Since it emerged sometime in the middle of the first millennium BCE the principal function of mathematical astronomy was to provide the most accurate possible predictions of the future positions of the main celestial bodies. This information was contained in the form of tables calculated with the help of the mathematical models, which had been derived by the astronomers from the observed behaviour of those bodies, the planets. The earliest Babylonian models were algebraic but were soon replaced by the Greeks with geometrical models based on spheres and circles. To a large extent it did not matter if those models were depictions of reality, what mattered was the accuracy of the prediction that they produced; that is the reliability of the associated tables. The models of mathematical astronomy were judge on the quality of the data they produced and not on whether they were a true reproduction of what was going on in the heavens. This data was used principally for astrology but also for cartography and navigation. Mathematical astronomy was a handmaiden to other disciplines.

Before I outline the history of such tables, a brief comment on terminology. Data on the movement of celestial bodies is published under the titles planetary tables and ephemerides (singular ephemeris). I know of no formal distinction between the two names but as far as I can determine planetary tables is generally used for tables calculated for quantitatively larger intervals, ten days for example, and these are normally calculated directly from the mathematical models for the planetary movement. Ephemeris is generally used for tables calculated for smaller interval, daily positions for example, and are often not calculated directly from the mathematical models but are interpolated from the values given in the planetary tables. Maybe one of my super intelligent and incredibly well read readers knows better and will correct me in the comments.

The Babylonians produced individual planetary tables, in particular of Venus, but we find the first extensive set in the work of Ptolemaeus. He included tables calculated from his geometrical models in his Syntaxis Mathematiké (The Almagest), published around 150 CE, and to make life easier for those who wished to use them he extracted the tables and published them separately, in extended form with directions of their use, in what is known as his Handy Tables. This publication provided both a source and an archetype for all future planetary tables.

The important role played by planetary tables in mathematical astronomy is illustrated by the fact that the first astronomical works produced by Islamic astronomers in Arabic in the eighth-century CE were planetary tables known in Arabic as zījes (singular zīj). These initial zījes were based on Indian sources and earlier Sassanid Persian models. These were quickly followed by those based on Ptolemaeus’ Handy Tables. Later sets of tables included material drawn from Islamic Arabic sources. Over 200 zījes are known from the period between the eighth and the fifteenth centuries. Because planetary tables are dependent on the observers geographical position most of these are only recalculation of existing tables for new locations. New zījes continued to be produced in India well into the eighteenth-century.

With the coming of the European translators in the twelfth and thirteenth centuries and the first mathematical Renaissance the pattern repeated itself with zījes being amongst the first astronomical documents translated from Arabic into Latin. Abū ʿAbdallāh Muḥammad ibn Mūsā al-Khwārizmī was originally better known in Europe for his zīj than for *The Compendious Book on Calculation by Completion and Balancing” (al-Kitab al-mukhtasar fi hisab al-jabr wa’l-muqabala)*, the book that introduced algebra into the West. The *Toledan Tables* were created in Toledo in the eleventh-century partially based on the work of Abū Isḥāq Ibrāhīm ibn Yaḥyā al-Naqqāsh al-Zarqālī, known in Latin as Arzachel. In the twelfth-century they were translated in Latin by Gerard of Cremona, the most prolific of the translators, and became the benchmark for European planetary tables.

In the thirteenth- century the *Toledan Tables* were superseded by the Alfonsine Tables, which were produced by the so-called Toledo School of Translators from Islamic sources under the sponsorship of Alfonso X of Castile. *The Alfonsine Tables* remained the primary source of planetary tables and ephemerides in Europe down to the Renaissance where they were used by Peuerbach, Regiomontanus and Copernicus. Having set up the world’s first scientific press Regiomontanus produced the first ever printed ephemerides, which were distinguished by the accuracies of their calculations and low level of printing errors. Regiomontanus’ ephemerides were very popular and enjoyed many editions, many of them pirated. Columbus took a pirate edition of them on his first voyage to America and used them to impress some natives by accurately predicting an eclipse of the moon.

By the fifteenth-century astronomers and other users of astronomical data were very much aware of the numerous inaccuracies in that data, many of them having crept in over the centuries through frequent translation and copying errors. Regiomontanus was aware that the problem could only be solved by collecting new basic observational data from which to calculate the tables. He started on such an observational programme in Nürnberg in 1470 but his early death in 1475 put an end to his endeavours.

When Copernicus published his *De revolutionibus* in 1543 many astronomers hoped that his mathematical models for the planetary orbits would lead to more accurate planetary tables and this pragmatic attitude to his work was the principle positive reception that it received. Copernicus’ fellow professor of mathematic in Wittenberg Erasmus Reinhold calculated the first set of planetary tables based on *De revolutionibus*. *The Prutenic Tables*, sponsored by Duke Albrecht of Brandenburg Prussia (Prutenic is Latin for Prussian), were printed and published in 1551. Ephemerides based on Copernicus were produced by Johannes Stadius, a student of Gemma Frisius, in 1554 and by John Feild (sic), with a forward by John Dee, in 1557. Unfortunately they didn’t live up to expectations. The problem was that Copernicus’ work and the tables were based on the same corrupted data as the *Alfonsine Tables*. In his unpublished manuscript on navigation Thomas Harriot complained about the inaccuracies in the *Alfonsine Tables* and then goes on to say that the *Prutenic Tables* are not any better. However he follows this complaint up with the information that Wilhelm IV of Hessen-Kassel and Tycho Brahe on Hven are gathering new observational data that should improve the situation.

As a young astronomer the Danish aristocrat, Tycho Brahe, was indignant that the times given in both the Alfonsine and the Prutenic tables for a specific astronomical event that he wished to observe were highly inaccurate. Like Regiomontanus, a hundred years earlier, he realised that the problem lay in the inaccurate and corrupted data on which both sets of tables were based. Like Regiomontanus he started an extensive programme of astronomical observations to solve the problem, initially at his purpose built observatory financed by the Danish Crown on the island of Hven and then later, through force of circumstances, under the auspices of Rudolph II, the Holy Roman German Emperor, in Prague. Tycho devoted almost thirty years to accruing a vast collection of astronomical data. Although he was using the same observational instruments available to Ptolemaeus fifteen hundred years earlier, he devoted an incredible amount of time and effort to improving those instruments and the methods of using them, meaning that his observations were more accurate by several factors than those of his predecessors. What was now needed was somebody to turn this data into planetary tables, enter Johannes Kepler. Kepler joined Tycho in Prague in 1600 and was specifically appointed to the task of producing planetary tables from Tycho’s data. Contrary to popular belief he was not employed by Tycho but directly by Rudolph.

Following Tycho’s death, a short time later, a major problem ensued. Kepler was official appointed Imperial Mathematicus, as Tycho’s successor, and still had his original commission to produce the planetary tables for the Emperor, however, legally, he no longer had the data; this was Tycho’s private property and on his death passed into the possession of his heirs. Kepler was in physical possession of the data, however, and hung on to it during the protracted, complicated and at times vitriolic negotiations with Tycho’s son in law, Frans Gansneb Genaamd Tengnagel van de Camp, over their future use. In the end the heirs granted Kepler permission to use the data with the proviso that any publications based on them must carry Tengnagel’s name as co-author. Kepler then proceeded to calculate the tables.

Put like this, it sounds like a fairly straightforward task, however it was difficult and tedious work that Kepler loathed intensely. It was not made any easier by the personal and political circumstances surrounding Kepler over the years he took to complete the task. Wars, famine, usurpation of the Emperor’s throne (don’t forget the Emperor was his employer) and family disasters all served to make his life more difficult.

Finally in 1626, twenty-six years after he started Kepler had finally reduced Tycho’s thirty years of observations into planetary tables for general use, now he only had to get them printed. Drumming up the financial resources for the task was the first hurdle that Kepler successfully cleared. He then purchased the necessary paper and settled in Linz to complete the task of turning his calculations into a book. As the printing was progressing all the Protestants in Linz were ordered to leave the city, Kepler, being Imperial Mathematicus, and his printer were granted an exemption to finish printing the tables but then Wallenstein laid siege to the city to supress a peasants uprising. In the ensuing chaos the printing shop and the partially finished tables went up in flames.

Leaving Linz Kepler now moved to Ulm where, starting from the beginning again, he was finally able to complete the printing of the *Rudophine Tables*, named after the Emperor who had originally commissioned them but dedicated to the current Emperor, Ferdinand II. Although technically not his property, because he had paid the costs of having them printed Kepler took the finished volumes to the book fair in Frankfurt to sell in September 1627.

Due to the accuracy of Tycho’s observational data and the diligence of Kepler’s mathematical calculations the new tables were of a level of accuracy never seen before in the history of astronomy and fairly quickly became the benchmark for all astronomical work. Perceived to have been calculated on the basis of Kepler’s own elliptical heliocentric astronomy they became the most important artefact in the general acceptance of heliocentricity in the seventeenth century. As already stated above systems of mathematical astronomy were judged on the data that they produced for use by astrologers, cartographers, navigators et al. Using the *Rudolphine Tables* Gassendi was able to predict and observe a transit of Mercury in 1631, as Jeremiah Horrocks succeeded in predicting and observing a transit of Venus for the first time in human history based on his own calculations of an ephemeris for Venus using Kepler’s tables, it served as a confirming instance of the superiority of both the tables and Kepler’s elliptical astronomy, which was the system that came to be accepted by most working astronomers in Europe around 1660. The principle battle in the war of the astronomical systems had been won by a rather boring set of mathematical tables, Johannes Kepler’s *Tabulae Rudolphinae*.

Wonderful summary of a complicated story. It’s amazing that Kepler managed to create the tables granted the tremendous disorder of his times: wide-spread plagues, the witchcraft craze, the Counter-Reformation, and the Thirty Years War. And that’s not to mention the fact that his principal patron, Rudolph II, was, to be charitable, one of the great eccentrics of history—his family decided that he was just plain nuts and had him replaced on the throne by another Hapsburg. The usurpation you alluded to was a family squabble. I’ve got a certain affection for Rudolph because he was a dead ringer for my uncle Ralph, but I suspect the family was right about him.

“Wonderful summary of a complicated story.” Worth repeating, extremely well written, compact and flows very easily. I open my blog to the world a bit more after reading some of you’re posts. Been fruitful, was looking through the work of a poet who stopped by to like something.

“i am mad as a storm to quarrel

and pick words from the floor of my thoughts”

Eddie Tay, The Mental Life of Cities

Thony, the distinction is really quite simply. Astronomical tables separate mean motions from equations. Ephemerides simply give you the true position of all planets at a single glance for days or small intervals of days. Astronomical tables are the basis for calculating Ephemerides.

PS: the mentioned ten-day intervals are more commonly encountered in Almanacs, which present true positions of planets in separate tables and often use goal-year methods to create a kind of cyclicity.

Yes, a nice article and I agree with the comments: It also seems true that there’s no standard terminology to point out the difference from the tables often called ephemerides, when one wants to refer to the other kind of tables, often just called (rather ambiguously) planetary tables, of which the Rudolphines were a great historical example. Besides the differences already pointed out between the two kinds, ephemerides or almanac positions are usually just lists of positions for specific dates+times — and they don’t generally enable other positions for quite different dates+times to be found. In contrast, the tables of the other sort give in principle a basis, often a complicated and indirect basis, for calculating positions for any given date+time, future or past.

This other sort of table seems to need a clear category-name for use when discussing them. Lansberg in 1632 used the name ‘perpetual’ tables for his offering, and he got hauled over the coals by Horrocks, and maybe also by others, for being presumptuous enough to claim by this that they would last for ever — where in fact they were already pretty inaccurate to begin with. (It could be that the word ‘perpetual’ was misunderstood, and that Lansberg only meant that the tables could be used for any date at will, just as those little ‘perpetual calendar’ gadgets can or could be set to any wanted combination of day- and month-names with date-numbers, without really suggesting that the gadgets themselves would be robust enough to last for ever.) Then again, this other kind of — fundamental? — tables was sometimes given the label ‘theory’. But that too sometimes attracted criticism, e.g. for implying somehow a greater content of theory than the tables contained, although such tables almost inevitably contain at least in summary the results of some kind of a theory. So then, what to call them, to give them a reasonable label and identification in general discussion, that tells them apart from just a list of positions for dates?

According to Curtis Wilson (“Predictive astronomy in the century after Kepler”), van Lansberge “assumed the exactitude of all the ancient observations, and attempted to devise a theory that would fit all of them.” So for example he included the (non-exixtent) trepidation of the equinoxes, that Tycho had ascribed (correctly) to faulty ancient observations. So “perpetual” was intended as a dig at tables that were less reverent to the ancients.

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The frontispiece seems to be in the style of the time, which also reminds me of the famous illustration from one of the Rosicrucian texts. Not that I’m saying there’s any link, but in this case an explication of the meaning of it would be interesting. I see the island pictured is the one Brahe and Kepler worked on, there are various astronomical instruments hanging up on the columns, and famous predecessors like Hipparchus and Ptolemy are pictured. Perhaps some of the other things, like the two front columns being of stone with different capitals, and the others of brick, is more of a convention of the time? Or is it related to the book being a work based in the real world?

The Temple of Urania depicted on the frontispiece is a heavily layered symbolic potted history of astronomy up to the publication of the Rudolphine Tables. It has been the object of several iconological and historical studies. See most recently http://www.ingentaconnect.com/content/shp/jhast/2014/00000045/00000001/art00001

Thanks – I see it is hot off the press!

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Love all things historical, especially science and math.

Just discovered your Rough Guides which satisfy both.

Coincidentally, I have just been comparing several accounts of Galileo’s “troubles” and been appalled at the inaccuracies that continue to be repeated.

“Hate” misuse of homophones, like principal and principle. (One of my students told me he had learned “The princiPAL is your PAL.” as a mnemonic. That’s good for spelling the “person,” but ignores the meaning of principal as “main, most important.” The “ple” ending ONLY refers to an ethic, as in “He lives by his moral principles.” Can you tell I get accused of being persnickety?

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This post seems to be implying that the Rudolphine tables were the primary factor leading to the acceptance of heliocentricity (as opposed to the acceptance of elliptical orbits within a heliocentric system.) However, it seems curious that the main rival to the Rudolphine tables in the 1630s were the heliocentric (but much less accurate) tables of Landsberg (This is explicitly stated in Russell 1964 https://arxiv.org/pdf/1301.1026.pdf and seems consistent with what I’ve read elsewhere.) Wouldn’t this tend to suggest that heliocentrism was already fairly popular before it became clear that Kepler’s tables were more accurate than anyone else’s?

There had been Copernican tables from very early on. The

Tabulae prutenicaefrom Erasmus Reinhold, the ephemeris from John Feild and the ephemeris from Johann Stadius. However, being all based on the same inaccurate Ptolemaic data they all suffered from the same inaccuracies. It is true that heliocentricity was gaining traction when Kepler published theTabulae Rudolphinaebut it still lagged behind the Tychonic system with diurnal rotation as most widely accepted astronomical system. The extremely high accuracy ofTabulae Rudolphinaecompared to everything that had gone before, including Lansberge, swung the argument in favour of heliocentricity and in favour of Kepler’s elliptical astronomy as opposed to Copernicus; they were regarded as rival systems. This was somewhat of a con as the accuracy owed more to the accuracy of Tycho’s observations than to Kepler’s modelThe claim that the accuracy of the Rudolphine tables was mostly due to having better observational data than everyone else seems surprising to me. Do you have a citation for this claim? Of course you can match any motion you like given enough epicycles, but did the non-elliptic models of the 17th century really have enough epicycles to match the Rudolphine tables even if the parameters had been set using perfect observations? Has anyone tried? Also, didn’t Longomontanus use pretty much the same data as Kepler for his Danish Tables? Finally, it surprises me that no one would have tried to improve the non-elliptic models with better observations in the 40+ years between the death of Tycho and the general acceptance of elliptic astronomy. After all, Horrocks only was able to use Kepler’s Model to predict the transit of Venus after improving it with his own observations.