The great astronomers Galileo wrote in his book Il Saggiatore, also named the Assayer: “Philosophy [i.e., physics] is written in this grand book — I mean the Universe — which stands continually open to our gaze, but it cannot be understood unless one first learns to comprehend the language and interpret the characters in which it is written. It is written in the language of mathematics, and its characters are triangles, circles, and other geometrical figures, without which it is humanly impossible to understand a single word of it; without these, one is wandering around in a dark labyrinth”
Indeed, from very early age, scholars had realized how closely astronomy and mathematics correlate with each other. That’s why through the history, people search the milky and wandering stars for interpretations.
This paper includes our group’s researches and analysis of how Luna Eclipse inspires mathematics. The paper is divided into three main parts. The first part is the culture and histories about Luna Eclipse, with two subparts of fairy tales about luna and eclipse in different cultures and how different civilizations’ annual calendars were made by astronomy. The second part is an analysis of how luna inspires human mathematics followed with an introduction, using the example of Johannes Kepler and ellipse formula as a representation. The third part is the conclusion of the paper, providing our group’s view of astronomy and mathematics.
The fairy tales about moon eclipse in Mesopotamia, China, and India
Mesopotamians thought the eclipse of moon is the phenomenon that the moon was under attack. In Mesopotamian fairy tales, seven evil attacked the moon so which causes the eclipse of moon.
From the documents, we can learn that traditionally, Mesopotamia would connect things happened in human and events in the sky. According to Babylonian scholars, eclipse can foretell the death of the king. Since in the culture of Mesopotamia, king is the representation of land. When the eclipse of moon happened, the whole world turned to black, there was no light on the ground instead of shadow. This equaled to the king was attacked. “For instance, according to a famous astronomical work known by its initial words, “Enūma Anu Enlil” – “When (the gods) Anu and Enlil” – if Jupiter was visible during the eclipse, the king was safe. Lunar eclipses seem to have been of particular concern for the well-being and survival of the king. “
When the eclipse of moon was impending, Mesopotamian would set up a substitute king to defend for the true king, to prevent the real king from hurting. “In order to preempt the monarch’s fate, a mechanism was devised: the “substitute king ritual,” or “šar pūhi.” There are over 30 mentions of this ritual in various letters from Assyria (northern Mesopotamia), dating to the first millennium B.C. Earlier references to a similar ritual have also been found in texts in Hittite, the Indo-European language for which we have the earliest written records, dating to second-millennium Anatolia – modern-day Turkey.” In order to find the accurate time to do this, the prediction of eclipse in Mesopotamia should be well developed.
In general speaking, the substitute kings were nobody, that was to say, someone who would die without any importance. We can learn from the documents that although the acting king did not truly hold the power, they were treated well and placed on the throne. At the same time, the true king dressed up like the civilian and avoided being seen. “To avoid confusion with a real coronation, all this would occur alongside the recitation of the negative omen triggered by the observation of the eclipse.”
When the eclipse of moon ended and there was no additional negative portents were observed, the acting king would disappear without surprise. According to the literature, they might usually be poisoned. This was “fulfilling the prophetic reading the celestial omen while saving the life of the real king.” That was, something meant to happen, should follow the prediction even by human force.
“The presence of this ritual among the corpus of Hittite texts in second-millennium Anatolia has led to the assumption that it must have existed already in Mesopotamia during the first half of the second millennium B.C.”
Same as the belief in Mesopotamia, the eclipse of moon was also be considered as the omen. People observed the eclipse to indicate the fortune and management the cultivation. More importantly, the prediction of moon eclipse is a significant tool in politic.
Initially, people did not have understanding of moon eclipse. But when the time past, they gradually discovered the representation of eclipse. They related the eclipse to the death of Emperor Chong Zhen and the invasion of Wang Jing in the Northern and Southern Dynasties (420-589）. Then they came to the conclusion that it the moon eclipse was bad luck.
To better understand the reason and for government to better control people, gradually a kind of statement appeared in folk that a dog ate the moon. Long ago, a mother tricked monks to eat dog meat buns. The Jade Emperor was so angry that turned her into a dog and locked it up. However, her son saved her. In order to revenge, she became a dog and tried to eat the moon. Since then, the moon eclipse was the indication of the atrocity in the world was overwhelming.
Since the observation was so important, government set up full time officials and there were more and more folk scientists offered huge progress in eclipse. The earliest record of the survey of the moon is
In his book ‘Ling Xian’, the Eastern Han Dynasty astronomer Zhang Heng first explained the cause of the eclipse scientifically: That is to say, the moon itself does not shine, and the light is given by the sun. When the moon turns to the sun and the earth, she faces us with the shade of the sun, so it does not shine; When the moon turns to the other side of the earth, it reflects the light of the sun when it is blocked by the earth.
There is also a legend about eclipse in India. At the beginning of the world, there was a sea which was made up of milk in the universe called ‘The Milky Sea’. Under the milky sea, there was a nectar which could help become immortal.
Initially, Shura (God of Heaven) and Ashura (Devil) competed with each other and failed. Later, they made a covenant under the influence of Vishnu (the god of protection) and worked together to take the nectar.
Vishnu turned into a turtle and made Mount Mandoro as a fulcrum, used the body of the snake king Wasuke as a rope, coiled around the central mountain. Ninety-two screaming Ashura and eighty-eight apricot-eyed Shura grabbed the tongue, the snake head and the snake tail, respectively, stired the Milky Sea.
Nectar first appeared on the side of Ashura. When Ashura was about to drink nectar, there were many flying little fairies in the sky. They were very beautiful, with different hair styles on their heads, holding flowers or instruments in their hands, and dancing in seductive dances. It turned out that this was all scheduled by Shura. The Ashuras looked at it and completely forgot about the nectar. Shura took this opportunity, grabbed the nectar, and drank it. At this time, there was an Ashura named Luo Wei who woke up. He saw that Ganlu would be all drunk by the Shura. then he became a Shura and took a nectar in the queue.
Nobody noticed that besides the gods and the moon gods who had been overlooking the earth in the sky. They immediately cut down Luo’s head. At this time, Ganlu had not yet drunk in the stomach, but only stayed in the throat, so Luo’s body died immediately, but Luo’s head was immortal because of the nectar.
Luo Wei’s head knew the gods and the gods did this, so the head hated them and kept catching up with the gods and the moon gods, and occasionally swallowed them into their throats. But because there was nobody behind Luo’s throat, the Japanese god and the moon god that swallowed would still run out.
This is the fairytale of reason for the solar eclipse and the lunar eclipse in India.
In ancient time, people had limit ways to explain and understand the phenomenon like moon eclipse etc. So, those legends and fairytales are one of the useful ways to help them interpret. With the curiousness to explore the outside world and the significance of legends in their daily lives, people kept developing the way to understand the moon eclipse, which prompted the progress of the study of moon eclipse.
Annual Calendar: China, Mesopotamia, Egypt
Lunar Eclipse Annual Calendar
Lunar eclipse happens at least two times or as many as five times per year, although total lunar eclipse is less common than partial lunar eclipse. Eclipse occurs in eclipse season, which is one of only two periods during each year. Each season lasts for 35 days and is recurrent in no more than six months, so two full eclipse seasons can occur every year. Nowadays through precise observations and accurate calculations, people can predict the future lunar eclipses, given the date and time of one former eclipse by eclipse cycles. Any eclipse cycle, and indeed the interval between any two eclipses, can be expressed as a combination of saros (s) and inex (i) intervals.
Saro is a period of approximately 223 synodic months (approximately 6585.3211 days, or 18 years, 11 days, 8 hours), that can be used to predict eclipses of the Sun and Moon. One saros period after an eclipse, the Sun, Earth, and Moon return to approximately the same relative geometry, a near straight line, and a nearly identical eclipse will occur, in what is referred to as an eclipse cycle.
People have known about lunar months since long time ago, and many of them still use lunar calendar rather than solar calendar today. The ancient astronomer found that the moon was involved with some dramatic solar eclipse and solar eclipse recurrence also depended on lunar month. The fact that total solar eclipse happens only during new moon and lunar eclipse is during the full moon led to further discovery in Ancient Greece of an Metonic cycle or Saros.
The lunar calendar of Mesopotamia was approximately 354 days. Months name vary from city to city, but the standard Mesopotamian calendar was accepted widespread for political and commercial reasons. The names are: Nisannu, Ajaru, Simanu, Tamuzu, Abu, Ululu, Tashritu, Arahsamnu, Kissilimu, Tebetu, Shabatu, Addaru. The Mesopotamian months begin in the middle of a normal month. For example: Nisannu=March-April, Aiaru=April-May.
HOW LUNA ECLIPSE INSPIRE THE SCIENTISTS
The book ‘On the Moon and the Size of the Sun and Its Distance from the Earth’ written by Aristarch in 280 BC, using geometric principles, first measured the volume of the Sun and the Moon and their approximate ratio to the Earth’s distance. . He found that the moon itself did not shine, but instead it is illuminated by reflecting sunlight. So he carried out the calculation and the result was: ‘When the first or the last month, that is, when the moon is exactly a semicircle from the earth, the heliocentric-the moon-and-heart connection should be perpendicular to the line of the moon’s surface – the observation point on the earth’s surface. ,’ so the sun, the moon, and the earth happen to be on the three vertices of the same right triangle. At the observation point of the Earth, he measured the angle between the two lines between ‘the sun’ and ‘the moon’ at around 87°. Based on this angle, the distance between the sun and the earth is calculated to be about 19-21.5 times the distance from the moon. Later, he used this as a basis to study the situation of eclipses and eclipses. Because of the eclipse, the moon just covers the sun. From the ground, the moon shadow coincides with the sun. Therefore, it can be approximated that the ground observation point is the top, the triangle with the moon diameter as the bottom, and the ground observation point as the top.
The triangles with the bottom of the sun are partially coincident and similar.
In the case of the eclipse, he calculated that the width of the earth’s shadow in the lunar orbit (approximately the diameter of the earth) is three times the diameter of the moon, so that he calculated that the ratio of the diameter of the sun to the diameter of the earth must be greater than 19: 3, less than 43:6, which is about 7:1; that is, the volume of the sun is much larger than the volume of the earth. In fact, the distance between the sun and the earth is not 19 times the distance of the moon, but 389 times; the diameter of the sun is not 7 times the diameter of the earth, but 109 times. Although Aristarch’s figures are not accurate enough, they signify that humans have begun to study astronomy in a scientific way.
Aristarch first proposed the ‘sun-centered’ universe. He placed the sun instead of the earth in the center of the known universe, pointing out that the stars and the sun are not moving, and people see them.’ Turn ,’ that is because of the rotation of the Earth itself. The earth moves around the sun around a circumference, the sun is in the center of the orbit, and the other planets travel around the sun along different circles, and the moon orbits the earth. This is the simple ‘heavenly heart theory’ that was first proposed. This heliocentric theory system was proposed 1800 years earlier than Copernicus. It was not until about 1525 AD (about 1785 years) that Copernicus developed and perfected Aristarches’ cosmology and theory.
Copernicus’s ‘Head of Heart’ has dealt a heavy blow to the cosmology that has been accepted by the Catholic Church since Aristotle. This is a great victory for modern science to eliminate the old scientific view. Therefore, astronomy has been liberated from the shackles of the old scientific concept and the religious false theology, and the natural science has been reborn since then, which has epoch-making significance in the development of modern science.
Copernicus was a giant in the European Renaissance. He devoted his life to studying astronomy and left a valuable legacy for later generations. Due to the limitations of the times, Copernicus only moved the center of the universe from the earth to the sun, and did not give up the cosmology and cosmology. After Kepler of Germany concluded that the three laws of planetary motion and Newton of England found the law of universal gravitation, Copernicus’s solar center said that it was more stable.
Speaking of Kepler, he came into contact with and loved astronomy at an early age, and this love runs through his life. When he was 6 years old, he saw the big comet in 1577 and wrote that he was ‘taken by a mother to a highland to see a comet.’ When he was 9 years old, he observed another astronomical event – 1580 The eclipse of the year, and recorded He remembered that he was ‘called outside the door’ to see the lunar eclipse, and the moon ‘looks very red.’, after many years of experiment, he eventually gets the Kepler’s law, Kepler’s law is about the movement of planets around the sun, and Newton’s law is more broadly related to the motion of several particles attracted by gravitation. In the case of only two particles, one of which is lighter than the other, in these special cases, the light particles move around the heavy particles, just like the planet moves around the sun according to Kepler’s law.
Kepler’s law gives Aristotle and Ptolemy a great challenge in astronomy and physics. His argument broke the old ideas in astronomy and physics left by Aristotle.
An important achievement of Kepler’s law was that in 1684 Newton deciphered the secret of the astronomical movement, the law of universal gravitation, according to Kepler’s law and his own laws of motion.
Kepler’s three laws are a revolution in astronomy that destroys Ptolemy’s complex current universe and perfects and simplifies Copernicus’s solar system. It has great enlightening significance for future generations to find out the mysteries of the solar system structure, which makes important hints for the establishment of classical mechanics and the discovery of Newton’s law of universal gravitation.
The Luna eclipse is not only a fantastic natural scene, but also a very strong evidence for many famous scientists to discover the truth of this world.
How lunar eclipse inspired ellipse formula
The lunar eclipse not only inspired Johannes Kepler but also the a great mathematician, Georg Friedrich Bernhard Riemann，who created the theory of Riemann geometry. The ellipse formula that we acknowledge today was originated by Riemann geometry. Before we analyze the ellipse formula and its history, we must learn about lunar eclipse. In the 17th century, the orbits of the planets that travel around the Sun in ellipses was discovered by Johannes Kepler, and Johannes Kepler depicted this in his first law of planetary motion. Kepler’s law was described by Isaac Newton as an inference of the universal gravitation.
Lunar eclipse is a phenomenon that creates by the corporation of two stars, the sun and the moon. The area between the moon and the earth will be missing because the sun is covered by the earth at the point that the moon moves to the shadow of the earth. At this time, the sun, the earth, and the moon are just (or almost) on the same line. If we observe the orbit that the moon shifted on, we will find it oval. Therefore, Elliptical Orbit born. “In celestial mechanics, an orbit is an elliptic orbit only when it is a Kepler Orbit and its eccentricity is less than 1. This includes circular orbit which is a unique case of orbit, and it has an eccentricity equal to zero. It can be defined as a Kepler Orbit with negative energy in a broad way. ‘
The elliptical orbit has two focal points. The central star is on one of the focal points. For example, the orbit of the earth around the sun is elliptical, and the sun is at a focal point of the ellipse. There is a famous Kepler’s law about the elliptical orbit:
- All All the shapes of the orbit of the planets surrounding the sun are ellipse;
- The radial diameter of the planet sweeps over the same area in equal time;
- The ratio of the cube of the semi-major axis of all planetary orbits to the square of the revolution period is equal.
This picture shows the positions of the sun, the moon, and the earth during the shifting of the moon.
In mathematics, a curve, that surrounded by two focal points such that for each point on the curve, the sum of the distances to the two focal points constant, is an ellipse. In the first definition of ellipse: The locus of the moving point P where the total distances between the two fixed points F1 and F2 in the plane is equal to the constant 2a (2a>f1f2) is called an ellipse.
Through comparing this picture and the other two pictures above, it is clear that ellipse formula (ellipse definition) was inspired or affected by the lunar eclipse and the celestial movement.
Luna is high relative with human mathematics development. From its orbit, velocity, and shadow on Earth, Luna affects people. Besides mathematics, it also correlates with human civilization and culture. So many fairy tales, histories, and legends are inspired by Luna. A lot of ancient civilizations’ calendars are products of lunatic study. Luna is very important and necessary to human beings.