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Near-Squares and Near-Cubes


1. Greek writers deal extensively with a type of numbers that does not occur at all in later mathematics: these are numbers of the form n(n + a), usually n (n + 1) and the form n 2(n + a), usually n 2(n +1). They are modifications of squares and cubes, and for this reason I call them near-squares and near-cubes. The Greek therm for them is paramekian numbers, that is, oblong numbes.

The only study dealing in general with them, by Oskar Becker, limits itself to clarify the terminology concerning what I call near-cubes. Nikomakos of Gerasa (Introduction to Math. II, 16, 3) declares that between the two well-defined entities, the cube and the” scalene” (a solid uven in all directions), there are parallelepiped numbers which are square in the surface. other writers distinguish two types of parallelepiped number: the plinthis and the dokis: the ttlivjic,” brick, building stone”, has a dimension a 2 b with a b, whereas the dokic,” plank”, has a dimension a 2 b with a b.

The terminology itself indicates the origin of this type of reckoning. Nobody, as far as I know, has explained the fundation of these entities in Greek mathematics.

2. It was Oppert who by considering the dimensions of Mesopotamian buildings, discovered the existence of near-squares as an empirical datum. He dealt with it in his first essay on metrology in 1872 and came back to it repeatedly up to an article of 1902, written shorthy before his death. He noted (Monatsb. Ak. Berlin, 1877, 742):

a) that usually the short side of the oblong has a length calculated by a round figure, whereas the other side adds to this round figure a sort of increment.

He sought for an explanation for this practice, but failed to find it; in his last essay on the subject (CRAI, 1902, 461), he reasorted to a mystical explanation:” Because of the superstition of Orientals, the absolute squares is not employed, but is always substituted for an oblong resembling a square.” But, even t though in this essay he followed the general tendency of the period in interpreting Mesopotamian mathematics as numerology and superstition, in earlier efforts he searched for a scientific explanation in the part dealing with Mesopotamian metrics, I shall mention instances of ziqquratsand temples calculated as near-squares.

Thureau-Dangin noticed that the equivalent of the Greek paramekian numbers can be found in Mesopotamian texts, where it is call basu (Sum. basi). He noticed it in studying tablet VAT 8521, but could not find any explanation. for the use of this entity. The tablet has been commented upon more than once by Thureau-Dangin and by Neugebauer, but they have not arrived at any specific interpretation. The tablet deals with several amounts of capital all lent at the standardrate of interest of 20% (12 sheqels to a mina) and calculates the interests in such a way that they be either the cubes of an integer. In my opinion, the explanation of this procedure must be found in the steremetric method of calculating volumes; the interest must be a unit of measure and a unit of measure is defined as a cube or as a near-cube. The capital and its interest must consist in a commodity, such as grain

The second problem states that the capital is 37|30 and the interest is 7|30, which is the cube of 30; this may mean, for instance that the capitalis 5 cubic cubits and the interest a cubic cubit, a cube with an edge of 30 fingers (that is, a kurru)

The .fourth problem states that the capital is 1|30 andthe interest is 18, which is the basu of 3 (= 3 x 3 x 2); if one reckons by fingers the capital is 1 2/3 qa and the interest is 1/12 qa or 5 sheqels (in tablet YBC 4669, the double of this amount, 1/6 qa , is descrived as a near-cube twice as high, that is , 3 x 3 x 4); if one reckons by units of six fingers the capital is 6 massiqtu of 60 qa and the interest is 18 qa; if one reckons by tenths of finger the capital is 36 sheqels and the interest is 1 4.5 sheqels.

The first and the third problem deal with cases in which the interest is a square and not a cube. In order to account for them I have to refer to a discovery of fundamental importance for numismatics, which I presented in my Harvad doctoral dissertation and which I have not been able to publish because, on account of the opposition to the old school of metrology, I have not been able to obain some empirical data that I consider essential to the completion of a proper paper. From the artheologycal material I have determined that in the ancient world one used as means of exchange gold and silver foil, which was not measured by weight, but evaluated by its surface extension. The pieces of gold and silver foild were originally round, since they represented the scale pan in the weighing of the sould (psychostasia); for this reason they were call Talavtov in Greek. In order not to waste material in cutting the pieces from a cheet of foil, they came to the most ofthen cut square. I have determined that the Greek invention of the coin was a development of these objects of metal foil; the earliest Greek coins combine the round and the square pattern, dince they are round with a square reverse impression. The artheologycal material shows that often one took a sheet of foil, usually a strip, and marked upon it with a repousse impression a series of squares.

The archeological material indicates that the currency of gold and silver foil was used as early as the Mycenean age. Great quantity of these objects has been found in in the kurgans of the Black Sea area. This type of currency continued to be used into the classical age and it is common in the Dark Ages of Europe; it was known as ttetalov or brattea.

In cuneiform documents the metal foil is called ruqqu. This term has been a great puzze to scholars; Neugebauer has somehow weeded out the welter of hypothesis by remarking:” The mention of the area, not volume, in connection with ruqqu seems to point to a flat object,” He has meved in the right direction, but not far enough. Tablet YBC 4669 states that a sheqels of silver is beaten into a ruqqu with a sideof 3 fingers. If a ruqqu with a surface of 9 square fingers has a weight of a sheqel of silver, it follows that is the made of foild. The surface is 1/100 of square cubit, which id very close to ¼ square meter. It is a piece of silver foild with a surface of about 25 cm2. Multiplying this figure by the specific gravity of silver 10.5, one obtains 262.5. By assuming a sheqel of 8.1 grams, the thickness is 0.30857 mm. If the sheqel is a double sheqel, the thickness is about 6/10 of mm. It is not by accident that a sheqel of ruqqu is calculated as 9 square fingers: this figure fits with the division of the sheqels into 180 grains, since a square finger would be worth 20 grains.

Having briefly explained what the ruqqu was and how it was used, it is possible to understand the first and third problem of tablet VAT 8521. In one case the capital is 8|20 and the interest is 1|40, that is, the square of 10, In the other case the capital is 3|0 and the interest is 36, that is the square of 6. If the thickness of the silver foild is the same as in the preceding document, the capital, in one case, is 10.000 grains or exactly a mina minus 1/24 (possibly a mina Stereometric netto) and the interest 2000 grains or almost exactly 11 sheqels, measured as a square with a side of 10 fingers. In the other case, the interest is a square with a side of 6 fingers or 4 sheqels, and the capital is 180 squares fingers or 20 sheqels.

Neugebauer makes one correct and important step in the understanding of the mathematical funtion of the basu, by noting that it has the purpose of allowing the solution of irrational roots. But he comments on the tablet YBC 4669, without noticing that it deals with near-cubes and it is actually the most important text on the subject. In this text, with whith I shall deal in detail in the part dealing with Mesopotamian metrics, all units of volume, from a quadruple talent down to aheqel, are defined either as cubes, or when this is not possible, as near-cubes. Every unit of volume has to be measured in fingers or in simple fractions of finger such as½ or 1/3; when it is not possible to define a unit of volume as a cube, it is defined as a cube raised or lowered in height. The q (double pint) is a cube with an edge of 6 fingers; but the quadruple talent of 60 qa is a near-cube of 24 x 24 x 22½ fingers, whereas a sixth of qa is a near-cube of 3 x 3 x 4.

The smallest unit, the sheqel (1/60 of qa) is a near-cube of 1 x 1 x 3½. In my opinion, this one text by itself should prove the contention of the old school that the ancients conceived of the units of volume as cubes; but it is true that the supporters of the new school have not been impressed by evidence even more univocal than this .

Neugebauer did not notice taht the near-square as an emperical datum had been already considered by Oppert . Oppert gives examples of texts giving the surface of buildings as near-squares. Neugebauer wants to deny the relation between Mesopotamian mathematics and current social practices; for this reason he overloocks all the evidence of concrete application of the basu. Furthermore, he prints, in the opening pages of MCT, tablets from Susa, first published by Father Scheil, in which there is calculated the surface of barious areas without calling to attention that the majority of the example are near-squares.

3. The near-square is of fundamental , importance in the study of ancient architecture. I have stated more than once that the main problem for ancient architects and landsurveyors was that of drawing the diagonals of squares.

Since the volue of the diagonal is expressed by the irrational quantity 2, several methods were used to approximate this value, as I have indicated; but there was a method to transform the diagonal into a near-square. For instance, a near square 20 x 21, has a diagonal of 29.

Other similar near-squares could be obtained by a simple application of the Theorem of Pythagores: the sum of the squares of the two sides of the near-square must be such that it is the square of an inter, or if this is not possible, an integer plus a simple fraction.

My remarks about the mathematics od near-squares and near-cubes, which have a geometric facies, must be linked with an important discovery made by E.M. Bruins in considering purely arithmetical procedures. He found that in Mesopotamia there was a method to approach the solution of irrational roots which consisted in adding to the denominator of a fraction one decimal point: for instance, the value tt = 3 1/7. may become 3 10/71. The principle is the same that applies to paramekian numbers.

With the exception of Oppert, scholars have never noticed the existence of buildings calculated as near-squares. Most ofhen when they have been confronted with near-squares, they have thought of some irregualarity or imprecision in the dimensions. Other times, the irregularity is overlooked and the building is reported as being square. The frecuent use of near-squares in architecture is concealed by the sircunstance that the ancient did not use it commonly in the case of square buildings, probably becauce the uneveness of the sides was considered unesthetic; but its use is most common in the case of a plan composed of two squares, because the oblong shape conceals the irregularity of the squares.

I may add that at times one not aim at obtaining a near-square that gives exactly a rational diagonal, but on eis satisfied with a very close appoximation. For instance, a near-square 50 x 49 can be considered to have a diagonal of 70. Sinse 49 2 = 2500, and 70 2 = 2900, he calculation would be perfect if the square of 49 were 2400.

I have observed that often a unit of surface that should meadure 50 square units, being one half of a unit of one hundred square units, actually measure 49. This happens because the relation between side and diagonal is calculating as 7. For instance, I have shown that Herodotos calculates the surface of the flanc of the Great Pyramid by a aroura that is 49/50 of a correct aroura. Such reduce units can be calculated as square with side and cuith an assumed diagonal 10.

It appears that in all square constructions one aimed at obtaining a diagonal expressed by a round figure. For instance, the Great Pyramid has a side of 440 cubits and a diagonal of 622.25397, with may have been computed as 622 ¼. I have reported that near-squares of the type 20 x 21 are most common, because they have a diagonal 29. The dimensions of the Second and Third Pyramid are perfect squares corresponding ti this near-squares; the Second Pyramid has a side of 410 cubits, and the Third Pyramid of 205 cubits. The diagonal of a square with a side 205 is only a trifle shorter than 290 (exactly 289.914). Up to now no explanation had been offered for the dimensions of the Second and Third Pyramid, except for the mysteriosophic ones.

4. The calculation by near-squares is documented also by Egyptian monuments of the first dynasties.

Lauer who is a careful observer of architectural details has come across the use of near-squares in his study of Zoser’s Complex. But having noticed that there were dimensions that needed an explanation, he turned to one that is highly objectionable. The wall that surrounds the entire complex has a dimension of 544.90 x 277.60 m. Laurer, having concluded from certain data that the particular royal cubit used in the Complex is a unit of 524 mm. gathers that the architect planned a square of 1040 x 530 cubits, He si surprised at the figure of 1040 cubits, whereas he would expect 1000, and suggests that 40 cubits represented the thickness fo the enclosure. But he contradicts this argument when he observes that the dimensions of Zoser’s Complex are those of the First Dynasty royal tombs at Naqadah, 54 x 27 m., increased ten times; he repeats this observation several times because it si essential to this theory about the development of architecture.

The explanation of the overall dimension of Zoser’s Complex is that it corresponds to two near-squares of 520 x 530 cubits, with a diagonal of 742.5. The reason for the choice of this diagonal is provided by Lauer when he observes that almost all dimensions are calculated by units of 5 cubits, but one finds dimensions of 33 1/3 (that is, 1/3 of 100) and of 123 cubits. This last dimension is cause of great puzzlement to

Lauer who resorts to a mystical explanation; but in reality 123.75 is 1/6 of the diagonal of the enclosure wall and the diagonal of a square of 175 cubits. In order to explain the dimension which I understands as 123 3/4 and he understands as 123 cubits, Lauer resorts to one of the worst devices of pyramidism, namely the magical meaning of a number composed of the first integers. The book of the pyramidite Ernste Binded, Die Agyptische Pyramiden als Zeugen vergangener Mysterienweisheit (Stuttgart, 1932), is ded cated to the magical number 123 which allegedly is found in the Pyramid of Gizah and explains the entire Greek theory of irrational numbers. This work is better informed about Greek thouht and more consequential than such book usually are, once one accepts the basic premise that the modern scientific attitude must be rejected; but on this issue I part way from most contemporary interpreters of Greek civilization. The pyramidite speculation about a number composed of the first integers has its start in an Arab legend mentioned by Greaves. This lengend, wchich, if I am correct, is first mentioned by the historian and geographer al-Masudi in the first half of the tenth century A. D., asserts that in the year 225 of the Hegira there was discovered in a nummy a Coptic manuscript written under Diocletian, which had been translated from a Greek text, in turn translated under King Philip from the original golden letters; this manuscript stated that, in preparation for the Deluge King Saurrid inscrebed on the walls of the three Pyramids of Gizah the principles and procedures of the arts and sciences, mathematics, geometry, medicine, and astrology, including the position of the planets at the end of the world. This legend which is one of the fundamental texts of pyramidism, states that the Arabic translation of the manuscript was made 4321 years after the construction of the pyramids. This is the final origen of Lauer’s explanation of the dimension of 123 cubits in Zoser’s Complex; the fact that one could fill a library shelf with writings on this number does not disprove that such speculations are dreams of sick men.

In may continue this series of examples of architectural near-squares by considering the data collected by Lauer, because, in spite of this occasional lapses into pyramidism, he is a superior observed of facts.

 
 

I The architect would planned a mastaba of 120 x 120 cubits

II The mastaba would have been enlarged to 120 x 136 cubits.

III A second 116 enlargement would have made it a near-square of 136 x 152 cubits

IV In the fourth plan the pyramidal form was conceived on a basis of 146 x 163 cubits. The data are not certain; Lauer derived them by adding 10 cubits to the sides of the preceding plan.

V In the fifth plan one side is 200 cubits and the other is” probably” 225.

VI In the sixth plan one dimension is 208 and the other is” probably” 231.

Lauer believes that the architect, Imhotep, changed his plan six times in the course of the execution of the project. This is possible, but it is not very convincing. Lauer has not tried to explain the figures, but they indicate that the architect was dealing with some sort of mathematical conundrum in the matter of basic plans. I suspect that Plan VI consists of a near-square 208 x 230 with diagonal 310. The preceding plan V may have been a near-square 200 x 223.5, with diagonal 300. Plan IV is probably a near-square 146.5 x 164 with diagonal 220. Lauer states that Plan III has a dimension of 136 x 152, but the only certain datum is that of 71.46 m. (compare 71.50 m. in Plan II) indicating a side of 136.5; hence I suggest that the near-square was calculated as 136.5 x 151.5 cubits with a diagonal of 204 . In my opinion, Plan IV is obtained by adding 10 and 12 cubits to the near-square of Plan III. From the dimension 136.5 cubits, one can gather that Plan II is a near-square 120 ¼ x 136.5 with diagonal 182. For Plan I, Lauer reports a side of 62.99m, which is something more than 120 cubits or 62.88m.; a square with a side of 120 1/6 cubits has a diagonal of 170. In conclusion, it seems that we meet with the series of diagonals 170, 182, 204, 220, 300, 310.

I have stated all that is necessary to illustrate the application of the mathematics of near-squares, but certainly my conclusions require a total reexamination of the architectural problem of the Step Pyramid. This is probably the single most importat problem in the evolution of architecture in the Old Kingdom.

Other examples of near-squares are provided by monuments of the age of Zoser. The mastaba of Bet Khallaf has a dimension of 84 x 43 m., probably a double near-square of 80 x 82 cubits (42.00 x 43 05 m.). with diagonal 112 3/7. The great rectangular excavation of Zaviet el-Aryan has a dimension 25 x 11.70 m.; possibly a double near- square of 24 x 22 with diagonal 34. The mastaba of Nebetka, the architect of which may have been Imhotep, measures 22. 70 x 10.55 m.; it is certainly a matter of near-squares of 20 x 21 2/3 cubits, with diagonal 29½ we can certain of the dimensions be cause they occurs also in the first Dynasty, Tomb 3038 seep 218.

For the pattern of a rectangle composed of a double near-square , the monuments of the First Dynasty provide at the first examination dozens of examples. These monuments are important in that they reveal in the architectural details a close connection with those of Msesopotamia; hence, we may ascribe a similar arigin to the Egyptian mathematics of near-squares.

The tomb of Hemaka, Chancellor of King Udimu, fourth or fifth king of the First Dynasty , has a dimension of 57.3 x 26 m. It appears composed of a double near-square of 50 50 x 110 110 = 57, 75 x 54 4/7 cubits, with diagonal 74. Walter B. Emery compares it with six other great mastabas of the same period, which have the following dimensions:

 
 

53.4 x 26.7 m.

42.0 x 16.1

48.30 =92 48.2 x 22.0

66=34, 650 34.3 x 15.7

35.7 x 15.15

32.25 x 12.75

I shall limit my analysis to some other monuments of the First Dynasty for which Emery provides a detailed report in Great Tombs of the First Dynasty. The Tomb of Aha, whom some identify with the first King Narmer, has the following dimensions:

 
 

North side 15.50 m.

South side 15.55

East side 41.50

West side 41.50

This building seems to have been calculated by the trimmed variety of the Egyptian royal cubit, 518 m. This would support the contention of Petric that the trimmed basic foot (Roman foot) was used in Egypt. The near-square would be 30 x 40 cubits (15.54 x 20.72 m.) with a diagonal of 50 cubits.

The dimensions of Tombs 3036 at Saqqarah are too uncertain to be interpreted in detail, but they indicate the use of near-squares;

 
 

North side 21.20 m.

South side 22.00

East side 41.00

West side 40.20

It is probable that Tomb 3471 has the same dimensions as the Tomb of Aha. Emery reports the following dimensions:

 
 

North side 15.15 m.

South side 15.05

East side 41.20

West side 41.30

Tomb 3111 can be clearly interpreted in tersm of Egyptian royal cubits of 525 mm.:

 
 

North side 12.10 m.

South side 12.00

East side 29.25

West side 29.10

It is a case of a double near-square of 23 x 28 cubits (12.075 x 14.70 m.), with diagonal 37. 37 2 = 1369.

Tomb 3338 has the following dimensions:

 
 

North side 14.00.

South side 13.80

East side 29.25

West side 29.10

If they are calculated in Egyptian royal cubits, one may think of near-squares 26 x 29 (13.65 x 15.225 m.) with diagonal 39 (exactly 38.95).

Tomb 3038 measures 22.7 x 10.55 m. in a first period and later was changed to the following dimensions:

 
 

North side 13.85

South side 13.75

East side 37.00

West side 36.80

One can easily recognize that the first plan consists of a double near-square 20 x 2 2/3 (10.50 x 11.37 m.), with diagonal 29½ The later plan is probably based on a near-square 26 x 35.5 (13.65 x 18.637 m. ) with diagonal 44.

5. The existence of buildings calculated by near-squares has never been noticed by Greek archeologist, even though this entity is important in Greek mathematics, but by the prevailing opinion of today, mathematics have nothing to do with the Greek view of the world. The existence of constructions that seemed irregular has been used by Dorpfeld to deny the appplicability of Newton’s method to Greek architecture.; Lepsius anticipated the havoc that would be created by Dorpfeld’s contentions, and when he was already paralized by the stroke that soon proved fatal, tried to ophold the cause of reason. But it was in vain, and other in the spirit of the new school, argued that the Greeks were inaccurate and caraless in measurements, an interpretation that agrees with the tendency to stress the irrational or chthonic-Dionysian aspect of Greek culture . But the so-called casualness of Greek architectural plants at close analysis proves to be the result of mathematical subtlety.

The Geek meeting halls provide such an evident example of near-swuares taht archeologists should have niticed it as a matter of course, except that they have been blinded to the facts by the” classical spirit”.

For the most typical one, the Ekklesiasterion of Prinene, the following accurate dimensions are reported:

 
 

South side 20.25 m.

West side 21.06

East side 21.18

One can be certain taht the plan is a near-square of the type 20 x 21, with diagonal 29. The dimensions are 60 x 63 natural barley feet (theoretically 20, 250 x 21, 262 mm.) The sixth century Bouleuterion of Athens, measuring 23.80 x 23.30 m., is to be explained as a near-square of 50 x 49 natural wheat cubits (theoretically 23, 912 x 23, 425 mm.) with diagonal 70. The even olden Boulenterion of Gortyna, with reported dimensions 28.70 x 34.00 m., may have been calculated as 90 x 107 7/16 (theoretically 28, 687 x 246 mm.), with diagonal 140. The same calculation can be applied to the several reconstructions of the Telesterionof Eleusis; the most certain dimensions are those of the one planned by the architect Iktinos, with dimensions 49.33 x 50.80m., which may correspond to a near-square of 160 x 165 1/3 artabic feet (theoretically 49, 247 x 50, 889 mm.) with diagonal 230. The same type of calculation applies to a similar construction of the late Roman Empire: the Senate Hall in rome, reconstructed by Diocletian, measures 17.60 x 25.20 m., that is, 60 x 85 Roman feet (calculating by the correct foot 17, 757 x 25, 155 mm) cuith diagonal 104. Earlier I have mentioned these dimensions to prove that it is not true that the Roman foot became shorter in the age of Vespasian. Thi building has been accurately measured during the Renaissance by Antonio Picconi da Sangallo (Antonio Sangallo il Giovani) and this data are important today to determine the value of the palmo mercantile of Rome.

Examples from the Hellenistic age are the Bouleuterion of Notion and the Bouleuterion of Herakleia on Latmos. The first, measuring 20.25 x 21.10 m., appears to be planned as 60 x 63 natural barley feet (theoretically 20, 250 x 21, 266 mm.), with diagonal 87. The second, measuring 22 x 15. 65 m., was planned as 70 x 50 trimmed barley feet (theoretically 22, 010 x 15, 722) with diagonal 86¼.

The calculation by near-squares applies to most Greek temples. A few consist of a single near-square, most of a double near-squares, and some, particularly in the archaic period, of a treble near-square.

The near-square of the type 20 x 21, with diagonal 29, occurs in the Treasury of Megara at Olympia: it measures 13.40 x 6.40 m., that is, a double near-square of 20 x 21 Olimpic feet (increased natural wheat feet) or theoreticaly 6, 412 x 6, 733. The Temple of Larisa in Aiolis is an earliy example (around 600 B.C.) in the form of a double near-square of 10 x 10½ artabic feet or theoretically 3, 077 x 3, 232 mm.; the reported dimensions are 6.25 x 3.25 m. An even older example of the same kind is the so-called Oikos of the Naxians at Vroulia (Rhodes) for which there are reported the dimensions 4.66, 4.70, and 8.38; it consists of a double near-square of 10 x 10½ trimmed lesser cubits or theoretically 4, 162 x 4, 374 mm. The following are examples of temples planned as a double near-square of 40 x 42. The Temple of Aphaia in Aigina (ca. 490 B.C.) measures 28.50 x 13.80 and is calculated in natural barley feet (hteoretically 28, 350 x 13, 500 mm.), whereas the contemporary Temple of Athena Pronaia at Delphoi measures 27.449 x 13.25 m. and is calculated in trimmed barley feet (theoretically 27, 967 x 13, 318 mm.).

At Delphoi a dimension 8.547 x 6.134 m. is reported for the Theasury of Siphnos, but a dimension of ca 8.55 x 6.25 m. is reported for the Treasury of Massalia. The two buildings have the same plan: a near-square of 27.5 x 20 artabic feet (8, 464 x 6, 155 mm.) with diagonal 34.

The later Temple of Athena at Cape Sunion, built after 470 B.C., measures 19.34 x 14.78 m. and has been planned as a single near-square of 50 x 65 trimmed basic feet (theoretically 14, 797 x 19, 364 mm.). One could repeat the examples by the score, but, taking one at random, I can mention the Temple of Athena Polias at Priene (end of fourth century B.C.), which has a stylobates of 37.20 x 19.55 and a cella of 29.48 x 11.84 m. The cella has been planned as a doulbe near-square of 50 x 40 trimmed basic feet (theoretically 14, 797 x 11, 837mm.); the stylobates seems to have been planned as a double near-square of 66 x 63 (19, 532 x 18, 645 mm.). Another example of the same period is the Temple of Athena Polias at Pergamon measuring 21.77 x 12.27 on the stylobates; the plan is a double near-square of40 x 35 artabic feet (theoretically 12, 312 x 10, 773 mm.).

The cella of the Temple of Apollo, costructed by the Alkmaionids at Delphoi, measures 15 x 6 m., and is calculated as a double near-square of 21 x 25 trimmed basic feet (theoretically 6, 215 x 7, 398 mm.) with diagonal 32 2/3; the overall dimensions are 58.0 x 21.65 m., that is, a double near-square of 50 x 74 or 29, 594 x 21, 899 (these figures indicate that these dimensions were reckoned on the euthynteria as it has been suggested), with diagonal 89 1/3 (exactly 89.308). The sixth century Temple of Aphrodite at Bassai, with dimensions 15.43 x 6.47, has the same proportions as the cella of Alkmaionid Temple, excep that the units are artabic feet (theorretically 6, 464 x 7, 695 mm.). The neighboring Temple of Artemis at Bassai measures 9.25 x 5. 74 m.; it too is calculated in artabic feet by a double near-square of 15 x 18 (theoreticaly 4, 617 x 5, 543 mm.), with diagonal 23½. The same pattern with the same units occurs in the eighth century temple of Dreros; it measures 9. 50 x 5.50 m. The same pattern is found again at the Temple of Themis at Rhamnos (ca. 500 B.C.), which measures 10.70 x 60.40; this would be an example of hybrid fooot, which is rather rare in Greece (theoretically 5, 250 x 6, 300 mm.).

Perhaps one is bound to deal with the dimensions of the Parthenon, about which one has written volumes. The plan seems to have been formulated by the increased artabic foot of 308.276 mm. The overall dimensions are calculated as a double near-square of 100 x 113 (theoretically 20, 828 x 34, 835 mm.) with diagonal150 1/12; the dimensions are reported as 30.86 x 69 51 m. The following data are reported for the cella:

 
 

East side 21.72 m.

West side 22.34 m.

North side 50.02

south side 59.83

It is a case of a double near-square of 70 x 96 or 97 feet (theoretically 21, 579 x 29, or 29, 903 mm.), with a diagonal of about 118.

The plan of older Olympeion in Athens (ca. 510 B.C.) deserves mention because of the occurrence of the factor 7 in the dimensions. It measures 107.70 x 42.90 mm, that is, a double near-square of 140 x 175 artabic feet (theoretically 43, 091 x 53, 864 mm.). 140 x 350

The so-called Megaron B of Thermon which belongs to the tenth century B.C., is an example of triple near-square.The dimensions have been estimated as 7.30 x 21.40m. possibly it could be a case of near-square of 20 x 21 hybrid feet (7, 350 x 7, 000 mm.). The seventh century temple at Neandria, measuring 21 x ca. 930 m., could also have been calculated in hybrid feet, by near-square of 20 x 26.5 (7, 000 x 9, 275 mm.) with diagonal 33 1/3.

The seventh century Heraion of Olympia mesures 40.62 x 10.72 m.; it may be a case of a quadruple near-square calculated as 20 x 21 natural barley cubits (10, 125 x 10, 613 mm.). At later times triple and quadruple near- squares are employed in the planning of the cella of temples.

It is necessary to call to attention that even thogh in a particular city a module of foot may be preferred, on emay expect to find examples of alll modules. At Olimpia one finds the employment of the Olympic foot, fooot, natural wheat foot calculated as 25/24 of artabic foot, but the sixth century Heraion is calculated by trimmed wheat foot; it measures 50.01 x 18.75 m. and it two near-squares of 60 x 80 feet (theoretically 18, 865 x 25, 155 mm.)

The general attitude of Greek archeologists on the matter of dimensions of buildings can be summarized by the remarks of Fernand Courby in his study of the Temples of Apollo at Delos. After propounding the problem:” Which measures has been used by the builders of the Temple? It is an embarassing question...”, he answers that” Greek workers usually were not concerned with impeccable with exactitude.” In the specific case that he is considering, the suggests that they used a foot varying between 314 and 316mm.; he adds that the data of Greek methrology are” too rare and too uncertain.” Now, the facts are just the contrary of what Gourby states: as far as linear metrology is concerned, one could hardly wish for a greater abundance of data; it many cases, one can draw conclusions with all reasonable certainty; the standards in general seem to be accurate to the tenth of millimiter, and the greatest difficulty usually is not created by the present state of the ruins, but by the inaccuracy fo most archeologists in reporting metric data, As soon as an archeologist shows concern with metric, data, the methrologist becomes simpliy swamped by the mass of material available for interpretation This is proved in the very case of Delos by the study of Rene Vallois on L’Architecture hellenique et hellenistique a Delos (Paris , 1944); in this study one could find one hundred examples of constructions for which one could try the calculation by near-squares I have presented. For instance, if one were to open Vol.I at pages 118 and 119, as I have done at random, one could see that the temples listed there can be analyzed in terms of near-squares For instance, the Temple of the Dioskouroi consists of a double near-square of 20 x 21 trimmed barley feet; the reported dimensions are 14 x 6. 70 m., whereas the theoretical ones are 13, 983 x 6, 659 mm., which proves that Greek workers were gar from being careless. I have mentioned one of the least important temples. One of the most important temples is the Temple of the Athenians, with dimensions 18.84 x 11.304 m.; it is an example of double near-square of 60 x 36, with diagonal 70, of which I have fiven deveral examples. The Temples is planned in trimmed wheat feet and has a theoretical size of 18, 866 x 11, 320 mm. Courby claims that one employed a foot shifting between 314 and 316 mm.

The calculation by near-squares applies also to Greek city planning. The calculation by near-square occursin city bloks, in the agorai, in the stoai and halls of any sort, and also in public fountains.

I shall discuss only the city plan of the Hill area of the city of Olynthos, because David

Robinson has built upon it some general deductions about Greek metrology. He recognizes the validity of Newton’s method, but finds that the width fo the streets connot be expressed in round figures. He assumes that it must be expressed at least in whole feet. He declares that only two types of foot may have been used: an Attic-Euboic foot of 295 or 296mm. and an Attic-Aiginetic foot of 328. Instances of the used of these feet are provided by several constructions;for thsi reason Robinsona asks the question of when and under which political circumstances one shifted from the first to the second unit. In my opinion, one used any module of foot as it was convenient. Since Robinson finds that neither foot applies perfectly to the general plan of the Hill area, he concludes that the surveyors were not accurate. In his opinion one cannot expect that the rods used by Olynthian surveyors were correct to the fraction of millimeter (Excavation at Olynthos, XIII, 47).

Vallois has observed that the dimensions of the city plan suggest a unit of about 500 mm.; he is perfectly correct, because teh unit is the natural barley cubit of 506.25 mm.

The Hill section was planned as divided by 13” streets” running East-West, and by four” avenues” North-South. All the streets and avenue A have a width varying between 4.90 and 5.04 m. Robinson observes wisely that the houses tend to encroach on the roads and that, hence, one must take the maximun width as the one intended to be 10 or 5, 062 mm. cubits; there are avenuesof 10, 12 and 14 cubits. The theoretical width of Avenue B is 14 cubits or 7, 087 mm.; Robinson reports a width between 6.90 and 7.02 m. The short side of the blocks is 70 cubits or theoretically 35, 437 mm.; Blocks are increased by the encroachment upon the roads. The long side of the blocks is 170 cubits, or theoretically 86, 066 mm.; reported dimensions vary between 86.10 and 86, 65 m.

Robinson measuredas 166.75m. a stretch corresponding to four blocks on the narrow side and to five strreets; he concluded that there cuos used an Attic-Euboic foot reduced to 295, 1mm. In my opinion the stretch corresponds to 330 natural barley cubits and has a theoretical length of 167, 062.5 mm. Hence the surveyors used units accurate to the fraction of millimeter.

The city blocks measure 70 by 170, 172, or 174, according to the width of the avenues. Robinson had come to the conclusion that only the narrowest avenues had been taken into account in the plan and thoght that von Gerkan in Griechische Stadtenalagen had been wrong in stating that wider main streets were taken into account by Greek city planners. Each block si divided into 10 lots; each lot has a side of 35 and a side varying between 34 and almost 35, depending on the width of the avenue. One finds here the usual reckoning of plots of land by the relation 5: 7 between side and diagonal. The lots are calculated as having a diagonal of 50 cubits. Hence they are½ of a square with a side of 50 cubits, that is, a fourth of an acre.As one would taday, one calculated each lot by a standard unit surface, which here is 1/8 of an acre; the acre is of the type used in Mesopotamia.

A principle similar to that followed by Robinson has been used by A. W. Parsons in interpreting the dimensions of the walls of Corinth, with the result that he too obtains approximate results. In a footnote he comments: (Corinth, III, Part 2, 291 n.3):” It is perhaps asking too much to demand precision in matters of this sort of a people whose outlook was that of a geometer and whose chief concern was with proportion.” It is refreshing to find such a scattered footnote. Parsons sees the necessary connection between.


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