Several historians of science have attempted to reconstruct the calculations involved in
On Sizes and Distances. The first attempt was made by
Friedrich Hultsch in 1900, but it was later rejected by
Noel Swerdlow in 1969.
G. J. Toomer expanded on his efforts in 1974.
Hultsch Friedrich Hultsch determined in a 1900 paper that the Pappus source had been miscopied, and that the actual distance to the Sun, as calculated by Hipparchus, had been 2490 Earth radii (not 490). As in English, there is only a single character difference between these two results in Greek. His analysis was based on a text by
Theon of Smyrna which states that Hipparchus found the Sun to be 1880 times the size of the Earth, and the Earth 27 times the size of the Moon. Assuming that this refers to volumes, it follows that : s = \sqrt[3]{1880} \approx 12+1/3 and : \ell = \sqrt[3]{1/27} = 1/3 Assuming that the Sun and Moon have the same apparent size in the sky, and that the Moon is 67 Earth radii distant, it follows that : S = \frac{s}{\ell} L \approx \frac{12+1/3}{1/3} (67+1/3) = 2491 + 1/3 \approx 2490 This result was generally accepted for the next seventy years, until
Noel Swerdlow reinvestigated the case.
Book 2 reconstruction (Swerdlow) Swerdlow determined that Hipparchus relates the distances to the Sun and Moon using a construction found in
Ptolemy. It would not be surprising if this calculation had been originally developed by Hipparchus himself, as he was a primary source for the
Almagest. Using this calculation, Swerdlow was able to relate the two results of Hipparchus (67 for the Moon and 490 for the Sun). Obtaining this relationship exactly requires following a very precise set of approximations. Using simple trigonometric identities gives : \ell = L \tan \theta \approx L \sin \theta and : h = \frac{\tan \varphi}{\tan \theta} \ell \approx \frac{\varphi}{\theta} \ell \approx \frac{\varphi}{\theta} L \sin \theta By parallel lines and taking
t = 1, we get : \ell + x = t + (t-h) = 2t-h \Rightarrow x\approx 2 - \left(\frac{\varphi}{\theta}+1\right) L \sin \theta By similarity of triangles, : \frac{t}{x} = \frac{S}{S-L} \Rightarrow S = \frac{L}{1-x} Combining these equations gives : S \approx \frac{L}{\left(\frac{\varphi}{\theta}+1\right) L \sin \theta - 1} = 1 \left/ \left( \left( \frac{\varphi}{\theta} + 1 \right) \sin \theta - \frac{1}{L} \right) \right. The values which Hipparchus took for these variables can be found in Ptolemy's
Almagest IV, 9. He says Hipparchus found that the Moon measured its own circle close to 650 times, and that the angular diameter of Earth's shadow is 2.5 times that of the Moon. Pappus tells us that Hipparchus took the mean distance to the Moon to be 67. This gives: According to Swerdlow, Hipparchus now evaluated this expression with the following roundings (the values are in
sexagesimal): : \ell \approx L \sin \theta \approx 0;19,30 and : h \approx \frac{\varphi}{\theta} \ell \approx 0;48,45 Then, because : \ell + h \approx \frac{\varphi}{\theta} L \sin \theta + L \sin \theta = \left(\frac{\varphi}{\theta}+1\right) L \sin \theta it follows that : S \approx L/(\ell + h - 1) \approx 67;20 / 0;8,15 \approx 489.70 \approx 490 Swerdlow used this result to argue that 490 was the correct reading of the Pappus text, thus invalidating Hultsch' interpretation. While this result is highly dependent on the particular approximations and roundings used, it has generally been accepted. It leaves open, however, the question of where the lunar distance 67 came from. Following Pappus and Ptolemy, Swerdlow suggested that Hipparchus had estimated 490 Earth radii as a minimum possible distance to the Sun. This distance corresponds to a solar parallax of 7', which may have been the maximum that he thought would have gone unnoticed (the typical resolution of the human eye is 2'). The formula obtained above for the distance to the Sun can be inverted to determine the distance to the Moon: : L \approx \frac{S}{\left(\frac{\varphi}{\theta}+1\right) S \sin \theta - 1} = 1 \left/ \left( \left( \frac{\varphi}{\theta} + 1 \right) \sin \theta - \frac{1}{S} \right) \right. Using the same values as above for each angle, and using 490 Earth radii as the minimum solar distance, it follows that the maximum mean lunar distance is : L \approx 1 \left/ \left( ( 2.5 + 1 ) \sin 0.277^\circ - \frac{1}{490} \right) \right. \approx 67.203 \approx 67\tfrac{1}{3} Toomer expanded on this by observing that as the distance to the Sun increases without bound, the formula approaches a minimum mean lunar distance: : L \approx 1 \left/\left( (2.5 + 1) \sin 0.277^\circ \right)\right. \approx 59.10 This is close to the value later claimed by Ptolemy.
Book 1 reconstruction (Toomer) In addition to explaining the minimum lunar distance that Hipparchus achieved, Toomer was able to explain the method of the first book, which employed a
solar eclipse. Pappus states that this eclipse was total in the region of the Hellespont, but was observed to be 4/5 of total in Alexandria. If Hipparchus assumed that the Sun was infinitely distant (i.e. that "the earth has the ratio of a point and center to the sun"), then the difference in magnitude of the solar eclipse must be due entirely to the parallax of the Moon. By using observational data, he would be able to determine this parallax, and hence the distance of the Moon. Hipparchus would have known \varphi_A and \varphi_H, the
latitudes of
Alexandria and the
Hellespontine region, respectively. He also would have known \delta, the
declination of the Moon during the eclipse, and \mu, which is related to the difference in totality of the eclipse between the two regions. : AH = \frac{t\operatorname{Crd} \angle AOH}{R} = \frac{t\operatorname{Crd}(\varphi_H - \varphi_A)}{R} Crd here refers to the
chord function, which maps an angle in degrees to the corresponding length of a chord of a circle of unit diameter. Also, R is the radius of the base circle that Hipparchus used to compute the chord function and t is the radius of the Earth. Since the Moon is very distant, it follows that \zeta' \approx \zeta. Using this approximation gives : \zeta = \varphi_H - \delta : \angle ZHA = 180^\circ - \angle OHA : \angle OHA = \frac{180^\circ - \angle AOH}{2} = \frac{180^\circ - (\varphi_H - \varphi_A)}{2} Hence, : \angle ZHA = 90^\circ + \frac{1}{2} (\varphi_H - \varphi_A) : \angle MHA = \theta = \angle ZHA - \zeta' \approx \angle ZHA - \zeta = 90^\circ - \frac{1}{2} (\varphi_H + \varphi_A) + \delta With AH and \theta, we only need \mu to get D'. Because the eclipse was total at H, and 4/5 total at A, it follows that \mu is 1/5 of the apparent diameter of the Sun. This quantity was well known by Hipparchus—he took it to be 1/650 of a full circle. The distance from the center of the Earth to the Moon then follows from D \approx D' + t. Toomer determined how Hipparchus determined the chord for small angles (see
Chord (geometry)). His values for the latitudes of the Hellespont (41 degrees) and Alexandria (31 degrees) are known from
Strabo's work on
Geography. To determine the declination, it is necessary to know which eclipse Hipparchus used. Because he knew the value which Hipparchus eventually gave for the distance to the Moon (71 Earth radii) and the rough region of the eclipse, Toomer was able to determine that Hipparchus used the solar eclipse of March 14, 190 BC. This eclipse fits all the mathematical parameters very well, and also makes sense from a historical point of view. The eclipse was total in
Nicaea, Hipparchus' birthplace, so he may have heard stories of it. There is also an account of it in
Strabo's Ab Urbe Condita VIII.2. The declination of the Moon at this time was \delta = -3^\circ. Hence, using chord trigonometry, gives : \theta = 54^\circ + \delta : AH = \frac{t\operatorname{Crd} 10^\circ}{R} \approx t\ \frac{600}{3438} : \mu = \frac{360 \cdot 60}{5 \cdot 650} : D' = \frac{\operatorname{Crd} 2 \theta \cdot AH}{\operatorname{Crd} \mu \cdot 2} = \frac{\operatorname{Crd} (108^\circ + 2 \delta) \cdot 600 \cdot 5 \cdot 650}{21600 \cdot 2 \cdot 3438} t Now using Hipparchus' chord tables, : \operatorname{Crd} (108^\circ + 2 (-3^\circ)) = \operatorname{Crd} 102^\circ \approx 2 \cdot 3438 \sin 51^\circ \approx 5340 and hence : D' = \frac{5340 \cdot 600 \cdot 5 \cdot 650}{21600 \cdot 2 \cdot 3438} t \approx 70.1 t \Rightarrow D \approx D' + t \approx 71.1 t This agrees well with the value of 71 Earth radii that Pappus reports. ==Conclusion==