# Centesimal angular measure

## Introduction

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.

## Conversions

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

Copyright © 2006, 2009, 2020 Andrew T. Young

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