What did our Forefathers see in a Microscope? 429 



that any other besides No. IV. was actually constructed. No. I. is 

 very similar to No. IV. in design, so also is No. II. to No. III., but 

 between the pairs there is a slight variation. The power of all 

 four is the same, but the aberrations in the order of their numbers 

 are proportional to the following figures : 3 ■ 75, 3 * 82, 1, and 9 ■ 6. 

 Therefore No. III. is the best, and No. IV. the worst. The radius 

 of the concave of No. IV. is 2*064; with this single exception 

 the radii of all the other mirrors is 2'0 in. The distances be- 

 tween the mirrors is as follows : 1 • 6, 1 ■ 0419, 1 ■ G558, and 1 ■ 6712. 

 The distances of the object from the concave mirrors are 1*7143, 

 1-7184, 1-7184, and 1-7719. The diameters of the concave 

 mirrors measure 1 • 1 in., and the convex ■ 45 in. These data 

 are sufficient to enable anyone either to make any of the four in- 

 struments or to calculate their aberrations. These four Microscopes 

 possess not only much brass and glass, but also mathematical 

 interest, consequently they have been largely copied in other 

 scientific works, but it does not appear that some of their more 

 salient features have ever been noticed. Thus, if Nos. II. and III. 

 are examined, it will be seen that No. II. has nearly four times as 

 much aberration as No. III. Wherein, then, does the variation lie 

 that causes this difference in the aberration ? A difference of ■ 0139 

 in a distance between the mirrors, which exceeds 1£ in., is the 

 only optical difference between these Microscopes, and it is hardly 

 probable that a quantity so small should have so profound an effect. 

 A minute examination of No. III. was then undertaken, and a 

 small slip was discovered in the learned doctor's work. In fig. 52 



Q VC 



Fig. 52. 



let E C be the axis, and A C the upper half of the concave mirror, 

 E C its radius 2 in., Q the object, and Q A a ray meeting the mirror 

 in A, from which point the perpendicular A V is dropped. We 

 have given, the angle AEV 15° 47', E A 2 in., and C Q 1-7184 in. 



Now A V = r sin AEV 



and CV=r(l -cos AEV) 



and VQ = CQ -CV 



Then tan A QV = -^ ^ 



and AQV= 18° 19' 11" -5 



the aperture of the lens being N.A. ' 314 ; Dr. Smith, however, gives 

 it as 18° 26' 6", or 6' 54" too much. In No. II. the error is the 



