64 PHILOSOPHICAL TRANSACTIONS. [aNNO 1684. 



Christiani Hugenii Astroscopia Compendiaria, TuhiOptici Molimine liherata; or, 

 the Description of an Aerial Telescope. N° l6], p. 608. 



This description of Huygen's aerial telescopes, of 100 or 200 feet in length, 

 by the contrivance, without a continued tube, of suspending the object glass 

 at the top of a long mast fixed in the ground, is described more fully in his own 

 works. But it was never found to be of any use, as it could not be brought 

 into practice, and has been superseded by the modern and powerful reflectors of 

 short lengths. 



A New and Easy Way of demonstrating some Propositions in Euclid. By Mr. 

 Ash, Member Phil. Soc. Dublin. N° 102, p. 672, 



As a further instance of the evidence of mathematical theorems, I believe it 

 not difficult to demonstrate any one of Euclid's, independently of the rest, 

 without precedent lemmas or propositions; as an essay of which, I shall here 

 subjoin some of the most useful, and on which the solution of most problems, 

 especially algebraical ones, depend, and those also in the most various and dif- 

 ferent parts of geometry, viz. concerning the properties of angles, circles, 

 triangles, squares, proportionals, and solids. The propositions which I shall 

 endeavour to demonstrate thus independently, shall be these, the 32d and 47th 

 of the 1st book; most of the 2d and 5th books; the 1st and 16th of the 6th, 

 with their corollaries. In order to demonstrate the 32d, I suppose it known 

 what is meant by an angle, triangle, circle, external angle, parallels, and that 

 the measure of an angle is the arch of a circle intercepted between its sides, 

 that a right angle is measured by a quadrant, and 2 right angles by a semicircle. 

 I say then, in fig. 5, pi. 1, that in the triangle ABC, the external angle BCE 

 is equal to the two opposite internal ones ABC, BAG. For let a circle be 

 drawn, C being the centre, and BC the radius, and let CD be drawn parallel 

 to AB; these two lines being always equidistant will both have the same incli- 

 nation to any 3d line falling on them ; that is, by the definition of angle, they 

 will make equal angles with it; for if any part of CD, for instance, did incline 

 more to BC than did AB, then they would not be parallel; it follows therefore 

 that the angles ABC, BCD, are equal; also BAC = DCE, because AE falls 

 upon two parallels; but the external angle BCE :=BCD-j-DCE, which was 

 before proved to be equal to ABC, BAC, Q. E. D. 



. Hence may be inferred as a corollary, that the three angles of every triangle 

 are equal to two right ones; for the angles ACB -|- BCE are measured by a 

 semicircle, and therefore equal to two right angles. Corollaries also from hence 



