Myths and Marvels of Astronomy
inches, it follows that the diagonal of the base contains 12,912 such
inches, and the two diagonals together contain 25,824 pyramid inches, or
almost exactly as many inches as there are years in the great
precessional period. 'No one whatever amongst men,' says Professor Smyth
after recording various estimates of the precessional period, 'from his
own or school knowledge, knew anything about such a phenomenon, until
Hipparchus, some 1900 years after the great pyramid's foundation, had a
glimpse of the fact; and yet it had been ruling the heavens for ages,
and was recorded in Jeezeh's ancient structure.' To minds not moved to
most energetic forgetfulness by the spirit of faith, it would appear
that when a square base had been decided upon, and its dimensions fixed,
with reference to the earth's diameter and the year, the diagonals of
the square base were determined also; and, if it so chanced that they
corresponded with some other perfectly independent relation, the fact
was not to be credited to the architects. Moreover it is manifest that
the closeness of such a coincidence suggests grave doubts how far other
coincidences can be relied upon as evidence of design. It seems, for
instance, altogether likely that the architects of the pyramid took the
sacred cubit equal to one 20,000,000th part of the earth's diameter for
their chief unit of length, and intentionally assigned to the side of
the pyramid's square base a length of just so many cubits as there are
days in the year; and the closeness of the coincidence between the
measured length and that indicated by this theory strengthens the idea
that this was the builder's purpose. But when we find that an even
closer coincidence immediately presents itself, which manifestly is a
coincidence _only_, the force of the evidence before derived from mere
coincidence is _pro tanto_ shaken. For consider what this new
coincidence really means. Its nature may be thus indicated: Take the
number of days in the year, multiply that number by 50, and increase the
result in the same degree that the diagonal of a square exceeds the
side--then the resulting number represents very approximately the number
of years in the great precessional period. The error, according to the
best modern estimates, is about one 575th part of the true period. This
is, of course, a merely accidental coincidence, for there is no
connection whatever in nature between the earth's period of rotation,
the shape of a square, and the earth's period of gyration. Yet this
merely accidental coincidence is very much closer than the other
supposed to be designed could be proved to be. It is clear, then, that
mere coincidence is a very unsafe evidence of design.
Of course the pyramidalists find a ready reply to such reasoning. They
argue that, in the first place, it may have been by express design that
the period of the earth's rotation was made to bear this particular
relation to the period of gyration in the mighty precessional movement:
which is much as though one should say that by express design the height
of Monte Rosa contains as many feet as there are miles in the 6000th
part of the sun's distance.[21] Then, they urge, the architects were
not bound to have a square base for the pyramid; they might have had an
oblong or a triangular base, and so forth--all which accords very ill
with the enthusiastic language in which the selection of a square base
had on other accounts been applauded.
Next let us consider the height of the pyramid. According to the best
modern measurements, it would seem that the height when (if ever) the
pyramid terminated above in a pointed apex, must have been about 486
feet. And from the comparison of the best estimates of the base side
with the best estimates of the height, it seems very likely indeed that
the intention of the builders was to make the height bear to the
perimeter of the base the same ratio which the radius of a circle bears
to the circumference. Remembering the range of difference in the base
measures it might be supposed that the exactness of the approximation to
this ratio could not be determined very satisfactorily. But as certain
casing stones have been discovered which indicate with considerable
exactness the slope of the original plane-surfaces of the pyramid, the
ratio of the height to the side of the base may be regarded as much more
satisfactorily determined than the actual value of either dimension. Of
course the pyramidalists claim a degree of precision indicating a most
accurate knowledge of the ratio between the diameter and the
circumference 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 know where the Scripture records
the sealing up of the great pyramid; but it is all but certain that
during the very time when the pyramid was being built astronomical
observations were in progress which, for their interpretation, involved
of necessity a continual reference to the ratio in question. No one who
considers the wonderful accuracy with which, nearly two thousand years
before the Christian era, the Chaldaeans had determined the famous cycle
of the Saros, can doubt that they must have observed the heavenly bodies
for several centuries before they could have achieved such a success;
and the study of the motions of the celestial bodies compels 'men to
trouble themselves' about the famous ratio of the circumference to the
diameter.
We now come upon a new relation (contained in the dimensions of the
pyramid as thus determined) which, by a strange coincidence, causes the
height of the pyramid to appear to symbolise the distance of the sun.
There were 5813 pyramid inches, or 5819 British inches, in the height of
the pyramid according to the relations already indicated. Now, in the
sun's distance, according to an estimate recently adopted and freely
used,[22] there are 91,400,000 miles or 5791 thousand millions of
inches--that is, there are approximately as many thousand millions of
inches in the sun's distance as there are inches in the height of the
pyramid. If we take the relation as exact we should infer for the sun's
distance 5819 thousand millions of inches, or 91,840,000 miles--an
immense improvement on the estimate which for so many years occupied a
place of honour in our books of astronomy. Besides, there is strong
reason for believing that, when the results of recent observations are
worked out, the estimated sun distance will be much nearer this pyramid
value than even to the value 91,400,000 recently adopted. This result,
which one would have thought so damaging to faith in the evidence from
coincidence--nay, quite fatal after the other case in which a close
coincidence had appeared by merest accident--is regarded by the
pyramidalist as a perfect triumph for their faith.
They connect it with another coincidence, viz. that, assuming the height
determined in the way already indicated, then it so happens that the
height bears to half a diagonal of the base the ratio 9 to 10. Seeing
that the perimeter of the base symbolises the annual motion of the earth
round the sun, while the height represents the radius of a circle with
that perimeter, it follows that the height should symbolise the sun's
distance. 'That line, further,' says Professor Smyth (speaking on behalf
of Mr. W. Petrie, the discoverer of this relation), 'must represent'
this radius 'in the proportion of 1 to 1,000,000,000' (or _ten_ raised
to power _nine_), 'because amongst other reasons 10 to 9 is practically
the shape of the great pyramid.' For this building 'has such an angle at
the corners, that for every ten units its structure advances inwards on
the diagonal of the base, it practically rises upwards, or points to
sunshine' (_sic_) 'by _nine_. Nine, too, out of the ten characteristic
parts (viz. five angles and five sides) being the number of those parts
which the sun shines on in such a shaped pyramid, in such a latitude
near the equator, out of a high sky, or, as the Peruvians say, when the
sun sets on the pyramid with all its rays.' The coincidence itself on
which this perverse reasoning rests is a singular one--singular, that
is, as showing how close an accidental coincidence may run. It amounts
to this, that if the number of days in the year be multiplied by 100,
and a circle be drawn with a circumference containing 100 times as many
inches as there are days in the year, the radius of the circle will be
very nearly one 1,000,000,000th part of the sun's distance. Remembering
that the pyramid inch is assumed to be one 500,000,000th part of the
earth's diameter, we shall not be far from the truth in saying that, as
a matter of fact, the earth by her orbital motion traverses each day a
distance equal to two hundred times her own diameter. But, of course,
this relation is altogether accidental. It has no real cause in
nature.[23]
Such relations show that mere numerical coincidences, however close,
have little weight as evidence, except where they occur in series. Even
then they require to be very cautiously regarded, seeing that the
history of science records many instances where the apparent law of a
series has been found to be falsified when the theory has been extended.
Of course this reason is not quoted in order to throw doubt on the
supposition that the height of the pyramid was intended to symbolise the
sun's distance. That supposition is simply inadmissible if the