The Study of Thomas Harriot's Manuscripts

Harriot Thomas , A briefe and true report of the newfound land of Virginia (London, 1588).

Harriot Thomas , Artis analyticae praxis (London, 1631).

For a modern account of this, see Scott J. F. , The mathematical work of John Wallis (London, 1938). Up to the present time Harriot's mathematics has been almost entirely judged from the Praxis, which was put together by his friends, including Torporley, Sir Thomas Aylesbury and Walter Warner: From his papers. See also Scriba C. J. , “Wallis and Harriot”, Centaurus, x (1964) 248–257.

“Mémoire sur la nouvelle planète Ouranus”, Mémoires de l'Académie de Bruxelles, v (1788) 22–48; “Etwas aus den, von Herrn von Zach in Jahr 1784 in England aufgefunden Harriotischen Manuscripten, vormemlich Original-Beobachtungen der beyden Kometen von 1607 und 1618”, Astronomischen Jahrbuchern, erster Supplement-Band (Berlin, 1793) 1–41; “Etwas von Hevelius und Harriot's Handschriften”, Monatliche Correspondenz, vii (Gotha, 1803) 36–60; “Beobachtungen des Uranus; … und Anzeize von den in England aufgefunden Harriotschen Manuscripten”, Astronomisches Jahrbuch für das Jahr 1788 (Berlin, 1785) 139–155; Correspondance astronomique, géographique, hydrographique et statistique du Baron de Zach (Genoa, 1822) 105–138; Allgemeine geographische Ephemeriden (Weimar, 1798) i, 230, 484, 635. Note that F. W. Bessel's first memoire was based on Harriot's observations, “Berechnung der Harriot‘schen und Torporley’ schen Beobachtungen des Cometen von 1607”, Monatliche Correspondenz, x (1804) 425–440. For a brief account of von Zach's life see Armitage A. , “Baron von Zach and his astronomical correspondence”, Popular astronomy, lvii (1949) 1–8.

Rigaud S. P. , “On Harriot's papers”, Royal Institution of Great Britain journal, ii (1831) 267–271; “On Harriot's astronomical observations contained in his unpublished manuscripts belonging to the Earl of Egremont”, Proceedings of the Royal Society, iii (1830–37) 125–126. See also Robertson A. , “An account of some mistakes relating to Dr Bradley's astronomical observations and Harriot's MSS”, Edinburgh philosophical journal, vi (1822) 313–318.

Rigaud S. P. , Supplement to Dr Bradley's Miscellaneous Works: With an account of Harriot's astronomical papers (Oxford, 1833). Some relevant correspondence appears in his Letters of scientific men of the seventeenth century (Oxford, 1841). Rigaud's papers in the Bodleian contain much on Harriot's life and works (Western MSS 26237, 26260, 26251, 26209, 26256, 26494).

Sixth report of the Historical Manuscripts Commission (London, 1867), 319.

Taylor E. G. R. , Tudor geography (London, 1930) paid particular attention to the influence of John Dee, who was acquainted with Harriot as the copy of Antonio de Espeio's El Viaie qve hizo Antonio de Espeio en el Anno de Cochenta y tres &c. &c. (Paris, 1587) in the British Museum shows by the inscription on the title page “Johannes Dee: Anno 1590. Januarij 29. Ex dono Thomas Hariot, Amici mei”. The Paris edition was a reprint by Hakluyt of the Madrid edition of 1586. Dee's name also features with some astronomical observations on an unfoliated sheet of Leconfield 241, 6b. Taylor considered the later period in her Late Tudor and early Stuart geography (London, 1934).

Taylor E. G. R. , “Hariot's instructions for Ralegh's voyage to Guiana, 1595”, Journal of the Institute of Navigation, v (1952) 345–350. Taylor E. G. R. Sadler D. H. , “The doctrine of nauticall triangles compendious”, Journal of the Institute of Navigation, vi (1953) 131–147.

Wright Edward , Certaine errors in nauigation … detected and corrected (London, 1599). 2nd enlarged edition, 1610, containing a translation of Rodrigo Zamorano's Compendio de la arte de nauegar (Seville, 1581), with which Harriot was acquainted.

Blundeville Thomas , His exercises (London, 1594), had quoted part of Wright's tables, with permission. There are slight discrepancies between the two published versions. To the intrinsic inaccuracies of Wright's and Harriot's early method must be added the uncertainties of the trigonometric tables that were relied on for the extraction of the secants. In 1594 Harriot was probably using Clavius C. , Theodosii Tripolitae sphaericorum libri III (Rome, 1586) as an erroneous value of a tangent he used agrees with the same value in Clavius' book. After 1596, the great Rheticus-Otho tables, Opus Palatinum (Neustadt, 1596), were available, with entries to 10 places and at 10″ intervals. B. Pitiscus' tables became available later, Canon triangulorum, 3rd edition (Frankfurt, 1612), with an English version in Ralph Handson's Trigonometric (London, 1614). John Dee's earlier work has survived in Ashmolean MS 242, where he gives tables of d. longs. on the 7 main rhumbs; see E. G. R. Taylor's edition of Wm. Bourne's A regiment for the sea (Cambridge, 1963), 415–433, for a printed version. This probably dates from 1558, and shows that Dee knew the general principles of the solution of the nautical triangle, together with (effectively) the meridional parts, which are identical to the d. longs. on the fourth rhumb. Dee's tables go up to 80° of latitude, and are tolerably accurate to near that latitude, which would be of interest to the Arctic explorers, such as John Davis, and others, such as the Muscovy Company, whom he advised on scientific matters. Even as early as 1511 and 1513, it is thought that the idea of a Mercator projection existed, as is shown by the maps on the case of a pair of sundials made by Erhard Etzlaub (J. Kenning, “The history of geographical map projections until 1600”, Imago mundi, xii (1955) 17).

There is the linguistic problem of asking if the calculation of logarithms is a use of logarithms, as well as that of asking whether the calculation of logarithmic tangents instead of the addition of secants is an integration of the secant function. In so far as, in the latter case, there is no hint of a general ‘integration’ process, but only a particular device for a particular problem, one may say that the process is the equivalent of an integration without implying that Harriot was consciously doing a calculus algorithm, whatever that might mean. The dangers of seeing these men's problems and methods with the light that was not available to them are well known, but must still be continually watched for and avoided.

Shirley J. W. , “The scientific experiments of Sir Walter Ralegh, the Wizard Earl, and the Three Magi in the Tower 1603–1617”, Ambix, iv (1949–1951) 52–66; “Sir Walter Ralegh's Guiana finances”, Huntington Library quarterly, xiii (1949) 55–69; “Binary Numeration before Leibniz”, American journal of physics, xix (1951) 452–454; “An early experimental determination of Snell's Law”, American journal of physics, xix (1951) 507–508.

Quinn D. B. , The Roanoke voyages 1584–1590 (London, 1955) 314–389; Raleigh and the British Empire (London, 1949); see also the Hakluyt Society reprint of Hakluyt's Principall navigations (Cambridge, 1965) 2 vols, with an Introduction by Quinn and R. A. Skelton.

Rigaud S. P. , Bradley supplement (op. cit.), 40–41.

Lohne J. A. , “Thomas Harriot (1560–1621), the Tycho Brahe of Optics”, Centaurus, vi (1959) 113–121; “A note on Harriott's scientific works”, Centaurus, vii (1961) 220–221; “The fair fame of Thomas Harriot”, Centaurus, viii (1963) 69–84; “Thomas Harriot als Mathematiker”, Centaurus, xi (1965) 19–45; “Dokumente zur Revalidierung von Thomas Harriot als Algebraiker”, Archive for history of exact sciences, iii (1966) 185–205. Lohne has also produced related works as follows, “Ballistik og bevegelselaere på Galileis tid”, Fra Fysikkens Verden, i (1964) 1–4; “Fermat, Newton, Leibniz und das anaklastische Problem”, Nordisk Matematisk Tidskrift, xiv (1966) 5–25; “Zur Geschichte des Brechungsgesetzes”, Sudhoffs Archiv, xlvii (1963) 152–172; “Regenbogen und Brechzahl”, Sudhoffs Archiv, xlix (1965) 401–415; “Drømmen om månen. Newtons eple”, Naturen, lxxxvii (1963) 159–174; “Isaac Newton: The rise of a scientist 1661–1671”, Notes and records of the Royal Society of London, xx (1965) 125–139.

Jacquot J. , “Thomas Harriot's Reputation for Impiety”, Notes and records of the Royal Society, ix (1952) 164–187. Kargon R. H. , “Thomas Hariot, the Northumberland Circle and early atomism in England”, Journal of the history of ideas, xxvii (1966) 128–136; Atomism in England from Hariot to Newton (Oxford, 1966).

Hakewill G. , An apologie or declaration of the power and providence of God in the government of the world, 3rd ed. (London, 1635) 301–302.

Harriot's views on the roots of equations are always said to be rather limited, see for example Cajori F. , “Revaluation of Harriot's Artis analyticae praxis”, Isis, xi (1928) 316–324; but this is a matter of what the editors made of his work. He found no difficulty, however, with complex roots, as may be seen in, for example, BM Add MS 6783, f. 157, where they are referred to as ‘noeticae radices’. It should be possible to rewrite the Praxis, as it were, or at least to see how it might have been done.

Kepler J. , Gesammelte Werke (Munich, 1938–1959), xv, xvi.

BM Add MS 6789, f. 449.

Rigaud in his Letters of scientific men, op. cit. (6), ii, 5, says that a copy of this was in Lord Macclesfield's collection.

Harriot retained an interest in the quadrature of the circle, and wondered if his work on the nautical triangle might lead to advances in the question, Leconfield 240, f. 230.

J. O. Halliwell prints this in his Collection of letters on scientific subjects (London, 1841), 109–116. Dr J. Landels of Reading is interesting himself in the question of what Torporley is attempting to do in this unfinished tract which breaks off in mid-sentence, and which appears to express dissatisfaction not so much with what Harriot had done as with what the editors of the Praxis had done to Harriot. However, it seems that Torporley, who was after all Harriot's mathematical executor, was more closely involved in the edition that had been previously thought. My own opinion is that in choosing Torporley for this duty, Harriot was choosing not only a friend of many years' standing, but one whose mathematical abilities were very considerable.

Seaton Ethel , “Thomas Hariot's secret script”, Ambix, v (1953–56) 111–114.

Of course, the longitude was not determinable with any certainty until about 1770, but the use of traverses with dead reckoning, in a way familiar to Harriot, would give estimates that were the best available.

This is used in the modern Traverse tables, e.g., Inman's nautical tables (London, 1957) latest edn., 26–100.

Harriot most likely used Rheticus' or Pitiscus' tables, op. cit. (n. 11), in the work about to be described.

This scale constant is important and although it occurs naturally in the addition of secants method, it does not arise in the fundamental formula of 1594, equation (7). The later work brings it in as a property of the logarithmic spiral.

Leconfield 241, 6b, p. 1.

Halley E. , “An easie demonstration of the analogy of the logarithmick tangents to the meridian line or sum of the secants: With various methods for computing the same to the utmost exactness”, Philosophical transactions, xvi (1695) 202–214, begins in a way very similar to Harriot's method, but claims originality for it himself, no doubt quite fairly, as Harriot's work was unpublished and quite unknown by that time. Halley said that the addition of secants was quite good enough for practical purposes. Henry Bond was writing in an edition of Richard Norwood's Epitome of navigation (London, 45), and may have made his conjecture by comparing graphs of the functions. Oughtred did not publish on the subject, but is quoted by Halley. Wallis J. , “Concerning the collection of secants; & the true division of the meridian in the sea-chart”, Philosophical transactions, xv (1685) 1193–1202; Collins J. , The marriners plain scale new plain'd (London, 1659); Gregory James , Exercitationes geometricae (London, 1668), 14–17; Barrow Isaac , Lectiones geometricae (London, 1670) 111. For Isaac Newton, see Whiteside D. T. , The mathematical papers of Isaac Newton (Cambridge, 1967), i, 466–467, 473–475, and plate iv.

The circle preserving property had been known since antiquity, and is used in, for example, the astronomer's astrolabe. But this is an independent property, although its proof in modern treatments is sometimes made to depend on the conformality theorem. Thus a statement in my “A note on Hariot's method of obtaining meridional parts”, Journal of the Institute of Navigation, xx (1967) 347–349, based on an uncritical reading of H. Michel's Traité de l'Astrolabe (Paris, 1947), needs amendment. (The fact that this Note was inadvertently printed from uncorrected proofs may make it more understandable.) Harriot's proof of conformality is at BM Add MS 6789, ff. 17–18.

Leconfield 240, ff. 439–440.

La géometrie (Leyden, 1637), 340–341.

Leconfield 240, ff. 211–214.

BM Add MS 6782, f. 67, “Interest vpon interest for 7 yeares”, expands (b + 1/n)⁷ⁿ/b^7n–2 by the binomial theorem, and takes the limit as n tends to infinity, namely, if b = 10, 100 + 70/1 + 49/2 + 343/60 + …, which turns out to be £207 7s. 6d.

BM Add MS 6782, “De numeris triangularibus et inde de progressionibus arithmeticis magisteria magna T.H.”, ff. 107–144, et seq.

Harriot's notation for tangent is a small circle with a small tangent line drawn above it, but I use the Greek letter tau here as being easier to print.

Leconfield 240, f. 161, has a note that I interpret in this way.

BM Add MS 6786, f. 99, is a sheet (one of many) of check-calculations, most probably in the hand of Harriot's assistant, Christopher Tooke, and dated Jan: 23. 1613/1614. If a reasonable rate of progress was being made with the arithmetical work, the tables should have been completed within a few months.

See W. R. Macdonald's translation of Napier's 1619 Constructio (Edinburgh, 1889), or Henderson James , Bibliotheca tabularum mathematicarum (Cambridge, 1926), 24–25. Edward Wright's 1616 edition of Napier's Descriptio (Edinburgh, 1614) reduces Napier's tables from 7 to 6 figures. Harriot himself knew that Napier's tables had errors in the last figure, see BM Add 6786, f. 84.

The calculations are at BM Add MS 6786, ff. 29 V., 149, 149 v., 160, 160 v.; 150–152 v.; 30–32 v.; 164; 33–34, 161, 161 v.; 2, 3, 29, 154, 154 v.; 6–7 v., 9, 9 v., 22 v.–23 v. The tables are at Leconfield 240, ff. 263–269.

After I had written this, Dr Whiteside drew my attention to Jakob Bernoulli's attack on the problem of this ‘spatium loxodromicum’ in Acta Eruditorum (1691), 282–8, which fitted in well with the further comment I had, rather speculatively put, namely “it is doubtful if there was anyone who could have solved this particular problem before the end of the century”.

See n. 23.

Intellectual origins of the English Revolution (Oxford, 1965). This contains a chapter on the influence of Harriot's first patron, Ralegh, and a section on Harriot himself (pp. 139–145). The statement (p. 142 n.) that Harriot wrote the fifth part of R. Hues' Tractatus de globis (London, 1594) is an error taken from C. Markham's edition (London, 1889). In fact. Hues says (p. 111) that he was looking forward to Harriot's forthcoming work on the origin, nature and use of rhumb lines, but that in the meantime he hoped that his account would suffice to introduce the topic as far as may be sketched out by globes (which is not very far).

The standard life is Stevens Henry , Thomas Hariot the mathematician, the philosopher and the scholar (London, 1900). Recent work by Shirley J. W. Quinn D. B. supplements this. A brief summary is given in my Harriot's work on mathematical navigation (London M.Sc. dissertation, 1967), which also describes his earlier work on navigation in some detail.

D. B. Quinn is, I understand, hoping to publish, with J. W. Shirley, a full account of this list and its implications, but at the moment one or two items remain unidentified. Hues' Tractatus has already been referred to and is not repeated here. Another work referred to by Harriot is Simon Forman's The grovndes of the longitude (London, 1591) of which there is a copy in the Bodleian which I have not seen. Harriot mentions “Hoodes Answere to the same”, but this has not been traced, despite the reference in Forman's Diary, given in J. O. Halliwell's A selection from the papers of Dr Simon Forman (London, 1843), 21–22, where it says “1591 … The 10. of Aprill I put the longitud in question, and the 21. dai I rod to London, and lai at Mullenaxes to teach him the longitude. … The 6. of Julii I put my bock of the longitude to presse. … The 22. of Novembre Mr Goodes [sic] bock came out against me” (MS Ashm. 208, fol. 17–67). I suspect that Halliwell mis-read “Mr Goodes” for “Mr Hoodes”, an easy mistake if one is reading Elizabethan handwriting in a hurry, and particularly when, as here, the writing is difficult, or so Halliwell says it is. I have not seen the MS myself and cannot offer a definite opinion on it.