for the Heptagon and Circle. 283 



to have set to work with rule and compasses to find coincidences. 

 They certainly used rule and compass in legitimate analysis; 

 only they seem to have required "a kind of mental contrivance 

 and construction to form a connexion between the data and 

 quaesita" *, which is not visible in^Kober's figure ; at least his 

 quaesitum is not demonstration. I think it is not unlikely that 

 the earliest mathematicians may have become acquainted with 

 the fraction, and not impossible that they may have constructed 

 Roberts figure. Meanwhile it must not be forgotten that in all 

 probability they could by approximation construct any angle they 

 pleased, although they might be unable to calculate the chord. 

 Bisection is quite elementary ; and with this power and a table 

 of the powers of two, nothing was easier than to construct a hep- 

 tagon with any desired degree of accuracy, with full knowledge 

 of the amount of error and with the power to correct it tenta- 

 tively. Surely if they were mathematicians at all they were 

 capable of a statement like this, 



2 ,, = 4096= (585x7) + 1 and \ - J| = ^error, 



1 4681 1 

 215= 32768= (4681x7) +1 and \ - |~| = ^ „ 



^ = 262144= (37449x7) + ! and \-^^^_„ 



and so on to any required degree of accuracy. For construction, 



1111111 1 . , . . 



7 = W + P + ¥ + 2^+2^+218 + ^218 • 



Or the construction might have been commenced with the side 

 of a hexagon or decagon. 



On this subject I would have consulted Pappus if the work 

 had been accessible to me : I only know from the article in 

 Hutton's" Dictionary, that in his 4th Book he shows how to 

 trisect an angle, and to divide a given arc or angle in any given 

 ratio. By some such method early mathematicians could have 

 described a heptagon • but something else was necessary to clothe 

 it with superstitious respect. 



I wish to submit to your consideration an hypothesis con- 



3927 

 nected with the Indian value of the circle ~r— =3*1416. I 



1250 



suspect that when they found themselves baffled in attempt- 

 ing to calculate the chords on successive subdivisions, they 

 resorted to a series of rational right-angled triangles (2#, 

 # 2 — 1, # 2 + l), constructed large figures, and hoped to find 



* Hutton's Mathematical Dictionary, under " Analysis," 



