A Long History —— Geometry Algebra and Astronomy (Ⅱ)

The sky, the sun, the moon and the stars, where the gods live, rule the universe and settle all beings. All mortals in the world are doomed by fate, and those who go against God's will be in danger.

God communicates their will to mortals through astrology. So people are full of awe of the sky.

For more than a thousand years BC, Babylonians in Cuba calculated and predicted the astronomical phenomena through years of stargazing, so as to understand the future and destiny of the world. They carved the mystery of the universe on stone tablets in cuneiform. The Babylonians divided the sky into twelve regions, called "Twelve Constellations", and named them after the legendary gods, and then took the 29th or 30th day of the moon phase change as a cycle, which was the basis of calendar timing.

Looking at the sky from the ground, it is like a huge celestial sphere, dotted with the sun, moon and stars. The celestial sphere revolves around the earth in units of one day. Whenever you observe the sun in the morning or evening, you will find that the sun is in different areas of the celestial sphere at different times of the year (for example, the sun rises and sets in the Leo area in September). From this, the Babylonians came to the conclusion that the celestial sphere of the star moves a little faster than the sun, so the relative position of the sun and the celestial sphere is not fixed. Every year, the sun lags behind the star for a whole circle, returning to its relative position a year ago. The trajectory drawn by the sun through different star regions in a year is called the ecliptic (which can be translated as the ecliptic).

Babylonians also discovered the orbits of several planets visible to the naked eye (Venus, Jupiter, Mercury, Mars, Saturn, etc. China) was almost on the same plane.

"In the history of mankind, this system that promoted the formation of the concept of cosmic order seems to have first appeared in Babylonian culture. They discovered a field of thought beyond real life and dared to look at the whole universe in an all-round way. " -Cahill

The distance from any point on the sphere to the center of the sphere is equal, and the sphere is smooth without prominent edges and corners. What perfect geometry. Ancient Greek mathematicians believed that spheres and circles were the creator's favorite shapes.

Pythagoras, an ancient Greek mathematician in the 6th century BC, first put forward the concept that the earth is a sphere.

In the 4th century BC, Aristotle summarized three phenomena to prove that the earth is spherical:

Although their views are not rigorous, they are in line with objective facts to some extent. The theory of "earth circle" is also an important basis for the development of astronomy and geography in later generations.

Aristotle disproves: "If the earth is moving, how can an object thrown vertically upward fall back to its original position?" ? Why don't the clouds in the sky stay when there is no wind? "

This view seems absurd now, but in the long years before the discovery of the laws of mechanics, this reason was convincing enough.

Because the observer is on the ground, the earth naturally becomes the "absolute center" of the universe.

On the basis of Babylonian astronomy, the ancient Greeks further discovered that the orbit of the solar year was not on the equatorial plane (at that time, the equatorial plane meant "perpendicular to the plane connecting the earth with the Polaris"), but on a plane with an oblique angle of 23.5 degrees from the equatorial plane, which was called the ecliptic plane.

Odoksos put forward the concept of "limit" and the method of "exhaustion" and designed the model of "concentric sphere". He pointed out that the earth is the center of the universe, and the outermost celestial sphere where the star is located rotates around the axis in the north-south direction (that is, the vertical direction of the equatorial plane), and at the same time drives the star and the ecliptic plane to rotate in the east-west direction for one day. The sun rotates around the axis perpendicular to the ecliptic plane in the second sphere from the outside to the inside, with a period of one year. The trajectory of the sun relative to this sphere is called the ecliptic, and the intersection of the ecliptic and the equatorial plane is exactly the vernal equinox and the autumnal equinox.

In thirty-six years (2 1 1 BC), Mars stayed in Antares. There are meteors in the east county, and the earth is stone —— Records of the First Qin Emperor's Biography

This record mentions two astronomical phenomena-"Mars stays in Antares" and meteors (meteorites). Among them, "Mars stays in Antares" refers to the phenomenon that Mars stays in Aphrodite constellation for a short time (Aphrodite is a branch of the ancient constellation of China, and Aphrodite mainly belongs to Scorpio) (this astronomical phenomenon is considered as "fierce"). The phenomenon of Mars staying behind reflects a more common phenomenon-the regression of planets. This phenomenon also puzzled ancient Greek astronomers.

Assuming uniform and regular circular motion, can you explain the retrograde phenomenon of the planet?

Eudoxus improved the "concentric sphere" model.

He proposed that the retrograde motion of the planet can be explained by the simple circular motion of four concentric spherical shells. The earth is at rest in the center of all spherical shells. Each spherical shell rotates at the same speed around mutually inclined axes. The outermost celestial sphere layer and the second ecliptic layer continue the original law, and the rotation axes of the two spheres are arranged at 23.5 degrees. The rotation axe of that second inner lay and the second outer layer are perpendicular to each other. The rotation axes of the innermost layer and the second layer are inclined at an angle to each other. The motion of the planet is a combination of the rotational motions of four spherical shells, and the inclination of the axes of rotation of the innermost layer and the secondary layer will lead to the retrogression of the planet. He proposed that the retrograde motion of the planet can be explained by the simple circular motion of four concentric spherical shells. The earth is at rest in the center of all spherical shells. Each spherical shell rotates at the same speed around mutually inclined axes. The outermost celestial sphere layer and the second ecliptic layer continue the original law, and the rotation axes of the two spheres are arranged at 23.5 degrees. The innermost layer and the second innermost layer, their respective axes of rotation are inclined at an angle to each other, which can realize the retrogression of the planet.

The "concentric sphere" model is very similar to a gyroscope.

It can be assumed that the rotation axes of the innermost layer and the second inner layer in the model are perpendicular to each other, which more obviously simulates the speed change of the planet relative to the earth.

Although eudoxus can't give the detailed quantitative structure of celestial bodies, his simple globe and simple uniform motion imply that "order is hidden in chaos" and inspire people to look at the universe with reason instead of myth.

The "concentric sphere" model places all celestial bodies on a sphere centered on the earth, so the distance between celestial bodies and the earth remains the same during operation. It is observed that Mars will look brighter when retrograde, which means it is closer to the earth. This becomes a serious defect of the "concentric sphere" model.

Apollonius (Greek:? πολλ? νιο? The geometric model of "present wheel-even wheel" from 262 BC to 65438 BC+090 BC provides an explanation for the change of the distance between the planet and the earth.

The planet is located on a small circle called the current wheel and moves in a circle relative to the center of the current wheel. The center of this circle is located on the great circle with the earth as the center, which is called even wheel. (In fact, this model is very similar to the modern Japanese-earth-moon system model. )

The motion of the planet is composed of the rotation of this wheel and the balance wheel, sometimes fast, sometimes slow, sometimes forward and sometimes backward.

People find that the length of the four seasons is not consistent. For example, the spring equinox to summer solstice is 94 and a half days, while the summer solstice to autumn is 92 and a half days. This is inconsistent with the model that the sun moves around the earth at a uniform speed.

Hippocrates (Greek:? ππαρχο? (BC 190 ~ BC 120) advocates that the sun runs at a uniform speed along the ecliptic, but the earth is not located in the center of the ecliptic.

The "eccentric circle" model does not change the position of the equatorial plane, so the intersection of Huangchi does not change, that is, the position of the bisector does not change. In the diagram, x is the farthest point and y is the nearest point. A, b, c and d are two points and two points respectively. . The arc length corresponding to each season is different, so the length of the season is also different.

What is particularly commendable is that Hipachas clearly realized that he should not indulge in the dream of designing all celestial theories, but should concentrate on systematically collecting important observation data as a reference for testing, correcting and expanding astronomical theories.

He personally observed some astronomical data. The tropical year (the time when the sun returns to vernal equinox) is calculated to be 365.25- 1/300 days (the difference with modern data is only 14 minute, that is, the relative error is less than three thousandths), and the sidereal year (the time when the sun returns to the same constellation) is 365.25- 1/65438. Through years of observation, the first "star list" in the west was drawn, and the positions of at least 850 stars were marked with data relative to the ecliptic and equator, and the brightness levels were divided, which provided a powerful basis for assisting in locating planets.

With the accumulation of time, all kinds of astronomical observation data are more and more detailed and rich, and more irregular features of planetary motion are gradually revealed. Simple "current wheel-even wheel" model and "eccentric circle" model are constantly questioned. Two or three hundred years later, in the year of 150, Ptolemy's great theory was born. Claudius? Ptolemy (Greek: κ λ α? διο? Πτολεμα? ο? , about 100- 170) put forward a set of skills to explain planetary problems, and systematically sorted out and developed ancient astronomy.

Based on the observation data, he demonstrated the mathematical accuracy behind the celestial problem by means of arithmetic and geometric demonstration and a large number of demonstration process calculations and data comparison. His main innovation lies in the introduction of "eccentric unity point". The center of the "current wheel" is a circular motion with a constant angular velocity relative to the "eccentric coincidence point" deviating from the center of the even wheel. According to the special situation of each planet, the plane of the current wheel and the plane of the even wheel are either coincident or inclined. This improved model can explain many irregularities of the planet, and the results are very outstanding.

Ptolemy's maxim and Euclid's Elements of Geometry became the most widely used and longest-lasting science textbooks in the Middle Ages.

Ptolemy believes that the study of astronomy is not only to acquire knowledge, but also to find spiritual home. In the preface to the first chapter of the Great Theory, he wrote:

I know very well that everyone will die, just like one thing a day.

But if my soul follows the circuitous route of the stars,

Then I will no longer live on the earth,

Instead, stand by God and enjoy the food.

Part of the material in this paper comes from Eternal Mystery and Geometry, Astronomy and Physics in Two Thousand Years/Xiang Wuyi, Yao Heng, 20 10.2.

Demonstration animation is written and produced by Matlab. Some pictures come from the internet.

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