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The Problem of Lineal Standards

Among the many reasons for which scholars since the fifteenth century have been trying to reconstruct the length of the ancient Roman foot, the most important one was that of having a standard by which to calculate the length of the degree of meridian. It was generally believed, with reason, that the ancients had an accurate calculation of the degree. But one could not interpret the statement that the degree is equal to 75 Roman miles, if the length of the foot was not exactly known. Most of Europe used as standard the Roman foot or units related by a simple fraction to the Roman foot, but the several samples of the Roman foot did not agree with each other. This results from the fact that the Roman foot already in ancient times existed in a correct version of 295.9454 mm., by my computation, and in two other versions slightly reduced; the smallest of these versions was used as standard in some parts of Europe. There was a version of the Roman foot which was known as geometric foot, since it was calculated as the edge of the cube containing 80 Roman librae. In principle the value of the Roman foot so calculated should have been absolutely reliable, but the Roman libra of the Middle Ages was heavier than the ancient one.

I have explained that ancient weights could be increased of a komma, (1/80). The correct weight of the Roman libra was 324 grams, but it could be legitimately increased to 328.2 grams. The medieval Roman libra usually approaches this maximum. To this value of the weight there corresponds a Roman foot of 297.675 mm. The length of the geometric foot, as indicated by writers of the fifteenth and sixteenth centuries, is certainly close to 297.5 mm.

Because it was used in the city of Rome, one gave particular authority to the palmo degli architetti used in that city. In Italy one called palmo the half cubit which is 3/4 of foot. In Rome one used also another palmo called mercantile, which is based on a barley foot (18/16 of Roman foot). The authority of the palmo architettonico was such that Kepler asked to be sent a sample of it, receiving instead a palmo mercantile, as it appears from his reckonings. Luca Petto in 1565 posted on the Palazzo dei Conservatori at the Capitol an inscription, with the length of the ancient Roman foot, as calculated by him, and that of the two palmi of Rome. One tried to use these markings as an absolute standard for Europe, but the lines were not cut on the stone with a sufficient precision of the ends to be true scientific standards. Moreover each observer measured the lines of Petto with his own standard, often rounding off the figures, so that the very fact that dozens of scholars reported the data of the Palazzo dei Conservatori increased the confusion, instead of dispelling it.

In 1810, at the moment of the adoption of the metric system in the Department of the Tiber, then annexed to France, the line of the palmo architettonico was found to be 99.042 Paris lines, indicating a corresponding foot of 132.0559 lines. As it happens in many similar operations of conversion of the ancient local measures in France, instructions were sent from Paris to the prefects to measure in meters, without providing a reliable standard of the meter. In the most fortunate cases one calculated by using the old toise (6 pieds de roi) of which good standards were available, and then converting the datum into meters at the ratio established in fixing the Paris meter, which was calculated as a fraction of the toise du Perou kept by the Academie des Sciences. In some other cases one did not measure anything, but simply reported figures based on the traditional conversion rate of the local measures to the toise. In some less fortunate cases one used poor standards of the meter, such as samples of the metre provisoire of 1793 (meridian circle of 5,132,430 toises according to the measure of 1740), instead of the final meter of 1799 (meridian circle of 5,130,740 toises, according to the measure performed between 1792 and 1798).

In the case of the standards of Rome the calculation was performed by the Jesuits of the Collegio Romano who used as reference the toise used in 1751 by the Jesuit Ruggiero Boscovich in his famous triangulation of Italy; the calculation was performed by competent hands by using a reference that is practically perfect. Boscovich had asked to be sent from Paris an exact copy of the toise du Perou, used in the operations for the calculation of the degree in Peru and in Lappland; later calculations proved that his toise, in spite of having been examined in Paris by Jean Jacques de Mayran, was too short by 8/78 of a line (about, 1/8500 too short). A close reading of the documents suggests that de Mayran may have known that the toisehe sent to Boscovich was not the toise du Perou. He opposed the motion of La Condamine by which the Academie des Sciences adopted the toise du Perou as the official standard; de Mayran demanded that his toise, which was 1/10 line shorter, be adopted because he had used it to calculate the length of the pendulum. The toise of de Mayran was later acquired by the astronomer Lalande who found it to be 11/12 of line too short. But this difference is almost irrelevant to our purposes.

Boscovich had tested the palmo of Petto with great care and found it to be 99.0 1/3 lines. One may accept this figure and correct it to read 99.02 so that the corresponding foot is 132.02 2/3. I am paying close attention to these details, because my figures further support the hypothesis of Guilhermoz that when one calculated the pied de roi of the toise du Perou, one calculated it as 12/11 of a Roman foot by the palmo architettonico. A pied de roi is divided into 12 inches of 12 lines, so that 11 inches are 132 lines. If the hypothesis of Guilhermoz is correct, the conventional basis of the Paris meter is the palmo marked by Petto.

According to my reckoning weights of the Renaissance and the following period are usually about 1/120 heavier than the corresponding ancient ones. The origin of the difference may be the sample of libra sent by Calif al-Mamun to Charlemagne. As I have determined, what was sent was a sample of the normal Mesopotamian mina of 486 grams (1 ½ Roman libra), but the sample may have been increased of a komma to be 492.1 grams. A study by Maurice Prou (Mem. Soc. Antiq. de France, 54 (1895), 20) concludes on empirical grounds that the original weight of the Carolingian libra was 491.179 grams. In Brussels there is preserved a sample weight, dating from the end of the thirteenth century marked as dormant du veritable poids de Troyes; since this piece weighs 368.8 grams, the Carolingian libra (4/3 pound Troy) would be 491.73 grams. Also in Brussels there is preserved a bronze Roman libra of the ninth on tenth century, marked Rodulfus negotiens, weighing 327.10 grams. and hence corresponding to a Carolingian libra of 490.65. The libra of Paris, which could be called Carolingian libra, at the moment of the adoption of the metric system was calculated as 489.506. This weight is based on a standard known as pile de Charlemange, still preserved at the Conservatoire des Arts et Metiers, which was manufactured in the fifteenth century. The practice of the medieval assizes of weights was to average the several standards available, and hence it is possible that one did average standards based on the ancient values with standards based on the Carolingian value, which was increased by 1/80.

2. In Paris the standard of the Roman foot was kept in the form of an aune equal to 4 feet. A sample was kept by the corporation of mercieres and one by the corporation of drapiers; the aune merciere kept in Rue Quiquenpoix was considered more authoritative, possibly because the roy des merciers had authority all over France and because the corporation manifested its activities at all important fairs. In Spain the Roman foot had to be exhibited at all fairs. At the time of introduction of the metric system the aune merciere was calculated as 1188.446 mm. (foot of 297.1115) and the aune drapiere as 1185.665mm. (foot of 296.416).

Prior to the establishment of the metric system, the unit considered to be the scientific standard of Europe was the pied de roi of Paris. French archeologists have determined that this unit occurs in pre-Roman and Roman southern France; I have determined that it occurs in the monuments of Athens. Some monuments of Athens, according to measurements reported by Dinsmoor, were planned by a unit whose value agrees with the value of the ancient pied de roi. Probably it was brought to Gaul by Greek colonists. I have discussed this in dealing with units of volume, since the reason for its existence most likely is the need to introduce a unit adjusted to the specific gravity of the wheat of Gaul and of Chersonese, the grain Athens would buy. Pliny states that the lightest wheat is that of Gaul and Chersonese, and the pied de roi was used in Athens and in France. It was introduced in order to have a unit equal to (1 1/3) basic talent netto, since 10 cubed:11 cubed = 1000:1331. The pied de roi is equal to 11/10 of Roman foot. In relation to this unit Guilhermoz, usually a most responsible and skillful scholar of medieval metrology, has been guilty of a serious blunder. He repeats that the foot was calculated as 12/11 of Roman foot, but the evidence he himself quotes proves that it was calculated as 11/10. Guilhermoz reports that the perche royale was equal to 22 Roman feet; he quotes evidence such as a report of a landsurvey of 1394 A. D. in which it is specified XXII piez pour la perche au pie du Chastelet. Calculated as 11/10 of the correct Roman foot of 295.9454 mm., the value of the pied de roi is 324.5399 mm.

In Paris one kept two authoritative samples of the pied de roi. One was kept at the bureau (Ecritoire) of the maitres jures des oeuvres de maconnerie et charpenterie de la ville de Paris. It was known as Toise de l’Ecritoire; Mersenne in La Verite des sciences describes this standard as tel que s’en sert en France et que je l’ai pries a l’Escritoire pres de St. Jacques de la Boucherie, auquel on a recours quand on veut ajuster les pieds. The other authoritative sample was kept at Chatelet, the seat of royal administration in Paris, where the toise du Chatelet was attached to a pillar of the courtyard. Sometimes before 1667, this sample was damaged by a bending of the pillar to which it was attached. In 1667 the newly established Academie des Sciences and Academie des Inscriptions were influential in causing the establishment of a new standard of the Chatelet, which too was posted in the courtyard, where in 1758 it was damaged by an accident similar to that had occurred to the older one. The older one remained in place until it was stolen in 1755.

It is a certain thing that the new toise du Chatelet was shorter than the older one: Jean Picard states that it was 5 lines shorter. The older one must have had a length similar to that of the Toise de l’Ecritoire, since only the introduction of the new toise caused a conflict on the question of the relative authority of the sample of the Chatelet and that of the Ecritoire. In the political climate of France at the time, one would not let a corporation use its standard against that sanctioned by royal authority: Nicolas de la Reynie, who had been appointed to the new position of lieutenant de police of Paris, prepared an order requiring the maitres macons to cause their sample to conform to the new toise. They tried to defend the authority of their sample; as a result a test was conducted, measuring the gallery of the Louvre which was found to be 220 toises 1 foot 2 inches 7 lines according to the Toise de l’Ecritoire, and 219 toises 9 inches 7 lines according to the new toise du Chatelet. Thereupon Colbert signed an ordinance commanding the shortening of the toise de L’Ecritoire. It is unfortunate that Guilhermoz, who consulted the official file on this affair, not only confused the problem, for the reason I have mentioned, but did not ascertain in situ the length in meters of the gallery that was measured. I wonder whether the gallery, which was finished in 1608 under Henry IV, had been intended to be 200 toises.

Three years after the establishment of the new toise du Chatelet Picard, assuming that he had an absolutely reliable standard, proceded to this famous Mesure de la terre. This was the datum for which Newton had waited anxiously, before announcing his theory of universal gravitation. Because of the use made of it by Newton, this test acquired universal fame. As Alfred Noyes puts it in the poem, Watchers of the Sky,

Newton withheld his hope
Until the day when light was brought from France,
New light, new hope, in a small glistening fact,
Clear-cut as any diamond; and to him
Loaded with all significance, like the point
Of light that shows where constellations burn.
Picard in France-all glory to her name
Who is herself a light among all lands-
Had measured earth’s diameter once more
With exquisite precision. To the throng,
Those few correct ciphers, his results,
Were less than nothing; yet they changed the world,
For Newton seized them and, with trembling hands,
Began to work his problem anew.

Shortly after the test of Picard to the north of Paris, Gian Domenico Cassini in 1682 measured the length of degree to the south of Paris and found a length lesser than that of Picard. The conclusion was that the earth was elongated at the poles, whereas Huygens and Newton had presumed that the effect of angular momentum would make it flattened at the poles. The theory of the elongation at the poles had been defended even before the test of Cassini by the Alsatian scholar of metrology Kaspar Eisenschmid; in 1691 in this Diatribe de figura telluris, by converting into Roman miles, according to this text of ancient metrology, the figures of the known measurements from Eratosthenes to Picard, he concluded that the earth was elongated at the poles. The controversy raged for more than half a century, involving vital issues of science, since the elongation at the poles would put in question basic assumptions of physics. The matter was settled only by the triangulation of Boscovich in Italy. Scholars became highly emotional about it and the general public took sides; when Maupertuis, after his expedition for the measurement of the degree in Lappland, in 1738 published De la figure de la Terre, defending the theory of polar flattening, it was stated authoritatively that in Paris il n’est presque personne, hommes et femmes, qui ne l’ait lu (de Mayran in Journal Helvetique, Sept. 1740). One had such confidence in the new toise du Châtelet that for almost half a century, nobody thought of questionning the length of the toise used by Picard. This is why the Cassinis could continue to defend the theory of polar elongation and Voltaire mocked Maupertuis as le grand aplatisseur.

Actually it was not difficult to ascertain the standard used by Picard, since he had taken as base a stretch of straight paved road between Villejuif and Juvisy. In 1729 Jacques Cassini, son of Gian Domenico, measured not only the base but also some of the trigonometric angles of Picard; he repeated the tests five times, always finding that the Picard’s figures were in excess. In 1754 the Academie des Sciences instructed eight members, divided into two independent groups of four, to repeat the measurements of Picard. The group of four in which were Cesar François Cassini, son of the Jacques, and Charles Etienne Camus, who was member also of the Royal Society, presented a report in which they properly stated that they had operated with “religious” care and found a length of degree 60 toises less than Picard. Boscovich finally realized that the reason why scholars had found the standard of the toise du Châtelet so reliable, was that almost all of them had used rules prepared by the engraver Langlois or his nephew Canivet, who had a private standard of their own. The standard used by Picard in his calculation of the degree was 1.14/1000 shorter than that of Langlois. In 1735 Langlois had constructed for the Academie des Sciences the toise du Perou.

Guilhermoz has tried to reconstruct the history of the standards of the toise, but he has been confused by the metrological error I have mentioned.

An ordinance of 1540 prescribes that the aune shall be 524 lines; this indicates that the pied de roi was calculated as 11/10 of the Roman foot of the aune. The ordinance simply rounds the figure of 523.63 2/3 lines. A sample of the aune merciere dated from 1554, was tested by the toise du Perou at different times by members of the Academie des Sciences and found to be 526 5/6. Hence, the aune merciere indicates a Roman foot of 297.113 mm. The value agrees with that ascertained at the time of the adoption of the final meter. Hence, one may conclude that before 1667 the pied de roi measured about 326.8 mm. Guilhermoz tries several methods to calculate it and arrives at a similar figure.

Mersenne reports that the cube of the pied de roi was usually calculated to contain either 70 ½ or 72 Paris librae, but he found that it contained 74. We may disregard his figure, because it must contain some error or some new evaluation of the libra. In the ancient world a cube of a pied de roi would have contained 70.987 reduced pints (Alexandrine sextarii) of 486 c.c. This amount was probably increased to the more convenient figure of 72 reduced pints or sextariior reduced to the figure of 70 pints or 105 Roman librae. Bude in De asse calculates the cubic foot of Paris as 72 chopines or setiers, which he identifies with the ancient sextarius. According to the figures given by Mersenne, a cubic foot of 72 Paris librae of 489.5 has an edge 327.864 mm., which is the length Guilhermoz and I have calculated for the pied de roi before 1667; a cubic foot of 70 ½ librae has an edge 325.597 mm., which is in almost perfect agreement with the value of 325.54 mm. in the ancient world. This indicates that one had preserved the ancient value. Possibly by measuring the gallery of the Louvre as equal to 2000 toises one would arrive at this value. The greater length of the pied de roi was adjusted to a cube of 72 librae.

When one reconstructed the toise in 1677, one must have known that the length equal to 72 librae was too long. Hence one recalculated it as equal to 70 librae; possibly one figured this new pied de roi as equal to 12/11 of the Roman foot of the aune merciere. Philippe de la Hire submitted to the Academie des Sciences an ancien instrument de mathematique, which he said had been the rule used in establishing the new toise; I have found that his numerous measurements of Roman monuments indicated that he used a foot about 1/1000 shorter than the standard of Langlois. He states that he found the aune merciere to be 527 ½ lines of his foot; if the pied de roi is 12/11 of foot of the aune, the figure should be 528.

Picard in his most famous report on La Mesure de la terre, which he performed three years after the establishment of the new pied de roi, states that the toise contains 641,326 grains of Paris (9216 grains to the libra), that is, 69.5983 librae. One may surmise what Langlois did in establishing his private standard: he corrected the length to make it equal to 70 librae. This would give a foot of 324.802 mm., which is a figure almost perfectly equal to the final value of 324.839, considering the extreme difficulty in calculating with modern scientific exactitude a foot from a weight. Whereas the pied de roi had been taken as 12/11 of the Roman foot of the aune merciere, he probably took as his standard 12/11 of the Roman foot indicated by the palmo of Rome.

One reason why the members of the Academie des Sciences and the Academie des Inscriptions in spite of their numerous studies of metrology, could not find the reason for what came to be called “Picard’s error,” is that Picard himself soon came to use instruments based on the standards of Langlois. In 1745 Camus and Jean Hellot found that a foot constructed by Butterfield (died in 1724), engineer of Louis XIV for the construction of mathematical instruments, for Picard and Auzout, did not differ from their standard. I have found that the measurements performed on Roman monuments by Picard and Auzout confirm the judgment of Camus and Hellot. Picard calculated the aune merciere as 526 4/5 which is almost exactly the length of 526 5/6 calculated by Camus and Hellot. Only La Hire preserved the original length of the toise of 1667; by comparing a value of aune merciere of 526 5/6 with the value of 527 ½, one obtains that the toise of 1667 was 1.12/1000 shorter than the toise du Perou. The astronomer C. Wolf, by reexamining several trigonometric operations, concluded that the “error of Picard” had been about 1.14/1000 (Annales de l’observatoire de Paris, 17 (1883), 1-78).

3. The difficulties created by the uncertainty of the toise stress the point that the greatest difficulty in metrological research was the lack of an absolutely fixed reference. Between the end of the fifteenth and the beginning of the sixteenth century the custom was introduced to refer to the rules portrayed on some monuments of ancient Rome, but the attending publicity caused sightseers to tamper with them. Another common device was that of printing in books the length of the intended unit as a line, but apparently printers were careless in following instructions and the drawings are usually most crude. The practice had been used in manuscripts, but was not appropiate for the printing press. In 1617 Willebrord Snell, who was the first to use triangulation in measuring the length of the degree of meridian, resorted to the desperate device of calculating the shrinking of the paper after printing. But it is enough to consider the total discrepancy between one edition and another of the same book, to realize that the method was inadequate. But it was not abandoned until the end of the seventeenth century, even though I have not found one single book in which the metric lines can be consider reliable.

The problem was complicated by the circumstance that most measures of Europe were related to the Roman ones, so that often one changed local standards as studies were published concerning the ancient Roman units. Hence, the reasearch which was intended to introduce uniformity of standards increased the uncertainty.

Either because England had a longer tradition of centralized authority or because one did not know how the English units were related to the Roman ones and hence they were not changed by antiquarians, English standards acquired the reputation of being the most stable in Europe. The English foot was certainly a most stable entity and probably would have been accepted as the scientific unit, except that one thought only of Roman units as being of universal nature. But Tito Livio Burattini, who, at the beginning of the seventeenth century was trying to establish a metro universale or metro cattolico as the basis of a decimal system of measures, considered the superiority of the English standard. The idea of a decimal system was being agitated at the time, and the English system is particularly adaptable to it. One may keep in mind that the present value of the pound aver depois is 7000 grains, whereas the pound Troy is 5760 grains; the Tower pound, which originally was known as sterling marc or livre de la Rochelle was fixed at 5400 grains. The correct ancient Roman libra of 324 grams is equal to 5000 grains. The grain Troy is today officially calculated as 0.06480 grams, but a more exact value according to some official sample weights is 0.064743 grams. Historically the grain Troy should be 1 ¼ Paris grain (9216 grains to the livre), which was calculated as 0.053115 grams when the metric system was introduced. But the main decimal feature in the English system is contained in the essential elements of the structure: the foot is the edge a cube containing 1000 ounces of water.

Later I shall have occasion to indicate that the basic English units are closely related to the ancient Egyptian ones, which were constructed according to decimal computation more than any other in the ancient world. It is worth noting that those who for the last one hundred years have written books and pamphlets to argue for the superiority of the English system of units over the metric one, often use the argument of the superiority of duodecimal computation over the decimal, assuming that English units are duodecimal. Some of these writers, out of hatred against the term decimal, use the term duodenal, which is indeed becoming to this visceral kind of thinking

All that was necessary to establish a decimal system in Europe at the beginning of the seventeenth century was to divide decimally the English foot, but, since all efforts up to that time had been directed toward the determination of the exact value of the ancient Roman foot as the international basis, it was necessary to establish the correct ratio between the English and the Roman foot. This was the task to which John Greaves applied himself. The English foot was known on the Continent as the London foot and the authoritative standard at the time was kept at Guild Hall, the seat of the municipal government. Greaves caused an expert craftsman to prepare a rule of 10 feet, each accurately divided into 2000 parts. which had to be the standards of his calculations. Provided with this instrument, he left for Italy, determined to settle the matter of the length of the Roman foot. He applied systematically all the methods which had been devised to calculate the ancient Roman foot, but in spite of the fact that A Discourse of the Roman Foot and Denarius (London, 1647) is a monument of sound scientific method, he could only confirm what others had found, namely, that the Roman foot existed in three varieties. He discounted the two longer varieties, which differ of less than half a millimiter and merged them into a single unit which he calculated as 970/1000 of English foot; he took as standard Roman foot that called pes Cossutianus by Renaissance scholars and calculated it as 967/1000. Reckoning by the English foot of 1824, the values are 296.2666 and 294.7426 mm. He measured the palmo of Petto as 732/1000, implying a foot of 976/1000 (297.486 mm.); he also found that by calculating from the libra one could obtain a foot about as long as the foot indicated by the palmo.

When Greaves saw that the several methods used to calculate the Roman foot gave slightly different results, indicating that there were in Rome slightly different versions of the foot, he tried to avoid the difficulty by leaving Rome for Egypt. There was a tradition, which had remained alive through the Middle Ages, that the Egyptians had invented the art of measuring and were masters of it. Hence, the problem became that of fixing the length of the Egyptian foot. Greaves was late proved right: to the Napoleonic expedition to Egypt there was attached a group of ingenieurs geographes for whom one of the main tasks was the determination of the Egyptian standard; thanks to the data collected by them it was possible to calculate that the Egyptian foot measures 300 mm., a datum which by consensus of all serious metrologists is the most certain datum in the entire field of ancient metrics. The absolute precision and reliability of this datum is an extremely valuable acquisition, since most metrologists of the nineteenth century have come to agree that the Egyptian foot is the basic unit of length of the ancient world.

Greaves applied to Roman buildings the method developed by scholars of the Renaissance, a method which Böckh called Newton’s method: it assumed that by studying the dimensions of any manufactured object, and a building in particular, one can determine the rule and the division thereof used by the manufacturer or builder; in particular, one can presume that usually the main dimensions of a building can be expressed as a simple multiple of the rule adopted in the construction. When he concluded that the several techiques used to calculate the Roman foot gave a figure varying between something less than 295 mm. and something more than 297, and that Newton’s method gave results varying between 296.26 and 294.74, he thought that he could avoid the difficulty by measuring the Great Pyramid of Gizah. This was a simple quadrangular building which obviously must have been planned with accuracy and about which one can presume that the sides were calculated as a round figure; furthermore, several ancient authors report its dimensions, so that one can interpret from the Pyramid the value of the standards they used. Greaves was also concerned about the fact that the repeated calculations of the Roman foot had been of no avail because each author had used a standard for which there was no absolute basis of comparison. He thought that if he could mark two points on the corners of the Great Pyramid, the segment so marked could be taken as the permanent reference standard by scholars.

When in 1638 Greaves left Italy for Egypt he found that Burattini had preceded him there for the purpose of setting the standard of this metro cattolico. Burattini had the advantage of having measured a great number Egyptian buildings and objects, and Greaves had that of being provided with a rule based on an English foot accurately divided into 2000 parts. Together they measured the Pyramids of Gizah, and when Greaves departed he left his rule with Burattini. As ill luck would have it, Burattini was robbed of his notes on this way back, so that history remembers him only as the one who suggested the name meter for the basis of the decimal metric system.

Greaves discovered that the Great Pyramid in its present state is not that monument of exact geometrical shape he could imagine before seeing it; the exact measurement of the sides of the Pyramid proved an arduous task which was brought to a final conclusion only in 1925. But Greaves found that the dimensions of the so-called King’s Chamber could be ascertained with greater certainty. The data about it which were published in Greaves’ Pyramidographia, were interpreted by Newton, who advanced metric research by two triumphant steps. He determined, mainly because of his familiarity with Talmudic writings, that, whereas normally the cubit is equal to 1 ½ foot and hence is 6 hands or 24 fingers, the Egyptian royal cubit is a unit of 7 hands or 28 fingers. He assumed that the King’s Chamber measures 20 x 10 such septenary cubits, and concluded that the cubit is equal to 1719/1000 of English foot. i.e., 523.9507 mm.

Newton’s calculation is amazingly accurate. The Egyptian foot can be evaluated as 525 mm., but there is a difference between the theoretical value of a standard and the concrete embodiment of it: it proves that the Great Pyramid was constructed by a rule of 524 mm. Petrie noticed that the blocks of the King’s Chamber have been shaken apart by earthquakes; calculating the original spacing of the blocks, he determined that the dimensions are based on a cubit of 524.053 +0.10 mm. This reckoning of Petrie proves that whereas in principle Newton’s method is most reliable, its concrete application must be surrounded by the greatest safeguards. Other details of the Great Pyramid indicated the use of a unit of 524 mm.

Newton was interested in determining the length of the Roman foot in order to calculate the circumference of the earch. As I have reported, the ancients had accurate estimates of the circumference of the earch, and so did geographers of the Middle Ages. The question had been confused by those writers of the Hellenistic age who, in order to deny the infinity of the universe, denied the validity of the calculations of the distance between Spain and the coast of China; this distance had been calculated with a good approximation, as proven beyond any doubt by the maps drawn before the age of Columbus, but the publication of Ptolemy’s geography threw European metrics into a state of confusion. I have pointed out that the genius of Columbus is a product of the development of empiricism among the Aristotelians of Italy; having concluded on empirical grounds that there was land to the West of Spain, he exploited the false arguments developed by Peripatetics and Stoics, in order to refute the correct calculations of mathematical geographers of his time. As a result of these developments the length of the Roman foot had become a matter of debate, and Newton did not know how to interpret the tradition that the degree has a length of 75 Roman miles (500 feet to a mile). This calculation is the same as that of 300,000 stadia for the circumference of the earth, reported by Archimedes; this stadion is composed of 300 cubits, the cubits, being 1 ½ Roman feet. I shall show that the correct value of the Roman foot is 295.9454 mm.; this implies a degree of 110.980m. Between degree 35 and degree 36, where at least some of the calculations were performed, the length of the degree is 110.947 m.

The amount of precision achieved by the ancients is indicated by the very fact that they calculated at degree 36, the degree they took as the axis of the inhabited world; they must have been able to notice the difference of about 1000 m. with the length of the degree at latitude 30, the latitude of the mounth of the Tigris and the Euphrates and of the southern tip of the Nile Delta. But Newton could not trust any of the slightly discrepant estimates of the length of the Roman foot. He hoped to be able to use the figure reported by Eratosthenes of 700 stadia to a degree, the stadion being equal to 300 Egyptian royal cubits. If the Egyptian royal cubit of Eratosthenes is calculated at its standard value of 525 mm., the degree of Eratosthenes is 110,250 m. This calculation is based on a round figure for the radius of the earth, that is, 40,000 stadia; since there are 40 such stadia in an hour of march, schoinos, the radius is 1000 schoinoi. The reason why Eratosthenes once reports the circumference of the earch as 250,000 stadia and at another time as 252,000 is that the ancients used concurrently a calculation of pi as 3 1/8 and 3 1/7. This proves that Eratosthenes is referring to a rough practical estimate he Great Pyramid of Gizah happens to be calculated by a cubit which is slightly short, 524 mm.

Newton almost hit the right figure in calculating the dimensions of the King’s Chamber, but he arrived at a different figure from Greaves’ data about the side of the Pyramid. Greaves measured the north side, but there must have been a great accumulation of sand against it, since he reported a length of 693 English feet, whereas the correct figure is slightly less than 760. Since Greaves’ data for other measurements of the Pyramid are accurate, one must wonder whether there has been some error in transcription; in a work of the astronomer Jakob Ziegler, published in 1536, this dimension, expressed in geometric feet, is correctly given. (Terrae Sactae quam Palaestinam nominant, Syriae, Aegypti doctissima descriptio (Strasbourg, 1532). The length of the bases is said to be 775 geometric feet; this indicates a foot of 297.5mm. (775 x 297.5 = 230,562mm.) Newton assumed that the side of the Pyramid had been calculated as 400 cubits, whereas it is 440, and derived from Greaves’ datum a cubit which is 1732/1000 of English foot (more exactly he should have said 1732.5/1000). If he had accepted this figure he would have obtained a cubit of 527.913m, which, multiplied by the figure of Eratosthenes, would give a meridian circle of 39,910 km. Hence, by a series of compensation of errors, Newton might have come close to the right figure.

According to Eratosthenes’ figure of 700 stadia to the degree and this evaluation of the cubit, the length of the degree would be 110.862m. by the English foot of 1824, the actual length of the degree between latitude 30 and 31 (the Pyramid is located at latitude 30). But Newton properly disregarded this calculation of the cubit, stating that he was certain that the Pyramid had been constructed by the cubit indicated by the King’s Chamber. According to this cubit equal to 1719/1000 of English foot, the measure of Eratosthenes, based on the length of the degree in Egypt, gives a meridian circle of 129,956 thousands of English feet. Since Newton positively accepts Greaves’ calculation of the Roman foot as 967/1000 of English foot, the datum of 75 Roman miles to the degree indicates a circle of 130,545 thousands of English feet. The calculation performed by Richard Norwood in 1635 between London and York, indicates a degree of 36,7196 English feet; this calculation, which was included in Norwood’s very popular Seaman’s Practice, indicates a circle of 130,971 thousands of English feet. It would be interesting to examine in detail Newton’s writings, to see whether this computation influenced him in suggesting the polar flattening of the earth. Newton was not certain that the rotation of a fluid body would tend to cause an equatorial swelling, this was proved by his pupil Calin Maclaurin. As I have reported, Eisenschmid used an argument of the sort just described to defend the polar elongation of the earth.

Before announcing theory of universal gravitation, Newton waited twenty years until he could avail himself of the calculation of Picard. From the degree of 50,060 toises of Picard, calculating by the toise du Perou, the meridian circle is 40,036.3 km.; this value attains the same precision that Newton could have obtained from the ancient one of 75 Roman miles to the degree, that is, 39,966.6 km., if he had known the exact value of the Roman foot.

The adoption of the new toise du Chatelet in 1667 marks a new phase in the study of ancient metrics and in the modern calculations of the dimensions of the earth. For the first time scholars had a reliable standard, which proved to be the private standard of Langlois. As I have stated, Langlois must have calculated the pied de roi as equal to the edge of a cube of 70 Paris livres. At the time of the establishment of the metric system, it was found that, calculating by the principles used in setting the standard of the kilogram, the pied cube of water at 4 degrees Celsius is equal 70 livres plus 223 grains (a grain is 1/9216 of livre).

The efforts of Picard, Gian Domenico Cassini, and Auzout, in calculating the Roman foot and the circumference of the earch, mark a new phase in the history of metrology, because they introduced the use of optical instruments and of the micrometer.

When in 1751-53 Boscovich performed his triangulation based on the Appian Way, definitely proving the polar flattening of the earth, on the same occasion he made clear the reason for the so-called “error of Picard” and also concluded that available empirical data indicate a Roman foot averaging 131 Paris lines (295.514 mm.). From that moment on, the length of the Roman foot became a matter of concern only for ancient scholars. The interest of cosmologists, astronomers and geographers in ancient metrology lasted up to the French revolution and died with Franz Xavier von Zach (1754-1832) and Charles Athenase Walckenaer (1771-1852).

4. Greaves’ study of Roman and Egyptian linear units in which he combined his insight into Roman archeology with his knowledge of Arabic scientific literature, was widely acclaimed in his time. But its value has been destroyed for contemporary scholarship, because at the end of the nineteenth century scholars such as Hultsch and Petrie accepted the statements contained in an inept paper written by Matthew Raper in 1760, “An Inquiry into the Measure of the Roman Foot,” Philos. Trans. 51 (1760), 774-823. In the new spirit of ancient scholarship one is less inclined to have patience with fine details that are often necessary in metrological research, and one looks with sympathy to those writings that, because of their irresponsibility, can solve problems by a direct and simple method. I have shown that the calculations of Greek metrology by Jakob Larsen are an extreme example of this contemporary tendency to make everything simple and totally false. But in the long run, such roughshod simplifications make scholarly problems abstruse and unsolvable by creating contradictions where they should not exist

Raper contends that the foot of Guild Hall was 3/1000 shorter than that of the Graham rule of the Royal Society. For our purposes we may consider the Graham rule identical with the British Parliamentary standard of 1824. The main argument of Raper is that Norwood calculated the degree as 367,196 English feet (111.921m. by the foot of 1824) by measuring between London (at Westminster the latitude is exactly 51° 30’, being 51° 28’ 38” at Greenwich Observatory) and York (which Norwood may have assumed to be exactly at latitude 54°, since the town is located at latitude 53° 58’); Raper on the basis of the recent tests of Peru and Lappland calculated that at latitude 52° 44’ the degree should be 57,438 toises (111,949 m. by the exact value of the Toise du Perou) and according to his reckoning of the foot of the toise as being 1065.4:1000 to the foot of the Graham rule, concluded that the foot of Guild Hall used by Norwood was 3/1000 too short. It is preposterous to question the validity of a standard of length on such a ground, since Norwood states that he calculated by measuring the road from London to York, deducting the deviations due to the curves along the way. Considering the method he followed or claims to have followed, it is a miracle that Norwood came as close as he did to the right figure. Actually, if we reexamine Raper’s figure for the degree by our standards, one finds that it agrees closely with Norwood’s. Following Raper’s line of thinking we should conclude that the Graham rule was 3/1000 too short. It is more preposterous that modern scholars should accept Raper’s argument without consulting an almanac to find that at the latitude 52° 44’ the degree is 111.282 m. Hence, according to the logic of the argument one should say that Greaves’ rule was 6/1000 too short.

Raper considers Greaves’ measurements of Roman monuments and tries to prove that they too indicate a foot which is about 2/1000 too short late. I shall examine in detail this part of the argument later, considering the empirical data but here I will limit must to a simple general consideration: Greaves in reporting his tests about the Roman foot rounded all figures to 1/1000 of English foot, so that a discrepancy of the range indicated by Raper can be easily explained, considering that Raper forced the figures to make them agree with the first calculation of a defect of 3/1000 in Greaves’ rule. The total futility of Raper’s argument is indicated by his using the fact that Greaves calculates the pied de roi according to the Toise du Châtelet as 1068/1000 of English foot, instead of 1065.4/1000 which according to Raper would be the relation between the pied de roi and the Graham rule (the pied de roi of the Toise du Perou is 1065.76/1000 of the English foot of 1824). Raper states that the Toise du Châtelet certainly was not changed in 1667, whereas the contrary was a well-known fact stated also by authors he does quote.

The bars kept at Guild Hall disappeared around the end of the seventeenth century; they are still used as the basic reference in the treatise of ancient metrology by Bernard, published in 1688. From this period on, one refers to other standards sanctioned by royal authority; one wonders whether the shift was due to the disappearance of the Guild Hall bars or to the policy, initiated by the Stuarts, aimed at reducing the power and the prestige of the municipal body of London. If there had been a notable shift in the length of the English foot, some of the several metrological essays of the time would have noted it. The Royal Society, founded in 1662, had set as one of its main aims to establish an absolutely fixed standard and cooperated in this endeavor with the Académie des Sciences and the Académie des Inscriptions, founded in the same decade; one would have noted a discrepancy of 3/1000 in the several calculations of the Roman foot published by these societies.

Raper obviously did not know or ignored the history of the Graham rule of the Royal Society. In 1742 George Graham constructed his rule by marking with E on a bar a point supposedly corresponding to the length of the yard of the Tower of London. Actually he followed a standard inscribed with the seal of Elizabeth I. Often in England measuring standards consisted of a metal bar that was placed inside a matrix also of metal; probably this practice had the purpose of allowing the reconstitution of the standard if the bar was stolen. Since the bar had to be lifted out of the matrix, there was a certain amount of play between the bar and its matrix. Graham chose a length intermediary between the bar and the matrix of the standard of Elizabeth I. There was raised some complaint that Graham had taken as standard the bar of the Tower, whereas the official standard of England should be that of the Exchequer. As a result of this criticism, the Royal Society proceeded to a survey of the standards available in London, whether royal or belonging to corporations such as those of the Clockmakers’ Company. This survey proves that the differences between the standards were such that they cannot be relevant to historical metrology. As a result of the survey, one marked on the Graham bar a point called Exch., for Exchequer standard, which relates as 36.0000:36.0075 to the mark E. Before this correction the bar had been sent to Paris where the Académie des Sciences had marked the length of the half toise. In 1758 Bird constructed a new bar taking as length a value intermediary between the length Exch. and the length E of the Graham bar, but closer to the latter as 36.000:36.0013. In 1824 by Act of Parliament the length of the Bird rule became the official length of the English foot (304.79974 mm). Since the English foot is no longer used as an instrument for scientific research, the custom prevailed very early in the United States and later in England according to the rules adopted by of the Board of Trade, in 1895 to calculate the English foot as a fraction of the Paris meter, making it 304.8 mm. This length has become official in the United States by Act of Congress in 1928, following an American practice of long standing. Since in current commercial and industrial practice the difference between the American and the English value has not created any major difficulty, it follows that the differences we are considering are less than what can be relevant for purposes other than modern scientific research.

In occasion of the survey conducted to check the Graham rule, on April 22, 1743 the President of the Royal Society, accompanied by six members, presented himself at Guild Hall where he was received and assisted by the officials in charge of the standards. The bars of Guild Hall were missing, but their matrices were available. It was found that a matrix of yard (36 inches) exceeded the Graham length E between 0.0434 and 0.0396 inches (the matrix was not perfectly square): a matrix of ell (45 inches) was compared with the length of the ell bar of the Exchequer and found to exceed it by 0.0444 or 0.0258 inches. Hence, one can conclude that the standard of Guild Hall did not differ from the English foot of 1824 in a manner relevant to the history of metric standards of the Renaissance and earlier.

But one could argue, with reason, that the bar of Greaves may be in fact have been different from the standard of Guild Hall. It is possible to test with certainty the value of the foot of Greaves. This may be done not by examining his figures for lengths of less than a foot, as it was done by Raper, but it can be done if one compares his figures for substantial distances, such as the measurements of the Great Pyramid of Gizah.

This calculation has been performed by C. Piazzi Smyth. When after 1860 the adoption of the metric system was being considered by the British Parliament, a part of the pyramidites and of the Anglo-Israelites argued that, if one had to make a concession to a system such as the French one, a superior standard would be the Pyramid Inch which is 1/500,000,000 of the polar diameter of the earh. The Great Pyramid of Gizah would have been calculated by this unit, which is about 1001/1000 of an English inch; the English inch would have deviated in the course of time from its original value that was that of the Pyramid Inch. In order to prove that the English Inch had become gradually shorter, the Astronomer Royal of Scotland, who was at the time the intellectual spokesman for the pyramidites, set to prove the opposite of what had been argued by Raper, namely, that the Standard of Greaves was somewhat longer than later English standards. He compared his measurements of the King’s Chamber and of the granite coffin which is placed in it, with the corresponding data of Greaves. Smyth arrived at the conclusion that the foot of Greaves is 1000.199/1000 of the foot of 1824. This proves that Greaves’ foot was for all reasonable purposes identical with the present English foot. The differences are amazingly small considering that Greaves rounded his figures to 1/1000 of foot, whereas Smyth calculated by 1/100 of inch. I have considered the measurements of Petrie, which were taken with extreme care, and have arrived at the following table of comparable data, expressed inches:

South side of chamber
West side of chamber
Inside width of coffin (West side)
Inside length of coffin (North side)

Considering the objects measured are not perfectly regular and the points of attack may differ, it is proved that the value of Greaves’ standard cannot justify a revision of his data.

I have belabored the question of Greaves’ rule with a care that may seem excessive, not only because important statements about the history of the Roman foot have been accepted from Raper’s essay, but because one of the main points of the new school of ancient metrology is that standards were not fixed with precision and changed from period to period. The instance of Greaves’ rule and the deduction of Raper are quoted as evidence that research such as mine is pointless, since one cannot expect any permanence or accuracy for standards set before the French Revolution.

5. Since the beginning of the eighteenth century there have been people who have proclaimed that English measures are derived from the Egyptian ones. Richard Cumberland, Bishop of Peterborough, who first considered this link at the end of the seventeenth century, had one reasonable argument that I shall consider below; but those who followed him and set the Anglo-Israelite and the pyramidite ideology, based themselves on the fact that Greaves and Newton had calculated the dimensions of the Great Pyramid in English feet. They quote Greaves and Newton as authorities, and a work apocryphally ascribed to Greaves is a sacred text to Anglo-Israelites and pyramidites.

Societies have been organized to prove the Egyptian origin of the English standards, magazines have been dedicated to this problem, and pyramidite books and articles continue to appear with regularity. The Anglo-Saxon Federation claimed seven million members in the United States at the beginning of the century; one can still hear in the United States a regular radio program dedicated to upholding the Anglo-Israelite view. The pyramidite theory is an important part of the creed of masonic societies. At the scholarly level, pyramidism has a great influence on Egyptian archeology, and there are Egyptian archeologists who are avowed pyramidites. After the discovery of the statue of the Sumerian regent Gudea, some shifted the pyramidite horizon to Mesopotamia and argued for the origin of the English standards from those of Mesopotamia. Some of these arguments have been accepted by the new school of metrology and are considered today established scholarship. Since the main purpose of the tendency is to link English measures with the Hebrew patriarchs, it does not matter whether the origin of English standards is found in Egypt or in Mesopotamia, but the Egyptian origin is the ancient and recognized version of the creed.

But all these efforts have not produced any study that can be considered responsible scholarship, even by a generous definition. Libraries have been cluttered and so have been the minds of ancient scholars, but the mountain has not produced even mice. It is myself, against whom those pyramidites I have had the occasion to meet in the course of the present research have expressed that personal rancor that only religious persuasion can produce, who has sucéeded in proving that there is some kernel of truth in the pyramidite doctrine.

Recent statements by responsible English historians have deplored the low state of the research on the history of English measures. The history of English measures appears wrapped in mystery. The reason for this is that the pyramidite conception is connected with Anglo-Saxon or Germanic racialism; hence, one has searched for the immediate antecedents of English measures among the Vikings, the Saxons, or similar ethnic groups. This approach has influenced the perspective of researchers who were objective in their intentions. A number of studies have been dedicated to the significance of the Sachsenspiegel for English standards; thanks to my beloved teacher Claudius von Schwerin, I feel at least as competent as these investigators in interpretating this fundamental text of Germanic Law and can declare that it is irrelevant to the issue.

I presume that any English researcher of the present period, being free from the fog of Nordic theories and from mystical religious tendencies, would grant that it is reasonable to look for the antecedents of English measures in the practices of the Roman Empire. But he would meet with a circular difficulty, because from pyramidite doctrines there has been developed in the field of ancient studies the new school of metrology, and as a result of this method the metrics of the ancient world have become an abstruse and uncertain topic.

At the beginning of the last century Böckh noted the existence of a unit he called the Babylonian-Egyptian great cubit, which is slightly longer than the Egyptian royal cubit of 525 mm. (7/6 natural basic cubit, 28 basic fingers). Jacques Frédéric Saigey, the mathematician who in 1820 founded with François Raspail the Annales des sciences d’observation with a most brilliant insight gathered that this longer cubit was calculated as 28 like the Egyptian royal cubit, but by artabic fingers, so that it become 537.404 mm. Böckh had noted that this Egyptian unit could be identified with a Mesopotamian unit.

In my study of Mesopotamian units I have determined that the typical cubit of literate Mesopotamia is the barley cubit, which is 499.408 mm. when trimmed. In cuneiform documents this unit is usually divided sexagesimally into 30 fingers, instead of 24. From the documents I have gathered that a times one used a great cubit composed of 32 such sexagesimal fingers (two feet of 16 fingers). This cubit of had already been noted by Oppert in considering the dimensions of Mesopotamian buildings. This has the advantage of relating as 9:5 to the Roman foot that was the main unit of predynastic Mesopotamia and remained a fundamental unit throughout the ancient world where one did not compute sexagesimally. In Mesopotamia one used two main units of surface: the ikû with a side of 180 barley cubits (surface 32,400 square cubits) and the acre with a side of 100 barley cubits. If the barley cubit is calculated as 32 sexagesimal fingers, instead of 30 (increased by 1/15), the acre becomes almost exactly 1/3 of the ikû, if this is calculated by the natural barley cubit, as it is usually.

This increased barley cubit, which I shall call great cubit, following Böckh, is mentioned on documents of the Old Babylonian period. But it séems that it is with the Persian Empire that one began to employ systematically units so conceived as to relate easily and with sufficient approximation to units used in different areas according to their different modules of the foot. This practice has obvious advantages for the fiscal administration of an empire and plays a great role in the Late Roman Empire.

In the Late Roman Empire there was used a unit of surface called iugerum castrense which has a side of 180 Roman feet. This unit is equal to 9/8 of regular Roman iugerum (240 x 120 feet) and is identical with the Mesopotamian acre calculated as a square with a side of 100 great cubits. According to a general practice of the ancient world, continued throughout the Middle Ages and the Renaissance, the area of fields was expressed also in units of seed according to a conventional seeding rate. The normal Roman iugerum castrense corresponds to 5 modii (16 sextarii each) of seed; the iugerum castrense corresponds to 5 modii castrenses of 18 sextarii each. The modius castrensis, which is the main unit for grains in the Edict of Diocletian, is equal to the most common Mesopotamian saton of 10 double (20 Alexandrine sextarii). The conventional rate of seeding in Mesopotamia was 4 such sata to an acre.

From the length of measuring rules one gathers that in Egypt the royal cubit had become lengthened to 28 artabic feet but was also often calculated as equal to the Mesopotamian great cubit of 532.401 mm. Hence the Egyptian aroura, square with a side of 100 cubits, becomes identical with the iugerum castrense and the Mesopotamian acre.

In the Roman Empire there is an itinerary unit called mivlion or milliarium equal to 3000 great cubits and hence to 5400 Roman feet (9/8 of Roman mile). The milion is divided into 8 stadia. If the stadia are divided as usual into 600 feet, one reverts to the normal unit of Mesopotamian reckoning, the barley foot.

If the great cubit is considered as an Egyptian unit of 7 hands or 28 fingers, one derives from it a foot of 4 hands which is slightly longer than the regular Egyptian foot of 300 mm. This increased Egyptian foot is the English foot. Such is the solution of a mystery about which one has poured rivers of ink.

Under the influence of the new school some scholars have thought that the English foot changed through the centuries. Other scholars have more soundly gathered that when Henry I (1100-1135) introduced as unit the yard (3 feet), to replace the cubit (1½ feet), and fixed its length by a bar of iron, this length must have been substantially the same as that of the bars constructed during the Renaissance and used to determine the English foot of 1824 A.D.

The English foot is 4/7 of Babylonian-Egyptian great cubit, and since this unit is 9/5 of Roman foot, the English foot is 36/35 of Roman foot. I have calculated the correct Roman foot as 295.9454 mm., and hence the English foot should be 304.401 mm. This length is amazingly close to the English foot as fixed by act of Parliament in 1824, according to the Bird bar of 1758 (304.79974 mm.).

The English mile is divided into 8 furlongs of 10 chains (or acres) each, as the milion was divided into 8 stadia of 10 plethra each. In the English units of length there is a peculiar break at the level of the chain, since the chain is divided into 66 feet instead of 60 and into 4 poles (perches, rods) of 161/2 feet. The English acre is a square of 660 by 660 feet; the acre, that is, the amount plowed in a day, is larger than the ancient one, because of the improvement in the construction of the plough. The chain is divided into units based on the factor 11; for this reason Edmund Gunter of Gresham College (1581-1625), suggested that the acre, called by him chain, be taken as a unit being divided into 100 units called links. Ever since, the Gunter decimal units have been used in surveying; they are the prescribed units for the surveying of public lands in the United States.

The existence of English lengths based on a unit of 11 feet parallels the existence of a perche royal in France composed of 20 pieds de roi or 22 Roman feet. Units related as 10:11 served several purposes, but in matters of landsurveying one must concentrate the attention on itinerary distances.

In the case of the English measures it is easy to see why there are units increased by a 1/10. The English foot increased by 1/10 can be considered equal to the barley foot, having a value between the trimmed and the natural barley foot. The barley foot is the unit from which the Babylonian–Egyptian great cubit was calculated and was a common unit of medieval Europe.

The relation of English linear units to the ancient ones is made clear by the following table:

Great cubits
Barley cubits
Increased Egyptian feet
Barley feet
Roman feet
English feet
English feet by 11/10

There is a small difference between the English mile and the milion. The present English mile is 1609.3 m.; according to my reckoning the theoretical English mile is 1607.2 m. The relation between milion and the English mile would be identical if the relation between Roman foot and English foot were 44:45. instead of 35:36.

It becomes possible to understand a legal prescription that has been a conundrum to English historians. It is reported that King Athelstan (924-940) prescribed that the King’s girth, that is, his mund (manus), shall extent 3 miles, 3 furlongs, 3 acres, 9 feet, 9 palms, 9 barleycorns (total 18,033 feet) from the gate of the burg of his dwelling. The same prescription was repeated by Henry I, the one who established the yard as a standard of length, except that in his case the number of acres in the figure is 9 instead of 3 (total 18,4251/2 feet). The figure quoted for Henry I is the correct one, since a difference 600:601 between 18,000 and 18,033 would not be a matter of concern. But if we consider the relation 44:45 that makes the milion equal to the Roman mile, the number of Roman feet in the King’s girth should be 18,409.1. Possibly the figure 18,000 of the English legislation was obtained only in order to use the picturesque formulation 3 miles, 3 furlongs, 3 acres, 9 feet, 9 palms, 9 barley corns.

I have pointed out that in the Middle Ages the Roman foot is used in a form called geometric foot, which I have calculated as 297.309 mm. (edge of the cube of 80 librae, the libra being increased by a komma). The geometric foot is multiplied by 45/44 and comes to 304.066 mm. which is close to a theoretical English foot of 304.401 mm. Calculating the King’s girth as 18,4251/2 geometric feet one obtained 5478.07 m., which well corresponds to 18,000 theoretical English feet or 5479.22 m. (the difference is about 4 feet).

A concrete embodiment of the geometric foot was the palmo architettonico of Rome. Petto cut on an inscription the length of the palmo so as to indicate a foot of 297.70 mm. (132 Paris lines). We know that at least as early as the seventeenth century the miles of Italian roads were calculated by the foot of Petto, and that during the Renaissance several local units were adjusted to the palmo. It seems that as soon as one began to speculate about the exact length of the Roman foot, the palmo architettonico of Rome was considered authoritative. Hence it is possible that when in 1496 there was cut the Exchequer yard of Henry VII, the earliest preserved sample of English foot, this unit was adjusted to the palmo of Rome. Since Guilhermoz suggested that the Toise du Pérou, basis of the Paris meter, was calculated by the palmo, if the standard of the modern English foot also goes back to the palmo, the two lineal standards of our time could be traced back to the palmo. It is necessary to measure English buildings dating from the Plantagenet and earlier period in order to reconstruct in detail the history of the English foot; it is also necessary to study buildings of medieval Rome in order to determine the origin of the geometric foot and its vicissitudes.

6. Since I have traced the origin of the English foot and determined thereby the units with which it is connected, the history of English units of volume and weight does not present any major difficulty.

A fundamental unit of the ancient world, as Hultsch has demonstrated, was the cube of the Egyptian foot of 300 mm., which is the basic talent brutto of 27,000 grams. Oxé calls it the talent of 1000 ounces, since it was equal to 1000 Roman ounces of 27 grams. It was divided into 60 basic minai brutto (Attic commercial minai) of 450 grams. Since the English foot is an increased Egyptian foot, according to the practice of the Hellenistic age, English units of volume and weight are the ancient basic ones adjusted according to the increase of the foot.

The main unit of the English system is the cubit foot, the firkin, of 1000 ounces. The ounce is increased over the Roman one according to the foot, being 28.350 grams by the present definition. The cubic foot is calculated as 62.5 pounds averdepois of 16 ounces, the ounce being 437.5 grains. At times, the ounce has been calculated by the round figure of 438 grains (as in the writings of Greaves) making the pound averdepois equal to 7008 grains, instead of 7000.

By the present calculation of the pound, one thousand ounces are a trifle more than a cubit foot, because when the units were regulated in 1824, one did not take into account the relation of the pound to the foot; one was concerned with making the pound averdepois, considered the only pound, equal to the round figure of 7000 grains. The legistation of 1824 was colored by a deliberate hostility to the French metric system and, hence, one did not define the units beginning from the foot. The foot was defined by the pendulum. But the legislation was based mainly on the report of a committee of the House of Commons appointed in 1758 “to enquire into the original standards of weights and measures in this kingdom.” On that occasion the metrologist Harris, King’s Assay Master of the Mint, having been consulted as an expert, declared that “a lineal standard should be the standard of all measures of capacity.”

Thomas Everard, a writer on metrology and an official of the excise, in 1696 found that according to the standards of the Exchequer a cubic foot was exactly 1000 ounces of water. The same result was obtained by a group of Oxford scholars in 1685. The only disagreement is represented by John Wybard who at the middle of the seventeenth century found 1000 ounces of pure rain or running water to be 1725.56 cubic inches, instead of 1728; but later writers, such as Jonas Moor and Everard, object that Wybard calculated the ounce by the rough relation 14:17 between the pound Troy and pound averdepois (the exact relation is 144:175). Hence a detailed study of the exact value of English weights would allow to determine exactly the small variations through history of the English foot; this length in turn should be tested on the monuments.

The cube of the correct Egyptian foot was divided into 60 basic minai of 450 grams. This unit, as the pound called averdepois, remained the English pound, even though the English foot is equal to the increased Egyptian foot.

Since the cube of the Roman foot it is 24/25 of the cube of the Egyptian foot and the theoretical English foot is 36/35 of Roman foot, the cube of the English foot should be 62.5354 pounds averdepois. Through the history of English measures one has calculated this cube as 62.5 pounds. The contemporary English definition of the units by which a cubit inch of distilled water at temperature 62 Fahrenheit, the barometer being at 30 inches, must weigh 252.326 grains, of which 7000 go to the pound, makes the cubit foot equal to 62.2885 pounds of water. The cube of the American foot by the same reckoning corresponds to 62.428 pounds of water. This proves the amazing stability of English units through more than a millennium of history. The reckonings that occur in the English system in turn had already been formulated in the Roman Empire and probably earlier.

The present pound averdepois is 453.59 grams; the increase over the basic mina brutto of 450 grams corresponds to the usual increase of about 1/120 of medieval units over the corresponding ancient ones (450 plus 1/120 is 453.75).

This increase corresponds to the increase of the present English foot over the theoretical English foot (36/35 of Roman foot). According to the present length of the foot, the pound should be 451.771 grams, but this figure must be increased inasmuch as the number of pounds in a cubit foot is less than 62.68.

The history of the submultiples of the English cubit foot has been influenced by the effort to make the number of pounds or pints in a cubic foot either 60 or 64. The pound usually has been kept stable around the figure of 62.5 pounds to the cubic foot, but the pint, which should be equal to the pound, has been calculated either as 1/60 or 1/64 of cubic foot. For this reason there are even today several types of pint. In the American system there are 59.8442 liquid pints in a cubic foot. The submultiples of the English foot have been influenced by their confusion with the very similar submultiples of the artaba (cube of the artaba foot of 307.796 mm.) The artaba is 9/8 of basic talent netto (cube of the Roman foot) and 27/25 of basic talent brutto (cube of the Egyptian foot ). If the English cubic foot (cube of the increased Egyptian foot) is calculated as 62.5 basic minai or pounds averdepois, it is 26/25 of the basic talent netto. The artaba contains 64.8 basic minai brutto.

In the Roman Empire, the basic talent netto (cube of the Roman foot, quadrantal of 80 Roman librae) was divided into 3 Roman modii of 16 sextarii or basic pints, whereas the artaba was divided into 3 modii castrenses of 18 sextarii or 20 Alexandrine sextarii (reduced pints). A Roman iugerum was sown with 5 Roman modii, whereas a iugerum castrense was sown with 5 modii castrenses; I have shown that the English foot is calculated as 1/175 of the side of the iugerum castrense.

One of the basic problems of ancient metrics was that of reconciliating a division of the cubic units by 60 with a division by 64. From the point of view of sexagesimal and decimal computation, it is easier to divide a cube by 60, but from the point of view of geometric construction it is easier to divide a cube into 8 small cubes and in turn divide these into 8 smaller cubes. This second procedure is exemplified by the division of the English cubic foot into 8 gallons of 8 pints or 8 pounds of water (wine). The gallon calculated as 1/8 cubic foot would be 221 cubic inches. The gallon of Guild Hall was 224 cubic inches, but most often the gallon has been so calculated as to make the cubic foot equal to 60 pints The American gallon of 231 cubic inches is based on a statute of Queen Anne defining the wine gallon as a cylinder with a diameter of 7 inches and a height of 6 inches (230.9070 cubic inches). A gallon such that 60 pints make a cubic foot should be 230.40 cubic inches. The wine gallon of Edward I (1272–1307) appears calculated as 2301/2 cubic inches.

Because cubic units could be divided either by 60 or by 64 there were in the ancient world units discrepant by a diesis (I shall show that the interval diesis of the musical scales derives from the units of volume). I have shown that the relation 15:16 is particularly important in Mesopotamia; the regular occurrence of this discrepancy in medieval measures has been stressed by several metrologists. A typical embodiment of this discrepancy was the marc sterling, which is 16/15 of Roman libra. In France this unit was called livre de la Rochelle, after the most important French harbor on the Atlantic; in England it was called Tower pound, because it was used as the monetary standard up to 1527 (when it was abolished by Henry VIII), and at the Tower of London there was located one of the principal mints. This unit was fixed in England at 5400 grains; 16/15 of an ancient Roman libra should be 5333.3 grains. The actual weight of the Tower pound is exactly related as 16/15 to the Roman libra increased by a komma, the libra of 328.05 grams that I have calculated as the medieval Roman libra. The ancient Roman libra of 324 grams is exactly 5000 grains.

Today there are slightly discrepant definitions of the grain, because the bill of 1824 made the pound averdepois equal to 7000 grains, whereas before it has been 7004. One usually reckons the grain as 0.0648 grams, even though it is legally defined also as 7000/7004 of this weight.

The pound Troy is based on a Roman ounce increased by 1/8; it should be 30.3750. The present ounce Troy is 31.104 grams (480 grains); considering the medieval libra of 328.05 grams, one would expect an ounce Troy of 30.7547. The last weight well corresponds to the Continental values of the once de Troyes; but in England, where it was essentially a foreign unit, the pound Troy was fixed at 5760 grains, that is, 16/15 of the Tower pound, in the great reordering of weights of 1587, under Elizabeth I. The pound Troy is theoretically 3/4 of the Alexandrine sextarius of 486 grams of water, which become the Carolingian libra (Paris livre). Since there are 60 Alexandrine sextarii in an artaba, the artaba contains 80 pounds Troy; hence, the pound Troy has the same relation to the artaba that the Roman libra has to the basic talent netto, which is 8/9 of artaba and contains 80 librae.

In England one identified the difference between Troy weights and averdepois weights with the difference between units of volume filled with wheat and filled with water (wine). This relation should be 4:5 but for the sake of convenience, in England as in the ancient world, this relation could be calculated as 5:6. The pound Troy is related to the pound averdepois as 144:175 in order to make the Troy pound 16/15 of the Tower pound; a ratio 4:5 between pound Troy and pound averdepois would be 140:175, and a ratio 5:6 would be 144:172.88. I have reported that Wybard calculated by the ratio 14:17.

The pound Troy is important today only because it defines the grain, the unit used for precious metals. According to the reform of Elizabeth I, the pound Troy is divided into 12 ounces of 20 pennyweights; the pennyweight is divided into 24 grains, whereas it used to be divided into 32. Numismatists have developed farfetched theories about the metrology of the pennyweight, the English denarius. The explanation is most simple: in the Carolingian sytem 240 denarii are to be struck from a pound of silver; in the English system 240 denarii are equal to a pound Troy.

A most important unit is the Winchester bushel, which originally was equal to 64 sextarii or 4 Roman modii (34.560 c.c.). This unit may have been legally established when Winchester was the political center of the Anglo-Saxon kingdom. The Winchester bushel is today the official unit of the United States (35.2393 c.c.), defined as 2150.42 cubic inches or 77.62701 pounds averdepois of water, being a cylinder with a diameter of 181/2 inches and a height of 8 inches. In 1696 when a bill was pending concerning an excise duty on malt, it was considered proper to define the bushel. Everard conducted a test before some members of the House of Commons and found that samples with the seal of Henry VII kept at the Exchequer indicated a bushel of 2145.6 cubic inches. Hence, at the time of Henry VII the bushel was still exactly the ancient unit of 4 modii. Everard suggested that the bushel be made of the size of 181/2 inches of diameter with a height of 8, for the sake of round figures. As a result the Winchester bushel, now the American bushel, became a unit of 2150.42 cubic inches.

The present Imperial bushel of 36,368 c.c. is the ancient artaba of 29.160 c.c. increased of 1/4 (36.450 c.c.), according to the specific gravity of wheat. In the ancient world wheat units were increased by 1/4, or by 1/5 in some calculations. In the United States the Winchester bushel when heaped must be 11/4 struck bushels. The Imperial bushel is 11/5 Winchester bushels. A statute of Henry VII defines the bushel as containing 8 gallons of 8 pounds Troy of wheat; hence a gallon (8 pounds averdepois) contains 10 pints of wheat. Local practices preserve even more than legislation the procedures and the units of the ancient world. For instance, I have found that the Clerk of the Peace of Bedford reported in 1854, on occasion of a Parliamentary enquiry on measures: “Wheat is usually sold by the load of five bushels, each bushel containing eight gallons; the average weight per bushel is 62 lb. averdepois; sometimes the seller guarantees a given weight per bushel... and makes good the deficit if any.”

The stability of English units indicates that ancient techniques allow to preserve correct standards through the centuries, and the millennia, provided there is a sufficiently effective government. In England the most important official.

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