July 24, 1903.] 



SCIENCE. 



109 



uniformly graduated scales, wliieh are es- 

 pecially useful iu problems involving plane 

 trigonometry, and a set of scales of merid- 

 ional parts for various latitudes, each scale 

 representing, say, 10 degrees of latitude. 

 In all eases the scales must be straight and 

 beveled on their edges. It is supposed that 

 we have also a T-square with a uniformly 

 graduated blade. 



RIGHT SPHERICAL TRIANGLES. 



By means of the board and scales we can 

 find such products as cot i X tan c by lay- 

 ing off cot b, according to the scale labeled 

 cot .T, along the base line of the protractor. 

 Let a straight edge, turning about the 



SINE RATIOS IN SPHERICAL TRLVNGLES. 



If two of the three given parts of a tri- 

 angle are opposites, the unknown part op- 

 posite the third given part becomes known 

 through the equality of the sine ratios ; viz., 



sin A : sin a = sin B : sin 6 = sin C: sin c, 

 or 



sin A sin 6 = sin B sin a, etc. 



The process of mechanically solving such 

 problems can be illustrated by taking a 

 particular case or problem; given A, a, B 

 to find b. 



Find a on the scale labeled 2 sin x and 

 direct the scale (pivoted to the board) 

 towards B as found on the margin of the 



pivot, be directed toward value c on the 

 margin. The T-square shows the line ex- 

 tending from the point where cot b was 

 laid off to the straight edge. The distance 

 is the product cot b X tan c. By reading 

 this distance on a scale cos x, or double this 

 distance on a scale 2 cos x we obtain a cer- 

 tain angle which is the value of the angle 

 J. of a triangle right-angled at B. Napier's 

 rules enable one to see at a glance what 

 product is required and how it is to be 

 read. Where tangents and cotangents are 

 involved, the application of this method is, 

 of course, somewhat restricted on account 

 of the length of the scales. 



board; this locates a certain point. Next 

 direct the scale towards A. Find on the 

 scale as now directed a second point whose 

 altitude is the same as that of the first. 

 This is done by sliding the graduated T- 

 square. The reading of the second point 

 on the scale labeled 2 sin x (still pivoted 

 to the board) is the required side b. 



RELATION BETWEEN THE THREE SIDES AND 

 ONE ANGLE. 



In treating the problem— given the three 

 sides to find an angle, or given two sides 

 and the included angle to find the remain- 

 ing side— it is important to consider two 



