Bo\v{s}kovi\'{c}'s Spherical Trigonometric Solution for Determining the Axis and Rate of Solar Rotation by Observing Sunspots in 1777

Damir Hr\v{z}ina; Davor Krajnovi\'c; Domagoj Ru\v{z}djak; Dragan Ro\v{s}a; Dragan \v{S}poljari\'c; Ivica Skoki\'c; Mirko Husak; Roman Braj\v{s}a

REVIEW 4 major objections 4 minor 26 references

Bo\v{s}kovi\'{c}'s Spherical Trigonometric Solution for Determining the Axis and Rate of Solar Rotation by Observing Sunspots in 1777

T0 review · 4 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read This paper derives, for the first time, the closed-form spherical-trigonometric equations behind Bošković's 1785 verbal method, and reproduces his 1777 solar rotation elements from three sunspot positions.

desk verdict A genuinely useful first derivation of Bošković's spherical-trig method, but the 'no approximation' and 10^-5 arcsec claims need to be walked back. read the letter →

arxiv 2507.20383 v1 pith:ORDDSAG5 submitted 2025-07-27 astro-ph.SR

Mirko Husak , Roman Brajša , Dragan Špoljarić , Davor Krajnović , Domagoj Ruždjak , Ivica Skokić , Dragan Roša , Damir Hržina This is my paper · ORCID

classification astro-ph.SR

keywords RuđerBoškovićsunspotpositionssolarrotationelementssphericaltrigonometryequatorinclinationascendingnodelongitudesiderealperiodhistoricalastronomy

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper claims that the spherical-trigonometric method Ruđer Bošković described in words in 1785 for finding solar rotation elements from three sunspot positions can be expressed as a complete, closed system of equations. The authors derive those equations and apply them to Bošković's own 1777 measurements, obtaining solar equator inclination $i = 6.80728^\circ$, ascending-node longitude $\Omega = 74.04774^\circ$, sidereal rotation period $T' = 26.806232$ days, and synodic period $T'' = 28.929403$ days. These values agree with his published planar solution ($i = 6^\circ49'$, $\Omega = 74^\circ03'$) and his sidereal period of 26.77 days at the arcminute level. The importance, if the derivation is right, is that all three rotation elements can now be extracted from just three spot positions in one procedure, with no iterative fitting.

What carries the argument

The load-bearing object is the configuration of eight oblique spherical triangles (plus the short-solution triangles △10 and △11) built from the northern ecliptic pole $P$, the northern solar-equator pole $P'$, and three sunspot positions $C$, $C'$, $C''$ in ecliptic coordinates. Sides such as $CC'$ and $P'C''$ are obtained by repeated cosine and cotangent rules; midpoints $E$ and $E'$ create right triangles through which $P'E'$ is found; and mirror symmetry of triangles △9 and △6 sets $P'C' = P'C''$. The final elements come from the side $PP' = i$ via the cosine rule, the angle at $P$ giving the node $\Omega$, and the angle at $P'$ combined with elapsed mean solar time giving $T'$.

What would settle it

Compute high-precision heliographic positions of a single sunspot at three times, apply these equations, and compare with an independent least-squares rotation-element fit over a longer track; if the three-position values differ from the fit by more than the measurement uncertainties, the rigid-parallel assumption is the source of the bias.

Watch

Extended reading notes

Core claim

The paper's central claim is that Bošković's §VII, №76–№81 description corresponds to an exact chain of spherical triangles linking the northern ecliptic pole, the northern solar-equator pole, and three observed positions of one sunspot in ecliptic coordinates. Solving the chain with cosine and cotangent rules yields the arc between the poles as the inclination $i$, the angle at the ecliptic pole as the node $\Omega$, and the angle at the equator pole divided by the elapsed mean solar time as the sidereal period $T'$. The reported results are $i = 6.80728^\circ = 6^\circ48'26.20337''$, $\Omega = 74.04774^\circ = 74^\circ02'51.87646''$, and $T' = 26.806232$ days, closely matching Bošković's planar solution and his sidereal period. The same equations over all twenty triples of his six positions produce a wide spread in $i$ and $\Omega$, so the choice of well-separated positions matters, and the paper concludes that the spherical solution is complete and gives all three elements without approximation.

Load-bearing premise

The chain assumes the spot stays at one solar latitude and the Sun rotates rigidly between the observations, so the pole-to-spot distance is the same for the first and third positions and the triangles on either side of the midpoint are mirror images.

Editorial extensions

If this is right

Three well-separated positions of one spot are enough to return $i$, $\Omega$, and $T'$ simultaneously from closed-form trigonometry, so no least-squares fit or initial guess is required.
Applying the new equations to Bošković's six-day 1777 sequence reproduces his published values ($i \approx 6^\circ49'$, $\Omega \approx 74^\circ03'$, $T' \approx 26.8$ d), strengthening the case that his verbal procedure was numerically complete.
Relative to the Carrington and Spörer elements for 1777, the spherical solution's errors are about 2–3% in $\Omega$, 2–7% in $i$, and under 1% in $T'$, making it a serviceable quick estimator for sparse historical data.
Because the same chain works for any three tracked surface features, the paper's stated generalization to other rotating bodies such as stars and exoplanets follows directly.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The derivation's rigid-parallel assumption means that differential rotation will leak into the inferred elements; the three period estimates in Table 4 (26.77, 26.81, 26.85 days) already bracket the effect, so feeding the equations modern latitude-resolved spot data would quantify the bias.
A sharper historical test would re-reduce Bošković's apparent-disk measurements with modern ephemerides before applying the new equations; any shift in $\Omega$ and $i$ away from his 1785 values would measure how much log-table rounding contributed to the original result.
The closed-form chain could be inverted symbolically to propagate measurement errors into $i$, $\Omega$, and $T'$ analytically, giving uncertainty estimates without Monte Carlo sampling.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The paper does what it promises: it gives the first explicit spherical-trigonometric equations for Bošković's 1785 method, applied to his 1777 sunspot positions. The math is classical and the worked example is transparent; I checked the key numbers and they reproduce Bošković's published rounded values. That is a real contribution to a narrow historiographic puzzle.

The soft spots are mostly about framing. The conclusion's claim that the solution is "closed equations without any approximation" is too strong. The derivation assumes the three sunspot positions lie on the same heliographic parallel and that the spot rotates rigidly (P'C = P'C' = P'C''). That is a model assumption, not a consequence of the spherical geometry. The paper's own Table 6 shows how much the answer depends on triple choice: across the 20 triples, i varies from 3.5° to 18.7°, and Ω from -41.2° to 87.8°. The chosen triple [136] gives the historically nice result, but other triples are far off. The paper describes Bošković's selection criteria in Section 4.2, which is honest, but it does not reconcile that sensitivity with the "no approximation" claim.

The 10^-5 arcsec precision claim in Section 4.3 is also unsupported. The inputs are arcminute-level (Bošković's published positions are rounded to arcminutes), so the many-digit agreement between the full and short solutions is internal arithmetic consistency, not measurement precision. The authors should either propagate errors from the inputs or explicitly say the precision refers only to the consistency of the equations.

Using the unpublished VFM method as confirmation is weak. "Paper in preparation" is not a citable validation. If VFM is essential, the authors need to publish it or at least provide a preprint; otherwise it should be demoted to a sanity check or removed. Table 6 is fine as exploratory, but the labels "near Carrington" and "near Spörer" are doing a lot of work.

There are also a few minor typographical issues (e.g., the footnote about №79 confusing CC′ and C′C′′), but nothing that undermines the derivation.

Bottom line: if the authors soften the precision and approximation claims, add a real error analysis, and either make VFM citable or drop it as validation, this is a solid historical reconstruction. It deserves a serious referee and, with revisions, publication. I would not cite it in my own work, but for anyone working on 18th-century solar astronomy or the history of spherical trigonometry, it will be the reference for Bošković's method.

Referee Report

4 major / 4 minor

Summary. The paper reconstructs the trigonometric spherical solution that Ruđer Bošković described verbally in his 1785 Opuscule II for determining the solar rotation elements (i, Ω, and the sidereal period T′) from three observed positions of the same sunspot. The authors derive a sequence of spherical-trigonometric equations for the full and the “short” solutions, apply them to Bošković's September 1777 sunspot positions (positions 1, 3, and 6), and obtain i = 6.80728° (≈6°48′26″), Ω = 74.04774° (≈74°03′), and T′ = 26.806 d (≈26.81 d), closely reproducing Bošković's published values from the planar-trigonometric solution. They also compare with a contemporary method (Roša et al. 2021) and with their own unpublished vector formalism method, and they examine the sensitivity of the result to the choice of triple via a 20-combination table.

Significance. If the reconstruction is correct, the paper closes a historical gap by turning Bošković's verbal description into a modern algorithm, and it appears to be the first complete equation set for that method. The derivation is self-contained and the worked example is internally consistent, with no free parameters, and it reproduces the arcminute-level published values—a concrete success that supports the historical claim. The main value is historical-methodological rather than a new solar-physics measurement. The paper's stronger claims (arcsecond-level precision, “without any approximation,” a single unambiguous procedure, and independent VFM confirmation) are not supported by the data or by the derivation as written, and these claims should be revised. With those revisions the paper would be a sound contribution to the history of solar rotation measurements.

major comments (4)

[§2.3, §2.4.1, Eqs. (26), (34), (43), (50)–(51); §4.2, Table 6] The derivation is exact only under the assumption that the three positions lie on a single heliographic parallel, i.e. P′C = P′C′ = P′C′′, combined with rigid rotation. This equality is asserted from the mirror-symmetry of triangles △9 and △6 (§2.4.1) and is not derived from the measurements. The conclusion's statement that the spherical solution consists of “closed equations without any approximation” is therefore unsupported: the trigonometric algebra is exact, but the physical model contains the constant-latitude, rigid-rotation approximation. The sensitivity of this assumption is visible in Table 6, where the VFM calculation over the 20 triples gives i between 3.512° and 18.696° and Ω between −41.197° and 87.811°, while Bošković's selected triple [136] gives i=6.807° and Ω=74.048°. The selection criteria in §4.2 (C1≈C6, C3 at minimum latitude) are precisely a search for a triple that approximately satisfies the same-parallel condition, so the method is not a general three-position inversion without additional assumptions.
[§4.3, Tables 1, 2, and 4] The claimed precision of 10⁻⁵ arcseconds is not supported by the input data. Table 1 lists ecliptic longitudes and latitudes to arcminutes, and the historical measurement process is far coarser; nevertheless Table 4 reports i = 6.80728° = 6°48′26.20337″ and Ω = 74.04774° = 74°02′51.87646″. A 10⁻⁵ arcsecond digit is about nine orders of magnitude finer than the arcminute input. The statement in §4.3 that the results were “calculated using high precision and closed equations without loosing precision” conflates arithmetic precision with measurement precision. The paper should propagate the input uncertainties and quote i, Ω, and T′ to a justified number of significant digits; the agreement with Bošković's values is then at the arcminute level, which is sufficient for the historical claim but not for the advertised precision.
[§4.1, Table 4] The VFM method used as a “contemporary method” check is unpublished (“paper is in preparation”). For the triple [136] it returns values identical to the spherical-trigonometric solution to eight significant figures (i=6.80727871°, Ω=74.0477436°, T′=26.8062322 d). If VFM is mathematically equivalent to the present derivation, this agreement is by construction and provides no independent confirmation; if it is not equivalent, the reader needs the VFM definitions to evaluate the comparison. The independent validation should rest on the comparison with Bošković's published values and with Roša et al. (2021); the VFM table should be moved to supplementary material or explicitly reported as an internal consistency check.
[§2.2–2.4, Eqs. (7), (30), (42), (55)] The equations use inverse trigonometric functions (arccos, arctan) without specifying branch selection. For example, Eq. (30) determines B′′−D from its cosine, which has two possible signs; Eqs. (42) and (55) have the same issue, and Eq. (7) assumes that P lies inside the angle CC′C′′ so that the two angles sum rather than subtract. A reader applying the method to arbitrary triples—as the Conclusion invites for exoplanets and stars—cannot reproduce the results without additional quadrant rules or case distinctions. The claim of a single closed-form procedure is therefore incomplete as stated.

minor comments (4)

[Table 2] The row for Eq. (6) appears twice with different values: first cos P C′′C′ = 0.107516 with angle 83.82785°, and later cos P C′′C′ = 0.108113 with angle 83.79345°. The second value is the one used in Eq. (28); the first appears to be a typographical error and should be harmonized.
[§4.2] The numerical values quoted for C1 and C6 (20°27′ and 22°45′) do not match the entries in Table 1 as the table is currently formatted (20°37′ and 22°45′, if read literally); please reconcile the text with the table.
[Eq. (37)] The quantity A = 365.25 days should be defined explicitly as the Earth's orbital (or tropical/sidereal) period used in converting between synodic and sidereal rotation periods; the present text introduces it only in a footnote-like parenthetical.
[Figure 5] The scanned historical page in Figure 5 is difficult to read in the arXiv version; a transcription of the relevant numbered paragraphs (№76–№81) would make the mapping from Bošković's text to the equations easier to verify.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the spherical-trigonometric reconstruction is self-contained and the historical validation is an external benchmark, not an input-output identity.

full rationale

The derivation chain is self-contained: equations (1)-(57) are spherical-trigonometric identities applied to the three observed ecliptic positions and mean solar times, and i, Omega, and T' are not used as inputs. No free parameter is fitted to Bošković's published values; the reproduction of his 1785 results is a consistency check of the reconstruction against an external historical benchmark, and the agreement with Carrington/Spörer values is an independent external comparison. The same-parallel/mirror-symmetry step (P'C = P'C' = P'C'', Sections 2.3 and 2.4.1) is a physical modeling premise of the rigid-rotation method, not a consequence of the outputs, so it creates a correctness/robustness limitation (visible in the Table 6 scatter) but not circularity. The unpublished VFM method is a same-author corroboration and is not load-bearing: the central reconstruction does not depend on it. The '10^-5 arcsec precision' statement refers to numerical internal consistency of the closed-form computation, not to validation accuracy, and is a precision-claim issue rather than a circular step. Therefore no significant circularity is found.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper introduces no new physical entities and fits no free parameters. The derivation rests on classical spherical trig and two physical model assumptions: constant heliographic latitude (rigid rotation) and uniform angular velocity. These are standard for analyzing historical sunspot observations, but they are model choices, not derived facts. The input data are historical and carry arcminute-level precision.

assumptions (4)

domain assumption The three sunspot positions lie on a single heliographic parallel (same heliographic latitude), i.e., P'C = P'C' = P'C''.
Invoked in Section 2.3 Equation 34 and the mirror symmetry in Section 2.4.1; if the spot migrates in latitude over the observation interval, the derived i, Omega, and T' are biased.
domain assumption The Sun's rotation is uniform between observations, so the angular travel around the solar axis is proportional to elapsed time (Equation 35).
Used to convert the angle CP'C'' into a sidereal period; differential rotation would make the period a function of latitude.
domain assumption The three positions are of the same physical spot and the ecliptic coordinates from Bošković's Tables are accurate to about an arcminute.
The method uses positions 1, 3, and 6 as the same sunspot; if identification is wrong, the output is meaningless. Also, Bošković's historical measurements carry unknown systematic errors.
standard math Standard spherical trigonometry (cosine rule, sine rule, cotangent rule) applies to the eight constructed triangles.
The entire derivation uses these identities; no formal verification is provided.

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Cite this review

Pith. "Pith review of Bo\v{s}kovi\'{c}'s Spherical Trigonometric Solution for Determining the Axis and Rate of Solar Rotation by Observing Sunspots in 1777." pith.science (2026). https://pith.science/paper/ORDDSAG5

@misc{pith2026250720383,
  author       = {Pith},
  title        = {Pith review of: Bo\vskovi\'c's Spherical Trigonometric Solution for Determining the Axis and Rate of Solar Rotation by Observing Sunspots in 1777},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ORDDSAG5}},
  note         = {Machine review of arXiv:2507.20383}
}

abstract

In September 1777 Ru\dj er Bo\v{s}kovi\'{c} observed and measured the sun-spot positions to determine the solar rotation elements. In 1785, among other methods, he described a trigonometric spherical solution for the determination of the position of the axis and rate of the solar rotation using three sunspot positions, but without equations. For the first time, we derive the equations that are applicable to modern computers for calculating the solar rotation elements, as they were described by Bo\v{s}kovi\'{c}. We recalculated Bo\v{s}kovi\'{c}'s original example using his measurements of sunspot positions from 1777 and the equations developed here, confirming his results from 1785. Bo\v{s}kovi\'{c}'s methodology of arithmetic means determines $i$, $\Omega$, and sidereal period $T'$ separately, while the planar trigonometric solution determines $i$ and $\Omega$ together. His spherical trigonometric solution calculates $i$, $\Omega$, and the sidereal period $T'$ in a single procedure. Keywords: Ru\dj er Bo\v{s}kovi\'{c}, Sunspots, Solar rotation, Spherical trigonometry

Figures

Figures reproduced from arXiv: 2507.20383 by the authors.

**Figure 1.** Positions of the sunspot: mean solar time T.M. and ecliptic coordinates lon.t and lat.B.t of the first sunspot, which Boscovich (1785b) observed and measured its positions on the apparent solar disk with astronomical telescope in 1777, and then he calculated T.M., lon.t, and lat.B.t from the measurements using trigonometry and logarithms in T ab.II., p167. Ruđer Bošković used astronomical telescope for observations … view at source ↗

**Figure 2.** Planar trigonometric solution: logarithmic calculation of Ω and i, using three sunspot positions 1, 3, and 6 from Tab. II. ( [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** Solar rotation periods calculated from six sunspot pairs of positions of one sunspot: sidereal T ′ = 26.77 days and synodic T ′′ = 28.89 days, calculated from T ′ (Boscovich 1785b, Tab. IX., Tab. X., and Tab. XI., p168). c:/users/public/documents/20240222 Feb. 22, 2024 17:56:38 [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

Figures from the paper (2 more)

**Figure 5.** Figure 5 [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]

**Figure 6.** Figure 6: Three sunspot positions (B, C, t), (B′ , C′ , t′ ), and (B′′, C′′, t′′) in ecliptic coordinates B, C and mean solar time t with poles P and P ′ make eight triangles for the trigonometric spherical solution by Bošković for solar rotation elements: the inclination of th…

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

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