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II. 
THE USE OF THE PROTRACTOR IN FIELD-GEOLOGY. 
By ALFRED HARKER, M.A., F.G.8. 
[COMMUNICATED BY PROFESSOR SOLLAS. | 
[Read Novemper 16; Received for publication NovemBeER 18, 1892; Published 
March 25, 1893. | 
In 1884 I showed that many of the little geometrical problems 
that arise in field-geology and mining can be solved graphically 
with the aid of a protractor. As these earlier notes have been 
found of service in some quarters, it may be useful to give the 
following simplifications and extensions of the methods there 
followed. 
The protractor employed is of the ordinary oblong form, and 
should be long in comparison with its breadth, so that the long 
edge may embrace as large an angle as possible, say 150°. This 
edge should be graduated in both directions from a zero-point 
in the middle. The distance from this zero-point of the mark 
corresponding to any angle will then be equal to the tangent of 
the angle, the breadth of the protractor being taken as unity, and 
this system of graduation may therefore be referred to as the 
tangent-scale. For some purposes it is convenient to have also a 
second system of graduation, in which the figures decrease both 
ways from 90° in the middle, and this we shall call, for a similar 
reason, the cotangent-scale. 
Another instrument that will be occasionally of use is an 
ordinary scale of equal parts, graduated so that its unit is equal to 
the breadth of the protractor. Used in conjunction with the 
1 Geological Magazine, Dec. 3, 1884, vol. i., p. 154. 
