Alexandria to Syene: What the Geometry Can Prove

01 · Reported modelA shadow angle becomes an angle at the Earth's centre

In Heath's account, at noon on the summer solstice the Sun casts no shadow at Syene, while a gnomon, the shadow-casting part of a sundial, casts a shadow at Alexandria. The argument takes the cities to lie on the same meridian, a north–south circle around the Earth, and treats the incoming solar rays as parallel. The Alexandrian angle is one-fiftieth of the whole meridian circle; the adopted distance between the cities is 5,000 stadia.¹

These conditions connect the observation to the geometry. On a spherical Earth, the two local verticals point towards the centre. With parallel sunlight and the stated alignment, the angular difference corresponds to the angle between those verticals. The surface interval therefore represents the same fraction of the meridian circumference.

The distance used in that proportion must represent the corresponding surface arc. A winding journey between two places and an arc along a meridian answer different questions; the number of stadia alone does not resolve the distinction. Heath's account supplies an adopted baseline, not a detailed survey record demonstrating how every part of it was obtained.¹ Once the model is declared, however, its arithmetic can be reproduced without choosing a modern length for the stadion.

02 · Worked exampleOne-fiftieth of a circle makes the multiplier fifty

A full circle contains 360 degrees. One-fiftieth is therefore 360 ÷ 50 = 7.2°. If a baseline d corresponds to an angle α, the same proportion gives C = d × 360 ÷ α, where C is the circumference. Degrees cancel against degrees; the result retains the baseline's length unit.

Insert the reported model's values: 5,000 × 360 ÷ 7.2 = 250,000 stadia. This reproduces the number Heath associates with Cleomedes' account. It checks the consequence of the stated inputs; it does not independently check their historical measurement or the spherical model's fit to the two places.¹

The distinction becomes visible when a second transmitted total enters the calculation. A result can be arithmetically consistent while its path through the historical record remains unsettled.

03 · ComparisonA total of 252,000 does not identify the changed input

Heath distinguishes Cleomedes' 250,000 from the 252,000 attributed to Eratosthenes by Strabo. He discusses a possible adjustment to obtain convenient subdivisions, including 700 stadia per degree. That is Heath's proposed explanation of a discrepancy, rather than a reason established by the multiplication itself.¹

We can test the arithmetic without deciding that historical question. The larger total is 0.8% above the smaller. Holding the angle at 7.2° requires a baseline of 5,040 stadia. Holding the baseline at 5,000 instead requires an angle of 50/7° ≈ 7.142857°. Both produce 252,000. These are alternative calculations constructed for comparison, not additional ancient observations.

There is one equation but two adjustable inputs. A final circumference fixes their ratio, not each input separately. The two derived alternatives therefore demonstrate a limit: the existence of a numerical reconciliation cannot identify the historical adjustment. That requires evidence beyond the total being reconciled.

04 · ExplorationVary the assumptions and keep the units visible

Try a different baseline or angle. Increasing the baseline increases the circumference in direct proportion. Increasing the angle while holding the baseline fixed makes the circumference smaller. Values entered here are model assumptions; the controls do not turn them into ancient measurements.

A small input change can be tested directly. From the default model, a 1% increase in the baseline gives 252,500 stadia. A 1% increase in the angle gives approximately 247,524.75 stadia, a decrease of about 0.99%. The two inputs affect the answer differently, even before a modern unit conversion is attempted.

05 · CalibrationA kilometre result adds another historical assumption

To convert a circumference in stadia into kilometres, multiply by an adopted metres-per-stadion value and divide by 1,000. Geometry has not supplied that value. It is an additional claim about the length standard attached to the count.

Heath offers a conditional reconstruction: if the relationship he cites from Pliny is accepted, and an Egyptian schoinos is taken as 12,000 royal cubits of 0.525 metres, then 40 stadia to that schoinos gives 300 cubits, or 157.5 metres, per stadion. Each link matters. The 157.5-metre value is a conclusion within that reconstruction, rather than a ruler recovered from Eratosthenes' calculation.¹

Using that conditional value, our arithmetic gives 250,000 × 157.5 ÷ 1,000 = 39,375 km, whereas 252,000 × 157.5 ÷ 1,000 = 39,690 km. The difference is 315 km. The multiplication is reproducible; choosing between the transmitted totals and establishing the unit remain historical questions. Neither kilometre result earns a claim of historical accuracy simply by looking close to a modern answer.

06 · PayoffKeep the calculation separate from the identification

The danger of working backwards from an expected Earth circumference is an old one. The Mensura article in William Smith's 1849 dictionary criticised attempts to infer different stadia from ancient circumference figures while first assuming the ancients knew the correct circumference. That criticism concerns the direction of the argument. It can be considered separately from the article's broader, contestable claim that early Greek stadia shared one standard.²

The Alexandria–Syene model gives us a practical sequence: establish the reported angular fraction and baseline, reproduce their proportional result, then examine the independent grounds for a physical unit conversion. If a different total is proposed, ask which inputs and assumptions would have to change and what evidence distinguishes those possibilities.

The achievement of the method lies in relating a local angular difference to a whole circumference. Its historical evaluation depends on more than a successful multiplication. For another example of the gap between a dimension and an inferred unit, continue with Petrie's Giza survey. For regional standards and reverse calculation, read There Was No Single Greek Foot; the length converter compares selected standards once their historical basis is declared.