340 A SHORT HISTORY OF SCIENCE 



and flattened in proportion to the distance of their objects from the 

 mirror. . . . Yet every straight line or plane in the outer world 

 is represented by a straight [ ? ] line or plane in the image. The image 

 of a man measuring with a rule a straight line from the mirror, would 

 contract more and more the farther he went, but with his shrunken 

 rule the man in the image would count out exactly the same number 

 of centimeters as the real man. And, in general, all geometrical 

 measurements of lines and angles made with regularly varying images 

 of real instruments would yield exactly the same results as in the 

 outer world, all lines of sight in the mirror would be represented by 

 straight lines of sight in the mirror. In short, I do not see how men 

 in the mirror are to discover that their bodies are not rigid solids and 

 their experiences good examples of the correctness of Euclidean 

 axioms. But if they could look out upon our world as we look into 

 theirs without overstepping the boundary, they must declare it to 

 be a picture in a spherical mirror, and would speak of us just as we 

 speak of them ; and if two inhabitants of the different worlds could 

 communicate with one another, neither, as far as I can see, would be 

 able to convince the other that he had the true, the other the dis- 

 torted, relation. Indeed I cannot see that such a question would 

 have any meaning at all, so long as mechanical considerations are 

 not mixed up with it. 



IMAGINARY NUMBERS. The solution of algebraic equations 

 had always been hampered by the seeming impossibility of per- 

 forming the inverse processes involved. The equation x + 5 = 

 could not be solved before negative numbers were known ; and the 

 equations 2 x = 5 and x z = 2 would be equally insoluble without 

 fractions and irrational numbers. Such equations as x 2 + 1 = 

 still remained a stumbling-block at the beginning of the nineteenth 

 century. Gauss first pierced the mystery and released algebra 

 from its traditional restriction, proving that an equation of any 

 degree has a corresponding number of roots of the form a + 6"^ 1 

 a discovery of far-reaching importance not merely for higher 

 mathematics but even for electrical engineering. 



That this subject [of imaginary magnitudes] has hitherto been 

 considered from the wrong point of view and surrounded by a mysteri- 

 ous obscurity, is to be attributed largely to an ill-adapted notation. 



