Regiomontanus

Who Made Triangles A Subject · Astronomer & Mathematician · 1436–1476

The astronomer who made trigonometry a subject in its own right in 1464, and who set up his own printing press because a hand-copied table is wrong by the third copy.

Who was Regiomontanus?

Johannes Müller was born in 1436 at Königsberg in Franconia; the Latin form of his birthplace gave him the name he is known by. He matriculated at Leipzig at eleven and at Vienna at fourteen, where he studied under Georg von Peuerbach and became his collaborator. When Cardinal Bessarion came to Vienna in 1460 seeking a better Latin abridgement of Ptolemy's Almagest, Peuerbach began one and died before finishing it. Regiomontanus completed it. The Epitome of the Almagest is clearer than Ptolemy, includes criticism Ptolemy would not have welcomed, and is the book from which Copernicus learned his astronomy. In 1464 he finished De triangulis omnimodis libri quinque — five books on triangles of every kind. What makes it matter is not any single result but the decision behind it: triangles are treated as a subject in their own right, plane and spherical, set out systematically, and not as an appendix to astronomy. It contains the law that in any plane triangle each side stands to the sine of its opposite angle in one and the same ratio. It was not printed until 1533, fifty-seven years after his death, and then it shaped European mathematics for a century. He travelled in Italy with Bessarion hunting Greek manuscripts, served the library of Matthias Corvinus in Hungary, and settled at Nuremberg in 1471 — chosen, he said, for its position and its craftsmen. There he built an observatory with the support of the citizen Bernhard Walther, and set up a printing press of his own. That was the unusual decision. Manuscripts copied by hand accumulate errors with every copy, and a table of numbers is exactly the kind of text where errors are invisible and fatal; a printed table can be corrected once and then be right for everybody. He printed Peuerbach's work, his own calendars, and ephemerides giving planetary positions which navigators carried to sea for decades. In 1475 Pope Sixtus IV summoned him to Rome to advise on reforming the calendar. He died there in July 1476, aged forty. Plague was blamed; so, in some accounts, was poison.

Major achievements

• Completed Peuerbach's Epitome of the Almagest, the book Copernicus learned from
• Wrote De triangulis omnimodis in 1464, making trigonometry a subject in its own right
• Stated the law relating each side of a plane triangle to the sine of its opposite angle
• Computed extensive tables of sines to a large radius
• Set up his own printing press at Nuremberg to publish astronomy without copyists' errors

Life in brief

• 1436 — Born at Königsberg: In Franconia; the name is the Latin form of the town.
• 1450 — Vienna: At fourteen, to study under Georg von Peuerbach.
• 1461–1464 — The Epitome: Completing Peuerbach's abridgement of the Almagest after his death.
• 1464 — De triangulis omnimodis: Five books on triangles as a subject of their own.
• 1471 — Nuremberg: An observatory and a printing press of his own.
• 1475 — Summoned to Rome: By Sixtus IV, to advise on the reform of the calendar.
• 1476 — Died at Rome: Aged forty; plague, or in some accounts poison.

Explore the life of Regiomontanus — five chapters

1. Vienna, at Fourteen — Peuerbach, and an unfinished commission
2. The Epitome — The book Copernicus learned from
3. Five Books on Triangles — 1464, and a subject cut loose from the sky
4. The Press at Nuremberg — Because a copied table decays
5. Summoned to Rome — The calendar, and forty years

Read the five-chapter life of Regiomontanus →

Begin

• Start a conversation with Regiomontanus — a dramatised, historically grounded AI portrayal
• Read the Historical Brief
• Explore the life of Regiomontanus — five chapters
• Find your exam topics · Meet all 208 figures