130 Sir W. R. Hamilton on Roberta Construction 



7. If we continue the construction, as Rober did, so as to 

 form an isosceles triangle, say ABC, with </> for its vertical angle, 

 and if we content ourselves with thousandths of seconds, the 

 angles of this triangle will be as follows : 



rA=0 = 25°42'51"-392 

 (F) \ B= 7 ^£ = 77° 8' 34"«304 



C = B = 77° 8'34"-304 



and we see that each base-angle exceeds the triple of the vertical 

 by only about an eighth part of a second, namely by that small 

 angle which occurs first in the system (D), and of which the 

 sine is the last number in the preceding system (C). And if 

 we compare a base-angle of the triangle thus constructed, with 



the base-angle — =77° 8' 34"-2857. . . of the true triangle, in 



which each angle at the base is triple of the angle at the vertex, 

 we find an error in excess equal nearly to 0"*018, or, more exactly, 



B-y=^-| = +0"-01807 61615 403, 



which amounts to less than a fifty -fifth part of a second, but of 

 which I conceive that all the thirteen decimals here assigned 

 are correct. And I suppose that no artist would undertake to 

 construct a triangle which should more perfectly, or so perfectly, 

 fulfil the conditions proposed. The problem, therefore, of con- 

 structing such a triangle, and with it the regular heptagon, by 

 right lines and circles only, has been practically solved by that 

 process which Rober believed to have been known to the ancient 

 Egyptians, and to have been employed by them in the architec- 

 ture of some of their temples — some hints, as he judged, being 

 intentionally preserved in the details of the workmanship, for 

 the purpose of being recognized, by the initiated of the time, 

 or by men of a later age. 



8. Another way of rendering conceivable the extreme small- 

 ness of the practical error of that process, is to imagine a series 

 of seven successive chords inscribed in a circle, according to the 

 construction in question, and to inquire how near to the initial 

 point the final point would be. The answer is, that the last 

 point would fall behind the first, but only by about half a second 

 (more exactly by 0"\506). If then we suppose, for illustration, 

 that these chords are seven successive tunnels, drawn eastward 

 from station to station of the equator of the earth, the last tun- 

 nel would emerge to the west of the first station, but only by 

 about fifty feet. 



