The Anomaly Outlived Its Planet

In 1859 Le Verrier found that 21 timed contacts of Mercury with the Sun’s disk fitted to within a second of arc once its perihelion was allowed to advance about 38 arcseconds a century faster than the known planets could make it. Run his own elimination on his own printed equations and get 38.44 against his 38.3. Then rerun the check that decided it, Newcomb’s 1882 reduction of the transits to 1881 (42.93 here, 42.95 printed), and plant Le Verrier’s excess into Newcomb’s own equations: the check finds it at 38.28, 13.0 jackknife standard errors clear of zero. The excess was real, and larger than claimed; the planet proposed to explain it, later called Vulcan, was never found.

Everything below is computed in your browser from three small files this page transcribed or collected. Numbers taken from a publication are marked as printed and cited where they appear; every other number is the page’s own arithmetic on those files, and the check at the bottom recomputes the page against itself while you read.

I · the claim, at full strength

Thirty-eight seconds that fitted everything

On Monday 12 September 1859 the Académie des sciences heard a letter from Urbain Le Verrier, director of the Paris Observatory, to the astronomer Hervé Faye. Le Verrier had spent years putting the theory of the Sun on a firmer footing, and with that done he had returned to Mercury. Its best record, he wrote, is its transits: when the planet crosses the Sun, the instant its disk touches the Sun’s edge from inside fixes the distance between the two centres to about a second of arc, and between 1697 and 1848 there were twenty-one such observations. They would not fit his tables. Then:

Mais, ce qui est remarquable, c’est qu’il a suffi d’augmenter de 38 secondes le mouvement séculaire du périhélie pour représenter toutes les observations des passages à moins d’une seconde près, et même la plupart d’entre elles à moins d’une demi-seconde.

U.-J. Le Verrier, Lettre à M. Faye sur la théorie de Mercure et sur le mouvement du périhélie de cette planète, Comptes rendus 49 (1859), pp. 379-383, p. 380

The page’s translation  But what is remarkable is that it was enough to increase the secular motion of the perihelion by 38 seconds to represent all the observations of the transits to within a second, and most of them even to within half a second.

The perihelion is the point of Mercury’s orbit nearest the Sun, and it turns slowly round the Sun because the other planets pull on Mercury. Le Verrier’s claim is that it turns faster than the known planets can make it turn, by about 38 arcseconds a century. The letter prints only the result. The work is in his Théorie du mouvement de Mercure, in volume V of the Annales of the Paris Observatory (1859), and on page 80 he prints the evidence entire: all 21 contacts, 13 in November and 8 in May, as equations of condition. Each equation says that a sum of small corrections to the adopted elements, each times a coefficient computed from the geometry of that transit, must cancel the difference between the observed and the tabular contact. There are fifteen unknowns (p. 79): corrections to Mercury’s mean longitude, mean motion, eccentricity and perihelion; to the annual changes of its eccentricity (e′) and of its perihelion (ϖ′, the claim); to the Sun’s four elements; to the masses of Venus (ν′) and of Mercury (ν); to the longitude of Mercury’s node (δθ); and to the distance of the centres at contact, κ in November and κ′ in May. To give the two seasons equal influence he multiplied every May equation by 13/8 (p. 79).

How he solved them

Not by least squares. For the unknowns he wanted he used “l’une des méthodes d’élimination précédemment exposées” (p. 81), Cauchy’s method: to find one unknown, multiply every equation by +1 or −1 so that the unknown’s coefficient is positive in each, add them all into one “équation propre”, solve it for that unknown, and substitute into every equation. He printed four such steps, (I) to (IV), for the mean longitude at the epoch, the mean motion, the perihelion and its annual motion. The last is the claim, as printed:

Then he fixed what the transits settle poorly (pp. 92-96). For the node, the meridian observations gave δθ = +12.4″ and the transits fitted best with −1.5″; “n’ayant aucune raison de choisir entre ces deux quantités”, having no reason to choose between them, he took the mean, +5.5″ (p. 92). For the distance of the centres he drew from his own (V) and (VI) a single mean relation between the corrections to the diameters of the Sun and the planet, which, with the Sun’s semi-diameter kept at 960″, gives κ = −0.11″ and κ′ = −0.14″ (p. 93). And “en négligeant la partie constante de ν′” he dropped the constant part of the Venus correction, −0.0228, holding the mass of Venus at its adopted value (p. 96). With e′ and the other small unknowns left at zero, his own (IV) gives 38.34 arcseconds a century, and he printed 0.383″ a year, 38.3 a century, the number he carried into equation (A) on p. 100.

The page’s finding, a reconstructionWith the plain sign rule at every step the page reproduces (I) to (III) closely, but only 5 of 8 printed values of (IV): its coefficient of κ comes out −0.0041 where Le Verrier printed +0.0081. Reversing one multiplier at the fourth step, on the November 1769 ingress (the equation whose reduced perihelion coefficient is the smallest positive one), reproduces all 8 of 8, with κ at +0.0081. The page tried every single and paired reversal at that step, 231 of them, and no other set does. This is the page’s inference from his printed coefficients, not something he wrote; the plain sign rule stays a choice above, and gives 38.01.

The anchor. Printed: 38.3. Computed by the page’s engine from the frozen record: 38.44. The gap is 0.14, inside the page’s tolerance of 0.5 arcseconds a century: perturbing every coefficient of the record within half a unit of its last printed digit moves the page’s value by 0.16 (one standard deviation), Le Verrier’s own (IV) differs from the page’s by his intermediate rounding, and his letter prints the integer 38. Noise like his own residuals moves the page’s value by 2.9 (the page’s formal standard error, which treats his adopted values as exact).

Not a pure transit result. The 38.3 depends on a node correction half taken from the meridian observations, on his averaging of the two diameters, and on keeping the mass of Venus as adopted. From the same 21 equations: his own transits-only values of κ, κ′ and ν′ (p. 82) give 40.16; keeping the constant part of ν′ gives 38.04; least squares with his p. 96 values gives 38.93. The nine combinations of method and fixed values the panel offers give between 37.63 and 41.36, and none comes near zero.

“À moins d’une seconde près.” At his adopted values the 21 contacts miss by at most 0.99″ (the May 1799 egress), root mean square 0.59″, May misses divided back by 13/8. With the perihelion held at the known planets’ pull and everything else re-solved, the largest miss is 6.29″ (the May 1753 egress) and the root mean square 2.43″. His own residual table on p. 82, printed at his transits-only values, is reproduced by the page within 0.08″ (largest at the November 1789 egress); one of its printed residuals, the November 1782 ingress, is −1.16″.

What he made of it

The first explanation he tested was a heavier Venus. The extra motion could come from the known planets only if “accroître la masse attribuée à Vénus du dixième au moins de sa valeur”, increasing Venus’s mass by at least a tenth, and the secular change of the obliquity of the ecliptic, measured from Bradley to his own day, argued against that (CRAS 49, p. 381). He did not choose:

Je n’ai nullement l’intention de décider d’une manière absolue entre ces hypothèses.

Comptes rendus 49 (1859), p. 382

The page’s translation  I have no intention whatever of deciding absolutely between these hypotheses.

Then, “pour fixer nos idées”, to fix ideas, he supposed a planet inside Mercury’s orbit, on an orbit little inclined to Mercury’s and, since the problem left it open, circular, and noted that its mass and distance trade against each other: “Pour une distance un peu inférieure à la moitié de la distance moyenne de Mercure au Soleil, la masse cherchée serait égale à celle de Mercure.” Why had no one seen so bright a body? “Toutes les difficultés disparaîtraient en admettant, au lieu d’une seule planète, l’existence d’une série de corpuscules circulant entre Mercure et le Soleil” (p. 382): a swarm of small bodies would do as well. The name Vulcan appears in neither this letter nor his 1860 reply below; the page says “later called Vulcan”. The claim this page tests is the excess, not the planet, whose fate is told separately below.

II · the deciding control

Newcomb’s re-reduction, 1882

Simon Newcomb, head of the American Nautical Almanac Office, published in 1882 a new discussion of every observed transit of Mercury from 1677 to 1881: new reductions of the old contact times, a weight for every contact, the masses he judged best, least squares throughout, and four transits observed since Le Verrier’s memoir, those of 1861, 1868, 1878 and 1881. His opening states the stakes:

The existence of this discrepancy, at least when the mass of Venus determined in other ways is employed, has been placed beyond doubt by observations of four transits since the publication of Leverrier’s work.

S. Newcomb, Discussion of Observed Transits of Mercury, 1677-1881, Astronomical Papers of the American Ephemeris, vol. 1, part VI (1882), p. 367

He did not simply repeat Le Verrier’s sum; he printed every step of his own. On pp. 457-458 are 58 equations of condition, internal and external contacts, each with its weight, 4 of them marked rejected; 54 equations from 21 transits remain. Their unknowns are not the elements themselves but the combinations a transit can measure: V for November transits and W for May transits, each a sum of corrections to Mercury’s and the earth’s longitudes, perihelia and eccentricities (p. 466), with V′ and W′ their changes per century. An extra perihelion motion shows up in exactly those two. On p. 459 he printed the normal equations, on p. 460 the solution, V′ = −2.63 and W′ = +1.84 with his k = 0 (uniform rotation of the earth), and on pp. 467-473 the chain from those two to the excess motion of the perihelion, p = 42.95 arcseconds a century:

It follows that the observed centennial motion of the perihelion of Mercury is greater by 43″ than the theoretical motion computed from the best attainable values of the masses of the planets.

Newcomb (1882), p. 473

The page’s value is 42.93 where Newcomb printed 42.95; with his fitted k it is 44.01 (solved freely, the page’s k is 0.292); restoring his four rejected contacts at unit weight, since he printed no weight for them (taking for the 1786 contact the second of his two printed values, which he preferred, and dropping the 1677 row he formed from two of the rejected ones), moves it to 42.92. His rejections do not make the result.

The page’s findings, about a printed page Forming the weighted normal equations from his 54 equations reproduces the 110 printed entries that do not involve k to within 0.004, except two, both in equation (8). First, (8) prints the coefficient of W′ as −6.156 while (6) prints the matching coefficient of S as +6.156; a normal matrix is symmetric, so one of them is wrong, and the page’s own sum gives the plus sign. Solving the printed equations as they stand gives V′ = −2.567 and W′ = +1.746; reading (8) with the plus sign reproduces every unknown on p. 460 within 0.006; changing (6) instead gives V′ = −2.561 and W′ = +1.815, which does not match. So Newcomb solved the right equation and the misprint is in the type. Second, the constant of (8) sums to −13.510 where p. 459 prints −12.510, a difference of 1.000. No single misread coefficient in the page’s transcription can produce a difference of exactly one in that sum alone, and the solution on p. 460 fits the printed constant slightly better (S = −0.036 against his −0.04, where the page’s constant gives −0.032), so the page reads it as a slip in forming that one sum, carried into his solution; it moves p by about one hundredth. Third, his closing pair of equations on p. 473, solved as printed, gives p = 42.954 and δn = +0.295, not the printed +0.37; no value of δn near that satisfies both at his p, and p is unaffected. None of the three is in his corrigenda on p. 484. All three are small slips on a printed page, and none touches his conclusion. The entries involving k differ by up to 1.4; they do not enter p at k = 0.

Could a heavier Venus explain it?

This was the question that mattered, and Newcomb answered it with his own printed expressions (p. 469): the theory’s secular changes of the two transit combinations as straight lines in ν′, the fractional correction to Le Verrier’s mass of Venus, 1/401,847. Drag it and the excess is recomputed from the page’s V′ and W′.

The excess vanishes at ν′ = +0.1553 from the page’s V′ and W′, and at +0.1554 from his printed ones, a Venus of 1/347,795 of the Sun, where Newcomb printed +0.1554 and 1/347,800 (p. 470). Against that stood the motion of Mercury’s node, which gives ν′ = −0.016, a Venus of 1/408,400 (p. 471, printed without an uncertainty), and the periodic perturbations, 1/396,000 (p. 472; ν′ = +0.015 ± 0.013 from his M = 10ν′ = +0.15 ± 0.13 at k = 0, of which he wrote, “We can, therefore, only attribute small weight to the result”). At those masses the excess is 45.04 and 36.89; at Le Verrier’s own mass it is 40.83. Every per cent of Venus is worth about 2.6 arcseconds a century of excess. Newcomb’s conclusion:

We must, therefore, conclude that the discordance between the observed and theoretical motions of the perihelion of Mercury, first pointed out by Leverrier, really exists, and is indeed larger than he supposed.

Newcomb (1882), p. 472

Le Verrier had made the same argument in 1859 with the obliquity of the ecliptic in place of the node. He died in 1877, before Newcomb’s reduction appeared; there is no reply from him to it.

III · the control on the control

Could Newcomb’s check have come back empty, or missed 38?

A check that could not have failed proves nothing by passing, and a check that could not have seen the effect proves nothing by missing it. The page tests both, with Newcomb’s own first-order theory (p. 467): an extra perihelion motion d raises the secular change of his first transit combination by 0.487d and of his second by 0.284d, so V′ moves by −0.487d and W′ by +0.284d. That shift goes into the absolute term of every November equation through its V′ coefficient and every May equation through its W′ coefficient, in a copy of his equations of condition, and the unmodified control runs on the copy.

The modern control is cited, not recomputed. Ranging to the MESSENGER spacecraft in orbit about Mercury measured the total precession of its perihelion as 575.3100 ± 0.0015 arcseconds a century (Park et al. 2017). Taking that published uncertainty on trust, Le Verrier’s 38.3 is about 25,500 times it; that ratio is context (grade B, the paper’s own sensitivity) and sets no cell of this page’s ledger.

IV · the claimant’s method, run on nothing

Could his elimination have made 38 out of noise?

Le Verrier’s own defence of his excess was that nothing smaller than enormous, systematic timing errors could remove it: the old observers would have had to make “des erreurs de plusieurs minutes de temps et variant même progressivement d’une époque à l’autre, chose impossible !” (CRAS 49, p. 380). The page puts that to the test with his own rows. Each null record keeps his 21 rows of coefficients exactly, replaces the constants by what his fitted model predicts with no excess at all, and adds noise drawn from his own 21 residuals with random signs, scaled up by 1.11, the square root of 21/17, because residuals left after fitting four unknowns run smaller than the errors behind them (and multiplied by 13/8 in May rows). Each record then goes through the same elimination and the same p. 96 values that produced 38.44.

Random errors of the size his residuals show cannot make his result: none of 10,000 does. What can is a systematic error per transit, the same for both contacts of a transit and independent from one transit to the next: at 2.0″ the procedure manufactures his excess in 0.07% of null records, at 10.0″ in 25.3%, and it crosses one in twenty at 4.2″, 7.1 times the root mean square of his residuals. The cheapest single culprit is 1832: every arcsecond of error in both its contacts moves the result by 3.27, so that transit alone would have to be wrong by 11.7″. These are arcseconds in the equations’ own units; the Annales do not print the conversion to time for every contact, so the page leaves them in arc beside his “plusieurs minutes de temps”.

V · the verdict, dated

What the record says, as of now

Sources for the verdict

1. Simon Newcomb, Discussion of Observed Transits of Mercury, 1677-1881, Astronomical Papers prepared for the use of the American Ephemeris and Nautical Almanac, vol. 1, part VI, pp. 363-487, Washington: Bureau of Navigation, 1882 (pp. 472-473). Internet Archive scan. The re-reduction that decided the excess.
2. A. Einstein, Erklärung der Perihelbewegung des Merkur aus der allgemeinen Relativitätstheorie, Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften zu Berlin, 1915, pp. 831-839. ADS scan. The explanation: “Die Rechnung liefert für den Planeten Merkur ein Vorschreiten des Perihels um 43″ in hundert Jahren, während die Astronomen 45″ ± 5″ als unerklärten Rest zwischen Beobachtungen und Newtonscher Theorie angeben. Dies bedeutet volle Übereinstimmung” (p. 839); on p. 831 he calls the motion “die von Leverrier entdeckte säkulare Drehung der Merkurbahn”.
3. R. S. Park, W. M. Folkner, A. S. Konopliv, J. G. Williams, D. E. Smith, M. T. Zuber, Precession of Mercury’s Perihelion from Ranging to the MESSENGER Spacecraft, The Astronomical Journal 153(3), 121, 2017, doi:10.3847/1538-3881/aa5be2. Total precession 575.3100 ± 0.0015 arcseconds a century.
4. A. Fienga and O. Minazzoli, Testing theories of gravity with planetary ephemerides, Living Reviews in Relativity 27, 1, 2024, doi:10.1007/s41114-023-00047-0 (CC BY 4.0): “astronomers identified the limitations of Newton’s theory through the anomalous advance of Mercury’s perihelion”.

VI · the second layer

Three further results

1 · How wrong would the old clocks have had to be?

From the null above: a systematic error of 4.2″ per transit, independent from transit to transit, lets Le Verrier’s procedure manufacture his excess one time in twenty; the cheapest single transit, 1832, would have to be off by 11.7″ on its own. His 21 contacts fit to 0.59″ once the excess is allowed. The quantity is citable, and answers his argument in arc, with his own rows.

2 · What Le Verrier’s own transits can tell apart

Hold the perihelion correction at any value, re-solve the rest by his method, and watch the fit. Inside his own model, on his masses and his adopted values, his 21 contacts barely separate a motion of 38 a century from one of 43, about the size Newcomb later found on other masses, other reductions and a longer span of years: holding it at 43 raises the root mean square miss only from 0.59″ to 0.64″. They do separate either from zero, which leaves 2.43″.

3 · The planet the sums required

The page’s own first-order secular theory for a body on a circular orbit in Mercury’s plane, inside its orbit: the perihelion advance it gives Mercury is Mercury’s mean motion times one quarter of the body’s mass (as a fraction of the Sun’s) times αb(α), where α is the ratio of the distances and b a Laplace coefficient computed in your browser. With the Mercury mass Le Verrier used, 1/3,000,000 of the Sun (Newcomb, p. 469), a body of Mercury’s mass must sit at 0.169 AU, α = 0.437, to give 38.3 a century: “un peu inférieure à la moitié”, as he wrote. At Lescarbault’s 0.1427 AU it needs 1.59 of those Mercury masses (3.2 of the modern ones), a world about 7,200 km across at Mercury’s density; the body Le Verrier inferred from the spot, one seventeenth of Mercury, supplies 1.42. The page refuses distances beyond Mercury’s perihelion distance, 0.3075 AU, where such an orbit would cross Mercury’s and the formula no longer applies. A calculation of another kind, a numerical integration of the planets with such a body added, reaches the same kind of answer for the modern excess: more than about three times Mercury’s mass on the orbit Le Verrier derived for Lescarbault’s body, and more than Mercury’s own mass even on the most favourable orbit it found (S. P. Pogossian, Vulcan and anomalous displacement of Mercury’s perihelion; we read the 2022 preprint of his 2023 paper in Astrophysics and Space Science). It concludes that a body of the mass Le Verrier inferred cannot explain the excess, and that one heavy enough would appear in the sky almost as large as Mercury.

The page’s result, in one sentence

Le Verrier’s own elimination on his own 21 equations gives 38.44 arcseconds a century against his printed 38.3; Newcomb’s 1882 control, recomputed from his equations of condition, gives 42.93 against his printed 42.95, and planted with Le Verrier’s 38.3 it recovers 38.28, 31.6 formal and 13.0 jackknife standard errors from zero.

The further result, and how far it goes

The citable results are the page’s: the multiplier set that reproduces Le Verrier’s printed (IV); the sign misprint and the constant of Newcomb’s equation (8), and his δn; the power of his control at the claimed size; and the systematic error per transit at which the claimant’s own procedure manufactures the claim. On what exists already: We searched the Artificial Wasteland corpus, the web (six search-engine queries), arXiv and GitHub on 2026-09-23 and did not find a public recomputation of Le Verrier’s 21 printed transit equations of condition that reproduces his eliminations by Cauchy’s method, or any note that Newcomb’s printed normal equation (8) carries a sign error in its W′ coefficient; the nearest we found, Constantin (2010, arXiv:1104.0548) and a public script built on its table, re-solve Le Verrier’s meridian equations of condition by least squares, and a 1993 paper by T. Inoue in Celestial Mechanics and Dynamical Astronomy, whose title says it looks for an excess motion of Mercury’s node in the observations Le Verrier used, may be nearer but could not be read. Two books we did not examine: N. T. Roseveare’s history, Mercury’s Perihelion from Le Verrier to Einstein (1982), and Trevor G. Underwood’s annotated translation, Urbain le Verrier on the Movement of Mercury (2021).

VII · the check

The check

What this page rests on, and what it chose

• Transcriptions. Le Verrier’s 21 equations were typed from 575 dpi crops of the scan of Annales t. V, p. 80, and every column is tested by his printed (I); his steps (I) to (VII), his residual table and adopted values from pp. 79-100. Newcomb’s 58 equations were typed from the page images of pp. 457-458 and tested by his printed normal equations; one printed sign in them (1868, contact II) is missing and is read as minus, which the normal equations confirm. Both files carry the printed page numbers.
• Choices, each yours to change. How Le Verrier’s equations are solved (his multipliers, the plain sign rule, least squares); which values are fixed before the perihelion is read off; which transits he keeps; Newcomb’s k; his rejections, and the unit weight given to restored contacts; the mass of Venus; the planted excess; the null’s noise and its systematic error; the perihelion motion held fixed; the inner planet’s distance.
• The page’s own readings. The 1769 multiplier (a reconstruction); the reading of Newcomb’s (8) constant as a slip carried into his solution; the anchor tolerance of 0.5; the 5 per cent agreement rule for (II) to (IV); the rule for calling the control able to confirm; Le Verrier’s arbitraries and e′ held at zero, as his p. 82 table and p. 96 formula leave them.
• What it refuses. A planted excess beyond ±100 arcseconds a century, outside Newcomb’s first-order regime; an inner planet beyond Mercury’s perihelion distance, 0.3075 AU, where its circular orbit would cross Mercury’s, or inside the Sun; a selection of transits that leaves fewer than five contacts, a pivot at zero, or one season only (at one node the perihelion’s motion cannot be told from the mean motion: the page’s formal standard error would be 206 arcseconds a century with the November transits alone and 227 with the May ones); an empty box; a significance for 38 against 43. A selection of both seasons whose three formal standard errors exceed the claim is computed but marked undetermined.
• Not redistributed. The ADS scans of the Annales and of Einstein’s paper are covered by the ADS terms; the page ships transcribed numbers and short quotations only.