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REVIEW 2 major objections 2 minor 56 references

The unreasonable effectiveness of the cathetus rule in ancient and modern optics

T0 review · 2 major / 2 minor · reviewed 2026-06-30 · grok-4.3

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Pith's one-line read The cathetus rule holds for sagittal images and enables astigmatism assessment via meridional ray tracing alone.

desk verdict The paper revives the cathetus rule as a meridional shortcut for sagittal images but asserts rather than derives its match to standard first-order astigmatism. read the letter →

arxiv 2606.28851 v1 pith:FIGGGCWU submitted 2026-06-27 physics.hist-ph physics.optics

Gavin R. Putland This is my paper · ORCID
classification physics.hist-phphysics.optics
keywords cathetusrulesagittalimageastigmatismmeridionalraytracingparaxialopticssphericalsurfaceshistoryofstigmatic
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

The paper traces the cathetus rule from Euclid through Ptolemy, Kepler, Tacquet, Barrow, and Newton, noting its historical disproof for tangential images outside the paraxial limit. It argues that the rule is equivalent to assuming a stigmatic image with a well-defined perpendicular, an assumption that holds approximately in first-order Gaussian optics. This validity for the sagittal image allows tracing only the chief ray from an off-axis object point through successive spherical surfaces, locating each sagittal image point on the rectilinearly extended chief ray by applying the rule at each surface. The result fills a gap in meridional ray tracing by permitting leading-order astigmatism evaluation without any rays outside the meridional plane.

What carries the argument

The cathetus rule, which places the image point on the perpendicular from the object point to the surface, applied to the sagittal image in the paraxial limit.

What would settle it

A paraxial ray-trace calculation through a spherical surface that shows the sagittal image point lying off the chief ray would falsify the claim.

Watch

Extended reading notes

Core claim

The cathetus rule is equivalent to the assumption that the image is stigmatic and the cathetus well defined. This narrow assumption is approximately true in the first-order (paraxial, Gaussian) analysis of lenses and mirrors. The validity of the rule for the sagittal image fills a critical gap in meridional ray-tracing through spherical surfaces: by tracing the chief ray from an off-axis object-point, then applying the cathetus rule to the successive surfaces, one can locate successive sagittal image-points on the chief ray (produced rectilinearly through surfaces as necessary), and hence assess astigmatism to leading order, without tracing any rays outside the meridional plane.

Load-bearing premise

The cathetus rule remains valid for the sagittal image formed at spherical surfaces to leading order, allowing the image point to be placed on the rectilinearly extended chief ray without additional non-meridional rays.

Editorial extensions

If this is right

  • Successive sagittal image points can be found on the chief ray after each reflection or refraction.
  • Astigmatism can be assessed to leading order using only rays in the meridional plane.
  • The rule applies to both reflection and refraction at spherical surfaces under the paraxial approximation.
  • Modern Gaussian optics expositions contain unacknowledged applications of the rule.

Reading between the lines

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

  • Lens design software could incorporate the rule for quick initial sagittal-focus estimates before full three-dimensional tracing.
  • The historical rule may offer computational shortcuts in other paraxial optical problems involving off-axis points.
  • Higher-order extensions of the method would require separate handling of tangential-image deviations.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 2 minor

Summary. The paper argues that the ancient 'cathetus rule' (image lies on the perpendicular from object point to surface) is equivalent to the assumption of a stigmatic image, holds approximately in the paraxial (Gaussian) limit, and was historically applied or salvaged for sagittal images (Newton) after disproof for tangential ones (Barrow). It further claims that applying the rule to successive sagittal images along a traced chief ray in meridional ray-tracing through spherical surfaces allows assessment of leading-order astigmatism without tracing non-meridional rays.

Significance. If the claimed equivalence and validity for sagittal images to leading order are established, the work offers historical insight into the development of image concepts from Euclid through Kepler, Tacquet, Barrow, and Newton, while identifying an unacknowledged shortcut in modern first-order optics. The absence of fitted parameters or invented entities strengthens the historical claims, but the modern application requires explicit linkage to standard results.

major comments (2)
  1. [Abstract and modern-application section] Abstract (final paragraph) and the section on modern application: the central claim that the cathetus rule 'fills a critical gap' by locating sagittal image-points on the chief ray to assess astigmatism to leading order requires an explicit derivation showing equivalence to the Coddington equations or the Seidel astigmatism term for off-axis points at spherical surfaces, even to O(field²); the manuscript asserts the validity for the sagittal image without providing the steps, error analysis, or comparison to standard formulas.
  2. [Section on paraxial/Gaussian analysis] The discussion of the paraxial equivalence: while the rule is stated to be 'equivalent to the assumption that the image is stigmatic and the cathetus well defined,' no explicit mapping is given between the geometric construction and the standard paraxial ray-transfer matrix or the condition for zero astigmatism in the sagittal plane.
minor comments (2)
  1. The historical narrative would benefit from explicit section headings or subsection numbering to separate the Euclid-to-Newton history from the modern ray-tracing claim.
  2. Notation for the cathetus and chief-ray extension should be defined once with a diagram or equation reference rather than relying on prose description alone.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive report. The two major comments correctly identify places where the manuscript asserts key equivalences without supplying the explicit derivations or mappings requested. We will revise the paper to include these, as detailed in the point-by-point responses below.

read point-by-point responses

  1. Referee: [Abstract and modern-application section] Abstract (final paragraph) and the section on modern application: the central claim that the cathetus rule 'fills a critical gap' by locating sagittal image-points on the chief ray to assess astigmatism to leading order requires an explicit derivation showing equivalence to the Coddington equations or the Seidel astigmatism term for off-axis points at spherical surfaces, even to O(field²); the manuscript asserts the validity for the sagittal image without providing the steps, error analysis, or comparison to standard formulas.

    Authors: We agree that the manuscript asserts the utility of the cathetus rule for locating successive sagittal images along the chief ray without furnishing the requested derivation. In the revised version we will add a dedicated subsection that derives the sagittal image location obtained by repeated application of the cathetus construction and shows, via a small-field-angle expansion of the exact reflection/refraction law, that this location coincides with the sagittal focal length given by the Coddington equations to O(field²). The same expansion will be compared term-by-term with the Seidel astigmatism contribution for a spherical surface; an explicit remainder estimate will quantify the leading-order error. This addition directly addresses the gap noted by the referee. revision: yes

  2. Referee: [Section on paraxial/Gaussian analysis] The discussion of the paraxial equivalence: while the rule is stated to be 'equivalent to the assumption that the image is stigmatic and the cathetus well defined,' no explicit mapping is given between the geometric construction and the standard paraxial ray-transfer matrix or the condition for zero astigmatism in the sagittal plane.

    Authors: The referee is correct that the manuscript states the equivalence to the stigmatic-image assumption without an explicit mapping to the paraxial formalism. We will insert a short derivation that begins from the geometric definition of the cathetus (the perpendicular from object point to surface) and shows that, under the standard paraxial approximations (small angles, neglect of higher powers), this construction is identical to the condition that the sagittal ray-transfer matrix maps the object height to an image height lying on the chief ray. The same steps will demonstrate that the resulting sagittal astigmatism vanishes identically in the Gaussian limit, thereby supplying the missing link between the ancient geometric rule and the modern ray-transfer matrix. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity; equivalence to paraxial assumption and modern application are independent reinterpretations

full rationale

The paper states that the cathetus rule 'is equivalent to the assumption that the image is stigmatic and the cathetus well defined' and that this 'narrow assumption is approximately true in the first-order (paraxial, "Gaussian") analysis', then observes that the rule's validity for the sagittal image 'fills a critical gap in meridional ray-tracing'. This is a historical and interpretive claim rather than a derivation chain in which any prediction reduces by construction to fitted inputs, self-citations, or renamed results. No equations are presented that force a result from the paper's own definitions, and the modern utility is obtained by applying the ancient rule to a new context without self-referential closure or load-bearing self-citation.

Assumptions & free parameters 0 free parameters · 1 assumptions · 0 invented entities

The central claims rest on standard paraxial optics definitions and the historical statements of the cathetus rule; no new free parameters or invented entities are introduced.

assumptions (1)
  • domain assumption The cathetus rule holds for the sagittal image at spherical surfaces to leading order.
    Invoked to justify placing sagittal image points on the rectilinearly extended chief ray.

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Cite this review

Pith. "Pith review of The unreasonable effectiveness of the cathetus rule in ancient and modern optics." pith.science (2026). https://pith.science/paper/FIGGGCWU

@misc{pith2026260628851,
  author       = {Pith},
  title        = {Pith review of: The unreasonable effectiveness of the cathetus rule in ancient and modern optics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FIGGGCWU}},
  note         = {Machine review of arXiv:2606.28851}
}
read the original abstract

The "cathetus rule" in optics alleges that the image of an object-point, formed by reflection or refraction at a surface, lies on the perpendicular ("cathetus") from the object-point to or through the surface. The first known statement of the rule, attributed to Euclid, was for a plane or spherical mirror. The rule was extended to refraction by Ptolemy.... Kepler was universally credited with the first disproof-and-salvage of the cathetus rule until 2018, when Benedetti's priority was exposed by Goulding. Kepler notwithstanding, the rule was reaffirmed by Tacquet for plane and spherical mirrors, except for the case in which the rays converge toward a point behind the eye; this became known as the "Barrovian case" because it troubled Barrow, in spite of his modern concept of an image. Barrow demolished the cathetus rule for the tangential image except in the paraxial limit, and Newton salvaged it for the sagittal image. The rule then seems to fade from history. But the rule is equivalent to the assumption that the image is stigmatic and the cathetus well defined. This narrow assumption is approximately true in the first-order (paraxial, "Gaussian") analysis of lenses and mirrors; and unacknowledged applications of the ancient rule can indeed be discerned in modern expositions of that subject. Moreover, the validity of the rule for the sagittal image fills a critical gap in meridional ray-tracing through spherical surfaces: by tracing the chief ray from an off-axis object-point, then applying the cathetus rule to the successive surfaces, one can locate successive sagittal image-points on the chief ray (produced rectilinearly through surfaces as necessary), and hence assess astigmatism to leading order, without tracing any rays outside the meridional plane.

Figures

Figures reproduced from arXiv: 2606.28851 by the authors.

Figure 1
Figure 1. This modern diagram, for locating the image I of an object-point O in a convex spherical mirror whose center of curvature is C, happens to agree with the ancient cathetus rule. In this case the cathetus is OC and the point of reflection is V. According to the rule, the image is at the intersection of the line of sight (through the point of reflection) and the cathetus. (Diagram by ‘Forna’ at Wikimedia Commons; publi… view at source ↗
Figure 2
Figure 2. Kitchen-bench reconstruction of Alhacen’s first experiment attempting to prove the cathetus rule for refraction. A vertical diameter and a sloping diameter are drawn on the base of a coffee mug. The vertical diameter appears to continue vertically into the water, showing that the image of the point of intersection is in the plane of the viewing position and the cathetus (vertical diameter). Alhacen would claim that … view at source ↗
Figure 3
Figure 3. Location of the image P ′ of an object-point P in a plane mirror B. In this special case the cathetus rule follows simply and rigorously from the law of reflection (although Ptolemy and Alhacen still cite the cathetus rule independently). (Original diagram by ‘MikeRun’ at Wikimedia Commons.) does Alhacen, who again shows that the line of sight intersects the cathetus as far behind the mirror as the image-point is in… view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: Isaac Barrow’s location of the tangential image Z of an object-point A seen by an observer at O due to refraction. The tangential image is the point of tangency between the refracted ray produced back from O, and the caustic (common tangent curve) of all the other prod…
Figure 5
Figure 5. Figure 5: Stigmatic image F2 of a virtual object-point F1 , formed by reflection in a convex hyperboloidal mirror with foci F1 and F2. Rays initially directed toward F1 are reflected through F2, including the undeviated ray F2F1, which is the cathetus. Thus the image-point is th…
Figure 6
Figure 6. Figure 6: Distances and angles for refraction at a spherical surface. (Diagram by the author.) the cathetus, hence the limit of the tangential image-point as the observation point approaches the cathetus (as shown by Barrow). The limiting position of I, by construction, is on th…
Figure 5
Figure 5. Figure 5: (p. 196) [PITH_FULL_IMAGE:figures/full_fig_p049_5.png]
Figure 7
Figure 7. Figure 7: Location of the image B ′ of an object-point B due to a thin lens. The (approximately) undeviated ray BOB′ plays the role of the ancient cathetus: the image may be taken to be at the intersection of this ray and any other refracted ray originating at B. (Diagram by ‘Ta…

Discussion (0). Continue with ORCID to comment.

Reference graph

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Reviewed June 30, 2026 · model on record in the stance chip above.