1434 (14 page)

Distance from one day to the next was calculated by the difference in the ship's latitude. There is a simple formula: Latitude equals 90—sun's max altitude ± declination. Declination tables of the sun are issued for each day of the year, so with the sun's altitude, the navigator can determine latitude. It's that simple.

However, this was not what Regiomontanus, Toscanelli, and Alberti were after. A few miles' difference (between 23°28' and 23°30') was in itself completely unimportant to Toscanelli. Instead, he, Alberti, and Regiomontanus were interested in the change in the sun's declination. A copy of that change can be seen in Needham's graph, by kind permission of Cambridge University Press. It shows the change in the sun's declination from 2000
B.C.
to the present day, determined by Greek and Chinese astronomers for the earlier measurements and by European astronomers for the later ones, ending with Cassini.

From this graph, we can see that Toscanelli's figure—23°30'—was recorded by the great Islamic astronomer Ulugh Begh also used 23°30' in his massive study completed in Samarkand in 1421—some fifty years before Toscanelli's measurement. (Regiomontanus's figure of 23°28' was determined by Cassini two hundred years after Toscanelli, so it would have been inaccurate had Regiomontanus used it.)

This is not some mathematical quibble. If the sun circled the earth, there would be no change in declination. A recognition of the change—the flatter the earth's trajectory, the smaller the declination—is tantamount to recognition that the earth revolves around the sun in an ellipse.

Their obsession with measuring the change in declination is evidence that Toscanelli, Alberti, and Regiomontanus understood that Aristotle and Ptolomy, who believed the sun revolved in a circle around the earth, were wrong. Consequently, Europeans who followed Toscanelli and Regiomontanus were basing their astronomy on a Chinese, rather than a Greek, foundation. This foundation also enabled Regiomontanus to produce tables to determine latitude in different parts of the world, which he published in 1474. Columbus and Vespucci used them, as described in chapter 21.

The exercises at Santa Maria del Fiore could be duplicated to observe the movement of the moon and produce equations of time of the moon. These, in turn, could be used in combination with the positions of stars to determine longitude (see chapter 4). Regiomontanus produced such tables, and Columbus and Vespucci used them to calculate longitude in the New World. Dias used them to determine the latitude of the Cape of Good Hope.

Each of the instruments Toscanelli used in his observations at Santa Maria del Fiore—camera obscura, gnomon, and clock—was used by Zheng He's navigators, as were the instruments Toscanelli used to determine the passage of the 1456 comet—Jacob's staff, clock, and torquetum. All of Toscanelli's discoveries—declination of the sun, obliquity of the ecliptic, passage of comets, ephemeris tables of the stars and planets—were contained in the 1408
Shoushi
astronomical calendar presented to the pope. They were copied and published in Europe by Regiomontanus in 1474.

In his letter to Columbus, Toscanelli said he had received “the most copious and good and true information from distinguished men of great learning who have come here in the Court of Rome [Florence] from the said parts [China].” In his letter to Canon Martins, Toscanelli described his long conversation with the ambassador from China who had visited the pope, and he cited the “many scholars, philosophers, astronomers and other men skilled in the natural sciences” who then governed China.

In my submission, Toscanelli must have obtained his copious new knowledge of astronomy from the “distinguished men of great learning” who had arrived in Florence from China.

Res ipsa loquitur!
“The thing speaks for itself.”

B
efore 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 observatory
¹²
and printing press
¹³
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.