London; Amsterdam: J. Windet and sold by S. Shorter; Jacob van Velsen for the author, ; 1672.
One of the most remarkable sammelbands from the Macclesfield library, containing the extremely rare first edition of the first published work on the ‘sector’, also called the ‘geometrical compass’ by Galileo who developed it independently in the late 1590s as an instrument for military engineering (although he did not publish an account of it until 1606). The volume also contains two exceptionally rare works by Mohr on Euclidean geometrical constructions..
One of the most remarkable sammelbands from the Macclesfield library, containing the extremely rare first edition of the first published work on the ‘sector’, also called the ‘geometrical compass’ by Galileo who developed it independently in the late 1590s as an instrument for military engineering (although he did not publish an account of it until 1606). “Hood’s sector was the first mechanical calculating device of general practical use to be published since the abacus of remote antiquity” (Stillman Drake, p. 17). “Although credit for the sector is often given to Galileo, it is clear that the instrument was well known and used in England before Galileo published his work on it” (Tomash & Williams, p. 1416). “The sector was one of the most familiar of mathematical instruments between the 17th and 19th centuries. It was however devised just before 1600 and was first published in 1598 by the English mathematical practitioner Thomas Hood. An independent version developed by Galileo Galilei in the 1590s was published early in the 17th century , and many other designs subsequently followed” (mhc.ox.ac.uk). OCLC lists no copies in North America, but we have located one (Folger), though it lacks the plates (present in this copy); ABPC/RBH list three copies (including Horblit and Kenney), all lacking the plates. Also included in this volume are three geometrical works by the Danish mathematician Georg Mohr that are so rare that they were thought to be lost until a copy of one of them, Euclides Danicus, was discovered in 1928. This work proves the ‘Mohr-Mascheroni’ theorem, according to which all geometrical constructions that can be carried out with ruler and compasses can, in fact, be carried out using compasses alone – it was proved independently by Lorenzo Mascheroni (1750-1800) 125 years after Mohr in his Geometria del Compasso. OCLC lists only one copy of Euclides Danicus in North America (Harry Ransom Center, University of Texas); only one other copy has appeared at auction. The present volume includes two further works attributed to Mohr, as rare as Euclides Danicus, also on geometrical constructions (no copies on OCLC). They are accompanied by three rare works by William Bedwell (1561-1632), two on architectural measuring instruments, the ‘carpenter’s rule’ and the ‘trigon,’ the third being the earliest published work on Tottenham, where Bedwell resided (now part of London but then a village to the north of the City). The final work in this volume, by Johannes Sturm (1507-89), is a contribution to the controversy which raged in the late sixteenth and early seventeenth century between Clavius, van Roomen, Viète, and Scaliger over the squaring of the circle.
HOOD, Thomas. The making and use of the geometricall instrument, called a sector. Whereby many necessarie geometricall conclusions concerning the proportionall description, and division of lines, and figures, the drawing of a plot of ground, the translating of it from one quantitie to another, and the casting of it up geometrically, the measuring of heights, lengths and breadths may be mechanically performed with great expedition, ease, and delight to all those, which commonly follow the practise of the mathematicall arts. London: Printed by Iohn Windet, and are to solde at the great North dore of Paules Church by Samuel Shorter, .
First edition of the first published work on the sector. “The sector, also known as the proportional, geometric, or military compass, was an analog calculating instrument used widely from the late sixteenth century until modern times … Requirements for extensive arithmetic calculation grew rapidly during the Renaissance and in the early years of the scientific and industrial revolutions. It soon became apparent to practitioners that calculation by hand, particularly the multiplication and division of large numbers, was both laborious and error-prone. It was small wonder that talented mathematicians and scientists sought to develop methods and mechanisms that would lessen the burden of computation while increasing accuracy” (Tomash & Williams, p. 1456).
Hood’s sector consisted of “a pair of flat rules hinged stiffly at one end and bearing identical scales engraved on the two arms, different on the two faces … It had three scales and was fitted with removable sights and a graduated quadrant, plumb line, and accessory graduated arm … Hood’s principal scale was one of equal linear divisions from pivot to end of either arm. On the other face he provided a scale which gave the side of various regular polygons inscribed in a circle of diameter equal to the separation of the ends, and another which gave the side of a square having an area which was an integral multiple of the area of a unit square. The enormous value of Hood’s sector for speedy mechanical approximation to a wide variety of commonest practical mathematical problems is obvious, and he explained these at great length in his book” (Drake, pp. 17-18).
“We know little of Thomas Hood (1556-1620) other than that he was the first mathematical lecturer for the City of London and gave public lectures there on topics such as the sector and other instruments. We do no know where or how Hood might have first learned of the instrument, but we presume that he learned of it through contacts with the military. In 1598, he published The making and use of the geometricall instrument, called a sector. With this book title he seems to have coined the English word sector (at least as it applies to a mathematical instrument). The book is well organized and contains useful diagrams, examples and exercises. It is obviously not a work created in haste, and this fact leads one to the conclusion that Hood must have been familiar with the sector for some time prior to 1598. Further, the book notifies the reader that Hood’s sectors were available for sale around 1594-1611 by the instrument maker Charles Whitwell, who engraved the illustrations for Hood’s book. Indeed, a sector signed by Whitwell bears the date 1597. Another sector from the same year, made by Robert Becket, has survived. Both instruments closely resemble the illustration in Hood’s publication” (Tomash & Williams, p. 1459).
“The work begins by describing the sector and its scales together with a short description of their use. These are followed by several chapters devoted to explaining individual operations. These range from performing simple multiplication (usually couched in terms of finding lines in certain proportions) to expanding and contracting figures, if given the radius of a circle to find the length of a chord of an angle, inscribing various figures inside squares and circles and similar basic functions performed with a sector” (ibid., p. 1416).
“In the same year (1598) there appeared at Venice a book on mathematical instruments written by G. P. Gallucci in which was illustrated a different kind of sector, having in common with Hood’s only the scales for construction of regular polygons … though Gallucci did not name the inventor, other evidence points to Guidobaldo del Monte, Galileo’s friend and patron. It had but two scales, one on either face, the second being designed to permit the division of a line into equal segments, just as the first was used to divide a circle into equal arcs. Guidobaldo’s sector was in no significant sense a calculating instrument; it simply gave direct mechanical solutions to two very common problems in drafting, designing, and instrument construction” (Drake, p. 18).
MOHR, Georg. Euclides Danicus, bestaende in twee deelen. Het eerste deel: handelt van de meetkonstige werckstucken, begrepen in de ses eerste boecken Euclidis: het tweede deel: geest aenleyding om verscheyde werckstucken te maecken als van Snijding, Raecking, Deeling, Perspective en Sonnewijsers … Amsterdam: Jacob van Velsen for the author, 1672.
Ibid. Compendium Euclidis curiosi: dat is, meetkonstigh passer-werck, hoe me meet een gegeven opening van een passer en een liniael, de werck-stucken van Euclides, ontbinden kan te samen gestelt door een Lief-hebber der selver konst. Amsterdam: J. Jansson van Waesberge, 1673.
S., J. D. [MOHR, Georg?]. Gegen-übung auf ein mathematisch Tractätlein, Compendium Euclidis Curiosi genant, worin nebst kurtzem Anweis um verscheidene Euclidische Aufgaben mit einer gegebenen Oeffnung des Zirkels noch auf andere Ahrt zu machen; zu mehrerem Nutzen wird vorgestellet eine kurze, iedoch grundrichtige Manier um den cörperlichen Inhalt einer Festung mit geringer Mühe aus zu rechnen Amsterdam: J. Jansson van Waesberge, 1673.
First editions of these three extremely rare works on Euclidean geometrical constructions. For the Greeks the straight line and the circle were the most perfect geometrical figures, and it was therefore of great importance to determine which geometrical figures could be constructed using straightedge and compass. Such questions remained of interest in the modern period. Indeed, one of the first discoveries made by Carl Friedrich Gauss was that a regular polygon with 17 sides is constructible using straightedge and compass. Later, in his great work Disquisitiones Arithmeticae (1801), Gauss determined exactly which regular polygons can be so constructed. A related question is whether the constructions that can be made by straightedge and compass can be made by straightedge alone, or compass alone. In Euclides Danicus, Mohr was the first to establish that all Euclidean constructions can be made using compass alone, while in Compendium Euclidis Curiosi he shows for the first time that such constructions can be made using straightedge and a compass with a single opening – this “was posed in the contests of the great Renaissance mathematicians” (DSB); it also contains mathematical problems related to fortification. Although the latter work was published anonymously, it has been established by Arthur E. Hallerburg that Mohr is its author. Hallerburg was not able to trace a copy of Gegenübung, and although this work has been ascribed to Mohr by Bierens de Haan and others, Andersen & Meyer have argued that its author, identified as ‘J.D.S.’ on the title page, is not Mohr: “The largest part of [Gegenübung] consists of calculations connected to fortification and is not related to Mohr’s theorems on fortification in Euclides Curiosus. Only the first five pages deal with constructions performed with a ruler and a fixed compass. In these pages J.D.S. offers alternative solutions to the problems Mohr had numbered IV, V. VI. VIII. XII and XIV.” However, Andersen & Meyer were unable to determine the identity of ‘J.D.S.’
Georg Mohr (1640-97) was born in Copenhagen, but spent much of his life in Holland (to which he travelled in 1662) before visiting England and France and then returning to Denmark. While in England he met Henry Oldenburg and John Collins. In a letter to Leibnitz (30 September 1675), Oldenburg writes of Mohr, of whom he says that he is ‘algebrae et mechanices probe peritus’, that he has recently left England for Paris and has left with John Collins “a certain work written in the Flemish tongue, a copy of which I was glad to communicate to you, because, according to Collins, the said Mohr asserted, this work … completes Cardan’s rules … and supplies roots of equations of the kind which are represented by surds” (Oldenburg, Correspondence XI, letter 2754). This work has not been identified, but if Mohr left Collins with one of his works it is surely possible that he also left him with copies of the geometrical works offered here. As the Macclesfield sale catalogue notes (Part 2, p. 12), “It is probably not unreasonable to suppose that anything published before 1683 [in the Macclesfield Library] belonged to him [Collins].” Does the presence of (Collins’ copy (?) of) Gegenübung in this sammelband together with Mohr’s other two works suggest that Gegenübung is also Mohr’s?
Euclides Danicus was published simultaneously, by the same publisher, in Danish and Dutch. “The obscurity that befell Mohr and his book can be attributed, in some degree, to the presentation of the material. In the body of the book, Mohr does not state the issue until the very last paragraph, although the lines are referred to as ‘imagined’ (gedachte). In the dedication to Christian V, he does say that he believes he has done something new, and on the title page the issue is explicitly stated. Still, it would be easy for an inattentive reader to misjudge the value of the book” (DSB). It is likely that the languages of composition and a small print run also played a part. J. Hjelmslev published a German translation in 1928, the year of its rediscovery. Compendium Euclidis Curiosi seems to have been published only in Dutch, although it was translated into English in 1677.
“Word of the discovery of Mohr’s book travelled quickly, and by 1929 there was an enthusiastic report on its contents by the eminent Berkeley historian, Florian Cajori, who also reported on Mohr’s contacts with Leibniz, whom Mohr met in 1676 … Any bibliophile has to be inspired by the discovery of the Mohr volume in a stack of ‘used books’, containing a geometrical theorem that would not become public for another 125 years. Needless to say, copies of Mohr’s book are exceedingly scarce. In 2005 a copy of the original (accompanied by the 1928 facsimile) appeared in the catalogue of a book auction house in San Francisco [PBA Galleries]. A Bay Area collector acquired it for the ridiculously low price of roughly $13,000” (Alexanderson).
It appears that all copies of Euclides Danicus are signed on the title page by the author with a flourish in authentication. Alexanderson suggests that “the flourish under Mohr’s signature follows a long Spanish tradition of certifying an author’s signature on documents, possibly a holdover from the Spanish Habsburg’s influence in the Low Countries that lasted into the early 18th century”.
OCLC lists one copy of Euclides Danicus in North America, and no copy of the other two works.
PILKINGTON, Gilbert [BEDWELL, William]. The turnament of Tottenham. Or, the wooing, winning, and wedding, of Ribbe, the reeu's daughter there. Written long since in verse, by Mr. Gilbert Pilkington, at that time as some haue thought parson of the parish. Taken out of an ancient manuscript, and published for the delight of others, by Wilhlm (sic.) Bedwell, now pastour there. [Including as second part:] A briefe description of the towne of Tottenham High Cross in Middlesex: together with an historical narration of such memorable things, as are there to be seene and obserued. Collected, digested, and written by Wilhelm Bedwell. London: J. Norton, 1631.
BEDWELL, William (1561-1632). Mesolabium architectonicum: that is, a most rare, and singular instrument, for the easie, speedy, and most certaine measuring of plaines and solids by the foote: necessary to be knowne of all men whatsoeuer, who would not in this case be notable defrauded: invented long since by Mr Thomas Bedwell Esquire: and now published, and the vse thereof declared, by Wilhelm [sic] Bedwell, his nephew, Vicar of Tottenham. London: J. N[orton]. for William Garet, 1631.
BEDWELL, William [SCHÖNER, Lazarus]. De numeris geometricis. Of the nature and proprieties of geometricall numbers. First written by Lazarus Schonerus, and now Englished, enlarged and illustrated with diuers and sundry tables and obseruations concerning the measuring of plaines and solids: all teaching the fabricke, demonstration and vse of a singular instrument, or rular, long since inuented and perfitted by Thomas Bedwell Esquire. London: R[ichard]. Field, 1614.
First editions of three very rare works by William Bedwell (1563-1632), English mathematician and Arabist. Educated at St. John’s College, Cambridge, Bedwell served as Vicar of All Hallows, Tottenham (known at the time as Tottenham High Cross) from 1607 until his death, and was the author of the first local history of the area, A briefe description of the towne of Tottenham High Cross in Middlesex. The poem, a burlesque upon the old feudal custom of marrying an heiress to a knight who vanquished all his opponents, was lent by George Wither. Thomas Pilkington was in fact the transcriber, not the author.
The other two works describe instruments devised by Bedwell’s uncle, Thomas Bedwell (d. 1595). “Bedwell’s ‘Carpenter’s rule’ was intended for … carpenter’s, surveyors, shipbuilders, and indeed merchants who had to measure timber but who had no academic knowledge of mensuration. In the text of the Mesolabium architectonicum, his reconstruction of the ‘little treatise’ which he had seen amongst his uncle Thomas’ papers and which he had tried to have published as early as 1602, Bedwell claimed that his object was to assist even ‘the meanest of understanding’ to measure a piece of timber accurately and to avoid being defrauded … The instrument of which Bedwell provided a diagram in his Mesolabium architectonicum was a ‘flat ruler or oblong parallelogram’, about two foot long and two and a half inches broad. On one side it contained ‘a scale of equall divisions’ of inches divided into halves, quarters, eighths and so on, and of an inch divided into 7, 11, 13, 17, 19, 23, ‘and other such equall parts.’ On the other side was ‘a scale of unequall divisions, serving for the measuring of Board and Tymber consisting of ‘two sortes of straight lines, the one Bevelling or Slanting, drawne askue from side to side. The other Parallell that is equidistant one from another running along the Ruler, from the one end toward the other. And therefore cutting those former, and dividing them into unequall portions, whereby not onely their sayd Quadrate or square measure is performed: But also all other whatsoever…’ In the rest of the treatise Bedwell gave instructions as to how planes and solids of various shapes could be measured with his instrument …
“The ‘Carpenter’s Rule’ was also presented in a different shape to the rectangular diagram with parallel and slanting lines. For Bedwell gave the same name to the Trigonum architectonicum, first published at his own expense in 1612 and reissued in De numeris geometricis in 1614 and at the end of Mesolabium architectonicum in 1631 … The principle of the triangular table is the same as that of the rectangular rule, but in the trigon the answer is given in numerals at the intersection of the columns corresponding to the known numbers which run, from 1 to 24, along the base and the side of the triangle …
“Bedwell’s presentation of his instrument did acquire an element of novelty when he appended the trigon to De Numeris geometricis and entirely transformed the mathematical treatise De Numeris figuratis by Lazarus Schöner (c. 1543-1607) into an introduction to the instrument devised by his uncle … Bedwell seems to have believed that that Schöner’s treatise, of which he retained little more than the propositions, was ideally suited for teaching ‘the nature and proprieties’ of ‘Geometricall Numbers’ which were essential for understanding Thomas Bedwell’s instrument. The emphasis in Bedwell’s preface is on application and simplification, and what Bedwell did was to take a selection of Schöner’s propositions on ‘geometrical numbers’ (Schöner himself chose to call them ‘figurate’, the term by which they are still known) and apply them to the measurement of boards, glass, cloth, wainscoting, and paving in feet and inches” (Hamilton, pp. 56-59). Schöner’s work originally appeared as an appendix to Pierre de la Ramée’s Arithmetices libri duo (1586).
OCLC lists four copies of Tottenham, two of Mesolabium and two of De Numeris in North America. ABPC/RBH list only one copy of Tottenham since 1980 (Christie’s, 6 July, 2000, lot 149, £4000); two copies of Mesolabium in the last 80 years (at least one of which was incomplete); and only the Kenney copy of De numeris (Sotheby’s, 1968).
STURM, Johannes (1507-1589). De accurata circuli dimensione et quadratura cum sylvula epigrammatum, aenigmatum, aliorumque Versuum de numeris, ad animum, partim instruendum, partim recreandum, inventis. Leuven: François Simon, 1633.
Sturm (1559-1650) studied at Leuven University, graduating Master of Arts before the age of 20, and on 12 June 1579 he was appointed to a benefice of the chaplaincy of St Margaret in Mechelen. In 1585 he was appointed to teach dialectics and metaphysics in Lily College, Leuven, while pursuing further studies in the Faculty of Medicine. In 1591 he graduated licentiate in medicine and was appointed to the university’s academic council. He audited the lectures of Adriaan van Roomen. In 1593 Sturm was appointed to the chair in mathematics vacated by Roomen, graduating doctor of medicine the same year. In 1603 he was appointed regent of Lily College, resigning in 1606 in order to marry Catherine van Thienen. After her death in 1619, Sturm took holy orders. He died in Leuven, and was buried in the church of St Kwinten.
“This is a strange and curious work in which Sturm presents elementary mathematical and geometrical problems and their solution – often in Latin verse. The work contains a section on squaring the circle in which Sturm gives values of π (one to over seventy places of decimals – so that it had to be printed vertically on the page because there would not have been enough room to accommodate it across the page)” (Tomash & Williams, p. 1249).
OCLC lists only five copies worldwide (all in Germany). ABPC/RBH list only one copy.
G. L. Alexanderson, ‘About the cover: two theorems on geometric constructions,’ Bulletin of the American Mathematical Society 51 (2014), 463-7. K. Andersen and H. Meyer, ‘Georg Mohr’s three books and the Gegenübung auf Compendium Euclidis Curiosi’, Centaurus 28 (1985), 139-144. Stillman Drake, Essays on Galileo and the History and Philosophy of Science, vol. 3, 2000. A. Hamilton, William Bedwell the Arabist: 1563-1632, 1985. E. Tomash & M. R. Williams, The Erwin Tomash Library on the history of computing, 2009.
Eight works in one volume, small 4to (185 x 130mm). Mohr, Euclides: pp. [iv], 36, with three folding plates, signed by the author in authentication. Mohr, Compendium: pp. , 5-24 with one folding plate. Mohr (?), Gegenübung: pp. , 5-24 with two folding plates. Bierens de Haan 15, p. 263 (giving Mohr as the author of Gegenübung and incorrectly giving the date of the Compendium as 1672). Bedwell, Tottenham: ff.  (lacking final blank), partly in verse, second part with separate dated title page on C1r with printer's fleur de lys device with motto ‘In Domino confido’; register is continuous. STC 19925; Upcott II, p. 587. Bedwell, Mesolabium: ff. , with two engraved plates. Harris 34; STC 1796; Taylor p. 346, 147. Bedwell, De numeris: pp. [vi], 82, with folding table titled ‘Trigonum atchitectonicum’ at end (lacking initial blank leaf). STC 21825. Sturm: pp. [xxiv], 72. Woodcut printer’s device on the title-page, with motto ‘Rebus in humanis fortuna volubilis errat’ in cartouche, six-line decorated woodcut initials. Poggendorff II, 1018; Sotheran 4631. Hood: ff. , 50, [1, errata], with two full-page plates. Black Letter, woodcut diagrams in text. STC 13695. Eighteenth-century mottled calf, spine gilt in compartments, red morocco lettering-piece, red edges (occasional cropping, upper joint weak). Preserved in a cloth folding box with black morocco lettering-piece..