Harxer—The Use of the Protractor in Field-Geology. 13 
latter, it becomes a scale of secants or cosecants, according as we 
_ take the angle on the first or second of the scales of the protractor. 
The two instruments, as thus combined, are shown in fig. 1, the 
details of the graduations being omitted. 
Fie. 1. 
The diagrams by which the various questions are solved are for 
the most part of the nature of “ gnomonic projections,” but they 
are drawn with the protractor alone, and the rationale of each 
solution is easily grasped without any mathematical knowledge. 
The plane of the paper is supposed to be horizontal, and it will be 
seen that the inclination to this of any inclined plane can be com- 
pletely represented in direction and amount by a straight line 
drawn from a fixed point O on the paper. For this purpose we 
take a plane parallel to the one in question, and passing through 
a fixed point C, not in the paper but beneath it, and at a distance 
below O equal to the breadth of our protractor. Through O we 
imagine a straight line drawn normal or perpendicular to the 
inclined plane and meeting the paper at the point P; then OP 
may be taken to represent the inclination to the horizon of the 
given plane. In practice this line is laid down on the diagram by 
simply laying the protractor on the paper with its zero-point at O 
and its edge in the given direction of inclination, and taking on 
the tangent-scale the given angle of inclination (fig. 2). To 
realize the explanation of the method, imagine the protractor 
_ placed instead in a vertical plane beneath the paper, its zero-point 
at O, and its graduated edge along OP; the centre from which 
the graduations radiate will then be at C, 4 point through which 
our inclined plane was drawn (compare fig. 3, supposed to be in a 
vertical plane, with fig. 2, the diagram actually used). In this 
