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A Note on Capt. Shortrede's Remarks in No. CXXIII. (Page 

 240 ) of this Journal, By S. G. T. Heatly, Esq. 



The subjects of geometry are not the creatures of arbitrary defini- 

 tion. We strive first to attain such a definite conception of them as 

 enables us to see how their properties follow from their nature : we 

 enunciate this conception as well as we can in words, and call it the 

 definition. But we cannot, however, enunciate the process of intuition 

 by which we are conscious of the necessary consequence of any the 

 most rudimentary property. We are compelled therefore to put down 

 this rudimentary property itself; it is termed an axiom. Hence the 

 indispensable appearance of axioms in a system of geometry. 



On these grounds I agree entirely with the position that, in mathe- 

 matical definitions, it is necessary to have a clear conception of the 

 idea, and then to use such words as will convey that conception to the 

 mind of another. It did not appear to me that a clear conception of 

 the idea of an angle is generally entertained ; and I endeavoured to 

 analyse the language commonly held on the subject, so as to detect the 

 peculiarity which impressed itself on the minds of various authors, and 

 to shew that the idea of an angle involved the conceptions — of surface 

 —of determinate extension in the direction of width — of indeterminate 

 extension in the direction of longitude. These are conceptions which 

 every one, sooner or later, finds floating in his mind clearly or obscurely, 

 and if they enable the student to perceive distinctly what he is about 

 when he is discussing angles, it is our business to place them before 

 him in the simplest and most direct form. 



The use made of the word direction arose from the habit of always 

 reducing geometrical magnitudes and positions along fixed axes, the 

 two axes being in this case (I need not say) one bisecting the angle, indi- 

 cating the direction of length — and one perpendicular to it, indicating 

 the direction of width. This appeared to me necessary to embody 

 distinctly the conceptions intended to be impressed. 



To the axiom I cannot conceive any objection raised : it is merely 

 an application of the principles of geometrical equality to angular 

 magnitude. The real "pinch and nip" (to use Colonel Thompson's 

 significant expression) lies in the perception of the truth that whatever 

 applies to an angular space, applies to its angle. This is the elemen- 



