4i4& Cambridge Philosophical Society. 



CAMBRIDGE PHILOSOPHtCAL SOCIETY. 



[Continued fi-om vol. vi. p. 73.] 

 Nov. 14, 1853. — A paper was read by Mr. Dobson on the Theory 

 of Cyclones. See Philosophical Magazine, vol. vi. p. 438. 



Also, on the Storm-tracks of the South Pacific Ocean. See Phi- 

 losophical Magazine, vol. vii. p. 268. 



A communication was made by Mr. C. C. Babington on the use 

 that has been made of the mode of growth to distinguish nearly 

 allied Species. 



Nov. 28. — A paper was read by Mr. Wedgwood on the Geometry 

 of the fu'st th»ee books of Euclid, synthetically demonstrated from 

 premises consisting exclusively of definitions. 



In a treatise* published by the author a few years ago, definitions 

 founded on relations of direction were indicated as exhibiting the 

 ultimate analysis of the conceptions of straightness and parallelism 

 in lines, and of planeness in surface ; and in proof of the adequacy of 

 these definitions as the basis of a cornplete system of geometry 

 without the aid of axioms or any other assumption whatever, they 

 were employed in demonstrating the principal propositions necessary 

 to place the student on the ground occupied by the definitions and 

 axioms of the ordinary system. If the basis thus built in underneath 

 the old foundations of the science had been complete in every nook 

 and corner, nothing more would have been required in order to rest 

 the entire demonstration on the single principle of definitions. So 

 long, however, as any step in the process, however subordinate, was 

 left to be supplied by others, there always would be room for sus- 

 picion that the assumption in reasoning which was speciously plas- 

 tered over in one place might be secretly undermining the system in 

 another. The reform, moreover, of the premises in geometry is a 

 problem on which such an infinity of thought has been spent, and 

 to which so many answers, more or less plausible, have been offered, 

 that nothing short of a complete exposition of a consistent scheme 

 of demonstration can be expected to carry comdction in the validity 

 of a fresh solution. The object of the present paper is accordingly 

 to complete the task undertaken in the foregoing publication by a 

 formal statement of the other definitions required in connexion with 

 those of straight and parallel lines and plane surface, and by a rigid 

 demonstration from these premises of the steps intervening between 

 those and the premises of the ordinary system ; and in additional 

 proof of the fundamental character of the proposed analysis, the de- 

 monstration is carried on through the geometry of the three first 

 books of Euclid by direct reasoning, without resort to the compara- 

 tively unsatisfactory method of ex ahsurdo proof, which, although 

 equally conclusive as to the necessity of the result, yet always leaves 

 a hankering in the mind for an answer why the case must be as the 

 demonstration shows that it cannot avoid being. 



* The Principles of Geometrical Demonstration deduced from the ori- 

 ginal conception of Space and Form. Taylor and Walton. 1844. 



