Mr. J. Walsh on the Twelfth Book of Euclid. 181 



the Encke Planet, be reconciled with the density and resist- 

 ance of our atmosphere, at the part traversed ; whereby the 

 theory of partially resisted planetary motion, in general, as 

 well as the height and constitution of our atmosphere, would 

 receive important illustration. 



In p. 350 of your 57th volume, I have endeavoured to point 

 out an important use which might immediately be made of 

 the shooting Stars, in accurately settling the longitudes of 

 places on Land : and I beg in conclusion here to mention, 

 the instantaneous vanishing of these shooting Stars, and of 

 satellitic Meteors, generally, on their passing the oxygenous 

 limit of our atmosphere, as phenomena, capable of being 

 pretty accurately observed, by two or three observers acting 

 in concert, at places rather distant, whose relative positions 

 were known, trigonometrically ; and to hint at the importance 

 which it might prove to science, to know the height of this 

 oxygenous limit, under the various circumstances of Pressure, 

 Temperature, Moisture, Electrical State &c. of the Air. 

 I am, gentlemen, 



Your obedient servant, 

 I lowland-street, July 3, 1824. John Farey. 



XXX. Observations on the Twelfth Book of Euclid. By 

 J. Walsh, Esq. 



" This I say to encourage those who are not far gone in these studies, to 

 use intrepidly their own judgement, without a blind or a mean deference 

 to the best of mathematicians, who are no more qualified than they 

 are, to judge of the simple apprehension or the evidence of what is 

 delivered in the first elements of the method." — Berkley, Defence of 

 Free Thinking in Mathematics." 



TT is true, the man who knows not A from B perceives as 

 -■■ clearly the first principles of geometry, as the most expert 

 analyst. It does not require the aid of deep science to per- 

 ceive that the whole is greater than its parts ; or that the line, 

 which is the shortest distance between two points, is a straight 

 line, that it does not bend more to any one side than it does 

 to any other. But some one, not satisfied with the immediate 

 evidence of his senses, that a straight line is shorter than any 

 curve line terminated by the same two points, thinks he can 

 prove tlit; proposition in this way: — He makes the straight 

 line the base of a triangle, having its vertex in the curve line, 

 then the sum of the two sides of the triangle is greater than 

 its base; in the same manner he makes the sides of the tri- 

 angle the bases of other triangles having their vertices like- 

 «i i in l he curve line, and so on ; then he thinks he has 



proved 



