Let us next consider the dimensions of the square base thus carefully placed in latitude 30o north, to the best of the builders' power, with sides carefully oriented. to be used for observing meridian transits | therefore could not have intended, as Proof the stars in order to determine sidereal fessor Smyth supposes, to have had the time; for close circumpolar stars, by rea- five-hundred-millionth part of the earth's son of their slow motion, are the least polar axis, as distinguished from any suited of all for such a purpose. As Pro- other, for their unit of length. But if they fessor Smyth says, in arguing against this made observations in or near latitude 30° suggested use of the star, "no observer in north, on the supposition that the earth is his senses, in any existing observatory, a globe, their probable error would exceed when seeking to obtain the time, would the difference even between the earth's observe the transit of a circumpolar star polar and equatorial diameters. Both diffor anything else than to get the direction ferences are largely exceeded by the range of the meridian to adjust his instrument of difference among the estimates of the by." (The italics are his.) It is precisely actual length of the sacred cubit, supposed such a purpose (the adjustment, however, to have contained twenty-five of these not of an instrument, but of the entire smaller units. And, again, the length of structure of the pyramid itself), that I have the pyramid base-side, on which Smyth suggested for this remarkable passage bases his own estimate of the sacred cubit, this "cream-white, stone-lined, long tube," has been variously estimated, the largest where it traverses the masonry of the measure being 9,168 inches, and the lowpyramid, and below that dug through the est 9,110 inches. The fundamental theory solid rock to a distance of more than three of the pyramidalists, that the sacred cubit hundred and fifty feet. was exactly one twenty-millionth part of the earth's polar diameter, and that the side of the base contained as many cubits and parts of a cubit as there are days and parts of a day in the tropical year (or year of seasons), requires that the length of the side should be 9,140 inches, lying between the limits indicated, but still so widely removed from either that it would appear very unsafe to base a theory on the supposition that the exact length is or was 9,140 inches. If the measures 9,168 inches and 9,110 inches were inferior, and several excellent measures made by practised observers ranged around the length 9,140 inches, the case would be different. But the best recent measures gave respectively 9,110 and 9,130 inches; and Smyth exclaims against the unfairness of Sir H. James in taking 9,120 as "therefore the [probable] true length of the side of the great pyramid when perfect," calling this "a dishonorable shelving of the honorable older observers with their larger results." The only other measures, besides these two, are two by Colonel Howard-Vyse and by the French savants, giving respectively 9,168 and 9,163.44 inches. The pyramidalists consider 9,140 inches a fair mean value from these four. The natural inference, however, is, that the pyramid base is not now in a condition to be satisfactorily measured; and assuredly no such reliance can be placed on the mean value 9,140 inches that, on the strength of it, we should believe what otherwise would be utterly incredible, viz., that the builders of the great pyramid knew "both the size and shape of the earth exactly." "Humanly, or by human science, finding it out in that age was, of It seems highly probable that, whatever special purpose the pyramid was intended to fulfil, a subordinate idea of the builders would have been to represent symbolically in the proportions of the building such mathematical and astronomical relations as they were acquainted with. From what we know by tradition of the men of the remote time when the pyramid was built, and what we can infer from the ideas of those who inherited, however remotely, the modes of thought of the earliest astronomers and mathematicians, we can well believe that they would look with superstitious reverence on special figures, proportions, numbers, and so forth. Apart from this, they may have had a quasiscientific desire to make a lasting record of their discoveries, and of the collected knowledge of their time. It seems altogether probable, then, that the smaller unit of measurement used by the builders of the great pyramid was intended, as Professor Smyth thinks, to be equal to the five-hundred-millionth part of the earth's diameter, determined from their geodetical observations. It was perfectly within the power of mechanicians and mathematicians so experienced as they undoubtedly were the pyramid attests so much to measure with considerable accuracy the length of a degree of latitude. They could not possibly (always setting aside the theory of divine inspiration) have known anything about the compression of the earth's globe, and course, utterly impossible," says Professor | Hipparchus, some nineteen hundred years Smyth. But he is so confident of the after the great pyramid's foundation, had average value derived from widely conflict- a glimpse of the fact; and yet it had been ing base-measures as to assume that this ruling the heavens for ages, and was revalue, not being humanly discoverable, was corded in Jeezeh's ancient structure." To of necessity "attributable to God and to minds not moved to most energetic forgethis divine inspiration." We may agree, fulness by the spirit of faith, it would apin fine, with Smyth, that the builders of pear that when a square base had been the pyramid knew the earth to be a globe; decided upon, and its dimensions fixed, that they took for their measure of length with reference to the earth's diameter and the sacred cubit, which, by their earth- the year, the diagonals of the square base measures, they made very fairly approxi- were determined also; and, if it so mate to the twenty-millionth part of the chanced that they corresponded with some earth's mean diameter; but there seems other perfectly independent relation, the no reason whatever for supposing (even if fact was not to be credited to the archithe supposition were not antecedently of tects. Moreover it is manifest that the its very nature inadmissible) that they closeness of such a coincidence suggests knew anything about the compression of grave doubts how far other coincidences the earth, or that they had measured a can be relied upon as evidence of design. degree of latitude in their own place with It seems, for instance, altogether likely very wonderful accuracy.* that the architects of the pyramid took the But here a very singular coincidence sacred cubit equal to one twenty-milmay be noticed, or, rather, is forced upon lionth part of the earth's diameter for their our notice by the pyramidalists, who chief unit of length, and intentionally asstrangely enough recognize in it fresh evi- signed to the side of the pyramid's square dence of design, while the unbeliever finds base a length of just so many cubits as in it proof that coincidences are no sure there are days in the year; and the closeevidence of design. The side of the pyr-ness of the coincidence between the measamid containing three hundred and sixty- ured length and that indicated by this five and a quarter times the sacred cubit theory strengthens the idea that this was of twenty-five pyramid inches, it follows that the diagonal of the base contains twelve thousand nine hundred and twelve such inches, and the two diagonals to gether contain twenty-five thousand eight hundred and twenty-four 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 It may, perhaps, occur to the reader to enquire what diameter of the earth, supposed to be a perfect sphere, would be derived from a degree of latitude measured with absolute accuracy near latitude 30°. A degree of latitude measured in polar regions would indicate a diameter greater even than the equatorial; one measured in equatorial regions would indicate a diameter less even than the polar. Near latitude 300 the measurement of a degree of latitude would indicate a diameter very nearly equal to the true polar diameter of the earth. In fact, if it could be proved that the builders of the pyramid used for their unit of length an exact subdivision of the polar diameter, the inference would be that, while the coincidence itself was merely accidental, their measurement of a degree of latitude in their own country had been singularly accurate. By an approximate calculation I find that, taking the earth's compression at one three-hundredth, the diameter of the earth, estimated from the accurate measurement of a degree of latitude in the neighborhood of the great pyramid, would have made the sacred cubittaken at one twenty-millionth of the diameter- equal to 24'98 British inches; a closer approximation than Professor Smyth's to the estimated mean probable value of the sacred cubit. the builders' 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 fifty, 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 five-hundred-and-seventy-fifth 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 | mated at 51° 50m. and 51° 52 1-4m., they 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 six-thousandth part of the sun's distance. 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. consider 50° 51m. 14.3sec. the true value, and infer that the builders regarded the ratio as 3.14159 to one. 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 31439228 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 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 termi-(" Budget of Paradoxes," p. 389), "that nated above in a pointed apex, must have been about four hundred and eighty-six 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 esti It is, however, almost impossible to mark any limits to what may be regarded as evidence of design by a coincidence-hunter. I quote the following from the late Professor De Morgan's "Budget of Paradoxes.' "" Having mentioned that 7 occurs less frequently than any other digit in the number expressing the ratio of circumference to diameter of a circle, he proceeds: "A correspondent of my friend Piazzi Smyth notices that 3 is the number of most frequency, and that 3 1-7 is the nearest approximation to it in simple digits. Professor Smyth, whose work on Egypt is paradox of a very high order, backed by a great quantity of useful labor, the results of which will be made available by those who do not receive the paradoxes, is inclined to see confirmation for some of his theory in these phenomena.' In passing, I may mention as the most singular of these accidental digit relations which I have yet noticed, that in the first 110 digits of the square root of 2, the num7 occurs more than twice as often as either 5 or 9, which each occur eight times, 1 and 2 occurring each nine times, and 7 occurring no less than eighteen times. ber so many great scholars in all ages should Be when the pyramid was being built astro- | round the sun, while the height reprenomical observations were in progress sents the radius of a circle with that which, for their interpretation, involved of perimeter, it follows that the height should necessity a continual reference to the ratio symbolize the sun's distance. "That line, in question. No one who considers the further," says Professor Smyth (speaking wonderful accuracy with which, nearly two on behalf of Mr. W. Petrie, the discoverer thousand years before the Christian era, of this relation), "must represent" this the Chaldæans had determined the famous radius "in the proportion of one to one cycle of the Saros, can doubt that they billion ” (or ten raised to power nine), “bemust have observed the heavenly bodies cause amongst other reasons ten to nine for several centuries before they could is practically the shape of the great pyrahave achieved such a success; and the mid." For this building "has such an study of the motions of the celestial bodies angle at the corners, that for every ten compels "men to trouble themselves" units its structure advances inwards on about the famous ratio of the circumfer- the diagonal of the base, it practically rises ence to the diameter. upwards, or points to sunshine" (sic)" by We now come upon a new relation (con- nine. Nine, too, out of the ten charactertained in the dimensions of the pyramid istic parts (viz., five angles and five sides) as thus determined) which, by a strange being the number of those parts which the coincidence, causes the height of the pyr- sun shines on in such a shaped pyramid, amid to appear to symbolize the distance in such a latitude near the equator, out of of the sun. There were 5,813 pyramid a high sky, or, as the Peruvians say, when inches, or 5,819 British inches, in the the sun sets on the pyramid with all his height of the pyramid according to the rays." The coincidence itself on which relations already indicated. Now, in the this perverse reasoning rests is a singular sun's distance, according to an estimate one singular, that is, as showing how recently adopted and freely used,* there close an accidental coincidence may run. are 91,400,000 miles or 5.791 thousand It amounts to this, that if the number of millions of inches,- that is, there are ap- days in the year be multiplied by one hunproximately as many thousand millions of dred, and a circle be drawn with a circuminches in the sun's distance as there are ference containing one hundred times as inches in the height of the pyramid. If many inches as there are days in the we take the relation as exact we should year, the radius of the circle will be very infer for the sun's distance 5,819 thousand nearly one billionth part of the sun's dismillions of inches, or 91,840,000 miles-tance. Remembering that the pyramid an immense improvement on the estimate which for so many years occupied a place of honor 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 Such relations show that mere numerin the evidence from coincidence-nay, ical coincidences, however close, have quite fatal after the other case in which a little weight as evidence, except where close coincidence had appeared by merest they occur in series. Even then they reaccident-is regarded by the pyramidalists 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 nine to ten. Seeing that the perimeter of the base symbolizes the annual motion of the earth I have substituted this value in the article "Astronomy," of the "British Encyclopædia," for the estimate formerly used, viz. 95,233,055 miles. But there is good reason for believing that the actual distance is nearly 92,000,000 miles. inch is assumed to be one five-hundredmillionth 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.* It may be matched by other coincidences as remarkable and as little the result of the operation of any natural law. For instance, the following strange relation, which introduces the dimensions of the sun himself, nowhere, so far as I have yet seen, introduced among pyramid relations, even by pyramidalists: "If the plane of the ecliptic were a true surface, and the sun were to commence rolling along that surface towards the part of the earth's orbit where she is at her mean distance, while the earth commenced rolling upon the sun (round one of his great circles), each globe turning round in the same time, then, by the time the earth had rolled its way once round the sun, the sun would have almost exactly reached the earth's orbit. This is only another way of saying that the sun's diameter exceeds the earth's in almost exactly the same degree that the sun's distance exceeds the sun's diameter." quire to be very cautiously regarded, see- | vertical height, that came out 5276 of the ing that the history of science records same inches. But the sum of those two many instances where the apparent law of heights, or the height taken up and down, a series has been found to be falsified equals one hundred inches; which length, as when the theory has been extended. Of elsewhere shown, is the general pyramid linear course this reason is not quoted in order representation of a day of twenty-four hours. And the mean of the two heights, or the to throw doubt on the supposition that the height taken one way only, and impartially to height of the pyramid was intended to the middle point between them, equals fifty symbolize the sun's distance. That sup- inches; which quantity is, therefore, the genposition is simply inadmissible if the hy-eral pyramid linear representation of only half pothesis, according to which the height was a day. In which case, let us ask what the already independently determined in an- entrance passage has to do with half rather other way, is admitted. Either hypothesis than a whole day? might be admitted were we not certain that the sun's distance could not possibly have been known to the builders of the pyramid; or both hypotheses may be rejected: but to admit both is out of the question. From The Examiner. GREEN PASTURES AND PICCADILLY. 66 BY WILLIAM BLACK. THE ADVENTURES OF A PHAETON, 99 66 THE On relations such as these, which, if really intended by the architect, would imply an utterly fatuous habit of concealing elaborately what he desired to symbolize, the pyramidalists base their belief that Considering the multitude of dimensions a mighty intelligence did both think out the of length, surface, capacity, and position, idolaters, in a primal age of the world, to work plans for it, and compel unwilling and ignorant the great number of shapes, and the vari-mightily both for the future glory of the one ety of material existing within the pyramid, true God of revelation, and to establish lastand considering, further, the enormous ing prophetic testimony touching a further number of relations (presented by modern development, still to take place, of the absoscience) from among which to choose, can lutely divine Christian dispensation. it be wondered at if fresh coincidences are being continually recognized? If a dimension will not serve in one way, use can be found for it in another; for instance, if some measure of length does not correspond closely with any known dimension of the earth or of the solar system (an unlikely supposition), then it can be under- AUTHOR OF stood to typify an interval of time. If, even after trying all possible changes of that kind, no coincidence shows itself (which is all but impossible), then all that is needed to secure a coincidence is that the dimensions should be manipulated a little. Let a single instance suffice to show how the pyramidalists (with perfect honesty of purpose) hunt down a coincidence. The slant tunnel already described has a transverse height, once no doubt uniform, now giving various measures from 47'14 pyramid inches to 47:32 inches, so that the vertical height from the known inclination of the tunnel would be estimated at somewhere between 52.64 inches and 52.85. Neither dimension corresponds very obviously with any measured distance in the earth or solar system. Nor when we try periods, areas, etc., does any very satisfactory coincidence present itself. But the difficulty is easily turned into a new proof of design. Putting all the observations together (says Professor Smyth), I deduced 47°24 pyramid inches to be the transverse height of the entrance passage; and computing from thence with the observed angle of inclination the CHAPTER XIII. FIVE-ACE JACK. We will now let Mr. Balfour and his young and charming bride go off together on their wedding-trip-a trip that ought to give them some slight chance of becoming acquainted with each other, though a certain profound philosopher, resident in Surrey, would say that the glamor of impossible ideals was still veiling their eyes and we will turn, if you please, to a very different sort of traveller, who just about the same time was riding along a cattletrail on the high-lying and golden-yellow plains of Colorado. This was Buckskin Charlie-so named from the suit of grey buckskin which he wore, and which was liberally adorned with loose fringes cut from the leather. Indeed, there was a generally decorative air about this herdsman and his accoutrements, which gave him a half-Mexican look, though the bright, sun-tanned complexion, the long lightbrown hair, and the clear blue eyes were not at all Mexican. There was a brass tip to the high pommel in front of him, round |