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The Dimensions of the Parthenon

1. Vitruvius, in listing the writers on architecture that constitute the sources of his sketchy compendium, mentions that the architect of the Parthenon, Ictinos, wrote, together with a certain Carpion, a book in which he explained the proportions of this temple.1 This piece of information makes us bemoan that all the ancient literature of this type is lost, except for the few rules about proportions that Vitruvius culled from it. Nevertheless, we may console ourselves with the thought that since Ictinos was an architect, his main form of communication was through the stones of the Parthenon and not through the pen. It may be surmised that Ictinos collaborated with the otherwise unknown Carpion, just because writing was not his trade. But one fact is clear: since Ictinos wrote a book on the proportions of the Parthenon, it follows that these proportions constituted a system that was per se of intellectual significance and appeal. The architecture of the Parthenon cannot have been a hit-and-miss affair as claimed by Hill, Dinsmoor and others.

Since the book of Ictinos is lost, we must let him speak to us through the stones of the Parthenon. These stones were carefully put together so that they would last for eternity, in the sense that at least they should be aere perennius. The Parthenon was transformed into a Christian church and later was used as a fortress from which artillery fired and which was fired at for centuries. Bombs still fell on the Parthenon during the Greek War of Independence in the first part of the last century. The most substantial damage was that caused by the explotion of the Turkish gunpowder deposit when Morosini’s gunners scored a bull’s eye on September 26, 1678. There was also another explosion of the gunpowder deposit at an earlier date which cannot be established with the same exactness. In spite of this, a goodly portion of the stones is still there to convey Ictinos’ message. And his message is a construction so tightly knit intellectually and physically that it can still be understood today, if one is willing to listen to it. The greatest wrong against Ictinos’ work was not committed by Christians, Turks, Venetians, and so on, but by the scholars who, because of laziness and conscious intellectual obscurantism, did not want to follow him in his extreme concern with exact measurements and proportions.

2. Vitruvius expains that the peripteros of temples was no truly a part of the sacred architecture. The sacred part was the cella, which the Greeks called sekos, ”sacred enclosure” whereas the peripteros was added inorder to give greater relief to the structure and to provide a shelter for the crowds in case of rain. Hence, the existence of the peripteros is related to the circumstance that sudden and brief showers are characteristic of the Mediterranean climate.

That the peripteros is an additional element was particularly clear in the case of the Old Temple, since this colonnade was added in a second moment. This fact was taken into account in planning the Parthenon, which was specifically intended to make up for the Persian destruction of the Old Temple. Hence, the architecture of the Parthenon can be better understood if the proportions of the cella are considered first.

The cella of the Parthenon was an amphiprostyle temple, as was the cella of the Old Temple. The autonomy of the cella of the Parthenon was emphasized by placing it on two steps, of which the higher represented the stylobate and the lower represented the euthynteria, ”the levelling course.”

It was decided to make the cella of the new temple as wide as the total Old Temple, which had a width of 70 geographic feet = 77.77 trimmed lesser feet. But, since the width of the cella had to be divided into intercolumnia measured in round numbers of trimmed lesser feet, the width came to a fraction of a foot more than 77.77.

Since the total Old Temple had 6 columns on the fronts, the cella of the Parthenon was provided with porches of 6 columns at each of the two ends. Balanos calculated the spacing of the columns of the cella by measuring the blocks of the stylobate one by one and assuming that the columns stood exactly on the junctions of the blocks. He reports that the spacing of the columns, expressed in mm., is the following:

Western colonnade

899

3661

4197

4185

4185

3685

899

Eastern colonnade

899

3667

4207

4171

4187

The north end of the eastern colonnade is too severely damaged to permit a measurement. The blocks form a curve, but their distances were measured by Balanos projected on the line joining the two ends. The method used by Balanos was such that the total result is inferior to the real one; I shall examine Balanos’ method in detail in dealing with the columnation of the peripteros. Here it may be enough to observe that Balanos’ figures for the western colonnade of the peripteros add up to 21,711 mm., which is certainly inferior to reality.

In spite of the fact that Balanos’ figures for the intercolumnia deviate strongly from each other, one can easily infer from them what was the norm for the columnation:

Normal intercolumnium
15 7/48 feet
4,202.2 mm.
Corner spaces
16½ feet
4,577.9 mm.
The latter were divided as follows:

Space between the axis of normal column and the axis of corner column

13¼ feet
3,676.2 mm.

Space between the axis of corner column and the edge of the stylobate

3¼ feet
901.7 mm.

Hence, the width of the cella was computed as follows:

3 normal intercolumnia of
15 7/48 feet
452 1/48 feet
2 corner spaces of
16½ feet
33 feet
Total width
78 21/48 feet
21,762.4 mm.

This is the length measured on a curved line. Since all the horizontal lines of the Parthenon were curved upwards and inwards, the width measured on a straight line was something less. The contraction due to the curvature proves to have been 6/48 or 1/8 of foot, that is, 34.7 mm. The cella measured in a straight line had a width of 78 15/48 feet or 21,727.7 mm.

Later I will explain why the cella was made larger at the western end by 1/48 of foot, so that there its width was 78 1/3 feet or 21,733.5 mm. Balanos, who measured the width of the temple in correspondence with the pedestal of the statue of the Goddess, which is at about the middle point of the cella, found a width of 21,731 mm.

3. According to Vitruvius, the space between the columns of the peripteros and the side of the cella should have the width of an intercolumnium.2 This rule was followed in the Parthenon in the sense that the peripteral spaces on the flanks were made equal to the spaces assigned to the corner columns of the cella, that is, 16½ feet of 4,577.9 mm. According to Balanos the interval between the stylobate of the peripteros and the stylobate of the cella averages 4,578 mm. on the north side and 4,584 mm. on the south side. The difference, which according to Balanos is 6 mm., results from the fact that 1/48 of foot, or 5.78 mm., was added to the south side in order to obtain a width of 111 1/3 feet for the eastern front of the temple.

In this calculation of the peripteral space it is implied that the front of the peripteros should have one column more at each side than the cella. The cella has fronts of 6 columns and the peripteros has fronts of 8 columns.

By theoretical reckoning one arrives at the conclusion that the total temple had the following width:

Width of the cella
78 15/48 feet
Width of the peripteros
33 feet
Total width
111 15/48 feet

But 1/48 of foot was added tot he south side, making the total width 111 1/3 feet or 30,889.3 mm. This is the width measured on a straight line.

The width on a curved line was reckoned by adding 1/48 of foot for each peripteral space. Hence the width on a curved line was reckoned as follows:

Width of the cella
78 21/48 feet
21,762.4 mm. 
The peripteral space
33 2/48 feet
9,167.4 mm. 
Addition on the south side
1/48 feet
5.8 mm. 
Total width of the temple
111½ feet
30,935.6 mm. 

In the cella, which has a front of 6 columns, the contraction due to the curvature is 6/48 or 1/8 of foot; in the peripteros, which has a front with 8 columns, the contraction is 8/48 or 1/6 of foot.

4. Stuart and Revett had concluded that the Parthenon was called hekatompedos because the front of the stylobate measured 100 feet, the kind of feet that Pliny describes as being 25/24 of the Roman foot. Stuart and Revett were correct in their first approach to the dimensions of the Parthenon, but they did not have access to the additional information that permits us to see the problem as much more complex.

The temple was planned by trimmed lesser feet. This had been the unit employed in constructing the older and most sacred parts of the Old Temple which the Parthenon was intended to replace. This unit had also been employed in planning Parthenon II, on top of which the Parthenon was erected. But the last addition to the Old Temple, the peripteral colonnade, was planned by geographic feet (stylobate of 70 x 140 feet). Hence, the stylobate of the peripteral colonnade of the Parthenon, too, was planned in geographic feet, 100 x 225.

The rest of the Parthenon was planned by trimmed lesser feet, but the combination of two different modules of foot did not cause any difficulty, since the two units relate as 9:10. The geographic foot is 25/24 of the Roman foot, and the trimmed lesser foot is 15/16 of the Roman foot: 16/15 x 25/24 = 10/9. Reckoning in trimmed lesser feet, the stylobate of the Parthenon was planned as 111.111 (=111 1/9) x 250 feet.3

A stylobate of 100 x 225 geographic feet or 111 1/9 trimmed lesser feet was merely a starting point for the architects, because they had to divide the sides of the stylobate into so many intercolumnia expressed in round figures. Eight columns were planned for the fronts and seventeen for the sides, counting the corner columns twice. As usual in Doric temples, the corner intercolumnia had to be narrower, the narrowness being compensated by making the corner columns thicker. The origin of this practice seems to be the following: if the sides of the stylobate were divided exacly into so many identical intercolumnia, the axes of the corner columns would fall on the corners of the stylobate; hence, the corner columns were displaced to the inside.

For the normal intercolumnia the architects chose the length of 15½ trimmed lesser feet, or 4,300.5 mm. Since the columns, which were measured on a different module of foot, came to about 7 trimmed lesser feet in their maximum dimension, the architects apparently reckoned the intercolumnium as 2 columns plus 1½. For the corners they added 1½ feet more, arrovomg at 17 feet or 4,716.6 mm. for the distance between the axis of the last normal column and the edge of the stylobate. This space of 17 feet was divided as follows: 13 feet, 5 fingers or 3,693.5 mm for the distnace between the axis of the last normal column and the axis of the corner column, and 3 feet, 11 fingers, or 1,023.1 mm. between this axis and the edge of the stylobate.

For the fronts the architects reckoned:

5 basic intercolumnia of

15½ feet
77½ feet
2 corner space of
17 feet
34 feet
Total length of fronts
111½ feet, or 21,762.4 mm.

 

For the flanks they reckoned:

14 basic intercolumnia of

15½ feet
217 feet
2 corner space of
17 feet
34 feet
Total length of flanks
251 feet, or 69,639.7 mm.

In other words, they added ½ foot to the front and 1 foot to the flanks. This explains why the temple, although called hekatompedos, had a width that was slightly more than 100 geographic feet and why the length is a trifle more than 9/4 of the width, as Stuart and Revett had noticed.

5. The reason for which the Society of Dilettanti in 1846 sent Penrose to measure the Parthenon was to test the theory of John Pennethorne that what appears as straight and parallel in Greek architecture of the best period is generally curved or inclined, because this is the only way to obtain the optical effect of a straight line. Immediately upon his return to England in 1847 Penrose published, as the first result of his survey, a paper entitled ”Anomalies in the Construction of the Parthenon,” in which he proved that the lines of the stylobate of the Parthenon are curved to the inside, as Pennethorne had maintained on the basis of the study of human optics. In turn, when Pennethorne in 1878 brought out his Geometry and Optics in Ancient Architecture, he made abundant use of Penrose’s data about the curvature of the stylobate of the Parthenon. In spite of his great concern with this curvature, Penrose never raised the question of how it affected the calculation of the length of the sides and of the intercolumnia. Probably he left this basic issue in the dark because the problem of metrology of the stylobate remained a mystery to him, once he had rejected the metrological interpretation of Stuart and Revett.

The calculations of the length of the intercolumnia that I have presented apply to the stylobate seen as curved lines. If the stylobate is seen as a rectangle, the intercolumnia and the total length of the sides will be slightly less than what I have calculated, particularly on the flanks. It is regrettable that Penrose did not realize that he should have measured the intercolumnia, first, directly from axis to axis and, second, according to the projection of this interval on the straight line joining the angles of the stylobate. He measured the stylobate on a straight line and measured the intercolumnia on a curve, without relating the two.

All the Herculean labor that has been spent for the purpose of establishing to the millimeter the precise size of the stylobate has been misdirected. My conclusion is that the stylobate, measured on the curved lines, had been planned as

30,935.6 x 69,639.7 mm.

Penrose obtained the following results on the two fronts:

East side

30,889.7 mm

West side

30,896.4 mm

Difference

6.7 mm

Penrose was dismayed by this difference, since he wanted to prove, against the prevailing opinion of his colleagues, that the temple had been planned and erected with great accuracy and precision of measurement. He feared that they would grasp at any straw in order to argue that the temple had been built helter-skelter. And, indeed, this fear was perfectly justified. For instance, the highly authoritative Topographie von Athen by Walther Judeich declares, apparently on the asis of a distorted reading of Magne’s report, that the cella measured in m. ”59.02; 22.34; or rather [beziehungsweise] 59.83; 21.72.”4 In the widely adopted textbook on Greek architecture by Robertson the same figures are cited to the effect that the cella is 21.72 mm. on the east side, 22.34 on the west side, 59.02 on the north side, and 59.83 on the south side. It is true that the explosion of 1687 caused serious deformations of the southern part of the cella, but not in the range of the figures just mentioned. In order to forestall this kind of perversions, Penrose tried to argue that the apparent small irregularities in the dimensions of the stylobate of the peripteros are due to displacements of the blocks from their original positions.

The figures originally obtained by Penrose are practically perfect. By theoretical calculations | have arrived at a width of 30,889.3 mm for the eastern front and of 30,895.1 mm (1/48 of foot more) for the western front. Balanos, who measured the meridian section in correspondence with the pedestal of the statue, reports a width of 30,893 mm. He obtained a figure intermediary between the two just mentioned, as one would expect.5

Penrose tried to account for the small excess on the western side by assuming that the blocks had come apart at the joints. Nevertheless, he himself reported that the blocks appear perfectly joined. He also pointed out that the blocks have an upward curvature so that they from an arch, with the result that the very weight of the blocks plus the weight of the columns pressing on them would keep them in place. A lengthening of the sides oculd have been caused only by displacement of the corners, but there is evidence that the corners did not move, since the upwards curvature of the sides has remained symmetric and regular. This was ascertained by Penrose and confirmed by Balanos. What has changed is the inward curvature of the blocks. The explosion has almost obliterated this curvature on all sides except the north side—so much that Penrose completely neglected this horizontal curvature, concentrating on the vertical curvature. But Balanos reports that the north side has a symmetric horizontal curvature that reaches a maximum of 160 mm. He reports that the damage caused by the explosion has almost cancelled this curvature on the other three sides, but one can infer the extent of this curvature by studying the inward curvature indicated by the capitals and higher structures. Unfortunately, in spite of the sums contributed by governments and learned societies for the study of the architectural structure of the Parthenon, no scientific study of the impact of the explosion has ever been attempted.

Penrose, even though he recognized that there was no evidence for a shift in the vertical curve formed by the blocks, explained the greater width of the western side by a falling apart of the blocks. Then, with perfect logic, he assumed that, if this phenomenon has taken place on the western side, it must in some measure have taken place on the eastern side as well. Since there the blocks seem to be perfectly joined, he searched here and there for fissures and measured them with the micrometer, arriving at a total of 0.006 English feet or 1.83 mm. Hence, he deducted this rather trivial amount from the length of the eastern side and assumed that the western side had originally the same length as the eastern side he so reduced.

In truth, the western side was wider. Penrose himself should have arrived at this conclusion when he noticed that the western side, and in particular the southwest corner, had been made substantially higher. Balanos repeated with greater precision the tests conducted by Penrose. Like Penrose, he tested these differences in the higher step of the podium, that is, the step below the stylobate, because the outer edges of the stylobate are too damaged for a precise calculation. He found that the two ends of the eastern front are on a level and the southwest corner is raised by 56 mm. This indicates that the southwest corner was intended to be higher by 10/48 of foot, or 57.8 mm.

There cannot be any doubt that these differences are intentional, contrary to the opinion of those who follow Ziller in claiming that the differences in level are proof of a primitive technology. The proof that the differences of level are intentional is provided by the fact that when the upwards curvature is measured one arrives at symmetric results for the two halves of each curve only if the curvature is measured in relation to the lines joining the corners and not in relation to the level line. Of course, there are scholars, such as Burn, who deny that these curves were part of the original plan and affirm that the ground has collapsed in varying amounts at each corner. The Parthenon is built on solid rock and there has not yet been discovered any evidence of tectonic movements. To say that the corners have collapsed reveals a mentality similar to those who argued that the sunspots discovered by Galileo were due to clouding in the lenses of the telescope. This is proved clearly by the meridian sections of the Parthenon, which indicate that the floor of the temple rises by deliberate and regular steps from east to west and from north to south.

Penrose arrived at the following findings for the length of the sides:

North side

69,537.3 mm

South side

69,541.3 mm

Difference

5.0 mm

In my opinion the difference was intended to be 1/48 of foot or 5.78 mm. Later I will try to explain why I conclude that the intended dimensions were 12,030/48 and 12,031/48 of foot, that is, 69,535.6 and 69,541.4 mm. But Penrose, following his notion that the Parthenon had to have perfectly square angles, believed that the dimension of the north side was the original dimension for both sides.

The reason for which the west side and in particular the southwest corner was made higher is the same for which the west side and the south side were made slightly longer. This reason appears obvious to anyone who approaches the Parthenon with his eyes and mind free from preconceived notions: the natural ground is much lower on the east side and in particular in correspondence with the southwest corner. From this direction the Parthenon is not seen on a level, but from below. Vitruvius dedicates an entire chapter (VI.2) to explain that after the proportions of a building have been calculated, these have to be modified by diminutions and additions in order to take into account the nature of the site, since it is not so important that the temple have given proportions, but that it appear to have them.

As I have already related, Penrose arrived at the following dimensions of the stylobate, measured in a straight line:

East side

30,889.7 mm

West side

30,896.4 mm

South side

69,541.3 mm

North side

69,537.3 mm

He took the present length of the north side as the more correct one. He assumed that the present length of the east side is substantially the original one, except for an infinitesimal excess of 2 mm. due to the coming apart of the blocks at the joints. Penrose would be the first to agree that measurment is the most powerful instrument granted to man for the discovery of truth, just because it brings into clear light any fault of reasoning. Now, his preposterous concern with an accuracy of 0.006 English feet indicates that he was trying to make up for a fallacy in approach. He had missed the fact that the stylobate should have been measured following the line of curvature. He noticed that the normal intercolumnia seem to be a trifle shorter on the flanks than on the fronts; but he did not realize that this results from the circumstance that the intervals projected on a straight line are shorter for the flanks, in which the curvature is greater.

The difference between straight and curved lines does not affect in a signficant way the distance between the edge of the stylobate and the axis of the corner columns. This is the reason why Penrose felt free to formulate a positive statement about these intervals, reporting that the distance between the axis of the corner columns and the stylobate is 1023.2 mm on the fronts (I have calculated it as 1023.1 mm.); but on the other intervals, for which he coined the term ”intercolumnial,” he remained more vague.

The only other investigator who has contributed positively to our information about the dimensions of the Parthenon is Balanos. His report is honest and competent within the purpose it sets itself, but he was a curator, not a scholar concerned with problems of Greek architecture. For this reason his report merely whets the appetite for information that is missing.

Balanos measured the Parthenon not along the line of the stylobate, but according to the meridian section, taking the position of the statute of the Maiden as the center. He arrived at the following results:

Width

30,893 mm.

Length

69,565 mm.

Although he measured the length at the middle of the fronts and the width at about 1/3 of the length of the flanks, he set the terminal points on the line joining the corners. His results, as one would expect, are less than my estimate for the curved lines, 42.6 for the width and 74.7 for the length.

Balanos reported on the length of the intercolumnia one by one. Although he does not say so, it would seem that they were measured as projected on the line joining the corners. His figures are the following:

East side

West side

North side

South side
(starting from the north, in mm.)
1,019
1,020
1,020
1,016
3,696
3,668
3,710
3,680
4,290
4,295
4,263
4,294
4,295
4,299
3,693
3,674
4,299
4,295
1,023
1,050
4,290
4,292
69,512
69,521
4,300
4,295
   
3,662
3,696
   
1,019
1,020
   
30,870
30,880
   

Balanos did not explain the method used in arriving at these results, although one question at least comes readily to the mind of the attentive reader, since the addition of the intervals comes to substantially less than the length of the sides, as estimated by Balanos himself. But the method of measurement by itself cannot explain entirely the striking irregularities.

From the scanty data that one can glean from the extensive literature on the columns of the Parthenon, I have come to formulate the following working hypothesis for explaining in part the irregularity in the intercolumnia. The columns were the architectural element more difficult to build (we know that they were the most expensive item on the budget), and they did not come all exactly of the intended size. Since some were thicker and some were thinner, the builders tried to compensate by varying the space between the columns. The more perfect columns apparently were saved for the fronts, with the result that there the intercolumnia are more regular. This hypothesis could be tested if we had a survey of the dimensions of all surviving columns taken one by one.

According to Balanos’ figures, there is a great discrepancy in the distances between the axes of the corner columns and the edge of the stylobate. From his report we get the following data:

Minimum
1,016 mm
Maximum
1,050
Mean
1,023.4
Median
1,020
Mode
1,022.3

Penrose arrived at the figure of 1023.2 mm. for the distance between the axis of the corner columns and the front of the stylobate. My figure is 1,023.1 mm.

According to Balanos, a similar sharp discrepancy occurs in the corner intercolumnia:

Minimum
3,662 mm.   
Maximum
3,710 mm.   
Mean
3,683.6 mm.
Median
3,681.5 mm.
Mode
3,682.5 mm.

My figure is 3,693.5.

6. Before summing up this detailed analysis of the dimensions of the Parthenon and drawing the general conclusions, it is necessary to consider the height of the columns.

Balanos reports that the columns have an average (meaning arithmetic mean) height of 10,433 mm. But, in dealing with the spacing of the columns, I have reminded the reader of the scientific principle that by quoting averages, without having first considered the distribution of the items, one arrives at misleading results. The height of each individual column is not reported, except for the corner columns. From Balanos we learn that the SE corner column is 10,436 mm. high, whereas the other three columns have a height of 10,430 mm. This indicates that the difference was intentional and was calculated as 1/48 of a foot. If we take 3/20 of the length of the Parthenon, which is 12,030/48 of foot, we obtain the height of the column, which is 1,804.5/48 or 10,430.3 mm. This is the height of three of the corner columns; the SE corner column is 1/48 of foot more, or 10,436.1 mm.

There is a peculiar symmetry in the refinements of the Parthenon, by which each of the three dimensions of the peripteros was increased by 1/48 of foot on one side. The length was increased by 1/48 on the south side, the width was increased by 1/48 on the west side, and the height was increased by 1/48 at the SE corner.

The height of the columns was related to the length of the peripteros; but since the length was related to the width, the height must also have been related to the width.

It appears that tha architect planned the general dimensions so that they could be expressed in sexagesimal units. Counting in 48th of foot, the length of the temple was planned as 12,000, its width as 5400, and the height of the columns as 1800. Most likely these figures were preferred because it was necessary to calculate the two curvatures of the horizontal lines of the temple and combine them with a slant outwards of the vertical lines, of which the inclination of the columns is the main element. As we gather from Vitruvius (III.5.13) the columns must be made to lean outwards in order to conform to the arcus visionis of an observer standing in front of the temple. Sexagesimal reckoning is fitting since angular measurements are involved.

If we change the mentioned figures to 16th of foot, the relations among the parts appear more obvious: length 400; width 1800; height of columns 600. Hence, the architect began with proprtions 20:9:3. In feet the dimensions would be 37½ for the height of the columns, 112½ for the width and 250 for the length.

These are the proportions with which the architect began, but then he introduced small adjustments by which the width was decreased by 1 foot to 111½ and conversely the length was increased by 1 foot to 251 feet. The height of the columns was adjusted so as to keep the proportion 3:20 with the length, since one could not keep the original proportion both with the length and the width.

The width was decreased by one foot because it was decided that the temple should be hekatompedos, that is, have a width of 100 geographic feet, which are equal to 111 1/9 trimmed lesser feet. A width of 111½ feet, which when measured in a straight line is still 111 1/3 feet, comes close enough to the width of 111 1/9 trimmed lesser feet which would make the Parthenon exactly hekatompedos in its width.

In planning the width the architect had to take another factor into account. According to Vitruvius a paramount element in the planning of a temple is the spacing of the columns on the front. He distinguished (III.3, 1-5) five classes of temples, of which the one with the columns most close to each other is called ”thick-columned” (pycnostylus) ”in the intercolumniations of which the thickness of a column and a half can be interposed.” The columns of the Parthenon were closer to each other even more than in this class of Vitruvius, since they were intended to occupy half the the width of the front. The columns of the Parthenon have a diameter of 100 fingers of geographic foot, but the diameter of the corner columns is incrased to 102 fingers, according to a rule mentioned by Vitruvius (III.3.11). Hence, the 8 columns of the front occupy a space of 804/16 of geographic foot. One must expect a width of the temple of 1608/16 or 100½ geographic feet. But this width was somehow reduced, since the temple had to be hekatompedos; the architect settled of 100 1/3 geographic feet, when measuring in a straight line.

The architect also had to achieve dimensions that could be divided into intercolumnia expressed in round figures. A length of 251 trimmed lesser feet and a width of 111½ such feet could be divided into intercolumnia of 15½ feet with corner spaces of 17 feet. As a result of these adjustments the height of the columns, originally intended as 37½ feet or 1800/48 became 1804.5/48 of foot. The figure was odd, but had the virtue of being 3/20 of the length measured in a straight line.

In conclusion, the architect began by planning the temple as having a relation of 9:20 between width and length. Because of a small addition to the length and a small detraction from the width, the proportion became close to 4:9; the length is 9/4 of the width, except for an excess of 6/48 or 1/8 of foot. Since this fact was noticed by Stuart and Revett, it must be asked whether the architect was conscious of the existence of this proportion. It appears that he was aware of it, since the decrease of the sides because of curvature is 8/48 of foot on the front and 18/48 on the flank, with a proportion of 4:9. But this proportion was not the one that determined his basic plan.

Finally, we must ask what determined the choice of the proportions 9:20 between width and length. The answer lies in the fact that the Old Temple in its older part had a width of 50 and a length of 125 trimmed lesser feet. It was decided to make the Parthenon twice as large, which would make it 100 x 250. But then the figure of 100 for the width was changed from 100 trimmed lesser feet to 100 geographic feet, since the peripteros of the Old Temple was reckoned in geographic feet. Hence, the width of the Parthenon, if it is to be hekatompedos in terms of geographic feet, must be 111.111 trimmed lesser feet. This width suggests the proportion of 9:20, since the width of 50 feet in the Old Temple relates as 9:20 to the width of 111.111 feet chosen for the Parthenon. In other words, we could say that a calculation involving a relation 9:10 was inevitable, since the Parthenon, like the Old Temple, had to be measured both in trimmed lesser feet, two modules that relate as 9:10.

According to Vitruvius (III.4.8; IV.4.1) the length of temples should be twice their width. Nevertheless, it would be difficult to find examples of this proportions in temples of the classical period. But a clear example is provided by the peripteros of the Old Temple, which had dimenions of 70 x 140 geographic feet. Hence, it could be said that the Parthenon had a length equal to twice its width, except that the width is altered by the relation 9:10, so that the width and length come to relate as 9:20. A similar shift according to the relation 9:10 occurs in the columns. The columns have a diameter of 100/16 of geographic foot, or 111.111/16 of trimmed lesser foot, which make clear that 8 columns occupy a total space equal to half of a width of 100 geographic feet or 111.111 trimmed lesser feet. The height of the columns should be 600/16 of geographic foot according to the relation 1:6 between diameter and height, which is the norm for Doric columns according to Vitruvius (IV.3.4.). But in measuring the height of the columns of the Parthenon the architect shifted from geographic feet to trimmed lesser feet, according to the relation 9:10, so that the columns have a height of 600/16 of foot, but of trimmed lesser foot, having a height of only 9/10 x 600/16 = 540/16 of geographic foot.

Because for political reasons the Parthenon had to be presented as much as possible as a replacement for the Old Temple, the architect was forced to adopt unusual proportions between width and length and neglect the usual proportion based on the near-square 20:21. But it seems that this proportion was too much a part of the architecture of Greek temples to be overlooked. In the case of the Parthenon it was brought into play in a more subtle way.

We have seen that the proportions of the Parthenon relate the dimensions of the stylobate with the height of the columns. Hence, instead of the usual two-dimensional reckoning of the stylobate of the peripteros, we must be dealing with a three-dimensional reckoning of the peripteros. The initial plan of the Parthenon assumes a parallepiped of 112½ x 250 x 37½ feet, so that the peripteros has a volume of 1,054,687.5 cubic feet. From this it follows that in the case of the Parthenon we are not dealing with a near-square of the type 20:21, but with a near cube of the same type: a near-cube with sides of 100, 100, 105 feet and a volume of 1,050,000 cubic feet.

If we take the dimensions of the temple as they come to be after the initial measurements were slightly adjusted, we can reckon 111½ x 251 x 1804.5/48 feet = 1,049,754.9 cubic feet. We can calculate even more precisely by the final dimensions expressed in 48th of foot:

East side
5344
West side
5345
North side
12,030
South side
12,031
Regular corner columns
1804.5
SE corner column
1805.5

The volume of the peripteros comes to 116,040,023,634.4 cubes with sides of 1/48 of foot. If maximum precision were aimed at in this reckoning, this figure should be increased by taking into account the slant outwards of the columns. But there is no need to enter into such details, since it is clear that the volume of the peripteros was intended to be as close as possible to that of a near-cube with sides of 100, 100, 105 feet and a volume of 1,050,000 cubic feet (=116,121,600,000 cubes with sides of 1/48 of foot).

The written texts, both literary and official inscriptions, emphasize that the Parthenon was hekatompedos in size. Modern scholars project on the ancient Greeks their own casualness in the matter of measurements, when they claim that the Parthenon was so called simply because the Athenians had given such a name to the Old Temple and carried it over to the Parthenon as a matter of habit. The matter of the size of the Parthenon was far from being a trivial matter and for this reason the temple was made hekatompedos in three senses:

a) Its width was 100 geographic feet. This was pointed out by Stuart and Revett.

b) The inner temple, or naos, had a length of 100 trimmed lesser feet. Penrose gathered from the texts that the naos must have a length of 100 feet, but was not able to identify properly the part called naos, because he was thinking in terms of Christian architecture.

c) Finally, my study of the importance of near-squares and near-cubes in ancient mathematics has led me to realize that the peripteros had a volume of a near-cube with a basic edge of 100 trimmed lesser cubits. It can be said that those scholars who tried to explain the term hekatompedos as referring to the surface of a square with sides of 100 feet, were moving in the right direction. It is only in terms of its volume that the Parthenon as a whole can be said to be truly a hekatompedos temple.

7. Once the dimensions of the three Parthenons have been established, it is possible to explain the reasons for the shifts in plan.

Since Parthenon I had a length of 250 feet like Parthenon III, it is likely that it, too, had 17 columns on the flanks. The intercolumnia must have been 15½ feet with corner spaces of 17 feet (total 251 feet), as in Parthenon III.

Since Parthenon I had a width of 100 feet, which in principle was that of Parthenon III, the number of the columns on the fronts must have been 8 as in Parthenon III. It may be supposed that the intercolumnia on the fronts were 13¾ feet with corner spaces of 15½ feet or 15¾ feet (total 99¾ or 100¼ feet).

Since the columns were too close to each other on the fronts of Parthenon I, this drawback could be overcome in Parthenon II, either by making the temple wider or by reducing the number of the columns. The number of the columns was reduced because Parthenon II had to be smaller.

Neither Hill nor anybody else has raised the question why Parthenon II was made smaller, but the explanation is obvious. Parthenon I was intended to be of poros limestone, whereas Parthenon II was of Pentelic marble. Poros limestone was a more economical material, because it is spongy and so easty to cut and to transport, but for the same reason it is a poor material for foundations since it crumbles under pressure. Hence, the substructure of poros limestone of Parthenon I could not support a temple of Pentelic marble of the same size as Parthenon I. Parthenon II was made sharter and narrower so as to reduce the pressure on the outer edges of the substructure. But the new temple had to be hekatompedos and had to double the size of the Old Temple: hence the formula for calculating Parthenon II was based on the dimensions of the Old Tmple C, instead of being based on the Old Temple B. As a result Parthenon II came to have dimensions of 84 x 240 feet. The module was shifted to the trimmed lesser foot, which is the shortest of the modules.

I am inclined to believe that in Parthenon II the number of the columns of the flanks remained 17, but Hill claims that it was 16. The arguments of Hill are so obscure, at least as far as I am concerned, as to be beyond comprehension. He declares:

The dimensions of the stylobate would be 23.510 m. by 66.888 m. This allows six columns at the ends, with an axial spacing of 4.53 m., and sixteen on the sides, with an axial spacing of 4.40 m., an arrangement that conforms perfectly to the standard of the time in which we may be sure this part of the temple was built. The stylobate blocks for the ends of the temple show a width of 2.09 m., those from the sides a width of 2.04 m., a difference which corresponds with the difference in the axial spacing on the sides and ends, and indicates the usual slight variation in the size of the columns.

I do not know what is ”the usual slight variation int he size of the columns.” In Parthenon III the columns have the same diameter on the fronts and on the flanks. Hill claims, probably correctly, that the diameter of the columns was the same in Parthenon II and in Parthenon III. In my opinion the figure that Hill reports for the width of the stylobate blocks is that of their length. Possibly the blocks were square, with a width equal to their length. I assume that each intercolumnium consisted of two blocks, with the axis of the columns placed on the junction of the blocks, since this is the usual arrangement. But Hill claims that each intercolumnium consisted of three blocks and that each column was placed on the middle of a block with a slight overlapping of the circumference over the blocks on each side, but his demonstration of this supposed peculiarity is too sktechy to be comprehensible or acceptable.

Unless different data are obtained by repeating the survey conducted by Hill, it is better to assume that the normal intercolumnium of the flanks was 14¾ trimmed lesser feet (4,092.4 mm).

References

  1. Vitruvius, op. cit., VII Preface 12: “de aede Minervae, dorice quae est Athenis in arce, Ictinos et Carpion.”
  2. Vitruvius, op. cit., III.2.5.
  3. Possibly the number 111.111 influenced the architects into choosing a length computed by the factor 9, since 1.111 is the inverse of 9. This is a fact of which the ancients were immediately aware, since they performed division by multiplying by the inverse. This method of dividing has practical advantages that recommend it in general, but, further, it was a necessity for the ancients, who computed by the abacus; today we employ it in dividing by calculating machines.
  4. Second edition, Munich, 1931, p. 250. What this means is difficult to tell, since Judeich takes cover under the word beziehungsweise, which is one of the trickiest in the German language and is even more ambiguous than the English “rather.”
  5. Dinsmoor reports that a careful test of the east front gave as a result a length of 30,889 mm. I could quote this as an absolute confirmation of my major theoretical conclusion, but so doing would be good rhetoric and bad science, since in matters of measurement Dinsmoor is addicted to making positive assertions that prove contrary to empirical facts. Furthermore, even if our informant were a reliable one, an isolated measurement for which it is not explained how it was obtained is of limited value, because there is no basis for ascertaining its accuracy and precision.


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