Some Clarifications of Harriot's Solution of Mercator's Problem

The fullest account of this work is given in my “Harriot's calculation of the meridional parts as logarithmic tangents”, Archive for the history of exact sciences, iv (1967–68), 359–413. A summary, with other material, may be seen in my “Harriot's unpublished papers”, History of science, vi (1967), 17–40. For the dating, see Archive, pp. 365–6.

I have in mind such tables as Ptolemy's table of chords (see I. [Bulmer-] Thomas , Greek mathematical works (London, 1941), ii, 412–45; and Napier's and Briggs's logarithm tables). The mathematical content of Harriot's tables is undoubtedly greater than that of the logarithm tables, but this is somewhat overshadowed by the more general interest and knowledge of logarithms, certainly until the advent of computers and fast desk calculators.

Taylor E. G. R. and Sadler D. H. , “The doctrine of nautical triangles compendious”, Journal of the Institute of Navigation, vi (1953), 131–47. Sadler's Part ii (“Calculating the meridional parts”) is pp. 141–7.

Of course, I omit any consideration of the adjustments necessary because of the earth being more of an oblate spheroid. See the International Hydrographic Bureau's Special Publication No. 21, Table des latitudes croissantes à 5 decimales … (Monaco, 1928). Boyer C. B. in his A history of mathematics (New York, 1968), 329, attributes use of the integral result to E. Wright whose important Certaine errors in navigation … appeared in London in 1599 (2nd ed., 1610). But this is a misunderstanding, as Wright added secants, as he himself says.

“Harriot's calculation”, op. cit. (ref. 1), 363.

See my “Harriot's earlier work on mathematical navigation: Theory and practice”, Thomas Harriot renaissance scientist (Oxford, 1974), 86–87.

It is typical of the general lack of explanation Harriot provides in his unpublished papers that he does not explicitly say that he obtained Equation (1) from the conformality result. In fact, the equation is given only as a numerical example (Leconfield ms 241, vib, p. 14). The conformality theorem is given at BM Add MS 6789, ff. 17–18, with the statement “Quod rumbus in planisphaerio nostro facit angulos cum meridiano quolibet aequales angulis factis a rumbo correlatio cum meridiano a sphaera”. The proof given demonstrates conformality in the general case, but the demonstration of Equation (1) from it has not been found in Harriot's surviving papers. A plausible reconstruction is given in “Harriot's calculation”, op. cit. (ref. 1), 368. For Lohne's account of the stereography, see ref. 20 below.

See “Harriot's calculation”, op. cit. (ref. 1), 368–74. This achievement of Harriot, which is accompanied by the quadrature, seems to be in danger of passing without notice, perhaps because it is recorded in an account mainly about something else. But, as will be remarked below, the reference to it in the Harriot entry of the Dictionary of scientific biography is liable to misinterpretation.

See “Harriot's calculation”, op. cit. (ref. 1), 378, 383–4.

No doubt one may say with the Marquise du Deffand that “La distance n'y fais rien; il n'y a que le premier pas qui coûte”. But St Denis's six-mile walk (carrying his head in his hand) was a continued repetition of his original first step. The stages of Harriot's construction, in contrast, bring in several fresh feats. In principle, I suppose, Harriot could have used Equation (1) followed by interpolation, which would have had to be extensive and diverse in its implementation. But this would not only have been rather messy and probably impractical, with unequal intervals and perhaps non-integral indices in Equation (1); it would not have been as systematic as the solution he eventually gave. And the scale factor would still have been missing.

The scale is brought in by calculating the ration of the radius vectors of the equiangular spiral of 45° for an interval of one minute of arc. This is not trivial either; see “Harriot's calculation”, op. cit. (ref. 1), 375–6, and 408–9.

Whiteside D. T. , The mathematical papers of Isaac Newton, i (London, 1967), 475, note 35.

This manuscript reference is rather incomplete. The folios mentioned give the conformality theorem. The remainder of the work is as Leconfield MS 241, vib, which comprises 40 numbered pages (some blank), Leconfield MS 240 (over 450 sheets), and BM Add MS 6786, ff. 1–217. There are near contemporary copies of some of the work at Harley MS 6002, ff. 35–42, and Harley MS 6001, ff. 19–24. The only relevant folios at BM Add MS 6789 are 17–18; the canon triangulorum nauticorum at ff. 7–16v. is simply a table of cosines as in a modern traverse table to convert distance into d.lat. (cf. Fig. 1).

Op. cit. (ref. 6), 83. The article as a whole is ch. 4, 54–90.

See ref. 1 above. The twenty years is from 1594 to 1614. There is no suggestion that the whole of this time was taken up with the work. Present-day knowledge of Harriot's activities gives him plenty of other pursuits during this time.

Essay review “In search of Thomas Harriot”, History of science, xiii (1975), 61–70. The footnote is no. 16, pp. 69–70.

I mean the one in op. cit. (ref. 16). The first one is the one quoted from his edition of Newton (ref. 12).

In both footnotes Dr Whiteside refers to the evaluation of the integral; in the first without special emphasis on the word evaluation, in the second with it in quotation marks. But, quite apart from this word, the word integral is misleading and anachronistic. Harriot's calculation of the meridional parts as logarithmic tangents is a direct calculation supplemented by interpolation for intervening values, and is based on the geometry of the sphere. It is not even a mental convenience or otherwise acceptable loose phrase to refer to this as an integration. Harriot's earlier calculation of the meridional parts by adding up secants would more acceptably be called an integration, as the modern name for such work is numerical integration. If the word integration is to be used in considering the direct method, then it could only properly occur in some such statement as that Harriot's method is a special one for the particular problem and in fact avoids any integration or summation process as far as secants are concerned (but not as far as infinite geometric series are frequently used). It is also true that there is, strictly, a misuse of modern set-theoretic terminology when Dr Whiteside refers to a table as a “dense ordered listing” (my italics).

“Harriot's calculations”, op. cit. (ref. 1), 376.

Lohne J. A. , “Thomas Harriot als Mathematiker”, Centaurus, xi (1965), 19–45. The first section is relevant, pp. 19–27.

In another criticism, Dr Whiteside complains that he sees nowhere in the article a definition of the “seven rhumb lines” on the globe. He evidently overlooked p. 57, lines 7–8, and p. 77, line 10.

Or any other. For full consideration of the shortcomings of the plane chart and the need for a corrected chart allowing for the actual convergence of the meridians, see Waters D. W. , The art of navigation in England in Elizabethan and early Stuart times (London, 1958) in several places, in particular pp. 64–70. The whole purpose of the Mercator chart was to provide a correction for the plane chart for the purpose of obtaining accurate course directions from the differences of longitude and latitude of the port and destination, and to construct a Mercator chart it was necessary to obtain the so-called meridional parts. Perhaps, to avoid another possible confusion, I should say that the so-called Mercator projection did not originate with Mercator and is not in the sense of elementary geometry a projection, but a rather complicated mathematical transformation. At this stage it would be pedantic not to call it after Mercator, but there is no evidence I am aware of that he attempted to work out the necessary tables. Until the publication in recent years of the work of Harriot, and Dee, the first calculations were thought to be those Edward Wright published in his Certaine errors in navigation … (London, 1599) but completed a few years earlier. See “Harriot's calculations”, op. cit. (ref. 1), 359–60.

They were also given on p. 80 of the book dealt with in Whiteside's review (see ref. 6 above).

Op. cit. (ref. 6), 75. These tables were not uncommon in the sixteenth century, and were usually accurate even before the theory of the nautical triangle on the sphere (defined to be bounded by a rhumb line, a meridian line, and a line of latitude) was properly understood. This is because the meridian line gives a direct measure of latitude, so that the plane triangle is applicable. But where the d.long. comes in it is no use expecting to be so lucky, as the departure has to be multiplied by the secant of the latitude to obtain it in each approximating triangle (see Fig. 4.11 of op. cit., 76).

History of science, vi (1967), 37–38, ref. 12 (see ref. 1 above), and Thomas Harriot renaissance scientist, 86–87 (see ref. 16 above).

Cf. May K. O. , “Historiographic vices. i. Logical attribution”, Historica mathematica, ii (1975), 185–7.