Observing with the Armillary Astrolabe

This name is used mainly for the instrument described in the Almagest. In modern terms, the Ptolemaic armillary astrolabe belongs to a class of the (zodiacal) armillary spheres.

Walther's observations were first published by Johannes Schöner in the Scripta clarissimi mathematici M. Ioannis Regiomontani (Nuremberg, 1544). For the analysis of observations of 1503–4 see Kremer R. L. , “Bernard Walther's astronomical observations”, Journal for the history of astronomy, xi (1980), 174–91.

Toomer G. J. (transl.), Ptolemy's Almagest (London, 1984), V.I, 217–19.

Rome A. , “L'Astrolabe et le météoroscope d'après le commentaire de Pappus sur le 5° livre de l'Almageste”, Annales de la Société Scientifique de Bruxelles, sér. A, Sci. math. Mémoires, xlvii (1927), 77–102, pp. 77–90. See also Dicks D. R. , “Ancient astronomical instruments”, The journal of the British Astronomical Association, lxiv (1954), 77–85, pp. 81–83; Rawlins D. , “An investigation of the ancient star catalog”, Publications of the Astronomical Society of the Pacific, xciv (1982), 359–73, pp. 372–3.

Ptolemy did not give the dimensions and proportion of the instrument. According to Pappus the outer diameter of the armillary astrolabe should be about 1 cubit (1 cubit = 44 cm approx). The sides of the cross-section of the ring should amount to 1/30 of the ring's diameter. Proclus claimed that the meridian ring of the instrument should be at least ½ cubit in diameter. The cross-section of the ring should be a rectangle with sides 1/30 and 1/48 of the ring's diameter. Cf. Rome, op. cit., 81–82. For the dimensions of Walther's armillary sphere, see Kremer, op. cit., 178.

The discrepancy between the instrumental and the actual obliquity of the ecliptic is a very important factor influencing the results of absolute observations, cf. Appendix B. According to Rome, op. cit., 89–90, fractions of a degree of ε were ignored in the makeup of the instrument, with the value of the obliquity taken from Proclus as 24°. This, however, does not seem to have been a general rule, cf. Neugebauer O. , A history of ancient mathematical astronomy (Berlin-Heidelberg-New York, 1975), 1034.

It should be noted there that the armillary astrolabe used in this study was made more with a view to teaching and demonstration than for actual measurement.

The construction of the armillary astrolabe does not allow the user to adjust the instrument in both places independently. This may considerably affect the proper setting of the instrument, cf. Rome, op. cit., 87.

Almagest, V.1, 219.

Almagest, VII.4, 339.

For the reference stars used in the Almagest see Pedersen O. , A survey of the Almagest (Odense, 1974), 236–7. Walther in his 1503–4 observations used α Tau and α Leo; see Kremer , op. cit., 183.

Almagest, V.1, 219.

See Dreyer J. L. E. , Tycho Brahe (New York, 1963), 348.

See Almagest, V.5, 227, n. 20.

Ptolemy describes an example of the method: The observation of Regulus on 23 February 139, Almagest, VII.2, 328.

This method of aligning the ecliptic ring was described by Pappus. He presented it as a competitor to Method B. Cf. Rome, op. cit., 87–88.

Almagest, V.1, 219. Let us note that it may be impossible to align rings 3 and 5 so as to put both of them in their own shadows. For detailed discussion see Appendix A.

Kremer , op. cit., 178. This is also the method described by Copernicus in De revolutionibus, II.14, with the parallactic ruler replaced by the quadrant.

The error of the longitude of the reference star was weighted by the factor (1 — tan ε tan β sin λ).

A conclusion of Newton's experiments “with a simplified model of an astrolable” is essentially the same. Newton does not give details of his instrument and experiments. Cf. Newton R. R. , The crime of Claudius Ptolemy (Baltimore, 1977), 145.

There are arguments against Ptolemy's authorship of the entire catalogue, such as the distribution of the fractions of degree in the star catalogue in the Almagest; cf. Newton, op. cit., 245–54, and the error waves test presented by Rawlins, op. cit.

Almagest, VII.2, 328. In Scripta there are 70 observations made by Walther with his Sun — Moon/Venus — planet/star procedure. A detailed analysis of these observations will be carried out by Richard Kremer and the present writer.

Newton , op. cit., 254.

A conclusion of Kremer's analysis of Walther's armillary observations is the opposite: “… the variance of errors is significantly greater between days than within a given day of observation” ( Kremer , op. cit., 187, n. 11).

Kremer , op. cit., 180, 184–5. Kremer has also analysed armillary observations of planets made between 1312 and 1316 by anonymous observer(s) in Paris. These observations are characterized by a mean relative error in longitude of −0°.4 and a standard deviation of the error of 0°.4. Cf. Kremer , op. cit., 180. The standard deviation of the longitude errors in the star catalogue of the Almagest is about 0°.4 ( Newton , The crime of Claudius Ptolemy, 216).

Thoren V. E. , “New light on Tycho's instruments”, Journal for the history of astronomy, iv (1973), 25–45, pp. 33–35.

Almagest, V.12, 244. The same problem was noticed by Tycho Brahe in his Astronomiae instauratae mechanica (1598): “The Copernican instrument [i.e. the parallactic ruler of Copernicus] … had holes, through which it is very difficult to observe the stars. There is the further disadvantage that the forward hole … has to be larger than the second one, for convenience in observing the stars through it, and in that case it must necessarily cover a certain, not very small, fraction of a degree, namely, at least one-eighth or one-tenth” (Tycho Brahe's description of his instruments and scientific work, ed. by Reader H. (Copenhagen, 1946), 46). For Tycho's description of his slit-sights, see ibid., 142–4.

Rome A. , Commentaires de Pappus et de Théon d'Alexandrie sur l'Almagest, I. Pappus d'Alexandrie: Commentaire sur les livres 5 et 6 (Rome, 1931), 8, n. 1.

Dreyer J. L. E. , “On the origin of Ptolemy's catalogue of stars. Second paper”, Monthly notices of the Royal Astronomical Society, lxxviii (1918), 343–9, p. 344; Newton R. R. , “On the fractions of degrees in an ancient star catalogue”, The quarterly journal of the Royal Astronomical Society, xx (1979), 383–94, pp. 391–2.

To compute (λ_obs — λ_R), differential relations between equatorial and ecliptic coordinates were used, under conditions ε = const and δ = const.

Figures 4 and 5 are constructed only for sunset, but diagrams can be easily transformed for sunrise, viz: Δ₁ (sunrise, λ) = −Δ₁ (sunset, λ + 180°). This is also true for Δ.

Formulae (5) and (6) are adapted from Britton J. , “On the quality of solar and lunar parameters in Ptolemy's Almagest”, Ph.D. diss., Yale University, 1967.

In the case of the instrument used in this study the offset of the ecliptic ring of 0°.2 produces the noticeable light-ribbon of 1mm on the inner part of the ring.

Kremer , op. cit., 179.

See Newton R. R. , “An analysis of the solar observations of Regiomontanus and Walther”, The quarterly journal of the Royal Astronomical Society, xxiii (1982), 67–93, pp. 85–86.