Error in statement of Tropical Year

There is a grievous error in Leroy Doggett's "Calendars" chapter from the Explanatory Supplement to the Astronomical Almanac. This same error also occurs in, Royal Greenwich Observatory pamphlet #48 (LEAP YEARS).

The Greenwich Royal Observatory pamphlet (henceforth G.R.O.#48) begins:

" The year is deefined as being the interval between two successive passages of the Sun through the vernal equinox. Of course, what is really occurring is that the Earth is going around the Sun but it is easier to understand what is happening by considering the apparent motion of the Sun in the sky."

" The vernal equinox is the instant when the Sun is above the Earth's equator while going from the south to the north. It is the time which astronomers take as the definition of the beginning of Spring. The year as defined above is called the tropical year and it is the year length that defines the repetition of the seasons. The length of the tropical year is 365.24219 days."

The first two paragraphs of R.G.O.#48 define the "tropical" year as the interval between vernal equinoxes and immediately thereupon give an incorrect value (365.24219 days) for the length of such a "tropical" year. The value given is not for the mean interval between vernal equinoxes but instead for a "mean tropical year" based on a "fictitious mean sun" as defined by Simon Newcomb et al. (e.g. "The elements of the four inner planets and the fundamental constants of astronomy", by Simon Newcomb, Supplement to the American ephemeris and nautical almanac for 1897. Washington, Gov't. print off., 1895.)

The length of a "real" tropical year depends upon which point of the tropical zodiac you choose to measure the year-length from (John Dee** emphasized this point as early as 1582 A.D.), and thus Newcomb's formula (or any updated version using atomic or dynamical time) cannot give you the vernal-equinox year since such formulae give an average over all points of the tropical zodiac! Values like these (365.2422 to the nearest ten-thousandth of a day) from such formulae have no more to do with the vernal equinox than they have to do with the fall equinox or for that matter the summer solstice or winter solstice or any tropical zodiacal point in-between! I have found only one modern analysis of the solar calendar which admits this fundamental fact ("Astronomical Appreciation of the Gregorian Calendar", 1949, in volume 2, #6, of Richerche Astronomiche Specola Astronomica Vaticana, by J. De Kort S.J.) but even here the value given for the length of the vernal-equinox year is incorrect (the jesuit De Kort comes up with 365.2423 days).

The result of these errors in current astronomical texts is to continue a centuries-long cover-up of the true value of the vernal-equinox year! This has importance for all Christian churches, all Persians (a.k.a. Iranians), and thus all historians of astronomy and calendars, since the major solar calendars are ostensibly deliberate attempts to keep the vernal equinox on the same date (or, in the Persian case, nearest the same midnight) of each year.

COMPLETELY ERRONEOUS ANALYSES of calendar accuracy IN GENERAL REFERENCE WORKS have resulted from this continuing scandal, e.g. SCIENTIFIC AMERICAN, May 1982, "The Gregorian Calendar"; THE DICTIONARY OF SCIENTIFIC BIOGRAPHY, p.324, in the article on "Al-Khayyami" a.k.a. Omar Khayyam; and THE ENCYCLOPAEDIA IRANICA, vol IV, pp670-672, on the Jalali and current Persian Calendar. Many works of scholarship such as these come to totally wrong conclusions about solar calendars, their accuracy, their possible reform and political history, because of pronouncements like this by the R.G.O. and other modern astronomical institutions.

The same mistake occurs in Leroy Doggett's chapter on Calendars in the current Explanatory Supplement to the Astronomical Almanac (1992). This publication is so authoritative that it is taken as "dogma" by astronomers and lay-persons alike. It is thus extremely important to correct errors like this in such respected works of reference, in order to counteract the pernicious spread of their influence. To ignore errors about such basic facts as the length of the year, will inevitably bring modern astronomical institutions into disrepute, just as ignoring technical errors in their "bible" brought the Catholic church into disrepute after the disgraceful Galileo affair.

I don't know whether Leroy Doggett himself was aware of the chaos he has helped to propogate but the fact that he states "The following expression ... IS USED for calculating the length ..." (my emphasis on "used") suggests that he may indeed have been aware of some problem. I asked an acquaintance of his (my friend, Prof. Gerald Hawkins, author of "Stonehenge Decoded") to let him know about this error, and Hawkins subsequently told me that Doggett finds it "very interesting" that the real vernal-equinox year-length is, in fact, 365 + 8/33 days to the nearest ten-thousandth of a day (365.2424 days) not 365.2422 days as a "Newcomb-style" formula implies ; a fact which can be verified most easily by referring to the work of Jean Meeus ("Astronomical Algorithms", 1991, Ch.26, where Meeus gives formulae from which four different lengths of the year can be calculated, for the two equinoxes and the two solstices).

Using the formulae given by Meeus, I have concluded that, the vernal-equinox year most probably first became 365.2424 days ("real" Universal days) when it dropped to that value in neolithic times. It leveled out at 365.2423 in the bronze age and returned to 365.2424 during the sixteenth century A.D.. It it is likely to continue as such (365.2424 real days) for some millenia to come. It should (according to formulae given by Stephenson and Houlden in 1986, for the changing length of the day) level out just above its current value of 365.24238 days around 3500 A.D. somewhere between 365.2424 and 365.2425. Extrapolating farther into the future is probably futile, though I will mention that R.R. Newton ("The moon's acceleration... 2 vols. 1984) seriously considered the possibility that the earth's rotational deceleration is coming rapidly to a halt, leaving open the possibility that the earth's rotation might in fact begin to accelerate if left to "natural" mechanisms in the forthcoming centuries.

Leroy Doggett's "Calendars" is republished on Lyle Huber's web-page ( and it is ironic that I did find good information about the Persian (and Jalali) calendars by following Lyle's link to "Calendarland" (at on to "Khayam", "Jalali Calendar" or "Persian Calendars" (at and, where, for the first time, I found Iranians who have not yet been deceived by the "great American Satan" into denigrating their own calendar because it follows a 33-year cycle instead of the American "astronomer's bible" (Expl.Suppl.Astr.Alm.). I hope this is because they have read, analysed and understood the errors, rather than adhering to different authority figures!

** John Dee's work includes e.g. MS Ashmole 1789,fols 1-40, datable to 1582 A.D., a very beautiful manuscript which like all Dee's work on reforming the calendar is unpublished but partly accessible through the Bodleian library at Oxford, England. There were rumors in 1982 A.D. that Prof. J. Heilbron at U.C. Berkeley was going to publish an edition of Dee's calendar reform treatise, but I have seen nothing from him yet. In the same year (1982) it was revealed that Clavius and Pope Gregory were made aware, prior to their inception of our current calendar in 1582 A.D., of the correctness of a 33-year leap-day cycle for the vernal equinox (see e.g. A. Ziggelaar S.J. in "Gregorian Reform of the Calendar" ed. Coyne, Hoskin, Pedersen, 1983). The Syrian Nestorian patriarch Na'amat allah, (who was Clavius' and Gregory's informant), like Dee, seems to have been unable, probably for reasons of "state security", to publish all that he knew about the correct way to reform the Christian Calendar!

Figure 1: Showing two fixed calendar years (Gregorian at 365.2425 and 33-year Khayyam-style at 365.242424..) compared to the Newcomb-style mean dynamical tropical year (labeled Wrong VE year) and to actual current Vernal Equinox year theories. The Dynamical curve is derived from Meeus, using unreal days of fixed length, pegged to the 20th.century day-length. The S&H curve is probably closest to the truth and combines the Meeus Dynamical formula with Stephenson and Houlden’s 1986 theory of the changing length of the day (see Meeus 1991, chapter 9). The R.R.N. curve is an unorthodox alternative, combining Meeus with (and illegally extrapolating) R. R. Newton’s model of the Earth’s rotational history ( see R.R. Newton, “The Moon’s Acceleration....” vol.2, 1984).

Simon Cassidy,
Oct.14 1996,  A.D.(a.k.a. TheDeeDate or Thuddite reckoning of the Erisian Calendar:-).

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