# How did Hipparchus measure the length of the tropical year?

My understanding is that he counted the number of days between every other equinox. The first year he would have counted 365 days, but after 4 years the tally would have reached 1461 days. Dividing this by 4 he would have had an average length of 365.25 days per year. Using records from older Babylonian and Greek astronomers he would have been able to achieve a higher accuracy. Am I on the right track?

Bonus question: how did he measure the lengths of the seasons?

• Right now I can't put my finger on documentary sources, but ancient methods of estimating year length usually depended on long-period observations, and I doubt whether Hipparchus used any short-period method, if only because of the appreciated uncertainties of individual observations. I'll try and get back with sources. Apr 2, 2023 at 15:51
• You are right. This is explained in my ans to this question: hsm.stackexchange.com/questions/15250/… You measure the interval in days between two equinoxes far apart, and divide by the number of intervals between the equinoxes. Apr 3, 2023 at 12:35
• There is also some evidence that Hipparchus used some Babylonian data, but the principle of measurement is the same. Apr 3, 2023 at 12:37
• Apr 3, 2023 at 12:39
• Thank you very much! Apr 5, 2023 at 7:16

A hindrance to answering this question is that we do not have any description from Hipparchus himself on how he derived his numerical results, whether for the length of the year or of the seasons. His only surviving work is a commentary on Aratus and Eudoxus that deals with star catalogues, and tells us nothing about Hipparchus's ideas or work on the sun or moon (see Swerdlow 1980). Nearly all else from Hipparchus is seen through the lens of quotations and secondhand descriptions given by Ptolemy in the Almagest, where quite naturally they are subject to Ptolemy's preferences about content and presentation.

The literature offers two possible lines of original derivation of the (estimated) (tropical) year-length adopted both by Hipparchus and by Ptolemy, of 365 ¼ days less 1/300 of a day, Both derivations yield the same result.

(1) One possible line of derivation, clearly available to Hipparchus (see below) comes from combining the month-length used as accurate by Hipparchus together with the Callippic lunisolar cycle (the use of which was also anciently ascribed to Hipparchus).

(2) Another line of derivation comes from the collection of observations and measurements of cardinal points of the sun, i.e. solstices and equinoxes, as set out in Ptolemy's Almagest (bk.3, ch.1).

Method (1) was suggested by Swerdlow (1980) as a possible origin of the Hipparchan year-length estimate. To summarise his description:-

Ptolemy's account in the 'Almagest' shows that Hipparchus used as accurate a synodic month-length

(in sexagesimal notation, where e.g. "29;31,50" means "29 + 31/60 + 50/3600") of:

• 29;31,50,8,20 days = 29 d + 12; 44,03,20 hr ≈ 29.530594(14) days.

Also, Censorinus (3rd-c. AD) wrote in 'De die natali' that 'Hipparchus intercalates 112 times in 304 years', meaning that Hipparchus used a lunisolar cycle where the 304 years include 112 years with 13 (lunar) months and the others with 12 months. On these figures, Hipparchus's cycle was based on 4 [Callippic] cycles, thus containing 304 = 4 x 76 years = 4 x 940 months.

The first of the possible derivations of Hipparchus's year-length then involves combining the month-length with the cycle, both of which, on the available historical evidence, were available to Hipparchus. The combination works in for example the following way:

The original Callippic cycle assumed 27759 days for its 940 months, implying a month-length of 29.530851(06) days. This is noticeably longer than the month-length used as accurate by Hipparchus. If instead he applied his accurate month-length to the Callippic-derived cycle of months attributed to him by Censorinus, he would have measured the 304-year cycle, of 4 x 940 months, as containing 111035 days (plus something less than an hour) (i.e. 1 day shorter than 4 of the original Callippic cycles). Dividing by 304 then gives each year of this period in days as follows :-
• 111035/304 ≈ 30,50,35; / 5,4;
• ≈ (to 2 places) 6,5;15,0 - 0;0,12 ≈ 365 ¼ d minus 1/300 d , which is indeed the year-length attributed to Hipparchus by Ptolemy. (No actual new observations would be needed to make this determination, although observations might be made to test or confirm a previously computed value, or to demonstrate how in principle such a quantity might be derived de novo.)

Swerdlow (1980), who published this possible Hipparchan derivation, acknowledged that Tobias Mayer had already made a similar suggestion in 1753 that the year-length of Hipparchus could have originated by combining the Callippic cycle of months with the Hipparchan month-length. In Mayer's description, there had been no particular reason to suppose that Hipparchus was interested in the Callippic cycle. Swerdlow's account cites ancient historical evidence from Censorinus to show that indeed Hipparchus was concerned with an extended form of the Callippic cycle. This would make the combination with his favoured month-length intrinsically more probable.

Method (2), i.e. the method of observing cardinal points of the sun, is the one described by Ptolemy in the 'Almagest'. The observations extended over 285 years according to Ptolemy, but if one considers only those that Hipparchus himself could have taken into account, the period over which the useful measurements extend is shorter. In order to consider this method, it is necessary to take account of peculiarities of Ptolemy's presentation, which provides the little that we indirectly know of what Hipparchus did of this kind. The description in the Almagest points out some problems of the method in accuracy, e.g. in the use of a ring mounted in the equatorial plane to detect equinoxes. A good account of the results is given in Pedersen and Jones (2011).

The procedures and results reported by Ptolemy have a centuries-long history of commentary and criticism, and the practical shortcomings and difficulties of the method of observing equinoxes and solstices have generated and reinforced suspicions about the results.

It may help consideration of the present question to mention that some recent authors have pointed out that it may be needless and unjustified to suppose that Ptolemy's descriptions were always or necessarily describing the first origin of his numerical results. His alternative intentions may have been for example:

(a) (as pointed out by Swerdlow (1980)) some of the measurements might be explicable on the basis that they were made to test or confirm a previously computed value, or to demonstrate how in principle such a quantity might be derived.

(b) Britton (1992) found on statistical treatment that some of Ptolemy's results were considerably more accurate than could have been expected given the small number of the observations and the intrinsic uncertainties involved in each. Britton's conclusion was that in some cases, Ptolemy must have relied on a larger store of data than the items he reported, and sifted from it the values that he reported. Britton conjectured that in the absence of a discipline of statistics, and constrained by his habit and possible need to describe his work in a geometrically rigorous way, Ptolemy had no way that he found satisfactory to justify to his audience the informal mthods he must have used to process the data to get his good results.

(c) A recent work (J Feke, 2018) studies Ptolemy's methods and motivations, emphasising that he was not a modern writer, and had philosophical commitments that might in principle cut across anything resembling a modern attitude to data. So there should perhaps be no expectation that data from Ptolemy occupies a place comparable to the place of data in a modern study.

The fragmentary state of the evidence prevents clear decision as between possible methods (1) and (2) for Hipparchus's way of deriving his value for the solar (tropical) year. The simplicity of method (1) and its dependence on established and approved results might make it seem the more likely first route to the result. The fact that the result of method (1) is just the same as the reported result of method (2) may in itself cast doubt on whether the steps taken in method (2) were truly independent of the result of method (1). (If method (1) was actually Hipparchus's, it raises the question why Ptolemy did not simply present it. A possible answer to that question is that Ptolemy's expressed plan for the Almagest was clearly to establish the solar theory and only then to go on to consider the moon. To make the tropical (solar) year appear to depend on lunar and lunisolar quantities as in method (1) would have run counter to that order of presentation.)

References:

J P Britton (1992), "Models and precision : the quality of Ptolemy's observations and parameters" (New York & London: Garland).

J Feke (2018), "Ptolemy’s Philosophy : Mathematics as a Way of Life" (Princeton University Press).

O Pedersen (rev.ed. with A Jones) (2011), ["A Survey of the Almagest, with annotation and new commentary by Alexander Jones"] (https://books.google.com/books?id=8eaHxE9jUrwC).

N M Swerdlow (1980), "Hipparchus's Determination of the Length of the Tropical Year and the Rate of Precession", Archive for History of Exact Sciences, Vol. 21(4) (1980): 291-309.

G J Toomer (1984) Ptolemy’s Almagest, translated and annotated by G J Toomer, (London: Duckworth)

• Thank you very much @terry-s Apr 5, 2023 at 7:16