13

THE FLORENTINE MATHEMATICIANS: TOSCANELLI, ALBERTI, NICHOLAS OF CUSA, AND REGIOMONTANUS

Before Toscanelli met the Chinese ambassador, Europe’s knowledge of the universe was based on Ptolomy.1 Ptolomy held that the planets were borne in revolving crystalline spheres that rotated in perfect circles around the earth, which was at the center of the universe. However, many European astronomers realized this did not square with their observations that planets have irregular paths. To resolve this conflict, medieval European astronomers introduced the notions of equants, deferants, and epicycles. Applying these peculiar explanations of planetary motion enabled astronomers to account for the irregular motion of the planets while holding fast to the belief that the heavens rotated around the earth.

To believe, on the other hand, that the earth was merely one planet among many revolving round the sun required a radical change in thought. This intellectual revolution was led by Nicholas of Cusa.2 Nicholas was born in 1401 on the River Moselle. He died in Umbria in 1464. His father, Johann Cryfts, was a boatman. In 1416 Nicholas matriculated at the University of Heidelberg, and a year later he left for Padua, where he graduated in 1424 with a doctorate in canon law. He also studied Latin, Greek, Hebrew, and, in his later years, Arabic.

While at Padua, Nicholas became a close friend of Toscanelli, who was also a student there. Throughout his life, he remained a devoted follower of Toscanelli, with whom he frequently collaborated on new ideas. At the height of his fame, Nicholas dedicated his treatise De Geometricis transmutationibus to Toscanelli and wrote in the flyleaf, “Ad pavlum magistri dominici physicum Florentinum” (To the Master Scientist, the Florentine Doctor Paolo).3

Nicholas had a huge and independent intellect. He published a dozen mathematical and scientific treatises; his collected works were contained in the Incunabula, published before 1476 and sadly, now lost. In his later life he believed that the earth was not the center of the universe and was not at rest. Celestial bodies were not strictly spherical, nor were their orbits circular. To Nicholas, the difference between theory and appearance was explained by relative motion. Nicholas was prime minister in Rome with great influence.

By 1444, Nicholas possessed one of the two known torquetums based upon the Chinese equatorial system.4 In effect, this was an analog computer. By measuring the angular distance between the moon and a selected star that crossed the local meridian, and by knowing the equation of time of the moon and the declination and right ascension of the selected star, one could calculate longitude.

During Nicholas’s era, the Alfonsine tables based on Ptolomy were the standard work on the positions of the sun, moon, and planets. Nicholas realized these tables were highly inaccurate, a finding he published in 1436 in his Reparatio calendarii.5 This realization led him to his revolutionary theory that the earth was not at the center of the universe, was not at rest, and had unfixed poles. His work had a huge influence on Regiomontanus—not least in saying, “the earth which cannot be at the centre, cannot lack all motion.”

Regiomontanus

Johann Müller was born in 1436 in Königsberg, which means “king’s mountain”—Johann adopted the Latin version of the name, Regiomontanus.6 The son of a miller, he was recognized as a mathematical and astronomical genius when young. He entered the University of Leipzig at age eleven, studying there from 1447 until 1450. In April 1450 he entered the University of Vienna, where he became a pupil of the celebrated astronomer and mathematician Peurbach.7 He was awarded his master’s degree in 1457. Peurbach and Regiomontanus collaborated to make detailed observations of Mars, which showed that the Alfonsine tables (based upon the earth being at the center of the universe) were seriously in error. This was confirmed when the two observed an eclipse of the moon that was later than the tables predicted. From that time, Regiomontanus realized as Nicholas of Cusa had done that the old Ptolemaic systems of predicting the courses of the moon and planets did not stand up to serious investigation. From his early life, again like Nicholas of Cusa, he started collecting instruments such as a torquetum for his observations. Although Regiomontanus was some forty years younger than Toscanelli, Nicholas of Cusa, and Alberti, he became part of their group in the late 1450s and early 1460s, when they used to meet at Nicholas’s house in Rome. There are numerous references in Regiomontanus’s writing to the influence Toscanelli and Nicholas of Cusa had on his work.8 Some of these will be quoted as we go along.

In 1457, at age twenty-one, Regiomontanus was appointed to the arts facility of the University of Vienna. The following year he gave a talk on perspective. He was now working on math, astronomy, and constructing instruments. Between 1461 and 1465 he was mostly in Rome; the following two years he seems to have disappeared—nobody knows where he went. In 1467 he published part of his work on sine tables and spherical trigonometry, and in 1471 he had constructed instruments and written scripta. In 1472 he published A New Theory of Planets (by Peurbach), and then in 1474 his own Calendarum and Ephemerides ab Anno tables.9 These two were his legacy—of monumental importance in enabling European mariners to determine latitude and longitude and their position at sea. He died in Rome on July 6, 1476, and a number of his works were published after his death.

Regiomontanus’s output after his master Peurbach died in 1461 (when Regiomontanus was twenty-five) up until his own death in 1476, at forty, was prodigious and mind-blowing. He was an intellectual giant, the equivalent of Newton or Guo Shoujing. Had he lived another thirty years, I believe he would have rivaled or eclipsed Newton. I have the greatest trepidation in attempting to do him justice, and have spent many sleepless nights trying to write this chapter—not least because I am not a mathematician.

We can reasonably start with his achievements, then go on to consider the possible sources he used and finally attempt to summarize his legacy. Doubtless critics will make the point that it is arrogant of me to even attempt to evaluate the achievements of such a brilliant figure—that such a task should be left to professional mathematicians. This is a fair point. In defense, I offer that I have spent years in practical astronavigation, using the moon, planets, and stars to find our position at sea, and should be qualified to recognize the huge strides Regiomontanus made in this science.

So here goes. In the course of fifteen years following Peurbach’s death, Regiomontanus provided first and foremost ephemeris tables—that is, tables of the positions of moon, sun, planets, and stars that were of sufficient accuracy to enable captains and navigators to predict when eclipses would occur, times of sunrise, sunset, moonrise, and moonset, the positions of planets relative to one another and to the moon. So accurate were these tables—for thirty years from 1475—that navigators could calculate their latitude and longitude at sea without using clocks. They could, therefore, for the first time, find their way to the New World, accurately chart what they had found, and return home in safety. With this and the Chinese world maps, European exploration could now start in earnest. And it did. Dias, for example, calculated the true latitude of the Cape of Good Hope using Regiomontanus’s tables.10 He reported this to the king of Portugal, who knew for the first time how far the captains had to travel south to get to the Indian Ocean. Regiomontanus’s ephemeris tables were 800 pages long and contained 300,000 calculations. Regiomontanus could be said to have been a walking computer on that account alone.

He had the energy and skill to devise and make a whole range of nautical and mathematical instruments, the two most fundamentally important being the clock (which was smashed on his death) and the equatorial torquetum.11 Regiomontanus’s torquetum has been described in chapter 4—it enabled him to transfer stars whose coordinates had been fixed by the Arab ecliptic method or by the Byzantine and Greek horizon method into Chinese coordinates of declination and right ascension, the system used down to our present day.

Of Regiomontanus’s designs, his observatory12 and printing press13 stand out for their practical use. Ephemeris tables could not have been produced to give accurate results had they not been printed. Similarly, Regiomontanus needed his observatory to check on the accuracy of the predictions in his tables. He made telescopes to see the stars; astrolabes to measure angles between stars, planets, and moon; portable sundials for gathering information on the sun’s height at different times of day and for different times of the year—even tables to enable bell ringers to forecast times of sunset and hence announce vespers.

The most astonishing discovery was Regiomontanus’s revolutionary idea (enlarging on Nicholas of Cusa’s) that the earth was not at the center of the universe, the sun was. And further, that the earth and planets circled the sun. This statement will perhaps create an uproar; so I present here my evidence.

First of all, Regiomontanus knew that the planetary system that had been in use in Europe since the time of Ptolomy—in which the earth was in the center and sun and planets rotated around it—did not work. The results of the Ptolemaic system were contained in the Alfonsine tables, which he and Peurbach had studied for years. The predictions contained in these tables were inaccurate. Adding equants, deferents, and other weird corrections failed to correct the errors.

Second, there is no doubt that Regiomontanus knew of Nicholas of Cusa’s work. Nicholas suggested that the sun was at the center of the universe and the earth and planets rotated around it. Regiomontanus describes planetary orbits: “What will you say about the longitudinal motion of Venus? It is chained to the Sun which is not the case for the three superior planets (Mars, Jupiter, Saturn). Therefore it has a longitudinal motion different from those three planets. Furthermore, the superior planets are tied to the Sun via epicyclic motions, which is not true for Venus.”14

Regiomontanus’s opinion that the sun is at the center of the universe is clearly expressed in folio 47v: “Because the Sun is the source of heat and light, it must be at the centre of the planets, like the King in his Kingdom, like the heart in the body.”15

Regiomontanus also had views on the orbital velocity of planets around the sun: “Moreover the assumption that Venus and Mercury would move more rapidly if they were below the Sun is untenable. On the contrary, at times they move faster in their orbits, at times slower.” This foreshadows Kepler.

Regiomontanus realized that the stars were at an almost infinite distance from the solar system: “Nature may well have assigned some unknown motion to the stars; it is now and will henceforth be very difficult to determine the amount of this motion due to its small size.”

He later refined this: “It is necessary to alter the motion of the stars a little because of the Earth’s motion” (Zinner).

The only possible motion of the earth relative to the stars is that around the sun, it cannot by definition refer to the circular motion of the earth around its own axis. This in my view is corroborated by Regiomontanus’s written comment alongside Archimedes’ account of Aristarchus’ assumption that the earth circles around an immobile sun, which is at the center of a fixed stellar sphere. Regiomontanus wrote:

“Aristarchus Samius” (Heroic Aristarchus)16

Unfortunately, Regiomontanus’s works after the date of this comment are missing.

It seems to me that Regiomontanus’s near obsession with measuring the change in the declination of the sun can only be understood if he had appreciated that the earth traveled in an ellipse around the sun and that the shape of this ellipse was changing with time. He wrote: “It will be beautiful to preserve the variations in planetary motions by means of concentric circles. We have already made a way for the sun and the moon; for the rest the cornerstone has been laid, from which one can obtain the equations for these planets by this table.”17

Before discussing Regiomontanus’s masterpiece, his ephemeris tables, we should attempt to address the $100,000 question—from where did he get his knowledge? Undoubtedly Regiomontanus studied Greek and Roman works extensively—Ptolomy for years and years, and he copied out Archimedes’ and Eutocius’ work on cylinders, measurements of the circle, on spheres and spheroids. Regiomontanus could read and write Greek and Latin fluently. He could also read Arabic. He had mastered a wide range of Arabic work, not least of which was al-Bitruji’s planetary theory. However, Regiomontanus adopted the Chinese equatorial system of planet and star coordinates; he rejected the Arabic, Greek, and Byzantine coordinate systems. He borrowed heavily from Toscanelli, including his and Alberti’s calculations of the earth’s changing ellipse around the sun, and he adopted Toscanelli and the Chinese measurement of the declination of the sun. His work on spherical triangles had been foreshadowed by Guo Shoujing’s. If Uzielli is correct, Regiomontanus collaborated with Toscanelli on drawing the map of the world that was sent to the king of Portugal—a map copied from the Chinese, something Regiomontanus must have known.

Regiomontanus repeatedly refers to Toscanelli’s work—on spherical trigonometry, declination tables, instruments, and comets. When doing so, he must have known of Toscanelli’s meetings with the Chinese—and of the enormous transfer of knowledge from them.

Regiomontanus also had intimate knowledge of Chinese mathematical work, which he acquired directly or through Toscanelli. Among that knowledge was the Chinese remainder theorem.

Regiomontanus’s Knowledge of Chinese Mathematics

Regiomontanus corresponded on a regular basis with Italian astronomer Francesco Bianchini.18 In 1463 he set Bianchini this problem: “I ask for a number that when divided by 17 leaves a remainder of 15; the same number when divided by 13 leaves a remainder of 11; the same number divided by 10 leaves the remainder of 3. I ask you what is that number” (GM translation of Latin).

Bianchini replied: “To this problem many solutions can be given with different numbers—such as 1,103, 3313 and many others. However I do not want to be put to the trouble of finding the other numbers.”

Regiomontanus answered: “You have rightly given the smallest number I asked for as 1,103 and the second 3,313. This is enough because such numbers of which the smallest is 1,103 are infinite. If we should add a number made up by multiplying the three divisions, namely, 17, 13, 10, we should arrive at the second number, 3,313, by adding this number again [viz 2210] we should get the third [which would be 5,523].”

Regiomontanus then drew in the margin:

image

It is obvious from Bianchini’s reply that he did not understand the Chinese remainder theorem (if he had, he would have realized how easy the solution was and not said, “I do not want to be put to the trouble of finding the other numbers.”

On the other hand, it is obvious that Regiomontanus had the complete solution to the problem—as the mathematician Curtze summarizes:19

“[Regiomontanus] knew thoroughly the remainder problem, the ta yen rule of the Chinese.”

The Ta-Yen rule is contained in the Shu-shu Chiu-chang of Ch’in Chiu-shao, published in 1247.20

It follows that Regiomontanus must have been aware of this Chinese book of 1247 unless he had quite independently thought up the Ta-Yen rule, which he never claimed to have done.

Regiomontanus’s knowledge of the Shu-shu Chiu-chang would explain a lot. Needham tells us that the first section of this book is concerned with indeterminate analyses such as the Ta-Yen rule.21 In the later stages of the book comes an explanation of how to calculate complex areas and volumes such as the diameter and circumference of a circular walled city, problems of allocation of irrigation water, and the flow rate of dykes. The book contains methods of resolving the depth of rain in various types and shapes of rain gauge—all problems relevant to cartographic surveying, in which we know Regiomontanus took a deep interest.

The implications of Regiomontanus knowing of this massive book, which was the fruit of the work of thirty Chinese schools of mathematics, could be of great importance. It is a subject beyond the capacity of a person of my age. I hope young mathematicians will take up the challenge. It may lead to a major revision of Ernst Zinner’s majestic work on Regiomontanus.

It seems to me we may obtain a snapshot of a part of what Regiomontanus inherited from the Chinese through Toscanelli (rather than through Greek and Arab astronomers) by comparing Zheng He’s ephemeris tables22 with Regiomontanus’s ephemeris tables.23

Regiomontanus’s tables are double pages for each month with a horizontal line for each day. Zheng He’s have one double page for each month with a vertical line for each day. On the left-hand side of each of Regiomontanus’s pages are the true positions of the sun, moon, and the planets Saturn, Jupiter, Mars, Venus, and Mercury, and the lunar nodes where the moon crosses the ecliptic. On the right-hand side are positions of the sun relative to the moon, times of full and new moon, positions of the moon relative to the planets, and positions of planets relative to one another. Feast days are given, as are other important days in the medieval European calendar.

Zheng He’s 1408 tables have an average of twenty-eight columns of information for each day (as opposed to Regiomontanus’s eight columns). Zheng He’s tables have the same planetary information as Regiomontanus’s—for Saturn, Jupiter, Mars, Venus, and Mercury, and also, like Regiomontanus’s, positions of the sun and moon. The difference between the two is that Zheng He’s gave auspicious days for planting seed, visiting Grandmother, and so on, rather than religious feast days. Zheng He’s have double the amount of information. The astonishing similarity between the two could be a coincidence—but the 1408 tables came first, printed before Gutenberg.

Zinner and others claim that Regiomontanus’s tables with 300,000 numbers over a thirty-one-year period were the result of using the Alfonsine (Greek/Arabic) tables amended by observation. If Regiomontanus’s tables were based on the Alfonsine tables, they would have been useless for predicting positions of sun, moon, and planets with sufficient accuracy to predict eclipses and hence longitude, as the Alfonsine tables were based on a wholly faulty structure of the universe, with the earth as its center and planets revolving round it.

Furthermore, Regiomontanus well knew that using the old Alfonsine tables would be useless. In his calendar for 1475–1531 he pointed out that in thirty of the fifty-six years between 1475 and 1531, the date of Easter (the most important day in the Catholic Church) was wrong in the Alfonsine tables. (Because of the sensitivity of this information it was omitted from the German edition of Regiomontanus’s calendar.) To base his ephemeris on tables he knew to be inaccurate would have been completely illogical. Regiomontanus had to use a new source.

Zheng He’s ephemeris tables, on the other hand, were based on Guo Shoujing—which relied on a true understanding of the earth’s and planets’ rotation around the sun as the center of the solar system. In my submission, Zinner’s claim that Regiomontanus’s tables were based upon his personal observations also breaks down because he did not have time to make the necessary observations. Regiomontanus died in 1475. His tables continued for another fifty-six years; one can see his amendments in red in the tables, and these cover only five of the fifty-six years.

I hope the accuracy of Zheng He’s and Regiomontanus’s ephemeris tables will be subjected to a test by the “Starry Night” computer program and compared with the Almagest ephemeris calculator (based on the Alfonsine tables), but this may not occur until the tables are translated and before this book goes to press. In the meantime we need a check into the accuracy of Regiomontanus’s tables in calculating eclipses, planetary positions, and longitude. If based upon Zheng He’s, they would work; if upon the Alfonsine tables, they would not.

Fortunately, Columbus, Vespucci, and others did use Regiomontanus’s ephemeris tables to predict eclipses, latitude, and longitude for years after Regiomontanus died.

Dias used the tables correctly to carefully calculate the latitude of the Cape of Good Hope at 34°22' on his voyage of 1487.24 Christopher Columbus and his brother Bartholomew were present when Dias returned and presented his calculations to the king of Portugal.25

Columbus used Regiomontanus’s ephemeris tables, as we know from tables that today are in Seville Cathedral with Columbus’s writing on them.26 Columbus referred to the ephemeris entry for January 17, 1493, when Jupiter would be in opposition to the sun and moon; he knew of Regiomontanus’s explanation of how to calculate longitude from a lunar eclipse. His brother Bartholomew wrote: “Almanach pasadoen ephemeredes. Jo de monte Regio [Regiomontanus] ab anno 1482 usque ad 1506.”27

Columbus’s first known calculation of longitude using Regiomontanus’s method of observing lunar eclipses (whose times Columbus obtained from the ephemeris tables) was on September 14, 1494, twenty years after Regiomontanus had entered the figures in the tables.28Columbus was on the island of Saya, to the west of Puerto Rico. (“Saya” on Pizzigano’s 1424 chart.) Regiomontanus explains how to calculate longitude by lunar eclipses at the front of the tables.

Using this explanation, through no fault of his own, Columbus used the wrong prime meridian (Cadiz) in his calculations rather than Nuremberg, which was Regiomontanus’s prime meridian. In his introduction to the ephemeris tables Regiomontanus does not mention this—one has to go to near the back of eight hundred pages to find this out. Columbus had another go on February 29, 1504, using the tables to predict a solar eclipse in Jamaica and to calculate longitude.29 He made the same understandable mistake again. Schroeter’s tables enable us to know the accuracy of Regiomontanus’s tables when predicting these eclipses on September 14, 1494, and February 29, 1504—delays of thirty minutes and eleven minutes respectively, and that twenty and thirty years after Regiomontanus had entered the figures—fantastic accuracy, which in my view demolishes the case that Regiomontanus’s ephemerides can have been based upon the Alfonsine tables, which got the date of Easter wrong thirty times between 1475 and 1531. Regiomontanus must have gotten his information from Toscanelli.

Vespucci used Regiomontanus’s ephemeris tables to calculate longitude on August 23, 1499, when the tables stated the moon would cross Mars between midnight and 1 A.M. Vespucci observed that at “1 1/2 hours after sundown the moon was slightly over one degree east of Mars and by midnight had moved to 5 1/2 degrees from Mars rather than in line with Mars at midnight at Nuremberg.”30 He incorrectly calculated the lunar motion compared to Mars and also used the wrong meridian—again Regiomontanus had not made this clear. In doing so he placed the wrong longitude for where he was (the River Amazon). Using the correct figures, in my view, demolishes the argument that Regiomontanus’s tables were based upon the Alfonsine tables. Likewise Columbus’s longitude errors almost disappear if he had used the correct zero point.

From the publication of Regiomontanus’s ephemeris tables in 1474, Europeans could for the first time calculate latitude and longitude, know their position at sea, get to the New World, accurately chart it, and return home in safety—a revolution in exploration.

Regiomontanus’s tables were improved upon by Nevil Maskelyne. These were published in 1767 and remained in use by Royal Navy captains and navigators well after Harrison’s chronometer was introduced.31

The great Captain Cook observed and calculated more than six hundred lunar distances to obtain the longitude of Strip Cove in New Zealand, and in 1777 he made one thousand lunar observations to determine the longitude of Tonga.32 Maskelyne’s tables were absorbed into the Nautical Almanac in which lunar-distance tables were incorporated until being phased out in 1907. (They were still in the library at Dartmouth when I learned navigation there in 1954.) With accurate instruments, the tables produced astonishingly good results. William Lambert reports (observations January 21, 1793) that without using clocks the longitude of the Capitol in Washington, D.C., was 76°46' by using the moon and Aldebaran; 76°54' on October 20, 1804, by using the Pleiades and the moon; 77°01' on September 17, 1811, by using an eclipse of the sun; 76°57' on January 12, 1813 by using Taurus and the moon.33 The true figure is 77°00' W.34 Hence five different methods, which could have been employed using Regiomontanus’s ephemeris tables by different people, gave a maximum error of 14'—around eight nautical miles without using clocks or chronometers. Harrison’s chronometer was useful but not essential in mapping the world.

Maps

Once Regiomontanus was able to calculate latitude and longitude, he could construct maps. He produced the first European map with accurate latitudes and longitudes in 1450. Its accuracy rivaled the Chinese map of 1137 which showed China mapped accurately with latitude and longitude and is held in the British Museum (Needham).

Regiomontanus was fully aware that he was remaking European astronomy. Zinner cites his drive to banish the errors of Ptolomy and centuries of misunderstanding:

He had in mind, as his life’s goal, the improvement of the planetary theory and planetary tables; he knew of their defects only too well. He wanted to have the best and most error-free editions of ancient manuscripts at the disposal of his contemporaries, so he intended to compose almanacs which represented celestial events in an errorless manner and which would be important aids for predictions and determination of positions…. He spoke of the sun as the king among the planets. He connected the three outer planets with the sun by means of epicyclic motion, whereas Venus was linked to the sun in other ways. Hence the special position of the sun was clear to him, in those days.

In addition, there came the realisation that the planetary tables were unsatisfactory. Later on, in his letters to Bianchini in 1463–64, he was quite clear about the fact that many of Ptolomy’s assumptions could not be correct, not only about the obliquity of the ecliptic but also about the paths of the planets themselves. If the planets really did move along epicycles, then their apparent diameters would have to change in a way that is completely contrary to observations.35

Just as the Aristotelian/Ptolomeic paradigm of the universe was shelved after 1434, so were Arabic methods of astronomy and astronavigation. The Arab system, with its azimuth star coordinate system and reliance on the ecliptic, had been brought to Beijing by Jamal ad-Din in 1269. It lasted only nine years. After Guo Shoujing was commissioned to produce the Shoushi calendar in 1276, he jettisoned the Arab ecliptic coordinates and built the simplified equatorial torquetum later used by Nicholas of Cusa and Regiomontanus.36

After the torquetum was introduced to Europe, astrolabes, on which Arabic and European astronomers had lavished all their mathematical art, passed out of favor. Guo Shoujing’s torquetum—forerunner of modern European instruments such as the astrocompass—lived on.

From there on, European astronomers followed Chinese methods.

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