Regiomontanus

How to use this course

Regiomontanus lived only forty years, yet he set the agenda for European astronomy for a century. He finished the best Latin guide to Ptolemy’s Almagest, wrote the first European book to treat trigonometry as a subject in its own right, computed sine and tangent tables to seven digits, and printed scientific books with a care that no one had attempted before. Copernicus learned his Ptolemy from Regiomontanus’s Epitome, and Tycho Brahe solved triangles with his methods.

Each technical lesson opens with a paragraph for the general reader, then gives statements in modern notation, derivations and exercises with folded solutions. A note on notation: medieval and Renaissance trigonometry worked with lengths of lines in a circle of a chosen radius \(R\), so Regiomontanus’s “sine of 30°” in a circle of radius 10,000,000 is the number 5,000,000. We write \(\mathrm{Sin}\,\theta=R\sin\theta\) for such a line and keep lowercase \(\sin\) for the modern ratio.

Much of what is known about his life comes from Ernst Zinner’s biography (1938, English translation 1990) and from his surviving letters. Where sources disagree, the text says so.

Timeline

Königsberg, Leipzig, Vienna, 1436–1457

Johannes Müller was born on 6 June 1436 in Königsberg, a small town in Lower Franconia, now in Bavaria. MacTutor places the birth in the nearby hamlet of Unfinden. It is a different Königsberg from the Prussian city of Kant. His father was a miller, and the name Müller simply means miller. In university registers he appears as Johannes Molitoris, Latin for the same thing. Contemporaries called him Johannes de Monte Regio or Johannes Germanus. The name Regiomontanus, a Latin rendering of “King’s Mountain”, became standard only after his death.

He was precocious. He studied at Leipzig from about the age of eleven, without taking a degree, and on 14 April 1450 enrolled at the University of Vienna. Vienna was then the strongest centre for astronomy in the German lands, with a tradition going back to John of Gmunden. Müller took his bachelor’s degree on 16 January 1452, at fifteen. University statutes did not allow a master’s degree before the age of twenty-one, so he waited, and received it in 1457. On 11 November 1457 he joined the arts faculty.

His teaching assignments show the range expected of a young master: perspective (optics) in 1458, Euclid in 1460, and Virgil’s Bucolics in 1461. Latin literature and mathematics were not separate careers in the fifteenth century, and his later prose, prefaces and polemics show a humanist’s training.

Peuerbach and the failing tables

At Vienna Müller became the pupil, then the colleague and friend, of Georg von Peuerbach, the leading astronomer of the university. Peuerbach’s Theoricae novae planetarum explained the geometry of planetary models more clearly than any earlier Latin text, and later became a standard university textbook.

The two men observed together. European astronomers predicted planetary positions with the Alphonsine Tables, compiled in Castile around 1270. Peuerbach and Müller found that the tables were wrong by observable amounts. Their observations of Mars, and of a lunar eclipse whose timing differed from prediction, convinced Müller that astronomy needed new observations, better models and accurate computation. That programme guided the rest of his life. They preferred eclipses and occultations of stars by the Moon because the moment of contact can be timed precisely.

The commission

In May 1460 Cardinal Johannes Bessarion arrived in Vienna as papal legate, seeking support for a crusade against the Ottomans. Bessarion was a Greek scholar from Trebizond who had settled in Italy and gathered a great library of Greek manuscripts. He disliked the recent Latin translation of Ptolemy’s Almagest by George of Trebizond and wanted a reliable guide to the work. Peuerbach did not read Greek well enough to translate it, so he began a condensed, modernized account from the Latin tradition, an Epitome. He also agreed to travel to Italy with Bessarion.

Peuerbach died on 8 April 1461, having finished six of the thirteen books. On his deathbed he asked Müller to complete the work. Müller took over both the book and the journey.

With Bessarion in Italy, 1461–1467

Müller reached Rome with Bessarion on 20 November 1461 and lived in his household for about four years. He learned Greek, copied and collated manuscripts, and met the leading Italian mathematicians, including Giovanni Bianchini of Ferrara and Paolo dal Pozzo Toscanelli of Florence.

He finished the Epitome of the Almagest in 1462 and dedicated it to Bessarion. It was more than a summary: it added later observations, corrected computations and made critical remarks. One of these, on the size of the Moon in Ptolemy’s model (Lesson 08), was read decades later by Nicolaus Copernicus.

In 1464 he wrote De triangulis omnimodis (“On triangles of every kind”), five books that develop plane and spherical trigonometry in the axiomatic style of Euclid. He also wrote a long letter to Bianchini, after 11 February 1464, reporting that he had found in Venice an incomplete manuscript of Diophantus’s Arithmetica and that he would translate it if the remaining books turned up. They never did, and his translation was never made. In the spring of 1464 he lectured at Padua on the ninth-century astronomer al-Farghani, opening with an oration on the history and value of the mathematical sciences.

Polemics

Müller wrote sharply. He criticized George of Trebizond’s commentary on Ptolemy in defence of the older commentator Theon of Alexandria, and attacked the errors of the widely used medieval Theorica planetarum attributed to Gerard of Cremona. He also rejected a claimed quadrature of the circle by Cardinal Nicholas of Cusa (Lesson 10). The polemics expressed a conviction he would act on in Nuremberg: corrupt texts and careless tables were the main obstacle to progress in astronomy.

From chords to sines: building the tables

For the general reader. Before calculators, every astronomical prediction required looking up trigonometric values in a printed or copied table. Ptolemy used a table of chords. Indian and Islamic astronomers replaced the chord with the half-chord, the sine. Regiomontanus computed sine tables with more digits and finer steps than any earlier European, and showed how to do it from first principles.

Chords and sines

In a circle of radius \(R\), the chord subtending a central angle \(\alpha\) has length \(\mathrm{Crd}\,\alpha=2R\sin(\alpha/2)\). Ptolemy took \(R=60\) and tabulated chords at half-degree steps in sexagesimal fractions. The half-chord of \(2\theta\) is \(\mathrm{Sin}\,\theta=R\sin\theta\), and the remaining part of the radius, \(\mathrm{Vers}\,\theta=R-R\cos\theta\), is the versed sine (see the figure at the top of the page).

Without decimal fractions, the way to obtain precision was to make \(R\) large, so that every value is an integer. Regiomontanus’s Compositio tabularum sinuum describes two tables: a sexagesimal one with \(R=6\,000\,000\) at intervals of one minute of arc, and, computed at Buda in 1468, a table with \(R=10\,000\,000\). The second is effectively a seven-place decimal table. Both were printed at Nuremberg in 1541.

Derivation and strategy

Relation 2 follows from \(1-\cos\theta=2\sin^2(\theta/2)\) multiplied by \(R^2\). Relation 3 is the addition formula, which Ptolemy proved for chords with his theorem on cyclic quadrilaterals. Starting from the exact values \(\mathrm{Sin}\,30^\circ=R/2\), \(\mathrm{Sin}\,45^\circ=R/\sqrt2\) and \(\mathrm{Sin}\,36^\circ\) (from the regular pentagon), repeated halving and addition give the sines of all multiples of \(3^\circ/4=45'\). No finite chain of square roots reaches \(1^\circ\). Ptolemy bridged the gap with an inequality between ratios of chords; later tables used interpolation or iteration. For small steps, the table is filled in by interpolation, with the error controlled because \(\sin\) is nearly linear over one minute of arc.

Plane triangles: De triangulis I–II

For the general reader. Given some sides and angles of a triangle, find the rest. Astronomers had done this for centuries inside their calculations. Regiomontanus collected the methods into a single book organized like Euclid’s Elements, with definitions, axioms and numbered theorems, so that trigonometry could be learned on its own.

Structure of the book

Book I gives definitions, axioms and 56 theorems on triangles that use only geometry and the table of sines, including the solution of right triangles. Book II begins with the law of sines and uses it to solve general plane triangles. Books III–V treat spherical triangles. The work was completed in 1464 and printed in Nuremberg in 1533. Each section ends with worked problems, and the preface tells the reader that the subject opens the door to all of astronomy.

Proof. Drop the altitude \(h\) from the vertex \(C\) to the line \(AB\). In the two right triangles so formed, \(h=b\sin A=a\sin B\), so \(a/\sin A=b/\sin B\). The same argument with another altitude gives the third ratio. (The common value is the diameter of the circumscribed circle, by the inscribed angle theorem.) Regiomontanus expressed the result with lines in a circle of radius \(R\): the sides are to each other as the sines of the opposite angles.

The altitude onto side \(a\) is \(b\sin C\). Book II gives this rule in words, as a method for finding the area of a triangle from two sides and the angle between them.

Algebra in a geometry book

Some problems in Book II cannot be solved by a straightforward chain of sines. Regiomontanus solved them with the “art of the thing and the square” (ars rei et census), the algebra of Arabic origin in which the unknown is the res (thing) and its square the census. Book II theorem 23 solves a trigonometric problem this way, the first such solution in the Latin West according to the Dictionary of Scientific Biography. The method is: name an unknown length, express the data as an equation, reduce to a quadratic, solve.

Spherical triangles and the cosine law

For the general reader. The sky looks like the inside of a sphere, and positions on it are measured by arcs of great circles. Three such arcs form a spherical triangle. Every question about when a star rises, how high the Sun stands at noon, or how far apart two cities are on the Earth is a spherical triangle problem. Books III–V of De triangulis are a systematic treatment of these problems.

Sources

Spherical trigonometry was developed by Menelaus in antiquity and transformed by Islamic astronomers, who introduced the spherical law of sines. Gerolamo Cardano later charged that Regiomontanus took much of his spherical material from the twelfth-century Andalusian astronomer Jabir ibn Aflah without credit. Historians agree that Jabir is an important source for Book IV, and disagree about how much this diminishes the work’s originality. The systematic arrangement and the cosine law of Book V are his.

On a unit sphere, a spherical triangle has sides \(a,b,c\) (arcs, measured as angles at the centre) and angles \(A,B,C\) at the vertices, with \(A\) opposite \(a\).

The two forms are equivalent: expand \(\operatorname{vers}(b-c)=1-\cos b\cos c-\sin b\sin c\) and simplify. This is the earliest known statement of the cosine law in Europe. He gave it only once and applied it sparingly; its full use came with later astronomers.

Proof with vectors

Put the vertex \(A\) at the north pole, \(\mathbf u_A=(0,0,1)\). The vertices \(B\) and \(C\) lie at polar distances \(c\) and \(b\), with longitudes differing by the angle \(A\):

The arc \(a\) between \(B\) and \(C\) satisfies \(\cos a=\mathbf u_B\cdot\mathbf u_C=\sin c\sin b\cos A+\cos c\cos b\). For small triangles, \(\cos x\approx1-x^2/2\) and \(\sin x\approx x\) turn the formula into the plane law \(a^2=b^2+c^2-2bc\cos A\).

Tangents and the Tabulae directionum

For the general reader. Astrologers and astronomers needed to know at what moment a point of the sky would cross the horizon or the meridian at a given latitude. The calculations involve the ratio of sine to cosine, our tangent. Regiomontanus computed a table of tangents for this purpose in Hungary in 1467, one of the first in Europe.

The tables

The Tabulae directionum profectionumque (“Tables of directions and profections”), computed in 1467 for Archbishop Vitéz and printed in 1490, served the astrological technique of primary directions, which needs the times at which points of the ecliptic reach given positions relative to the horizon. The work contains a table of tangents at one-degree intervals with \(\mathrm{Tan}\,45^\circ=100\,000\), and in its tenth problem proposes a sine of \(90^\circ\) equal to 100,000. The tables are computed for latitudes up to \(60^\circ\). He also devised a system for dividing the sky into twelve astrological houses, later known by his name and widely used in Europe.

As a line, \(\mathrm{Tan}\,\theta=R\tan\theta\) is the segment of the tangent to the circle at the end of the radius, cut off by the extension of the other radius (see the figure at the top). From a sine table one obtains \(\mathrm{Tan}\,\theta=R\,\mathrm{Sin}\,\theta/\mathrm{Cos}\,\theta\).

Derivation. In the triangle formed by the celestial pole, the zenith and the star, the sides are \(90^\circ-\varphi\), \(90^\circ-\delta\) and the zenith distance \(z\), with the angle \(H\) at the pole. The cosine law of Lesson 06 gives \(\cos z=\sin\varphi\sin\delta+\cos\varphi\cos\delta\cos H\). At rising, \(z=90^\circ\), which gives the formula. The product of two tangents is why a tangent table saves work.

The Epitome and Ptolemy’s Moon

For the general reader. Ptolemy’s Almagest was the most advanced astronomy available in 1460, but it is long and difficult. The Epitome condensed it into thirteen books, reworked the computations and added criticism. One remark in it pointed to a flaw in Ptolemy’s theory of the Moon that Copernicus later set out to repair.

Ptolemy’s second lunar model

Ptolemy placed the Moon on an epicycle whose centre moves on an eccentric deferent. The deferent’s centre itself revolves around the Earth, so that the epicycle is pulled toward the Earth at quadrature (half Moon) and pushed away at conjunction and opposition. This “crank” reproduced the second inequality of the Moon’s motion, now called evection. In units where the deferent radius is 60, Ptolemy’s parameters are: epicycle radius \(5;15\) (that is, \(5\tfrac14\)) and eccentricity \(10;19\).

Computation. At syzygy the epicycle centre is at distance 60, and the Moon’s distance ranges from \(60-5;15\) to \(60+5;15=65;15\). At quadrature the epicycle centre is at the deferent’s nearest point, distance \(60-2\times10;19=39;22\), and the Moon’s least distance is \(39;22-5;15=34;07\). So

Regiomontanus stated the difficulty clearly. Copernicus read it as a student in Italy, and his own lunar theory in De revolutionibus replaced the crank with a second epicycle, keeping the distance variation small.

The Epitome as a book

Printed in Venice in 1496, the Epitome was for most sixteenth-century readers the way into Ptolemy. It is shorter than the Almagest, uses sines in place of chords, updates parameters from medieval observations, and frequently adds alternative proofs.

Parallax, eclipses and longitude

For the general reader. To find how far away something in the sky is, look at it from two places and measure how its direction shifts. In January 1472 Regiomontanus applied this idea to a bright comet. He also published tables that let travellers compare the time of an eclipse with the time predicted for Nuremberg and so find how far east or west they were.

The comet of 1472

A bright comet, now catalogued as C/1471 Y1, was visible from Christmas Day 1471 until 1 March 1472 (Julian calendar). Regiomontanus and Bernhard Walther followed it from Nuremberg, measuring its positions relative to stars with a Jacob’s staff. Their positions are accurate enough for modern astronomers to identify the comet. Aristotle had taught that comets are atmospheric, and Regiomontanus worked within that view, but he tried to measure its distance from its parallax. He concluded that it was at least about 8,200 miles (13,120 km) away. The estimate was off by orders of magnitude, yet the method was right. Tycho Brahe’s parallax measurement of the comet of 1577, which placed it beyond the Moon, carried this programme through.

The second formula is the law of sines in the triangle formed by the Earth’s centre, the observer and the object.

Eclipses and longitude

The Ephemerides, printed in 1474, gave daily positions of the Sun, Moon and planets for 1475–1506, with eclipses. A lunar eclipse is seen at the same absolute moment from everywhere on the night side of the Earth. If the predicted Nuremberg time of an eclipse is \(t_N\) and a traveller observes it at local time \(t_L\), then the longitude difference is \(15^\circ\) per hour of \(t_N-t_L\). Christopher Columbus, stranded on Jamaica, used the Ephemerides prediction of the lunar eclipse of 29 February 1504. The practical difficulty was the poor accuracy of clocks and of contact timings.

A maximum problem and a false quadrature

For the general reader. Where should you stand to see a statue on a pedestal as large as possible? Too close, you look up at it edge-on; too far, it shrinks. Regiomontanus posed this question in a letter of 1471, an early example of an extremum problem in European mathematics. In the same spirit of exact geometry he exposed an invalid construction for squaring the circle.

The rod

The problem, as Regiomontanus put it, asks at which point on the ground a vertically suspended rod appears largest. Let the rod occupy heights \(a<b\) above eye level, and let the observer stand at horizontal distance \(x\). The rod subtends the angle \(\theta(x)=\arctan(b/x)-\arctan(a/x)\).

Proof with algebra. By the subtraction formula, \(\tan\theta=\dfrac{(b-a)x}{x^2+ab}=\dfrac{b-a}{x+ab/x}\). Maximizing \(\theta\) means minimizing \(x+ab/x\), and by the inequality of arithmetic and geometric means \(x+ab/x\ge2\sqrt{ab}\), with equality exactly when \(x=\sqrt{ab}\).

Proof with geometry. Points \(P\) on the eye-level line from which the segment is seen under angle \(\theta\) lie on circles through the rod’s ends. The angle grows as the circle shrinks, and the smallest circle that still meets the line is tangent to it. By the tangent–secant theorem, the tangent length from the foot of the rod satisfies \(x^2=a\cdot b\).

Cusanus and the circle

Nicholas of Cusa, cardinal, philosopher and amateur geometer, proposed several constructions claiming to square the circle and sent work on the subject to Peuerbach. Regiomontanus answered with a critique showing that the ratio of circumference to diameter implied by Cusa’s construction fell outside the bounds that Archimedes had proved. In private he was scornful of the cardinal’s geometry. The episode shows his standard: a claim in geometry must agree with established, proved results.

Hungary and Nuremberg, 1467–1475

By 1467 Regiomontanus was in Hungary, in the service of János Vitéz, archbishop of Esztergom and a humanist patron, who was founding a university at Pressburg (Bratislava), the Universitas Istropolitana. There he computed the Tabulae directionum, working with the Polish-born astronomer Martin Bylica of Olkusz, and built instruments. He then moved to the court of King Matthias Corvinus at Buda, made an astrolabe for the king, collated Greek manuscripts for the royal library, and in 1468 computed the decimal sine table. He also wrote the Tabulae primi mobilis (tables of the first motion), dedicated to the king, which gave ready-made solutions of spherical triangles for astronomers.

Nuremberg

In 1471 political turmoil in Hungary ended his position, and he chose Nuremberg as his home. In a letter of 4 July 1471 he wrote that he had chosen it as his permanent home. The city had skilled instrument makers and was a centre of trade and communication. The council granted him residence that November. A wealthy merchant, Bernhard Walther, became his pupil and partner and financed an observatory, a workshop for instruments, and a printing press in his house.

His press was one of the first devoted to science. The first book was Peuerbach’s Theoricae novae planetarum (1472), with diagrams more numerous and clearer than those in manuscripts. Others followed: the Astronomica of Manilius, the Kalendarium in Latin and German with calendars for 1475–1531, and the Ephemerides for 1475–1506. In 1474 he issued a broadside listing the books he had printed and those he planned: Ptolemy, Euclid, Archimedes, Theon, Witelo, and his own commentaries and treatises. He corrected typographical errors by hand before books left the shop and wrote that printing is useful only for properly corrected editions. Only a fraction of the programme was completed.

Rome, death and legacy

In 1475 Regiomontanus left Nuremberg for Rome. A Nuremberg chronicle reports that Pope Sixtus IV summoned him to work on reforming the calendar, whose drift against the seasons had become obvious. The report is plausible but not documented in papal sources, and the story that he was to be made bishop of Regensburg is doubtful. His last Nuremberg observation is dated 28 July 1475; Walther’s own observations began five days later and continued until his death in 1504.

Regiomontanus died in Rome on 6 July 1476, aged forty (some sources give about 8 July). The Tiber had flooded in January, and plague followed; this is the likely cause. A later rumour, repeated by Pierre Gassendi in his biography of 1654, claimed that the sons of George of Trebizond poisoned him in revenge for his attacks on their father. Historians give the rumour little weight. Walther kept his manuscripts and instruments; after Walther’s death they were dispersed, and in 1509 the painter Albrecht Dürer bought Walther’s house.

Legacy

• Trigonometry: De triangulis omnimodis, printed in 1533, made trigonometry an independent mathematical discipline in Europe. Copernicus and Tycho Brahe used it. His seven-place decimal tables pushed astronomers toward decimal computation; Simon Stevin cited them in arguing for decimal fractions.
• Astronomy: the Epitome was the standard guide to Ptolemy and directed Copernicus to the weakness of the lunar theory. His insistence on new observations and better tables anticipates Tycho.
• Printing: his Nuremberg press set a standard of accurate scientific publishing. Later Nuremberg editors, above all Johann Schöner and the printer Johannes Petreius, published his manuscripts; Petreius also printed De revolutionibus in 1543.
• Heliocentrism: some writers have suggested he considered a moving Earth, citing a manuscript note on Aristarchus and a later report by Georg Hartmann. The evidence is thin, and the course does not rely on it.
• Memory: Hartmann Schedel’s Nuremberg Chronicle (1493) pictures him holding an astrolabe. A lunar crater and Nuremberg’s public observatory bear his name.

Connections

People linked to Regiomontanus across this collection. A name without a link has no page here yet. See Regiomontanus in the network of all 337 lives →

Teachers and advisors

• Georg von Peuerbach 1450-1461

Students

• Bernhard Walther 1471-1475

Influenced by

• Claudius Ptolemy
• Nasir al-Din al-Tusi
• Jabir ibn Aflah

Influenced

• Nicolaus Copernicus
• Simon Stevin
• Tycho Brahe
• Johann Schöner

Colleagues and collaborators

• Basilios Bessarion 1460-1467
• Giovanni Bianchini
• Martin Bylica 1467-1471
• Paolo dal Pozzo Toscanelli

Rivals

• George of Trebizond
• Nicholas of Cusa

Also mentioned in this life

• Archimedes
• Aristarchus of Samos
• Aristotle
• Gerolamo Cardano
• Diophantus of Alexandria
• Euclid

Fields

• astronomy 50 lives
• trigonometry 9 lives

On CoursesHub

• Astrophysics for Beginners

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Johann Müller Regiomontanus, MacTutor History of Mathematics.
2. Regiomontanus, Wikipedia (with references).
3. Regiomontanus, Johannes, Complete Dictionary of Scientific Biography, via Encyclopedia.com.
4. H. K. Strick, Regiomontanus, MacTutor.
5. Whipple Museum, Cambridge, Regiomontanus and books and Regiomontanus and mathematical techniques.
6. E. Zinner, Regiomontanus: His Life and Work, trans. E. Brown, North-Holland, 1990 (German original 1938, 2nd ed. 1968).
7. P. Gassendi, Vita Regiomontani, appended to his life of Tycho Brahe, 1654.

Mathematics and astronomy

1. B. Hughes, Regiomontanus on Triangles (Latin text and English translation of De triangulis omnimodis), University of Wisconsin Press, 1967.
2. Regiomontanus, Epytoma in almagestum Ptolomei, Venice, 1496.
3. G. Van Brummelen, The Mathematics of the Heavens and the Earth: The Early History of Trigonometry, Princeton, 2009.
4. G. Van Brummelen, Heavenly Mathematics: The Forgotten Art of Spherical Trigonometry, Princeton, 2013.
5. N. M. Swerdlow, Regiomontanus on the critical problems of astronomy, in Nature, Experiment, and the Sciences, Kluwer, 1990.
6. J. A. Jervis, Cometary Theory in Fifteenth-Century Europe, Reidel, 1985.
7. Regiomontanus’ angle maximization problem, Wikipedia; H. Dörrie, 100 Great Problems of Elementary Mathematics, Dover, 1965.