The Rise of “the Mathematicals”: Placing Maths into the Hands of Practitioners—The Invention and Popularization of Sectors and Scales

  • Joel S. SilverbergEmail author
Conference paper
Part of the Proceedings of the Canadian Society for History and Philosophy of Mathematics/La Société Canadienne d’Histoire et de Philosophie des Mathématiques book series (PCSHPM)


Following John Napier’s invention of logarithms in 1614, the remainder of the sixteenth century saw an explosion of interest in the art of mathematics as a practical and worldly activity. Mathematics was no longer the exclusive realm of scholars, mathematicians, astronomers, and occasional gentlemen. Teachers of mathematics, instrument makers, chart makers, printers, booksellers, and authors of pamphlets, manuals, and books developed new audiences for the study of mathematics and changed the public’s perception of the status and aims of mathematics itself. The inventions of mathematical instrument makers facilitated the rapid expansion of sophisticated mathematical problem solving among craftsmen and practitioners in areas as diverse as navigation, surveying, cartography, military engineering, astronomy, and the design of sundials.


Pivot Point Magnetic Compass Transverse Distance Spherical Triangle Similar Triangle 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. Ellis, A. (1956). The Penny Universities: A history of the coffee-houses. London: Secker & Warburg.Google Scholar
  2. Euclid. (1956). The thirteen books of Euclid’s elements (2nd ed.) (T. L. Heath, Trans.). New York: Dover (Original work published 1908).Google Scholar
  3. Euler, L. (1748). Introductio in analysin infinitorum. Lausanne: M. M. Bousquet.Google Scholar
  4. Finck, T. (1583). Thomae Finkii Flenspurgensis Geometriae rotundi libri XIIII. Basileae: Per Sebastianum Henricpetri.Google Scholar
  5. Galilei, G. (1606). Le Operazioni Del Compasso Geometrico, Et Militare. Padova: In Casa dell’Autore: Per Pietro Marinelli.Google Scholar
  6. Gunter, E. (1623). De sectore et radio. The description and vse of the sector in three bookes. The description and vse of the cross-staffe in other three bookes. For such as are studious of mathematicall practise. London: William Jones.Google Scholar
  7. Hood, T. (1598). The making and vse of the geometricall instrument, called a sector Whereby many necessarie geometricall conclusions concerning the proportionall description, and diuision 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 vp geometrically, the measuring of heights, lengths and breadths may be mechanically perfomed with great expedition, ease, and delight to all those, which commonly follow the practise of the mathematicall arts, either in suruaying of land, or otherwise. Written by Thomas Hood, doctor in physicke. 1598. The instrument is made by Charles Whitwell dwelling without Temple Barre against S. Clements church. London: Published by I. Windet, and are to sold at the great North dore of Paules Church by Samuel Shorter, 1598, 1598.
  8. Johnston, S. (2005). History from below: Mathematics, instruments, and archaeology.
  9. Lillywhite, B. (1963). London coffee houses; a reference book of coffee houses of the seventeenth, eighteenth, and nineteenth centuries. London: G. Allen and Unwin. ISBN: 7800542238 9787800542237.Google Scholar
  10. Napier, J. (1614). Mirifici logarithmorum canonis descriptio ejusque usus, in utraque trigonometria; ut etiam in omni logistica mathematica, amplissimi, facillimi, & expeditissimi explicatio. Authore ac inventore, Ioanne Nepero, Barone Merchistonii, ©. Scoto. Edinburgi: Andreæ Hart.
  11. Napier, J. (1618). A description of the admirable table of logarithmes with a declaration of the most plentifull, easie, and speedy use thereof in both kinds of trigonometry, as also in all mathematicall calculations. Inuented and published in Latine by that honourable Lord John Nepair, Baron of Marchiston, and translated into English by the late learned and famous mathematician, Edward Wright. With an Addition of an Instrumentall Table to finde the part proportionall, inuented by the Translator, and described in the end of the Booke by Henry Brigs Geometry-reader at Gresham-house in London. All perused and approued by the Author, & publishes since the death of the Translator. London: Simon Waterson.Google Scholar
  12. Napier, J., & Briggs, H. (1619). Mirifici Logarithmorvm Canonis Constrvctio Et eorum ad naturales ipsorum numeros habitudines: una cum appendice, de aliâ eâque praestantior logarithmorum specie condenda quibus accerer. Edinburgi: Hart.Google Scholar
  13. O’Connor, J. J., & Robertson, E. F. (2006). London coffee houses and mathematics. MacTutor History of Mathematics archive, March 2006.
  14. Pitiscus, B. (1600). Bartholomaei Pitisci Grunbergensis Silesii Trigonometriae Sive De dimensione Triangulorum libri quinque: item problematum variorum … nempe … geodaeticorum, altimetricorum, geographicorum, gnomonicorum et astronomicorum libri decem: cum canone triangulorum. Augusta Vindelicorum: Mangerus.Google Scholar
  15. Ramus, P. (1569). Arithmeticae libri duo: geometiae septem et viginti. Basileae: per euserium episcopium et Nicolai fratris haeredes.Google Scholar
  16. Ramus, P., & Kempe, W. (1592). The Art of Arithmeticke in whole numbers and fractions … Written in Latin by P. Ramus: And translated into English by William Kempe. B.L. Richard Field for Robert Dextar dwelling at Pauls Church yard at the signe of the brasen serpent, London.Google Scholar
  17. Regiomontanus, J. & Hughes, B (Trans., Ed.). (1967). Regiomontanus: On triangles. De triangulis omnimodis. Madison: University of Wisconsin Press. ISBN: 0299042103 9780299042103.Google Scholar
  18. Rheticus, G. H., & Otho, V. (1596). Opus palatinum de triangulis a Georgio Joachimo Rhetico coeptum. L. Valentinus Otho, … consummavit. — Georgii Joachimi Rhetici libri tres de fabrica canonis doctrinae triangulorum. — Georgii Joachimi Rhetici de Triquetris rectarum linearum in planitie liber unus. Triquetrum rectarum linearum in planitie cum angulo recto magister est matheseos. - Georgii Joachimi Rhetici de Triangulis globi cum angulo recto. — L. Valentini Othonis, … de Triangulis globi sine angulo recto libri quinque quibus tria meteoroscopia numerorum accesserunt. — L. Valentini Othonis, … Meteoroscopium numerorum primum, monstrans proportionem singulorum parallelorum ad aequatorem vel meridianum. — Georgii Joachimi Rhaetici Magnus canon doctrinae triangulorum ad decades secundorum scrupulorum et ad partes 10 000 000 000, recens emendatus a Bartholomaeo Pitisco, … Addita est brevis commonefactio de fabrica et usu hujus canonis quae est summa doctrinae et quasi nucleus totius operis palatini … excud. M. Harnisius, Neostadii in Palatinatu.Google Scholar
  19. Stewart, L. (1999). Other centres of calculation, or, where the royal society didn’t count: Commerce, coffee-houses and natural philosophy in early modern London. The British Journal for the History of Science, 32(02), 133–153.CrossRefGoogle Scholar
  20. Taylor, K. (2011). Vernacular geometry: Between the senses and reason. BSHM Bulletin: Journal of the British Society for the History of Mathematics, 26(3), 147–159.
  21. Taylor, K. (2013). Reconstructing vernacular mathematics: The case of Thomas Hood’s sector. Early Science & Medicine, 18 (1 and 2), 153–179.Google Scholar
  22. Van Brummelen, G. (2013). Heavenly mathematics: The forgotten art of spherical trigonometry. Princeton: Princeton University Press. ISBN: 9780691148922 0691148929.Google Scholar
  23. Viète, F. (1579). Canon mathematicus seu ad triangula: Cum Adpendicibus. Paris: Mettayer.Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Emeritus of MathematicsRoger Williams UniversityBristolUSA
  2. 2.ProvidenceUSA

Personalised recommendations