Abstract

Edme Mariotte in the seventeenth century attributed halos to tiny ice prisms in the atmosphere. Christiaan Huygens attributed them to tiny spheres or cylinders. The two seemingly incompatible theories largely agree in their predictions for the common halos. This article explains why.

© 2008 Optical Society of America

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References

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  1. E. Mariotte, Oeuvres de Mr. Mariotte, Vol. 1, pp. 272-281 (Pierre Vander, 1717).
  2. C. Huygens, Oeuvres Complètes de Christiaan Huygens, Vol. 17 (Martinus Nijhoff, 1932). Available at http://gallica.bnf.fr/.
  3. We often refer to Mariotte's halo-maker as a prism, but we usually ignore all but the entry and exit faces, in which case the “prism” is really an idealized wedge having only two faces, which are half-planes.
  4. W. Tape and J. Moilanen, Atmospheric Halos and the Search for Angle x (American Geophysical Union, 2006).
  5. A. Bravais, “Mémoire sur les halos et les phénomènes optiques qui les accompagnent,” Journal de l'École Royale Polytechnique, 31 Cahier 18, 1-280 (1847).
  6. W. Tape, E. Seidenfaden, and G. P. Können, “The legendary Rome halo displays,” Appl. Opt. 47, H72-H84 (2008).
    [PubMed]
  7. Cassini, “Observation de deux Paraselenes, & d'un Arc-en-Ciel dans le crepuscule,” Histoire de l'Académie royale des sciences avec les mémoires de mathématique et physique 10, 400-402 (1730). The article is originally from 1693.
  8. C. Huygens, Relation d'une Observation faite a la Bibliotheque du Roy, à Paris, le 12 May 1667... (Jean Cusson, 1667). An English version is in .
  9. C. Huygens, B. de Volder, and B. Fullenius, Christiani Hugenii Zelemii, dum viveret, Toparchae. Opuscula Postuma,... (Apud Cornelium Boutesteyn, 1703). Available at http://fermi.imss.fi.it/. An English version is in .
  10. R. Smith, A Compleat System of Opticks in Four Books... (Cornelius Crownfield, Cambridge, 1738). The discussion of halos is in Chap. 11 of Book 2, pp. 199-238.
  11. P. Musschenbroek, Institutiones Physicae Conscriptae in Usus Academicos (Apud S. Luchtmans et filium, 1748). The halo discussion is pp 677-689 and Table XXVIII.
  12. T. Stack, “An Account of a Tract Intituled, Jo. Friderici Weidleri Commentatio de Parheliis Mense Januario Anni 1736...,” Phil. Trans. R. Soc. 41, 459-465 (1739-1741). Available at http://www.jstor.org/stable/i206906.
    [CrossRef]
  13. J. Priestley, The History and Present State of Discoveries Relating to Vision, Light and Colours (J. Johnson, 1772). Halos are treated on pp. 596-630 and in Figs. 141-148.
  14. T. Young, A Course of Lectures on Natural Philosophy and the Mechanical Arts (J. Johnson, 1807). Vol. 1, pp. 443-444 and Plate XXIX, and Vol. 2, pp. 303-308.
  15. G. Venturi, Commentarj sopra la Storie e le Teorie dell' Ottica (Pe' fratelli Masi, 1814).
  16. C. Huygens, “An Account of the Observation, Made by the Philosophical Academy at Paris, May 12 1667 ...” Phil. Trans. R. Soc. 5, 1065-1074 (1670). Available at http://www.jstor.org/stable/i206870.

2008 (1)

1847 (1)

A. Bravais, “Mémoire sur les halos et les phénomènes optiques qui les accompagnent,” Journal de l'École Royale Polytechnique, 31 Cahier 18, 1-280 (1847).

1730 (1)

Cassini, “Observation de deux Paraselenes, & d'un Arc-en-Ciel dans le crepuscule,” Histoire de l'Académie royale des sciences avec les mémoires de mathématique et physique 10, 400-402 (1730). The article is originally from 1693.

1670 (1)

C. Huygens, “An Account of the Observation, Made by the Philosophical Academy at Paris, May 12 1667 ...” Phil. Trans. R. Soc. 5, 1065-1074 (1670). Available at http://www.jstor.org/stable/i206870.

Bravais, A.

A. Bravais, “Mémoire sur les halos et les phénomènes optiques qui les accompagnent,” Journal de l'École Royale Polytechnique, 31 Cahier 18, 1-280 (1847).

Cassini,

Cassini, “Observation de deux Paraselenes, & d'un Arc-en-Ciel dans le crepuscule,” Histoire de l'Académie royale des sciences avec les mémoires de mathématique et physique 10, 400-402 (1730). The article is originally from 1693.

de Volder, B.

C. Huygens, B. de Volder, and B. Fullenius, Christiani Hugenii Zelemii, dum viveret, Toparchae. Opuscula Postuma,... (Apud Cornelium Boutesteyn, 1703). Available at http://fermi.imss.fi.it/. An English version is in .

Fullenius, B.

C. Huygens, B. de Volder, and B. Fullenius, Christiani Hugenii Zelemii, dum viveret, Toparchae. Opuscula Postuma,... (Apud Cornelium Boutesteyn, 1703). Available at http://fermi.imss.fi.it/. An English version is in .

Huygens, C.

C. Huygens, “An Account of the Observation, Made by the Philosophical Academy at Paris, May 12 1667 ...” Phil. Trans. R. Soc. 5, 1065-1074 (1670). Available at http://www.jstor.org/stable/i206870.

C. Huygens, Relation d'une Observation faite a la Bibliotheque du Roy, à Paris, le 12 May 1667... (Jean Cusson, 1667). An English version is in .

C. Huygens, B. de Volder, and B. Fullenius, Christiani Hugenii Zelemii, dum viveret, Toparchae. Opuscula Postuma,... (Apud Cornelium Boutesteyn, 1703). Available at http://fermi.imss.fi.it/. An English version is in .

C. Huygens, Oeuvres Complètes de Christiaan Huygens, Vol. 17 (Martinus Nijhoff, 1932). Available at http://gallica.bnf.fr/.

Können, G. P.

Mariotte, E.

E. Mariotte, Oeuvres de Mr. Mariotte, Vol. 1, pp. 272-281 (Pierre Vander, 1717).

Moilanen, J.

W. Tape and J. Moilanen, Atmospheric Halos and the Search for Angle x (American Geophysical Union, 2006).

Musschenbroek, P.

P. Musschenbroek, Institutiones Physicae Conscriptae in Usus Academicos (Apud S. Luchtmans et filium, 1748). The halo discussion is pp 677-689 and Table XXVIII.

Priestley, J.

J. Priestley, The History and Present State of Discoveries Relating to Vision, Light and Colours (J. Johnson, 1772). Halos are treated on pp. 596-630 and in Figs. 141-148.

Seidenfaden, E.

Smith, R.

R. Smith, A Compleat System of Opticks in Four Books... (Cornelius Crownfield, Cambridge, 1738). The discussion of halos is in Chap. 11 of Book 2, pp. 199-238.

Stack, T.

T. Stack, “An Account of a Tract Intituled, Jo. Friderici Weidleri Commentatio de Parheliis Mense Januario Anni 1736...,” Phil. Trans. R. Soc. 41, 459-465 (1739-1741). Available at http://www.jstor.org/stable/i206906.
[CrossRef]

Tape, W.

W. Tape, E. Seidenfaden, and G. P. Können, “The legendary Rome halo displays,” Appl. Opt. 47, H72-H84 (2008).
[PubMed]

W. Tape and J. Moilanen, Atmospheric Halos and the Search for Angle x (American Geophysical Union, 2006).

Venturi, G.

G. Venturi, Commentarj sopra la Storie e le Teorie dell' Ottica (Pe' fratelli Masi, 1814).

Young, T.

T. Young, A Course of Lectures on Natural Philosophy and the Mechanical Arts (J. Johnson, 1807). Vol. 1, pp. 443-444 and Plate XXIX, and Vol. 2, pp. 303-308.

Appl. Opt. (1)

Histoire de l'Académie royale des sciences avec les mémoires de mathématique et physique (1)

Cassini, “Observation de deux Paraselenes, & d'un Arc-en-Ciel dans le crepuscule,” Histoire de l'Académie royale des sciences avec les mémoires de mathématique et physique 10, 400-402 (1730). The article is originally from 1693.

Journal de l'École Royale Polytechnique, 31 Cahier (1)

A. Bravais, “Mémoire sur les halos et les phénomènes optiques qui les accompagnent,” Journal de l'École Royale Polytechnique, 31 Cahier 18, 1-280 (1847).

Phil. Trans. R. Soc. (2)

C. Huygens, “An Account of the Observation, Made by the Philosophical Academy at Paris, May 12 1667 ...” Phil. Trans. R. Soc. 5, 1065-1074 (1670). Available at http://www.jstor.org/stable/i206870.

T. Stack, “An Account of a Tract Intituled, Jo. Friderici Weidleri Commentatio de Parheliis Mense Januario Anni 1736...,” Phil. Trans. R. Soc. 41, 459-465 (1739-1741). Available at http://www.jstor.org/stable/i206906.
[CrossRef]

Other (11)

J. Priestley, The History and Present State of Discoveries Relating to Vision, Light and Colours (J. Johnson, 1772). Halos are treated on pp. 596-630 and in Figs. 141-148.

T. Young, A Course of Lectures on Natural Philosophy and the Mechanical Arts (J. Johnson, 1807). Vol. 1, pp. 443-444 and Plate XXIX, and Vol. 2, pp. 303-308.

G. Venturi, Commentarj sopra la Storie e le Teorie dell' Ottica (Pe' fratelli Masi, 1814).

C. Huygens, Relation d'une Observation faite a la Bibliotheque du Roy, à Paris, le 12 May 1667... (Jean Cusson, 1667). An English version is in .

C. Huygens, B. de Volder, and B. Fullenius, Christiani Hugenii Zelemii, dum viveret, Toparchae. Opuscula Postuma,... (Apud Cornelium Boutesteyn, 1703). Available at http://fermi.imss.fi.it/. An English version is in .

R. Smith, A Compleat System of Opticks in Four Books... (Cornelius Crownfield, Cambridge, 1738). The discussion of halos is in Chap. 11 of Book 2, pp. 199-238.

P. Musschenbroek, Institutiones Physicae Conscriptae in Usus Academicos (Apud S. Luchtmans et filium, 1748). The halo discussion is pp 677-689 and Table XXVIII.

E. Mariotte, Oeuvres de Mr. Mariotte, Vol. 1, pp. 272-281 (Pierre Vander, 1717).

C. Huygens, Oeuvres Complètes de Christiaan Huygens, Vol. 17 (Martinus Nijhoff, 1932). Available at http://gallica.bnf.fr/.

We often refer to Mariotte's halo-maker as a prism, but we usually ignore all but the entry and exit faces, in which case the “prism” is really an idealized wedge having only two faces, which are half-planes.

W. Tape and J. Moilanen, Atmospheric Halos and the Search for Angle x (American Geophysical Union, 2006).

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Figures (8)

Fig. 1
Fig. 1

(a) The Huygens cylinder. (b) End view showing paths of some sun rays through the cylinder. The least deviated ray (for a fixed sun and fixed cylinder) is tangent to the core.

Fig. 2
Fig. 2

Relation between the Huygens cylinder and the equivalent Mariotte wedge. The ray path—the same in all three diagrams—is the minimum deviation ray path (for fixed sun) through the fixed cylinder, and it is also the minimum deviation ray path through the wedge as the wedge rotates in the plane of the paper.

Fig. 3
Fig. 3

Geometric formulation of Snell’s law. (a) Incident and refracted rays S and R in media having refractive indices n 1 and n 2 . The line segment N 1 is normal to the boundary between the two media. (b) Corresponding light points S and R on spheres having radii n 1 and n 2 and common center O. The vectors S O and R O are in the directions of S and R. Snell’s law is equivalent to requiring that the line segment S R be parallel to N 1 and not penetrate the inner sphere.

Fig. 4
Fig. 4

(a) Ray path through a vertical Mariotte prism, seen from above, together with line segments N 1 and N 2 normal to the entry and exit faces. The segments N 1 and N 2 are in the plane of the paper (horizontal), but the ray path here is not. (b) The corresponding vee—the configuration of light points S, R, and H of the entry, internal, and exit rays. The spheres have radii 1 and n, the refractive indices outside and inside the prism. The outer sphere (gray) has been cut away to expose part of the inner sphere. (c) The same vee but seen from above in order to emphasize its relation with the ray path: The vectors S O , R O , and H O are in the directions of the entry, internal, and exit rays. The segments S R and R H are parallel to N 1 and N 2 . Here the points S, R, and H are in the plane of the paper, but O, the common center of the spheres, is not. (d)–(f) Similar but for the Huygens cylinder.

Fig. 5
Fig. 5

Comparison of the Mariotte and Huygens parhelia. In each case the sun is at S and the (right) parhelion is the red circular arc P T . The sunward edge P of the parhelion is the same for both. The vees shown are those associated with the points P (orange) and T (green). Sun elevation Σ = 30 ° .

Fig. 6
Fig. 6

Illustrating that the upper and lower tangent arcs consist of rotated parhelia. (a) Upper and lower tangent arcs arising in the Mariotte prism. The two heavy red curves are the portions of the tangent arcs that arise in prisms whose axes point in the horizontal direction A. (b) The same two red curves, but emphasizing that they are rotated parhelia. Before rotation, these two parhelia would look like those in Fig. 5a. But S in that figure is not the same as S here; the angular distance between S and the zenith in Fig. 5a is the same as that between S and A here.

Fig. 7
Fig. 7

The 22 ° circular halo of Huygens. It is an annular region centered at the sun S. Each “spoke” of the halo is a rotated Huygens parhelion, one of which (the heavy curve) is shown in more detail at the right. The two vees shown are for minimum and maximum deviation for the Huygens sphere.

Fig. 8
Fig. 8

Diagram of Huygens [8] showing ray paths through his halo-making cylinder. Huygens recognized that the triangular prism ACA was closely related to his cylinder, but it apparently did not occur to him that such prisms—not the cylinders—might be making the halos. (Courtesy Leiden University Library).

Equations (3)

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m = sin α 2 ,
n = n 2 n 1 ( Bravais refractive index ) ,
n 2 = n 2 sin 2 Σ n 1 = cos Σ

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