The author wishes to thank the Wellcome Trust and the UIUC Campus Research Board for Travel Awards related to this research. Thanks also go to the faculty and staff of the Wellcome Trust Centre for the History of Medicine at UCL and of the Wellcome Library for their unfailing courtesy during the author's visit as a Research Associate, and to Professor Dan Grayson, UIUC Department of Mathematics, for his advice on mathematical questions.

To facilitate the identification of the various editions discussed in this paper, the numbers assigned to them in the following catalogues have been included: English Short-Title Catalogue: (ESTC); the Short-Title Catalogue of English Books printed in England, Scotland, and Ireland, and of English books printed abroad, 1475–1640: (STC); and the Short-Title Catalogue of Books printed in England, Scotland, Ireland, Wales and British America, and of English books printed in other countries, 1641–1700: (Wing).

¹ See, for example, E G R Taylor, The mathematical practitioners of Tudor and Stuart England, Cambridge, for the Institute of Navigation at the University Press, 1967, passim; J A Bennett, ‘The challenge of practical mathematics’, in Stephen Pumfrey, Paolo L Rossi and Maurice Slawinski (eds), Science, culture and popular belief in Renaissance Europe, Manchester and New York, Manchester University Press, 1991, pp. 176–90.

² Note that this is only one of many possible answers that will solve the problem correctly using alligation alternate.

³ Jonas Moore, Moores arithmetick, London, printed by Thomas Harper for Nathaniel Brookes, 1650 (Wing M2563). See note 42 below.

⁴ Ibid., p. 187.

⁵ Frances Willmoth, Sir Jonas Moore: practical mathematics and Restoration science, Woodbridge, Suffolk, Boydell, 1993, p. 73.

⁶ Andrew Wear, Knowledge and practice in English medicine, 1550–1680, Cambridge University Press, 2000, pp. 92–5.

⁷ Ibid., p. 93.

⁸ Ibid., p. 94, citing William Harrison, The description of England, ed. Georges Edelen, New York, Dover Publications, 1994, p. 266 (text based on Harrison's An historical description of the Island of Britain, London, 1587).

⁹ Wear, ibid., p. 95, citing James Primrose, Popular errours or the errours of the people in physick, trans. Robert Wittie, London, printed by W Wilson for Nicholas Bourne, 1651, pp. 24–5.

¹⁰ Wear, ibid., pp. 100–1.

¹¹ Nicholas Culpeper (tr.), A physical directory: or, Translation of the dispensatory made by the Colledge of Physitians of London, and by them imposed upon all the apothecaries of England to make up their medicines by whereunto is added, the vertues of the simples, and compounds, 2nd, much enlarged, ed., London, printed by Peter Cole, 1650, p. 56 (Wing C7541).

¹² Chapter on ‘The education of the apprentices’, in Cecil Wall, H Charles Cameron and E Ashworth Underwood, A history of the Worshipful Society of Apothecaries of London, vol. 1: 1617–1815, London, Published for the Wellcome Historical Medical Museum by the Oxford University Press, 1963, pp. 76–90, on p. 77. See also Juanita Burnby, ‘“An examined and free apothecary”’, in Vivian Nutton and Roy Porter (eds), The history of medical education in Britain, Clio Medica, 30, Amsterdam and Atlanta, Rodopi, 1995, pp. 16–36, p. 18: the apprentice “was taught how to dispense a physician's prescription, how to compound the pharmacopoeial preparations, and how to recognize the drugs, both compound and simple, then in use”.

¹³Tentamen medicinale: or, an enquiry into the differences between the dispensarians and apothecarys. Wherein the latter are prov'd capable of a skilful composition of medicines, and a rational practice of physick.… By an apothecary, London, printed, and sold by John Nutt, 1704, pp. 126–7 (ESTCt20981).

¹⁴ Mordechai Feingold, The mathematicians' apprenticeship: science, universities and society in England, 1560–1640, Cambridge University Press, 1984, pp. 24–5.

¹⁵ John Securis, A detection and querimonie of the daily enormities and abuses co[m]mitted in physick concernyng the thre parts therof: that is, the physitions part, the part of the surgeons, and the arte of poticaries, [Londini, In aedibus Thomae Marshi], 1566, A6v–A7r (STC 22143).

¹⁶ John Henry, ‘Doctors and healers: popular culture and the medical profession’, in Pumfrey, Rossi and Slawinski (eds), op. cit., note 1, above, pp. 191–221, on pp. 208–9. On the medieval theories of Arnald of Villanova and their context, see Michael R McVaugh, ‘The two faces of a medical career: Jordanus de Turre of Montpellier’, in Edward Grant and John E Murdoch (eds), Mathematics and its applications to science and natural philosophy in the Middle Ages: essays in honour of Marshall Clagett, Cambridge University Press, 1987, pp. 301–24 (especially pp. 303–6). John Dee cited Arnald (and other predecessors) in his Mathematicall præface to Euclid (1570), see note 30 below.

¹⁷ Feingold, op. cit., note 14 above, p. 116.

¹⁸ Robert Record, The grou[n]d of artes: teachyng the worke and practise of arithmetike, moch necessary for all states of men. After a more easyer [et] exacter sorte, then any lyke hath hytherto ben set forth: with dyuers newe additions, London, R Wolfe, 1543 (STC 20797.5).

¹⁹ Francis R Johnson and Sanford V Larkey, ‘Robert Recorde's mathematical teaching and the Anti-Aristotelian movement’, Huntington Library Bulletin, 1935, 7: 59–87, p. 62.

²⁰ Robert Record, The ground of artes teachyng the worke and practise of arithmetike, Imprinted at London, by Reynold Wolff, 1552, p. 7 (A4r) (STC 20799.3). The first edition was dedicated to “Rychard Whalleye Esquyer”; later, when a new dedication was addressed to the young King Edward, the dedication to Whalleye was rewritten as a preface to the reader.

²¹ Ibid., U6r.

²² Ibid., U6r–v, U7r, X1r, X3v ff., X2v.

²³ Ibid., X2v–X3r, X6r.

²⁴ Robert Record, The whetstone of witte whiche is the seconde parte of Arithmetike: containyng thextraction of rootes: the cossike practise, with the rule of equation: and the woorkes of surde nombers, Imprinted at London, by Ihon Kyngstone, 1557 (STC 20820).

²⁵ Ibid., A1; sig. b2.

²⁶ Nicholas H Clulee, John Dee's natural philosophy: between science and religion, London and New York, Routledge, 1988, pp. 85–6. Joy Easton enumerated most of the minor changes made by Dee, but does not mention the quoted paragraphs: Joy B Easton, ‘The early editions of Robert Recorde's Ground of artes’, ISIS, 1967, 58: 515–32, and particularly pp. 529–32.

²⁷ Robert Record, The grounde of artes: teaching the worke and practise of arithmetike, both in whole numbres and fractions, after a more easyer and exacter sorte then any like hath hitherto been sette forth: made by M. Robert Recorde doctor of physik, and now of late ouerseen & augmented with new & necessarie additions, Imprinted at London, by Reginalde VVolfe, 1561, Z1v (STC 20800).

²⁸ Robert Record, The grounde of artes teaching the perfecte vvorke and practise of arithmetike, … augmented by M. Iohn Dee. And now lately diligently corrected, [and] beautified with some new rules and necessarie additions: and further endowed with a thirde part, of rules of practize, abridged into a briefer methode than hitherto hath bene published: with diverse such necessary rules, as are incident to the trade of merchandize. Whereunto are also added diuers tables [and] instructions … By Iohn Mellis of Southwark, scholemaster, [London], imprinted by I Harrison, and H Bynneman, 1582 (STC 20802).

²⁹ Dee added a poem at the very end of the 1582 volume, “To the earnest Arithmetician” (2Y6v), which advised the student to study Euclid. Although Dee may seem simply to be advertising his recent edition of Euclid, in doing so he was also being true to Record, who had said that arithmetic was a prerequisite for the more advanced study of geometry (and after that, astronomy), and who had written his own texts to teach those advanced arts.

³⁰ Euclid, The elements of geometrie of the most auncient philosopher Euclide of Megara. Faithfully (now first) translated into the Englishe toung, by H. Billingsley … With a very fruitfull præface made by M. I. Dee, specifying the chiefe mathematicall scie[n]ces, what they are, and wherunto commodious: where, also, are disclosed certaine new secrets mathematicall and mechanicall, vntill these our daies, greatly missed, Imprinted at London, by Iohn Daye, 1570 (STC 10560). Most widely available in a facsimile edition: John Dee, The mathematicall præface to the elements of geometrie of Euclid of Megara (1570), with an introduction by Allen G Debus, New York, Science History Publications, 1975.

³¹ There are two main branches, “Principall” and “Derivative”: of the Principall kinds are “Arithmetike” and “Geometrie”, and these are each subdivided into “Simple” and “Mixt” species. Derivative mathematical applications may be related to the Principalls under the terms of “Arithmetike vulgar” (such as “Arithmetike of Proportions” or “Arithmetike Circular”) and “Geometrie vulgar”, either relating to things at hand (as by “Mecometrie”, “Embadometrie” and “Stereometrie”) or at a distance (such as “Geodesie”, “Geographie”, or “Stratarithmetrie”); or they may have their own names, such as “Perspective”, “Astrologie”, “Navigation” and sixteen others, some quite arcane, such as “Archemastrie,—Which teacheth to bring to actuall experience sensible, all worthy conclusions, by all the Artes Mathematicall purposed: and by true Naturall philosophie, concluded”.

³² Dee, Præface, in Euclid, op. cit., note 30 above, *2v.

³³ “R.B.” is likely to refer to the controversial Roger Bacon, and possibly the manuscript was in Dee's possession. In Bacon's Radix mundi there is a chapter called ‘Of the differences of the medicine, and proportions used in projection’. See the translation prepared by William Salmon in his Medicina practica, or, Practical physick shewing the method of curing the most usual diseases happening to humane bodies…: to which is added, the philosophick works of Hermes Trismegistus, Kalid Persicus, Geber Arabs, Artesius Longaevus, Nicholas Flammel, Roger Bachon and George Ripley…: together with a singular comment upon the first book of Hermes, the most ancient of philosophers, London, printed for T Howkins, J Taylor, and J Harris, 1692 (Wing S434). (See Table of Contents and pp. 585–642.) Salmon, who will be discussed later in this article, says he translated the Radix mundi from a “manuscript out of the library of a learned man, and our particular friend, a Doctor of Physick, who set a great value upon it, and not undeservedly” (Preface B3v).

³⁴ Dee, Præface, in Euclid, op. cit., note 30 above, *3r.

³⁵ Ibid., *4v.

³⁶ Chapters on alligation were normally followed only by chapters on the “Rule of Falsehood” or of “False Positions”, which was carried out by supplying any reasonable figure to a problem, solving the problem using this figure, and then using the result as the basis for calculation using the “Golden Rule”.

³⁷ Humphrey Baker, The well spryng of sciences whiche teacheth the perfecte woorke and practise of arithmeticke, Imprinted at London, by Ihon Kyngston for Iames Rowbothum; 1568, leaves 148r–152r (STC 1210).

³⁸ Thomas Hylles, The arte of vulgar arithmeticke both in integers and fractions, … Newly collected, digested, and in some part deuised by a welwiller to the mathematicals, Imprinted at London, by Gabriel Simson, 1600 (STC 14040.7). The chapters on alligation are on ff. 192r–208r (2L4r–2N4r).

³⁹ Ibid., leaf 204r (2M8r).

⁴⁰ Edmund Wingate, Arithmetique made easie, in tvvo bookes. The former, of naturall arithmetique: containing a perfect method for the true knowledge and practice of arithmetique, according to the ancient vulgar way, without dependance vpon any other author for the grounds thereof. The other of artificiall arithmetique, discovering how to resolve all questions of arithmetique by addition and subtraction. Together with an appendix, resolving likewise by addition and subtraction all questions, that concerne equation of time, interest of money, and valuation of purchases, leases, annuities, and the like, London, printed [by Miles Flesher] for Phil. Stephens and Chr. Meredith, 1630 (STC 25849).

⁴¹ It appears that “Alligation Partiall” is nothing more than doing a reduced proportional calculation based on the differences enumerated during an alligation alternate procedure; while “Alligation Totall” involves an increased proportional calculation.

⁴² Jonas Moore, Moores arithmetick: discovering the secrets of that art in numbers and species. In two bookes: the first, teaching (by precept and example) the ordinary operations in numbers, whole and broken … the second, the great rule of algebra in species, resolving all arithmeticall questions by supposition: with a canon of the powers of numbers, London, printed by Thomas Harper for Nathaniel Brookes, 1650 (Wing M2563). While the imprint date is 1650, Moore's ‘Epistle to the Reader’ is dated 30 October 1649, and in it he says that “it is now two years since this peece was delivered to be printed” (A5v).

⁴³ Willmoth, op. cit., note 5, above, p.73.

⁴⁴ See ibid., pp. 42, 18.

⁴⁵ The second part, or book, of Moore's Arithmetick was devoted to Algebra in species, that is, to more difficult and theoretical mathematical laws and applications. The dedication on Aa2 reads, “To his much honoured Friend, Iohn Bathurst, Doctor of Physicke, One of the Fellowes of the Colledge of Physitians of London, a judicious Favourite of best meriting studies. For the encouragement of his eldest son, Christopher Bathurst, An early and hopefull proficient in the Arts Mathematicall, and all other Literature. The Author maketh this Second Book publique, and Dedicateth the Same” (Aa2).

⁴⁶ Elizabeth Land Furdell, The royal doctors, 1485–1714: medical personnel at the Tudor and Stuart courts, Rochester, NY, University of Rochester Press, 2001, p. 146.

⁴⁷ Moore, op. cit., note 42 above, p. 151.

⁴⁸ Ibid., p. 169.

⁴⁹ Ibid., pp. 171, 173, 174, 178, 181, 182, 186.

⁵⁰ Ibid., p. 170.

⁵¹ Ibid., p. 178. For translation of the Latin, see note 53 below.

⁵² Daniel Sennert, The institutions or fundamentals of the whole art, both of physick and chirurgery, divided in to five books, London, printed for Lodowick Lloyd, 1656 (Wing S2535).

⁵³ Daniel Sennert, Nine Books of physick and chirurgery, London, printed by J M for Lodowick Lloyd, 1658 (Wing S2537). The Latin passage cited by Moore is here translated: “for if the vertue of the medicine be weake, tis to be strengthned [sic] by mixture with more vehement, if any faculty be deficient, tis to be mixt” (p. 410).

⁵⁴ Ibid., pp. 412–13.

⁵⁵ Moore, op. cit., note 42 above, p. 169. Other medical writers besides Sennert exhibited similar designs. For example, Philip Barrough, The method of physick containing the causes, signes, and cures of inward diseases in mans body, … whereunto is added, the form and rule of making remedies and medicines, which our physicians commonly use at this day; with the proportion, quantity, and names of each medicine, London, printed by Abraham Miller, and are to be sold by John Blague and Samuel Howes, 1652 (Wing B921). Barrough enumerates the ingredients for a wide variety of medicines, but does not explain how calculations were to be done in producing compounds. It was left to the medical astrologer-practitioner, William Salmon, to outline the details of the procedures (see below).

⁵⁶ Jonas Moore, Moor's arithmetick, London, Printed by J G for Nath. Brook, 1660, A6r (Wing M2564).

⁵⁷ Edmund Wingate, Arithmetique made easie, or, A perfect methode for the true knowledge and practice of natural arithmetique according to the ancient vulgar way without dependence upon any other author for the grounds thereof, London, printed by J Flesher for Phil Stephens, 1650 (Wing S2997).

⁵⁸ Ibid., Preface, sig. A3v.

⁵⁹ Moore, ‘The Epistle to the Reader’, op. cit., note 42 above, A6.

⁶⁰ Kersey also supplied geometrical proofs for other operations, for example, “a Geometricall demonstration of the Rule of False, by two Positions” (Wingate, op. cit., note 57 above, p. 337).

⁶¹ Ibid., pp. 319, 323.

⁶² This according to a note in the British Library's copy, which also indicates that the sixth volume included “the first impression of Fermat's maxima and minima, tangents, etc.”

⁶³ Pierre Herigone, Tome second du Cours mathematique, contenant l'arithmetique practique: le calcul ecclesiastique: & l'algebre, tant vulgaire que specieuse, avec la methode de composer & faire les demonstrations par le retour ou repetition des vestiges de l'Analyse, A Paris, chez Simeon Piget, 1644, pp. 99–102.

⁶⁴ Kersey in Wingate, op. cit., note 57 above, p. 323.

⁶⁵ Ibid., pp. 325–6, 329, 333.

⁶⁶ Ibid., p. 336.

⁶⁷ George Shelley was responsible for the 11th to 17th editions, appearing from 1704 to 1740; James Dodson had a hand in preparing the 18th and 19th editions for the press in 1751 and 1760 respectively.

⁶⁸ Thomas Willsford, Willsfords arithmetick, naturall, and artificiall: or, decimalls. Containing the science of numbers.… Made compendious and facile for all ingenious capacities, viz: merchants, citizens, sea-men, accomptants, &c., London, printed by J G for Nath. Brooke, 1656 (Wing W2874).

⁶⁹ Ibid., p. 250; S5v.

⁷⁰ Ibid., p. 261; T3.

⁷¹ Robert Record, Records arithmetick: or, The ground of arts; teaching the perfect work and practice of arithmetick, London, printed by James Flesher, and are to be sold by Joseph Crawford, 1662 (Wing R646). See Willsford's postscript ‘To every young Arithmetician, or Practitioner in numbers, who shall peruse these Bookes’, pp. 534–6.

⁷² James Hodder, Hodder's arithmetick, London, 1661, 3rd ed. 1664.

⁷³ John Newton, The art of natural arithmetick, in whole numbers and fractions vulgar and decimal, in a plain and easie method suteable to the capacity of children, for whom it is chiefly intended. In which the multiplication and division of numbers of several denominations, and the rule of alligation are more fully explained, than in any treatise of this nature as yet extant in the English tongue, London, printed by E[van] T[yler] and R[alph] H[olt] and are to be sold by Rob Walton, 1671 (Wing N1051B). Full title information from ESTCr225478.

⁷⁴ Newton's arithmetic book was republished after his death as The compleat arithmetician: or, The whole art of arithmetick, vulgar and decimal in a plain and easie method, suitable to the meanest capacity, in which the multiplication and division of numbers of several denominations, and the rule of alligation are more fully explained than in any treatise of this nature, yet extant, London, printed for John Taylor and Christopher Browne, 1691 (Wing N1054). The quotation is from ‘The Epistle to the Reader’, A6v. At my request, Jennifer Schaffner, Reference Librarian at the William Andrews Clark Memorial Library, UCLA, kindly examined the text of the 1671 edition and found it to be identical to that quoted (private communication).

⁷⁵ William Salmon, Synopsis medicinae, or A compendium of astrological, Galenical, & chymical physick. Philosophically deduced from the principles of Hermes and Hippocrates. In three books. The first, laying down signs and rules how the disease may be known. The second, how to judge whether it be curable or not, or may end in life or death. The third, shewing the way of curing according to the precepts of Galen and Paracelsus, London, printed by W Godbid, for Richard Jones, 1671 (Wing S454).

⁷⁶ Ibid., pp. 497, 499, 501, 502.

⁷⁷ William Salmon, Seplasium. The compleat English physician; or, The druggist's shop opened. Explicating all the particulars of which medicines at this day are composed and made. Shewing their various names and natures, London, printed for Matthew Gilliflower and George Sawbridge, 1693 (Wing S452).

⁷⁸ Charles Webster, The great instauration: science, medicine and reform, 1626–1660, London, Duckworth, 1975, p. 265.

⁷⁹ See ibid., pp. 191 and 198–202.

⁸⁰ Webster's and Ward's pamphlets are reproduced with an extensive introduction in Allen G Debus, Science and education in the seventeenth century: The Webster-Ward debate, London, Macdonald, 1970. See also Taylor, op. cit, note 1 above, p. 97.

⁸¹ John Wilkins and Seth Ward, Vindiciae academiarum, Oxford, printed by Leonard Lichfield … for Thomas Robinson, 1654, pp. 35–6, 47; in Debus, ibid., pp. 229–30, 241.

⁸² Webster, op. cit., note 78, above, p. 307.

⁸³ See Sir George Clark, A history of the Royal College of Physicians of London, Oxford, Clarendon Press for the Royal College of Physicians, 1964, vol. 1, p. 308 and note 3.

⁸⁴ For a discussion of the debate and a list of the relevant pamphlets, see Harold J Cook, ‘Henry Stubbe and the virtuosi-physicians’, in Roger French and Andrew Wear (eds), The medical revolution of the seventeenth century, Cambridge University Press, 1989, pp. 246–71.

⁸⁵ Taylor, op. cit., note 1, above, p. 96.

⁸⁶ Benjamin Donn, Mathematical essays; being essays on vulgar and decimal arithmetick, London, printed for W Johnston, P Davey and B Law, 1758 (ESTCt96887).

⁸⁷ Ibid., p. xvii. Donn is quoting the preface of Thomas Morgan's Philosophical principles of medicine, in three parts (1725; enlarged ed., 1730). He then goes on to quote at length from such authors as Hermann Boerhaave (d. 1738), James Keill (d. 1719), and Richard Mead (d. 1754). Indeed, the majority of Donn's preface is little more than a pastiche of lengthy quotations from these and other authors.

⁸⁸ Ibid., p. 204.

⁸⁹ Ibid., p. 206.

⁹⁰ For example, Daniel Fenning, in The schoolmaster's most useful companion, and scholar's best instructor in the knowledge of arithmetic, London, printed for the author and sold by S Crowder, 1765 (ESTCn36277), has a brief section on the composition of medicines (pp. 132–4) which explains the Galenic qualities in bodies in an entirely conventional manner. Fenning dedicates his work to “the school-masters of Great Britain and Ireland, and to other teachers of youth in arithmetic” (p. iii). Set up as a dialog between a “Tutor or Master and his young Pupil or Scholar”, this work certainly does not address the adult audience of practitioners as did the arithmetic writers of the 1650s.