On the making of Ptolemy’s star catalog

Abstract

The assumption that Ptolemy adopted star coordinates from a star catalog by Hipparchus is investigated based on Hipparchus’ equatorial star coordinates in his Commentary on the phenomena of Aratus and Eudoxus. Since Hipparchus’ catalog was presumably based on an equatorial coordinate system, his star positions must have been converted into the ecliptical system of Ptolemy’s catalog in his Almagest. By means of a statistical analysis method, data groups consistent with this conversion of coordinates are identified. The found groups show a high degree of agreement between Hipparchus’ and Ptolemy’s data. The value of the obliquity of the ecliptic underlying the conversion is estimated by adjustment and statistically agrees with Ptolemy’s value of this parameter. The results allow the assumption that Ptolemy’s coordinates were determined from Hipparchus’ coordinates by an accurate star globe or even by calculation. For a calculative derivation of ecliptical coordinates from equatorial ones, possible calculation methods are discussed considering the mathematics of the Almagest.

Appendices

Adjustment computation

For the adjustment computations, matrix calculus is employed. The condition equations (25), (26) of the star data under consideration can be written in the form ${\mathbf {0}} = {\mathbf {h}}({\mathbf {x}}, \overline{{\mathbf {l}}}={\mathbf {l}}+{\mathbf {v}})$, where ${\mathbf {0}}$ is a zero vector and the right sides of the condition equations are composed in vector ${\mathbf {h}}$ with elements $h_j$. ${\mathbf {h}}$ constitutes a function of vector ${\mathbf {x}}$ of the unknown parameters and of vector $\overline{{\mathbf {l}}}$ of the adjusted values of the stochastic variables; ${\mathbf {l}}$ and ${\mathbf {v}}$ are vectors of the values of the stochastic variables and their residuals. In the present case of one unknown parameter, ${\mathbf {x}}=(\epsilon )$ is a scalar. Linearization of the equation system yields ${\mathbf {0}} = {\mathbf {h}}({\mathbf {x}}^{0}, \overline{{\mathbf {l}}}^{0}) + {\mathbf {A}}\cdot ({\mathbf {x}}-{\mathbf {x}}^{0}) + {\mathbf {B}}\cdot (\overline{{\mathbf {l}}}-\overline{{\mathbf {l}}}^{0})$, where ${\mathbf {x}}^{0}$ and $\overline{{\mathbf {l}}}^{0}$ are initial values of ${\mathbf {x}}$ and $\overline{{\mathbf {l}}}$ or the results of the preceding iteration step, respectively, matrix ${\mathbf {A}}=\partial {\mathbf {h}} / \partial {\mathbf {x}}$ at point ${\mathbf {x}}^{0}$ and matrix ${\mathbf {B}} = \partial {\mathbf {h}} / \partial \mathbf {\overline{{\mathbf {l}}}}$ at point $\overline{{\mathbf {l}}}^{0}$. For the first iteration step, $\overline{{\mathbf {l}}}^{0} = {\mathbf {l}}$ is used. In the present case, ${\mathbf {A}}$ is a vector with elements $\partial h_j/\partial \epsilon$. The weights $p_i$ (27) of the stochastic variables constitute the diagonal elements of the diagonal weight matrix ${\mathbf {P}}$. In the following, the example of star No. 197 is regarded, for which $\delta _\mathrm {H}=40^{\circ }00'$, $\lambda _\mathrm {P}'=32^{\circ }10'$ and $\beta _\mathrm {P}=30^{\circ }00'$ are available. In stochastic model 1 (28), $\lambda _\mathrm {P}'$ and $\beta _\mathrm {P}$ (category 3) are stochastic variables but not $\delta _\mathrm {H}$ (category 1). ${\mathbf {B}}$ has the two elements $\partial h_j/\partial \lambda _\mathrm {P}'$ and $\partial h_j/\partial \beta _\mathrm {P}$ in the intersection of row j for the condition equation of $\delta _\mathrm {H}$ and of the two columns for $\lambda _\mathrm {P}'$ and $\beta _\mathrm {P}$; the other elements of this row and these columns are zero. For $\lambda _\mathrm {P}'$ and $\beta _\mathrm {P}$, the resolutions $a=10'$ and $15'$ are assumed (cf. Appendix B) leading to the a priori standard deviations $\sigma _i=5'$ and $7.5'$ and $p_i=1/5^2$ and $1/7.5^2$. In stochastic model 2 (29), $\delta _\mathrm {H}$, $\lambda _\mathrm {P}'$ and $\beta _\mathrm {P}$ are stochastic variables so that ${\mathbf {B}}$ has an additional column for $\delta _\mathrm {H}$ with the element $\partial h_j/\partial \delta _\mathrm {H}$ in the related row. The $\sigma _i$ of $\delta _\mathrm {H}$, $\lambda _\mathrm {P}'$ and $\beta _\mathrm {P}$ are $\sigma$, $\sigma /\cos \beta _\mathrm {P}$, $\sigma$ yielding $p_i=1/\sigma ^2$, $\cos \beta _\mathrm {P}^2/\sigma ^2$, $1/\sigma ^2$.

The parameter solution of an iteration step is $\Delta {\mathbf {x}} = {\mathbf {x}}-{\mathbf {x}}^{0} = -{\mathbf {Q}}_\mathrm {x} {\mathbf {A}}^\top {\mathbf {N}}^{-1} {\mathbf {w}}$ with ${\mathbf {N}} = {\mathbf {B}} {\mathbf {P}}^{-1} {\mathbf {B}}^\top$, ${\mathbf {Q}}_\mathrm {x}=({\mathbf {A}}^\top {\mathbf {N}}^{-1}{\mathbf {A}})^{-1}$, ${\mathbf {w}} = {\mathbf {h}}({\mathbf {x}}^{0}, \overline{{\mathbf {l}}}^{0}) + {\mathbf {B}}({\mathbf {l}}-\overline{{\mathbf {l}}}^{0})$. The variance–covariance matrix of ${\mathbf {x}}$ is $s_0^2{\mathbf {Q}}_\mathrm {x}$ with $s_0$ (31). The residuals are ${\mathbf {v}}={\mathbf {P}}^{-1}{\mathbf {B}}^\top {\mathbf {k}}$ with ${\mathbf {k}}=-{\mathbf {Q}}_\mathrm {k} {\mathbf {w}}$ and ${\mathbf {Q}}_\mathrm {k} = {\mathbf {N}}^{-1}({\mathbf {I}} - {\mathbf {A}} {\mathbf {Q}}_\mathrm {x}{\mathbf {A}}^\top {\mathbf {N}}^{-1})$ (${\mathbf {k}}$: Lagrangian multipliers, ${\mathbf {I}}$: identity matrix). The diagonal elements of the redundancy matrix ${\mathbf {R}}={\mathbf {Q}}_\mathrm {v}{\mathbf {P}}$ with ${\mathbf {Q}}_\mathrm {v}={\mathbf {P}}^{-1}{\mathbf {B}}^\top {\mathbf {Q}}_\mathrm {k}{\mathbf {B}}{\mathbf {P}}^{-1}$ are the redundancy numbers $r_i$, which are used in the test statistics $T_i$ (32) and $T'_i$ (33). ${\mathbf {x}}={\mathbf {x}}^{0}+\Delta {\mathbf {x}}$ and $\overline{{\mathbf {l}}} = {\mathbf {l}} + {\mathbf {v}}$ are used for ${\mathbf {x}}^{0}$ and $\overline{{\mathbf {l}}}^{0}$ in the linearization of the next iteration step.

In subset 1B, for example, the iterative solution of the adjustment yields $v_\lambda =-1'$ and $v_\beta =-4'$ for No. 197 and in subset 2A $v_\delta =2'$, $v_\lambda =-1'$ and $v_\beta =-2'$. $T_i$ (1B) and $T'_i$ (2A) of these residuals have small absolute values of about 0.5 and lie below the deliberately low thresholds for gross errors of about 2.0 (resulting from the significance level of 5 %).

Data resolution

For stochastic model 1 (28), the resolutions a of the coordinate values under investigation are chosen as follows.

Table 5 Frequencies $n_m$ of the fractions of degree m of the ecliptical coordinates under investigation

Table 5 shows the frequencies $n_m$ of the fractions of degree m of $\lambda _\mathrm {P}'$ and $\beta _\mathrm {P}$. In the case of $\beta _\mathrm {P}$, the frequencies of $m=10'$, $20'$, $40'$ and $50'$ are the lowest; for them, $a=10'$ is assumed. $a=20'$ probably does not occur because $n_{20}$ and $n_{40}$ are not inflated. For $m=15'$ and $45'$, $a=15'$ is assumed. The inflate of $n_{30}$ is explicable by the occurrence of $a=15'$ and $30'$ in addition to $a=10'$. Since, however, the number of $\beta _\mathrm {P}$ with $a=30'$ seems to be low, $a=15'$ is used for $m=30'$. $n_{0}$ shows a strong inflation. Assuming $a=10'$ and $15'$ and rejecting $a=20'$, the inflate can be caused by $a=30'$ and $1^{\circ }$ or by coordinates whose fractions of degree got lost. Since $a\ge 30'$ would lead to a strong downweighting in the adjustment, $a=15'$ is set for $m=0'$.

In the case of $\lambda _\mathrm {P}'$, $m=10'$ and $50'$ have the lowest frequencies; for them, $a=10'$ is applied. $n_{20}$ and $n_{40}$ are inflated. If longitudes with $m=15'$ and $45'$ were transformed by a precession correction of $+2^{\circ }40'$, then the resulting $m=55'$ and $25'$ were probably rounded to $0'$ and $20'$ (Newton 1977, pp. 251–252), and their back-transformation to $\lambda _\mathrm {P}'$ by Eq. (4) yields $20'$ and $40'$. Thus, $n_{20}$ and $n_{40}$ may be based on longitudes with $a=15'$. Since, however, the transformation produced an error of $5'$ and since values with $a=20'$ cannot excluded completely, $a=20'$ is set for $m=20'$ and $40'$. Since the inflation of $n_{30}$ corresponds to those of $n_{20}$ and $n_{40}$, $a=30'$ is rejected and $a=15'$ is used for $m=30'$. For $m=0'$, $a=15'$ is set for the same reasons as in the case of $\beta _\mathrm {P}$ above. The only abnormal $m=35'$ of No. 434 gets $a=10'$.

The weighting of $\alpha _\mathrm {H}$ and $\delta _\mathrm {H}$ with respect to their fractions of degree concerns category 2 only. The chosen a for the occurring m in category 2b are: $m=30'$, $45'$: $a = 15'$; $m=0'$: $a = 30'$. The data of category 2a are possibly less accurate and therefore get larger a: $m=30'$: $a=30'$; $m=0'$: $a=1^{\circ }$.

Statistical tests

Test statistic of the individual test of stochastic variables with known accuracy for gross errors is the standardized residual

$$\begin{aligned} T_i = \frac{v_i}{\sigma _i \sqrt{r_i}}, \end{aligned}$$

(32)

where $r_i$ is the redundancy number of the ith stochastic variable (cf. Ettlinger and Neuner 2020). If the accuracy of the stochastic variables is unknown, the externally studentized residual

$$\begin{aligned} T'_i = \frac{v_i \sqrt{p_i}}{s'_0 \sqrt{r_i}}, \quad s'_0=\sqrt{\frac{S - v_i^2p_i/r_i}{f-1}} \end{aligned}$$

(33)

(e.g., Jäger et al. 2005, p. 193) is used for the test (for S and f see Eq. 31). If the errors of the stochastic variables are normally distributed, $T_\mathrm {i}$ and $T'_\mathrm {i}$ follow a standard normal distribution and a $t_{f'}$-distribution with the degree of freedom $f'=f-1$, respectively. In order to reject data with large errors with regard to their accuracy, a large significance level of 5 % is applied in these two-sided tests. If in different condition equations equal maximal test statistics occur, which exceed the preset limit, then the equation with the maximal squared sum of its test statistics is removed in the data snooping procedure.

Furthermore, the residuals of the adjustment are tested collectively for normal distribution by the Kolmogorov–Smirnov test (see e.g., Sachs 1984, pp. 330–332). The condition for this test that the residuals have the same standard deviation and are uncorrelated is not fulfilled. In particular, the residuals of a condition equation are fully correlated. Therefore, an approximate method according to Wolf (1997, p. 311) is applied, which tests the approximately homogenized residuals (32) for standard normal distribution. The test requires the standard deviations $\sigma _{i}$ of the stochastic variables, which are not given but estimated in the application of stochastic model 2 (29). In this case, $s_i$ (31) is used for $\sigma _{i}$. Thus, the test becomes more conservative (e.g., Sachs 1984, p. 331), which is, however, acceptable. The significance level used is $5\,\%$.

Test statistic of the t-test of a parameter for a significant deviation from a given value $x_0$ is

$$\begin{aligned} T = |x - x_0|/s_x \end{aligned}$$

(34)

(e.g., Koch 1999, p. 283), where x is the estimate of the parameter and $s_x$ is the standard deviation of x from the adjustment. If $x_0$ is valid, T follows a $t_f$-distribution. The significance level applied in this two-sided test is $5\,\%$.