Xi^A^^-'^^On a Solution of the Theory of Parallels, 371 



decomposed by alkaline substances in a similar manner as gly- 

 oxylic acid, a method for the production of all the acids homo- 

 logous to oxalic acid is given, viz. 



2(C3H6 04) + 3KH0 = C3H5K03 + C3H2K204+4H20. 



The results of the examination of the aldehydes, produced by 

 the action of nitric acid on alcohol along with the glyoxylic acid, 

 will be communicated in another paper. 



Queenwood College, near Stockbridge, Hants. 



XLV. On a Solution of the Theory of Parallels, from the Defi- 

 nitions of Euclid, without deviating from the ordinary Principles 

 of Geometrical Proof By J. P. Hennessy*. 



IN my remarks on Professor Stevelly^s paper, I ventured to 

 call attention to the fact that Euclid's definition of a square 

 bore an analogy to the definition of parallels in which an equality 

 of angles to a constant quantity is made the difiierentia. A con- 

 sideration of the origin and nature of this analogy has led me to 

 perceive, that in the former definition, as well as in the latter, 

 an assumption is made, which, when legitimately employed in 

 connexion with other data furnished by Euclid, leads to a strictly 

 geometrical solution of the doctrine of parallels. 



It has often been remarked that Euclid's definition of a square 

 was partly based on what has been called an unnecessary assump- 

 tion. This so-called unnecessary assumption is said to consist 

 in the statement, that a four-sided equiangular and equilateral 

 figure has its angles all right angles ; inasmuch as that this pro- 

 perty of the figure, instead of being implied in its definition, 

 might have been made the subject of subsequent demonstration. 



One writerf, who aspired to make geometry a perfectly exact 

 science, seems to have been so impressed with this idea, that 

 he thought it necessary to remove the definition of a square from 

 its usual position among the premises of Euclid, and to place it 

 between the 34th and 35th propositions of the first book. But 

 whilst the assumption that Euclid has made has frequently been 

 noticed, its legitimate consequences have not attracted any atten- 

 tion. These consequences appear, however, to be of no small 

 importance. 



The various attempts to solve the theory of parallels may be 

 divided into three classes. (1) Those which essentially consisted 



* Communicated by the Author. 



rf^ ^ .'iqiL;4iil t Colonel Perronet Thompson. 



