[1]Aubrey’s Brief Lives, ed. A. Clark, Vol. II, Oxford, 1898, p. 106.
[2]“To the English Gentrie, and all others studious of the Mathematicks, which shall bee Readers hereof. The just Apologie of Wil: Ovghtred, against the slaunderous insimulations of Richard Delamain, in a Pamphlet called Grammelogia, or the Mathematicall Ring, or Mirifica logarithmorum projectio circularis” [1633?], p. 8. Hereafter we shall refer to this pamphlet as the Apologeticall Epistle, this name appearing on the page-headings.
[3]Companion to the [British] Almanac of 1837, p. 28, in an article by Augustus De Morgan on “Notices of English Mathematical and Astronomical Writers between the Norman Conquest and the Year 1600.”
[4]New and General Biographical Dictionary (John Nichols), London, 1784, art. “Oughtred.”
[5]Rev. Owen Manning, History of Antiquities in Surrey, Vol. II, p. 132.
[6]Skeleton Collegii Regalis Cantab.: Or A Catalogue of All the Provosts, Fellows and Scholars, of the King’s College . . . . since the Foundation Thereof, Vol. II, “William Oughtred.”
[7]Aubrey, op. cit., Vol. II, p. 107.
[8]Rigaud, Correspondence of Scientific Men of the Seventeenth Century, Oxford, Vol. I, 1841, p. 5.
[9]Aubrey, op. cit., Vol. II, p. 110.
[10]Ibid., p. 111.
[11]Op. cit., Vol. II, p. 132.
[12]Mr. William Lilly’s History of His Life and Times, From the Year 1602 to 1681, London, 1715, p. 58.
[13]Rigaud, op. cit., Vol. I, p. 60.
[14]Aubrey, op. cit., Vol. II, p. 107.
[15]Rigaud, op. cit., Vol. I, p. 16.
[16]Owen Manning, op. cit., p. 132.
[17]New and General Biographical Dictionary (John Nichols), London, 1784, art. “Oughtred.”
[18]Op. cit., Vol. II, p. 110.
[19]Rev. Owen Manning, The History and Antiquities of Surrey, Vol. II, London, 1809, p. 132.
[20]Op. cit., Vol. II, 1898, p. 111.
[21]Budget of Paradoxes, London, 1872, p. 451; 2d ed., Chicago and London, 1915, Vol. II, p. 303.
[22]The full title of the Clavis of 1631 is as follows: Arithmeticae in numeris et speciebvs institvtio: Qvae tvm logisticae, tvm analyticae, atqve adeo totivs mathematicae, qvasi clavis est.—Ad nobilissimvm spectatissimumque invenem Dn. Gvilelmvm Howard, Ordinis qui dicitur, Balnei Equitem, honoratissimi Dn. Thomae, Comitis Arvndeliae & Svrriae, Comitis Mareschalli Angliae, &c filium.—Londini, Apud Thomam Harpervm. M.DC.XXXI. In all there appeared five Latin editions, the second in 1648 at London, the third in 1652 at Oxford, the fourth in 1667 at Oxford, the fifth in 1693 and 1698 at Oxford. There were two independent English editions: the first in 1647 at London, translated in greater part by Robert Wood of Lincoln College, Oxford, as is stated in the preface to the 1652 Latin edition; the second in 1694 and 1702 is a new translation, the preface being written and the book recommended by the astronomer Edmund Halley. The 1694 and 1702 impressions labored under the defect of many sense-disturbing errors due to careless reading of the proofs. All the editions of the Clavis, after the first edition, had one or more of the following tracts added on:
- Eq.=De Aequationum affectarvm resolvtione in numeris.
- Eu.=Elementi decimi Euclidis declaratio.
- So.=De Solidis regularibus, tractatus.
- An.=De Anatocismo, sive usura composita.
- Fa.=Regula falsae positionis.
- Ar.=Theorematum in libris Archimedis de Sphaera & cylindro declaratio.
- Ho.=Horologia scioterica in plano, geometricÈ delineandi modus.
The abbreviated titles given here are, of course, our own. The lists of tracts added to the Clavis mathematicae of 1631 in its later editions, given in the order in which the tracts appear in each edition, are as follows: Clavis of 1647, Eq., An., Fa., Ho.; Clavis of 1648, Eq., An., Fa., Eu., So.; Clavis of 1652, Eq., Eu., So., An., Fa., Ar., Ho.; Clavis of 1667, Eq., Eu., So., An., Fa., Ar., Ho.; Clavis of 1693 and 1698, Eq., Eu., So., An., Fa., Ar., Ho.; Clavis of 1694 and 1702, Eq.
The title-page of the Clavis was considerably modified after the first edition. Thus, the 1652 Latin edition has this title-page: Guilelmi Oughtred Aetonensis, quondam Collegii Regalis in Cantabrigia Socii, Clavis mathematicae denvo limata, sive potius fabricata. Cum aliis quibusdam ejusdem commentationibus, quae in sequenti pagina recensentur. Editio tertia auctior & emendatior. Oxoniae, Excudebat Leon. Lichfield, Veneunt apud Tho. Robinson. 1652.
[23]Rigaud, op. cit., Vol. II, p. 476.
[24]See, for instance, the Clavis mathematicae of 1652, where he expresses himself thus (p. 4): “Speciosa haec Arithmetica arti Analyticae (per quam ex sumptione quaesiti, tanquam noti, investigatur quaesitum) multo accommodatior est, quam illa numerosa.”
[25]Oughtred, The Key of the Mathematicks, London, 1647, p. 4.
[26]Clavis 1694, p. 19, and the Clavis of 1631, p. 8.
[27]See for instance, Oughtred’s Elementi decimi Euclidis declaratio, 1652, p. 1, where he uses A and E, and also a and e.
[28]See Christophori Clavii Bambergensis Operum mathematicorum, tomus secundus, Moguntiae, M.DC.XI, algebra, p. 39.
[29]Christophori Clavii operum mathematicorum Tomus Secundus, Moguntiae, M.DC.XI, Epitome arithmeticae, p. 36.
[30]See F. Cajori, “The Cross × as a Symbol of Multiplication,” in Nature, Vol. XCIV (1914), p. 363.
[31]See Elementi decimi Euclidis declaratio, 1652, p. 2.
[32]See Johannis Wallisii Operum mathematicorum pars prima, Oxonii, 1657, p. 247.
[33]Clavis of 1631, chap. xix, sec. 5, p. 50.
[34]We have noticed the representation of known quantities by consonants and the unknown by vowels in Wingate’s Arithmetick made easie, edited by John Kersey, London, 1650, algebra, p. 382; and in the second part, section 19, of Jonas Moore’s Arithmetick in two parts, London, 1660, Moore suggests as an alternative the use of z, y, x, etc., for the unknowns. The practice of representing unknowns by vowels did not spread widely in England.
[35]Philosophical Transactions, Vol. XIX, No. 231, London, p. 652.
[36]Ibid., Vol. XIX, p. 56.
[37]There are two title-pages to the edition of 1632. The first title-page is as follows: The Circles of Proportion and The Horizontall Instrument. Both invented, and the vses of both Written in Latine by Mr. W. O. Translated into English: and set forth for the publique benefit by William Forster. London. Printed for Elias Allen maker of these and all other mathematical Instruments, and are to be sold at his shop over against St. Clements church with out Temple-barr. 1632. T. Cecill Sculp. In 1633 there was added the following, with a separate title-page: An addition vnto the Vse of the Instrvment called the Circles of Proportion. . . . . London, 1633, this being followed by Oughtred’s To the English Gentrie etc. In the British Museum there is a copy of another impression of the Circles of Proportion, dated 1639, with the Addition vnto the Vse of the Instrument etc., bearing the original date, 1633, and with the epistle, To the English Gentrie, etc., inserted immediately after Forster’s dedication, instead of at the end of the volume.
[38]The complete title of the English edition is as follows: Trigonometrie, or, The manner of calculating the Sides and Angles of Triangles, by the Mathematical Canon, demonstrated. By William Oughtred Etonens. And published by Richard Stokes Fellow of Kings Colledge in Cambridge, and Arthur Haughton Gentleman. London, Printed by R. and W. Leybourn, for Thomas Johnson at the Golden Key in St. Pauls Church-yard. M.DC.LVII.
[39]Jer. Collier, The Great Historical, Geographical, Genealogical and Poetical Dictionary, Vol. II, London, 1701, art. “Oughtred.”
[40]Rigaud op. cit., Vol. I, p. 82.
[41]A. De Morgan, Budget of Paradoxes, London, 1872, p. 451; 2d ed., Chicago, 1915, Vol. II, p. 303.
[42]E. Gunter, Description and Use of the Sector, the Crosse-staffe and other Instruments, London, 1624, second book, p. 31.
[43]F. Cajori, “On the History of a Notation in Trigonometry,” Nature, Vol. XCIV, 1915, pp. 642, 643.
[44]A. von BraunmÜhl, Geschichte der Trigonometrie, 2. Teil, Leipzig, 1903, pp. 42, 91.
[45]H. Hankel, Geschichte der Mathematik in Alterthum und Mittelalter, Leipzig, 1874, pp. 369, 370.
[46]M. Cantor, Vorlesungen Über Geschichte der Mathematik, II, 1900, pp. 640, 641.
[47]This matter has been discussed in a paper by F. Cajori, “A History of the Arithmetical Methods of Approximation, etc., Colorado College Publication, General Series No. 51, 1910, pp. 182-84. Later this subject was again treated by G. EnestrÖm in Bibliotheca mathematica, 3. Folge, Vol. XI, 1911, pp. 234, 235.
[48]See F. Cajori, op. cit., p. 193.
[49]See William Oughtred’s Key of the Mathematicks, London, 1694, pp. 173-75, tract, “Of the Resolution of the Affected Equations,” or any edition of the Clavis after the first.
[50]A. De Morgan, op. cit., p. 451; 2d ed., Vol. II, p. 303.
[51]See F. Cajori, History of the Logarithmic Slide Rule, New York, 1909, pp. 7-14, Addenda, p. ii.
[52]Rigaud, op. cit., Vol. I, p. 12.
[53]The New Artificial Gauging Line or Rod: together with rules concerning the use thereof: Invented and written by William Oughtred, London, 1633.
[54]W. Oughtred, Apologeticall Epistle, p. 13.
[55]Quarterly Journal of Pure and Applied Mathematics, Vol. XLVI, (1915), p. 169. In this article Glaisher republishes the “Appendix” in full.
[56]Aubrey, op. cit., Vol. II, 1898, p. 108.
[57]Wood’s Athenae Oxonienses (ed. P. Bliss), Vol. IV, 1820, p. 247.
[58]Wood, op. cit., Vol. II, p. 445.
[59]Rigaud, op. cit., Vol. I, pp. 33, 35.
[60]Rigaud, op. cit., Vol. I, pp. 16, 26.
[61]Rigaud, op. cit., Vol. I, p. 66.
[62]Ibid., Vol. I, p. 9.
[63]Rigaud, op. cit., Vol. II, p. 475.
[64]Ibid., Vol. II, p. 471.
[65]J. W. L. Glaisher, “On Early Logarithmic Tables, and Their Calculators,” Philosophical Magazine, 4th Ser., Vol. XLV (1873), pp. 378, 379.
[66]Rigaud, op. cit., Vol. I, p. 65.
[67]Rigaud, op. cit., Vol. I, p. 87.
[68]King’s Life of John Locke, Vol. I, London, 1830, p. 227.
[69]Exercitationum Mathematicarum Decas prima, Naples, 1627, and probably Cataldus’ Transformatio Geometrica, Bonon., 1612.
[70]Rigaud, op. cit., Vol. II, pp. 477-80.
[71]Miscellanies: or Mathematical Lucubrations, of Mr. Samuel Foster, Sometimes publike Professor of Astronomie in Gresham Colledge in London, by John Twysden, London, 1659.
[72]The Works of the Honourable Robert Boyle in five volumes, to which is prefixed the Life of the Author, Vol. I, London, 1744, p. 24.
[73]The letter is printed in John Wallis’ De algebra tractatus, 1693, p. 206.
[74]See La Correspondance de Descartes, published by Charles Adam and Paul Tannery, Vol. II, Paris, 1898, pp. 456 and 457.
[75]H. Bosmans, S.J., “La premiÈre Édition de la Clavis Mathematica d’Oughtred. Son influence sur la GÉomÉtrie de Descartes,” Annales de la sociÉtÉ scientifique de Bruxelles, 35th year, 1910-11, Part II, pp. 24-78.
[76]Ibid., p. 78.
[77]Vincent Wing, Harmonicon coeleste, London, 1651, p. 5.
[78]Seth Ward, In Ismaelis Bullialdi astronomiae philolaicae fundamenta inquisitio brevis, Oxford, 1653, p. 7.
[79]John Wallis, Elenchus geometriae Hobbianae, Oxford, 1655, p. 48.
[80]An Idea of Arithmetick, at first designed for the use of the Free Schoole at Thurlow in Suffolk. . . . . By R. B., Schoolmaster there, London, 1655, p. 6.
[81]The Miscellanies: or Mathematical Lucubrations, of Mr. Samuel Foster . . . . by John Twysden, London, 1659, p. 1.
[82]Moor’s Arithmetick in two Books, London, 1660, p. 89.
[83]Isaac Barrow, Euclidis data, Cambridge, 1657, p. 2.
[84]Francisci Dulaurens Specima mathematica, Paris, 1667, p. 1.
[85]ElÉmens des mathÉmatiques, Paris, 1675, Preface signed “J. P.”
[86]Nouveaux ÉlÉmens de gÉomÉtrie, Paris, 1692 (permission to print 1684).
[87]Ozanam, Dictionnaire mathÉmatique, Paris, 1691, p. 12.
[88]Analyse des infiniment petits, Paris, 1696, p. 11.
[89]Petro Nicolas, De conchoidibus et cissoidibus exercitationes geometricae, Toulouse, 1697, p. 17.
[90]R. P. Bernard Lamy, ElÉmens des mathÉmatiques, Amsterdam, 1692 (permission to print 1680).
[91]Nouveaux ÉlÉmens de gÉomÉtrie, 2d ed., The Hague, 1690, p. 304.
[92]W. W. Beman in L’intermÉdiaire des mathÉmaticiens, Paris, Vol. IX, 1902, p. 229, question 2424.
[93]John Collins, The Mariner’s Plain Scale New Plain’d, London, 1659, p. 25.
[94]James Gregory, Optica promota, London, 1663, pp. 19, 48.
[95]Philosophical Transactions, Vol. III, London, p. 868.
[96]William Leybourn, The Line of Proportion, London, 1673, p. 14.
[97]Elementa geometriae . . . . a Gulielmo Sanders, Glasgow, 1686, p. 3.
[98]Cocker’s Decimal Arithmetick, . . . . perused by John Hawkins, London, 1695 (preface dated 1684), p. 41.
[99]Joseph Raphson, Analysis Aequationum universalis, London, 1697, p. 26.
[100]E. Wells, Elementa arithmeticae numerosae et speciosae, Oxford, 1698, p. 107.
[101]John Ward, A Compendium of Algebra, 2d ed., London, 1698, p. 62.
[102]Plain Elements of Geometry and Plain Trigonometry, London, 1701, p. 63.
[103]George Shelley, Wingate’s Arithmetick, London, 1704, p. 343.
[104]A Synopsis of Algebra, Being a posthumous work of John Alexander of Bern, Swisserland. . . . . Done from the Latin by Sam. Cobb, London, 1709, p. 16.
[105]John Craig, De Calculo fluentium, London, 1718, p. 35. The notation A:B::C:D is given also.
[106]Trigonometry, 2d ed., Edinburgh, 1724, p. 11.
[107]MÉthode pour la mÉsure des surfaces, la dimension des solides . . . . par M. CarrÉ de l’acadÉmie r. des sciences, 1700, p. 59.
[108]Application de l’algÈbre À gÉomÉtrie . . . . Paris, 1705.
[109]ElÉmens de la gÉomÉtrie de l’infini, by M. de Fontenelle, Paris, 1727, p. 110.
[110]Eclaircissemens sur l’analyse des infiniment petits, by M. Varignon, Paris, 1725, p. 87.
[111]Application de la gÉomÉtrie ordinaire et des calculs diffÉrentiel et intÉgral, by M. Robillard, Paris, 1753.
[112]TraitÉ de gÉomÉtrie thÉorique et pratique, new ed., Paris, 1764, p. 15.
[113]Recherches sur les courbes À double courbure, Paris, 1731, p. 13.
[114]Analyse des infiniment petits, by the Marquis de L’Hospital. New ed. by M. Le FÈvre, Paris, 1781, p. 41. In this volume passages in fine print, probably supplied by the editor, contain the notation a:b::c:d; the parts in large type give Oughtred’s original notation.
[115]The tendency during the eighteenth century is shown in part by the following data: Jacobi Bernoulli Opera, Tomus primus, Geneva, 1744, gives B.A::D.C on p. 368, the paper having been first published in 1688; on p. 419 is given GE:AG=LA:ML, the paper having been first published in 1689. Bernhardi Nieuwentiit, Considerationes circa analyseos ad quantitates infinitÈ parvas applicatae principia, Amsterdam, 1694, p. 20, and Analysis infinitorum, Amsterdam, 1695, on p. 276, have x:c::s:r. Paul Halcken’s Deliciae mathematicae, Hamburg, 1719, gives a:b::c:d. Johannis Baptistae Caraccioli, Geometria algebraica universa, Rome, 1759, p. 79, has a.b::c.d. Delle corde ouverto fibre elastiche schediasmi fisico-matematici del conte Giordano Riccati, Bologna, 1767, p. 65, gives P:b::r:ds. “Produzioni mathematiche” del Conte Giulio Carlo de Fagnano, Vol. I, Pesario, 1750, p. 193, has a.b::c.d. L. Mascheroni, GÉomÉtrie du compas, translated by A. M. Carette, Paris, 1798, p. 188, gives v3:2::v2:Lp. Danielis Melandri and Paulli Frisi, De theoria lunae commentarii, Parma, 1769, p. 13, has a:b::c:d. Vicentio Riccato and Hieronymo Saladino, Institutiones analyticae, Vol. I, Bologna, 1765, p. 47, gives x:a::m:n+m. R. G. Boscovich, Opera pertinentia ad opticam et astronomiam, Bassani, 1785, p. 409, uses a:b::c:d. Jacob Bernoulli, Ars Conjectandi, Basel, 1713, has n-r.n-1::c.d. Pavlini Chelvicii, Institutiones analyticae, editio post tertiam Romanam prima in Germania, Vienna, 1761, p. 2, a.b::c.d. Christiani Wolfii, Elementa matheseos universae, Vol. III, Geneva, 1735, p. 63, has AB:AE=1:q. Johann Bernoulli, Opera omnia, Vol. I, Lausanne and Geneva, 1742, p. 43, has a:b=c:d. D. C. Walmesley, Analyse des mesures des rapports et des angles, Paris, 1749, uses extensively a.b::c.d, later a:b::c:d. G. W. Krafft, Institutiones geometriae sublimoris, TÜbingen, 1753, p. 194, has a:b=c:d. J. H. Lambert, Photometria, 1760, p. 104, has C:p=BC²:MH². Meccanica sublime del Dott. Domenico Bartaloni, Naples, 1765, has a:b::c:d. Occasionally ratio is not designated by a.b, nor by a:b, but by a, b, as for instance in A. de Moivre’s Doctrine of Chance, London, 1756, p. 34, where he writes a, b::1, q. A further variation in the designation of ratio is found in James Atkinson’s Epitome of the Art of Navigation, London, 1718, p. 24, namely, 3..2::72..48. Curious notations are given in Rich. Balam’s Algebra, London, 1653.
[116]Chr. Clavii Operum mathematicorum tomus secundus, Mayence, 1611, Algebra, p. 39.
[117]Invention nouvelle en l’algÈbre, by Albert Girard, Amsterdam, 1629, p. 17.
[118]La gÉomÉtrie et pratique gÉnÉrale d’icelle, par I. Errard de Bar-le-Duc, IngÉnieur ordinaire de sa MajestÉ, 3d ed., revised by D. H. P. E. M., Paris, 1619, p. 216.
[119]Novae geometriae clavis algebra, authore P. Jacobo de Billy, Paris, 1643, p. 157; also an Abridgement of the Precepts of Algebra. Written in French by James de Billy, London, 1659, p. 346.
[120]Miscellanies: or Mathematical Lucubrations, of Mr. Samuel Foster, Sometime publike Professor of Astronomie in Gresham Colledge in London, London, 1659, p. 7.
[121]Quarterly Jour. of Pure and Applied Math., Vol. XLVI (London, 1915), p. 191.
[122]Pietro Cossali, Origine, trasporto in Italia primi progressi in essa dell’ algebra, Vol. I, Parmense, 1797, p. 52.
[123]In Is. Bullialdi astronomiae philolaicae fundamenta inquisitio brevis, Auctore Setho Wardo, Oxford, 1653, p. 1.
[124]John Wallis, Algebra, London, 1685, p. 321, and in some of his other works. He makes greater use of Harriot’s symbols.
[125]Euclidis data, 1657, p. 1; also Euclidis elementorum libris XV, London, 1659, p. 1.
[126]John Kersey, Algebra, London, 1673, p. 321.
[127]E. Wells, Elementa arithmeticae numerosae et speciosae, Oxford, 1698, p. 142.
[128]Cocker’s Decimal Arithmetick, perused by John Hawkins, London, 1695 (preface dated 1684), p. 278.
[129]Th. Baker, The Geometrical Key, London, 1684, p. 15.
[130]Richard Sault, A New Treatise of Algebra, London (no date).
[131]Richard Rawlinson in a pamphlet without date, issued sometime between 1655 and 1668, containing trigonometric formulas. There is a copy in the British Museum.
[132]F. Dulaurens, Specima mathematica, Paris, 1667, p. 1.
[133]J. Milnes, Sectionum conicarum elementa, Oxford, 1702, p. 42.
[134]Cheyne, Philosophical Principles of Natural Religion, London, 1705, p. 55.
[135]J. Craig, De calculo fluentium, London, 1718, p. 86.
[136]Jo. Wilson, Trigonometry, 2d ed., Edinburgh, 1724, p. v.
[137]Commercium Epistolicum, 1712, p. 20.
[138]C. Le Paige, “Sur l’origine de certains signes d’opÉration,” Annales de la sociÉtÉ scientifique de Bruxelles, 16th year, 1891-92, Part II, pp. 79-82.
[139]Gravelaar, “Over den oorsprong van ons maalteeken (×),” Wiskundig Tijdschrift, 6th year. We have not had access to this article.
[140]H. Bosmans, op. cit., p. 40.
[141]Claudii Ptolemaei . . . . annotationes, BÂle, 1551. This reference is taken from the EncyclopÉdie des sciences mathÉmatiques, Tome I, Vol. I, Fasc. 1, p. 40.
[142]Due Correction for Mr. Hobbes. Or Schoole Discipline, for not saying his Lessons right. In answer to his Six Lessons, directed to the Professors of Mathematicks. By the Professor of Geometry. Oxford, 1656, pp. 7, 47, 50.
[143]Oughtred, Apologeticall Epistle, p. 27.
[144]J. Edleston, Correspondence of Sir Isaac Newton and Professor Cotes, London, 1850, pp. 279-92.
