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Roman and Egyptian Foot

1. The arguments used by the new school of metrology to assert that it is not true that the ancients derived the units of volume and weight from the cube of the units of length, are so manifestly contrary to known historical facts and to textual evidence, that their universal acceptance by scholars of the twentieth century is a most striking phenomenon. I have analyzed in detail the development of new views of ancient culture by which the facts ascertained by metrologists of what is now called the old school, have to be rejected a priori as contrary to the spirit of ancient civilization. There remains still that in spite of the prejudices developed in the academic world, metrologists of the old school had such a strong case that they should have succeeded in refuting the new dispensation. Reason even though unarmed against the forces of emotional distortion, can still prevail if it relies on its own forces to the very end: reason must be totally rational. The old school was not able to present a case sufficiently strong because it had failed to solve some of the basic problems it had set for itself.

One problem was set by Newton and remained unsolved, namely, the relation between the Roman foot of 296 mm. and the Egyptian foot of 300 mm.

From the end of the fifteenth century to the early nineteenth century, the list of those who Investigated the length of the Roman foot reads like a Who’s Who of European scholarship, both In the humanities and in the sciences. In spite of the efforts of the most learned classical scholars and Orientalists and of the best mathematical minds, such as Kepler, Riccioli, Mersenne, Newton, Boscovich, and Laplace, It was thought that the result had been for a good part negative. In truth, they had obtained perfectly correct evaluations from the empirical data, but could not discover the principle that links them.

I have solved the inner rationale of ancient and medieval units of length, and by implication, of all units of measure, by discovering two facts:

a) that there were four fundamental types of foot related as 15:16:17:18,
b) that each of these types existed in two varieties related as the cube root of 24/the cube root of 25.

I will show that one of the greatest stumbling blocks in the research was the notion, based on a false interpretation of the Bible, that up to the time of Archimedes the ancients had calculated pi as equal to 3. Measures were constructed so as to obtain mechanically pi=31/8, pi = 3 1/7, and other even nicer values.

The following table sums up the essence of my contribution to the metrology of linear units.

Corresponding cubit
Lesser foot (15 basic fingers) trimmed natural 277.4489 mm. 281.50 mm 416.173 mm. 421.875 mm.
Basic foot  (16 basic fingers) trimmed natural 295.9454 mm. 300.0 mm. 443.928 mm. 450.0 mm.
Wheat foot (17 basic fingers) trimmed natural 314.4419 mm. 318.750 mm. 471.662 mm 478.250 mm..
Barley foot (18 basic fingers) trimmed natural 332.9384 mm. 337.50 mm. 499.408 mm. 506.250 mm.

In practice standards of length were defined up to the tenth of millimeter, but, since they were calculated as the edges of cubes corresponding to the units of volume and weight, their theoretical definition was much more precise.

2. By the end of the sixteenth century, it had been determined that the Roman foot exists in three varieties of increasing length to which, there was given the name of :

pes Cossutianus
pes Statilianus
pes Aebutianus

It was further determined that if one calculates the Roman foot from the quadrantal, the cube equal to 80 librae of water, and the libra is calculated from the sample weights, one obtains a foot that goes from the length of the pes Aebutianus to a maximum of about 131.8. Paris lines (297.3428 mm.). All that scholars of the seventeenth and eighteenth century could do was to check and recheck the empirical data, arriving always at the same results.

In discussing standards of weights, I have pointed out that the riddle of the edge of the quadrantal, which defied the genius of the best physicists, was solved in 1909 by Decourdemanche when he noted that all ancient units of volume end weight can exist in a form increased as 80:81 (discrepancy komma in my terminology), Segré discovered the theoretical reason for the existence of this discrepancy—even though he considered its application only in the metrology of Greek papyri. The cube of the foot relates to the cube of the cubit as 1:3 3/8, since (1½)3 =3 3/8, but in practice this relation was simplified as 1:3 1/3, with a resulting discrepancy komma. Hultsch discovered that all ancient units of weight were related to the kite (basic sheqel in my terminology), which is 1/10,000 of the cube of the Egyptian normal cubit of 450 mm. (basic load brutto), 91,125 But he did not calculate the basic sheqel as 9.1125 grams, giving to it instead a value intermediary between 9,0 and 9.1125 grams. Thereby he obscured the essential point that the basic sheqel can be calculated also as 1/3000 of the cube of the Egyptian foot of 300 mm.; the unit of 27.000 grams I call basic talent brutto and Oxé calls talent of 1000 ounces. The Attic and the Roman system of measurer were anchored to the foot and not to the cubit, and, hence, reckoned by the basic sheqel of 9.0 grams, assuming an Attic monetary mina of 432 grams (48 basic sheqels) and a libra of 324 (36 basic sheqels); but these units could be legitimately increased up to 437.4 and 328.05 grams, producing a foot increased from 295.9454 up to a maximum of 297.1734 mm. The latter is substantially the medieval geometric foot and the Roman foot calculated by Grienberger.

The existence of a libra increased by a komma has been a formidable impediment in the process of reconstructing the exact value of the Roman foot, since the approach which should have been most certain mathematically, that of calculating the edge of the quadrantal, led to unacceptable results. For this reason the investigation of Father Grienberger, which was the most rigorous from the point of view of scientific method, was not as influential as it should have been, because its numerical conclusions were rejected by most; this work is remembered only because it contains the first published formulation of the relation between equilibrium and center of gravity. For the same reason two works which approach the problem purely on the basis of the interpretation of empirical evidence, have not received all the credit they should: I mean the treatises of Greco-Roman metrology by Hussey and Paucker, which Böckh finds outstanding for their “thoroughness and thoughtfulness,” qualities that later became rare in the field of ancient studies. These two works must be given particular weight because they appeared independently almost at the same time, one in Russia in 1835 and the other in England in 1836, and they arrived at the same conclusion. Hussey arrived at a Roman foot of 11.6496 English inches (0.9108 feet) and Paucker to one of 11.650 (295.910 mm.). It is amazing that from such data as the dimensions of buildings and the distance of Roman milestones, one did arrive at such a precise figure (by the mathematical rationale I have determined the figure 295.9454 mm.). This is a great tribute to the refinement of the method developed through three and a half centuries of steady pursuit of the problem; but unfortunately, whereas these studies contributed to the development of the techniques of observation and interpretation of empirical data in the physical sciences, their lesson was in great part lost for ancient historical research.

Paucker, while claiming with. reason that he had achieved “the definitive determination of the foot and the libra,” granted that many elements remained uncertain. The nature of the problem was clearly stated by Böckh when in 1838 he settled for the calculation of the Roman foot by Wurm as 131.15 Paris lines (295.8521 mm.), but observed that certainly there were shorter varieties of Roman foot.

Today the main empirical problem can be considered solved, since Sir Charles Walston discovered the iron bar of the Argive Heraion that indicates a Euboic mina of exactly 405 grams. I have shown in my doctoral dissertation that this was the Paris standard of Greece; from this value one can derive right away an Attic monetary mina of 432 grams and a Roman libra of 324 which indicate a Roman foot of 295.9454 mm. (edge of the amphora of 60 minai and 80 librae). I shall show that the Sumerian bronze bar of Nippur, which is about two millennia older, is the most important standard discovered in Mesopotamia: it indicates with absolute precision an Attic monetary mina of 432 grams and an Italic foot of 274,7314mm. (edge of the cube of 48 minai or 4/5 of amphora or basic talent netto, weight of the amphora filled with wheat). It is most saddening to consider that these objects that allow to solve a problem that has intrigued the best minds of Europe for four centuries, have been totally disregarded: the bar of Nippur was neglected, and Sir Charles Walston was vilified as "a Jew who did not know his place" when he tried to stress the importance of the object he had discovered. Since the correct value of the Roman foot can be established with absolute certainty, it becomes even more necessary to find the reason for the existence of modified values of it.

3. A first step towards the discovery of the reason why there should be modified units was made by Petrie. Petrie asked the question why there should be an Egyptian royal foot of 7 palms next to a regular one of 6 palms, and properly answered that the existence of a septenary unit had the purpose of allowing an easy reckoning of the relation between side and diagonal of a square as 7:10. Petrie did not notice that this was not the only septenary unit and that in the ancient world there was a series of septenary units. He should have been aware of the circumstance that Newton correctly guessed that the Egyptian royal cubit Is a unit of 7 palms, by considering the mention in the Bible of a cane of 7 cubits. Petrie did not realize that the calculation by septenary units had also the purpose of calculating the relation between diameter and circumference of a circle according to the formula 7:22. I shall show that the calculation of the diagonal was most important in land surveying and in building, but methods had been found to render the ratio of diagonal to the side s rational number through the mathematics of near-squares, whereas the problem of squaring the circle could not be solved but by properly adjusted units of measure. The problem of squaring the circle was connatural to the organization of measures, since the units of volume were calculated as cubes, but measuring vessels were constructed as cylinders.

Petrie reasoned that in Egypt one calculated the relation between side and diagonal of a square as 7:10, but that, since the reciprocal of the square root of two equals 0.7071068, there must have been some method to correct the approximation. This method would be to measure the diagonal by a Roman foot of 297 which is 1/100 less than an Egyptian foot of 300 mm. in principle the Roman foot can have been so used, but the evidence that one used the Roman foot in Egypt is far from conclusive. The joint use of the Egyptian foot and of the Roman foot is certain in Greece and in Rome, so that the calculation suggested by Petrie would have been possible, but I have not noticed any object or construction suggesting that it was used. However, the discovery of Roman rules based on a foot of 300 mm. is startling; it is difficult to think of any practical purpose for them, if it is not that suggested by Petrie. A Roman foot of 296.985 mm. would have provided a perfect diagonal when used together with a rule of 300 mm. Unfortunately the number of published rules of about 300 mm. found in Italy can be counted, on the fingers of one hand, so that one cannot tell whether they were made slightly shorter than 300 mm. with a purpose, A number of slightly shortened units of 300 mm, used together with a foot of 296 mm. would prove that one reckoned as Petrie suggests.

Petrie did not consider that not only the diagonal of a square equals the side multiplied by the square root of two, but also the side equals one half the diagonal multiplied by the square root of two. The method of calculating the diagonal by constructing a right triangle with sides 5 and hypotenuse 7, could be used by considering as hypotenuse either the diagonal or the side of the square.

Before Petrie the problem of diagonals had been studied more thoroughly by Girard in his Mémoire sur les mesures agraires des anciens égyptiens, written as part of the scientific report of the Napoleonic expedition to Egypt. Girard remarks that one of the fundamental problems for ancient surveyors was that of constructing a square double of another: for instance, Roman surveyors reckoned by actus (120 x 120 feet), by its double the iugerum (120 x 240), and by the double of the iugerum, the heredium (240 x 240). In order to double a square field they used the theorem of Pythagoras by which the square constructed on the diagonal is the double of the square. In Egypt one used for surveying a rod of 5 cubits, so that the agricultural unit with a side of 100 cubits, called aroura by the Greeks, was measured by 20 rods. The use of the rod of 5 cubits suggested by Girard, has found confirmation in the use of units of 5 cubits in Pyramids of Gizah and in Zoser’s Complex. Girard remarks that the diagonal of the aroura (practically equal in surface to the Roman iugerum: both are the amount of land plowed in a day) was measured by an equal number of rods of 7 cubits, so that the double aroura has a side of 20 rods of 7 cubits; the rod of 7 cubits is well known from the Bible and from Mesopotamian documents. The double aroura in turn could be duplicated by using a rod of 10 cubits. Girard aptly notices that one measures the aroura with 20 rods of 5 cubits and that its diagonal can be measured either with 20 rods of 7 cubits or with 28 rods of 5 cubits. In substance he is pointing out that the relationship 10:7 can be reversed: one can obtain the diagonal by multiplying the side by \/2 but one can also obtain the side by multiplying by V2 the semidiagonal. I have come to the conclusion that one averaged the two methods considered by Girard: by averaging the result of the relation 5:7 between side and diagonal with the relation 5:7 between semidiagonal and side. One obtains for V2 a value of 99/70 = 1.4142857 which is an excellent approximation of 1.4142136.

The problem of a duplication of in area is discussed in logistic terms by Vitruvius (IX, praef., 8). He considers as unit of surface a square with a side of 10 feet and then asks how it can be duplicated; the points out that it is necessary to calculate the diagonal of the square with side 10, a problem that cannot be solved by arithmetic, since 14 is too little and 15 is too much. Hence, he commends that the problem be solved by direct measurement the diagonal.

Girard and Petrie did not know that the relation 5:7 between side and diagonal of a square is also mentioned in the Mishnah. The Mishnah, however, considers this an approximation valid for practical purposes and mentions so the relation 1:1.41 below I shall show that the figure 4l was also used in Egypt. The Mishnah also supports Girard`s contention that the calculation of diagonals is important for the duplication of square areas. The rabbis discuss the geometry of a square with a surface of 5000 square cubits, the space within which one is lowed to carry things on the Sabbath. It is obvious at this square is the double of a square of 2500 square cubits (square with a side of 50 cubits); since the rabbis point out how difficult it is to calculate a square with a side \/5000, they define this square through its diagonals, as having a diagonal of 70 cubits and a fraction (Erub. 2, 5) and they refer to it as “the square of 70 cubits.” That this is double unit is proved by the fact that often they call it the area of two seah, following the general ancient practice of defining areas through the seeding rate.  

A direct confirmation of Girard’s interpretation is provided by Egyptian mathematical texts dealing with the size of buildings, in which units of 5 cubits are used for the main dimensions and units of 7 cubits occur in the less fundamental ones (Papyrus Anastasi I, 14, 2 and 14, ff).

My contention is that one obtained the right diagonal by averaging two calculations by the simple relation 7:10. It seems to ale that a proof of this may be the extraordinarily accurate calculation of the diagonal as 1|24|51|10 or 1.89470/216000 = 1.41421286, found in cuneiform tablets of the Old-Babylonian period. Since this value is found in texts that refer to practical calculations, one should wonder why there is adopted a value that is accurate to a degree superfluous for any practical application. Neugebauer who has published this startling datum (MCT, p.43) observes that tablets of the Seleucid period employ the more practical value 1J2.5 or 34/24 = 1.41666. In my opinion one adopted the value 1.41421286 in sexagesimal computations, because one had arrived at the value 1.4142857 by the simple method of averages I have described, Neugebauer suggests that the value of the square root of two, which he has brought to attention, was obtained by alternating the arithmetic and the harmonic cleans of previously obtained approximations; this method could be an expansion of the method I have described. Surveyors who did not compute sexagesimally could obtain diagonals accurate for all practical purposes merely by measuring with units related by the factor 7.

A clear illustration of the use of the Roman foot as a septenary unit is provided by builder’s markings on the Roman aqueduct of Bologna. Some Egyptian archeologists have paid attention to the marks made by builders In measuring constructions, obtaining thereby absolutely certain evidence for the units of measure used, but scholars of Roman archeology have completely neglected this datum. The only exception I know of is due to the alertness of a French scholar, E. Pélagaud, who happened to be in Bologna in 1879, when the Roman aqueduct was being cleaned and restored in order to supply the city. He noted that for a great length the inside mortar coating of the aqueduct is marked by lines made with a sharp metal point. On one wall the marks are spaced 295 mm., whereas on the other they are spaced 43-35 fourteen units of the first type correspond to ten units of the second type. There are numbers written every ten units; at one point there is an inscription I.C.X.M.P, which Pélagaud reads as incipit caput decem millium pedum. A report was made to the Academie des Inscriptions (Mem. Ac. Inscr., VII (l879), 154-l58), but I do not know of any metrologist taking notice of it. Pélagaud thinks that the unit of 413 mm. is a local unit of Roman Bononia, but one can recognize that we are dealing with the cubit of the foot I call lesser foot and is usually called Italic or Oscan by archeologists. This unit was in general use throughout pre-Roman Italy and was used in early Rome; it is the unit usually employed in Etruscan constructions. It reappears in Italy after the Fall of the Empire, indicating that in spite of Roman legislation trying to enforce the official standard, it never went out of use. This foot occurs in a natural variety of 261.25 mm. and a trimmed variety of 277.4489; for reasons that I shall explain, it frequently occurs in a doubly trimmed variety of 274.7314. It is unfortunate that Pélagaud did not try to ascertain the dimensions with greater precision than the millimeter, but from a cubit of 413 mm. one can derive a foot of 275 1/3, which appears to be a doubly trimmed lesser foot of a theoretical length of 274.7314, with a cubit of 412.0971. In order to have a Roman foot such that 14 equal 10 such cubits, the foot must measure 294.355 mm; I shall show that by definition the Italic foot (trimmed lesser foot) is 15 fingers of Roman foot (trimmed basic foot), but here instead of the relation 15:16, there is applied the relation 14:13, obtained by using a doubly trimmed lesser foot and a slightly shortened Roman foot. A slightly shortened trimmed basic foot is also indicated by some buildings of Mesopotamia; the doubly trimmed basic foot is embodied in the bronze bar of Nippur. The Roman foot was the standard unit of predynastic Mespotamia.

The general use of the combination of units of the aqueduct of Bononia, is indicated by a little ivory box (10.5 cm wide) with silver hinges found in Pompeii, and described by Matteo delta Corte (Monumenti antichi, 2b (1929), 6-99). Della Corte has gathered that the box belonged to a surveyor and was used by him to carry his instruments into the field; but in my opinion, the markings on it have no practical usefulness and the box is an objet de vertu, reproducing in small scale a box of the kind that could have been used to carry a groma or any other form of transit. On the lid there is a pattern of two lines forming an angle, intersected by a set of three parallel lines. Della Corte thinks that they are a portable sundial with which the surveyor measured time in the field, but actually they are a proportional compass (circinnus). It would be important to know how they were spaced, but this datum has not been ascertained; as I have reported, a number of proportional compasses has been noticed by epigraphists, but none has been published, as far as I know. On the front panel and on the back panel of the box there is drawn a pattern of five parallel lines which had the purpose of comparing different units of length, a purpose which must have been also that of the circinnus on the lid. Della Corte reports the markings of only three of the lines on the front panel; it is regrettable that he did not try to establish the markings of the other seven lines, even though these are partly damaged, because they might provide extremely valuable evidence for Roman metric practices. He reports that the first line has a mark on a point 24.5 mm. from the beginning; this is an uncia (1/12) of a foot of 294 mm. The second line is marked at 18.5 and at 37.2 mm.; these are digiti (1/16) of a foot of 296 or 297.6 mm. The approximation of the markings indicates that the box merely imitates the scheme of a real pattern. Up to this point the markings correspond to those commonly found on the faces of metal foot rules; these are commonly quadrangular with a face divided into digiti, one into unciae, one into palmi (¼) and a blank face. But the marking of the third line, which is at 34.0 mm., and which Della Corte cannot explain, is an uncia of a cubit of 412 mm., the second unit, of the aqueduct of Bononia.

My conclusion is that a shortened Roman foot, which is the pes Cossutianus, could be used as a septenary unit in combination with an Italic cubit, trimmed and doubly trimmed; but for reasons that I shall explain presently the main reason for the existence of the pes Cossutianus was the calculation of the relation between diameter and circumference.

4. Segrè, when he explained that the discrepancy 80:81 exists in order to simplify to 1:3 1/3 the relation between cubic foot (talent) and cubic cubit (load), noted that a further simplification is obtained by introducing the artaba. The artaba is a cubic foot unit increased by 1/6 over the cube of the Roman foot (basic talent netto), so that 3 artabai make a cubic cubit. The edge of the artaba is the artabic foot of 307.796 mm. Since an artaba is 9/8 of a basic talent netto, the artabic foot relates to the Roman foot as the cube root of nine:cube root of eight = 1.0400419:1. But numerous authors of the Roman period state that the Roman foot is 24/25 of artabic foot. Petto noted that some of his samples of Roman foot indicate a foot which is more than 24/25 of what he called Greek foot. As Böckh noted, this ratio can be obtained either by increasing the artabic foot or by decreasing the Roman foot. The evidence indicates that both methods were used. A good number of Athenian buildings is calculated by the artabic foot, so that in the second half of the seventeenth century, when the first reports from. Athens became available, it was possible to determine the length of the artabic foot with an excellent approximation, Bernard calculated the Greek foot as 1010 5/12/1000 of English foot; calculating by the English foot of 1824, this would be 307.983 mm. Hultsch on the basis of the dimensions of Greek buildings calculates the artabic foot, either as 307.6 or as 308.2 mm. I calculate the correct unit as 307.796 mm. and the increased one as 308,277. The most famous embodiment of the second unit is the one hundred foot front of the Parthenon.

The artabic foot is a unit particularly connected with sexagesimal reckoning, because the artaba contains 60 reduced pints (Alexandrine sextarii) of 486 c.c., the normal units of volume and weight in Mesopotamia, and 3 artabai make a load of 160 pints. But the artabic foot became particularly important as itinerary unit, since 600 stadia of 600 feet make a degree. The usual decempeda, akaina, kanón, the surveying cane of 10 feet of Sumerian origin, has the exact length of a second of degree. Thirty stadia make a Persian parasang or hour of march. It is probably for its convenience as a sexagesimal unit, well related also to units of the Attic-Roman type, that the artaba and the artabic foot became the standard units of the Persian Empire. In the Roman Empire one continued to use the artabic foot for itinerary measurements: the Roman mile is considered equal to 8 stadia of 600 artabic feet, There was a stadion of 600 Roman feet, but when Roman authors speak of stadion they usually mean the artabic stadion equal to 625 Roman feet. Distances at sea were always calculated by artabic stadia; the Antonine Itinerary calculates land distances by millia passuum, but sea distances by artabic stadia. The artabic unit fit perfectly into the sexagesimal division of the meridian, giving a degree divided into 600 stadia of 600 feet (circumference of 216,000 stadia), but it also fits perfectly into sexagesimal computation as used in cuneiform documents, in sexagesimal accounts one usually reckons by stadia of 720 units: a stadion of 720 artabic feet is contained 500 tines in the degree and 180,000 times in the circumference of the earth. The circumference of 180,000 stadia is mentioned by Strabo and Ptolemy.  

Since the artabic foot is so closely connected with the length of the degree, one winders which artabic foot was used in the reckonings. The correct artabic foot of 307.796 mm. corresponds to a degree of 110.806 m., which is the average length of degree in Egypt (it is the length of degree between latitude 27 and 28). The increased artabic foot of 308.277 mm. corresponds to a degree, of 110.980 m., which is almost the degree of 110.956 at latitude 36. It would be extremely important to know whether a distinction was made in the calculation of itinerary distances between one type and the other of artabic foot. One wonders whether when Roman authors refer to the artabic foot as Alexandrine or Ptolemaic foot, they intend to refer specifically to the shorter one. It would be significant if in Egypt one used only the correct form.  

The relation 24:25 between Roman foot and artabic foot was also obtained by shortening the Roman foot to 295.484. This is the unit known as pes Statilianus. Several author calculated this unit as 131.0 Paris lines (295.514 mm.). La Condamine, after having measured several buildings and rules in his Voyage d’Italie, concluded that 131.0 was the length of the Roman foot, and Boscovich arrived at the same result as average by his own observations and by considering the several data reported by Father Diego Revillas. Revillas (Acc. Cortona, III (l741), p.III calculated the pes Statilianus as 131.0 5/6 Paris lines. Hultsch follows Raper, who in turn follows Greaves, in merging the pes Statilianus with the pes Aebutianus; but Hultsch himself indicates the existence of a foot of 295.5 next to a foot of 296.

The only point that remains uncertain is whether the pes monetalis, the standard of which was kept at the Temple of Juno Moneta under the watch of the Augurs, and which is described by Latin writers as 24/25 of Ptolemaic or Alexandrine foot is a pes Aebutianus or a pes Statilianus. If a survey of sample weights were to prove that those with the marking EX CAPITOLIO or EXACTUM AD CAPITOLIUM indicates a libra of 322.83 grams, one could conclude that the pes monetalis is the pes Statilianus.

 Ancient authors report that the interval between milestones was divided into stadia, at a rate of 8 or 1/3 stadia to the mile: in the first case the stadia were artabic stadia of 625 Roman feet, and in the second case they were Roman stadia. The ratio between Roman and artabic stadion is 24:25. According to Ploutarchos (Gaius Gracchus, 7 ) the lex sempronia viaria prescribed that the division of the roads into stadia be marked by stones; one cannot tell whether the statement that there are “eight stadia and something more “ to the mile, is a formulation of the law or of Ploutarchos. Polybios (III, 39) reports that the stretch of road from Narbo to the Rhone river “has now been carefully surveyed (bebématistai) and marked at the rate of eight stadia”; according to this statement the only markers were milestones. One cannot gather whether the miles were based on the pes Aebutianus and on the increased stadion; they should have had this length to conform with the length of the degree on the northern shore of the Mediterranean. It would be a very startling result, if it could be determined that one used the shorter artabic stadion and mile on the southern shore of the Mediterranean and the longer ones on the northern shore. This would be the proof that the units were adjusted to the length of the degree at different latitudes.

Unfortunately practically nothing is known about the length of the Roman miles. Several scholars of the seventeenth and eighteenth century have been vitally interested in testing the interval between Roman milestones, but their evaluations cannot be trusted because of the uncertainty of their methods of surveying and the doubt about the units used. In the nineteenth century, when reliable triangulations and topographiical maps were established, archeologists ceased being interested in collecting metric data. Furthermore a great number of ancient roads and because archeologists thought well to remove milestones to lapidary collections, without thinking that a milestone removed from its location is practically destroyed as archeological evidence. There is only one significant report on the matter of milestones: Luigi Camina caused a survey to be performed of the distance between markers XLII and XLVI of the Appian Way, when these markers were no longer in place, but apparently one could determine from the foundations where they had been. According to the report of Ingegner V. Minottini the four miles have an average length of 1478 m., but he states also that two of the miles have a length of 1480; from this I draw the necessary conclusion that two of the miles have a length of 1476 m. The shorter interval indicates a pes Cossutianus of 294.4, whereas the longer interval indicates a foot of 296 mm. The earlier surveys of the same interval between marker XLII and XLVI would indicate a foot of the shorter variety; but these surveys were performed before the introduction of absolutely reliable standards. For the same reason one cannot give too much importance to the commission that prepared the Gregorian Calendar, to the effect that he tested the milestones of the Via Latina from Rome to Albano and found them in agreement with the foot of Petto, pes Cossutianus.

A survey was conducted by father Angelo Secchi, in 1885 of the interval between mile IV ( near the tomb of Caecilia Metella ) and mile XI of the Appian way;the purpose was to test the exactitude of the calculations of Boscovich in 1751, since he took this stretch as base for his triangulation of Italy. The measurements of Boscovich, which proved the polar flattening of the earth, had been object of dispute for a century, because Boscovich had used a toise which proved not to be an exact copy of the Toise du Perou he had asked for. On this occasion Canina asked Secchi to measure the length of the miles, and the report, states that the miles have a length of 1481.750 m., but I could not find in Secchi’s report any statement to the effect that there were milestones in location. On the contrary, it was found a practically hopeless task to ascertain the terminal points of Boscovich’s base. But Canina draws from Secchi’s report that the Roman foot had a length of 296.35mm., and that this length was adjusted to the length of the degree at the latitude of Rome. Actually the foot corresponding to the degree at latitude 42 is 296.122 mm. There is good evidence of the use of a foot of 296.2, slightly longer than the common foot of 296, but this longer foot must be related wit h a libra slightly heavier than 324 grams. Cagnazzi having found foot rules from Pompeii indicating a foot of about 296.2, observed that their length corresponds to the samples of libra found in the same city.

The modern neglect of the available information about Roman Linear units, is indicated by the case of the markers of Pisco Montano. The direction of the Appian Way is mainly determined by the promontory of Terracina: the road moves from Rome towards the sea, passes between the mountain and the sea at Terracina, and then turns inland. Where the Appian Way curves there is a mighty rocky pinnacle called pisco Montano; in the age of Trajan or thereabout, this rock was cut for a height of 128 feet in order to straighten the course of the road. The height of the cut is indicated by markers spaced every ten feet. The numismatist and archeologist Antoine Mongez, who prepared the plan for the introduction of the decimal monetary system in France in 1792, having heard these markers mentioned by Ennio Quirino Visconti, cased the Academie des Inscriptions to ask for a survey. The survey was performed by Ingegner Scaccia in 1813 with the help of a transit. The results were the following, beginning with the highest preserved marker:


3.103.96 m
XXXX to L 2.960.96
L to LXX 5.874.52 (marker LX could not be seen from observation point)
LXX to LXXX 2.913.56
LXXX to XC 2.940.46
XC to C 2.950.96
C to CX 2.950.56
CX to CXX 2.852.36
Average foot 293.191 mm.

Mongez explained the irregularities by the contractor’s intention to defraud. Letronne quoted the measurements as evidence that the Roman foot had a value of 295 mm. The marker: are not all of’ the same size and they are not even in a vertical line; obviously one was not too careful in placing them, but the builders must have calculated carefully and have known exactly that the height was 128 feet. Marker CXXIIX was found in 1911 below the present level of the road, but the archeological report, while providing a photograph of it together with the next marker CXX, does not ascertain the distance, which could have been easily done (Notizie degli scavi, 1911, 100 ) Apparently the extreme irregularity found by Scaccia is due to the fact that he used a transit ; Canina unfairly charges this report with carelessness. A survey conducted at the request of Canina by Ingegner Minottini with the more primitive method of applying a chain gives the following results:

XXX to CXX 26.608 m., foot of 295.6
C to CXX 5.900 m., foot of 295.0
CX to CXX 2.949 m., foot of 294.9

This survey indicates that by using a chain that follows the bends of the rock one proceeds as the Roman measurers did. If one examines the data, one can draw the conclusion that marker CXX is out of place; and in fact under it there is cut a door, so that one can suppose that it was moved to make way for the door. The measurement of the distance between this marker and marker CXXIIX should correct this error.

The significant fact is that even though millions of people pass in front of these markers each year, nobody in the last century has thought to inquire about their metrology, In Forma Italiae (Regio I, Vol. I, 210), Giuseppe Lugli becomes lyrical about the great interest of archeologists in this monument, but mechanically reports that the markers are calculated by a foot of 296 mm. It is to be noted that Lugli is almost unique among contemporary scholars of Roman archeology in having some interest in. metrology. A proper survey of the markers of Pisco Montano together with a correct statistical analysis, could determine the length of the Roman foot used.

5. Nobody has explained why there were units related as 24:25. The importance of these units is indicated by a passage of the Mishnah (Kel. 17.9): The measure of the cubit of which they have spoken applies to the cubit of the middle size. There were two cubits by the Palace of Susa, one at the north-eastern corner and another at the southeastern corner. That at the north-east was longer than the cubit of Moses by half a finger; that at the north-east was longer than the other by half a finger; thus it was one finger longer than the cubit of Moses." The measures placed at the Eastern Gate of the Second Temple of Jerusalem where there was a picture of the Palace of Susa (Midd. 1,3), most likely to indicate that the standards were copies of those kept at the royal palace of Susa as official standard of the Persian Empire, In my opinion the text is concerned with three cubits; the longest is the official standard of the Persian Empire, the artabic cubit of 471.66 mm.; the medium one is an Egyptian cubit of 450 mm. which the Persian were obliged to consider and which is the basic foot of the ancient world (I shall show that later Hebrew units are calculated by this unit); the cubit of Moses is a Roman cubit, which would be a unit of 443.226, if the artabic cubit is of the correct variety and if the difference is exactly one finger. Between the cubit of Moses and the artabic foot there is a relation 24:25, and, hence, the difference is a finger; the statement that the intermediary Egyptian foot differs of half a finger from the other two, is an approximate one. The Temple was built by the Egyptian cubit (natural basic cubit), but at the time the Mishnah was written calculations were made by the cubit of Moses, which is identical with the Roman one (trimmed basic cubit), since it is often called Italian measure (e.g., Kel. 17, 11).

The rabbis offer a pious explanation for the existence of two standards at the temple: "And why was there ordained a larger cubit and a smaller cubit? So that the craftsmen might undertake their tasks according to the measure of the smaller cubit and fulfill them according to the measure of the larger cubit, and thereby escape the guilt of sacrilege. But most likely the true explanation is t hat the artabic foot and cubit were legally prescribed in the Persian Empire. I shall show that there was a trimmed wheat foot increased from the correct value of 318.75 mm. to 320.606, known as Olympic foot, since it is embodied in the stadion and other buildings of Olympia, which relates as 25:24 to the artabic foot.

The function of units related as 24:25 is to allow an easy reckoning of the relation between diameter and circumference of a circle. If one assumed pi = 3, as it was customary, and multiplied by 3 the length of the diameter expressed in Roman feet, the result calculated in artabic feet gives a value pi = 31/8. If one takes, as an example, the circumference of earth of 216,000 artabic stadia, one can calculate mentally the radius as 36,000 Roman feet, I tried to submit a paper on this discovery to the Archeological institute of America, before the publication of the cuneiform tablets of Susa that gives a clear textual confirmation of my interpretation.

The ancients used not only the value pi= 3 1/8, but also the more accurate value 3 1/7. This last value was obtained by using septenary units, which allow to reckon the ratio between diameter and circumference by the logistic formula 7:22. I have shown that the Roman foot was used as a septenary unit when it had the reduced length of 294.355 mm., l5/l4 of the doubly trimmed lesser foot. But this Roman foot could be used to square the circle in a more direct manner, since it relates almost as 21:22 to the increased artabic foot, If one measures the diameter by this foot and the circumference by the increased artabic foot, reckoning by pi = 3, one obtains a value 3.14188, which is even more accurate than 3 1/7. The expert of measures who figured this process achieved by extremely simple means a feat of mathematical accuracy. Since the doubly trimmed basic foot was used in Sumerian times and building calculated by the artabic foot are found in preliterate Mesopotamia, it is likely that the discovery was known to the Sumerians. But it is a fact that it was generally accepted in Roman times, since there is a great number of measuring rules that fit exactly this dimension.

When Petto in 1573 gathered a group of famous antiquarians in order to obtain their endorsement of his calculation of the Roman foot, one of his major arguments was that of the five rules he considered, three agreed with each other. His argument was obviously that if rules do agree with each other, they prove thereby to be accurate pieces, This principle of Petto is an excellent one, since one finds highly different standards of accuracy in Roman rules. One must not forget that Dörpfeld warned archeologists to keep in mind that in a store of mechanical supplies of Athens he found for sale meters differing from each other as much as 4 mm, (1/250). On the basis of the rules and other data Petto established a length which was marked on a stone of the Palazzo dei Conservatori. It can be calculated as 130.5 or 130.6 Paris lines (294.386 or 294.611 mm.). Riccioli and Gosselin calculate the pes Cossutianus as 130.6 Paris lines. Greaves took this foot as the standard Roman one and calculated it as something more than 293.968 mm. George Shuckburgh, when he tried to establish the English standard in 1798, took the average of several Roman rules including one of the British Museum, and arrived at 11.6063 inches of his standard (294.795 mm); his English foot measures 304.794. mm. When Barthelemy was sent to Italy by the King, of France in order to investigate the matter of the Roman foot, he settled for the length of 130.6 Paris lines. He attached particular significance to a bronze foot recently found by Francesco de Ficoroni and placed in the Vatican Library which seems to be 295.070 mm.; he chose a figure slightly below this length. It is to be noted that Abbe Barthelemy, since he considered himself mainly a linguist and a man of letters in spite of his acquaintance with Greek and Oriental mathematics, asked for the assistance of Father François Jacquier, who had made Newton known in France, in order to have the help of an experimental physicist.

Later Jacquier aided Boscovich in his trigonometrical operation, in occasion of which there was calculated a Roman foot of 131.0 Paris lines; this indicates how the objective evidence indicates both a foot of 130.6 and 131.0 Paris lines.

Those who have considered mainly the evidence afforded by foot rules have come to figures close to my value 294.355 for the pes Cossutianus. There must be a few hundreds of Roman rules of bronze and iron scattered through museum and private collections, but a few dozens at most have been mentioned in print. Apparently the metal rules are not as accurate as the bone ones; these were more precise, as indicated by the division into fine lines, but archeologists seldom care to report about small bone objects. The bone pieces are usually so accurate that even a fragment provides a valuable datum. One of the few reports on Roman rules is that of Luca de Samuele Cagnazzi, concerning six rules from Pompei preserved at the Museo Nazionale of Naples, which have the following lengths:  

I 294.435
II 294.432
III 291.45
IV 294.439
V 296.30
VI 296.20 (of this only one half is preserved)


The agreement of rules I, II, and IV is amazing. One can trust Cagnazzi’s data because he did not assign any particular significance to these figures, and averaged all lengths together. If three rules agree that closely it means that they are intended to be precise pieces; they differ of 1/10 of mm. from my calculated value of the pes Aebutianus as 294.355. One should reexamine these objects and one should take accurate tests of other rules, in order to ascertain which value of pi was achieved. Assuming that one used the three rules of Naples together with an artabic foot having exactly the length 308.277 mm., the value of pi would be 3.14103.

A small list of rules found in France has been compiled by H0ron de Villefosse; more than half agree with the length of the pes Cossutianus. At Mount Châtelet near Joinville there was found a foot of 295 mm. and the broken half of a foot measuring l47.3; at the Museum of Besançon there is the broken half of a foot of 147.0; in the Forest of Maulevrier near Caudebec there was found a rule of 292.5; a rule of 294. from Apt near Vaucluse is at the Museum of Lyons, and the same length is reported for a rule found at Mirebeau-sur-Béze (Côté-d’Or), in Germany, at Weissenberg there was unearthed a foot of 293.5. Also in Germany, at the site of Castrum Lamiacum, there were found the two bronze ends of a measuring pole, probably a decempeda, on which there is carried on the division into inches; the marking is rough, but since it is according to the Roman standard, about 294.4 mm., and about the pes drusianus used by the Romans in Germany, about 330.76, one can rest assured that the figures are substantially; reliable, since the relation must be 16:18. In order to obtain the right relation either the shorter unit should be 294.01 or the longer unit 331.2.

The foot of 294.355 corresponds to a mina of 425.073 grams. This fact is highly significant, since I have determined that a mina of about 425 grams is the basis of the coinage of Alexander the Great and his successors, and that in the second century B.C., this mina was adopted by Athens. With the introduction of New Style coins, the Attic system of weight standards underwent a revolution as radical as Solon’s reform: coinage became based on the pheidonian units instead of the basic ones. In discussing the evidence provided by Athenian inscriptions, I shall discuss the relation between the mina of 425 grams and the pes Cossutianus. On such occasion I shall also point out that the present welter of theories about the history of the Roman denarius, results from the fact that numismatists know that the denarius is identical with the Athenian drachma, but do not know that the drachma in question has a metric basis completely different from that of the Solonian drachma.

In conclusion one can draw the following table of feet used in Rome:

295.945 mm. correct trimmed basic foot, pes Aebutianus, 24/25 of increased artabic foot (308.276 mm.)
295.416 pes Statilianus, 24/24 of correct artabic foot (307.796 mm.
294.355 pes Cossutianus, 3/pi of increased artabic foot, 15/14 of doubly trimmed lesser foot (274.731 mm.)

5. Having determined the length of the Roman foot, it is possible to cope with the most baffling problem of the entire field of metrological research, that of the relation between Roman and Egyptian foot. But before dealing with this problem, it is necessary to dispose of some serious distortions that have crept into the discussion of the problem. Confronted with the elusiveness of this central problem, the best of scholars have resorted to desperate explanations.

Newton understood that the Roman foot must be a modification of the Egyptian foot, but could not ascertain the mathematical principle linking them. He resorted to a non-mathematical explanation: the Roman foot developed because units of length become shorter in the course of time. But Boscovich and Maire, in their report on the calculation of the meridian, in order to explain why the medieval Roman foot is longer than the ancient Roman foot assumed as self-evident that units op length become longer in the course of time. The two Jesuit scholars submit two possible explanations of the alleged process of lengthening: one is that the accumulation of rust makes standards longer; the other in that craftsmen in copying standards tend to cut them in excess, knowing that such an error can be corrected by filing, whereas one cannot correct an error in defect. The two Englishmen, Newton and Maire, and the Italianate Boscovich, considered that there were small differences among the standards of t heir own time, and assumed that such variations would accumulate in the course of time. That small variations existed and were difficult to avoid is proved by the unfortunate experience of Boscovich and Maire; they relate how they took great pain to procure a toise attested by French experts to be an exact replica of the Toise du Perou, employed to measure the meridian in Lappland and Peru, but it was later demonstrated that the standard sent from Paris to Rome was 7/78 of a line too short. This discrepancy, which is relevant for modern calculations, caused a scientific controversy about Boscovich’s data on the polar flattening that lasted a century, until Father Secchi repeated the same triangulation, starting from the same base on the Appian Way.

The fact that Boscovich and Maire were thrown off balance by the difference between ancient and medieval Roman foot, reflects the fact that they did not know that the Roman libra or any other ancient weight could be increased of a komma. As I have said, since the calculations of another famous Jesuit, Grienberger, had made clear that Roman foot and libra did not r elate well, one had lost confidence in the process of defining units of length through the units of weight. If Newton, Boscovich and Maire had assumed units of length defined by the units of weight, they could not have presumed that small variations of the length in excess or defect can accumulate in the course of time.   Small variations are possible, but not a steady accumulation of them in one direction. The difference between ancient Roman foot and medieval Roman foot can be traced back to the standard of Mesopotamian mina (1/2 Roman libra) sent by Calif al-Mamun to Charlemagne; the differences of standards created by this unit, heavier of about a komma than earlier standards, created by the seventeenth century an unjustified distrust in the stability of all standards of length. The reform of the Toise du Châtelet increased the uncertainty, because the old toise continued to be used together with the new one, and further the new standard was not fixed as stably as one had intended, with the result that it had to be reconstituted as the Toise du Perou.

The uncertainty of standards became evident when one compared the calculation of the length of degree performed by Picard and Gian Domenico Cassini reckoning by the new toise; it was gathered that if they had used the same standard, their data indicated that the earth is elongated at the poles, whereas Huygens and Newton had guessed that centrifugal force causes a polar flattening. Metrologists and physicists debated the issue for about half a century, until Boscovich settled it. But before Boscovichs demonstration that the degrees of meridian are not of the same length, the very difference among the several calculations of the length of the degree increased the distrust in the standards of length. Newton did not accept the Calculation of the length of degree performed by Norwood between York and London in 1635: Norwood had calculated 367,196 English feet by a standard which must be that of Guild Hall; reckoning by the English foot of 1824, his figure is 111.921m., which happens to be 663 m. more than the length of the degree at the latitude of Greenwich Observatory. Below I shall show that the standard of Guild Hall cannot have been appreciably different from later English standards. The remarks by Newton and by Boscovich and Maire, had the result of confusing not so famous metrologists who followed them. Uzielli, who is an excellent scholar of Italian Renaissance metrics and who has the merit of having discovered the key to the solution of the Columbian Problem, put forth the ludicrous theory that Roman standards are shorter than the corresponding ones of Athens because the ropes used as standard in Athens became shorter in the more humid climate of Rome.

Hultsch exploited to the full the theory of progressive shortening: from the Egyptian royal cubit of 525 mm. there would have been formed, by taking 3/5, a foot of 315, which imported by the Greeks, was gradually reduced to an artabic foot of 308 and then to the foot of 297; this unit would have acquired in Rome a value between 296 and 295,5, shortened to 294.2 in the age of Titus.

The only support for this theory is provided by a paper submitted by Matthew Raper to the Royal Society in 1760; this paper is actually the only irresponsible essay on ancient metrology written in the eighteenth century, but Hultsch declares it to be the best treatment of the problem of the Roman foot. Raper assumes that the Toise du Châtelet established in 1667, has the same length of the preceding standard, in spite of the fact that the very French writers whose names he mentions, clearly state the contrary. Starting from this false assumption, he sets to prove that the foot of Guild Hall used by Greaves was 3/1000 too short in relation to the Graham rule of the Royal Society. The Yard of Guild Hall had been lost by the time of Raper, but, since it was the official standard of the City of London, a discrepancy of the sort would have been noticed by earlier writers. Raper echoed without understanding them expressions of doubt about the value of the English foot used by Norwood in calculating the length of the degree of meridian; this doubt was understandable as long as it was not proved that the degree is longer the closer one goes to the poles. Greaves calculated the pied de roi as 1068/1000 of the English foot of Guild Hall; he had received a copy of the Toise du Châtelet prepared under the personal inspection of Claude Hardy, who not only was a famous mathematician but also a judge at the Châtelet, the seat of royal justice for the City of Paris. In the following generation Bernard, one of the greatest scholars of ancient metrology who ever lived, defined the same English foot (pes Curiae Londinensis) as 1066/1000 of the reformed pied de roi. This proves that Burattini was correct when he stated, in the age of Greaves, that the English standards were the most reliable of Europe. Furthermore, Raper did not know that, in the very transactions of the Royal Society (Philos. Trans., 1736, 262-266), there is a report of Martin Folkes who with the Graham standard repeated some of the measurements performed in Rome by Greaves and found substantial agreement. Our certainty is increased by the circumstances that the tests of Roman monuments performed by Father Revillas were based on the esattissimo piede inglese that Folkes left with him upon returning to England from Italy.

Having established on a false argument that the foot of Guild Hall was different from the later English standards, Raper interprets this conclusion as evidence of a great uncertainty of all standards of length, including the ancient ones.

Greaves had combined the pes Statilianus and the pes Aebutianus into one unit, but these units were not too important to him, since he considered the pes Cossutianus to be the true Roman foot. But after the studies of La Condamine and Boscovich it was no longer possible to assume that the correct Roman foot was shorter than 131.0 Paris lines ( 295.514. mm.); hence Raper reinterpreted Greaves figures making the Roman foot equal to 970/1000 of English foot (about 295.65 mm.; 295.6557 mm, by the foot of 1824). In order to eliminate the datum of Greaves based on the pes Cossutianus, Raper constructs a piece of historical fiction. The fire set in 69 A.D. by the partisans of the Emperor Vitellius to the Temple of Jupiter Capitolinus, would have destroyed the Roman standards; the new Emperor Vespasian would have established a new standard of foot as less than 965/1000 of English foot (about 294.35 mm.), by neglecting the relation between length and weight.

Scholars of the early nineteenth century observed how unfounded was Raper’s contention that the pes Cossutianus had become the Roman standard in the age of Vespasian, but this doctrine was accepted by Hultsch. It is sufficient to rnention Trajan’s Column. And the Hall of the Senate, rebuilt by Diocletian, which are certainly based on a foot of 296 mm. Hultsch adopts Raper’s figures to defend that the Roman foot had a length of either 296 or 295.5 mm. Until it was shortened to 294.2 in the age of Titus. Hultsch uses Raper’s theory of the destruction of the standards by the Vitellians to substantiate a general theory of progressive shortening, whereas Raper had spoken only of uncertainty of the standards.

Hultsch presented the theory of progressive shortening in his Metrologie, which he published as a general survey of the field, before having adequately investigated all aspects of ancient metrics; the first edition was a weak work, but even in the far superior second edition there are parts that remain questionable. In his later works Hultsch substantially repealed the theory of progressive shortening by proving that all ancient units of weight are related to the Egyptian kite and t hat all units of volume and weight can be derived from the cube of the Egyptian normal cubit of 450 mm. But these products of the mature thought of Hultsch are never quoted, whereas the new school exploited to the full the theory of progressive shortening in order to prove that ancient standards were only approximately fixed.

As I have said, Petrie in order to prove t hat the Roman foot is derived from the Egyptian foot of 300 mm. through the calculation of the diagonal, has to argue that the former is a unit of 297 mm. In order to prove this contention he follows Hultsch: the foot was 297 mm. In Greece, but was imported into Italy as 296, becoming 294.6 in Rome. But Petrie defends as a general principle the opposite theory of progressive lengthening, because “measures of length become longer by repeated copying.” One of the main pieces of evidence for the phenomenon of progressive; lengthening would be the Italian mile which is longer than the Roman mile, but the Italian mile is longer because it is based on a medieval Roman foot of about 297.5 mm. Boscovich and Maire report that it was customary to mark the miles of Italian roads by referring to the palmo degli architetti kept at the Capitol.

Petrie realized that units of length are fixed with extreme precision and appear very stable, but could not explain this precision by referring to the units of weight. If he had accepted the link between weights and length, there would not have been any problem, since he submitted as one instance of precision of weights that a group of Arab sample weights of the eighth century A.D. differ from each other of not more than a third of gram. Since it is easy to compare and preserve weights, and the units of length vary in the inverse cubic ratio of the weights, it is easy f or supporters of the old school to explain the precision and permanence of standards of length. Petrie was forced to present the absurd theory that the length of the Egyptian royal foot was determined by the length of the pendulum that swings 100,000 times in a day at latitude 30° (latitude of Memphis). This pendulum of 740.57 mm. Is the diagonal of a square the side of which is the Egyptian royal cubit of 523.62. Petrie was truly a man endowed with supreme skill as an observer and classifier of empirical data, but as a theorist he never was able to free himself from the influence of his father who directed him to the study of Egyptology and metrology in order to uphold the pyramidite cause. In one of his weak moments, Petrie also intimates that the Egyptians had the telescope; as a result there is today in the United States a particular conventicle of pyramidites dedicated to prove that the telescope was used in Egypt. In general followers of Petrie have gone back to the purity of pyramidite faith, as exemplified by A. E. Berriman in his Historical Metrology (London, 1953) and in his recent article in the Journal of Egyptian Archeology (41 ( 1955 ) , 48-50 ) . This proves how careful one must be in separating the gold from the lead in Petrie’s writings.

In order to prove that the Egyptian royal foot originally had a length of 523.62 mm. determined by the pendulum, Petrie introduced the theory of progressive lengthening. The length would have become 524 mm., in the Fourth Dynasty, when the Great Pyramid was built, to arrive at the value of 525 in the following Fifth Dynasty, But Petrie himself presented evidence of the use of a cubit of 525 mm. in the First Dynasty and also in predynastic times. It is true that the Great Pyramid was constructed by a cubit of about 524 mm., but, unless one is a pyramidite, there is no reason to believe that the standard of this construction must be taken as the official standard of Egypt. For pyramidites, the Great Pyramid is even more than the official standard of Egypt; it was erected by divine dispensation to be the standard prescribed for mankind; it is usually understood that it is the standard used in Creation and the English standard, According to some pyramidites the great pyramid should also prove that the Fahrenheit thermometric scale, used in Anglo-Saxon countries, is the only one in agreement with divine will.

In order to clarify the problem it is necessary to specify the impact of pyramidite notions, I have traced the origin of the pyramidite belief to the polemic of Bishop Cumberland against Hobbes about the existence of absolute ethical standards; it appears in full form in a book published in 1704 and which is apocryphally ascribed to Greaves, as I have ascertained. The pyramidite exploited to the full and distorted the metrological ideas of Greaves and Newton. Pyramidite ideas become an essential part of the lore of masonic societies; an outcome of them is the Great Seal of the United States, consisting of a pyramid surmounted by the eye of God. I have shown that pyramidites ideas are linked with the doctrine of Anglo-Israel, the contention that England or the United States is the true Israel. It is in great part because of pyramidite and Anglo-Israel beliefs that the French metric system was not adopted in England and in the United States: one cannot reject a metric system ordained by God, as proved by the Great Pyramid, and which proves t hat those who use it are the chosen keepers of the divine standards.

A special aspect of the pyramidite ideology is the problem of the pendulum. After Burattini was robbed of the not he had taken in Egypt, he suggested that his metro cattolico should be based on the length of the pendulum that beats the second. This idea of Burattini was implemented by Jonas Moor, one of the original members of the Royal Society, who calculated that the metrum catholicum equal to 1/3 of the length of the pendulum that beats the second at the latitude of London, to be 4/3 of palmo of Genova (1089 /1000 of English foot or 331.9268 mm., reckoning by the English foot of 1824;that is essentially a trimmed barley foot. The pendulum that beats the second at the latitude of London reduced to sea level has a length of 995.1806 mm. For a period the standard of Moore was considered the solution to the problem of the fixed standard of length; but soon it was found out that this method of establishing a standard, first suggested by Gabriel Mouton, on the basis of Riccioli’s calculations, while perfect in theory, was most difficult to apply in practice, since the length of the pendulum not only varies according to latitude, but is substantially affected by the elevation above sea level, the presence of dense landmasses, and the difference between physical and mathematical pendulum. For this reason, the standard of the new Toise du Châtelet imposed itself as the scientific standard of Europe. However, when the Royal Society was established in 1662, one of the first tasks it sets to itself was that of determining the length of the pendulum that beats the second; to this purpose John Evelyn, one of the founders, in his trip to Italy, not only took the usual measurernents of standards of the Roman foot, but also of the braccio of Florence and of the braccio of Bologna, used respectively by Galileo and Riccioli in the calculation of the pendulum. But under the influence of the Academie des Sciences and the Academie des Inscriptions, established in the same decade as the Royal Society, the matter of fixed standard was settled in France by reforming the pied de roi, so as to make it 12/11 of Roman foot. As result of this, the issue whether the standard should be determined by the pendulum or by a bar of a given length, acquired a nationalistic tinge. Ever since the opponents of the French metric system proclaimed the superiority of the standard determined by the length of the pendulum. When the English foot was fixed by act of Parliament in 1824, it was legally defined by the pendulum, even though in f act it is determined by the Westminster bar, copy of the Bird bar of the Royal Society. The issue has been so charged with emotions in the United States that the length of the American foot was not officially fixed up to 1928; when it was proposed in Congress that it should be calculated as a fraction of the Paris meter, as it was already done in practice, the lobby of the opponents of the metric system became very active, but not being as powerful as in earlier times, obtained as a compromise that the American foot be fixed as a fraction of the Paris meter and also by the wave length of sodium light. In their blessed ignorance the lobbyists did not know that, while condemning the works of French atheism, immorality, and political radicalism, they were advocating a standard of measure conceived by Jacques Babinet, a pupil of those who established the Paris meter. It was a sham battle, since the wave length unit may be of use only in highly precise scientific calculations for which one has always used the decimal metric system. The entire issue is best summed up by some pamphlet published by the American institute of Weights and Measures in the 30’s, under the sponsorship not only of the heads of some major industrial and business enterprises, but also of the President of the Carnegie Institute of Technology, in which it is explained that the adoption of the French system is advocated by three classes of undesirables: political radicals, foreign born, and scientists.

It was the intention of the Founding Fathers of the United States to adopt a system like the French metric system and a clause was included in the Constitution to this intent; but Jefferson, who probably was influenced by Masonic ideology, opposed the adoption of the French metric system with the argument that one should adopt a standard based on the pendulum. Ever since, the point of view of Jefferson has been upheld by religious fundamentalists and by nativists. This strange sequence of ideas explains how the notion that the Egyptian standard of length was based on the pendulum was upheld by Petrie.

Even Lehmann-Haupt, the soundest metrologist of this century, argued that the basis of the ancient system of measures is the Mesopotamian cubit, calculated as half the length of the pendulum that beats the second at latitude 30 (latitude of the mouth of the Tigris and Euphrates). It happens by chance that the Mesopotamian foot, which according to ray reckoning is 332.9384 mm. in its trimmed variety, has about the same length as the metrum catholicum of Jonas Moor; but the chance is not so extraordinary when one considers that historically the edge of the Winchester bushel, which is now the American bushel, is determined by the length of that foot. By stretching the figures, as it has been done by Petrie and Lehmann-Haupt, one can easily connect any ancient standard with the pendulum. Already Rome de Lisle in 1789 had suggested, as a passing thought that the Roman foot may have been calculated as half of the length of the pendulum that beats the half second.

Since pyramidite ideas are so pervasive that they influence even those Egyptologists who denounce them as mere humbug, it is necessary to insist that the use of a royal cubit of 524 mm. in the Great Pyramid has no particular significance. Since Egyptian units of weights are subject to the influence of the discrepancy komma, it should not be surprising to find specific embodiments of the royal cubit that are about two millimeters less or more than the correct value of 525 mm.; later, I shall discuss the specific problem of the exact determination of the weight of the kite, between the minimum of 9.0 grams and the maximum of 9.1125 grams. It happens that the Great Pyramid was calculated by a standard of 525 mm, as was the Pyramid of Zoser in the Third Dynasty, but it can be proved that a standard of 525 mm. was in use at the time the Great Pyramid was built.

Pyramidites have measured again and again the granite coffin of the King’s Chamber, since according to them it is a unit of measure; it has been often argued that this was not the outer casing of the royal sarcophagus, but a trough similar in function as standard of measure to the Brazen Sea of Solomon’s Temple. Petrie reports the internal volume of the coffin. The coffin was not a unit of measure, as it is indicated by the fact that it was roughly cut. Petrie himself reports that the entire side was cut by one stroke of a huge saw; the saw at times was backed up after it had dented the stone as much as one inch out of plomb. In spite of the lack of symmetry in the object, one can calculate with which unit it was measured, thanks to the accurate reports available. The lateral walls of the coffin must have been calculated as 2 hands of 75 mm. each, 1/6 of a cubit of 525, since they have the following dimensions:

The height was the dimension calculated with the greatest care since the huge stone lid had to fit perfectly; it is 1049.27 mm. and it has been calculated as two cubits or 1050 mm. The width is 977. 90 or 13 hands (75 x 13 = 975); the length is 2276.34 and was calculated as 31 hands (75 x 31 = 2275). In my opinion there is nothing of a mysterious meaning in the size of the coffin, but its size proves that, while the Pyramid was built using a rule of 524 mm., the workshop that prepared the granite casing of the sarcophagus used a rule of 525. A well documented occurrence of this value is provided by the measurement of the north side of the Fifth Dynasty pyramid of Neuserre at Abydos, based on a cubit of 525,07 mm.

Lehmann-Haupt too felt that in some way he had to dispose of the relation between the Roman foot and the Egyptian foot; he disposed of it in the worst possible way; by denying the existence of the problem. He combined the two units into an intermediary one varying between 297 and 298.8; he ascribed a similar variation to all other types of foot and cubit, since he had proved from the textual evidence that all ancient units of length are interrelated. In order to arrive at these results for the Roman and the Egyptian foot, he had to make the typical cubit of Mesopotamia a unit varying between 490 and 496 mm., whereas according to my calculations the barley cubit is 499.407 trimmed and 506.25 natural. By these values Lehmann-Haupt made the units of length entities that could not be determined exactly by the unit of weight, even though he argued that such relation existed. In order to establish the mathematical basis of the units of length he resorted to the same theory as Petrie, that of the pendulum: the correct value of the Mesopotamian cubit is the half of length of pendulum that beats the second at latitude 30, To this pendulum of 992.33 there would correspond a Babylonian cubit of 491.16, and a foot of 330.8, from which by taking 9/10 one forms the Egypto-Roman foot of 297.7 mm, The evidence of this value of the Roman foot is the medieval Roman foot and the Italian mile. In some of his works Lehmann-Haupt presents the theory of the pendulum as tentative, whereas in others he is more positive. By submitting such a theory, he contributed to discredit the old school of metrology when he had remained its last defender; but actually the theory of the pendulum indicates that he too had abandoned in substance the position that unit s of length and unit s of weight are interrelated. Lehmann-Haupt reports that Hermann von Helmholtz considered with favor his theory of the pendulum, but the great physicist could not know that Lehmann-Haupt’s figures were in contradiction with the data collected in four centuries of application of Newton’s method to ancient buildings. Petrie at least tried to remain faithful to the evidence obtained by Newton’s method; the great positive contribution of Petrie is that he did not share the new school’s condemnation of Newton’s method and for more than forty years steadily applied it. I shall show that the mathematical rationale of ancient units of length produces figures that agree with the empirical data collected by Petrie.

6. Measurement is that part of science that is nearest to religion, since in this area the scrupulous testing of right and wrong is the issue: one could say that measurement is the nemesis of any incorrect scientific theory and procedure. For this reason throughout history religious movement s have appropriated to themselves the symbols of measurement, as expression of the absolute they are searching. But the history of metrology proves that one cannot study measures except in an atmosphere free from any outside consideration; it appears here most clearly that science only can be a law to science. Lehmann-Haupt has been the most responsible scholar of ancient measures in this century, but he came short of a great achievement because he made two concessions to tendencies of religious nature.

I have pointed out how he accepted from the pyramidites the notion that ancient standards of length were determined by the pendulum, implying that the people of Mesopotamia knew of the isochronism of the oscillations before Galilei. The pyramidite movement, which can be traced back to gnostic ideas, has the main purpose of deflating the importance of biblical religion, by stressing that God revealed to the Egyptians the secrets of science. Some verses of the poem The Hierophants by  Richard K. Haight (quoted in George R.Gliddon, Ancient Egypt, 12th ed., (Philadelphia,1848), 31) can be quoted as a summation of this outlook:

We will once more repeat, that though Moses did write
That in the beginning God said, Let there be light;,
All the wisdom he spake was but Egypt’s old lore,
Thence he learned all he knew, there ‘twas taught long before.
Though Moses was learned in the wisdom of yore,
Diospolitan craft, and Heliopolite lore;
Yet in those latter days, the blind wisdom of man,
No more saw the spirit of Jehovah’s great plan.
But when Inspiration was vouchsafed him at last,
Then the bright light of Truth flashed full o’er the past;
hen mystic tradition received explanation,
The symbolical page became Revelation.

When Mesopotamian culture was discovered one tried to use it for the purpose one had used the Egyptian one, as exemplified by the tendency “Babel and Bible.” Yielding to this tendency, Lehmann-Haupt. Transferred the use of the pendulum from the Egyptians to the Mesopotamians. The French excavations of Lagash were conducted at the moment in which the great political issue in France was the struggle of the republicans to reduce the influence of religious orders on education. One thought that the discovery of the remainders of Sumerian civilization could be used to deal a blow to Jesuits and monarchists. Jules Ferry, who as Ministre de l’Education Publique et des Beaux-Arts was leading the moderate Republicans in their fight for what they considered liberal religion, exploited to the full the archeological findings. As a result of this political argument, the two statues of the Sumerian regent Gudea amount received a startling of publicity and they are still today the two best known Sumerian objects. They should have. Proved that the measuring standards of the Bible were of Sumerian origin; actually this is not less true for Israel than for any other country of the ancient world, but in order to belittle before the general public the importance of biblical revelation, one lied to argue that the rule or Gudea was a unit of sixteen fingers like the foot of the Bible and t hat this rule was in some way a sort of Paris meter of the ancient world. Léon Heuzey was appointed Curator of Oriental Antiquities at the Louvre by Ferry, and reserved to himself the study of the rule of Gudea, whereas r there was at the Louvre a curator, Heron de Villefosse, who was an outstanding expert of metrology; one wonders whether Heuzey was so innocent when, against the opinion of metrologists, he reported that the rule of Gudea consists of sixteen fingers. Lehmann-Haupt was able to resist this distortion, which became a dogma for the new school, and recognized that the rule consists of fifteen fingers, one half of a barley cubit divided sexagesimally into thirty fingers; but he accepted the contention that the rule of Gudea should be given paramount importance as metrological evidence. Obviously a rule cut on a statue is not likely to have been intended to be an example of scientific precision. Lehmann-Haupt, on the basis of the rule of Gudea and of the length of the pendulum that beats the second, made the barley cubit too short and as a result distorted all other data. The Egyptian foot by being made too short and forced to coincide with. The Roman foot at a length intermediary between the two. Lehmann-Haupt was engaged in a struggle against the new school, defending that units of volume and weight were a function of length, but by making two single concessions to the opposite current, he destroyed his entire argument.

The new school grows from pyramidite notions and their development into the “Babel and Bible” tendency. This appears clearly in the book Ideal Metrology in Nature, Art, Religion, and History (Dorchester, Mass., 1908) written by the pyramidite H. G. Wood. While stating that “there seems to be good foundation for believing that the old European systems of metrology were derived from the cubit of Egypt, which was the cubit of Noah, Moses, and Ezechiel” (p.78), he adds “Gudea’s rule proves the antiquity of the inch rule preserved by the Anglo-Saxon race” (p.87). Such were the tendencies to which Lehmann-Haupt gave way in a limited degree, but a limited quantity of such poison is enough to kill any scientific endeavor.

Lehmann-Haupt considered the greatest achievement of his scholarly life to have discovered that most ancient and medieval units of volume and weight exist in two varieties related as 24.25. Oxé recognized the importance of this discovery when he formulated his general structuring of units of volume and weight by starting from the distinction between netto and brutto units. I have noted that this distinction corresponds to a most important musical interval, the interval leimma, the Pythagorean semitone. Viedebantt recognized the importance of the discovery when he claimed credit for it and bitterly turned against his mentor Lehmann-Haupt because this latter insisted on the his priority; from that moment Viedebantt became the most active supporter of the new school and denounced the value of his own earlier achievements, including the gathering of data illustrating the ratio 24:25.

If Lehmann-Haupt had not blurred the issue of the distinction between Roman and Egyptian foot, he would have realized that he had in his hands the solution of a problem that had tormented metrologists since Newton’s time. If there are units of volume and weight related as 24:25, it follows that there must be units of length related as the cube root of 24 divided by the cube root of 25. This is the ratio between the Egyptian foot of 300 mm. and the correct Roman foot of 295.9454 (pes Aebutianus).

There is a talent of 27.000 equal to 3000 basic sheqel of 9 grams or 1000 Roman ounces of 27 grams (basic talent brutto), equal to 60 Attic commercial minai of 450 grams or 50 basic pints of 540 c.c.; this unit is a cube with an edge of 300 mm. There is an Attic monetary talent of 24.920 grams (basic talent netto) equal to 80 librae of 324.grams, 60 Attic monetary minai of 432 grams, and 48 basic pints; this unit is a cube with an edge of a foot of 295.9454 mm.

All units of length, except the artabic foot and cubit, which is a special derivation of the Roman foot, exist in two varieties related as the cube root of 24 divided by the cube root of 25, which I call trimmed and natural versions.

7. The fact that the Roman foot and the Egyptian foot are nothing but two versions of the same unit explains why in Greece and Rome there was used also a foot of 300 mm., next to the foot of 296. In Greece, the evidence for a foot of 300 mm. is provided by the dimension of constructions, but in Rome it is provided by measuring standards.

Scholars of the sixteenth century mention the existence in the Church of Santi Apostoli in Rome of a porphyry column marked in Greek letters POD (theta) nine feet’, which indicated a foot longer than the Roman foot. This type of reference standard was not unique in ancient Home, since Guillaume Philandrier (1505-1565) mentions that in Via Lata, the present Corso, there was lying another column of porphyry, unfortunately broken, marked POD IB “twelve feet.” Petto reports that as a young man he saw the column of Santi Apostoli, but when he went to examine it for his study on the Roman foot, he could no longer find it. Giovanni Bartolommeo Marliani (died about 1560) mentions it in a later edition of his popular traveller’s guide to Rome, Topographia antiquae Romae; he states that the column indicates a foot slightly longer (parvum ampliorem) than the length of Roman foot he adopts in reporting the extensions of ancient Rome. The length of the two feet is printed in some editions of the book; by comparing the two one can conclude that the foot of Santi Apostoli could be an Egyptian foot of 300 mm. Raphael Holinshed in his First Volume of the Chronicles of England (London, 1577; Book III, Ch.22) prints the length of the foot of Santi Apostoli next to that of the Roman foot of Porzi, the English foot, and the French foot; since the values are 290, 282, 300 and 326 mm., one can rest assured that, taking into account the inaccuracy of drawing and the shrinking of the paper, the foot of Santi Apostoli was about 300 mm. Philandrier reports that he adopted as length of the Roman foot the one suggested by Porzi, because it agrees with the rule of the monument of Statilius, and that he measured the foot of Santi Apostoli and found it longer of a ninth of an uncia, or 1/108. But the famous Jesuit scholar of constitutional law, Juan de Niariana (1536-1624.), in De ponderibus et mensuris, states that the foot of Santi Apostoli is 1/48 longer than the pes Cossutianus. If the latter foot were calculated as 293.88 mar., the former foot would be 300 mm. If Philandrier assumed a Roman foot of Porzi slightly longer than 297 mm., one would also arrive at a foot of 300 mm.

Since scholars do not expect to find a foot of 300 mm. in Rome, there are very few reports about it, In the town of Este in the Po Valley, there was discovered a fragment of bone foot, accurately divided into tenths of finger; the fragment is broken along the palm line and measures 75 mm. (Notizie degli scavi, 1906, 174). This is certain evidence of a foot of 300 mm. Bone rules are much more accurate than metal ones, as indicated by the fine divisional lines; even today one would use a rule of this sort for precision drawing. Even a broken part provides good evidence, but archeologists seldom report the unearthing of small bone objects. Hence, our information about Roman rules depends almost exclusively on bronze, pieces, put one should ask about many of them whether they were actually used as measuring instruments. It has been recognized that most of the Egyptian rules preserved in museums are highly decorated ceremonial pieces; their non-metric function at times is indicated by the inaccuracy of the divisions. Typical is the ceremonial cubit of wood covered with gold foil given by King Amenophis II to his royal architect; the cubit is inscribed with references to events that have nothing to do with measurement. One speaks of ceremonial cubits in Egypt, but one has never tried to define their precise nature. Such an investigation would be useful, also because many of the Roman foot rules too may have been symbolic objects, having some sort of religious meaning. That there is a problem of this kind was noted by L. Fröhlich in reporting the discovery, near Brugg in Switzerland, of two excellently preserved bronze rules measuring 294.8 and 292.8 mm. (Anz. Schweiz. Altertumskunde, IX (1907), 42). These rules are of the usual type, folding at the middle, with three of the four faces marked respectively by divisions into palmi, unciae and digiti, but the divisions are as much as 3 mm. off the right point. Such irregular bronze foot rules are most common.

The religious meaning of the measuring rule is indicated by the report of Uzielli that on many medieval manuscripts there is marked with a line the standard length of the body of Christ, obviously assuming the assumed normal height of six feet; in medieval and Renaissance Italy particular efficacy was ascribed to prayers based on the misura del corpo di Cristo, to be repeated as many times as there are subdivisions of the length. These data collected by Uzielli, throw some light on the meaning of a rule of bronze inscribed with the monogram of Christ found in a cremation tomb near Trier in Germany, dating around 300 A.D. This particular object has a length of 300 mm., with a margin of error of less than one millimeter (Festschrift Ox , 147-152). It is divided into 12 unciae varying between 23.1 and 26.5 mm. In his report Ludwig Hussong excluded that it could represent an Egyptian foot, with the false argument that the object was found far away from Egypt.

In the location of the ancient Thibilis, in Algeria, there was found a mensa ponderaria on which by initiative of an aedile there were inscribed the lines corresponding to mensurae structoriae et fabriles reported by archeologists to be 50.5, 50, and 30 cm. (Bull. archeol. du Comité, 1919, 75). The first two units represent the natural and trimmed version of the barley cubit (506.25 and 499.4 mm.); the third unit is an Egyptian foot of 300 mm. This monument confirms not only that in Rome one used the natural version of the unit called Roman foot, but also that in the ancient world all the units of length formed a system and were used concurrently. In the not too distant town of Cuicul, on another mensa ponderaria there is marked a line described as being “about 52 cm.” (op. cit., 1913, 494); most likely  this is an Egyptian royal cubit of 525 mm., corresponding to the foot of 300 mm.

Petrie claims that the Roman foot, trimmed basic foot in my terminology, was also used in Egypt, next to the natural version. Repeatedly he asserts that there are Egyptian measuring rods in which the fingers are calculated as Roman fingers (hence, 18.5 mm. instead of 18.75 mm.), leaving a space at the end of the rod; but he has never described in detail a rod so constructed. The only certain evidence is the quadrangular bronze rod of Torino, on which a side is a regular Egyptian cubit of 28 natural basic fingers, and two other sides are marked with trimmed and artabic fingers. But this rule may belong to the Hellenistic period or the period of the Persian domination. Petrie claims that he found details of constructions calculated by trimmed fingers, but he doss not submit detailed numerical evidence. It is quite possible that he is correct, considering that in Greece and Rome there were used both the natural and the trimmed basic foot, but unfortunately Petrie confuses a scientific issue with the superstition of the pyramidites according to whom there was an entity called the Pyramid inch, which should be a divinely ordained ancestor of the English inch. It is desirable to examine again some of the evidence cited by Petrie.

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