Repeating circle

Tobias Mayer's reflecting circle had demonstrated the principle: measure the same angle multiple times around a graduated circle, and the random errors in the scale divisions will tend to cancel when you average the results. But Mayer's instrument required resetting between each measurement—a slow process that limited how many repetitions were practical. Jean-Charles de Borda, a French naval officer and mathematician, redesigned the instrument to make repetition mechanical, automatic, and fast.

Borda's repeating circle, developed in the 1770s and refined by 1784, allowed the observer to accumulate multiple angle measurements without returning to zero. The telescope could be locked to the target, the circle rotated to a new position, and another measurement taken—all without releasing the angle. After ten or twenty repetitions, the accumulated reading was divided by the number of observations to get the average. The mechanical action eliminated the setup time that had made Mayer's method tedious.

The French Navy sponsored Borda's work because precise navigation remained unsolved. The longitude problem—determining east-west position at sea—required either accurate chronometers or accurate astronomical observations. Borda's circle offered the latter: by repeating lunar distance measurements, navigators could achieve precision that single observations could never match. The French, having lost their North American empire partly due to British naval superiority, invested heavily in anything that might improve their own navigation.

But the repeating circle found its greatest application on land. In 1792, as revolutionary France sought to rationalize all measures, the Academy of Sciences dispatched two astronomers to measure the meridian arc from Dunkirk to Barcelona. Jean-Baptiste Delambre surveyed northward; Pierre Méchain surveyed south. Their task: determine the exact length of the meridian quadrant so that the new unit of length—the meter—could be defined as one ten-millionth of that distance.

The survey took seven years. Revolution, war, and political turmoil interrupted constantly. Méchain was imprisoned in Spain; Delambre worked through regions torn by civil conflict. Both relied on Borda's repeating circles for the triangulation measurements that chained together the survey's reference points. Each angle was measured multiple times, the accumulated reading divided by the repetitions, the random errors smoothed away. The precision was unprecedented—and necessary, because a small angular error at one station would propagate through the entire chain.

The repeating circle defined the meter that still governs commerce and science today. The survey's results were slightly wrong—Méchain discovered discrepancies he concealed rather than report—but the meter was fixed at the calculated value and remains essentially unchanged. Every measurement in the metric system traces back to angles measured with Borda circles in the 1790s.

The instrument had a longer legacy in geodesy, the science of measuring the Earth's shape. Through the 19th century, Borda circles and their successors enabled the great triangulation surveys that mapped continents. The Survey of India, the mapping of Africa, the positioning of national boundaries—all depended on precise angle measurement that only repeating instruments could provide. Borda's insight that mechanical repetition could overcome instrumental error became foundational to precision measurement of all kinds.