I’ve been fascinated by astrolabes for a very long time, roughly 20 years. It was this avocation that led to my interest in sundials and, because they share museum space, my interest in clocks. When I lived in Rockford, Illinois, I would haunt the Time Museum, an institution that produced the most beautiful book on astrolabes. Adler Planetarium in nearby Chicago has one of the best astrolabe collections in the entire world, producing another beautiful book solely on Western astrolabes and a gorgeous book on antique scientific instruments in general. None of these provide the mathematical details of astrolabe design beyond a description of stereographic projection, and indeed this kind of detailed information is rarely found. The Astrolabe, a new book by James E. Morrison, is an absolutely unique and wonderful book on the mathematics needed to create accurate, beautiful designs of astrolabes, quadrants and other related instruments. I can’t recommend it enough to those who share the interests of this blog.
The Analemmas of Vitruvius and Ptolemy
Have you ever had to calculate the positions of astronomical objects? Orbital calculations relative to an observer on the Earth require derivations and time-consuming solutions of spherical trigonometric equations. And yet, these kinds of calculations were accomplished in the days prior to the advent of calculators or computers!
For example, to find the zenith angle (angle to overhead) and azimuth (angle from North) of the sun at any day and time of the year for any location on Earth, the laws of spherical trigonometry produce the formulas below. Here the solar declination δ is a function of the solar longitude λ and ecliptic angle ε as shown in the figure to the left.
These calculations can be automated today—but did I mention that these solutions were found before electronic calculators?
… or slide rules, or logarithms?
… or trigonometric formulas?
… or even algebra??
In fact, Vitruvius (ca. 50) and Ptolemy (ca. 150) provided mathematical and instrumental means of calculating the sun’s position for any hour, day, and observer location by the use of geometric constructions called analemmas (only indirectly related to the figure-8 analemma on globes). An important application of analemmas was the design of accurate horizontal and vertical direct and declining sundials for any observer location. These analemmas are awe-inspiring even today, and as the study of “Descriptive Geometry” has disappeared from our schools they can strike us as mysterious and wondrous inventions!