39 



Some observations are lastly added concerning the nature and 

 situation of the ciliary processes in various animals; also on the 

 nature of the marsupium nigrum of birds, and the horseshoe-like 

 appearance in the choroid of fishes ; both which have improperly 

 been termed muscular, the former being a mere duplicature of a 

 membrane which may be unfolded ; and in the latter the whole mass 

 being evidently of an uniform texture, the fibrous appearance which 

 has misled some former observers being the effect of transverse fis- 

 sures, or cracks, which may easily be mistaken for filaments. 



The lecture concludes with a few observations on the bony scales 

 of the eyes of birds, to which the author denies any concern in 

 changing the focus of the eye ; and on a cavity observable in the 

 eyes of some insects which has been supposed to be in some measure 

 subservient to this purpose. 



On the necessary Truth of certain Conclusions obtained by Means of 

 Imaginary Quantities. By Robert Woodhouse, A.M. Fellow of 

 Caius College. Communicated by the Rev. S. Vince, A.M. Plumian 

 Professor of Astronomy in the University of Cambridge. Read 

 January 8, 1801. {Phil. Trans. 1801,;?. 89.] 



The object of this paper is to show, that we maybe assured of the 

 justness and accuracy of conclusions obtained by means of imaginary 

 quantities, without verifying such conclusions by separate investiga- 

 tions, or without inferring their truth from analogy. In the first part 

 the author premises at some length certain arguments, to show that 

 the operations with impossible quantities must have a logic equally 

 strict and certain with the logic that appertains to real quantities, 

 and that the aid obtained by these quantities would be perfectly use- 

 less if such conclusions rested only on the frail basis of analogy. 



The author proceeds next to show that operations with imaginary 

 quantities are by no means mechanical, but that they are conducted 

 according to the rules of strict and rigorous logic ; and that, although 

 strictly speaking no proposition concerning them can be true or false, 

 yet, after the demonstrations of certain formulae for real quantities, 

 demonstrations with impossible quantities may be legitimately and 

 logically conducted. The series, for instance, for the development 

 of an exponential, when the exponent is an impossible quantity, can 

 never, independently of certain arbitrary assumptions, be duly esta- 

 blished ; and yet, when the exponent is the sign of a real quantity, 

 the formula for the development may be rigorously proved. With 

 regard to demonstration, it is shown, as in the case of real quantities, 

 it actually proceeds by a series of transformation, each proved to be 

 the same as the foregoing, not by any arguments grounded on the 

 properties of real quantities, but by reference to the forms certain 

 abridged symbols are made to represent, and to the nature of the 

 operations directed to be performed with certain signs. 



After thus establishing the principle by which operations with 

 imaginary characters are regulated, the author shows its efficacy and 



