A Statement by the Editors In response to Adam Becker and Undark.

European astronomers of the sixteenth century encountered two competing systems of mathematical astronomy. On the one hand, in the ancient system of Ptolemy, the earth was the immobile center of the universe. Constrained by Aristotelian physical principles to move in uniform circular motion, the moon, sun, and seven known planets orbited the earth. “Absolutely all phenomena, are in contradiction,” Ptolemy writes, “to any of the alternate notions that have been propounded.” On the other hand, the new theory of Nicolaus Copernicus which, while still committed to uniform circular motion, argued that by placing the sun at the center instead, the apparent retrograde motion of the planets could be accounted for with greater mathematical simplicity and elegance.

Copernican theory encountered resistance on theological and philosophical grounds. Other opponents pointed to the evidence of the senses: the earth was perceptibly not in motion. But what about the predictive power of Copernican theory? Whatever its philosophical or physical flaws, was it at least superior to the Ptolemaic system in calculating the motions of celestial objects?

Accounting for the observed phenomena had long been a primary goal of astronomy. When Copernicus published De revolutionibus orbium coelestium (On the Revolutions of the Heavenly Spheres) in 1543, it appeared with an anonymous preface now attributed to its publisher, the German theologian Andreas Osiander. In an attempt to forestall criticism of the heliocentric system, Osiander argued that it was impossible to know what was truly taking place in the heavens; the best one could hope for was a theory that was able to match the observational data.

[I]t is the job of the astronomer to used painstaking and skilled observation in gathering together the history of the celestial movements, and then … to think up or construct whatever causes or hypotheses he pleases such that, by the assumption of these causes, those same movements can be calculated from the principles of geometry for the past and for the future too.

Because Copernicus relied on the Aristotelian requirement that the planets move in circles, his theory was not capable of predicting planetary positions with perfect accuracy. Only with Johannes Kepler’s discovery of elliptical planetary orbits could the heliocentric system consistently predict planetary motion.

Between Copernicus and Kepler stood Tycho Brahe, whose meticulous observational accuracy enabled Kepler’s breakthrough. Tycho was not only committed to avoiding error in observation, he also rejected the then-standard practice of observing planets at only a few key points in their orbits.

Throughout his working life, Tycho attempted to verify or disprove astronomical theories on empirical, observational grounds. Indeed, Tycho claimed that his own concern with observation was stimulated by watching the conjunction of Jupiter and Saturn in 1563, a conjunction that had been predicted by both the Ptolemaic theory and the Copernican, and that, he later reported, “agreed with neither … although the agreement with the latter was better than with the former.”

It is Tycho’s systematic comparisons between the predictive powers of the Ptolemaic and Copernican systems that are the subject of this essay.

Tycho’s Notebooks

Published in Denmark in the early twentieth century under the editorship of his biographer, the astronomer John Dreyer, Tycho’s observation notebooks are the only extensive surviving record of comparisons between the predictive powers of the Ptolemaic and Copernican systems made during the sixteenth century.

The notebooks contain a record of all Tycho’s observations made between 1564 and 1601. Within this vast body of work, the specific instances in which Tycho made a direct comparison between his own observations and the predictions of Ptolemy and Copernicus form the basis for the analysis that follows.

Tycho first compared the predictions of Ptolemy and Copernicus for planetary positions on May 1, 1564, when he was just seventeen years old, a year after the conjunction of Jupiter and Saturn. Tycho’s final comparison was recorded on January 9, 1601, a little less than a year before his death at the age of 54 on October 24, 1601.

This observational span coincides with the latter half of the period between the publication of De revolutionibus in 1543 and Kepler’s Astronomia nova, describing the discovery of his first two laws of planetary motion, published in 1609. Tycho’s notebooks thus provide an excellent picture of the observational situation during the period in which the theories of Ptolemy and Copernicus were the two competing models of planetary astronomy.

Before examining Tycho’s observational data, a brief word about the process by which the predictions were calculated may be instructive. Prior to the advent of modern computers, predictions for the positions of the planets were, generally speaking, obtained from a three-step process. The starting point was mathematical astronomical theory, either according to Ptolemy, as described in his Almagest, or according to Copernicus, as described in his De revolutionibus. Astronomical tables, such as the Alfonsine tables and Prutenic tables, providing the data necessary for computing planetary positions, were then prepared from the general theory and copied or published separately. The tables presented certain values computed for astronomically significant functions. The correction from a mean planetary position to a true position, according to the distance from the planet’s apogee is an example. From these tables, another more detailed table known as an ephemeris was then computed, providing the actual predicted positions of the planets for a particular year, day, and time.

The Alfonsine tables were computed in the thirteenth century using the astronomical theory in the Almagest. The original version was written in Castilian using astronomical data compiled by a team of astronomers in Toledo under the patronage of King Alfonso X. Their calculations begin with the year 1252. A Latin edition was prepared in Paris during the 1320s, the first printed edition of which appeared in 1483. The Prutenic tables were computed from De revolutionibus by the German astronomer Erasmus Reinhold, and were first published in 1551.

On May 1, 1564, the longitudes of Venus, Mars, Saturn, and Jupiter were the subject of Tycho’s first recorded comparison between his own observations and the predictions of Ptolemy and Copernicus.

Planet Observed Stadius Carello Venus 25˚ 22˚ 54′ (-3˚ 6′) 24˚ 3′ (-57′) Mars 14˚ 12˚ 52′ (-1˚ 8′) 11˚ 20′ (-2˚ 40′) Saturn 0˚ 27˚ 36′ (-2˚ 24′) 29˚ 53′ (-0˚ 7′) Jupiter 3˚ 1˚ 30′ (-1˚ 30′) 0˚ 50′ (-2˚ 10′)

Tycho compared his observations to two ephemerides, those of Johannes Stadius and Giovanni Battista Carello, published in 1556 and 1557 respectively. The ephemeris of Stadius is based on the Prutenic tables computed from De revolutionibus, while the ephemeris of Carello was based on the Alfonsine tables computed from Almagest. In this comparison, the Ptolemaic theory was superior for Venus and Saturn, and the Copernican theory superior for Mars and Jupiter.

It should be noted that in his observational notebook Tycho recorded only the observed and predicted longitudes. He did not explicitly compute the difference between observations and predictions. This is noted in parentheses.

Five years after this first comparison, Tycho visited the Bohemian astronomer Cyprian von Leowitz, a professor of astronomy and mathematics in the town of Lauingen. According to Tycho, von Leowitz told him that, in his opinion, the predictions of Copernicus agreed better with observations of the superior planets and solar eclipses, while Ptolemy’s predictions were more accurate for lunar eclipses and the positions of the inferior planets. Von Leowitz, as our analysis shows, was correct about the relative superiority of the two theories in predicting the longitudes of the planets.

The Data from Tycho’s Notebooks

Tycho’s longitude and latitude comparisons for the individual planets are presented in a series of tables over the following sections.

Tycho variously used ephemerides based on the Alfonsine and Prutenic tables, or more usually, computed the predicted positions directly from the tables. Any prediction from an ephemeris based on the Alfonsine tables, or directly calculated from them by Tycho, is labeled “Ptolemy.” Similarly, any prediction based on the Prutenic tables or De revolutionibus is labeled “Copernicus.” For example, the figure of +1˚ 50½′ which is found under the label Mercury and Copernicus on October 25, 1585, indicates that the prediction of Copernicus was +1˚ 50½′ ahead of Tycho’s observation. Ptolemy’s prediction, on the other hand, was +2˚ 58′ ahead. Thus Copernicus was superior by 1˚ 7½′ in this instance.

Each table notes only the difference between Tycho’s actual observations and the predictions. As noted earlier, Tycho did not generally compute the differences between the predictions and his observations, except toward the end of his life.

The median superiority for each planet is provided at the bottom of each table. Rather than the mean, which would have required comparing all the observations over the entire period, the median is likely a much better estimate for how Tycho would have perceived the relative superiority of the two theories.

Longitude Comparisons

Table 1a. Mercury Longitude Comparisons

Date Ptolemy Copernicus Net Ptolemy Net Copernicus 1586 Oct 25 +2˚ 58′ +1˚ 50 ½′ 1˚ 7 ½′ 1586 Oct 29 +0˚ 59′ +1˚ 35′ 36′ 1586 Oct 31 -0˚ 2′ +1˚ 12′ 1˚ 10′ 1586 Nov 1 -0˚ 28 ½′ +1˚ 5′ 36 ½′ 1586 Nov 2 -0˚ 56′ +0˚ 56′ Tie Tie 1586 Nov 4 -1˚ 42 ½′ +0˚ 39 ½′ 1˚ 3′ 1586 Nov 7 -2˚ 50 ¼′ +0˚ 15′ 2˚ 35 ¼′ 1586 Nov 8 -3˚ 9′ +0˚ 11′ 2˚ 58′ 1586 Nov 10 -3˚ 38′ +0˚ 5′ 3˚ 33′ 1586 Nov 11 -3˚ 57 ¼′ -0˚ 5′ 3˚ 52 ¼′ 1589 Mar 24 -1˚ 1′ -0˚ 20′ 41′ 1589 Mar 27 +0˚ 4′ 37′′ 0˚ 20′ 23′′ 15′ 46′′ 1589 Mar 28 +0˚ 32′ 47′′ -0˚ 8′ 13′′ 24′ 34′′ 1589 Mar 31 +2˚ 15′ 4′′ +0˚ 16′ 34′′ 1˚ 58′ 30′′ 1591 Feb 16 -12′ 13′′ +52′ 37′′ 40′ 24′′ 1591 Feb 17 -8′ 15′′ +41′ 45′′ 33′ 24′′ 1591 Feb 18 -1′ +31′ 30′′ 30′ 30′′ 1591 Feb 19 +11′ +24′ 20′′ 13′ 20′′ 1591 Feb 20 +23′ 52′′ +16′ 52′′ 7′ 1591 Feb 21 +42′ +14′ 28′ 1591 Feb 22 +1° 7′ 50′′ +16′ 50′′ 51′ 1591 Feb 26 +3° 22′ 54′′ +39′ 54′′ 2˚ 43′ 1592 Feb 3 +34′ 15′′ +10′ 45′′ 23′ 30′′ 1595 Aug 30 -1° 59′ 10′′ -16′ 10′′ 1° 43′ 1595 Sep 1 -1° 47′ 30′′ -33′ 30′′ 1° 14′ 1595 Sep 2 -1° 33′ 30′′ -35′ 30′′ 58′ 1595 Sep 3 -1° 19′ -38′ 41′ 1595 Sep 4 -1° 1′ -36′ 25′ 1595 Sep 6 -14′ 30′′ -26′ 30′′ 12′ 1595 Sep 11 +2° 15′ -1′ 2° 14′ 1595 Sep 22 +18′ 15′′ +44′ 15′′ 26′ 1601 Apr 29 +61 -36′ 25′ Median 30 ½′ 1˚ 5′

Table 1b. Venus Longitude Comparisons

Date Ptolemy Copernicus Net Ptolemy Net Copernicus 1564 May 1 -57′ -3˚ 6′ 2˚ 9′ 1564 Dec 10 +2′ +40′ 38′ 1564 Dec 14 -16′ +17′ 1′ 1568 Nov 1 -18′ -22′ 3′ 1568 Nov 9 +5′ -28′ 23′ 1578 Oct 23 +1˚ 36 ½′ -2˚ 48 ½′ 1˚ 12′ 1585 Sep 23 +2 -12′ 10′ 1585 Oct 11 -6 ⅓′ -16 ⅚′ 10½′ 1589 Apr 5 -1˚ 4′ +1˚ 48′ 44′ 1589 Apr 8 -1˚ 3′ +1˚ 44′ 41′ 1589 Apr 13 -1˚ 0′ 20′′ +1˚ 48′ 40′′ 48′ 20′′ 1589 Apr 15 -0˚ 57′ 20′′ +1˚ 43′ 40′′ 46′ 20′′ 1589 Apr 20 -0˚ 56′ 5′′ +1˚ 35′ 55′′ 39′ 50′′ 1589 Apr 22 -1˚ 5′ 10′′ +1˚ 21′ 50′′ 16′ 40′′ 1589 Apr 23 -0˚ 48′ 50′′ +1˚ 34′ 10′′ 46′ 20′′ 1589 Apr 24 -0˚ 45′ 30′′ +1˚ 33′ 30′′ 48′ 1589 Apr 26 -0˚ 43′ 30′′ +1˚ 30′ 30′′ 47′ 1591 Sep 27 -31′ 43′′ -31′ 13′′ 30′′ 1591 Sep 30 -32′ 10′′ -27′ 10′′ 5′ 1591 Oct 2 -38′ -29′ 9′ 1591 Oct 9 -36′ 19′′ -19′ 19′′ 17′ 1591 Oct 10 -38′ 24′′ -19′ 24′′ 19′ 1591 Oct 16 -42′ 30′′ -12′ 30′′ 30′ 1591 Nov 10 -1° 26′ 20′′ +31′ 40′′ 54′ 40′′ 1591 Nov 11 -1° 22′ 45′′ +39′ 15′′ 43′ 30′′ 1595 Sep 22 +9′ 45′′ -39′ 15′′ 19′ 20′′ 1596 Jun 11 -28′ -31′ 3′ 1596 Jun 20 -36 ⅚′ -1˚ 23 ⅚′ 47′ 1596 Jun Elong. off

9 days * Elong. off

8 days * 1-2 days 1600 Feb 22 +29′ 2′′ +23′ 25′′ 5′ 37′′ Median 38′ 18′

* Tycho observed that the predicted day of maximum elongation was incorrect by 8 and 9 days for the predictions of Ptolemy and Copernicus respectively.

Table 1c. Moon Longitude Comparisons

Date Ptolemy Copernicus Net Ptolemy Net Copernicus 1586 Oct 24 -40′ -50′ 10′ 1587 Jan 6 -20′ -17′ 3′ 1587 Aug 4 +3′ +28′ 25′ 1587 Aug 5 0′ +16′ 16′ 1587 Aug 17 +25′ +7′ 18′ 1587 Aug 18 +44′ +20′ 24′ 1591 Aug 23 -12′ 47′′ +1′ 13′′ 11′ 34′′ 1593 Mar 15 -47′ -8′ 39′ 1594 Dec 10 +42 ¼′ +16 ¼′ 26′ 1594 Dec 11 +1° 14 ⅓′ +35 ⅓′ 39′ 1594 Dec 14 +56 ⅔′ +43 ⅔′ 13′ 1594 Dec 15 +39 ⅙′ +28 ⅙′ 11′ 1594 Dec 20 -1° 4 ½′ -1° ½′ 4′ 1594 Dec 21 -43 ½′ -1° 2 ½′ 19′ 1594 Dec 22 -14 ¾′ -53 ¾′ 39′ 1594 Dec 23 -⅚′ -36 ⅚′ 36′ 1594 Dec 26 +10 ⅔′ +34 ⅔′ 14′ 1594 Dec 27 +20′ +57′ 37′ Median 21′ 15′

Table 1d. Sun Longitude Comparisons

Date Ptolemy Copernicus Net Ptolemy Net Copernicus 1583 Jan 26 +22′ 54′′ -28′ 36′′ 5′ 42′′ 1583 Mar 26 +19′ 15′′ -20′ 45′′ 1′ 30′′ 1583 Mar 27 +19′ 30′′ -20′ 30′′ 1′ 0′′ 1583 Mar 28 +15′ 34′′ -22′ 23′′ 6′ 49′′ 1583 Apr 2 +19′ -19′ Tie Tie 1583 Apr 4 +38′ +19′ 19′ 1583 Apr 5 +18′ -19′ 1′ 1583 Apr 9 +18′ -18′ Tie Tie 1583 Apr 14 +18′ -17′ 1′ 1583 Apr 15 +18 -16′ 2′ 1583 Apr 16 +17′ -16′ 1′ 1583 Apr 18 +16′ 38′′ -15′ 22′′ 1′ 16′′ 1583 Apr 19 +16′ 37′′ -15′ 23′′ 1′ 14′′ 1583 Apr 20 +16′ 37′′ -15′ 23′′ 1′ 14′′ 1583 Apr 25 +15′ -15′ Tie Tie 1583 May 5 +13′ -12′ 1′ 1583 May 6 +13′ -12′ 1′ 1583 Aug 15 -1′ -1′ Tie Tie 1583 Sep 5 -½′ -2 ½′ 2′ 1583 Sep 8 Dead on -4′ 4′ 1583 Sep 11 -½′ -4 ½′ 4′ 1583 Sep 23 +1′ -6′ 5′ 1583 Sep 29 +1 ½′ -6 ½′ 5′ 1583 Oct 1 +1 ⅓′ -6 ⅔′ 5 ⅓′ 1583 Oct 10 +2 ⅓′ -8 ⅔′ 6 ⅓′ 1583 Oct 12 +3 ½′ -8 ½′ 5′ 1583 Oct 15 +3 ⅓′ -10 ⅔′ 7 ⅓′ 1583 Oct 20 +4 ⅓′ -11 ⅔′ 7 ⅓′ 1583 Oct 24 +6′ 5′′ -12′ 55′′ 6′ 50′′ 1583 Oct 26 +5′ 15′′ -12′ 45 7 ½′ 1583 Oct 30 +6′ 15′′ -13′ 45′′ 7 ½′ 1583 Nov 7 +8′ -16′ 8′ 1583 Nov 8 +8′ 10′′ -15′ 50′′ 7′ 40′′ 1584 Aug 23 -2′ 31′′ -3′ 5′′ 34′′ 1584 Sep 14 +1′ 28′′ -3′ 35′′ 2′ 7′′ 1584 Oct 3 +6′ 32′′ -4′ 13′′ 2′ 19′′ 1584 Oct 7 +9′ 55′′ -2′ 44′′ 5′ 11′′ 1585 Mar 9 +15′ 40′′ -30′ 43′′ 15′ 3′′ 1585 Mar 13 +15′ 28′′ -29′ 30′′ 14′ 2′′ 1585 Mar 15 +15′ 36′′ -28′ 25′′ 12′ 49′′ 1585 Mar 17 +14′ 20′′ -29′ 30′′ 15′ 10′′ 1585 Apr 15 +9′ 36′′ -25′ 39′′ 16′ 3′′ 1585 Apr 26 +6′ 49′′ -22′ 33′′ 15′ 44′′ 1586 Mar 9 +15′ 8′′ -30′ 53′′ 15′ 45′′ 1586 Mar 10 +15′ 50′′ -30′ 23′′ 14′ 27′′ 1586 Mar 11 +12′ 40′′ -33′ 30′′ 20′ 50′′ 1586 Mar 17 +4′ 22′′ -29′ 12′′ 24′ 50′′ 1586 Mar 19 +14′ 17′′ -29′ 2′′ 14′ 45′′ 1586 Apr 1 +12′ 19′′ -27′ 21′′ 15′ 2′′ 1586 Apr 13 +10′ 33′′ -25′ 57′′ 15′ 24′′ 1586 Apr 16 +9′ 25′′ -24′ 48′′ 15′ 23′′ 1586 Apr 18 +8′ 38′′ -24′ 24′′ 15′ 46′′ 1586 Apr 19 +8′ 43′′ -24′ 3′′ 14′ 20′′ 1586 Apr 27 +6′ 7′′ +36′ 11′′ * 30′ 4′ 1586 May 7 +3′ 46′′ -21′ 17′′ 17′ 31′′ 1586 May 8 +3′ 26′′ -22′ 4′′ 18′ 38′′ 1586 May 11 +5′ 3′′ -19′ 28′′ 14′ 25′′ 1586 May 18 +1′ 35′′ -19′ 36′′ 18′ 1′′ 1586 May 25 +1′ 20′′ -16′ 43′′ 15′ 13′′ 1586 May 29 -1′ 45′′ -18′ 0′′ 16′ 15′′ 1586 Jul 23 -3′ 1′′ -7′ 11′′ 4′ 10′′ 1586 Jul 24 -0′ 30′′ -4′ 8′′ 3′ 38′′ 1586 Jul 27 -0′ 3′′ -3′ 7′′ 3′ 4′′ 1586 Aug 4 +0′ 36′′ -2′ 3′′ 1′ 27′′ 1586 Sep 4 +2′ 21′′ -1′ 18′′ 1′ 3′′ 1586 Sep 24 +5′ 56′′ -3′ 17′′ 2′ 39′′ 1586 Oct 10 +10′ 50′′ -2′ 52′′ 7′ 58′′ 1587 Mar 9 +0˚ 15′ 44′′ -0˚ 32′ 10′′ 16′ 34′′ 1587 Mar 10 +0˚ 15′ 34′′ -0˚ 32′ 4′′ 16′ 30′′ 1587 Mar 17 +0˚ 14′ 14′′ -0˚ 31′ 13′′ 16′ 59′′ 1587 Sep 12 +0˚ 1′ 22′′ -0˚ 3′ 36′′ 2′ 14′′ 1588 Mar 27 +0˚ 12′ 38′′ -0˚ 28′ 34′′ 15′ 56′′ 1601 Mar 19 +15′ -36′ 21′ Median 18′ 1′ 14′′

* The sudden decrease in accuracy for the Copernican prediction on April 27, 1586 is likely due to a calculation error. Note that the error is negative for both the immediately preceding and following observations.

Table 1e. Mars Longitude Comparisons

Date Ptolemy Copernicus Net Ptolemy Net Copernicus 1564 May 1 -2˚ 40′ -1˚ 8′ 1˚ 32′ 1568 Nov 1 -1˚ 8′ +53′ 15′ 1570 Apr 26 -1˚ 29′ +14′ 1˚ 15′ 1578 Sep 15 -3˚ 28′ -3 ¼˚ 3′ 1580 Nov 17 * -59′ -32′ 27′ 1582 Dec 30 -5′ +4′ 1′ 1584 Nov 13 -1˚ 34′ 15′′ +37′ 45′′ 57′ 30′′ 1585 Jan 14 -3˚ 26½′ +2˚ 18¼′ 1˚ 8¼′ 1585 Jan 22 * -3˚ 32¼′ +2˚ 16¾′ 1˚ 15½′ 1585 Jan 24 -3˚ 31¾′ +2˚ 5½′ 1˚ 16¼′ 1585 Jan 31 -3˚ 39′ +2˚ 20′ 1˚ 19′ 1586 Sep 23 -34′ +28′ 6′ 1591 Apr 29 +2° 33′ 35′′ -4′ 10′′ 2° 33′ 25′′ 1591 May 12 +3° 18′ 10′′ +9′ 3° 9′ 10′′ 1591 May 13 +3° 22′ +10′ 20′′ 3° 19′ 40′′ 1591 May 19 +3° 45′ 15′′ +15′ 30′′ 3° 29′ 45′′ 1591 May 21 +3° 49′ +15′ 35′′ 3° 33′ 25′′ 1591 May 24 +3° 56′ 40′′ +12′ 3° 44′ 40′′ 1591 May 29 +4° 8′ 40′′ +6′ 10′′ 4° 2′ 40′′ 1591 Jun 6 +4° 21′ -8′ 4° 13′ 1591 Jun 8 +4° 19′ 18′′ -10′ 42′′ 4° 8′ 36′′ 1591 Jun 10 +4° 15′ 30′′ -16′ 30′′ 3° 59′ 1591 Jun 11 +4° 15′ -18′ 3° 57′ 1591 Jun 12 +4° 14′ -22′ 3° 52′ 1591 Jun 28 +3° 42′ -50′ 2° 52′ 1591 Jul 17 +2° 29′ -1° 17′ 1° 12′ 1591 Sep 26 +2° 16′ 45′′ -13′ 15′′ 2° 1′ 30′′ 1591 Dec 10 +1° 25′ 30′′ -31′ 30′′ 54′ 1593 Jul 21 +4° 58′ 15′′ -2° 57′ 15′′ 2° 1′ 1593 Jul 22 +4° 58′ 15′′ -2° 57′ 15′′ 2° 1′ 1593 Jul 30 -3° 42′ 30′′ -3° 42′ 30′′ Tie Tie 1593 Jul 31 +5° 6′ 30′′ -3° 42′ 30′′ 1° 24′ 1593 Aug 10 * +5° 18′ 30′′ -4° 7′ 30′′ 1° 11′ 1593 Aug 17 +5° 8′ 55′′ -4° 18′ 5′′ 50′ 50′′ 1593 Aug 18 +5° 8′ 55′′ -4° 18′ 5′′ 59′ 10′′ 1593 Aug 20 +5° 2′ -4° 28′ 34′ 1593 Aug 21 +5° 2′ 30′′ -4° 28′ 30′′ 34′ 1593 Aug 22 +5° 0′ 45′′ -4° 31′ 15′′ 29′ 30′′ 1593 Aug 23 +4° 58′ 40′′ -4° 33′ 20′′ 25′ 20′′ 1593 Aug 24 -2′ 50′′ † -5° 33′ 50′′ 5° 17′ 1593 Aug 29 +4° 43′ 35′′ -4° 33′ 25′′ 10′ 10′′ 1593 Dec 10 +1° 24′ -15′ 1° 9′ 1593 Dec 19 +1° 22′ 10′′ -12′ 50′′ 1° 9′ 20′′ 1594 Dec 10 -4′ -14′ 10′ 1595 Sep 7 +50′ -1° 30′ 40′ 1595 Sep 30 +9′ 30′′ -1° 46′ 30′′ 1° 37′ 1595 Oct 12 -1′ 30′′ -1° 56′ 30′′ 1° 55′ 1595 Oct 16 -5′ 45′′ -1° 55′ 45′′ 1° 50′ 1595 Oct 25 -7′ -1° 59′ 1° 52′ 1599 Oct 20 -45′ 57′′ +16′ 3′′ 29′ 54′′ 1599 Oct 21 -47′ 25′′ +16′ 35′′ 30′ 50′′ 1599 Oct 24 -54′ 41′′ +16′ 19′′ 38′ 22′′ 1599 Oct 25 -55′ 20′′ +17′ 40′′ 37′ 40′′ 1599 Oct 27 -59′ 10′′ +17′ 50′′ 41′ 20′′ 1599 Oct 29 -1˚ 1′ 10′′ +19′ 50′′ 41′ 20′′ 1600 Mar 2 ‡ -2° 59′ 46′′ +1° 25′ 14′′ 1° 34′ 32′′ 1600 Mar 9 ‡ -2° 51′ 4′′ +1° 12′ 56′′ 1° 38′ 8′′ 1600 Mar 15 ‡ -2° 49′ 30′′ +57′ 30′′ 1° 52′ 0′′ 1600 Mar 16 ‡ -2° 47′ 53′′ +56′ 7′′ 1° 51′ 46′′ Median 1° 37′ 1˚ 16 ¼′

* In opposition. † Likely an error in recording the Ptolemaic prediction. Counted here as a very superior prediction for Ptolemy. ‡ Tycho computed the difference between his observations and the predictions of Ptolemy and Copernicus.

Table 1f. Jupiter Longitude Comparisons

Date Ptolemy Copernicus Net Ptolemy Net Copernicus 1564 May 1 -2˚ 10′ -1˚ 30′ 40′ 1568 Jan 19 -1˚ 31′ -3′ 1˚ 18′ 1583 Aug 6 +33′ -12′ 21′ 1583 Aug 28 35′ -14′ 30′′ 20′ 30′′ 1583 Sep 5 +34 ½′ -13′ 0′′ 21½′ 1583 Sep 6 +31 ¾′ -14 ¼′ 17½′ 1583 Sep 10 +31′ -15′ 16′ 1583 Dec 11 +38 ½′ -7 ½′ 31½′ 1584 Aug 20 +30′ 29′′ -32′ 31′′ 2′ 2′′ 1586 Sep 23 +16′ +33′ 17′ 1586 Dec 21 -16′ -25′ 9′ 1589 Jan 30 -1˚ 29′ +4′ 1˚ 25′ 1589 Feb 21 -1˚ 27′ +11′ 1˚ 16′ 1590 Mar 15 * -1° 45′ 54′′ +24′ 59′′ 1° 20′ 55′′ 1590 Mar 17 * -1° 45′ 59′′ +32′ 1′′ 1° 13′ 58′′ 1590 Jul 11 * -1° 52′ 17′′ -12′ 44′′ 1° 39′ 33′′ 1591 Apr 14 -1° 46′ 10′′ +24′ 30′′ 1° 22′ 60′′ 1591 Apr 20 † -1° 45′ 15′′ +23′ 45′′ 1° 11′ 30′′ 1591 Apr 22 -1° 45′ 50′′ +25′ 20′′ 1° 20′ 30′′ 1591 Apr 23 ‡ -1° 45′ +25′ 20′′ 1° 19′ 40′′ 1591 Apr 24 -1° 42′ +27′ 1° 17′ 1591 Apr 26 -1° 43′ +26′ 1° 17′ 1591 Apr 29 ‡ -1° 46′ 30′′ +24′ 30′′ 1° 22′ 1591 Apr 30 -1° 47′ 13′′ +23′ 47′′ 1° 23′ 26′′ 1591 Jun 12 -1° 46′ 20′′ +19′ 40′′ 1° 26′ 40′′ 1592 May 16 -1° 29′ +17′ 1° 12′ 1592 May 18 -1° 30′ 40′′ +15′ 20′′ 1° 15′ 20′′ 1593 Jun 29 † -51′ 50′′ +8′ 10′′ 43′ 40′′ 1594 Oct 28 +1′ 10′′ -11′ 50′′ 10′ 40′′ 1596 Jul 25 +1˚ 44 ⅓′ +1˚ 38 ⅔′ 5⅔′ 1596 Aug 8 +21 ⅔′ -41 ⅓′ 19 ⅔′ 1596 Oct 12 +11 ⅓′ -49 ⅔′ 38 ⅓′ 1596 Oct 15 +17 ⅙′ -49 ⅚′ 32 ⅔′ 1598 Dec 20 1˚ 4′ at most 1˚ 4′ at most Tie Tie 1599 Oct 20 -34′ 41′′ +16′ 3′′ 18′ 38′′ 1599 Oct 21 -42′ 51′′ -15′ 51′′ 27′ 1599 Oct 24 -44′ 28′′ -19′ 28′′ 25′ 1599 Oct 25 -43′ 31′′ -17′ 31′′ 26′ 1600 Feb 3 † § -60′ 29′′ -4′ 29′′ 56′ 1600 Feb 4 § -60′ 48′′ -4′ 28′′ 56′ 20′′ 1600 Feb 5 § -60′ 29′′ -4′ 29′′ 56′ 1600 Mar 15 § -58′ 21′′ -3′ 21′′ 55′ 1601 Jan 6 -1° 29′ 40′′ +9′ 40′′ 1° 20′ Median 17′ 1° 12′

* Tycho compared tables calculated by the German astronomer Michael Maestlin with the Prutenic tables and the Alfonsine tables on March 15, March 17, and July 11, 1590. The less accurate of the two Copernican predictions was used. † In opposition. ‡ Two different values for the same day were given for the accuracy of Copernicus, differing by 4′ on April 23, 1591 and 2′′ on April 29, 1591. § Tycho computed the difference between his observations and the predictions of Ptolemy and Copernicus.

Table 1g. Saturn Longitude Comparisons

Date Ptolemy Copernicus Net Ptolemy Net Copernicus 1564 May 1 -0˚ 7′ -2˚ 24′ 2˚ 17′ 1568 Jan 19 +52 -5′ 47′ 1568 Nov 1 +13′ +2′ 11′ 1568 Nov 19 +11′ -1′ 10′ 1582 Aug 19 +46′ 44′′ +16′ 44′′ 30′ 1583 Aug 6 +1˚ 26′ +24′ 1˚ 2′ 1583 Aug 28 +1˚ 17′ 47′′ +18′ 47′′ 59′ 1583 Sep 5 +1˚ 17′ 7′′ +16′ 7′′ 1˚ 1′ 1583 Sep 6 +1˚ 15 ⅙′ +14 ⅙′ 1˚ 1′ 1583 Sep 10 +1˚ 13′ +12′ 1˚ 1′ 1585 Sep 23 +1˚ 59′ 50′′ +4′ 50′′ 1˚ 55′ 1585 Sep 26 * +1˚ 58′ 45′′ +2¾′ 1˚ 56′ 1585 Oct 10 +1˚ 42′ -2′ 1˚ 40′ 1586 Sep 23 +4˚ 25′ +4′ 4˚ 21′ 1586 Dec 31 * +2˚ 26′ +0.5′ +2˚ 25½′ 1587 Oct 23 * +2˚ 41′ 0′ 2˚ 41′ 1588 Nov 10 * +2˚ 56′ 25′′ -0˚ 5′ 38′′ 2˚ 50′ 13′′ 1591 Oct 25 +2° 53′ 20′′ +3′ 20′′ 2° 50′ 1591 Dec 4 +58′ 30′′ -1′ 30′′ 57′ 1591 Dec 5 +2° 57′ 10′′ -1′ 30′′ 2° 55′ 40′′ 1591 Dec 6 +2° 56′ 30′′ -4′ 30′′ 2° 52′ 1591 Dec 9 * +2° 57′ 30′′ -3′ 30′′ 2° 54′ 1591 Dec 10 +2° 57′ 4′′ -3′ 56′′ 2° 53′ 8′′ 1591 Dec 16 +2° 56′ -5′ 2° 51′ 1591 Dec 21 +2° 55′ 40′′ -7′ 20′′ 2° 48′ 20′′ 1591 Dec 24 +2° 52′ 20′′ -9′ 10′′ 2° 41′ 10′′ 1591 Dec 25 +2° 51′ 40′′ -8′ 50′′ 2° 42′ 50′′ 1592 Dec 29 * +2° 48′ 55′′ -5′ 2° 43′ 55′′ 1594 Jan 14 * +2° 37′ 15′′ +5′ 15′′ 2° 32′ 1594 Dec 10 +2° 15′ 40′′ +14′ 40′′ 2° 1′ 1595 Jan 24 +2° 13′ 50′′ +4′ 50′′ 2° 9′ 1600 Feb 4 † +40′ 16′′ +42′ 16′′ 2′ 1600 Feb 5 † +39′ 29′′ +42′ 29′′ 3′ 1600 Feb 6 † +38′ 43′′ +42′ 29′′ 2′ 46′′ 1600 Feb 22 † +40′ 16′′ +45′ 16′′ 5′ 1600 Feb 23 † +40′ 2′′ +45′ 2′′ 5′ 1600 Apr 7 † +37′ 37′′ +39′ 37′′ 2′ 1600 Apr 10 † +36′ 32′′ +40′ 32′′ 4′ 1600 Apr 11 † +39′ 13′′ +41′ 32′′ 2′ 19′′ 1600 Apr 12 † +39′ 8′′ +41′ 13′′ 2′ 5′′ 1600 Apr 13 +38′ 42′′ +40′ 42′′ 2′ 1600 Apr 14 * † +37′ 10′′ +39′ 10′′ 2′ 1600 Apr 17 † +36′ 29′′ +39′ 29′′ 3′ 1600 Apr 18 † +35′ 52′′ +38′ 52′′ 3′ 1600 Apr 20 † +34′ 32′′ +37′ 32′′ 3′ 1600 Apr 27 † +30′ 28′′ +34′ 28′′ 4′ Median 3′ 2° 1′

* In opposition. † Tycho computed the difference between his observations and the predictions of Ptolemy and Copernicus.

Latitude Comparisons

Table 2a. Mercury Latitude Comparisons

Date Ptolemy Copernicus Net Ptolemy Net Copernicus 1586 Oct 25 -0˚ 35′ -1˚ 30 ¼′ 55 ¼′ 1586 Oct 29 -0˚ 34′ -1˚ 13 ½′ 39 ½′ 1586 Oct 31 -0˚ 31 ½′ -0˚ 59 ½′ 28′ 1586 Nov 1 -0˚ 30 ½′ -0˚ 52 ½′ 22′ 1586 Nov 2 -0˚ 28 ¾′ -0˚ 48 ¾′ 20′ 1586 Nov 4 -0˚ 25 ½′ -0˚ 44 ½′ 19′ 1586 Nov 7 -0˚ 20′ -0˚ 34′ 14′ 1586 Nov 8 -0˚ 18′ -0˚ 28 ⅔′ 10 ⅔′ 1586 Nov 10 -0˚ 19 ½′ -0˚ 19 ½′ Tie Tie 1586 Nov 11 -0˚ 17′ -0˚ 12′ 5′ 1589 Mar 24 -0˚ 51′ -0˚ 36′ 15′ 1589 Mar 27 -0˚ 54′ 15′′ -0˚ 36′ 15′′ 18′ 1589 Mar 28 -0˚ 56′ 28′′ -0˚ 36′ 28′′ 20′ 1589 Mar 31 -0˚ 55′ 24′′ -0˚ 24′ 24′′ 31′ 1591 Feb 16 -46′′ +14′ 14′′ 13′ 28′′ 1591 Feb 17 -8′ 15′′ +5′ 15′′ 3′ 1591 Feb 18 -13′ 7′′ -1′ 17′′ 11′ 50′′ 1591 Feb 19 -18′ -8′ 10′ 1591 Feb 20 -22′ 30′′ -14′ 30′′ 8′ 1591 Feb 21 -26′ 45′′ -14′ 45′′ 12′ 1591 Feb 22 -33′ 30′′ -16′ 30′′ 17′ 1591 Feb 26 -42′ 8′′ -20′ 48′′ 21′ 20′′ 1592 Feb 3 +4′ ½′ +11 ½′ 7′ 1595 Aug 30 -48′ 30′′ -29′ 30′′ 19′ 1595 Sep 1 -52′ 40′′ -23′ 40′′ 29′ 1595 Sep 2 -53′ 40′′ -24′ 40′′ 29′ 1595 Sep 3 -54′ 30′′ -23′ 30′′ 31′ 1595 Sep 4 -55′ 50′′ -23′ 50′′ 32′ 1595 Sep 6 -46′ -21′ 35′ 1595 Sep 11 -48′ 30′′ -3′ 30′′ 45′ 1595 Sep 22 -43′ -57′ 14′ Median 19′ 19′

Table 2b. Venus Latitude Comparisons

Date Ptolemy Copernicus Net Ptolemy Net Copernicus 1564 Dec 10 -46′ -2˚ 14′ 1˚ 28′ 1564 Dec 14 -2′ -1˚ 22′ 1˚ 20′ 1568 Nov 1 -0˚ 10′ -0˚ 24′ 14′ 1568 Nov 9 -16′ -1˚ 20′ 1˚ 4′ 1578 Oct 23 -32 ½′ +39 ½′ 7′ 1585 Sep 23 Dead on -39′ 39′ 1585 Oct 11 +5′ +44′ 39′ 1589 Apr 5 -0˚ 12′ +0˚ 36′ 24′ 1589 Apr 8 -0˚ 10′ 50′′ +0˚ 40′ 10′′ 29′ 20′′ 1589 Apr 13 -0˚ 0′ 40′′ +0˚ 48′ 20′′ 47′ 40′′ 1589 Apr 15 -0˚ 10′ 5′′ +0˚ 42′ 55′′ 32′ 50′′ 1589 Apr 20 -0˚ 12′ 30′′ +0˚ 46′ 30′′ 34′ 1589 Apr 22 -0˚ 16′ 20′′ +0˚ 44′ 40′′ 28′ 20′′ 1589 Apr 23 -0˚ 10′ 0′′ +0˚ 51′ 0′′ 41′ 1589 Apr 24 -0˚ 7′ 50′′ +0˚ 51′ 10′′ 43′ 20′′ 1589 Apr 26 -0˚ 7′ 20′′ +0˚ 52′ 40′′ 45′ 20′′ 1591 Sep 27 -1° 13′ 53′′ -47′ 53′′ * 26′ 1591 Sep 30 -1° 2′ 10′′ -42′ 10′′ 20′ 1591 Oct 2 -1° 2′ 40′′ -48′ 40′′ 14′ 1591 Oct 9 -51′ -47′ 4′ 1591 Oct 10 -52′ 30′′ -49′ 30′′ 3′ 1591 Oct 16 -35′ 12′′ -43′ 12′′ 8′ 1591 Nov 10 +26′ 15′′ -59′ 45′′ 33′ 30′′ 1591 Nov 11 +33′ 50′′ -50′ 10′′ 16′ 20′′ 1595 Sep 22 -31′ -19′ 15′ 1600 Feb 22 -16′ 13′′ -31′ 13′′ 15′ Median 17′

* This figure is taken from Tycho’s table for Venus on page 156 of volume 12 of his collected works. On page 153 of the same volume, Tycho gives values which constitute an error of -2° 47′ 53′′ rather than -47′ 53′′. The latitude of Venus cannot have changed by -2° between September 27, 1591 and September 30, 1591. Hence the figure -47′ 53′′ is used here.

Table 2c. Moon Latitude Comparisons

Date Ptolemy Copernicus Net Ptolemy Net Copernicus 1580 Jan 31 -2 ¾′ +1 ¼′ 1 ½′ 1586 Oct 24 -57′ -55′ 3′ 1587 Jan 6 -57′ -58′ 1′ 1587 Aug 4 +40′ +41′ 1′ 1587 Aug 5 +41′ +42′ 1′ 1587 Aug 17 -49′ -48′ 1′ 1587 Aug 18 -48′ -48′ Tie Tie 1591 Aug 23 -50′ 34′′ -46′ 50′′ 3′ 44′′ 1593 Mar 15 +9 ¼′ +26 ¼′ 17′ Median 1′ 17′

Table 2d. Mars Latitude Comparisons

Date Ptolemy Copernicus Net Ptolemy Net Copernicus 1568 Nov 1 -0˚ 10′ -0˚ 3′ 7′ 1570 Apr 26 -17′ +10′ 7′ 1574 Mar 20 -20′ -45′ 25′ 1580 Jan 17 -1 ½′ -1 ⅗′ 6′′ 1584 Nov 13 -17′ 24′′ -36′ 24′′ 19′ 1585 Jan 14 -10 ¼′ -15′ 4 ¾′ 1585 Jan 22 -12 ¾′ -11 ¾′ 1′ 1585 Jan 24 -12 ⅔′ -12′ ⅔′ 1585 Jan 31 -12′ -4 ½′ 7 ½′ 1586 Sep 23 -36′ -43′ 6′ 1591 Apr 29 +5′ 15′′ -54′ 45′′ 49 ½′ 1591 May 12 +11′ 30′′ -1° 8′ 30′′ 58′ 1591 May 13 +3′ 30′′ -1° 18′ 30′′ 1° 15′ 1591 May 19 +24′ 15′′ -1° 9′ 45′′ 45 ½′ 1591 May 21 +18′ 40′′ -1° 13′ 46′ 20′′ 1591 May 24 +14′ 40′′ -1° 5′ 41′ 20′′ 1591 May 29 +21′ 45′′ -57′ 25′′ 35′ 40′′ 1591 Jun 6 +38′ 30′′ -56′ 30′′ 18′ 1591 Jun 8 +46′ -54′ 8′ 1591 Jun 10 +55′ 30′′ -49′ 30′′ 6′ 1591 Jun 11 +53′ -53′ Tie Tie 1591 Jun 12 +53′ 30′′ -53′ 30′′ Tie Tie 1591 Jun 28 +25′ 30′′ -55′ 30′′ 30′ 1591 Jul 17 +37′ -57′ 20′ 1591 Sep 26 -9′ 45′′ -16′ 15′′ 6′ 30′′ 1591 Dec 10 +16′ 15′′ +42′ 30′′ 26′ 15′′ 1593 Jul 21 -17′ 15′′ -16′ 15′′ 1′ 1593 Jul 22 -17′ 15′′ -16′ 15′′ 1′ 1593 Jul 30 -20′ 15′′ -6′ 15′′ 14′ 1593 Jul 31 -20′ 15′′ -6′ 15′′ 14′ 1593 Aug 10 -18′ 52′′ +11′ 8′′ 7′ 44′′ 1593 Aug 17 -10′ 7′′ +23′ 53′′ 13′ 46′′ 1593 Aug 18 -10′ 7′′ +23′ 53′′ 13′ 46′′ 1593 Aug 20 -4′ +16′ 12′ 1593 Aug 21 +24′′ +31′ 24′′ 31′ 1593 Aug 22 +5′ 10′′ +34′ 10′′ 29′ 1593 Aug 23 +7′ 30′′ +32′ 30′′ 25′ 1593 Aug 24 +10′ 30′′ +31′ 30′′ 21′ 1593 Aug 29 +29′ 45′′ +28′ 45′′ 1′ 1593 Dec 10 45′′ +29′ 15′′ 28 ½′ 1593 Dec 19 -3′ -1′ 2′ 1594 Dec 10 -11′ 15′′ -7′ 15′′ 4′ 1595 Sep 7 -8′ +1° 45′ 1° 37′ 1595 Sep 30 -9′ +1° 5′ 56′ 1595 Oct 12 -9′ +1° 15′ 1° 6′ 1595 Oct 16 -10′ 40′′ +1° 11′ 20′′ 1° 0′ 20′′ 1595 Oct 25 -6′ 50′′ +54′ 10′′ 47′ 20′′ 1599 Oct 20 -15′ -35′ 20′ 1599 Oct 21 -15′ 34′′ -35′ 34′′ 20′ 1599 Oct 24 -14′ 45′′ -33′ 45′′ 19′ 1599 Oct 25 -14′ 20′′ -34′ 20′′ 20′ 1599 Oct 27 -14′ -34′ 20′ 1599 Oct 29 -15′ 40′′ -34′ 40′′ 19′ 1600 Mar 2 -21′ 51′′ -2′ 51′′ 19′ 1600 Mar 9 -22′ 15′′ -2′ 15′′ 20′ 1600 Mar 15 -18′ 53′′ -10′ 53′′ 8′ 1600 Mar 16 -18′ 45′′ -9′ 45′′ 9′ Median 25′ 7′

Table 2e. Jupiter Latitude Comparisons

Date Ptolemy Copernicus Net Ptolemy Net Copernicus 1583 Aug 6 +21′ +7′ 14′ 1583 Aug 28 +21′ +10′ 11′ 1583 Sep 5 +22 ⅙′ +10′ 12 ⅙′ 1583 Sep 6 +22′ 36′′ +9½′ 13′ 6′′ 1583 Sep 10 +22′ +10′ 12′ 1583 Dec 11 +11 ⅓′ +5 ⅓′ 6′ 1584 Aug 20 34 ½′ +38 ½′ 4′ 1586 Sep 23 +15′ +34′ 19′ 1586 Dec 21 +3′ +31′ 28′ 1589 Jan 30 +12 -4′ 8′ 1589 Feb 21 +13′ No error 13′ 1590 Mar 15 +26′ 45′′ +22′ 51′′ * 2′ 54′′ 1590 Apr 17 +1° 25′ 20′′ +21′ 57′′ * 1° 3′ 23′′ 1590 Jul 11 +5′ 53′′ +6′ 11′′ * 18′′ 1591 Apr 14 +27′ 30′′ +33′ 30′′ 6′ 1591 Apr 20 † +30′ 20′′ +37′ 20′′ 7′ 1591 Apr 22 +28′ 36′′ +36′ 36′′ 8′ 1591 Apr 23 +28′ +37′ 9′ 1591 Apr 24 +29′ 36′′ +38′ 36′′ 9′ 1591 Apr 26 +30′ 20′′ +37′ 40′′ 7′ 20′′ 1591 Apr 29 +28′ 10′′ +37′ 10′′ 9′ 1591 Apr 30 +28′ +37′ 9′ 1591 Jun 12 +43′ 30′′ +45′ 30′′ 2′ 1592 May 16 ‡ +28′ 15′′ +45′ 15′′ 7′ 1592 May 18 ‡ +28′ 45′′ +45′ 45′′ 7′ 1593 Jun 29 † -12 +4′ 8′ 1594 Oct 28 +7′ 50′′ -9′ 10′′ 1′ 20′′ 1599 Oct 20 -6′ 38′′ -22′ 50′′ 16′ 12′′ 1599 Oct 21 -5′ 39′′ -22′ 39′′ 17′ 1599 Oct 24 -4′ 24′′ -22′ 48′′ 18′ 24′′ 1599 Oct 25 -6′ 5 -23′ 17′′ 17′ 12′′ 1600 Feb 3 † ‡ +2′ 51′′ -20′ 39′′ 17′ 48′′ 1600 Feb 4 ‡ +2′ 13′′ -21′ 2′′ 18′ 49′′ 1600 Feb 5 ‡ +1′ 47′′ -21′ 13′′ 20′ 26′′ 1600 Mar 15 ‡ -3′ 46′′ -22′ 46′′ 19′ 1601 Jan 6 +20′ 15′′ +6′ 15′′ 14′ Median 9′ 12′

* In 1590, Tycho compared his observations with the Copernican predictions, firstly, as calculated by Michael Maestlin and, secondly, as listed in the Prutenic tables. Maestlin’s predictions were slightly more accurate and are the ones used in the above table. The predictions from the Prutenic tables were: March 15, +22′ 53′′; March 17, +24′ 33′′; July 11, +6′ 36′′. † In opposition. ‡ Tycho computed the difference between his observations and the predictions of Ptolemy and Copernicus.

Table 2f. Saturn Latitude Comparisons

Date Ptolemy Copernicus Net Ptolemy Net Copernicus 1568 Jan 19 -11′ -15′ 4′ 1568 Nov 1 -0˚ 15′ -0˚ 22′ 7′ 1583 Aug 6 +10′ -24′ 14′ 1583 Aug 28 +8′ 23′′ -24′ 37′′ 16′ 14′′ 1583 Sep 5 +8′ 27′′ -23′ 33′′ 15′ 6′′ 1583 Sep 6 +8 ⅓′ -23 ⅔′ 15 ⅓′ 1583 Sep 10 +8′ -24′ 16′ 1585 Sep 23 * +14′ Dead on 14′ 1585 Oct 10 +15 ⅓′ ⅓′ 15′ 1586 Sep 23 +13 +11′ 2′ 1591 Oct 25 +4′ 40′′ +53′ 10′′ 48′ 30′′ 1591 Dec 4 -6′ +59′ 30′′ 53′ 30′′ 1591 Dec 5 +6′ 30′′ +59′ 54′′ 53′ 24′′ 1591 Dec 6 +5′ +58′ 20′′ 53′ 20′′ 1591 Dec 9 * +5′ 45′′ +59′ 15′′ 53′ 30′′ 1591 Dec 10 +5′ 15′′ +58′ 45′′ 53′ 30′′ 1591 Dec 16 +2′ 45′′ +57′ 45′′ 55′ 1591 Dec 21 +4′ +59′ 55′ 1591 Dec 24 +4′ 5′′ +58′ 5′′ 54′ 1591 Dec 25 +4′ 4′′ +58′ 4′′ 54′ 1592 Dec 29 * -2′ +43′ 41′ 1594 Jan 14 * -1′ 30′′ -43′ 30′′ 42′ 1594 Dec 10 -3′ 30′′ -49′ 30′′ 46′ 1595 Jan 28 +50′′ -49′ 40′′ 48′ 50′′ 1600 Feb 4 † +5′ 40′′ -20′′ 5′ 20′′ 1600 Feb 5 † +2′ 57′′ -2′ 33′′ 24′′ 1600 Feb 6 † +1′ 54′′ -4′ 6′′ 3′ 12′′ 1600 Feb 22 † +4′ 24′′ -2′ 6′′ 2′ 18′′ 1600 Feb 23 † +3′ 16′′ -2′ 44′′ 32′′ 1600 Apr 7 † +11′ 54′′ +5′ 54′′ 6′ 1600 Apr 10 † +12′ 50′′ +5′ 50′′ 7′ 1600 Apr 11 † +13′ 55′′ +7′ 55′′ 6′ 1600 Apr 12 † +13′ 16′′ +7′ 16′′ 6′ 1600 Apr 13 † +13′ 11′′ +7′ 11′′ 6′ 1600 Apr 14 * † +12′ 45′′ +6′ 45′′ 6′ 1600 Apr 17 † +12′ 46′′ +7′ 46′′ 5′ 1600 Apr 18 † +13′ 4′′ +8′ 4′′ 5′ 1600 Apr 20 † +12′ 53′′ +7′ 53′′ 5′ 1600 Apr 27 † +12′ 56′′ +7′ 56′′ 5′ Median 46′ 7′

* In opposition. According to Tycho, Saturn was in opposition on September 26, 1585, a date for which he compared longitude, but not latitude. † Tycho computed the difference between his observations and the predictions of Ptolemy and Copernicus.

Longitude Comparison Summaries

Table 3a. Mercury Longitude Summary

Year Observations Ptolemy Copernicus Ties 1586 10 3 6 1 1589 4 1 3 1591 8 4 4 1592 1 1 1595 8 3 5 1601 1 1 Totals 32 11 20 1

Table 3b. Venus Longitude Summary

Year Observations Ptolemy Copernicus Ties 1564 3 2 1 1568 2 2 1578 1 1 1585 2 2 1589 9 9 1591 8 8 1595 1 1 1596 3 2 1 1600 1 1 Totals 30 19 11

Table 3c. Moon Longitude Summary

Year Observations Ptolemy Copernicus Ties 1586 1 1 1587 5 2 3 1591 1 1 1593 1 1 1594 10 5 5 Totals 18 8 10

Table 3d. Sun Longitude Summary

Year Observations Ptolemy Copernicus Ties 1583 33 20 9 4 1584 4 2 2 1585 6 6 1586 24 21 3 1587 4 4 1588 1 1 1601 1 1 Totals 73 54 15 4

Table 3e. Mars Longitude Summary

Year Observations Ptolemy Copernicus Ties 1564 1 1 1568 1 1 1570 1 1 1578 1 1 1580 1 1 1582 1 1 1584 1 1 1585 4 4 1586 1 1 1591 16 16 1593 15 1 13 1 1594 1 1 1595 5 5 1599 6 6 1600 4 4 Totals 59 7 51 1

Table 3f. Jupiter Longitude Summary

Year Observations Ptolemy Copernicus Ties 1564 1 1 1568 1 1 1583 6 6 1584 1 1 1586 2 2 1589 2 2 1590 3 3 1591 9 9 1592 2 2 1593 1 1 1594 1 1 1596 4 3 1 1598 1 1 1599 4 4 1600 4 4 1601 1 1 Totals 43 7 35 1

Table 3g. Saturn Longitude Summary

Year Observations Ptolemy Copernicus Ties 1564 1 1 1568 3 3 1582 1 1 1583 5 5 1585 3 3 1586 2 2 1587 1 1 1588 1 1 1591 10 10 1592 1 1 1594 2 2 1595 1 1 1600 15 15 Totals 46 16 30

Latitude Comparison Summaries

Table 4a. Mercury Latitude Summary

Year Observations Ptolemy Copernicus Ties 1586 10 8 1 1 1589 4 4 1591 8 1 7 1592 1 1 1595 8 1 7 Totals 31 11 19 1

Table 4b. Venus Latitude Summary

Year Observations Ptolemy Copernicus Ties 1564 2 2 1568 2 2 1578 1 1 1585 2 2 1589 9 9 1591 8 3 5 1595 1 1 1600 1 1 Totals 26 20 6

Table 4c. Moon Latitude Summary

Year Observations Ptolemy Copernicus Ties 1580 1 1 1586 1 1 1587 5 3 1 1 1591 1 1 1593 1 1 Totals 9 3 5 1

Table 4d. Mars Latitude Summary

Year Observations Ptolemy Copernicus Ties 1568 1 1 1570 1 1 1574 1 1 1584 1 1 1585 4 2 2 1586 1 1 1591 16 13 1 2 1593 15 8 7 1594 1 1 1595 5 5 1599 6 6 1600 4 4 Totals 57 38 17 2

Table 4e. Jupiter Latitude Summary

Year Observations Ptolemy Copernicus Ties 1583 6 6 1584 1 1 1586 2 2 1589 2 2 1590 3 1 2 1591 9 9 1592 2 2 1593 1 1 1594 1 1 1599 4 4 1600 4 4 1601 1 1 Totals 36 24 12

Table 4f. Saturn Latitude Summary

Year Observations Ptolemy Copernicus Ties 1568 2 2 1583 5 5 1585 2 2 1586 1 1 1591 10 10 1592 1 1 1594 2 2 1595 1 1 1600 15 1 14 Totals 39 22 17

Overall Comparison Summary

Legend O Number of Observations P Ptolemy superior C Copernicus superior T Tie between Ptolemy and Copernicus P% Ptolemy % C% Copernicus % T% Tie %

Table 5a. Longitude Comparisons Summary

O P C T P% C% T% Mercury 32 11 20 1 34.38 62.50 3.13 Venus 30 19 11 0 63.33 36.67 0 Moon 18 8 10 0 44.44 55.56 0 Sun 73 54 15 4 73.97 20.55 5.48 Mars 59 7 51 1 11.86 86.44 1.69 Jupiter 43 7 35 1 16.28 81.40 2.33 Saturn 46 16 30 0 34.78 65.22 0

301 observations in total

The superiority of Copernicus for the superior planets (Mars, Jupiter, Saturn) is striking. For Mars and Jupiter in particular, Copernicus was vastly superior. For the superior planets, when taken together, from a total of 148 comparisons Copernicus was superior in 116 comparisons (78%) and Ptolemy in 30 (20%).

The picture for the inferior planets is less clear-cut. Ptolemy and Copernicus essentially tied for Mercury and Venus. When the Sun is included, from a total of 135 comparisons Ptolemy was superior in 84 (62%) and Copernicus in 46 (34%).

The predictions of Copernicus were narrowly superior for the Moon. In the years 1578, 1580, 1581, and 1584 (all years before he began comparing lunar longitudes), Tycho compared his own observed times of lunar eclipses with the predictions of Copernicus and Ptolemy. In all four cases, as can be seen in the following section, the predictions of Copernicus were clearly superior.

Table 5b. Latitude Comparisons Summary

O P C T P% C% T% Mercury 31 11 19 1 35.48 61.29 3.23 Venus 26 20 6 0 76.92 23.08 0 Moon 9 3 5 1 33.33 55.55 11.11 Mars 57 38 17 2 66.66 29.82 3.50 Jupiter 36 24 12 0 66.66 33.33 0 Saturn 39 22 17 0 56.41 43.59 0

198 observations in total

The superior planets again yielded a clear winner for latitude comparisons, although it was Ptolemy, rather than Copernicus. For the superior planets combined, from a total of 132 comparisons Ptolemy was superior in 84 comparisons (63.64%) and Copernicus in 46 (34.85%). This is a smaller margin than for the superior planet longitude comparisons, but Ptolemy’s predictions were nonetheless clearly superior.

As was the case with the longitude comparisons, Copernicus was superior for Mercury and Ptolemy for Venus. However, when the moon is included, from a total of 66 comparisons, Ptolemy was superior in 34 (51.52%) and Copernicus in 30 (45.45%).

Lunar Eclipse Timing Comparisons

Tycho also compared the predictions of Ptolemy and Copernicus for the timing of lunar eclipses. The results are presented in the following tables.

Table 6a. September 15, 1578

Time Stadius * Alfonsine Beginning +10 mins +1 hr 10 mins Middle +8 mins +58 mins End +5 mins +45 mins

* Published in 1557, the ephemeris of Stadius was based on the Prutenic tables computed from De revolutionibus.

Table 6b. January 31, 1580

Time Copernicus Alfonsine Beginning -30 mins +1 hr 8 mins Middle -32 mins +1 hr 2 mins End -34 mins +56 mins

Table 6c. January 19, 1581

Time Prutenic Alfonsine Beginning -44 mins +1 hr 19 mins Middle -34 mins +1 hr 9 mins End -34 mins +59 mins

Table 6d. August 15, 1581

Time Prutenic Alfonsine Beginning +21 mins +34 mins Middle +21 mins +34 mins End +21 mins +26 mins

Table 6e. November 7, 1584

Time Alfonsine Peuerbach * Prutenic Maestlin † Start +1h 26m +58m +2m +1h 41m Begin Umbria +1h 28m +1h +2m +1h 44m Middle +1h 21m +53m +2m +1h 38m Begin End +1h 14m +46m +2m +1h 32m End +1h 17m +48m +2m +1h 35m

* A collection of tables of eclipse calculations prepared by the Austrian astronomer Georg von Peuerbach was published as Tabulae Eclipsium in 1514. † The German astronomer Michael Maestlin edited a new edition of the Prutenic tables that was published in 1571.

Table 6f. September 6, 1587

Time Maestlin Cyprianus * Beginning +3 mins +32 mins Middle +3 mins +23 mins End +3 mins +14 mins

* Cyprian von Leowitz's Tabulae Peuerbachii Alphonsiane was published in 1556.

Conclusion

In recent decades there has been debate as to whether the Copernican theory in its original (pre-Kepler) form was in fact superior in predictive power to the Ptolemaic system. Prior to the 1960s, the consensus held that Copernicus was indeed superior to Ptolemy. In The Copernican Revolution, published in 1957, Thomas Kuhn noted:

[Erasmus] Reinhold’s Prutenic Tables [based on Copernicus] … for most applications, were measurably superior to the old. They were not, of course, completely accurate; Copernicus’ mathematical system was intrinsically no more accurate than Ptolemy’s … But most comparisons displayed the superiority of Reinhold’s work … Therefore, if the decision between the Copernican and the traditional universe had concerned only astronomers, Copernicus’ proposal would almost certainly have achieved a quiet and gradual victory.

Just three years later, Kuhn’s assessment had changed significantly. In The Structure of Scientific Revolutions, he wrote:

[F]or the planets, Ptolemy’s predictions were as good as Copernicus’ … In fact, Copernicus’ theory was not more accurate than Ptolemy’s … Until Kepler, the Copernican theory scarcely improved upon the predictions of planetary position made by Ptolemy.

As we have shown, “scarcely improved upon” is not quite correct. Still, during the intervening years, the consensus has remained aligned with Kuhn’s later view. Copernican scholar Owen Gingerich, writing in 1973:

It is a curious fact that until the systematic observations of Tycho Brahe, there was relatively little way to distinguish between the accuracy of the Alfonsine [based on Ptolemy] and Prutenic Tables … [W]hy the Prutenic Tables were so widely adopted is a topic as yet unexplored.

It should be noted that Gingerich was not discussing the comparisons made by Tycho. Rather, Gingerich was essentially agreeing with the later assessment of Kuhn.

If, as Kuhn suggested in The Structure of Scientific Revolutions, Copernicus was not superior to Ptolemy in predictive power, why then, historians of science have asked, was Copernicus accepted by astronomers and Ptolemy rejected? For the past fifty years the answer given by historians has been that the Copernican theory was more elegant and more intellectually satisfying than that of Ptolemy.

In a paper for the Journal of Economic Literature published in 1990, N. Gregory Mankiw, an economics professor at Harvard University, wrote:

Compared to the then prevailing geocentric system of Ptolemy, the original Copernican system was more elegant and, ultimately, it proved more useful. But at the time it was proposed and for many years thereafter, the Copernican system did not work as well as the Ptolemaic system. For predicting the positions of the planets, the Ptolemaic system was superior.

These remarks on Copernicus and Ptolemy were intended as a parable for the study of macroeconomics, but Mankiw’s claim that Ptolemy was superior is surprising, to say the least. The standard scholarly sources at this time were Kuhn’s The Copernican Revolution and the work of Gingerich.

This explanation poses a problem. If scientists can choose one theory over another on purely aesthetical or philsophical grounds, the implication is that in this case, at least, astronomers were not motivated by observation. If so, why shouldn’t politics or religion also be allowed to trump observation? The battle between the due massimi sistemi del mondo (two chief world systems), as Galileo Galilei memorably described them, was the most important in scientific history. If observation did not play an essential role in this battle, then what role remains for experiment in any future campaign? It should be noted that three of the world’s leading cosmologists, Paul Steinhardt, George Ellis, and Joseph Silk, expressed their alarm in 2014 at the claim that theories such as string theory and inflation theory, which have no experimental support, should nevertheless be accepted as true.

During the twentieth century, historians of science began downplaying the idea that experiment was the foundation of science. In a book first published in 1939, the French historian Alexandre Koyré claimed that Galileo didn’t conduct any of the experiments reported in his books. Instead, according to Koyré, Galileo’s deductions relied on aesthetic considerations. In a famous 1973 paper, the Galilean scholar Stillman Drake bemoaned the acceptance of such notions:

More than a decade has elapsed since Thomas Settle published a classic paper in which Galileo’s well-known statements about his experiments on inclined planes were completely vindicated. Settle’s paper replied to an earlier attempt by Alexandre Koyré to show that Galileo could not have obtained the results he claimed in his Two New Sciences by actual observations using the equipment there described. … Koyré’s paper was reprinted years later in book form without so much as a note by the editors concerning Settle’s refutation of its thesis. And the general literature continues to belittle the role of experiment in Galileo’s physics.

Claims that seek to diminish or deny the predictive superiority of Copernicus also tend to disparage the role of experiment in scientific development.

As we have seen, however, sixteenth-century astronomers had a good and purely observational reason to adopt the Copernican theory. Reasons other than observation need not have played any role whatsoever.

The analysis offered here has demonstrated that Tycho compared the predictions of planetary positions made by Copernicus and Ptolemy with his own observations and indeed found, on balance, Copernicus to be superior. In fact, Tycho’s own data refute the claim, made by many historians of science, that overall the Copernican theory was not superior to the Ptolemaic. Tycho never accepted the idea that the earth moved around the sun. His own compromise was to place the earth at the center of the universe with the sun and moon orbiting it, as Ptolemy had, and at the same time, to place the other planets in orbit around the sun, following Copernicus. For planetary motion, Tycho evidently favored Copernicus. Surely his own comparisons of the predictive power of Copernicus versus Ptolemy factored into this conclusion.