During the 1519–1522 Magellan circumnavigation, the fleet's astronomer and pilot Andrés de San Martín made six shipboard longitude determinations by a simplified lunar-distances procedure. A 2024 quantitative reconstruction (Gaspar, Journal of Navigation 77(1):124–150) used the JPL DE431 ephemerides and an error-signature propagation model to identify his method and explain why two of the six measurements (San Julián, +0.6°; Suluan, −2.0°) were an order of magnitude more accurate than anything else in the 16th century — attributing the accuracy to fortuitous cancellation of large ephemeris, parallax, and geometric errors. That reconstruction leaves a precise open core: the exact celestial aspects actually observed at Suluan and at the three lost observations (Río de la Plata, Gulf of San Matías, Pacific Ocean) are undetermined; the numerical results of the three lost measurements are unrecovered; whether San Martín carried the 1496 Leiria or the 1502 Venice edition of Zacut's almanac is undecidable from the observations themselves (both give identical values); the skill-versus-luck attribution has not been tested for robustness to the error model's assumptions; and the question whether San Martín and Magellan were aware of the method's unreliability, i.e. how close this expeditionary practice came to a systematic, generalizable shipboard longitude method, remains open.

Background

Determining longitude at sea was the central unsolved problem of early modern navigation. Longitude difference between two places equals the difference in local time, so a navigator who could compare local time (from a Sun or star altitude) with time at a reference meridian (published in an astronomical almanac) could fix longitude. The lunar-distance method — measuring the angular distance between the Moon and the Sun or a planet and comparing it with tabulated values — was proposed in print by Johannes Werner in 1514, but before accurate ephemerides and instruments (the octant, the Nautical Almanac, the chronometer — developments of the 18th century) it was, in the words of the standard history of the method, at best a throw of the dice: 16th-century ephemerides carried roughly 0.55° RMS error in lunar ecliptic longitude (the coordinate measured along the ecliptic, the Sun's apparent annual path), which alone translates to about 15° of longitude error (Gaposchkin & Haramundanis 2007; de Grijs 2020).

On the Magellan expedition (1519–1522), the Spanish crown's astronomer Andrés de San Martín — a trained pilot described by biographers as an astronomer of genuine but land-level-unattained competence (Mena García 2020) — made at least six longitude measurements, reported only second-hand by chroniclers (Herrera, Barros, Castanheda, Albo, Pigafetta) because his original logbooks are lost. Two were extraordinarily accurate (San Julián, 17 April 1520, from a solar eclipse: +0.6°; Suluan Island, 16 March 1521, from a Moon–Sun configuration: −2.0°); one was catastrophically wrong (Rio de Janeiro, 17 December 1519: −231.5°); three are known only by date and place with their numerical results lost (Río de la Plata, 1 February 1520; Gulf of San Matías, 24 February 1520; Pacific Ocean, 23 December 1520). The longitude of the Spice Islands relative to the Treaty of Tordesillas demarcation line between the Spanish and Portuguese hemispheres was the expedition's central political stake, which is why these readings mattered to contemporaries.

Gaspar (2024) supplied the first quantitative reconstruction. Using NASA/JPL's DE431 long-term ephemerides (modern tables giving positions of the Moon and planets to sub-arcminute precision millennia into the past) as a baseline sky, empirically based instrument errors (period-faithful replica marine astrolabes and quadrants give altitude readings with ~15.9′ standard deviation; period local-time determinations ~9 min), and a numerical error-signature model — the probability distribution of total longitude error obtained by propagating each error component (almanac equation errors, lunar parallax, geometric-reduction approximations, instrument noise, local time) — the paper concluded that San Martín used a simplified lunar-distances method (treating measured altitude differences between the Moon and a reference body directly as ecliptic-longitude differences, without parallax correction); that the Rio catastrophe came from applying the Moon–Jupiter relative motion in reverse; that the two accurate results were fortuitous cancellations among large errors; that the three lost observations most plausibly used an azimuth-substitution variant (compass azimuth differences in place of ecliptic-longitude differences, worth tens to over a hundred degrees of geometric error); and that everything was done with pilot instruments and two Zacut almanacs. The reference almanacs were Abraham Zacut's Almanach Perpetuum — the standard Iberian astronomical tables, printed in five editions (Leiria 1496; Venice 1498, 1502, 1525, 1528), the 1498 edition using Toledo and the others Salamanca as reference meridian). One of San Martín's two almanacs used the Salamanca meridian; the other he attributed to Regiomontanus's printer Liechtenstein, which the paper argues was the misattributed 1498 Venice Zacut (Toledo meridian), following Chabás and Goldstein's observation that Regiomontanus's tables were folded into that edition.

Problem Statement

Complete the historical and quantitative reconstruction of Andrés de San Martín's shipboard longitude program (1519–1521) — the earliest documented series of lunar-distance longitude determinations at sea — by settling the specific questions that Gaspar (2024) leaves open, all in service of one objective: a full evidentiary and reliability account of what the expedition's astronomer actually did, observed, and could trust. Concretely, determine:

  1. Observed aspects. Establish which celestial configuration San Martín actually observed at Suluan (the paper's Moon–Sun choice at ~13:13 local time rests on the incompatibility of the Moon–Mars alternative within its error model) and what procedure and body pairs produced the three lost measurements at the Río de la Plata (1 February 1520), the Gulf of San Matías (24 February 1520), and in the Pacific Ocean (23 December 1520), for which the azimuth-substitution variant is an abductive hypothesis.
  2. Missing numerical results. Recover, from newly identified documentary evidence if any survives, the longitude values San Martín obtained for the three lost measurements (whose numerical results are not preserved in the surviving second-hand record).
  3. Almanac edition. Determine whether the Salamanca-meridian Zacut Almanach Perpetuum San Martín carried was the original 1496 Leiria edition or the 1502 Venice re-edition — the observations cannot decide this because both editions give identical tabulated values for all six observations, so only external evidence (fleet inventories, book provenance, marginalia) can.
  4. Skill-versus-luck attribution. Test whether the paper's conclusion — that the San Julián (+0.6°) and Suluan (−2.0°) accuracies were fortuitous cancellations of large almanac, parallax, and geometric errors rather than products of observational discipline — is robust to the error-signature model's assumed component magnitudes and dependency structure. The source itself cautions against dismissing San Martín's work as sheer luck, describing him as "an exceedingly rare combination of a capable astronomer and a knowledgeable mariner"; the role of his practical judgment (method selection, almanac cross-checking including detection of the Toledo edition's typographical errors, plausibility screening) remains unquantified.
  5. Systematic-practice assessment. Assess, from the primary record, whether San Martín's and Magellan's expeditionary practice constituted or approached a systematic, generalizable shipboard longitude method, and whether the two men were aware of the method's unreliability. For this assessment, treat a practice as systematic only if it is set out as reproducible rules that another pilot could apply with a stated error expectation, and as generalizable only if it is not restricted to a single observation; "approached" then denotes the degree to which the documented practice satisfies these properties.

All five sub-questions are required of a complete account, and each is resolved on its own terms, independently of the others. Item 1 is settled by identifying the observed celestial configurations and the procedures and body pairs used. Item 2 is settled by recovering the lost longitude values, or by a well-supported finding, from the surviving record, that they cannot be recovered. Item 3 is settled by a bibliographic determination of the Zacut edition, or by a well-supported finding that no external evidence decides it. Item 4 is settled by the robustness analysis of the error-signature model. Item 5 is settled by a reasoned verdict against the criterion stated in it, and does not depend on whether items 1–4 return positive or negative findings.

References

  1. Gaspar, R. (2024). San Martín's accurate longitude measurements on Magellan's circumnavigation: luck or mastery? The Journal of Navigation, 77(1), 124–150. doi:10.1017/S037346332400016X. https://doi.org/10.1017/S037346332400016X
  2. Gaspar, J. A. (2025). Magnetism, Navigation, and Cartography in the Early Modern Period: An Elusive Relationship. Cartographica, 60(2), 45–61. doi:10.3138/cart-2024-0033. https://doi.org/10.3138/cart-2024-0033
  3. Gaspar, R. (2026). The Revolution of Magellan: How 500-Year-Old Rocket Science Changed the World. Springer. doi:10.1007/978-3-032-10797-8. https://doi.org/10.1007/978-3-032-10797-8
  4. Mena García, C. (2020). Conocer y dominar los astros. El piloto Andrés de San Martín y la expedición de Magallanes/Del Cano. Temas Americanistas, 44, 197–231. doi:10.12795/Temas-Americanistas.2020.i44.08. https://doi.org/10.12795/Temas-Americanistas.2020.i44.08
  5. de Grijs, R. (2020). A (not so) brief history of lunar distances: lunar longitude determination at sea before the chronometer. Journal of Astronomical History and Heritage, 23(3), 495–522. doi:10.3724/SP.J.1440-2807.2020.03.04. https://doi.org/10.3724/SP.J.1440-2807.2020.03.04