (https://hal.inria.fr/inria-00543938/document).



Small 4to (179 x 139mm), [Description and use of the sector:] pp. [ii, additional engraved title: “Description and use of the sector, the crosse-staffe and other instruments. For such as are studious in mathematicall practise. London: Printed by William Jones and are to be sold by Edmund Weaver, 1624”], [ii, letterpress title], [viii], 143, [1, blank], 216, with full-page engraved plate of the sector following letterpress title, A4 with errata and volvelle for p. 60, numerous woodcut headpieces and illustrations in text. [Canon Triangulorum:] pp. [94], [2, blank], [16]. Contemporary blind-ruled calf, later spine label (extremities rubbed one corner worn). Sector: brass, 455mm (fully extended). Engraved with scales on both sides and with additional hinged strut contained within (also engraved with scales on both sides). The two limbs of the sector are indented so that the leg can be adjusted at an angle. Signed ‘Elias Allen fecit’ on the circular hinge. Very good condition: surface marked and pitted but all engraving legible and hinge working smoothly. Housed in a custom-made clamshell box.

Item #5913

Price: $65,000.00