4 C. Barns — /Spiral Goniometry in its 



prolonged and rotated in opposite directions until their initial 

 radii subtend the 'same angle as the sector of the index. The 

 limiting case is again reached when the sector vanishes. The 

 reading is here made from the center outward. If however 

 in figure 1, the area AEA' had been tilled with concentric 

 circles and the area AD' A' left blank, the readings for both 

 positive and negative angles would be made from the circum- 

 ference inward. It is in the direction of further progress to 

 obtain two such readings simultaneously. 



4. Repetition. — In figure 3 the spirals terminating the ends 

 of the partial circles on the dial are t no longer symmetric, but 



are so circumstanced that one spiral may be brought into coin- 

 dence with the other by rotating it about the center a stated 

 number of degrees. In such a case if the reading of positive 

 angles be from without inward, the reading of negative angles 

 will be from the center outward. It is clear also that angles 

 of any magnitude may be read by proportionately increasing 

 the size of the dial. The same result may, however, be reached 

 within a reasonably small area of dial by methods of repeti- 

 tion such as the following ; and these methods are particularly 

 appropriate for the present purposes, because they incidentally 

 afford the means for reducing the error of reading to the 

 smallest possible value. §6. 



In figure 3 let the spirals be 90° apart, and let each admit of 

 a direct/ reading (as above shown) between 0° and 180°. Con- 

 formably with this arrangement let the index BB' be a full 

 circle with a right-angled sector B'C'D cut out of it. The 

 initial position of the sector on the dial is indicated by the 

 dotted lines A'Cd. If therefore the index rotates on the dial 

 counter-clockwise, the measurement is made in virtue of the 

 spiral Cba, and the partial circles will gradually emerge from 

 behind the index until with 180°, the ends of all are visible. 

 On further rotation the circles beginning with the innermost 



