hy Means of Graphical Methods, 283 



A list of measurements made with the small circle protractor 

 is given in the table on the foregoing page. The average of 

 the errors is 11', with errors of more than a quarter of a degree 

 in only three instances. In figure 30, the angle between the 

 great circles J,c and 5,j? determines v (compare figures 6 and 

 8), which was found to measure 29° 25'. The angle i^, plotted 

 from unity on the a axis, as shown in figure 32, gives another 

 method for determining the length of the c axis ; in this case 

 G was measured as 0*545, calculated 0*550. 



Conclusion. 



In the foregoing pages the attempt has been made to indi- 

 cate not only the methods of plotting problems in the several 

 systems, but, also, to give an idea of the accuracy of the results 

 thus obtained. In judging the accuracy of the results it must 

 be kept in mind that the scale employed was quite small, the 

 engraved ch'cle being only 14^"^ (5|- in.) diameter. Each prob- 

 lem presented was worked out from fundamental measure- 

 ments, as would have been the case if done by numerical cal- 

 culation, and the results indicate that fur all practical pur- 

 poses graphical methods are in every way satisfactory. The 

 lengths of axes have frequently been obtained correctly to the 

 third place of decimals, and never varied more than one in the 

 second place, and measurements of arcs and angles by the 

 stereographic protractors have been reasonably close to the 

 calculated values in all cases. After becoming familiar with the 

 principles involved in dealing with the projection — and they 

 are not difficult to understand — any problem may be worked 

 out without the use of prescribed formulas, or tables of any 

 kind, and this the writer believes is one of the great advan- 

 tages of the method. Then, too, it is a very simple step to 

 proceed from a graphical to a numerical solution, for the prin- 

 ciples involved are identical, and it is only necessary to apply 

 the formulas of spherical and plane trigonometry in order fco 

 obtain the desired results by calculation. The writer has never 

 made use of formulas for solving even the most complex prob- 

 lems of crystallography, other than those needed for the gen- 

 eral cases which arise in dealing with spherical and plane 

 triangles, and it is his belief that the too general use of pre- 

 scribed formulas is, if anything, a hindrance to true progress 

 in crystallography. Graphical methods will also be found to 

 have very decided value as giving a ready means of checking 

 the results of numerical calculations. During the past two 

 years, since they have been in use, numerous cases have arisen 

 where mistakes in calculation have been made, at times 

 amounting to less than a degree, but they have been almost 



