Stated in a few words, the experiment of Eratosthenes was this.
His geographical studies had taught him that the town of Syene lay directly south of Alexandria, or, as we should say, on the same meridian of latitude. He had learned, further, that Syene lay directly under the tropic, since it was reported that at noon on the day of the summer solstice the gnomon there cast no shadow, while a deep well was illumined to the bottom by the sun.
A third item of knowledge, supplied by the surveyors of Ptolemy, made the distance between Syene and Alexandria five thousand stadia. These, then, were the preliminary data required by Eratosthenes. Their significance consists in the fact that here is a measured bit of the earth’s arc five thousand stadia in length. If we could find out what angle that bit of arc subtends, a mere matter of multiplication would give us the size of the earth. But how determine this all-important number? The answer came through reflection on the relations of concentric circles.
If you draw any number of circles, of whatever size, about a given centre, a pair of radii drawn from that centre will cut arcs of the same relative size from all the circles. One circle may be so small that the actual arc subtended by the radii in a given case may be but an inch in length, while another circle is so large that its corresponding are is measured in millions of miles; but in each case the same number of so-called degrees will represent the relation of each arc to its circumference. Now, Eratosthenes knew, as just stated, that the sun, when on the meridian on the day of the summer solstice, was directly over the town of Syene. This meant that at that moment a radius of the earth projected from Syene would point directly towards the sun.
Meanwhile, of course, the zenith would represent the projection of the radius of the earth passing through Alexandria. All that was required, then, was to measure, at Alexandria, the angular distance of the sun from the zenith at noon on the day of the solstice to secure an approximate measurement of the arc of the sun’s circumference, corresponding to the arc of the earth’s surface represented by the measured distance between Alexandria and Syene.
The reader will observe that the measurement could not be absolutely accurate, because it is made from the surface of the
earth, and not from the earth’s centre, but the size of the earth is so insignificant in comparison with the distance of the sun that this slight discrepancy could be disregarded.
The way in which Eratosthenes measured this angle was very simple. He merely measured the angle of the shadow which his perpendicular gnomon at Alexandria cast at mid-day on the day of the solstice, when, as already noted, the sun was directly perpendicular at Syene. Now a glance at the diagram will make it clear that the measurement of this angle of the shadow is merely a convenient means of determining the precisely equal opposite angle subtending an arc of an imaginary circle passing through the sun; the are which, as already explained, corresponds with the arc of the earth’s surface represented by the distance between Alexandria and Syene. He found this angle to represent 7
degrees 12′, or one-fiftieth of the circle. Five thousand stadia, then, represent one-fiftieth of the earth’s circumference; the entire circumference being, therefore, 250,000 stadia.
Unfortunately, we do not know which one of the various measurements used in antiquity is represented by the stadia of Eratosthenes. According to the researches of Lepsius, however, the stadium in question represented 180 meters, and this would make the earth, according to the measurement of Eratosthenes, about twenty-eight thousand miles in circumference, an answer sufficiently exact to justify the wonder which the experiment excited in antiquity, and the admiration with which it has ever since been regarded.
{illustration caption = DIAGRAM TO ILLUSTRATE ERATOSTHENES’
MEASUREMENT OF THE GLOBE
FIG. 1. AF is a gnomon at Alexandria; SB a gnomon at Svene; IS
and JK represent the sun’s rays. The angle actually measured by Eratosthenes is KFA, as determined by the shadow cast by the gnomon AF. This angle is equal to the opposite angle JFL, which measures the sun’s distance from the zenith; and which is also equal to the angle AES–to determine the Size of which is the real object of the entire measurement.