Which of the following did Eratosthenes NOT need or use in his attempt to calculate the Earth’s circumference in the 3rd century BCE?

Question

BlackTom AI · Accepted Answer

The angular difference of the Sun’s rays between two cities equals the fraction of the Earth’s circumference between them, so Eratosthenes compared a shadow angle in Alexandria with the no-shadow condition in Syene to estimate the whole circumference.

1) Relevant knowledge points and concepts
- Parallel solar rays at Earth’s scale imply the difference in solar zenith angles between two locations equals the central angle subtended at Earth’s center by the arc between them.
- Measuring a shadow gives that angular difference; estimating distance along the surface between the two locations gives the arc length.
- Circumference = (360° / measured central angle) × arc length.

2) Complete step-by-step reasoning
- He observed that at local noon on the summer solstice the Sun was directly overhead at Syene (no shadow, Sun reflected at well bottom) while in Alexandria a vertical stick cast a measurable shadow.
- He measured the stick’s shadow to give an angle of about 7.2° between the Sun’s rays and the vertical at Alexandria.
- He estimated the surface distance between Alexandria and Syene from caravan travel to be about 5,000 stadia.
- Using the angular fraction, he scaled up to the full circumference.

3) Intermediate calculation steps
- Fraction of full circle = 7.2° / 360° = 0.02.
- Reciprocal = 1 / 0.02 = 50.
- Circumference = distance × 50 = 5,000 stadia × 50 = 250,000 stadia (≈ modern ~39,000–40,000 km depending on stadion conversion).

4) Why the correct answer is correct
- Eratosthenes did not need any estimate of Earth’s diameter or reliance on Sun–Moon relative positions. His method used only solar rays, shadow angle, distance along the surface, and the assumption of nearly north–south alignment. Diameter estimates from lunar observations are unrelated to his technique.

5) Why the other options are incorrect
- Knowing Alexandria was roughly due north of Syene: needed to assume the arc between them followed a meridian.
- Estimating the distance by caravan travel: required to get arc length.
- Observing Syene’s no-shadow condition: critical calibration point.
- Measuring the shadow angle in Alexandria at that same time: essential measurement to compute the angular fraction.

6) Conclusion
The statement he did NOT need is the estimate of the Earth’s diameter based on Sun–Moon positions. This matches the provided correct answer.

Which of the following did Eratosthenes NOT need or use in his attempt to calculate the Earth’s circumference in the 3rd century BCE?单项选择题

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