256 Mr Meikle on the 



fore the first isosceles triangle. Nor could the radiants, or 

 lines drawn from A, ever begin to decrease without continu- 

 ing to do so ; because otherwise, an increase behoved to 

 occur in the angles ABC, BCD, &c., where the radiants were 

 shortest. 



Construct a triangle GCD every way equal to ABC ; then 

 since angle DCG exceeds BCD, the equal lines AC, DG, cut 

 each other in H, and unequally too ; thus DH exceeds CH, 

 because opposite a greater angle, and so the remainder GH 

 is less than AH. Hence, the triangle AHD exceeds GPIG, 

 and, therefore, the triangle ACD exceeds GCD or ABC. 

 Again, because the angles BCD, CDA, respectively exceed 

 CDE, and CDG, or BCA, the angle ADE is less than ACD ; 

 and, consequently, it may be shewn in the same way that the 

 successive areas continually increase, the length of the radi- 

 ants making the triangles great when they approximate to 

 being isosceles. 



The value of an angle in the series ABC, BCD, &c., is at 

 first 150°, but on approximating to two angles in a great or 

 isosceles triangle, it will, as was noticed above, be less than 

 the minute sum of angles in a large triangle. Now between 

 150° and this small value, there may obviously be an angle 

 of every different intermediate value, and each placed in the 

 order of its magnitude. If, then, a small rate of decrease in 

 those angles would cause the radiants to stretch out to an 

 unlimited distance (for since the contrary has neither been 

 proved, nor ought to be assumed, this case must be provided 

 for), and since it is quite evident that a great decrease would 

 ultimately make the radiants also decrease in length, there- 

 fore these two results would be so different as to be running 

 into directly opposite extremes ; and so there must still be 

 some intermediate rate which would neither cause the radi- 

 ants to stretch out indefinitely nor yet to decrease. Hence, 

 about A as a centre, and with a radius greater than any of 

 the radiants, a circle could be described which would include 

 the whole series of triangles, and consequently insure their 

 going round the centre A ; otherwise, an infinite number of 

 their increasing areas would not nearly cover the finite area 

 of the circle. 



