Math Horizons - Volume XXV, Issue 3, February 2018
Albrecht Dürer’s Celestial Geometry
2018-02-21 00:24:45
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STEPHEN LUECKING
Albrecht Dürer (1471–1528), noted Renaissance printer and painter, twice left his native Germany for sojourns to Italy, once from 1494 to 1495 and again from 1505 to 1507. During those years his wide-ranging intellect absorbed the culture and thinking of noted artists and mathematicians.
Perhaps the most important outcome of these journeys was his introduction to scientific methods. His embrace of these methods went on to condition his thinking for the rest of his life. One notable consequence was Dürer’s abandonment of astrological subject matter—a big seller for a printer and publisher such as himself (figure 1)—in favor of astronomy.
Collection of Staatliche Graphische Sammlung, Munich
Astronomy was not to be a casual interest. Just before his second trip to Italy, Dürer published De scientia motus orbis, a cosmological treatise by the Persian Jewish astronomer Masha’Allah ibn Atharī (ca. 740–815 CE). Since Masha’Allah wrote the treatise for laymen and included ample illustrations, it was a good choice for introducing Europeans to Arabic astronomy.
In 1509 Dürer purchased the entire library of Regiomontanus (1436–1476 CE) from the estate of Nuremberg businessman Bernhard Walther. Regiomontanus was Europe’s leading astronomer, a noted mathematician, and a designer of astronomical instruments. Walther had sponsored Regiomontanus’s residency in Nuremberg between 1471 and 1475. Part of Walther’s largesse was to provide a print shop from which Regiomontanus published the world’s first scientific texts ever printed.
Star Map
In 1515, Dürer and Austrian cartographer and mathematician Johannes Stabius produced the first map of the world portraying the earth as a sphere. Afterward, Stabius proposed continuing their collaboration by publishing a star map—the first such map published in Europe (figure 2). Their work relied heavily on data assembled by Regiomontanus, plus refinements from Walther.
The star map required imprinting the threedimensional dome of the heavens onto a twodimensional surface without extreme distortions, a task that fell to Stabius. He used a stereographic projection. In this method, rays originate at the pole in the opposite hemisphere, pass through a given point in the hemisphere, and yield a point on a circular surface (figure 3).
The principles of stereography date back to the Greek astronomer Hipparchus (190–120 BCE), but were first written about in the second century when the Greco-Roman astronomer Claudius Ptolemy (100–168 CE) wrote Planispherium. This text and his better-known Almagest, written to espouse his theory of planetary motion, became the basis for the development of the astrolabe, a device used to measure the inclination of celestial bodies (figure 4). This development peaked in the Muslim world during the ninth century because of the efforts of scientists like Masha’Allah.
As the son of a goldsmith, Dürer’s exposure to stereographic projection would have been by way of the many astrolabes being fabricated in Nuremburg, then Europe’s major center for instrument makers. As the 16th century moved on, the market grew for such scientific objects as astrology slipped into astronomy. Handcrafted brass instruments, however, were affordable only to the wealthy, whereas printed items like the Dürer-Stabius maps reached a wider market.
Epicycles and Deferents
To the ancient mind it was a given that all celestial motion must be perfectly circular. But the observed paths of the planets, as seen from Earth, were anything but circular. To the contrary, planets would appear to stop and then loop backwards—called retrograde motion—before proceeding forward.
In Almagest Ptolemy explained this retrograde movement by describing the path of the planets as generated by circles moving upon circles (figure 5). This then ensured the perfect motion of the circle to guide the paths of the planets.
In particular, such a path is obtained by following a fixed point on a circle (called the epicycle) as the center of the circle moves along the circumference of another circle (called the deferent), as in figure 6. A host of unique curves are possible by varying the radii of the two circles.
In his 1525 book Die Messerung (On Measurement), Dürer presents an instrument of his own design used to draw these and other more general curves (figure 7). This compass for drawing circles upon circles consisted of four telescoping arms and calibrated dials. An arm attached to the first dial could rotate in a full circle, a second arm fixed to another dial mounted on the end of this first arm could rotate around the end of the first arm, and so on.
As a trained metalsmith, Dürer possessed the expertise to craft this complex tool. Precision calibration and adjustable arms allowed its user to plot an endless number of curves by setting the length of each telescoping arm and determining the rate at which the arms turned. This, in eff ect, constituted manual programming by setting the parameters of each curve plotted.
The example given in On Measurement is a curve known to mathematicians as the limaçon of Pascal (figure 8), although Dürer published the curve some 100 years before Étienne Pascal (father of Blaise) (1588–1651).
Dürer used two arms of his compass to draw this curve. Each time he rotated the interior arm in 30-degree increments, he simultaneously rotated the external arm in 30-degree increments in the same direction. He continued in this way until both arms revolved one full time. Interpolating the plotted points into a smooth curve completed the construction. The curve approximates one rotation of the planets Mars and Venus, which may have been the artist’s rationale for constructing it.
From Albrecht Dürer, Underweysung der Messung, 1525
One needs only two arms to draw the limaçon, but Dürer’s tool features four arms. This was because one can draw curves with more than two circles. Imagine replacing the planet in figure 6 with a third pivot point around which the planet then orbited. Three such points and a three-armed tool were needed, for instance, to draw the path of Mercury as determined by Ptolemy. Dürer added a fourth arm in anticipation of even more complex curves describing the motion of celestial bodies not yet discovered.
From Albrecht Dürer, Underweysung der Messung, 1525
Of course, the days of Ptolemy’s geocentric model of planetary motion were numbered. At the same time that Dürer was drawing complex curves with his compass, Nicolaus Copernicus (1473–1543) was developing the heliocentric model, which would shortly be the dominant view of astronomy. Nevertheless, we must admire the beautiful art, mathematics, and craftsmanship that infused Dürer’s work.
Stephen Luecking, a widely exhibited artist, is former chair of the art department at DePaul University and cofounder of the graphics program in the university’s computer science school.
10.1080/10724117.2018.1424456
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