How Eratosthenes measured the Earth's circumference

Question

When Eratosthenes measured the Earth's circumference, he provided $250,000$ stadia as its measure equal to about $25,000 $miles. Since the radian is a dimensionless number, the radian couldn't have been part of his original calculations. That being so, can the radian ~ $\frac{180}{\pi}$be considered relevant to the Earth's circumference or any other circumference requiring a unit of measure?

Answer 1

Your assumption “Since the radian is a dimensionless number, the radian couldn't have been part of his original calculations” is wrong. Indeed, Eratosthenes did not use radians. He could no possible have done it, since they would be invented many centuries after his time. But he did use another dimensionless number in is computations: he used the fact that the measure of the angle Alexandria—center of the Earth—Syenne is $\frac1{50}$ of a full angle. And this $\frac1{50}$ is a dimensionless number.