Roman and Egyptian
Foot1. 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 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, 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 The following table sums up the essence of my contribution to the metrology of linear units.
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 :
It was further determined
that if one calculates the Roman foot from the In discussing standards
of weights, I have pointed out that the riddle of the edge of the The existence of a 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 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 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 ( 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 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 The general use of the combination
of units of the aqueduct of Bononia, is indicated by a little ivory
box (10.5 cm wide) My conclusion is that a
shortened Roman foot, which is 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 artabic foot is a unit
particularly connected with sexagesimal reckoning, because the artaba
contains 60 reduced pints (Alexandrine 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 The only point that remains
uncertain is whether the 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
( 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 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 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:
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:
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 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 The ancients used not only
the value 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 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
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 A small list of rules found
in France has been compiled by H0ron de Villefosse; more than half agree
with the length of the 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 In conclusion one can draw the following table of feet used in Rome:
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 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 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 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
Scholars of the early nineteenth
century observed how unfounded was Raper’s contention that the
pe Hultsch presented the theory
of progressive shortening in his 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 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 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 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 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
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 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 ( 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 ( 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. ( 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 In the location of the
ancient Thibilis, in Algeria, there was found a 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. |