A. HarJier — Graphical Metlwds in Field Geology. 157 



Use of the Protractor. 



In accordance with the foregoing we require some convenient 

 means of laying down at once on a diagram a length proportional to 

 the tangent of any given angle. Such a means is furnished by a 

 common protractor of the oblong form, graduated along a straight 

 edge. This instrument serves not only as ruler, scale and protractor, 

 but also as a rough table of tangents and cotangents. For the last 

 purpose it is convenient to have it graduated with a second set of 

 figures in addition to those usual on protractors, the second set 

 increasing both ways from zero at the middle point Z of the straight 

 edge. Take the breadth of the protractor OZ as unity ; if the point 

 P corresponds to say 35° reckoned from Z, then the angle ZOP is 

 35° and ZP is tan 35° (Fig. 5). It will be seen that for high angles 

 (for angles greater than 60° in the figui'e) the application of this 

 principle is less ready : for instance to lay down tan 70°, it is 

 necessary to dot in the positions of and Q and produce the lines 

 ZP and OQ to meet in B, then ZR is the required length. The 

 longer the protractor is in proportion to its breadth, the more 

 degrees will be marked on the straight edge directly, but since the 

 scale on which the tangents are represented depends on the breadth 

 OZ, this should not be too small; say double the dimensions of 

 Fig. 5. There would be some advantage in having and ^not in 

 the middle, but at one end of the protractor. 



In the constructions which follow, a straight line will be said to 

 represent the dip of any given strata when it is drawn from a fixed 

 line Z — 



(1) in the direction of the said dip, like the arrows on a common 



geological map, and 



(2) of length corresponding to the amount of the dip, that is, the 



length given on the edge of the protractor from zero to the 



proper degi"ee-mark. 

 In this way the observations of the compass and clinometer are 

 graphically recorded by one stroke, the protractor being used for 

 the former purpose in the usual way, and for the latter in the manner 

 described above. Similarly the slope of the ground or the inclination 

 of any axis may be indicated, both in direction and amount, by a 

 line on the diagram, the lines being always drawn from a fixed point 

 of reference, Z. 



Practical Applications. 



(i.) Given the direction and amount of full dip, to find the appai'ent 

 dip in any given direction. 



In Fig. 6 draw ZA to represent the full dip, ZB in the other given 

 direction, A B perpendicular to it ; then ZB, the part cut ofi", repre- 

 sents the apparent dip in magnitude as well as direction, and the 

 amount of apparent dip may be read off by applying the edge of the 

 protractor. The proofs of this and the two following constructions 

 are evident from the second of the equations (a). 



(ii.) Given the apparent dip in one direction, and the direction 

 of full dip, to find the amount of the latter. 



