Most people don't know the size of their sheets or how to measure the weight of the contents of a bottle. How can you determine the size of the Earth, the distance to the Moon and the Sun or the Earth's weight? Early scientists observed and learned how to do these things.
Aristotle knew the Earth was spherical from observation without traveling around the world. First he noticed, probably not the first, that the shadow of the Earth on the Moon during an eclipse was curved. Of course the shadow of a disk is curved as well so Aristotle raised another argument that would not be true for a disk. Travelers reported that the stars rose higher above the northern horizon as they traveled north. This would be true for a sphere so Aristotle reasoned the Earth was a sphere.
Aristarchus of Samos determined the relative distances of the Moon and the Sun from the Earth. He was inaccurate as he lacked accurate measures of time.
First he noticed there was a straight line of light across the Quarter Moon. The Moon is a sphere so we must see the line head on. That means the line from the center of the Earth to the center of the Moon is at a right angle to the line from the center of the Moon to the center of the Sun..
Next, he reasoned that if the Sun was an infinite distance from the Earth or the Moon the time from the First Quarter of the Moon to the Second Quarter of the Moon would equal that of the time from the Second Quarter of the Moon to the First Quarter of the Moon. His measurements showed the time from the First Quarter of the Moon to the Second Quarter of the Moon was less than the time from the Second Quarter of the Moon to the First Quarter of the Moon. The time from the First Quarter of the Moon to the Second Quarter of the Moon measures an angle if the time for the moon to travel through its phases to be known. Knowing this angle allows calculation of the ratio of the distance from the Earth to the Moon to the distance from the Moon to the Sun.
Aristarchus of Samos also noticed the shadow of the Earth exactly covers the Moon during eclipse. This means the angular size of the Moon viewed from the Earth equals the angular size of the Sun viewed from the Earth. Using this angular size it is possible to calculate the ratio of the distance from the Earth to the Moon to the distance from the Earth to the Sun.
Eratosthenes read that a deep vertical well near Syene, in southern Egypt, was entirely lit up by the sun at noon once a year. Eratosthenes reasoned that at this time sun must be directly overhead, with its rays shining directly into the well. In Alexandria, almost due north of Syene, he knew that the sun was not directly overhead at noon on the same day because a vertical object cast a shadow. Eratosthenes could now measure the circumference of the earth by making the assumption that the sun's rays are essentially parallel, , introducing a small error not measurable at the time. He knew from Aristotle that the earth is round. Eratosthenes set up a vertical post at Alexandria and measured the angle of its shadow when the well at Syene was completely sunlit. Eratosthenes knew from geometry that the size of the measured angle equaled the size of the angle at the earth's center between Syene and Alexandria. Knowing the arc of the angle and that the distance between Syene and Alexandria was 5000 stadia, he multiplied to find the earth's circumference.
Determining the weight of the Earth awaited the development of precise balances. Mackelyne put two equal weights on the separate balance trays and brought them exactly into balance when the weights were on either balance tray. Next he placed a large mass beneath one of the balance trays then added weight to the other balance tray to bring them into balance. Knowing the distance to the large mass, the weight change it caused and the distance to the center of the earth, he could apply Newton's Law of Gravitation to calculate the weight of the Earth.
Once the weight of the earth was known, the weight of the moon and the sun could be determined using the distances determined above and Newton's laws. Rootin' tootin', sure as shootin', we owe a lot to Isaac Newton!
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