ZOOLOGY AND BOTANY, MICROSCOPY, ETC. 449 



measuring the dilieclral angle of microscoj)ical crystals depends on the 

 principle that the angle of inclination of two planes can be easily calcu- 

 lated when the positions have been determined of six points in space, of 

 which three (not collinear) lie in one of these planes. For the measure- 

 ment of the rectangular co-ordinates of these points, Wertheim used (1) 

 an eye-piece with cross wires ; (2) a fine division on the head of the 

 micrometer screw of the Microscope ; and (3) an object-stage movable 

 by screws in two directions at right angles. 



Dr. S. Krysinski considers that the practical application of this 

 method can lead to no accurate measurements. In the first place the 

 X and y co ordinates cannot be exactly determined, since they are 

 measured by turning the screws on the movable stage, by which no pre- 

 cision can be obtained. The addition of stage micrometer screws would 

 render the instrument too costly and complicated. To avoid this 

 difficulty he proposes to use the screws only for producing the movement, 

 and to effect the measurements by means of the eye-piece. For this 

 purpose the eye-piece micrometer of Hartnack has been modified in the 

 following way. 



On the eye-piece fitting, about 12 mm. from the lower end, is a metal 

 drum, 55 mm. in diameter and 10 mm. in height, which consists of two 

 cylinders rotating in one another, of which the under is rigidly connected 

 with the lower end of the eye-piece, while the upper, connected with the 

 upper part of the eye-piece, is movable on the under part. In this under 

 part are fixed the cross wires, and in the upper, just above the cross 

 wires, the micrometer scale. This scale, movable to and fro in a guide by 

 means of a projecting screw, consists of a right-angled triangle, of which 

 one of the sides containing the right angle is exactly ten times as long as 

 the other, and is divided into 100 equal divisions, with each division 

 mark perpendicular to it, and equal in length to the shorter side. It 

 follows from this construction that the segment of the division line cut 

 off between the hypotheuuse and the long side is equal to a tenth part 

 of the corresponding segment of the long side. By means of two indices 

 on both parts of the drum and catch-spring, care is taken that the long 

 side of the scale can be brought at once into a position parallel or at 

 right angles to one of the cross wires. Lastly, on the periphery of the 

 drum is a corresponding vernier. 



If the size of the microscopical object to be measured does not exceed 

 the value of ten divisions, the measurement is effected by first bringing 

 the long side of the scale into exact coincidence with an edge of the 

 object, and then by means of the screw parallel to the long side pushing 

 the scale along until its hypothenuse cuts the object in the diametrically 

 opposite point. The division mark of the scale passing through this 

 point of contact then gives directly the length of the object. This kind 

 of measurement, which the author distinguishes as " Einkeilung," is 

 executed on any given point of the field of view. When, however, the 

 diameter of the object exceeds the length of ten divisions the long side 

 must be brought into coincidence with it, and the length read off directly. 

 On account of the unreliability of the table supplied by opticians, the 

 author strongly insists that a table of values of the scale divisions should 

 be independently made out for each objective. 



The author theu describes in what way with this instrument the 

 X and y co-ordinates of a point in space can be easily and simply 

 determined, and then the z co-ordinate by means of the micrometer-screw 



