1842.] On the Theory of Angular Geometry. 233 



In Col. Peyronnet Thomson's* Geometry without Axioms, these un- 

 limited spaces are for the first time distinctly enunciated. The fourth 

 edition of that work contains the following paragraphs : — 



" The latest innovation has been the assertion, that an angle (or 

 " the thing spoken of by Geometers under that item, whether they knew 

 " it or not) is a plane surface." Pref. page x. 



" The plane surface (of unlimited extent in some directions, but limit- 

 " ed in others) passed over by the radius vectus in travelling from one 

 " of the divergent straight lines to the other, is called the angle between 

 " them." " Hence," adds the Colonel, " angles are compared together 

 " by their extension sideways only, without reference to the greater or 

 " smaller length of the straight lines between which they lie." 



After making this decided step however, Colonel Thomson stops ; the 

 definition is registered in his Book of Nomenclature, but he establishes 

 the properties of angles by the old criterion of supposition. Not only 

 indeed does the definition remain a dead letter, but the gallant radical 

 reformer in Geometry as in Politics, 



*' Astonished at the sound himself had made," 



virtually doubts its correctness, when at page 14, reviewing the proof 

 of M. Bertrand, he says, " All references to the equality of magnitude 

 " of infinite areas are intrinsically paralogisms." 



The edition in question of " Geometry without Axioms," was reviewed 

 in the 13th No. of the Journal of Education, in an article which betrays 

 the sparkling pen of Professor De Morgan. The part relative to angles 

 is noticed thus : " His is the first work, which we know, in which this 

 " idea (that of a plane surface) is fairly brought before the beginner. 

 " We suspect he is quite right, and that in the extension of the term 

 " equal to unlimited figures which coincide in all their parts, will be 

 "found the ultimate resting point of the theory of parallels. Had our 

 " author stuck close to his definition, the demonstration of Euclid's axiom 

 " given by M. Bertrand, ought to have been sufficient. " After noticing 

 the neglect of Colonel Thompson to make any use of his definition, as 

 well as his attack on unlimited spaces, the reviewer proceeds : " We 

 " wonder therefore that the definition should have been inserted, for it 

 " is in the definition only, and the difficulty which a beginner must find 



* The well known Editor of the Westminster Review, and author of* the Corn 

 Law Catechism. 



