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XXI. On the Theory of Parallel Lines. 

 By Mr. Henry Meikle. 



To the Editors of the Philosophical Magazine arid Journal. 

 Gentlemen, 



"V^OUR correspondent X. Y., who, in the Philosophical 

 A Magazine for September 1844, has been pleased to take 

 notice of my paper on parallel lines, inserted in the Edinburgh 

 Philosophical Journal for April 1844', alleges that the reason- 

 ing depends on the assumption, that no triangle can have the 

 sum of its angles inappreciably small. Now if this is really 

 supposed to need any proof, the following I presume will be 

 quite satisfactory. 



If the sum of the angles of a triangle could ever be so small 

 that it could not be multiplied to exceed .any given angle, 

 then an infinite number of copies of that triangle could be 

 placed around a point, and yet these would all lie within a 

 single circle whose radius is their longest side. Consequently, 

 whilst owing to the minuteness of the sum of the angles, the 

 area of each included triangle would be a maximum, or the 

 greatest which any triangle could have*, the sum of an infi- 

 nite number of such areas would be infinite; and yet the 

 whole of them would not nearly cover the finite area of the 

 circle, which is absurd. 



Your very obedient Servant, 

 Maitland Street, Edinburgh, Henry MEIKLE. 



June 20, 1846. 



completely transparent throughout one half and the other portion of an opake 

 white. Under the microscope, however, I found that they presented some 

 difference, appearing to consist almost entirely of irregular fragments of 

 ice, which presented numerous dark points, such as are seen in fig. 1. I 

 clearly detected numerous bubbles of air in the stone previous to its fusion, 

 some perfectly spherical, others elongated like parti- y\o 3. 



cles of mercury on a plain surface. The spherules of 



ice so constantly met with in the other observations 1 ^ 



were very scarce, and each contained a globule of air ( >v 



placed near its circumference, as in fig. .'3. The hail- 

 stone, which was partly transparent, was found on the 

 transparent side to be free from globules of air or 

 spherules of ice, with all the characters of a transpa- 

 rent particle of ice; the opake part presented the 

 same appearances as the other hailstones, of which about seven or eight 

 were examined. 



* This, in the circumstances, is an obvious inference from my second 

 proposition, which X. Y. admits to be satisfactorily demonstrated, namely, 

 that if in one triangle the sum of the angles differed from two right angles, 

 so it would in every triangle; the difference would always have the same 

 sine, and (as in spherics) be proportional to the area of the triangle. 



V0 



