Centesimal angular measure


Originally, the metric system was intended to replace all  existing units, including those of time and angle. Along with the renamed months of the year, the French revolutionaries wanted to introduce a decimal division of the day, and of angles.

The traditional units proved too difficult to replace. Nevertheless, the new ones were used in many works of great importance — notably, Laplace's Mécanique Céleste . And, as Nathaniel Bowditch chose to retain this usage in his heavily annotated English translation of it, we occasionally still need to cope with those revolutionary units.

The centesimal units

In the centesimal system, a right angle is divided into 100 centesimal degrees; each centesimal degree, into 100 centesimal minutes; and each centesimal minute into 100 centesimal seconds. (Centesimal degrees are also known as grads , grades , or gon .)

This creates a treacherous difficulty for readers who are familiar with the ordinary sexagesimal system, because the degrees (in particular) are almost  the same size as the usual ones. There is not enough difference for the change of units to be immediately obvious — a problem exacerbated by Laplace's use of the usual terms (degrees, minutes, and seconds) without the qualifier “centesimal”.

On the other hand, the discrepancy amounts to about a factor of two at the minutes, as centesimal minutes are (9/10)×(6/10) or 0.54 as large as minutes of arc. And it's about a factor of 3 in the seconds; so when Laplace remarks that the refraction at 50° zenith distance is 186″,728 (see p. 530 of Bowditch's version), or that (as on p. 477 of Bowditch's translation) the horizontal refraction is about 6500″, the modern reader immediately has a sense that something is wrong.


It's easy enough to convert, once you're aware of the need to, and know the simple conversion factors. And the UNIX units utility even offers conversions to and from “centesimalminutes” (and seconds) and the usual arcminutes, degrees, etc. Of course, typing those long names is a nuisance.

But it becomes still more of a nuisance if you want to convert some long mixture of degrees, minutes, and seconds from one system to the other.

So, here's a simple JavaScript converter. Just type in the values in one set of boxes, and read their equivalents in the other units.

corresponds to

of arc


Copyright © 2006, 2009, 2020 Andrew T. Young

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