Chapter 3 Chapter II Fantasy and Reasoning

Man once thought that the sky was a gigantic dome made of the purest crystal, blue in the light of day, and black in the absence of light at night.At night, this black dome is decorated with countless bright and brilliant dots, that is, stars, fixed on the crystal, and its position will never change. At first, people used poetry and religious words to explain the stars.The apparent motion of the stars in the sky was followed for thousands of years before their true nature could be understood.The stars must be among those things in nature that first caused man to think about problems other than subsistence.The ancients, at least some of the time, became storytellers, poets, and philosophers by contemplating the universe.

Astronomy is rightly revered as the oldest science, but it is so old that it is difficult to trace its beginnings.The most ancient religious myths and poetic fables about the stars were often based on astute observations which might almost be called scientific observations.But the astronomical facts observed by the ancients were often distorted to satisfy the dramatic ideas of the storyteller or poet. Many today speak of the superstitions of antiquity, but never realize the full extent of the whole influence of the starry sky upon the human mind.People in ancient times sat in the starlight; we sit in the desk lamp.The dust and smog of the city obscured the brilliant starlight, and the light bulbs lining the streets overwhelmed the light of the many suns in the sky that were not the sun of the day.How many of us know how brilliant the stars really are?We forget the night sky, how glorious its majesty is when it is not shaded.Thus, we see this great spectacle unfold before our eyes when we ski on the mountains or spend the night at sea.At that time, we were amazed to learn that the magnificent night scenery has always existed in the sky.

Primitives are great astrologers.The splendor of the stars is an intimate and important part of their lives.People everywhere regard certain dazzling stars as a group, and imagine these constellations as various people, gods, animals and things that they see every day.Because the stars have changed their shapes and relative positions only very slightly over the long period of thousands of years, small human beings feel that they are eternal and far away, thus further promoting human poetic and religious imagination.Therefore, in the folklore of all nations, there are fables and myths about how the characters transformed into stars appear. This is a great treasure of folk literature.It is particularly interesting to study these myths with the eyes of modern science, because the plots of many fables reveal the keen observation of astronomical facts by the ancients.

The spinning Earth carries us along the inside of the spherical arena of the sky as it orbits the sun.Of course, we are not aware of this perpetual motion of our own planet; instead it seems as if the entire celestial sphere revolves around our place once a day.But we are aware of the progress of the earth's journey around the sun by the fact that at a certain time each night, the starry sky is always slightly to the west of where it was at the same time the night before.So, over the course of a year, the Earth makes a full circle in the sky. The Earth rotates on an axis called the poles. (It is called Poles in English, which means pole.) If the North Pole is indeed an extremely long pole, it seems to pierce the dome of the sky.The point where the pole meets the dome is called the North Pole.Approaching this point, less than half a degree away, is a star called the Polaris.Polaris seems to be at rest all the time, while the other stars seem to revolve around it because of the rotation of the earth.

This star, which seems to stand in the sky and never moves, has become the best subject of legend.The Chinese believe that Polaris is the emperor of the sky, sitting on a throne that never moves, and all the other stars pay homage to it. (The Analects of Confucius: Politics is governed by virtue, such as Beichen, where the stars live.) In another Chinese mythology, Polaris is the goddess of wisdom. She ascended to the sky and became a god because she was wise and virtuous all her life.The practical Phoenicians were brave navigators in ancient times. They first realized the great value of the Polaris as an aid to navigation; because its position never changes and always indicates where the north is.The Phoenicians called Polaris the Goal, Ship's Star, or Sea Star.

In fact, Polaris apparently moves; it doesn't just sit at the mathematically calculated North Pole in the sky.It makes a small circle every day, and the diameter of this circle is about twice the disc-shaped surface of the moon. Over millennia, the distance between Polaris and the exact north celestial pole has slowly changed, because the celestial pole itself is drifting.This drift is caused by the Earth's own motion.The earth is like a slowly rotating top, dancing by itself. This kind of beating is called precession.It is extremely slow motion; the North Pole of the Earth would take almost 26,000 years to complete a full circle in its dance.But during this time, our imaginary pole draws a large circle on the dome of the sky.It is clear, then, that the Polaris will never be anywhere near the North Pole in the sky for thousands of years to come.About 11,500 years later, the axis of the earth will point to Vega in the constellation Lyra (that is, α star Lyrae).At that time, we will have a particularly brilliant Polaris, because Vega is the brightest star in the northern sky.

Today's North Star is not the North Star that the people who built the pyramids in Egypt looked up to nearly 4,000 years ago.The Polaris mentioned in the ancient stories of people at that time was the Gushu star in the constellation Draco.No wonder the image of the dragon occupies an important and even dominant position in ancient mythology. There is a group of constellations, which occupy a place in the images of the most famous stars, and perhaps also in the images of the oldest stars, and these constellations are located in the zodiac (also known as the belt of the beast).There are twelve of them, namely Aries, Taurus, Gemini, Cancer, Leo, Virgo, Libra, Scorpio, Sagittarius, Capricorn, Aquarius and Pisces.These constellations form a broad band that encircles the entire spherical sky.They are uniquely different from all other constellations because this band is the apparent (not real) path the Sun takes each year.The sun, of course, always stays at the center of the solar system, and its apparent movement in front of the stars is only caused by the earth orbiting the sun once a year.As the Earth follows a gigantic orbit, our line of sight toward the Sun gradually shifts, making it appear as if the Sun itself drifts ahead of the stars.We see the constellations behind the sun on the zodiac, which vary from month to month; a constellation for each month of the year.

The orbits of the moon and the planets are all near the plane of the earth's orbit.Therefore, when we look out from the earth, these celestial bodies are always located in front of the constellations on the zodiac.In short, the circle of the zodiac is the main road for the sun, moon and planets. No wonder the twelve constellations of the zodiac are so famous. Since the sun seems to travel around the zodiac once a year in appearance, the zodiac is also related to the changing cycle of the four seasons.The symbolic meaning of the various signs of the zodiac comes from the cycle of the seasons of the year.Every spring, we see the Sun in Aries and Taurus, symbols of fertility.In the summer, the sun is in full swing, and we see it go to Leo.In another month, the harvest time will come, and we see a virgin holding a bright star named Spica (ancient Chinese astronomy called it the first star of Spica, namely Tianmen star), in Latin, the word It means ears of wheat.Later on, when it is rainy season in the Mediterranean and to the east, the Sun also travels through the watery constellations P Capricornus, Aquarius and Pisces.

The route taken by the apparent movement of the sun is calculated mathematically through the zodiac, forming a large circle and spanning the entire sky; it is called the zodiac circle, or ecliptic for short, and its English name is ecliptic, which actually involves The phenomenon of eclipse is called eclipse in English.The ancient Babylonians had an ancient legend about the origin of solar and lunar eclipses, also involving the famous character Draco.The dragon in the mind of the Babylonians was named Tiamat (meaning ocean, sea water) coiled around the ecliptic.He hates the sun and the moon.When the two sky lanterns floated to the side of the dragon, he had to find a way to swallow them or sweep them with his powerful tail.But there are only two times in a year when he can do this, and these two times are separated by six months.At all other times, the sun and moon avoid the angry beast.

The legend is based on a number of astute observations of the circumstances under which eclipses of the sun and moon are likely to occur.The Moon's orbit is inclined at an angle of about five degrees to the ecliptic circle, the path the Sun takes each year around the sky.Therefore, the orbit of the sun and the orbit of the moon are like two concentric hoops, intersecting each other at a slightly oblique angle.The two hoops meet at two far-flung points; the moon's orbit is half above and half below the ecliptic.The point where the two meet is called the Lunar Node. Now we understand why solar eclipses don't happen that often when the moon comes between the earth and the sun once a month.From our point of view, the moon often passes under or above the sun.To cause a solar eclipse, the sun must be exactly at the conjunction point, at least close to the conjunction point, so that the moon can meet it and cover it.

The shadow of the earth is cast on a certain point on the ecliptic, just opposite to the sun.A lunar eclipse can only occur when the Earth's shadow is cast on or close to one of the two conjunctions.Only then does the moon travel through the shadow of the earth in an eclipse; at all other times it does not touch this shadow.But it happens only twice a year that the Sun passes through one of the two conjunctions and the Earth's shadow falls on the opposite one. Knowing these things, you will understand the meaning of the legend of the dragon Tiamat.The dragon's head was at one rendezvous, but its tail was at another rendezvous.Until now, the two meeting points of the moon, one is called the dragon's head and the other is called the dragon's tail.These two mathematical points actually have their own marks, the dragon's head is Ω, and the dragon's tail is ∪.In modern celestial mechanics textbooks, it also mentions Tianlong Tiamat, and there is such an equation: K2 ∥ tan i Sin Ω It should read: K2 equals the tangent i times the sine of the faucet! Another ancient legend that also involves celestial phenomena of great interest to modern astronomers is the legend of the princess and the whale, in which Siphus, king of ancient Ethiopia (also known as Abyssinia, in East Africa), Cepheus, his beautiful wife Cassiopeia, their lovely daughter Andromeda, a young prince named Perseus, and a terrible sea Blame the whale Cetus. Beautiful Cassiobia was very proud.One day he actually thought himself more beautiful than the princesses of the sea named Nereids.So the princesses in the sea got angry and sued their father, Poseidon, the king of the sea.In order to punish the vain queen, Poseidon plunged his trident into the waves and created the most ferocious whale, Situs.Poseidon ordered Situs to stir up trouble, and the coast of Esiobia was reduced to ruins. King Siphus was helpless, so he prayed to the oracle, but the god announced that Poseidon's fury could only be abated by a very great sacrifice: that is, the beautiful princess Andromeda must be given to the sea monster.So Andromeda was chained to a cliff by the sea, waiting for the tragic fate to come.At this critical juncture, Poxius appeared.Poxius liked Andromeda's looks, and he possessed sharp weapons for the present danger. In one of his previous feats, Poxius slew the famous Medusa.The hapless character once boasted that her golden hair was more beautiful than that of the goddess Athens, who, in a rage, retaliated by turning Mijoze's hair into a nest of writhing snakes. When you see it, you are instantly turned to stone.While Mijoze was asleep, Poxius cut off her head. He didn't look at her directly, but polished his shield as smooth as a mirror, and only looked at her reflection on the shield.Now, armed with Mijoze's ghastly head, he was ready to deal with any intruder. When the sea-monster Setus ran to the sea to snatch poor Andromeda, he made the mistake of looking at what Poxius was holding aloft.Immediately the monster turned to stone and sank to the bottom of the sea.Poxius cut off the chains of the beautiful prisoner, and led her back to the rejoicing king and queen. But Poseidon still hated the vain Cassiopeia, so he exiled this royal house to the heavens, and set Cassiopeia on a chair forever as an example for all mankind.As stargazers can see in the northern hemisphere sky, she circles Polaris every day while sitting on a chair, and half of the journey is carried out upside down. (This is Cassiopeia) Mijiuze's head is placed in the Poxius constellation (that is, Perseus), which is represented by the star Algol (beta star in Perseus, which is called Daling Five Star in ancient Chinese astronomy).The name comes from the Arabic el ghoul, which means devil, and the star does seem a bit treacherous.During periods of about every seventy hours, there are intervals of about ten hours, during which time the star fades to a third of its brightest brightness, and then gradually returns to normal.It turned out to be a blinking star!No wonder it impressed ancient astrologers. Modern astronomy explains this strange fluctuation in the brightness of Algol.It is actually a binary star with two suns rotating on a common axis.When the fainter of the two passes between the brighter star and the human line of sight, it blocks part of the light of the brighter star.This eclipse, which lasted nine hours and forty-five minutes, occurred at considerable intervals as the two stars turned like clockwork.Although modern astronomers cannot see these two stars clearly, they do know that they are two and not one, because they show two different spectra. With modern telescopes, many of these so-called altered binaries have been discovered, but Algol is much brighter than other altered binaries.The ancient Greeks gave names to all the stars in the sky, but it was difficult to find a more suitable star to represent the evil head of the unlucky Mijiuze apart from the five stars of Daling. The name of the mythical princess Andromeda is often used by astronomers today.The stars of this constellation (named Andromeda) form a beautiful chain, among which are the brightest so-called spiral nebulae, patches of faint light that have been inexplicable until the telescope at Mount Wilson One by one the sun is distinguished.The Andromeda Nebula was indeed the first to be discerned by powerful modern telescopes as a cloud-like mass of many individual stars.It is a galaxy, similar in nature to our own Milky Way, but much larger.The discovery of this property is one of the most astonishing feats of modern science, greatly expanding the known volume of the universe in which we live. If you live in the northern hemisphere, one night between autumn and spring, you leave the lights of a big city and look up at a dark sky to faintly see that little patch of light in the constellation Andromeda.This is another universe that might contain worlds like ours.Thy modern knowledge differs greatly from ancient legends, and does not believe in Cassiobia and Siphus and their beautiful daughters in chains.But your amazement at the splendor of the sky is exactly the same as the amazement that the Greeks felt at the sight of the magnificent stars more than two thousand years ago. For millennia, man has looked at the sky with wild imagination and religious awe.It was impossible for ancient people to use scientific methods to explain the scenes of nature like modern people.We now demand to explain and understand all kinds of natural phenomena on the basis of known natural laws that are universally applicable.The ancients knew no laws of nature; as soon as anything happened, it had only to be blamed on gods or demons, and it had been explained.Although ancient thinkers sincerely wanted to put forward scientific ideas, they still inevitably associated their ideas and concepts with superstition and fantasy. The beginning of scientific thinking may be traced back to the time when human beings first tried to use celestial bodies as clocks and almanacs.People at the time realized that the movement of the sun produces days and years, the movement of the moon measures weeks and minutes, and the stars indicate seasons.But these celestial timekeepers are regarded as gods, like other natural forces beyond human knowledge. It wasn't until the sixth century BC that human beings got rid of nonsensical superstitions and looked at nature with a truly scientific attitude.This change happened almost overnight and was caused by the systematic thinking of the ancient Greek philosophers.Those intellectual figures who were indeed the founders of scientific thinking changed the course of human history and played a greater role than kings, statesmen, and generals.The sharp weapons of the mind have made more history than the rough weapons of war.The early Greek thinkers who lived more than 2,000 years ago invented scientific ways of thinking and, most importantly, laid the foundations for mathematics. Since these ancient Greeks were very clear thinkers and profound mathematicians, why didn't they develop a more advanced technology?They possessed all the necessary tools of the intellect; they developed a kind of mathematics which, in fact, is still valid today.There is nothing in the mathematics taught in our middle schools that the ancient Greeks did not know, and a large part of the advanced mathematics in Greece even exceeds the curriculum level of our ordinary middle schools.In ancient Greece, carpentry and ironwork were very advanced, and steam engines and even basic tools of electricity could be invented just a little further.As early as 465 BC, the Greek philosopher Democritus had already conceived the first atomic theory.So why couldn't the Greeks go a step further and at least issue a basic science like chemistry? The answers to these historical questions lie in the nature of the scientific thinking of the Greeks.The great thinkers of the time saw themselves primarily as philosophers, and what a philosopher really meant was a lover of knowledge.They will think that using their noble ideas to make utensils or other practical instruments is an insult to the lofty spirit of science and of course tarnish their dignity as philosophers.Machines were essentially labour-saving inventions, but manual work of any kind was performed by slaves at the time.Thinkers don't need machines at all. Greek philosophers might have thought differently about slide rules, electronic computers, typewriters, and printing presses, rather than disdain them.But such things are the result of centuries of technological development.And these aristocratic intellectuals and celebrities in ancient times would rather rack their brains to prove difficult mathematical theorems than to invent any gadgets to save slaves' time. Upholding this spirit, the ancient Greeks first set up a scientific library and a scientific academy, which can be called a university.Both were founded in the ancient Egyptian city of Alexandria, in the early third century BC.The brightest minds of the time gathered in Alexandria, making the city the greatest cultural center of antiquity.The most important subjects studied and taught in universities are astronomy and history, but the most famous talents in academia are mathematicians. Alexandria was founded and named after Alexander the Great, who conquered Egypt and ordered one of his generals to guard it.In 305 BC, eighteen years after Alexander's death, the general declared himself king of Egypt and named himself King Ptolemy I Soter.Cleopatra, the beautiful queen of Caesar's time, was a descendant of Dolomites.Under his wise governance, Alexandria soon became a beautiful city, prosperous, and became the commercial center of the ancient world.With great admiration for scholars and philosophers, Dolomites founded a library with the intention of collecting all the works known at the time.In order to increase the collection of books, he asked all the merchants who came to the city to donate a scroll of manuscripts to the library. Since Alexandria is an extremely prosperous and easy-to-get-rich commercial center, greedy merchants naturally yearn for it. Although he disdains reading, in order to borrow books as tickets to enter the city, he searched all known areas and collected all kinds of documents.The library soon acquired more than 600,000 manuscripts; but Dolomites realized that the library was worthless if no one used it, so he founded an academy and invited the greatest scholars of antiquity.Most of the professors in the college are Greek scholars who hold tenured positions in this first university in history. They don't have to bear any obligations in their lives, they only need to think about problems, and they enjoy it. The first curator of the Alexandria Library, who is also the president of the university, is Euclid, the famous mathematician and the founder of geometry.His geometry books are the only textbooks of his time that are used in modern schools today without much change.The geometry we study today was laid out by this thinker as early as 2,200 years ago. Euclid was succeeded as librarian and rector of the university by the great thinker Eratosthenes.He was an astronomer, mathematician, and geographer, and he became famous throughout the ages for his amazing feat of measuring the size of the earth.This measurement was made in the third century B.C., before anyone in the world suspected that the earth was spherical.The details of the measurements are worth mentioning, for they represent a perfect, albeit naive, scientific thought. There is a city in Egypt called Sienni, about five hundred and sixty miles south of Alexandria, where there is a deep well.Eratosthenes knew that at noon on the longest day of the year in Cyone, the sun was just overhead.At this moment, the sun shines directly into the deep well, and the sun is reflected on the water at the bottom of the well.Eratosthenes Denise was a mathematician who knew angles and circles and believed that the earth was a sphere, so he thought about the fact that the sun was shining at the bottom of the well and made a plan.He measured the length of the shadow cast by an obelisk in Alexandria, at the time and day when the sun was still on the top of Erni.After finding out the length of the shadow and measuring the height of the obelisk, this Greek genius was able to calculate the volume of the earth! The graphics on this page show how easy this is.We may assume that Eratos Deneus drew a diagram of the obelisk and its projection.He draws a line from the top of the tower to the center of the earth, draws a line from the top of the tower to the tip of the shadow of the tower on the ground, and then measures the angle formed by the intersection of these two lines.This angle is seven and one-fifth of a degree.Since the sun was on top of the city of Sienni at that time and had no shadow, the relevant angle was zero degrees.From this it follows that the distance between Cyennes and Alexandria is 560 miles, equal to seven and one-fifth degrees of the circumference of the earth, assuming the earth is spherical.So, how many miles should three hundred and sixty degrees equal to the entire circumference of the earth?Eratosthenes Denise divided three hundred and sixty by seven and one fifth (that is, seven point two).Get fifty, multiply by five hundred and sixty, and the number is 28,000 miles, which is the total length of a circle around the earth along the equator in the middle of the earth. Because of inaccurate measurements of the distance between Cyene and Alexandria, Eratosthenes calculated a result of 3,000 miles more than the approximate figure of 25,000 we admit today.But this error in no way detracts from the intellectual merit of Eratos Denise.With the power of his mind alone, he measured the size of the earth, while at that time only a small part of the earth, the Mediterranean area, was known to him and to people in general.Eratosthenes Denise foreshadowed the scientific achievements of the millennia that followed. But a famous story from the University of Alexandria shows how these brilliant Greeks utterly despised the practical application of scientific knowledge.The person involved in this story may be the greatest mathematician in ancient times. He is Archimedes, nicknamed the Great Thinker.Eratos Denise deeply admired Archimedes' mathematical genius and invited him to leave his hometown Syracuse, the Greek colony in Sicily, to take up a teaching position at the University of Alexandria.The great Syracuse accepted the invitation.The exact circumstances of what happened to him in Alexandria are not known, but the information we have obtained at least shows the attitude of many famous and powerful scholars there. We must first understand that Archimedes was the lost lamb of ancient mathematics.He is eager to apply the principles of mathematics to practical applications. Today, the world knows that he invented many great machines, but he does not know much about his major contributions to pure mathematics.Soon after he arrived in Egypt, he invented the so-called water screw, which relieved the great labor of people and livestock from pumping water from the Nile River to the bank to irrigate the fields.The main part of this ingenious thing is a large wooden spiral mounted inside a long wooden cylinder.The bottom of the cylinder extends into the Nile River, and as long as the screw is turned, the water can be lifted to the top of the cylinder, and the water flows continuously.The working method of this water screw is very similar to the screw in the meat grinder, which pushes the meat to the knife edge to grind and squeeze out.The water spiral is propelled by a water wheel, which itself is propelled by the current of the river.Thus, the majestic Nile was forced to do its own work, feeding water into Egypt's innumerable irrigation canals.What's more worth mentioning is that although the water pump of Archimedes was invented more than two thousand years ago, Egyptian farmers are still using it today. Archimedes had to play some mechanical gadgets outside of the course, which is likely to arouse objections from other professors.But they must have been shocked when Archimedes came up with an answer to one of the hardest problems in mathematics of the time, that of how to calculate the volume of circular objects such as cones, cylinders and spheres.Archimedes conjectured: If the above three shapes are normal and their bases and heights are equal, then the volume ratio of the three must be one to two to three.He asked a local carpenter to make wooden models of these three objects for him to test his idea, and it worked.There are three cones, a semicircle, and a cylinder in the whole set, the bases of which are equal in size and height. With these props in hand, Archimedes invited his colleagues who were teaching at the time to a lecture, suggesting that he would announce an amazing mathematical discovery. When he read the title of his thesis, On the Volume of Circular Objects, his colleagues immediately became tense.For many years, they all tried to solve this problem, but they couldn't.Did this brash kid have an answer? Archimedes announced the result first, writing down the simple ratio: Cone ∥ 1 Semicircle ∥ 2 Cylinder ∥ 3 The audience fell silent.Everyone leaned forward, wanting to know how he could prove this astonishing claim.But Archimedes didn't come up with a long mathematical calculation, only took out his wooden props and a balance.He first weighed three cones and one cylinder on both sides of the balance, completely balanced.Then remove the two cones and replace them with a semicircle, and the balance remains balanced.Finally, he weighed two cones and a semicircle on both sides of the balance, and the weights were exactly equal again. But instead of cheers, Archimedes evoked a stern silence. A fourteen-year-old boy stood up at last.He was Apollonius of Pulong (place name), a famous mathematician who taught at the university when he was only a little over ten years old.He has long been known for his work on a set of curves and cones in advanced mathematics.He described the strange properties of these curves and gave them names: ellipse, parabola, and hyperbola.These astonishing achievements earned him a place on the university faculty as a teenager.But Apollonius stood up and said this: Mr. Principal, professors!I propose the permanent expulsion of Archimedes from the University of Alexandria, for defilement of the pure spirit of mathematics with filth. Archimedes could not stand in Alexandria, for he committed the worst crimes against the spirit of mathematics.Mathematical proofs can never be faked by experiments and can only be verified by pure reasoning. Archimedes had to return to his hometown of Syracuse.He continued his creative career in his hometown, discovered the principle of levers, invented pulleys, and proposed the theorem of floating bodies, which the world calls Archimedes' theorem.In his later years, he actually theoretically solved the problem that made him lose his teaching position at the University of Alexandria: he calculated the value of the number π (sound school) for the first time, and described its strange properties. ratio of the diameter.With it, the volume of a circular object can be calculated mathematically without resorting to a balance. But Archimedes never lost his taste for practical inventions.He was the first famous scientist in history to be enlisted to fight for his motherland. He invented a very powerful ballista, which fired a large number of stones at the Roman soldiers who besieged the city of Serracle under the command of General Marcellus.Archimedes built a lever-operated hook that hoisted Roman ships and threw them overboard to shatter them.He installed a huge parabolic mirror to concentrate the sun's rays and ignite the sails of enemy ships. Archimedes had this vision of practical invention, which is very close to our modern thinking.Even his mathematical papers read as if they were written by modern people.But it would be a mistake to think that Archimedes was just a genius inventor.He was a true Greek at heart, true to the mathematical spirit of the Greeks, and in fact he agreed with those who drove him out of the University of Alexandria.He never considered the ingenious weapon he had invented to be worthy of the true scientific spirit.He just called it the geometry of the game. The Roman biographer Plutarch, who lived about 1,900 years ago, wrote of Archimedes' admirable scientific attitude: He was of the highest spirit, the deepest soul, and the most profound scientific knowledge; therefore, although these inventions gave him the honor of being a superman of wisdom, he refused to send any written works on these subjects; Considering the mechanics of purpose and arts as base and unseemly, he devoted his whole inclination and aspirations to pure reflections, unmixed with the vulgar necessities of life; In research, the superiority of which is considered by all to be unquestionable, the only thing in which there can be any doubt is whether it is the beauty and greatness of the subject under study, or the precision and method of proof, which are most worthy of our admiration. Woolen cloth? Archimedes had to die.None of his ingenious weapons could hold back the superior force of Rome, or prevent the final fall of his native city.Marcellas had ordered the old eccentric to be captured alive, but a Roman soldier found him, not knowing who he was.At that time, Archimedes was still drawing geometric figures on the sand table in his yard, and was busy answering a geometric problem. When the soldiers strode on his figures, Archimedes shouted: Don't step on my circle!The soldier was furious and raised his knife; Archimedes only asked him to move slowly in order to complete his mathematical proof; the soldier chopped it down with a knife. Archimedes had already engraved some patterns on his tombstone to represent his greatest achievements; the figures were a sphere and a cylinder, the volumes of which he first calculated.After all, he was a greater mathematician than all those who looked down on him. Although the lofty ideas of these ancient Greek scholars are worthy of our admiration, their scientific research methods are severely restricted.After they had developed the basis of science, mathematics, which was developed solely by the power of the mind, they stopped making progress.In fact, if we want to understand and thus master the universe, we must use experiments.The laws of nature cannot be understood by mere abstract thinking; they must be discovered and finally proved by experiment. Fifteen hundred years after the death of these great classical philosophers, in the Renaissance era, a man appeared whose intellect was as excellent as the ancient philosophers, but in a different way: he was the famous Italian scientist Galileo.伽利雷（Galileo Galilei），舉世公認為是倡導實驗方法的人，他的倡導之功，超過同時代的任何人。這不是說別人就都沒有作過實驗。十三世紀的英國神秘家兼科學家羅哲．培根（Roger Bacon）進行過實驗性的研究，既巧妙又多種多樣。鍊金術士們，特別是其中登峰造極的偉大的帕拉塞爾薩斯（Paracelsus），都是實驗家。活動不休的文藝復興時代的彫刻家和繪畫家們，秉著他們史無前例的真實感和對大自然的今世感，不但用他們的鑿刀、油彩和畫筆來作實驗，還用上了透視法則。這些法則使藝術家能夠在一片平版的畫布上創造遠近的印象和三因次（長、寬、高）的幻覺。他們甚至利用了機械的觀測裝置，使藝術對象變形以便產生遠近的幻覺。著名的德國畫家阿爾布列赫特．杜萊爾（Albrecht Durer）曾創作大批有趣的木刻，透露繪畫方面運用的這種引起幻覺的技巧。他作了一個觀測架，是他的工具，用來瞄準所要畫的對象身上和個部位，圖畫快要完成時，打把架子轉到旁邊去。 因此，伽利略所代表的趨向實驗科學的新風氣，只是當時絕大多數有創造性的人所採取的新觀點的一方面而已。它出現於彫刻和繪畫中；還出現於音樂和建築中，事實上也出現於哲學中。人類的理解已轉向於現實的世界。這就是現代科學的開端。 我們回顧往昔，覺得進展很大。實驗方法解決了多麼了不起的天才！但若沒有創造數學的人首先表演純粹推理的力量的人的精神，這是絕對不可能的。