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arxiv: 2606.28851 · v1 · pith:FIGGGCWUnew · submitted 2026-06-27 · ⚛️ physics.hist-ph · physics.optics

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

Gavin R. Putland This is my paper

Pith reviewed 2026-06-30 08:45 UTC · model grok-4.3

classification ⚛️ physics.hist-ph physics.optics
keywords cathetus rulesagittal imageastigmatismmeridional ray tracingparaxial opticsspherical surfaceshistory of opticsstigmatic image
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The pith

The cathetus rule holds for sagittal images and enables astigmatism assessment via meridional ray tracing alone.

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

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.

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.

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.

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.

Where Pith is reading between the lines

These are 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.

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.

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.

Figures

Figures reproduced from arXiv: 2606.28851 by Gavin R. Putland.

Figure 1
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 Comm… view at source ↗
Figure 2
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 c… view at source ↗
Figure 3
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-p… view at source ↗
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 produced refracted rays from the same object-point in the same plane of refraction. Point K is the image location given by the old cathetus rule… view at source ↗
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 the intersection of the cathetus with any other reflected ray. (Diagram by ‘Episcophagus’ at Wikimedia Commons.) 3.2 Axis The convenience of c… view at source ↗
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 the cathetus, salvaging the cathetus rule as an approximation for small angles; and because the limit is a stationarity, the deviation from th… view at source ↗
Figure 5
Figure 5. Figure 5: (p. 196) [PITH_FULL_IMAGE:figures/full_fig_p049_5.png] view at source ↗
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 ‘Tamasflex’ at Wikimedia Commons.) (produced if necessary) from the same object-point. In short, the approximately undeviated ray plays the rol… view at source ↗
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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

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

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.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 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.

axioms (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.

pith-pipeline@v0.9.1-grok · 5861 in / 1329 out tokens · 43993 ms · 2026-06-30T08:45:42.594791+00:00 · methodology

discussion (0)

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Reference graph

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    and as the “auxiliary axis” in their Figs. 3F & 3H (pp. 51, 53). More constructions reminiscent of the cathetus rule, with the ray through the center of the lens in the role of the cathetus, can be found in their Figs. 4F, 4G, 4H (for each lens), 4I (ditto), and 7B, and in (e.g.) Figs. 5.23, 5.24, and 5.29 of Hecht (2017, pp. 172, 176). For an object-poin...

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    as is often incorrectly asserted in the literature

    240 Jenkins & White, 1976, p. 169, Eqs. (9p), 2nd eq. (for refraction) and p. 111, 2nd eq. (for reflection), citing Monk, 1963, pp. 424–6. 241 Conrady (1992), pp. 409–10. §4. Off-axis astigmatism 53 characteristic function.242 None of these sources uses the word cathetus or refers to the cathetus rule. Corresponding expressions for the distance of the tan...

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    In our Figure 6, thesagittal plane after refraction is the plane perpendicular to the plane of the diagram and containing the ray PI

    is similarly misleading; the sagittal focal lineS should be along the auxiliary axis—that is, parallel to the incoming rays (the object-point being at infinity). In our Figure 6, thesagittal plane after refraction is the plane perpendicular to the plane of the diagram and containing the ray PI. If we leave the sagittal plane fixed and rotate the point of ...

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    244 Born & Wolf, 2002, p

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    To the first order

    Earlier on the same page, Born & Wolf themselves may seem to have asserted what they now deny. But the exculpatory words are “To the first order”; for a thin pencil, if the distance between the focal lines measured along the central ray is first-order, then the obliquity of either focal line to the central ray is second-order. 54 Putland, The unreasonable...

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    R. Bacon (ed. J. Combach), 1614,Perspectiva, Frankfurt: Wolfgang Richter for Anton Humm; google.com/books?id=Cn6k7IC-yaMC. 56 Putland, The unreasonable effectiveness of the cathetus rule. . . I. Barrow, 1669,Lectiones XVIII, Cantabrigiæ in scholis publicis habitæ; in quibus opticorum phænomenωn genuinæ rationes investigantur, ac exponuntur, London: Willia...

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    Once Snell breaks down: From geometrical to physical optics in the seventeenth century

    O. Darrigol, 2012, A History of Optics: From Greek Antiquity to the Nineteenth Century, Oxford. E.J. Dijksterhuis (ed.), 1955, The Principal Works of Simon Stevin, vol. 1, Amsterdam: Swets & Zeitlinger. F.J. Dijksterhuis, 1999,Lenses and Waves: Christiaan Huygens and the Mathematical Science of Optics in the Seventeenth Century (doctoral thesis), Universi...

    work page doi:10.1080/00033790.2022.2026479 2012