Sense of Scale

Euclidean geometry enabled the Greeks to develop a sense of scale for the size of the earth and heavens. For example, Eratosthenes estimated the size of the earth by noting that at noon on midesummer day, the sun shone directly to the bottom of a well in a town called Syene.

However, about 500 miles to the north, the midsummer sun made an angle of about 7.5 degrees with the vertical. On this basis, Eratosthenes concluded that the earth's circumference C satisfies

or that C is about 24,000 miles.

This ancient estimate leads to the approximation "R = 4000 miles" and provides a basis for asking one of the most fundamental "sense of scale" questions concerning our environment: How many acres of arable land are there per person on earth?

Archimedes' formula for the surface area of a sphere

leads to S = 4 * pi * 4000² or about 200 million square miles. Since the earth's oceans account for 3/4 of this area and about11% of the earth's land is arable, we arrive at a "back of the envelope" estimate of about 6 million square miles, or 4 billion acres, of arable land (there are 640 acres/square mile).

Dividing this equitably among the world's 6 billion human inhabitants would provide each person with about 2/3 of an acre.

While it is possible to grow a lot of potatoes on 2/3 acre, an American diet and life style seems to call for about 2 acres of arable land per person. Additionally, American agriculture requires about 10 Calories of stored energy (primarily fossil fuels) to produce 1 Calorie of food.

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Another interesting "sense of scale" calculation is to determine the impact of universal peroleum consumption at the American level of about 2.5 gallons/day or 22 barrels/year. Supplying 6 billion people with 2.5 gallons/day would require 15 billion gallons/day or about 130 billion barrels/year (there are 42 gallons to a barrel). At this rate, the world's "proven oil reserves" of about 900 billion barrels would be exhausted in 7 years.

Economists are likely to remind us that market forces will prevent any such scenario from actually taking place. Rather than answering the question "will the world run out of oil," such calculations provide a quantitative starting point for thinking about such questions.