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The more direct application of this theory to the design of the Mausoleum will be explained as we proceed, but in the meanwhile it may be asserted that without it many of the dimensions of this celebrated monument might for ever have remained matters of dispute. With its assistance there is scarcely one that may not be ascertained with almost absolute certainty.

Another and quite distinct set of ratios was discovered by Colonel Howard Vyse and his architect Mr. Perring, in their explorations of the Pyramids of Egypt. They found, for instance, in the Great Pyramid that the distance

They also found that the length of the base line was to this dimension in the ratio of 8 to 5, making it 448 cubits or 767?424 feet English exactly. With these two dimensions all the other parts of so simple a figure follow as a matter of course.

The bearing of this also on the Mausoleum will be seen in the sequel, though a much more complicated system of ratios was of course necessary either to such a building or to even the very simplest Greek temples.

There is one other point which must be carefully attended to in any attempt to restore the Mausoleum, which is the ratio between Greek and English measures. Those quoted by Pliny are in the former, of course; those obtained by the excavations are in the latter; and every result is vitiated and worthless without due attention to the difference.

The length of a Greek foot may be attained most directly by comparison with the Roman. From the researches of the best antiquaries as summed up by Niebuhr, the length of the Roman foot was ?972 English--a result confirmed by Mr. Penrose's careful independent investigation. Now, as it is known that the ratio between the Greek foot and the Roman was as 25 to 24, we arrive at the result of 101?25 English feet equal to 100 Greek.

Mr. Penrose obtained a slightly different result from his measurement of the upper step of the Parthenon. The front was known or assumed to be exactly 100 Greek feet; it gave 101?341, or about one inch in excess in 1200. As the flanks were to the front in the ratio of 4 to 9, this ought to have given 228?019. It was found to be 228?166, or nearly two inches in excess. But, on the other hand, it is admitted that the term Hecatompedon in Greek authors seems always to apply to the Naos and not to the step; and this, as measured by Mr. Penrose, including the transverse wall, gave 101?222, or a little under the other--the mean between the two being almost exactly identical with the measure derived from the Roman foot. In consequence of this the preference will be given throughout the following pages to the ratio of 101?25, or 101 ft. 3 in. English, as being equal to 100 Greek feet.

Turning from this to the measurement of the steps of the Pyramid, which, as mentioned above, is one of the most important elements for the restoration which have been brought to light by the recent excavations, we find their dimensions quoted throughout by Lieut. Smith, Mr. Pullan, and Mr. Newton as 1? 9??, or 21 inches English for the wider, and 1? 5??, or 17 English inches for the narrower step. The first thing that strikes one on considering this is, that it is a most wonderful coincidence that these dimensions should come out so exactly in English measures, without any fraction either way. On any moderate calculation of chances the odds are at least 100 to 1 against this being the case. The suspicion that there is an error somewhere is confirmed by observing that, though so very nearly in the ratio of 4 to 5, they are not exactly so; but if we try with the lower number we find 4 : 5 :: 17 : 21?25, or within the minutest fraction of 21 Greek inches. If we adopt 17?01 English inches for the shorter, we have 21?2625, or exactly 21 Greek inches, for the latter.

The determination of this point was so essential that I have carefully measured all the angle and roofing stones I could get access to in the Museum, and find that, as nearly as can be ascertained, the dimension of 17 inches is correct; but the longer one is, it may be, 2/10ths--it may be 3/10ths--of an inch in excess. Any one can verify this for himself; but I am so convinced of its correctness by my measurements, that I shall use the longer step as a dimension of 21 Greek, or 21?2625 English, inches.

The well known tablets at Mylassa, given in B?ckh, prove incontestably that Mausolus acknowledged himself a satrap of Artaxerxes as late as 355, or only two years before his death. If it is contended that he afterwards emancipated himself from the Persian yoke--of which there is no proof--it is by no means clear that he did not commence his own tomb himself some time before his death. At least it is nearly certain that no other man ever had a tomb of any great magnificence who did not in his lifetime take measures to secure its erection.

All this does not, it is true, prove that the Babylonian cubit was used in Caria; but it makes it so probable that it may have been that there will be nothing shocking in calling the length of the longer step by this name; and as this measure was the modulus of the whole building, and occurs over and over again, it will be convenient, and avoid circumlocution, if--of course, without prejudging the fact--we call the measure of 21 Greek inches as equal to 1 Babylonian or Halicarnassean cubit. If it could be proved that such a measure was never known in Caria, this would not in the least affect the result. All that is wanted here is a name which shall express a measure of 21 Greek inches. If any other can be suggested it will answer equally well. But it seems necessary that some definite term should be used in the sequel; and, till some other is found, I may perhaps be allowed to employ this.

Next in importance to the steps of the Pyramid, for the purposes of restoration, are the fragments of the Cymatium which were discovered in the excavations. Of these some six or seven were found, and on each was either a Lion's head covering the joint, or the mark of a Lion's head on the further edge of the stone next the joint.

Each of these pieces was, like the steps of the Pyramid, 21 inches, or 1 cubit, in length; and, according to the evidence we now have, the Lions' heads were consequently spaced 2 cubits, or 3 feet 6 inches, from the centre of one to the centre of another.

The interest of this measurement lies in the certainty that the inter-columniation was somehow commensurate with it. The usual arrangement in Greek architecture would have been that there should be one Lion's head over the centre of each column, and one half-way between. This certainly was not the arrangement here, as the columns, which are 3 ft. 6 in. Greek, or exactly 2 cubits in width, in their lower diameter, would then have been only one diameter apart.

It has been suggested that, as the Lions' heads are so unusually close, the pillars may have been so arranged that one column had a Lion's head over its centre, and those on each side stood between two Lions' heads--thus making the intercolumniation 8 ft. 9 in. The first objection that occurs to this view is, that it is unknown in any other examples; that it is contrary to the general principles of the art, and introduces an unnecessary complication; and is, therefore, unlikely. But the great objection is, that it cannot be made to fit in with any arrangement of the Pyramid steps. Let it be assumed, for instance, that the thirty-six columns of the Pteron were so arranged as to give an uneven number each way, so as to have eleven intercolumniations on one side by seven on the other; this would give a dimension of 96 feet 3 inches by 61 feet 3 inches from centre to centre of the angle columns, to which it would be impossible to fit the Pyramid, assuming, from the evidence of the steps, that its sides were in ratio 4 to 5, or nearly so at all events. If, on the contrary, it is assumed that there were 10 intercolumniations by 8, this would give a dimension of 87?6 by 70; and adding 2 ft. 9 in. each way, which we shall presently see was the projection of the first step of the Pyramid beyond the centre of the angle column, we should have for its base 93 feet by 75 feet 6 inches, within which it is impossible to compress it, unless we adopt a tall pyramid, as was done by Mr. Cockerell and Mr. Falkener before the discovery of the pyramid steps, or unless we admit of a curvilinear-formed pyramid, as was suggested by myself. With the evidence that is now before us, neither of these suggestions seems to be for one moment tenable; and as we cannot, with this intercolumniation, stretch the dimensions of the Pteron beyond what is stated above, it must be abandoned.

Advancing 1 cubit beyond this, we come to 6 cubits, or 10 feet 6 inches Greek, as the distance from the centre of one column to the centre of the next; and the Lions' heads then range symmetrically, one over each pillar, and two between each pair.

At first sight there seems to be no objection to the assumption that one plain piece of the Cymatium may have been inserted between each of the pieces to which were attached the Lions' heads, or the impress of them. It is true none were found; but as there could be only one plain piece in three, and as only six or seven fragments were found altogether, the chances against this theory are not sufficient to cause its rejection. The real difficulty is, that a Lion's head exists on a stone 1 cubit from the angle; and, unless the architects adopted a different arrangement at the angles from what they did in the centre, which is, to say the least of it, extremely improbable, it cannot be made to fit with the arrangement. If one plain piece had been found, it would have fixed the distance between centre and centre of column at 10 ft. 6 in. absolutely. As none, however, were found, or at least brought home, we must look for our proofs elsewhere.

The first of these is a very satisfactory one, on the principle of definite proportions above explained. As we have just found that six pyramid steps, or 6 cubits, are equal to one intercolumniation, so six intercolumniations, or 36 cubits, is exactly 63 Greek feet--the "sexagenos ternos pedes," which Pliny ascribes to the cella or tomb; it is further proved that this was not accidental, by our finding that twice the length of the cella, or 126 Greek feet, or 72 cubits, is, or ought to be, the total length of the building, measured on its lowest step. This, as before mentioned, Mr. Newton quotes, in round numbers, as 127 feet English; but as neither he nor any of those with him had any idea that any peculiar value was attached to this dimension, they measured carelessly and quoted loosely. My own conviction is, that it certainly was 127 ft. 6-3/4 in. English, which would be the exact equivalent of 126 Greek feet. At all events, I feel perfectly certain that the best mode of ascertaining the exact length of the pyramid step would be to divide this dimension, whatever it is, by 72.

Returning to the Pteron: if the columns were ranged in a single row--and no other arrangement seems possible with the evidence now before us--there must have been eleven columns on the longer faces and nine at the ends, counting the angle columns twice, and consequently a column in the centre of each face. This, at least, is the resultant of every conceivable hypothesis that I have been able to try. No other will, even in a remote degree, suit the admitted forms and dimensions of the pyramid: it is that adopted by Lieutenant Smith and Mr. Pullan; and, according to the evidence before us, seems the only one admissible.

Adopting it for the present, the first difficulty that arises is that 10 intercolumniations at 10 ft. 6 in. give 105 feet; to which if we add as before 5 ft. 6 in., or twice 2 ft. 9 in., for the projection of the first step of the pyramid beyond the centres of the columns, we have 110 ft. 6 in., a dimension to which it is almost impossible to extend the pyramid; and, what is worse, with a cella only 63 feet in its longest dimension, it leaves 21 feet at either end, from the centre of the columns to the wall, a space which it is almost impossible could be roofed by any of the expedients known to the Greeks; and the flanks are almost equally intractable. It was this that rendered Lieutenant Smith's restoration so unacceptable. He boldly and honestly faced the difficulty, and so far he did good service, and deserves all praise. Mr. Pullan's expedient of cutting 6 inches off each intercolumniation is not so creditable, nor is the result much more satisfactory.

After trying several others, the solution appears to me to lie in the hypothesis that the angle columns were coupled,--or, in other words, half an intercolumniation apart from centre to centre.

Should it be asked if there are any other examples of this arrangement, the answer must probably be that there are not; but there is also no other building known with a pyramidal roof, or which, from its design, would so much require strengthening at the angles. The distance between the columns and the front must necessarily be so great,--the height at which they are placed is so considerable,--and the form of the roof so exceptional, that I feel quite certain any architect will admit that this grouping together of the angle columns is aesthetically an improvement.

Although this arrangement may not be found in any Ionic edifice, it is a well-known fact that in every Doric Temple the three columns at the angles are spaced nearer to each other than those intermediate between them, either in the flanks or front. The usual theory is that this was done to accommodate the exigencies of the triglyphs. It may be so, but the Greeks were too ingenious a people to allow any such difficulty to control their designs if they had not thought it an improvement to strengthen the angles of their buildings. We may also again refer to the Lion Tomb at Cnidus , where the angle intercolumniations are less than the centre ones, for no conceivable reason but to give apparent strength to that part.

The proof, however, must depend on how it fits with the other parts.

Taking first the flanks, we have 8 whole and 2 half intercolumniations, equal to 94 feet 6 inches Greek, or 48 cubits, or just once and a half the length of the cella; which is so far satisfactory. At the back of the gutter behind the cymatium there is a weather mark which certainly indicates the position of the first step of the pyramid, and, according to Mr. Pullan's restoration of the order, this mark is 2 ft. 8-1/2 in. beyond the centre of the columns. As there are a great many doubtful elements in this restoration, and as, from the fragmentary nature of the evidence, it is impossible to be certain within half an inch or even an inch either way, let us, for the nonce, assume this dimension to be 2 ft. 9 in. Twice this for the projection either way, or 5 ft. 6 in., added to 94 ft. 6 in., gives exactly 100 Greek feet for the dimension of the lowest step of the pyramid. So far nothing could be more satisfactory; but, if it is of any value, the opposite side ought to be 80 feet,--or in the ratio of 5 to 4.

On this side we have 6 whole and 2 half intercolumniations, or 73 ft. 6 in.,--to which adding, as before, 5 ft. 6 in. for the projection of the step, we obtain 79 feet! If this is really so, there is an end of this theory of restoration on a system of definite proportions; and so for a long time I thought, and was inclined to give up the whole in despair. The solution, however, does not seem difficult when once it is explained. It probably is this: the steps of the Pyramid being in the ratio of 4 to 5, or as 16?8 in. to 21 inches Greek, the cymatium gutter must be in the same ratio, or the angle would not be in the same line with the angles of the steps or of the pedestals, or whatever was used to finish the roof. In Mr. Newton's text this dimension is called 1 ft. 10 in. throughout; according to Mr. Day's lithographer it is 1??88, which does not represent 1 ft. 10 in. by any system of decimal notation I am acquainted with. According to Mr. Pullan's drawing it scales 2 feet. From internal evidence, I fancy the latter is the true dimension. Assuming it to be so, and that it is the narrowest of the two gutters, the other was of course as 4 is to 5, or as 2 feet to 2 feet 6 inches, which gives us the exact dimensions we are seeking, or 6 inches each way. This I feel convinced is the true explanation, but the difficulty is that, if it is so, there must be some error in Mr. Pullan's restoration of the order. If we assume that we have got the wider gutter, the other would be 19?2 in., which would be easily adjusted to the order, but would give only 4?8 in. each way, or 1-2/10 in. less than is wanted. It is so unlikely that the Greeks would have allowed their system to break down for so small a quantity as one inch and one-fifth in 40 feet, that we may feel certain--if this difficulty exists at all--that it is only our ignorance that prevents our perceiving how it was adjusted. If it should prove that the cymatium we have got is the larger one, and that consequently this difference does exist, the solution will probably be found in the fact of the existence of two roof stones, with the abnormal dimensions quoted by Mr. Pullan as 10-1/2 inches and 9 respectively. It may be they were 9?? and 10???2, which would give the quantity wanted. But, whatever their exact dimensions, it is probable that they were the lowest steps of the pyramid; and, if the discrepancy above alluded to did exist, they may have been used as the means of adjusting it. Be all this as it may, I feel convinced that whenever the fragments can be carefully re-examined, it will be found that the exact dimension we are seeking was 80 Greek feet.

There is another test to which this arrangement of the columns must be submitted before it can be accepted, which is, the manner in which it can be made to accord with the width of the cella.

The first hypothesis that one naturally adopts is that the peristyle should be one intercolumniation in width, in other words that the distance between the centres of the columns and the walls of the cella should be 10 feet 6 inches. Assuming this, or deducting 21 Greek feet from the extreme width we have just found above of 73 feet 6 inches, it leaves 52 feet 6 inches for the width, which is a very reasonable explanation of Pliny's expression, "brevius a frontibus." It is also satisfactory, as it is in the proportion of 5 to 6, with 63 feet, which is Pliny's dimension, for the length of the cella. But the "instantia crucis" must be that it should turn out--like the longer sides--just one half the lower step, or rock-cut excavation. What this is, is not so easily ascertained. In his letter to Lord Stratford de Redcliffe, of 3rd April, 1857, Mr. Newton calls it 110 feet; in the text it is called 108; while Lieut. Smith, who probably made the measurement, calls it 107 . The latter, therefore, we may assume is the most correct. If the above hypothesis is correct, it ought to have been 106?31 English or 105 Greek feet, which most probably was really the dimension found; but as it did not appear to the excavators that anything depended upon it, they measured it, as before, carelessly and recorded it more so.

In the meanwhile, therefore, we may assume that the width of the cella was 52 feet 6 inches, or 30 Babylonian cubits. The width of the lower step on the east and west fronts was 105 Greek feet, or 60 cubits exactly.

Of course this is exactly in the proportion of 5 to 6 with the longer step, which, as we found above, was 72 cubits or 126 Greek feet; and this, as we shall presently see, was the exact height of the building without the quadriga, the total height being 80 cubits or 140 Greek feet.

Having now obtained a reasonable proportion for the lower step of the Pyramid, 100 by 80 Greek feet, the remaining dimensions are easily ascertained.

Mr. Pullan, using the nearly correct measure of 17 English inches for the shorter step, obtained 32 feet 6 inches English for the spread of the pyramid in one direction. It need hardly be remarked that when there were 24 joints, and each stone sloped slightly backwards instead of having its face perpendicular to its bed, it is impossible now to attain any minute accuracy in this dimension; but 32?5 ft. English is so nearly 32 Greek feet that we may fairly assume that that was the dimension intended, the difference being very slightly in excess of one inch.

In the other direction Mr. Pullan obtained 39? 11-1/2?? English; but as it is impossible, for the reasons just stated, to ascertain to half an inch what this dimension really was, we may assume this to be 40 English feet; and as Mr. Pullan used the erroneous measurement of 21 English instead of 21 Greek inches, we at once obtain 40 Greek feet for the spread in the longer direction, or again in the ratio of 4 to 5.

This leaves a platform on the summit of 20 Greek feet by 16, on which to erect the pedestal or meta, which is to support the quadriga. The question is,--is it sufficient?

According to Mr. Pullan's drawings , the group measures 15 feet English in length by 13? 6?? across, and 12? 6?? from the extreme hoof on one side to that on the other. This, however, hardly accords with the facts stated in the text. It is stated at page 162, that the horses measure each 3 feet 6 inches across the chest, which alone makes 14 feet, supposing them to stand with their shoulders touching each other. Between the two central horses was the pole, which may have measured 9 inches, and as it could hardly be supported otherwise, if of marble, probably touched the shoulder of the horse on either side; and, allowing the same distance between the two outer horses, we get 16? 3?? English, or, as near as may be, 16 Greek feet for the extreme width of the group. This, however, is probably overstating the matter; 3? 6?? seems an extreme measurement, in so far as I can ascertain. There is no proof that they were all so, and 6 inches is sufficient for the width between the outer horses. This dimension may therefore be stated as between 15 and 16 Greek feet. The width of the plinth would be less than either, for a horse stands considerably within his extreme breadth, and I need hardly say that anywhere, but more especially at such a height as this, a sculptor would bring the hoof as near the edge of the plinth as possible. In the Museum, there is one hoof of one of the chariot-horses placed within 2 inches of the edge of the stone on which it stands; but this does not seem to have been an outside stone; though the same dimensions would be ample if it were. There is no difficulty, therefore, in this dimension; the plinth probably may have been 15 Greek feet, which would allow 6 inches either way for the projection of the step.

Before leaving the pyramid, there is one little matter which requires adjustment. Two steps were found differing from the others, and measuring 9 inches and 10-1/2 inches in width respectively. Mr. Pullan places these at the top of the pyramid, where it appears they must have made a very unpleasing break in the uniformity of the lines. I fancy they were the lowest steps of all.

As will be observed from the diagram the lowest step of the pyramid is buried to half its height in the gutter behind the cymatium; and with that projecting 2 feet beyond, it could not be seen anywhere within

In so far as any accordance with Pliny's dimensions is concerned, the height of the pyramid steps is not of the smallest consequence. Whatever is added to the pyramid must be taken from the meta; whatever is taken from the meta, which there is nothing to govern, must be added to the pyramid. What its height really was, can only be ascertained when some system of definite proportions for the vertical heights of the building shall have been satisfactorily settled, which, as will be explained farther on, is rather difficult to establish absolutely, though easy to fix within certain tolerably narrow limits.

With regard to the vertical heights, there is absolutely no difficulty in making them agree with those found in Pliny. The pyramid,--"in metae cacumen se contrahens,"--was 25 Greek cubits, or 37 ft. 6 in. The order was the same in height exactly, and if we choose to assume that the expression "pyramis altitudine inferiorem aequavit" referred to the pteron as the "lower part," it comes out correctly. If we add to the pyramid the quadriga, estimating that at 13? 9??, we have 51? 3??, and taking the same quantity for the basement, we have

or exactly the dimensions found in Pliny.

All this is so clear and so satisfactory, that there the matter might rest. There is no real necessity to look further, were it not that one or two peculiarities come out in the investigation which seem worthy of being noted.

In restoring the basement, after making its entablature of such proportions as seemed to me most appropriate, I was surprised to find, on applying a scale, that I had obtained exactly 37 ft. 6 in. for the height from the ground line to the soffit over the piers. Though I have tried several other dimensions since, this seems so appropriate that, as very little depends on it, we may allow it to stand.

Assuming this, therefore, we find the height dividing itself into three portions, each of which was 37 ft. 6 in., and two which seem to be 13 ft. 9 in. each. But if this were so, we come to the difficulty that there is no very obvious rule of proportion between these parts, which there certainly ought to be. Even if we add the two smaller ones together we obtain 27 ft. 6 in., which, though nearly, is not quite in the ratio of 3 to 4 to the larger dimension of 37 ft. 6 in. If we add to the first 9 inches we get the exact ratio we require; but by this process increase the height of the building by that dimension, which is impossible.

The explanation of the difficulty may perhaps be found in the fact that the order overlaps the pyramid nearly to that extent, as is seen in the diagram It is by no means improbable that the architects made the pyramid 37 ft. 6 in. from the bottom of the bottom step,--as they naturally would,--and measured the order to the top of the cymatium; and consequently these two dimensions added together did not make 75 feet, but 74 ft. 3 in., or something very near to it.

There is a curious confirmation of this in another dimension which must not be overlooked. At page 24 we found the extreme length of the building to be 126 feet, or 72 Babylonian cubits. This ought to be the height; and so it is, to an inch, if we allow the quadriga to have measured 14 Greek feet. Mr. Newton, it is true, makes it only 13 ft. 3 in. English, but it was necessary for his theory of restoration to keep it as low as possible; and, though it may have been only that height, there are no data to prevent its being higher, nor indeed to fix its dimensions within the margin of a foot. Considering the height at which it was seen, there is everything to confirm the latter dimension, which has besides the merit of being exactly one-tenth of the total height of the building.

From these data we obtain for the probable height of the different parts of the building the following:--

or exactly 80 Babylonian cubits, which is probably the dimension Hyginus copied out, though either he or some bungling copier wrote "feet" for "cubits," just as the lithographers have altered all Mr. Pullan's decimals of a foot into inches, because they did not understand the unusual measures which were being made use of.

There is still another mode in which this question may be looked at. It appears so strange that the architects should have used one modulus for the plan and another for the height, that I cannot help suspecting that in Satyrus's work the dimensions were called 21 Babylonian or 25 Greek cubits, or some such expression. The difference is not great , and it seems so curious that Greek cubits should have been introduced at all that we cannot help trying to find out how it was.

In the previous investigation it appeared that the only two vertical dimensions obtained beyond those quoted by Pliny which were absolutely certain were 126 feet or 72 cubits for the height of the building, and 8 cubits or 14 feet for the quadriga. Now, if we assume thrice 21 cubits for the height, we have 63 cubits, and this with 8 cubits for the quadriga, and 9 for the entablature of the basement, making together 17 cubits, complete the 80 we are looking for. In other words, we return to the identical ratios from which we started, of 17?? and 21??, if these figures represented in inches the dimensions of the steps, as they are always assumed to be by Messrs. Newton, and Pullan, and Smith. If it were so, nothing could be more satisfactory; but, to make the ratio perfect, the last dimension, instead of 9 cubits, ought to be 8?8; so that we should get a total of 4 inches too short, instead of being in excess, as it was by the last calculation.

It would, of course, be easy to apportion this as one inch to each of the four parts; but that is inadmissible in a building planned with such exactitude as this, and I therefore merely state it in order to draw to it the attention of some one cleverer at ratios than I am, confessing that I am beaten, though only by an inch.

Having now obtained all the dimensions of the building, except the 411 feet as the "totus circuitus" mentioned by Pliny, to which we shall come presently, the next point is to explain the architectural peculiarities of the structure.

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