Reflecting circle

The octant, invented by John Hadley in 1731, had transformed marine navigation by allowing sailors to measure the angle between the sun or stars and the horizon even on a rolling deck. Its double-reflection principle—using two mirrors to superimpose celestial and horizon images—eliminated the errors that plagued earlier instruments. But the octant had a fundamental limitation: it could only measure angles up to 90 degrees. For lunar distance navigation, the key to determining longitude at sea, larger angles were often needed. More importantly, any single measurement contained errors from imperfect mirrors, divided scales, and human observation. The octant offered no way to cancel these errors out.

Tobias Mayer, a German astronomer and cartographer at the University of Göttingen, saw a solution in 1752. Instead of an arc covering one-eighth of a circle, he constructed an instrument spanning the full 360 degrees. This was not mere extravagance. The complete circle enabled a revolutionary error-reduction technique: the same angle could be measured multiple times at different positions around the circle, and the results averaged. Random errors in the scale divisions, which might bias a single reading, would tend to cancel when measurements were taken at different points. The instrument embodied a statistical insight decades before formal error theory was developed.

The technical execution required precision that pushed contemporary instrument-making to its limits. The graduated circle had to be divided with extreme accuracy—any systematic error in the scale would not average out. The mirrors needed precise mounting and adjustment. The telescope sights demanded optical quality. Mayer worked in Göttingen, which was becoming a center for astronomical and geodetic work under the patronage of the Hanoverian kings who also ruled Britain. The university's connections to British science and instrumentation helped make such precision work possible.

Mayer's circle was designed specifically for his lunar distance tables—a method for determining longitude by measuring the angular distance between the moon and reference stars. The method required angular measurements accurate to within a few arc-seconds, far beyond what ordinary instruments could achieve. By enabling multiple observations to be averaged, the reflecting circle brought such precision within reach. Mayer submitted his lunar tables and instrument design to the British Board of Longitude, which eventually awarded his widow a prize after his early death in 1762.

The French astronomer Jean-Charles de Borda refined the concept into the repeating circle in the 1770s, adding mechanical features that made repeated measurements faster and more convenient. Borda's instrument became the standard for the French geodetic surveys that would define the meter as a fraction of the Earth's circumference. The chain of triangulation measurements that stretched from Dunkirk to Barcelona, determining the length of the meridian arc, relied on repeating circles descended from Mayer's design.

The reflecting circle demonstrates how an instrument can embody a mathematical principle. The idea that errors can be reduced by repetition and averaging is now fundamental to experimental science, but in 1752 it was not obvious how to build such statistical reasoning into physical apparatus. Mayer's full circle made the insight tangible: more measurements, from different parts of the scale, produced better results. The precision demands of 18th-century astronomy and navigation drove the development of instruments that would, in turn, enable the precise measurements underlying modern science.