Halley's lines, checked

Halley’s lines, checked

Georeferencing Edmond Halley’s 1701 chart of the variations of the compass and measuring its lines against gufm1

A state-of-understanding report · M4 · r3 (25 Oct 2026): cold-read findings applied (epoch resolution, circularity, 50°S west end, attributions)

Errata note. Errata entries M4-E01 and M4-E02 apply to this essay's thesis and to the title of leg 2: Halley's lines agree most closely where his own observations may have informed the model, and the best-fit years carry the same caveat. See the Errata page. Author: Two · Dynamic Geo-Temporal Science and Systems Analyst (AI), Temporal Data Dynamics. Text as delivered in report r6 (essay r3, numbers filled from the study's files).

Abstract

Edmond Halley’s New and Correct Chart Shewing the Variations of the Compass (London, 1701) draws lines through places where a compass needle points the same number of degrees east or west of true north. I treated the chart as a measurement and checked it against gufm1, the model of the Earth’s main field that Jackson, Jonkers and Walker (2000) rebuilt from historical observations. First I fitted the chart’s engraved graticule. It is a Mercator projection (rms 0.24° at 91 line crossings; a plain grid misses by 1.48°) with London as prime meridian, and its coastal longitudes are good to 1.1° rms. Then I read 349 points along Halley’s solid lines and one verified dotted line, plus the 36 labelled crossings of his own “Degrees of Variation” scale on 50°S. Against gufm1 for 1700 the misfit is 1.27° rms overall and 0.43° in the tropics. On 50°S it is within about 0.25° from 38°W to 20°W, beside the icebergs and track Halley drew, and misses by -2.2° at the west end and +2.8° east of 10°W, where his track does not go. Repeating the comparison for every year from 1650 to 1800 puts the minimum at 1705, resolved to about five years, so the chart dates itself to the first years of the 1700s. By 1720 the misfit has doubled, and by 1744, when Mountaine and Dodson issued a renewed chart, it is 5.1°. One caveat governs every number: it is plausible, though I could not confirm it, that Halley’s own observations are inside gufm1, so part of the agreement and of the dating may be the chart agreeing with itself.

Thesis. Halley’s lines match a modern reconstruction where he sailed, date themselves to the first years of the 1700s, and were out of date within a generation.

The object and its stake

The scan was supplied by the operator at turn 23 (2200 × 2576 pixels). Its copy, holder and state are not identified (H01). The cartouche gives the full title and the claim this study tests: the variations “as Observed in ye Year 1700”. The legend panel at left explains the lines in Halley’s words: the curve lines show at one view where the variation is the same, the numbers give degrees east or west, and the double line “passing near Bermudas and the Cape de Verde Isles” is where the needle stands true. The chart was published in 1701. Wakefield says it was issued in February (H07); dealers’ catalogues say it was presented to the Royal Society on 4 June 1701 (H06) and that the Society thanked Halley that day (H18). The two need not conflict.

I take the stake to be the date the chart claims for itself, observed 1700, because every measurement below compares the chart with the Earth’s field at a date. Publication in 1701 and the later reissues (c.1713 and 1722 per H18) do not change what was observed.

Halley’s data came from two Atlantic voyages of the Paramore, 1698–1700. Thrower gives the survey’s extent as 50°00′N to 52°24′S (H04, p. 597); Stern puts the ship at 52°40′S on 1 February 1700 among icebergs (H08). The two latitudes disagree, so I used the chart’s own evidence instead: it draws icebergs at about 52°S between 40°W and 30°W, and a dashed track that crosses 50°S near 27°W.

Leg 1 · A true Mercator, pinned to London

What I did. I found the graticule lines in the scan with a Hough transform and refined each line to sub-pixel precision along its length. I numbered them from the chart’s own labels: the equator is graded 20–10–0–10–20, the zero falls on a meridian engraved “The Meridian of London”, and parallels run every 10°. I intersected them and fitted two projections to the 91 crossings, each with an affine transform to absorb scan rotation and paper shear: Mercator (x linear in longitude, y linear in the Mercator function of latitude) and the plain grid.

What I found. Mercator fits to 0.24° rms (4.1 pixels); the plain grid misses by 1.48° and its errors swing systematically with latitude (Figure 1, left). The Mercator scales are isotropic: 16.81 pixels per degree of longitude, 16.80 per Mercator degree. The 10°S and 20°S parallels sit 5–8 pixels south of the fit, an engraving or paper irregularity of about 0.4°. Reading the tropics off the fit gives 23.6°N and 23.9°S.

I then read the charted positions of 20 coastal features and compared them with modern coordinates (Figure 1, right; Appendix D). Longitudes are good to 1.08° rms (median −0.38°). Tristan da Cunha is the outlier, 3.55° too far west. Latitudes are good to 0.31° rms. Two features I could not identify at the scan’s resolution (Cape Cod; the eastern mouth of the Strait of Magellan) are excluded and listed.

Why it matters. Halley’s lines are drawn in this frame. Because the frame is a regular Mercator pinned to London, I can turn every point on a line into a latitude and longitude, which is what the comparison in Leg 2 needs.

Leg 2 · Right where he sailed

What I did. Halley drew one line per degree. Every tenth line is solid, the zero line is double, and the rest are dotted. A guided tracer followed the solid lines well; on the dotted lines it ran onto coasts, parallels and rhumb lines. I therefore kept only the solid lines (0°, 10° and 20°, east and west), one dotted line I could verify by eye along part of its length (5°W, near 38°N between 61°W and 37°W), and the 50°S “Degrees of Variation” scale, where Halley labelled every crossing from 25° east to 10° west. Each trace was overlaid on the scan and checked; stretches where the tracer coasted without seeing a line were dropped. That leaves 349 traced points and 36 scale crossings.

For each point I computed the declination that gufm1 gives for 1700.0. The coefficients come from pmagpy 4.5.2, and I synthesised the field with my own code, which reproduces IGRF 2020 to better than a millionth of a degree. As a provenance check, gufm1 gives London −7.47° (7.5° west) for 1700, and the CALS3k.4 coefficients in the same package agree with it to 0.36° rms over the chart.

What I found. The misfit (gufm1 minus Halley) is 1.27° rms; two thirds of points are within 1°. Figure 2 shows every point on the chart. By region (Figure 3):

• Tropics, 23.5°S–30°N: 0.43° rms, most of it a constant offset (mean -0.37°). That is about the size of the 0.46° error gufm1 assumes for a single pre-1800 reading (as relayed from Jackson et al., H03), though one is a line’s misfit and the other a reading’s error.
• North of 30°N: 1.21° rms. The 20°W line in the Labrador Sea, off Newfoundland, is the worst stretch (mean +2.1°).
• South of 23.5°S: 1.55° rms west of 10°W and 1.34° east of it.

The 50°S scale is the cleanest test, because Halley numbered every crossing (Figure 4). From 38°W to 20°W the chart is within about 0.25° of gufm1, right beside the icebergs and track he drew. West of that it misses by -2.17° at 52°W; five of those western crossings were read from label positions (Appendix B). East of 10°W the error grows steadily, to +2.8° at 15°E. Over the whole scale the rms is 1.43°, worse than the chart’s overall figure. His track does not go there, and the chart’s no-variation line there sits about 3° west of the model’s.

Why it matters. The chart agrees best with the model where Halley took readings himself and worst where he joined his lines across water he did not cross, subject to the circularity set out below.

Leg 3 · A chart with a shelf life

What I did. I repeated the whole comparison against gufm1 for every year from 1650 to 1800.

What I found. The misfit is lowest at 1705 (1.10°), against 1.27° at 1700 (Figure 5); north of the tropics alone it is lowest at 1704, south of them at 1707. gufm1 is tabulated every five years and I interpolate linearly, so the curve can only bend at those years: the minimum is resolved to about five years. The table is keyed by the lower bound of each five-year branch; if each entry stands for the middle of its branch, the minimum moves to 1707.5, not earlier. Either way the chart’s own content puts it within a few years of the cartouche’s 1700. The dating is exposed to the same circularity as Leg 2. Moving away from that minimum, the misfit doubles by 1720 (2.6°), reaches 5.1° by 1744 and 6.2° by 1756.

Why it matters. Halley knew the variation drifted; his 1692 paper explains the change by a slowly turning inner sphere (H10). The chart was therefore a dated product. Mountaine and Dodson made the point in 1757: by the summary of their paper I read (H11), they redrew the curves for 1744 and again for 1756 on Halley’s Mercator frame, because the variation keeps changing.

What this does not show

• Circularity. gufm1 was built from historical logs. Jackson et al. (2000) mention Halley’s surveys, and their data reach back through Van Bemmelen’s 1899 compilation, so inclusion of his readings is plausible. I read the paper only as a summary from a non-institutional copy (H03), and a summary’s silence is weak evidence. If his readings were used, part of the agreement (above all in the tropics and on 50°S) and part of the dating are the chart agreeing with itself. The comparison then measures consistency, not independent accuracy.
• Coverage. I traced the solid lines, one dotted line and the 50°S scale, not all 1° lines. The no-variation line has a gap between the Cape Verde Islands and 12°N, under lettering and the compass rose.
• Positions versus lines. I compare in the chart’s graticule frame. If Halley placed his readings relative to coasts that are themselves 1° out in longitude, part of the misfit is position error, not variation error. At the median gradient (0.68° of declination per degree), a 1° position error is worth about 0.7°.
• Georeference. The fit’s 0.24° shifts the model’s declination by about 0.17° at the median gradient.
• Model. gufm1 is smoother than the real field before 1800 (H03). Its error at a point is not given, and the largest misfits here are where data were thinnest.

What is new here, and what is not

I found no published comparison of this chart’s lines with gufm1 in what I could reach (three searches; NOVELTY_H). The method is not new: Mandea, Korte, Soloviev and Gvishiani (2010) assessed Humboldt’s magnetic charts against gufm1 in the same way (H12). Two studies of how Halley built the chart, Murray (2012, H13) and an Imago Mundi article (2017, H14), could not be read in full here and may overlap. I have not claimed a first.

The chart is often called the first published isoline map of any phenomenon (Thrower, H04, p. 597). Thrower also notes two earlier manuscript depth charts (Bruinsz 1584; Ancelin 1697). Borri is said to have drawn lines of equal variation around 1620 on a chart now lost (H06). The wording therefore matters: first published, first printed, first surviving.

Appendix B · Uncertainty budget

Appendix C · Decisions and walls

• Dotted 1° lines were not traced, because the tracer could not follow them reliably; the one exception is a 5°W stretch I verified by eye (28 points). This was recorded rather than hidden.
• The zero line near Bermuda was retraced from a better seed after the first seed tracked a parallel line; the rejected trace is in the workspace, not in the measurements.
• Four sources were behind walls I did not go around: the Murray thesis PDF (robots disallowed), the Imago Mundi article (fetch failed at site), a HAL record (bot protection) and the Royal Society publisher page (not on the allowlist).
• The research agent’s dossier was read as a WebFetch summary throughout. Quotations it relays are marked as relayed; I re-opened Thrower and Mandea et al. myself.