18 Mr. John W. Gordon [Feb. 17, 



this central pencil is used to throw upon the screen the magnified 

 shadow — which can be seen from every part of this theatre — of a toilet 

 comb. The coarser teeth throw separate and well-defined shadows, so 

 that in this part of the picture the image is well resolved. But the 

 fine teeth are too fine to yield a resolved picture, and in this part of 

 the image, therefore, the structure is entirely lost. But now, if 

 between the comb and the source of light I introduce the blade of 

 a table knife, with its straight edge held parallel to the teeth of the 

 comb, you will observe that the edge of the shadow thrown by the 

 knife is blurred by a diffraction fringe, and that where this diffraction 

 fringe serves as a background to the picture, the images of the coarse 

 teeth become much stronger than before, and the fine teeth are 

 represented by a fully resolved image. Exactly the same effect is 

 produced in the microscope by the artifice of introducing such a 

 diffraction fringe into the background of the picture which you want 

 to exhibit with improved definition. Figs. 5 and 6 are photographs 

 of the shadows exhibited in this experiment.* 



Assuming the best possible image to be formed of the object 

 exhibited upon the stage, there still remains the problem of seeing the 

 image so provided. To see an image is not quite the same thing as 

 to see a material object, for the object can be seen from many points 

 of view ; the image, if it be an aerial image, will be visible only 

 through a limited angle, and when the image is highly magnified as 

 well as aerial, this limited angle is very limited indeed. Mere limita- 

 tion of this angle of view is of no great consequence to the user of an 

 optical instrument, so long as the beam emitted by the instrument is 

 large enough to fill the pupil of the observer's eye. But Lagrange 

 proved that, in the case of a telescope, the diameter of the emergent 

 pencil is proportioned inversely to the magnifying power of the in- 

 strument and directly to the diameter of the object-glass. In 1874 

 Helmholtz extended Lagrange's theorem to the microscope, showing 

 that in that instrument also the emergent pencil of light has a diameter 

 inversely proportional to the magnifying power. But as the micro- 

 scope cannot be fitted, like the telescope, with an object-glass many 

 inches in diameter, it is not possible to expand the transmitted beam 

 of light by the expedient of using a large object-glass. Helmholtz 

 showed that in the case of a microscope it is the angle under which 

 light from the object is received at the front face of the objective 

 which determines the breadth of the transmitted beam, and he modi- 

 fied Lagrange's formula accordingly, so as to express the law that 

 the emergent pencil of light is proportional directly to the " numerical 

 aperture," or, as he called it, the normal magnifying power, and in- 

 versely to the actual magnifying power of the instrument. 



♦ In the photographs which have been prepared for the illustration of this 

 paper, a thick wire has been substituted for the knife-blade, by which means 

 two diffraction fringes (one at the upper, the other at the lower edge of the 

 wire) are obtained in place of one. 



