142 S. L. Penfield — Stereographic Projection. 



de Janeiro, pages 123 and 124, results are often within a quarter 

 of a degree of the truth, seldom more than half a degree out. 



Conclusion. — Starting with the very simple idea of making 

 a protractor with stereographically projected degrees upon its 

 base line, one possibility after another has presented itself, 

 involving, as stated at the outset, no new mathematical princi- 

 ples, but leading, as the writer believes, to very important 

 results. The main features of this article are the development 

 of the scales shown in figure 3, by means of which it is pos- 

 sible to make stereographic projections very quickly and accu- 

 rately; and the discovery of the transparent stereographic 

 protractors. The writer can not learn that any one has ever 

 made use of such protractors, and accordingly has made appli- 

 cation for patents to control their manufacture and sale. Up 

 to the present time, protractors have been made to conform to 

 a circle of 14 cm diameter only, but any demand for protractors 

 and scales based on a larger circle can be easily satisfied. 

 With accurately engraved stereographic plates and protractors, 

 based on a circle of 12 or 18 inches diameter, for example, 

 very accurate plotting and measuring could be done. Many 

 have expressed surprise that on a circle of only 14 cm diameter, 

 representing a whole hemisphere, such accurate measurements 

 can be made as those cited in this article. It may be stated, 

 however, that the method in itself is absolutely exact, and 

 errors are wholly due to inability to work accurately on so 

 small a scale as the one adopted, and with plates and pro- 

 tractors graduated to degrees only. Based on a circle of 14 cm 

 diameter, a stereographically projected degree at the periphery 

 and at the center is represented by distances of about 1-2 and 

 0*6 mm , respectively. Of these distances, one ought to be able 

 to estimate about one tenth of the former and one quarter of 

 the latter ; hence, a point carefully located near the periphery 

 ought not to be more than one tenth of a space, or six minutes, 

 out of the way, and the chances are that it will be correct 

 within three minutes, while near the center a point ought to 

 be located at least within a quarter of a degree and probably 

 within ten minutes of the truth. The possibilities as thus 

 stated correspond closely with actual tests of the method. If 

 the plotting is of a simple nature and made near the periphery, 

 and especially if the measurement can be made with one read- 

 ing of the protractor, the result is rarely more than 6' and 

 averages less than 4' from the truth. If, on the other hand, 

 the plotting is done near the center of the circle, and measure- 

 ments necessitate two readings of the protractor, errors amount- 

 ing to a quarter of a degree must be expected, but the average 

 will be less than 10 / from the truth. 



