26 The Three Sections, Tmigencies, 



these conditions, but part of one, it follows that, in all 

 inquii'ies in which the theorem is used, the results must be 

 defective in generality. 



The complete theorem which should replace it is as 

 follows : — 



If aa a.nd bb are fixed straight lines tlu'ough fixed points 

 A and B, should cc and dd be any other pair of straight 

 lines through A and B, making the angle cc right to aa equal 

 to the angle bb I'ight to dd, then "^dll the locus of the inter- 

 section of cc and dd be a rectangular hyperbola through the 

 fixed points A and B. 



Indeed, a due consideration of the requirements of a com- 

 plete logic, or of the laws of nature, will show that the im- 

 provements are necessary to the explicit enunciations of 

 implied operations, and confer precision and generality on 

 most important theorems; and are therefore, in so far as these 

 are understood, a correct step in the advancement of pure 

 science. 



Finally, I think it right to remark that, in indicating- 

 opposite formations of magnitudes, I have purposely avoided 

 the terms positive and negaiive, as so many meanings are 

 given to these words by metaphysicians and others. Besides, 

 the introduction of rigid and left renders the language more 

 elegant, and often affords important advantages in allowing 

 us to decide, according to circumstances, whether right or 

 left should be indicated 1)y plus or minus. 



Martin Gardiner, C.E. 



DEFINITIONS. 



1 . If a straight line, which we may conceive produced to in- 

 finity in its primitive directions, be supposed to become rigid, 

 and one point of it to be permanently fixed, the rigid line 

 being otherwise capable of movement in any plane in which 

 it may lie, then it is evident that there are but two ways of 

 revolving the line in this plane ; one being by means of a 

 '^ right " rotation, and the same as that in which the hands of 

 a Avatch move if the dial-plate be towards us, in the plane ; 

 and the other being by means of the contrary, or " left" ro- 

 tation. 



2. If AxV and BB be two straight lines, and I their point 

 of intersection, then the ''angle lA rigid to B," means the 

 angle formed at I by a rigid line having I as a fixed pivot, and 



