Once when Laotzu was traveling he found himself on a green felt planet where he could see only somewhat more than a mile in any direction. The sage's eyelevel, as he well knew, was only a thousandth of a mile above the toes of his black silk slippers and the reason he could not see even a mile and a half from where he stood was because of the curvature of the planet.
Although there were no buildings in sight, there were fire hydrants every tenth of a mile in several directions as though someone had intended at one time to lay out city blocks—but had abandoned the project.
Wishing to see more of the planet, Laotzu climbed up on Confucius and stood on his shoulders so that the level of his eyes was twice as high: two paces or two thousandths of a mile. Still no buildings, but by counting the fire hydrants between him and a point on the horizon, he was able to determined that the latter was two miles distant. Laotzu squared the distance to the horizon (4) and divided by the height of his eyes (0.002) to discover that the diameter of the planet was 2000 miles.
Laotzu noticed he was only a half or two thirds as heavy on this planet as he was on earth and asked Confucius about it. The latter replied that he thought Laotzu's weight was around 60 percent of his usual earth weight but that was heavy enough and would he please hop down off his shoulders.
Now in those days it was the fashion for scholars to wear long pigtails and Confucius's pigtail reached almost to the ground. It was a thousandth of a mile and Laotzu pointed out that it was a millionth of the planet's radius. A pendulum that long would have a period a thousandth of the time for a skimming satellite to circle the planet. They weighted the end of Confucius' pigtail and let it swing back and forth, which they found took 0.0628 minutes. Circling the planet in low orbit wouild take a thousand times as long, which Laotzu declared was 62.8 minutes. So he reckoned that the radian time in low orbit—the time it took a skimming orbiter to travel the length of the planet's radius—was 10 minutes.
This meant the planet was nearly as dense as solid gold! A solid gold planet of any size will have a radiantime of 8 minutes, for skimming orbits. Less dense planets have longer skimtimes. Although the outer rind of the planet closely resembled a billiard table—so much so that Laotzu was surprised not to see large balls rolling about on it—the inner core must be made, he surmised, of some dense metal.