The ordered regiment of the minus sign: Off-beat mathematics in Harriot's manuscripts

References

• B.L.Add.Mss. 6788 f. 468 – f. 468 .
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• Notably in Rafaele Bombelli (1526–1572) L'Algebra. Parte maggiore dell'arithmetica divisa in tre libri Bologna 1572 77 77 Also see Luca Paciuolo (1445–1514), Summa de arithmetica … (Venice 1494; 1523), f. 112 v, 113; this work ‘contained the mathematical knowledge of the time’ and set the Italian tradition of writing for a wider public on such subjects. Modern views are summarised by S. A. Jayawardene. ‘The influence of practical arithmetics on the Algebra of Rafael Bombelli’, Isis 64 (1973), 510–523 (esp. p. 512); compare M. Cantor, Vorlesungen über Geschichte der Mathematik, vol. 2 (2nd ed. Leipzig, 1913), ch. 57, ‘Luca Paciuolo’, 316–344, esp. p. 319.
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• Cajori . 1917 . A history of elementary mathematics 244 – 244 . New York lists the first serious investigations into fundamentals of algebra to include this question
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• B.L.Add.Mss. 6783 45 – 45 . 6784, ff. 263, 293; 6785, 384 v, 386, 388 v; 6786, ff. 217, 461, 480; 6787, f. 4; 6789, ff. 425–438 v. Further, in the Harriot collection at Petworth House: Leconfield 241/3, ff. 79–87, where f.82 headed ‘8.) Apendicula 2da.’ is a virtual postscript to the well-known signed letter of 4 March 1611 (B.L.Add.Ms. 6789, ff. 429–430, printed in J. O. Halliwell's Collection of letters illustrative of the progress of science … (London, 1841), 48) and shows the typical numeral shapes of Lower's hand particularly well.
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• The dictionary of national biography 1893 34 mentions Lower ‘of Treventy (d. 1615)’ only in the article on his cousin the dramatist of the same name (1600?–1662) on p. 206: Information from all sources was being collected by S. P. Rigaud (1774–1839) still in 1835 (Bodleian Rigaud Ms.35, ff. 314 ff) additional to his Note I (pp. 68–70) in his Account of Harriot's astronomical papers, the Supplement to Bradley's miscellaneous works (Oxford, 1833: reprinted London and New York, 1972).
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• See Thomas Harriot, Renaissance scientist Shirley J.W. Oxford 1974 for many mentions of Lower listed in the index, and associated bibliography.
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• Yates , Frances . 1936 . A study of ‘Love's labour's lost’ 93 – 94 . Cambridge gives extracts of Lower's letter of ‘the longest day of 1610’ showing all these characteristics in reference to astronomical matters. An even greater paradox is touched on, that of Giordano Bruno (1548–1600), self-styled Nolanus: ‘that the eye being placed in anie parte of the univers the appearence would be still all one as to us here’. (From B.L.Add.Ms.6789, ff. 425 r and v, 426; reproduced in full in Rigaud (footnote 6), 25–26, and in Henry Stevens, Thomas Harriot and his associates (London, 1900). 116–118.)
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• B.L.Add.Ms. 6785 ff. 386 v – ff. 386 v . 387, 388
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• 1489 . Johann Widmann's German Arithmetic Leipzig discussed at length by Cantor (footnote 2), 228–251), gave the signs in print which are found in some earlier manuscripts, and first only the minus sign, this in another German text (Dresden Codex 80). For a report on these sources, see J. W. L. Glaisher, ‘On the early history of the signs + and -’, Messenger of mathematics, 51 (1921–1922) (the whole number)
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• The statement is true for all works from 1550 to 1650 listed by Augustus de Morgan (1806–1871) in his Arithmetical books from the invention of printing to the present time London 1847
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• Viète , Francois . 1591 . In artem analyticen Tours 1540–1603 Praeceptum III: ‘quod fit ex adfirmata A in G negatam manet negatum, quod est nimium negare minuereve’, concluding with: ‘in compensationem dum B negata ducitur in G negatam factum est adfirmatum’. See Opera mathematica (ed Fr.à Schooten: Leyden, 1646; repr. Hildesheim and New York, 1970), 5–6.
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• See B.L.Add.Ms. 6785 386 v 386 v Harriot's change to small letters is habitually mentioned as bringing him much closer to modern usage; but this is somewhat accidental. Harriot tended to write all letters small.
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• Viète initiated this at the end of his Supplementum geometricae, ex opere restituae mathematicae analyseos seu Algebra nova Tours 1593 Opera (footnote 13), 240–257. The notation was finally established in a work there referred to, but published only posthumously by A. Anderson (1582–1620?), Viète's De Recognitione et emendatione aequationum (Paris, 1615); Opera (footnote 13), 84–158. For example, Opera, p. 88: ‘A cubus + B in A quad. aequetur B in D quad.’ stands for the general equation which we should write x 3 + bx 2 = c. However, small-print numerical examples revert to the earlier N, Q, C etc. for our x, x 2, x 3 etc.
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• ‘Harriot is the first who occasionally places a negative term by itself on one side of an equation Cajori A history of elementary mathematics New York 1917 232 232
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• ‘Stifel erfasst das Wesen der negativen Zahlen vollständig’ Hofmann J.E. Geschichte der Mathematik I. Von den Anfängen bis zum Auftreten von Descartes , 2nd ed. Berlin 1963 131 131 commenting on the Arithmetica integra of Michael Stifel (1487–1567)
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• ‘Von negativen Gleichungswurzeln will Harriot nichts wissen’ Cantor L'Algebra. Parte maggiore dell'arithmetica divisa in tre libri Bologna 1572 792 792 The general verdict until quite recently was based exclusively on Harriot's Artis analyticae praxis (posth., London, 1631), the only portion of his algebra printed and all that was known to the commentators concerned.
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• Rigaud . 1893 . Dictionary of national biography , 34 : 52 – 52 . and Plate 5. The original is Petworth Ms. 241/1, f. 14.
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• Sion College, Torporley Ms.Arc L 40.2/L.40 35r – 55v . Miscellanea A part of the Harriot originals is listed in its proper sequence in my Shirley (footnote 7), 106, footnote 48.
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• B.L.Add.Ms. 6784 f. 401 v – f. 401 v .
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• B.L.Add.Ms. 6785 f. 383 – f. 383 .
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• Recorde , R. 1557 . The Whetstone of Witte London Stifel (footnote 17). See Cajori, A history of mathematical notations (2 vols, La Salle, Ill., 1928), vol. 1, 165. Compare Charles Hutton (1737–1823), Tracts on mathematical and philosophical subjects (3 vols., London, 1812), vol. 2, Tract 33, ‘History of algebra’, p. 238.
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• B.L.Add.Ms. 6782 f. 277 – f. 277 . Christoph Clavius (1537–1612) introduced Stifel's notation in Italy (Cajori (footnote 23), 151). He was a German Jesuit who became Cardinal in Rome. His Opera mathematica appeared in five volumes (Mainz, 1611–1612). Harriot refers to these elsewhere (B.L.Add.Ms. 6784, f. 39) and also to the earlier publication of his Geometrica practica (Rome, 1604). Here his Algebra (Rome, 1608) was more nearly implicated.
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• B.L.Add.Ms. 6787 f. 242 – f. 242 . r and v
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• B.L.Add.Ms. 6787, f. 241 v. For the cypher, see Seaton Ethel Thomas Harriot's secret script Ambix 1953–1956 5 111 114 It was the first attempt to decode the cypher in Sion College Torporley manuscripts, and probably requires revision and emendation in places.
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• B.L.Add.Ms. 6787 f. 245 – f. 245 . The title is found again B.L.Add.Ms. 6789, f. 450, in the second list of ‘Papers of Mr Harriots delivered to Mr Torperley’, the only item without indication of numbers of pages or ‘a bundell’. It might have been just this sheet, and ff. 247, 248 which are on the subject. See my ‘Nathaniel Torporley and the Harriot manuscripts’, Annals of science, 25 (1969), 340–349 (B16 on p. 349).
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• The principle here followed by Harriot was first put into practice by Simon Stevin (1548–1620) for the location and step by step approximate evaluation of a root in his Arithmétique Leyde 1585 This is pointed out by Cantor (footnote 2), 628–629, citing Vol. 1 of Stevin's collected works edited in 1634 by Albert Girard (1595–1632). The Arithmétique is now available in reprographic edition by D.J. Struik, with his notes in English, in The principal works of Simon Stevin (ed. E. Crone, E. J. Dijksterhuis, R. J. Forbes, M. G. J. Minnaert and A. Pannekoek), vol. 2B (Amsterdam, 1958).
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• Harriot . 1631 . “ Artis analyticae praxis ” . In Exegetice numerosa 117 – 167 . London posth Francisci Vietae De numerosa potestatum purarum et adfectarum ad exegesin resolutione tractatus (ed. M. Ghetaldi: Paris, 1600); Opera (footnote 13), 162–228.
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• B.L.Add.Ms. 6785 f. 397 v – f. 397 v .
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• B.L.Add.Ms. 6785 ff. 390 – 400 . (f. 392 v)
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• B.L.Add.Ms. 6785 ff. 140 – 142 . where the relevant resolution of a quadratic equation is ‘according to the auncient manner as in Salignacus or Stevin &c.’ (f. 142). This is one of the rare batches of Harriot papers with verbal text potentially important for dating, particularly as it has Harriot's literal notation for equations combined with Recorde's equality sign instead of his own (see Section 7). Bernard Salignacus, a little-known pupil of Pierre de la Ramée (Petrus Ramus, 1515–1572), is named also in the title of the diagram heading this batch (f. 239); ‘De Mesolabio B. Salignaci’
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• B.L.Add.Ms. 6782 ff. 310 – 312 . Stevin (footnote 28), 331 is Harriot's reference here.
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• B.L.Add.Ms. 6783 f. 121 – f. 121 .
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• B.L.Add.Ms. 4396 f. 90 – f. 90 . r and v. The reference is to Sir Thomas Aylesbury (1576–1657) (see Stevens (footnote 8), 189–190). See also in Shirley (footnote 7) my ‘Henry Stevens and the associates of Thomas Harriot’, 91–106 (pp. 98, 99); and Jean Jacquot, ‘Harriot, Hill, Warner and the new philosophy’, 107–128 (where, however, the description on p. 108 of Warner as ‘ten years younger than Harriot’ is a slip, and the idea (p. 109) that he was ‘his disciple’ altogether unjustified).
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• See Shirley J.W. The scientific experiments of Sir Walter Ralegh, the Wizard Earl, and the three Magi in the Tower, 1603–1617 Ambix 1949–1951 4 52 66 (esp. pp. 56–57). See also my ‘The study of Thomas Harriot's manuscripts. 1. Harriot's will’, History of science, 6 (1967), 1–16 (esp. p. 14, footnote 37)
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• See Pepper J.V. A letter from Nathaniel Torporley to Thomas Harriot Brit. j. hist. sci. 1967 3 285 290 esp. p. 288, for the main evidential references. We might add Harriot's heading for a trigonometrical theorem in B.L.Add.Ms. 6782, f. 483: ‘A proposition of Vietas delivered by m Thorperly, but no demonstration’
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• Hofman . 1963 . Geschichte der Mathematik I. Von den Anfängen bis zum Auftreten von Descartes , 2nd ed. 137 – 138 . Berlin in reference to the Latin edition: Euclidis Elementorum libr. xv. Una cum scholijs antiquis. A Federico Commandino Urbinate nuper in latinum conversi, commentariisque quibusdam illustrati (Pisa, 1572). It is mentioned cursorily by Cantor (footnote 2), 553, who comments fully on that of Clavius (footnote 24) two years later in 1574, which saw at least four new editions during, his life and many more later
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• Heath , T.L. 1908 . The thirteen books of Euclid's Elements , 2nd ed. Cambridge Cambridge, 1925: repr. New York 1956), Introduction, 104–105.
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• 1575 . De gli Elementi d'Euclidi libri quindici. Con gli scholii antichi. Tradotti prima in lingua latina da M. F. Commandino … & con commentarij illustrati, et hora dell'istesso trasportati nella nostra vulgare, & da lui riveduti Urbino The relevant passage occurs on f.162, and was unchanged in the second edition (Pisa, 1619).
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• Commandinus . 1572 . Euclidis Elementorum libr. xv. Una cum scholijs antiquis. A Federico Commandino Urbinate nuper in latinum conversi, commentariisque quibusdam illustrati f. 149 – f. 149 . Pisa ‘qui vero ex quarta propositione secundi id quadratum sibi inveniendum proponunt, coguntur dicere si minus per minus multiplicetur produci plus. quod verum non esse primus animadvertit Hieronymus Cardanus non solum mathematicus, sed et Philosophus, ac medicus Prestantissimus, ut apparet in libro de regula aliza, quem nuper edidit. Verum quoniam ex eorum operatione error non sequitur, hoc ipsis condonandum est’
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• Commandinus . 1572 . Euclidis Elementorum libr. xv. Una cum scholijs antiquis. A Federico Commandino Urbinate nuper in latinum conversi, commentariisque quibusdam illustrati f. 149v – f. 149v . Pisa ‘ac recentiores ad hoc inveniendum utuntur 1. theoremate; et ob id asserunt si minus per minus multiplicetur, produci plus, sed non recte, cum utendum sit theoremate secundo; neque enim 8, et ℞12 sunt partes unius rectae lineae; immo vero 8 est tota linea, et eius pars ℞12, et similiter dicendum de 4 minus ℞3. ex ipsorum tamen operatione nullus sequitur error’.
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• B.L.Add.Ms. 6785 ff. 153 – 156 . in particular, f, 156
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• Cantor , M. 1907 . Geschichte der Mathematik , 3rd ed. Vol. 1 , 263 – 263 . Leipzig
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• B.L.Add.Ms. 6784 f. 321 v – f. 321 v .
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• Torporley . Ms.Arc L 40.2/L.40 (Miscellanea) ff. 35r – 55v . in particular, f.35v
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• Harriot . 1631 . Artis analytivae praxis 7 – 11 . London posth The original is slightly re-arranged, separating addition and subtraction, which Harriot treated together on f. 322. There are a few other alterations (and some misprints).
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• The version in Rukeyser Muriel The traces of Thomas Harriot New York 1970 151 151 is not entirely accurate
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• B.L.Add.Ms. 6785 f. 384 v – f. 384 v .
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• The date is in blacker ink, and the faintly visible 8 looks as if sanded out with 1 substituted. Rukeyser The traces of Thomas Harriot New York 1970 151 151 gives the year as 1598 without comment. This question will be taken up in section 7
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• F. 383 ff. 383 – ff. 383 . v and 384 r are blank
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• B.L.Add.Ms. 6785 f. 392v – f. 392v . see text to footnote 31
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• B.L.Add.Ms. 6782 ff. 362 – 374 v . In ‘Thomas Harriot's reputation for impiey’, Notes and records of the Royal Society, 9 (1952), 164–187, Jean Jacquot attempts a description (pp. 177–179), summarised in his (footnote 36). 115.
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• Sion College Torporley Ms. Arc L 40.2/L.40, ff. 1–34 (f. 33v). See my Nathaniel Torporley's “Congestor analyticus” and Thomas Harriot's “De triangulis laterum rationalium” Annals of science 1977 34 393 428 (p. 425)
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• Dawson , G.E. and Kennedy-Skipton , L. 1968 . Elizabethan handwriting 1500–1650 London Preface, vii. quoting from Muriel St. Clare Byrne, ‘Elizabethan handwriting for beginners’, Review of English studies, 1 (1925), 198–209. Details of palaeographical history and terminology in the sequel are drawn mainly from the above work.
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• See Johnson Charles Jenkinson Hilary English court hand A.D. 1066 to 1500 Oxford 1915 14 15
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• See Johnson Charles Jenkinson Hilary English court hand A.D. 1066 to 1500 Oxford 1915 15 15 example No. 7
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• In Dawson Elizabethan handwriting 1500–1650 London 1968 Plate 32, is almost the only example there showing an epsilon e (in Dorset's signature); the date is 1607.
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• B.L.Add.Ms. 6786 ff. 375 v – 376 .
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• B.L.Add.Ms. 6788 f. 417 v – f. 417 v .
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• The secretary e finally disappeared from Harriot's writing after he had changed to the epsilon form of e, as may be seen in his record of Ralegh's speech from the scaffold on 23 October 1618 (o.s.) B.L.Add.Ms. 6789 f. 533 f. 533 reproduced in facsimile as Plate 1 in Shirley (footnote 7))
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• Seaton . 1953–1956 . Thomas Harriot's secret script . Ambix , 5 : 111 – 114 . figure 3; from B.L.Add.Ms. 6789, f. 266. Miss Seaton misread the year as 1602. The same date is given without ambiguity on f. 268
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• Andrews , J.H. 1978 . Irish maps , The Irish Heritage Series, 18 Dublin Plate 4
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• National Maritime Museum, Dartmouth Collection, Map No. 38 (misprinted 39 in the catalogue); Surveys for the Commissioners. 1589. Munster.
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• See Tjomas Harriot, Renaissance scientist Shirley Oxford 1974 20 22
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• Dartmouth Collection National Maritime Museum, Dartmouth Collection, Map No. 29
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• Wallace . March 1979 . Private communication March , 7 Double letters ‘ou’, ‘th’ and ‘sh’ are represented by a single symbol
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• The inscription is transcribed in full in the Catalogue National Maritime Museum, Dartmouth Collection, Map No. 38
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• Dawson . 1968 . Elizabethan handwriting 1500–1650 19 – 19 . London the plates show only this one example in Plate 47B, a stylised version of Harriot's here
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• Hatfield House Cecil Papers , 114 ff. 40 – ff. 40 . 41; microfilm M. 485/24, University Microfilms: in the British Library Manuscript Room, where vols. 113–115 are on one spool
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• The later letter is given in Shirley Tjomas Harriot, Renaissance scientist Oxford 1974 29 29
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• Wien. Nat.-bibl. Cod. 10703, Bl. 381–382: eigenhändig. Reprinted in Kepler J. Gessammelte Werke Caspar Max Munich 1957 15 365 368
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• Kepler . 1957 . Gessammelte Werke Edited by: Caspar , Max . Vol. 15 , 348 – 352 . Munich A German translation is in M. Caspar and W. van Dyck, Kepler in seinen Briefen (Munich and Berlin, 1930), 268–269.
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• Kepler . 1958 . Gessammelte Werke Edited by: Caspar , Max . Vol. 16 , 172 – 173 . Munich See Wien (footnote 86), Bl, 384–385 for the Harriot letter.
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• Notably the folio already mentioned as reproduced by Rigaud in facsimile (footnote 19 and text), the last of sixteen all in the same style, with the first fourteen an elaboration of Viète's De numerosa potestatum purarum atque adfectarum ad exegesin resolutione tractatus Paris 1600 174 189 Opera (footnote 13), 163–228 (here pp. 195–210).
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• B.L.Add.Ms. 6785 f. 239 – f. 239 . ‘Simon Stevin, Novemb: 25. 1612’
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• B.L.Add.Ms. 6782 f. 481 – f. 481 . ‘1600. August. 29.♀.’ with a pure secretary g
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• Information on watermarks has been obtained mainly from Briquet C.M. Les filigranes , facsimile ed. Amsterdam 1968 4 vols
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• B.L.Add.Ms. 6787 f. 242 – f. 242 . see text to footnote 25
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