Regiomontanus

Chapter 1 · Vienna, at Fourteen

Peuerbach, and an unfinished commission

1436 – 1461 · Königsberg in Franconia · Leipzig · Vienna

Johannes Müller was born in 1436 at Königsberg in Franconia, a small town whose Latin name — royal mountain — gave him the name history uses. His father was a miller.

He matriculated at Leipzig at eleven and moved to Vienna at fourteen, which was then the best place in Europe to study astronomy. There he met Georg von Peuerbach, ten years older, and the relationship shaped everything after it. They observed together, computed together, and Regiomontanus took his master's degree at twenty-one and began teaching.

In 1460 Cardinal Bessarion arrived in Vienna on a diplomatic mission. Bessarion was a Greek who had come west at the fall of Constantinople, and his project was the rescue of Greek learning: he wanted a better Latin version of Ptolemy's Almagest than the wretched translations then circulating, which had come through Arabic and been mangled twice.

Peuerbach agreed to produce an abridgement. He did not read Greek and had to work from the Latin, which he complained about constantly. In 1461, with the work part-finished, he died, and on his deathbed asked Regiomontanus to complete it.

Regiomontanus went to Italy with Bessarion, learned Greek, and finished it — from the Greek, which changed what the book could be.

Why this matters

The transmission of Greek science westward depended on individual scholars learning the language and correcting centuries of copying.

A deathbed commission and a language to learn first. What would you ask him?

Ask Regiomontanus

• “What was Vienna like for astronomy in the 1450s?”
• “What did Peuerbach teach you?”
• “Why did Bessarion want a new version of Ptolemy?”
• “How much difference did reading the Greek make?”
• “What was it like to be handed an unfinished book by a dying man?”

Chapter 2 · The Epitome

The book Copernicus learned from

1461 – 1463 · Rome · Venice · Padua

The Epitome of the Almagest is an abridgement, and calling it that undersells it considerably.

Ptolemy's Almagest is thirteen books, dense, and by the fifteenth century wrapped in layers of commentary and mistranslation. What Peuerbach began and Regiomontanus finished is shorter, clearer, and organised for use: the propositions in order, the reasoning tightened, the computations made checkable.

It also does something an abridgement is not supposed to do. It criticises. Regiomontanus points out where Ptolemy's models disagree with observation — most famously, that Ptolemy's theory of the moon requires the moon's apparent size to vary by nearly a factor of two over a month, which anybody can see it does not. That criticism is in the Epitome, and Copernicus read it.

How far that particular observation started Copernicus thinking is argued about, but the connection is direct: the Epitome was printed in 1496 and Copernicus, then a student at Kraków and Bologna, owned and annotated a copy. It was the standard textbook of advanced astronomy for a century.

Regiomontanus spent these years in Italy with Bessarion, hunting manuscripts, learning Greek, and observing. He was, by his mid-twenties, one of the few people in Europe who could read Ptolemy in the original and compute with him fluently.

Why this matters

A summary that criticises its source can do more than the original: Copernicus learned Ptolemy from a book that told him where Ptolemy failed.

An abridgement that argued with the book it abridged. What would you ask him?

Ask Regiomontanus

• “What did you change in Ptolemy, and why?”
• “Why does the moon's size cause a problem for Ptolemy?”
• “Should an abridgement criticise its original?”
• “What was hunting Greek manuscripts in Italy actually like?”
• “Did you expect anybody to build on the criticism?”

Chapter 3 · Five Books on Triangles

1464, and a subject cut loose from the sky

1464 · Venice · Hungary

In 1464 Regiomontanus finished De triangulis omnimodis libri quinque — five books on triangles of every kind — and the decision behind it matters more than any single result in it.

Until then, what we call trigonometry existed only as the computational apparatus of astronomy. Chords, and later sines, were tabulated because astronomers needed them to find where planets were. Nobody treated triangles as a subject.

Regiomontanus did. The five books set out, systematically and in order, what can be determined about a triangle from what is given: the plane cases first, then the spherical. It contains the rule that in any plane triangle, each side stands to the sine of the angle opposite it in one and the same ratio — the sine rule, which the whole of the ambiguous-case difficulty later hangs on. It contains spherical results equivalent to the cosine rule.

It is written in words, as everything was: he has no algebraic notation, no decimals, no negative quantities. A proposition is stated, a case is worked, and a table is cited.

And then it sat in manuscript. It was not printed until 1533, fifty-seven years after his death — a delay that says something about how precarious the transmission of a book still was. Once printed it became the foundation of European trigonometry, read by Copernicus, by Rheticus, and by everybody who came after.

Why this matters

Deciding that triangles deserve a book of their own is why trigonometry is taught as a subject rather than as an astronomical technique.

Five books about triangles, and none of them about stars. What would you ask him?

Ask Regiomontanus

• “Why write about triangles rather than about the heavens?”
• “What is the rule relating a side to its opposite angle?”
• “How do you write mathematics with no algebra and no decimals?”
• “Does knowing two sides and an angle always fix a triangle?”
• “How does it feel that the book waited fifty-seven years to be printed?”

Chapter 4 · The Press at Nuremberg

Because a copied table decays

1471 – 1475 · Nuremberg

In 1471 Regiomontanus settled at Nuremberg, and he explained the choice: it lay conveniently at the centre of Europe, its instrument makers were the best available, and its merchants' correspondence would let him exchange letters with anybody.

With the backing of a wealthy citizen, Bernhard Walther, he built an observatory and a workshop. And he did something no scholar had done: he set up his own printing press.

The reason is specific and it is the most modern thing about him. A text copied by hand acquires errors. In a work of prose the errors are usually visible, because the sentence stops making sense. In a table of numbers they are invisible: a digit changes and the page looks exactly as authoritative as before. Astronomical tables were copied and recopied, and by the fifteenth century nobody could say which figures in a given manuscript were the author's.

Printing breaks the chain. Set the type once, correct it once, and every copy is the same copy. Regiomontanus understood this as a scientific instrument rather than a commercial one, and he issued a prospectus listing some forty-five works he intended to print — Greek and Latin classics of astronomy and mathematics, his own tables, and Peuerbach's.

He printed his own calendar and a set of ephemerides giving planetary positions for the years to 1506. Navigators carried them; Columbus had a copy and, in a famous incident in Jamaica in 1504, used its prediction of a lunar eclipse to frighten the local people into supplying his crew.

He completed only a handful of the forty-five before he left for Rome.

Why this matters

The idea that a scientific table must be printed rather than copied — because copying silently corrupts numbers — is the beginning of reliable published data.

A scholar who became a printer to stop his numbers rotting. What would you ask him?

Ask Regiomontanus

• “Why set up a printing press yourself?”
• “What goes wrong with a table copied by hand?”
• “What were you planning to print in the forty-five?”
• “Who actually used your ephemerides?”
• “Why Nuremberg, of all places?”

Chapter 5 · Summoned to Rome

The calendar, and forty years

1475 – 1476 · Nuremberg · Rome

The Julian calendar was wrong, and everybody knew it. Its year is about eleven minutes longer than the real one, and by the fifteenth century the accumulated error had shifted the equinox — and with it the date of Easter — by around ten days. Fixing it required both astronomy and authority.

In 1475 Pope Sixtus IV summoned Regiomontanus to Rome to advise on the reform. It was recognition of the highest kind: he was the leading computational astronomer in Europe, and the problem was exactly his.

He went. He died there in July 1476, aged forty. The cause is usually given as plague, which was in Rome that summer. Some later accounts allege poisoning by the sons of George of Trebizond, whose translation of Ptolemy Regiomontanus had attacked in print with real savagery; there is no evidence for it and it makes a better story than a fact.

The calendar was not reformed for another hundred and six years, until Gregory XIII did it in 1582.

What he left was scattered: the observatory at Nuremberg passed to Walther, who went on observing for another three decades; the press stopped; De triangulis waited until 1533; the manuscripts were dispersed and some are still lost. And yet the Epitome went into Copernicus's hands, De triangulis founded a subject, and the ephemerides went to sea.

He was forty, and he had reorganised trigonometry, corrected Ptolemy for the man who would replace him, and worked out that scientific data has to be printed.

Why this matters

A short life, mostly of unfinished projects, that nevertheless reset two subjects — and a reminder how much depends on who lives long enough.

Summoned by a pope, dead within a year. What would you ask him?

Ask Regiomontanus

• “What was actually wrong with the calendar?”
• “Why did the reform need a pope as well as an astronomer?”
• “Were you poisoned, or was it plague?”
• “What would you have printed if you had lived?”
• “Which of your works did you expect to matter most?”

What Regiomontanus changed

De triangulis omnimodis, finished in 1464, made trigonometry a subject in its own right rather than an appendix to astronomy — which is why the sine rule is taught as a fact about triangles. The Epitome of the Almagest, with its criticisms of Ptolemy, is the book Copernicus learned astronomy from. And the press at Nuremberg embodied the realisation that numerical data must be printed rather than copied, because copying corrupts numbers invisibly — the beginning of reliable published tables.

A debate that continues

De triangulis was not printed until 1533, so how far it shaped sixteenth-century mathematics directly rather than through later intermediaries is debatable. The allegation that he was poisoned by the sons of George of Trebizond has no evidence behind it. How much his criticism of Ptolemy's lunar theory contributed to Copernicus's thinking is argued about, though the textual connection is certain.

Keep exploring — ask Regiomontanus

• “Why is a corrupted number more dangerous than a corrupted sentence?”
• “When is criticism part of a summary's job?”
• “How much does a subject depend on somebody deciding it is a subject?”

Related lives

• Claudius Ptolemy — Who Tabulated the Heavens
• Johannes Kepler — Discoverer of Planetary Motion
• Luca Pacioli — Who Printed The Merchant's Arithmetic

Related themes

Trigonometry · Printing and scientific accuracy · Astronomy before Copernicus