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Art. IX. — Improvements in Fundamental Ideas and Elementary 

 Theorems of Geometry . By Martin Gardiner, Esq., GE. 



[Read before the Institute 13th July and 3rd August, 1859.] 



Many eminent mathematicians have decided on the necessity 

 of introducing a systematic motional philosophy into the 

 higher Geometry. 



Monsieur M. A. Chaucy, as far back as 1846, brought the 

 subject before the Academy of Sciences of Paris, in a paper 

 entitled " Memoire sur les avantages que presente dans la 

 Geometrie Analytique Femploie de facteurs propres a indiquer 

 le sens dans lequel s'effectuent certaines mouvements de 

 rotations, et sur les resultats construites avec les cosinus des 

 angles que deux systemes d'axes forment entre eux." 



In this memoir, after explaining what has been done in 

 his algebra as to the legitimate restrictions of notation, he 

 says : — " This expedient, already generally adopted by 

 geometers, has caused the disappearance of the uncertainty 

 which the interpretations of certain formulas presented, and 

 the contradictions which symbolical calculations seemed to 

 present." 



Dr. August Wiegand, professor in the University of Halle, 

 in a note to the editors of the Mathematician, in 1847, 

 entitled " Generalization of the leading operations of Arith- 

 metic in reference to Geometry," shows, by a motion of 

 rotation, how we may find rational interpretations for what 

 are known as imaginary expressions. 



Something of a similar method is given in De Morgan's 

 Double Algebra, published in 1849 ; and more extended ideas 

 concerning these and kindred subjects are the ground-work 

 of Sir William Hamilton's celebrated Quaternions. 



Writers on Trigonometry have indeed been universally 

 compelled to take cognizance of directions on straight lines 

 and formations of angular magnitudes ; and Professor Chasles, 

 in his " Geometrie Superieure," shows, beyond all doubt, 

 that great advantages result from introducing such concep- 

 tions into pure geometrical reasonings. 



All this is well known to mathematicians, and many are of 

 opinion that a more liberal motional philosophy should be 

 formally introduced into the common primary elements. This, 

 to some persons, may appear an unnecessary innovation; 



