cumference of a circle; and the angle of the only casing stone
measured being diversely estimated at 51 deg. 50' and 51 deg. 52-1/4',
they consider 50 deg. 51' 14.3" the true value, and infer that the
builders regarded the ratio as 3.14159 to 1. The real fact is, that the
modern estimates of the dimensions of the casing stones (which, by the
way, ought to agree better if these stones are as well made as stated)
indicate the values 3.1439228 and 3.1396740 for the ratio; and all we
can say is, that the ratio really used lay _probably_ between these
limits, though it may have been outside either. Now the approximation of
either is not remarkably close. It requires no mathematical knowledge at
all to determine the circumference of a circle much more exactly. 'I
thought it very strange,' wrote a circle-squarer once to De Morgan
(_Budget of Paradoxes_, p. 389), 'that so many great scholars in all
ages should have failed in finding the true ratio, and have been
determined to try myself.' 'I have been informed,' proceeds De Morgan,
'that this trial makes the diameter to the circumference as 64 to 201,
giving the ratio equal to 3.1410625 exactly. The result was obtained by
the discoverer in three weeks after he first heard of the existence of
the difficulty. This quadrator has since published a little slip and
entered it at Stationers' Hall. He says he has done it by actual
measurement; and I hear from a private source that he uses a disc of
twelve inches diameter which he rolls upon a straight rail.' The
'rolling is a very creditable one; it is as much below the mark as
Archimedes was above it. Its performer is a joiner who evidently knows
well what he is about when he measures; he is not wrong by 1 in 3000.'
Such skilful mechanicians as the builders of the pyramid could have
obtained a closer approximation still by mere measurement. Besides, as
they were manifestly mathematicians, such an approximation as was
obtained by Archimedes must have been well within their power; and that
approximation lies well within the limits above indicated. Professor
Smyth remarks that the ratio was 'a quantity which men in general, and
all human science too, did not begin to trouble themselves about until
long, long ages, languages, and nations had passed away after the
building of the great pyramid; and after the sealing up, too, of that
grand primeval and prehistoric monument of the patriarchal age of the
earth according to Scripture.' I do not