MATHEMATICAL INVESTIGATIONS. 241 



altered in shape. In the experiment of the bent pole, for 

 example, the part which is beneath the water seems bent 

 upward, but not otherwise altered that is, the rays of 

 light being all refracted in coming out of the water, with- 

 out any change in their relative positions, but only in the 

 direction in which the whole system enters the eye, the 

 submerged pole is altered in position only, and not in 

 shape. 



Of course, as there is absolutely no limit to the forms of 

 curved surfaces, nor to the changes in the character and 

 condition of the different beams and pencils of light falling 

 upon them, the phenomena resulting both from reflection 

 and refraction are infinite in number and variety. To un- 

 derstand the subject fully, in all its possible ramifications, 

 must, of course, transcend the power of the human mind; 

 for the circle of phenomena widens and expands in every 

 direction, and the facts run into an infinity of complicated 

 details, where, of course, the finite powers of the human 

 mind can not follow them. The study has, however, been 

 carried very far by mathematical opticians ; and the inves- 

 tigations which they have made, and the engravings which 

 have been executed to illustrate the results, fill volumes. 

 It would not be possible, in such a work as this, to give 

 even a summary of these results. 



There is, however, both in the case of reflection and re- 

 fraction, a simple general principle on which all these re- 

 sults depend, and which it is important that every one 

 should have in mind. This principle is in each case the 

 key to all the phenomena, however complicated, and it en- 

 ables us, if we possess it, to understand a great many won- 

 derful effects taking place every day around us whicli 

 would be otherwise mysterious and unintelligible. I shall 

 explain these two principles as well as I can, first in com- 

 mon language, and then give the mathematical expression 

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