398 Mr. E. A. N. Pochin : 



be used in the ordinary way for making calculations. The 

 highly accurate values, usually published in book form, are 

 not obtained by direct measurement, but by an indirect 

 process which does not at present concern us ; and their use 

 is chiefly restricted to those who desire great precision. For 

 the most part, however, we do not require this extreme 

 accuracy, and continue to work direct from the spiral. The 

 radial lines are cut off to a circle about 0, and the appropriate 

 length of each is written against it. Two such circles are 

 connected by a pivot through the poles, and constitute one 

 form of the familiar " watch calculator," by means of which 

 the desired angles may readily be added or subtracted: — 

 here is a model of this useful appliance. It should be 

 noted that the graduations are not necessarily made from 

 the spiral in fig. 2, but are, like this model, more usually 

 derived from a spiral drawn with the roller set at about 

 68^° — (tan" 1 27r/log e 10). In this position, the roller will be 

 ten inches from 0, after sweeping through 360°. An im- 

 portant advantage arises from this change, to which we shall 

 refer. 



By rolling one of the dials along a straight piece of wood 

 and copying off the graduations, as each comes into contact, 

 we obtain a straight logarithmic scale ; and two of these, 

 when combined, form the device known as a slide-rule. 



Analysis. 

 So far, the accuracy of our conclusions has a purely con- 

 structional basis, and it is very desirable to apply the crucial 

 test of analysis, and satisfy ourselves that no residual errors 

 exist, which might have eluded our most careful measurements. 

 Fig. 3 is a reproduction, on a larger scale, of the spiral 

 shown in fig. 2. The angle AO# has been drawn equal to 

 one radian, OA being one inch, or unity. This angle is 

 supposed to be divided into one million equal parts by lines 

 OB, 00, OD ... which meet the curve at B, C, D .... Also 

 A6 is drawn perpendicular to OB, Be to 00, &c. 



Now, from the mechanical construction of the curve, it is 

 clear that the spiral is, at all points, uniformly inclined at 

 45° to the radius vector ; for the tracing roller was clamped 

 at that inclination to the steel rod. It is also evident that we 

 may, without material error, regard the portions of the spiral 

 AB, BC, &c. as straight lines, and also consider OA equal to 

 Ob, &c. 



Accordingly A = 1 



0B = 0b + bB 

 = 0A + A6. 



