Simpler Methods in Crastallography. 355 



until they touch the prolongation of mO, nO\ sign these 

 points respectively jp, q. Then the pairs vip, nq are points 

 in the circumference of the great circle required. Erect 

 perpendiculars upon the middle of the chords mp, nq, and 

 produce them until they meet. The point thus determined 

 is the centre required. 



70. In conclusion, it only remains for me to remark that 

 this very elementary and somewhat lengthy description is 

 not put forward for the use of mathematicians, who already 

 know (or can easily do so, if they choose) how to work out 

 all these problems by methods much neater, briefer, and, 

 in some cases, more precise, than those advanced here. 

 Many subjects have been intentionally left unreferred to, 

 simply because their complexity might cause confusion at 

 the outset. After doing a little practical work on the lines 

 here laid down, the student will gradually feel his way to 

 dealing with maps of crystals belonging to the Anorthic 

 System, and to draw even these by geometrical methods and 

 PROPORTIONAL MEASUREMENT, such as I have dealt with in 

 the simpler cases here described. 



71. Lastly, it will be observed that I have rarely made 

 use of angular measurements. That has been intentional. 

 In Crystallography we are concerned with proportions and 

 directions much more than with angular measurements. Any- 

 one who will glance at the Tables at the end will see that 

 these factors in Crystallography are certainly not com- 

 mensurate with those commonly employed in angular 

 measurements in general. They belong, in fact, to a dif- 

 ferent order, which has its basis in the molecular constitu- 

 tion of the crystal. To lose sight of this fact is, I think, to 

 close our eyes to the gradually increasing light which is 

 rendering new paths in molecular science visible to our view. 



The Maps appended to this — a gnomonogram of Quartz, 

 a c projection of Barytes, and both I and c projections of 

 Orthoclase — will exemplify the projections described in 

 the paper. They are part of a set drawn for Heddle's 

 " Mineralogy of Scotland," and are inserted here through the 

 courtesy of Mr Alex. Thoms, of St Andrews, and the 

 publishers of the work, the Messrs Douglas, of Edinburgh. 



