Further Remarks on the Theory of ■parallel Lines. 251 



whether all this be in its proper place, every one must agree 

 in admiring the writer's profound learning. 



4. Every attempt to overcome the difficulty about parallel 

 lines, uniformly leads to one conclusion ; proving that it is 

 insuperable by a direct process of reasoning. New definitions ; 

 new postulates; every mode of investigation that can be devised ; 

 only place the same difficulty in various aspects. The cause 

 lies in the imperfect nature of the definition of a straight line; 

 and, as in other similar cases, we cannot hope for success in 

 this research, except by having recourse to the indirect me- 

 thod of demonstration. 



In this Journal for March 1822, the foundation of an exact 

 theory of parallels is laid, by proving, in an indirect manner, 

 that the three angles of a triangle are equal to two right an- 

 gles. It is shown, first, that the sum of the angles cannot be 

 greater than two right angles ; and, secondly, that it cannot 

 be less. The first proposition is made out by transforming any 

 proposed triangle successively into a series of others, so that the 

 sum of the angles of every triangle shall be the same, while 

 one angle continually diminishes till at length it is less than any 

 given angle however small. Thus the sum of the three angles 

 of the first triangle approaches without limit to the sum of 

 two angles of another; and as the latter sum is always less 

 than two right angles, the former sum cannot exceed the same 

 quantity. We have lately seen a theory of parallel lines by a 

 professor of the mathematics at Basle*, which, like the attempt 

 in this Journal, follows the indirect mode of demonstration; 

 and further resembles it in proving the proposition here 

 spoken of by the same procedure. But in demonstrating that 

 the angles of a triangle cannot be less than two right angles, 

 the Professor at Basle follows Legendre in the first editions 

 of his geometry, a new postulate being added for the sake of 

 rigorous accuracy. We conceive that this is a blemish in the 

 theory ; because, when the indirect method of reasoning is 

 employed, the demonstrations should be effected without any 

 gratuitous assumption. In this respect the advantage is in 

 favour of the proof of the same proposition given in the Num- 

 ber of this Journal already cited, since it proceeds upon the 

 admitted principles of geometry. We shall briefly sketch an 

 outline of the reasoning. If a quadrilateral figure be divided 

 into any number of triangles that have their angles upon the 

 sides of the figure, or at points within the figure ; then, be- 

 cause all the angles at a point on one side of a straight line, 



* Nova Theoria de Parallelarum RectarumProprietatibus, Auctore Danicle 

 Ilubero, Basileense, in Acad, patria Mathem. Professore, et Bibliotliecario. 

 Basileae, 1823. 



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