By Björn Björklund

In 1888 Francis Penrose, an English architect who became the surveyor of St Paul’s Cathedral, measured the length of the upper step of the front of the temple called the Parthenon on the Acropolis in Athens. An ancient Greek tradition held that the breadth of this temple in Greek feet was exactly one hundred feet, a hecatompedon. Greek ekatom means “one hundred” and pedon means “feet”.

Penrose wrote “The result of my own measurements of the breadth of the Parthenon, on the top of the upper step, is 101.341 feet on the east front … ” (Penrose, Francis. An investigation of the principles of Athenian architecture: or the results of a recent survey conducted chiefly with reference to the optical refinements exhibited in the construction of the ancient buildings at Athens — London, 1888, page 11). He measured in English feet, not Greek feet.

“The principal dimensions are as follows :— Measured. Front, on the upper step … 101.341 …” (page 12).

“The breadth, 101.34 is exactly a second of latitude at the equator. This is remarkable, and would have been a happy coincidence for Gosselm in his attempt to establish an identity between ancient measures and the size of the globe” (page 12).

“The dimension above given is perhaps a sufficient approximation for a general statement, but there can be little doubt but that the front of the temple, which was always accessible for reference as a standard, was the true Hecatompedon in point of exact measurement” (page 13).

A length of 101.341 English feet is equal to 30.8887 metres. Taking this to be the length of a second of latitude we multiply by 60 to find the length of the corresponding minute: 1853.3242 metres.

“Francis Cranmer Penrose FRS (1817 – 1903) was an English architect, archaeologist, astronomer and rower. He served as Surveyor of the Fabric of St Paul’s Cathedral, President of the Royal Institute of British Architects and Director of the British School at Athens” (Wikipedia on Francis Penrose).

He was a precise man and a perceptive one in thinking that the Greek hecatompedon was equal in length to a second of latitude. But he did miss the extraordinary significance of the length of the upper step of the front of the Acropolis: it was not exactly equal to a second of latitude at the equator, as Penrose thought: the length of a second of latitude at the equator is 30.7150 metres. The hecatompedon was equal to a second of latitude on a sphere with the authalic radius of Earth. This is a sphere with the same surface area as the Earth.

Penrose must have known the authalic radius of Earth because in 1866 the English surveyor Colonel Alexander Ross Clarke published his first derivation of the Earth ellipsoid, from which the authalic radius and the authalic circumference could be calculated. Clarke published further derivations of the Earth ellipsoid in 1878 and 1880 and in 1887 was awarded the Gold Medal of the Royal Society for his work in determining the figure of the Earth.

The International Union of Geodesy and Geophysics states that “For the Earth, the authalic radius is 6,371.0072 km” (Wikipedia on Earth radius). This is the radius of a sphere with the same surface area as Earth.

Let us do the maths: The authalic diameter is 12,742,014.4 metres. Pi times the diameter gives the circumference: 40,030,218.8310 metres. Dividing the circumference by 360 gives the length of a degree of latitude on a sphere with the same surface area as Earth: 111,195.0523 metres. Dividing by 60 gives the length of a minute of latitude: 1,853.2509 metres. Dividing by 60 gives the length of a second of latitude: 30.8875 metres. This is 0.0012 metre or 1.2 millimetres shorter than the length measured on the hecatompedon by Penrose. He measured to a thousandth of an English foot, which is equivalent to 0.3 millimetre.

Penrose wrote: “With regard to the difference of .022 between the breadths of the two fronts [east and west], even wooden measuring rods are liable to a variation at least as great as this, from changes in the moisture of the atmosphere. I found during my measurements at Athens that some deal rods which I made use of in some of the measurements, and was accustomed to verify by occasionally comparing them with a known length on the building, varied as much as 1/4300 of their length = .023 foot in 100 feet; so that if wooden measures were used by the Greeks, and we suppose that an interval of several hours elapsed between the setting out of the east end, and that of the west, during which time there had been a sudden hygrometric change, there would have arisen as much difference as exists, notwithstanding the utmost desire to render the Hecatompedon exact” (page 12).

One assumes that Penrose measured the length of the upper step of the front (east end) of the Parthenon on a cool day in shadow, not in sunlight (and therefore in the afternoon) to minimise an error that might be caused by expansion of the marble in hot sunlight.

Now we compare the length of the minute of latitude derived from Penrose’s measurement of the upper step on the front of the Acropolis with the length of the minute derived from the modern value for the authalic radius of the Earth: the former is 1853.3242 metres, the latter is 1,853.2509 metres. The difference is 0.0733 metre or 73.3 millimetres (less than a hand’s breadth in a nautical mile). The length of a minute of latitude at the Equator is 1842.9 metres.

What Penrose should have said was that the breadth of the hecatompedon of the Acropolis was “exactly equal to a second of latitude on a sphere having the same surface area as the Earth”.

The Greeks could not have found the length of a second of latitude on a sphere of the same surface area as the Earth because it was impossible for them to accurately measure the equatorial circumference of the Earth and thus find the equatorial radius. Both the polar radius and the equatorial radius are required for the calculation of the authalic radius of the Earth (Wikipedia on Earth radius).

It is interesting to note that “Both the United States and the United Kingdom used an average arcminute, specifically, a minute of arc of a great circle of a sphere having the same surface area as the Clarke 1866 ellipsoid. The authalic (equal area) radius of the Clarke 1866 ellipsoid is 6,370,997.2 metres (20,902,222 feet). The resulting arcminute is 1853.2480 metres (6080.210 feet)” (Wikipedia on Nautical mile). The United States used this arcminute or nautical mile until 1954 and the United Kingdom until 1970.

The Length of The Stade

The length given for the Greek stade varies greatly depending on the literary source, but several authors have written that it was about 185 metres.

“Scholar of Greek antiquity Carl Friedrich Lehmann-Haupt claims the existence of at least six different stades. To the contrary, astronomer and historian Dennis Rawlins makes the following claim.

“That 1 stade = 185 meters (almost exactly 1/10 nautical mile) is well established.

“The 185 meter stade, as claimed by Rawlins earlier, is the most commonly accepted value for the length of the stade used by Eratosthenes in his measurements of the Earth. This is so because a great number of authors from the first century CE onward make reference to the fact that 1 Roman mile is equal to 8 stades. History tells us that the Roman mile is equal to 5000 Roman feet, each of which is just short of the familiar English foot.

“The exact difference between the Roman foot and the English foot is uncertain, but if 1 Roman foot is taken to be approximately 11.65 English inches, then one Roman mile is approximately equal to 1479 meters.  Taking 1/8 of this Roman mile gives the length of 1 stade as approximately 184.8 meters. Again, this length corresponds to one of Lehmann-Haupt’s six stades. He refers to this most frequently accepted stade as the ‘Italian’ stade” (Eratosthenes and the Mystery of the Stades – How Long Is a Stade? https://www.maa.org/press/periodicals/convergence/eratosthenes-and-the-mystery-of-the-stades-how-long-is-a-stade).

In fact if 1 Roman foot is 11.65 English inches then one Roman mile is 58,250.0000 English inches; we multiply by 25.4 and get 1,479,550.0000 millimetres or 1,479.5500 metres. We divide by 8 and find that the length of the stade is 184.9438 metres.

But if in hypothesis we take the length of the Roman foot as 11.674 English inches we find that one Roman mile is 58,370.000 English inches = 1,482,598.0000 millimetres = 1,482.5980 metres. We divide by 8 and find that the length of the stade is 185.32475 metres. If there are 10 stades to the nautical mile we find that the length of the nautical mile derived from this stade is 1853.2475 metres. The length of a minute of latitude derived from the modern value for the authalic radius of the Earth is 1,853.2509 metres. The difference is 3.4 millimetres. We therefore propose that the length of the Roman foot was 11.674 English inches = 296.52 millimetres.

“We have no ancient measures by which to determine the length of the Greek foot, but we have the general testimony of ancient writers that it was to the Roman in the ratio of 25 : 24” (A Dictionary of Greek and Roman Antiquities, edited by William Smith and Charles Anthon, under the heading Pes, page 763).

But we have always had such an ancient measure: we can find the length of the Greek foot from Penrose’s measurement of the length of the hecatompedon: we divide 30.8887 metres by 100 and multiply by 1000 and find 308.89 millimetres.

We divide this length of the Greek foot, 308.89 millimetres by our proposed length of the Italian foot, 296.52 millimetres and find 1.0417. We divide 25 by 24 and find 1.0417.

©Björn Björklund 2024