Hostname: page-component-586b7cd67f-gb8f7 Total loading time: 0 Render date: 2024-11-23T17:03:57.327Z Has data issue: false hasContentIssue false
  • English
  • Français

Pedro Nunes' Discovery of the Loxodromic Curve (1537). How Portuguese Sailors in the Early Sixteenth Century, Navigating with Globes, had Failed to Solve the Difficulties Encountered with the Plane Chart

Published online by Cambridge University Press:  23 November 2009

W. G. L. Randles
W. G. L. Randles
Affiliation:
(Maison des Pays lbériques, Université de Bordeaux III)
Get access Rights & Permissions [Opens in a new window]

Extract

In the early sixteenth century, Portuguese navigators on voyages outside the tropics ran into an unexpected problem in navigating with the plane chart since the latter took no account of the convergence of the meridians. Voyages between Brazil and the Cape of Good Hope were accomplished much faster than the chart led them to forsee. To overcome this problem, were Portuguese pilots of the first half of the sixteenth century using globes to set their courses with on great circle courses? The general opinion is that they were not. Yet the mathematician Pedro Nunes (1502–78) describes in a short treatise entitled Tratado que ho Doutor Pedro Nunez fez sobre certas duvidas na navigação (published in 1537, but probably written in 1534) how the navigator Martim Afonso de Sousa, on his return from the east coast of Brazil in 1530–2, had asked him to solve several problems of navigation encountered on the voyage.

Keywords

1. History.Loxodromes.3. Rhumb lines.
Type
Research Article
Information
The Journal of Navigation , Volume 50 , Issue 1 , January 1997 , pp. 85 - 96
Copyright
Copyright © The Royal Institute of Navigation 1997

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

NOTES AND REFERENCES

1 Cf. Randies, W. G. L., ‘From the Mediterranean portulan chart to the marine world chart of the Great Discoveries: the crisis in cartography in the sixteenth century’, in Imago Mundi, vol. 40 (1988), pp. 111–114.Google Scholar
2 In Nunes, Pedro, Obras, Lisbon, 1940, vol. 1, pp. 159–174.Google Scholar
3 Cf. Fontoura da Costa, A., A Marinharia dos Descobrimentos, Lisbon, 2nd ed. 1939, p. 219.Google Scholar
4 Pedro Nunes, op. cit. p. 159.Google Scholar
5Lopes de Sousa, Pero, Diário de Navegaçào de Pero Lopes de Sousa (l530–32), Prefácio do Commandante A. Teixeira da Mota, Leitura do Doutor Jorge Morais Barbosa. Ed. Agênda Geral do Ultramar, Lisbon, 1968.Google Scholar
6 Rodolfo Guimaräes, Sur la vie et l'œuvre de Pedro Nunes, Coimbra, 1915, p. 26 (original in French). In a letter dated 8 April 1985, Professor Luis de Albuquerque wrote to us that it was obvious to him that Martim Afonso de Sousa could not have been using a contemporary chart and that it was probable that he was carrying a globe in spite of their use being prohibited by the King.Google Scholar
7Coelho, J. Ramos (ed.), Alguns Documentos da Torre de Tombo, Lisbon, 1892, pp. 138–139. (‘… defendemos que nao facam nenhuns mestres das cartas de marear, nem outros alguns oficiais nenhumas pomas grandes nem pequenas, de pouco, nem muito, porque nao queremos que se facam em maneira alguma …’)Google Scholar
8 The term ‘quadratic chart’ is used by Nunes, Pedro, Tratado… em defensam da carta de marear, Lisbon, 1537, in Idem, Obras, vol. 1, pp. 176–177. Cf. Luís de Albuquerque's very pertinent remarks that Portuguese nautical charts of the sixteenth century were not in fact true quadratic charts. Luís de Albuquerque, Ciência e experiência nos Descobrimentos Portugueses, Biblioteca Breve, Lisbon, 1983, pp. 15–19, and Idem. ‘Considerações sobre a carta portulano’, in Revista da Universidade de Coimbra, vol. 31 (1984), pp. 19–22.Google Scholar
9 Cf. the analysis of this problem in W. G. L. Randies, art. cit. (n. I), pp. 112–113.Google Scholar
10Fernandes de Enciso, Martin, Suma de Geographia, Seville, 1519, sign, b.ix (v°).Google Scholar
11 João de Lisboa, Livro de Marhinaria, ed. Rebelo, Brito, Lisbon, 1903, pp. 197–198. Marcel Destombes has drawn attention to a chart in the Top Kapu Sarayi Museum in Istanbul which he attributes to Pedro Reinel and which he dates to between 1522 and 1524 and which was probably presented at the Badajoz Conference in 1524. It shows the southern hemisphere with converging meridians, just as João de Lisboa declared should ideally be the case. The chart was, however, probably prepared for diplomatic purposes rather than for navigational use. Cf. Marcel Destombes, ‘L’Hémisphère austral en 1524; une carte de Pedro Reinel à Istanbul’, in Marcel Destombes, Selected Contributions to the History of Cartography and Scientific Instruments (eds) Gunter Schilder, Pieter van der Krogt and Steven de Clerq, Utrecht/Paris, 1987, pp. 175–184, (with illustration of the chart).Google Scholar
12 Left out of consideration here is the phenomenon of magnetic variation, a separate problem never touched on by Nunes in his works. In the sixteenth century it was recognized but not clearly understood.Google Scholar
13 Nunes appears to be imagining a ship heading home from the east coast of Brazil (Martim Afonso de Sousa) on a north-easterly course to cross the equator.Google Scholar
14 Pedro Nunes, ‘Tratado […] sobre certas duvidas de navegação’, in op. cit. ed. cit. vol. 1, p. 161. ‘Mas posto que o circulo grande sobredito nos encaminhe ao oriente equinocial e se represente pello leste da [a]gulha: e quern pello tal cirulo for: va ter ao dito oriente equinocial: não auemos porem de cuidar: que quem por elle for yra a leste. porque tanto que por elle andar achara que o leste da agulha não vay na proa do seu navio. e andando espaço de caminho: em que esta deferenca possa sentir achara que vay ja per outro rumo. E por tanto o que gouerna sem entender o porque o faz: emenda logo de principio sua navegação: se quer jr em huma mesma altura. E he isto per tal maneyra que se gouernassemos a leste: e atassemos o gouernalho de sorte: que nenhuma mudança fezesse: e o mar fosse tarn tranquillo: que nenhuma cousa nos embargasse nossa navegação: e per cima de tudo isto ho vento nos fauorecesse: como quisessemos: e corresse pera aquel parte onde vay enderecado o leste da agulha: todauia se assi andasemos notauel espaço de caminho e oulhasemos a agulha: achariamos que imos fora de leste.’Google Scholar
15 In his Portuguese text, Nunes believed that the pole would be reached, but in his Latin text of 1566, he wrote that the ship would get closer and closer, but never touch it. Pedro Nunes, ‘Tratado…em defensam da carta de marear’ in Obras, vol. I, p. 184, and Pedro Nunes, Opera, Basel, 1566, p. 173.Google Scholar
16 Cf. the exhaustive treatment of the mathematical definition by d'Hollander, Raymond, ‘Historique de la Loxodromie’, In Mare Liberum, no. 1, Lisbon, 1990, pp. 29–69.Google Scholar
17 Cf. the remark of Joaquim de Carvalho in his introduction to Pedro Nunes, [Defensão do Tratado da Rumação do Globo para a Arte de Navegar, s.d.], Coimbra, 1952, p. xx.Google Scholar
18Nunes, Pedro, ‘Tratado…em defensam da carta de marear’, in Obras, vol. 1, p. 183.Google Scholar
19Nunes, Pedro, Obras, vol. 1, pp. 167—68.Google Scholar
20 Idem, Obras, vol. 1, p. 183.Google Scholar
21 Cf. Bk IV (1608), pp. 85–87. Cf. also Herman n Wagner, ‘Gerhard Mercator und die ersten Loxodromen auf Karten’ in Annalen der Hydrographie und Maritimen Meteorologie, Bd. 43, Heft VII-IX, Berlin, 1915, p. 302.Google Scholar
22 Pedro Nunes, ‘De regulis & instruments ad varia rerum tarn maritimarum quam ccelestium apparentias deprehendas, ex Mathematicis disciplinis, Liber II’, in Opera, Basel, 1566.Google Scholar
23 This is a reference to Book I of his Latin text, which is a translation of his earlier treatises in Portuguese, ‘Certas dúvidas da navegação’ and ‘Tratado…em defensam da carta de marear’;, both published in Lisbon in 1537.Google Scholar
24 The late Professor Pierre Costabel emphasized to us the importance of the sequential nature of the two movements.Google Scholar
25Nunes, Pedro, ‘De iis quae praemitti debent ad ducendum eas lineas in globo quas nautae rumbos appellant’, in ch. 2 of Book II of Pedro Nunes, Opera, Basel, 1566, p. 158.Google Scholar
26Pedro, Nunes, ‘Tratado sobre certas duvidas da navegação’ in Obras, vol. 1, p. 167.Google Scholar
27 Geber (Jabir ibn Aflah) (1st half of the 12th century), De astronomia libri IX, Latin translation by Gerard de Cremona, 2nd half of the 12th century. First published by Peter Apian, Nuremberg, 1535. Cf. Bk I, Prop. XIII (p. 11), (‘I declare that in a triangle formed by the arcs of great circles, the proportion of the sine of each of the sides to the sine of the arc of the angle subtended by that side is constant’).Google Scholar
28 Diogo de Sá, De Navigatione, Paris, 1549, f. 80 r°-v°. The late Professor Luis de Albuquerque had prepared a Portuguese translation of Diogo de Sá's book which I believe has remained unpublished. He was kind enough to make available to me a photocopy of the typescript. The above translation is, however, made directly from the Latin and I take full responsibility for it.Google Scholar
29 One of the scholastic arguments (not however invoked by Diogo de Sá) against the existence of the loxodromie curve, was that the force of gravity (Aristotelian gravity) acting from the centre of the Earth, would pull on the ship to keep it on a great circle course (!).Google Scholar
30 Pedro Nunes [Defensão do Tratado da Rumação do Globo para a Arte de Navegar], Subsídios para a História da Filosofia e da Ciência em Portugal, IV, publicados por Joaquim de Carvalho, Coimbra, 1952.Google Scholar
31 Pedro Nunes, op. cit. p. 31.Google Scholar
32Pedro, Nunes, Traité de Navigation, n. d., Ms Fr 1338 (Fonds Colbert), Bibliothéque Nationale, Paris.Google Scholar
33 Michel Coignet, Instruction Nourelle des poincts plus excellents et nécessaires touchant I'art denaviguer, Antwerp, 1581, ch. IIII, ‘Des cartes marine s et de ce qu'en dépend’, pp. 16–26.Google Scholar
34 Idem, p. 26.Google Scholar
35 Cf. the English edition: A Learned Treatise of Globes, both celestial and terrestrial, London, 1638, re-edited by Clements Markham, Hakluyt Society, 1st series, vol. 79, London, 1889. A French edition appeared in Paris in 1618.Google Scholar
36 The ‘precedent proposition’ stated that: ‘A great circle drawne through the vertex of any place that is in any whit distant from the Equator cannot cut diverse meridians at equal Angles’.Google Scholar
37Hues, Robert, A Learned Treatise of Globes…, Hakluyt Society edition, p. 130.Google Scholar
38Wright, Edward, Certaine errors of Navigation, London, 1599, reprint Amsterdam, 1974, preface.Google Scholar
39 E. Wright, op. tit. ch. 1.Google Scholar
40 Idem, ch. II.Google Scholar
41Wagner, Hermann, ‘Gerhard Marcator und die ersten Loxodromen auf Karten’, in Annalender Hydrographie und Maritimen Meteorologie, vol. 43 (1915), pp. 299–311;Google Scholar
Bensaude, Joaquim, Histoire de la Science Nautique Portugaise, Geneva, 1917, pp. 78–85;Google Scholar
Wagner, Hermann, ‘Die loxodromische Kurve bei G. Mercator. Eine abwehr gegenüber Senhor Joaquim Bensaude’, in Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen, Phil.-Hist. Classe Berlin, 1917, pp. 254–264, andGoogle Scholar
Bensaude, Joaquim, Les Légendes Allemandes sur l'histoire des Découvertes Maritimes Portugaises, Part I, Geneva, 1917–20, pp. 19–44; Part II, Coimbra, 1925–7.Google Scholar
42 Pedro Nunes, Opera, Basel, 1566, Livre II, ch. 26. ‘Proposition globum rumbis delineare’.Google Scholar
43 The map, in the Germanisches National Museum, Nuremberg, is reproduced in Köberer, Wolfgang, Das rechte Fundament der Seefahrt, Hoffmann und Campe, Hamburg, 1982, Plate 28. Cf. alsoGoogle Scholar
Drecker, Joseph, ‘Ein instrument, eine Karte und eine Schrift des Nürnberger Kartographen und Compostmachers Erhardt Etzlaub’, in Annalen der Hydrographie und Maritimen Meteorologie, vol. 45 (1917), pp. 217–224 andGoogle Scholar
Hammer, Ernst, ‘Die Mercator projection und Erhardt Etzlaub’, in Petermann's Mittheilungen, vol. 63 (1917), pp. 303–304.Google Scholar