Pedro Nunes and Solar Retrogression

In northern latitudes we are used to the Sun rising in the East, following a smooth and even course through the southern sky and setting in the West. The idea that the compass bearing of the Sun might reverse seems fanciful. But in 1537 Portuguese mathematician Pedro Nunes showed that the shadow cast by the gnomon of a sun dial can move backwards.


Pedro Nunes (1502–1578). Portuguese postage stamp issued in 1978.

Nunes’ prediction was counter-intuitive. It came long before Newton, Galileo and Kepler, and Copernicus’ heliocentric theory had not yet been published. The retrogression was a remarkable example of the power of mathematics to predict physical behaviour.

Pedro Nunes

Pedro Nunes, one of the greatest mathematicians of his time, is best known for his contributions to navigation and to cartography. He introduced the loxodrome curve, which he called the rhumb line (See Last Post).

Nunes was aware of the link between solar regression and the biblical episode of the sun dial of Ahaz (Isaiah, 38:7–9). However, what he predicted was a natural phenomenon, requiring no miracle. It was several centuries before anyone claimed to have observed the reversal (Leitão, 2017).

Condition for Solar Retrogression

Leitão, who has made a detailed study of Nunes’ works, reviewed the method used by him. Nunes demonstrated the retrogression using spherical trigonometry. This was long before Newton’s laws or differential calculus were available. A simpler derivation, using elementary differential calculus, will appear elsewhere (Lynch, 2017). An expression is found for the azimuth of the Sun as a function of the time. For reversal to occur, the derivative of this function must vanish.

The vanishing of the derivative leads to the delightfully simple and elegant equation

\displaystyle \cos\lambda_S = \frac{\tan\phi_O}{\tan\phi_S} \ \ \ \ \ (1)

where {(\lambda_S,\phi_S)} are the longitude and latitude of the Sun and {(0, \phi_O)} the longitude and latitude of the observer. Clearly, there will be an azimuth at which the derivative vanishes only if the right hand side is less than unity, that is, if

\displaystyle \phi_O < \phi_S   \ \ \ \ \ (2)

This means that retrogression will be seen only if the observation point is between the Equator and the latitude of the Sun. In particular, it must be in the tropics.

An Example of Retrogression

We consider the daily path of the Sun at the time of the Summer solstice ({\phi_S=23.5^\circ}N) for observations from a point within the tropics ({\phi_O=20^\circ}N). We plot the Sun’s course during the morning, as seen from the tropical observatory, in the Figure below.  The angular coordinate is the azimuth or compass bearing. Radial coordinate is the zenith angle. SR denotes Sunrise. The azimuth at sunrise is close to {65^\circ}. It increases to around {77^\circ} by mid-morning and then decreases to zero at Noon.


Path of the Sun at the Summer solstice for a observation point at 20N

For the specific values {\phi_O=20^\circ} and {\phi_S=23.5^\circ}, the condition (2) above gives the turning longitude as {\lambda_S=33.17^\circ} corresponding to an azimuth of {77.4^\circ} and an elevation of {59.1^\circ}. This is the retrogression phenomenon.


• Leitão, Henrique, 2017: A brief note on the power of mathematics: Pedro Nunes and the retrogradation of shadows. Proceedings of Recreational Mathematics Colloquium V — G4G (Europe), Ed. Jorge Nuno Silva, pp.~45–52.

• Lynch, Peter, 2017: Pedro Nunes and the Retrogression of the Sun. To appear (PDF on request from

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