1434 (15 page)

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:

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 tables
²²
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.
²⁸
Columbus 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: