Science,English,and math 4th

【Pythagoras and a secret art on numbers】

《A discovery by Pythagoras》

We easily imagine that Pythagoras and his disciples were surprised and impressed with the fact very much. A musical interval which means difference two sounds between high and low. We the human being thought it was beautiful automatically, and it was expressed with simple integral numbers.

They had a feeling as if they had discovered a trick made by the God, didn’t they? If they thought the numbers which the God used for the trick, it means the integral number, was the words from the God, it may be a matter of course.

【Pythagoras and a secret art on numbers】

《Adding the numbers on the date of its birth and telling the fortune》

Actually Pythagoras and his disciples learned to think the source from everything was the number soon and to have a faith in the integral numbers themselves as if they had been the God. It developed into the secret art on numbers by Pythagoras. What was it?

The secret art gave meaning numbers from 1 to 10. The secret art on numbers is one of fortune telling like the fortune telling in Western and the art of divination. Each of meanings on the secret art of numbers in modern era...

【Pythagoras and secret art on numbers】

《Adding number on the date of its birth and telling the fortune》

Each meaning on the number defined by the modern secret art in the modern era is different, according to its school. The meaning done by Pythagoras and his disciples was like the next.

1 reason 2 woman 3 man 4 justice, truth 5 marriage 6 love and soul 7 happiness 8 essence and love 9 ideal and ambition 10 holy number

The most general way of telling fortune using the secret art on numbers by Pythagoras is adding all the numbers on the date of one’s birth, and we got a number...

【Pythagoras and secret art on numbers】

《Adding all the numbers on the date of one’s birth and telling fortune》

...and we got a number in two-digit, and still more adding the number of each digit and thinking of the meaning on the last number.

For example, there is the one who was born on the 18th day June in 1974, 1＋9＋7＋4＋7＋1＋8 =37⇒3＋7=10. It means the holy number and the book says it means the perfect, or cosmology.

We can use the number and calculate. For example, 2 ＋ 3 = 5 , a man plus a woman is equal a marriage. 2 × 3 ＝ 6, multiple a man by a woman is equal a love. 4 ＋ 5 = 9...

【Pythagoras and the secret art on numbers】

《Adding all the numbers on the date of one’s birth and telling the fortune》

...and 4 ＋ 5 = 9 means justice adds to marriage is equal an ideal.

The author said it was a good device, wasn’t it? And said he wants us to try to have enjoy other combination.

Needless to say we don’t always have to take the meaning each number on the secret art by Pythagoras as very serious, but if we success in being familiar with the numbers like that, it will be meaningful.

《Why was his disciple murdered?》

While the numbers had been deified, Hippasus who was one...

【Pythagoras and secret arts on numbers】

《Why was his disciple murdered?》

While the numbers had been deified, Hippasus who was one of disciple of Pythagoras was aware something important. He discovered he couldn’t express an oblique side of a right angle isosceles 2等辺 triangle with any numbers which they discovered until then.

The numbers means rational numbers which are expresses with fraction of rational number on both a denominator and a numerator.

Besides, ironically it turned out that the existence of the number was clear by the Pythagoras theorem.

When hearing the report from...

【Pythagoras and secret arts on numbers】

《Why was a disciple of Pythagoras murdered?》

When hearing the report from Hippasus, Pythagoras and all his entire disciples tried to verify and reached a conclusion that the report from Hippasus was right.

There is another version of the story which holds that when hearing it, Pythagoras was very frightened at it, and he ordered all his disciples not to leak out the existence of the numbers anyone, and to kill Hippasus....

We call the numbers which don’t belong to the rational numbers irrational numbers at present. As to the irrational number...

【Pythagoras and secret arts on numbers】

《Why was his disciple murdered?》

If it was the fact, why did Pythagoras have to murder his disciple then?

As to an irrational number, for example when trying to express the square root of seven with numerical numbers, it’s 1.141421356...as irregular numbers continue in decimals forever, so expressing its number exactly is hard, but the irrational number exists definitely in our world as the length of the oblique side on the right isosceles angle triangle, of which triangle is between the same length of two sides.

On the other hand, in the era of...

【Pythagoras and his secret art on numbers】

《Why was his disciple murdered?》

On the other hand an era of the Ancient Greece was the time when various facts had been verified exactly by math for the first time. The author said he’s sure that lots of the people thought there was nothing more definite than math.

Pythagoras was in the center of the time, so he couldn’t forgive the existence of the irrational number which he couldn’t understand its precise value.

To be exact, the number which continued irregular ones decimals forever must not have existed for the great mathematician, for...

【Pythagoras and secret arts on numbers】

《Why was his disciple was murdered?》

...for it was against his sense of beauty because math should be simple for him.

By the way, it proved that irrational numbers are by far more than the rational numbers at present.

《An unexpected relation between mathematicians and a tune》

Pythagoras and his disciples invented musical scales like what is called do re mi fa singing method from the discovery on a relation between musical interval which was resonant wonderfully and mysterious integral numbers.

Putting musical scales from the first one, do to the...

【Pythagoras and secret arts on numbers】

《An unexpected relation between mathematicians and tunes》

Putting various sounds between the first do to the last do from the musical scales according to the degree on the sounds. We call its rule a tune.

There was a musical interval of do and so when the proportion on the weight of the hammers was 3 to 2, and as being based on the musical interval, Pythagoras and his disciples invented tunes.

To tell the truth, a way of specifying tunes is various. There have never been perfect tunes until at present in other words.

The perfect tunes mean that...

【Pythagoras and secret arts on numbers】

《Unexpected relation between mathematicians and tunes》

The perfect tunes mean that when several sounds resonate at the same time, a pleasure of the resonating and the one when listening to as melody are consistent.

If trying to invent the tunes, we need knowledge on geometric progression 等比数列, or radical root, so there is a deep relation between mathematicians and tunes.

In fact, Kepler and Euler left original tunes, and Japanese mathematician named Genkei Nakane 中根元圭 who was born in Edo era invented tunes divided into 12 octaves. It’s temparment.

【Pythagoras and secret arts on numbers】

《An unexpected relation between mathematicians and tunes》

Temperament means 平均律, though I can express it in English, but to my sorrow I have none of knowledge on it. First of all, as to the music, all we have to do is enjoying ourselves, so we don’t need theory which is hard to understand, I’m sure.

【Math used to be music, or astronomy?】

《An origin of a word on math》

The author said he was going to express on the origin of the word of math.

数学 was used as translation from mathematics in a foreign book written in China in the 19th century for the...

【Math used to be music or astronomy?】

《An origin on a word of math》

The word of 数学 was used as a translation from a foreign book written in China in the 19th century for the first time.

The word was used in 英和対訳袖珍辞書 which was issued the first orthodox English Japanese dictionary in 1862.

How should we translate the word of math? It seemed that there had been various opinions then.

As to its meaning, arithmetic resembles math, and it’s frequently translated into 算数, but it’s not accurate.

Arithmetic is one of fields on math in which study on the calculation using four basic operation.

【Math used to be music and astronomy?】

《An origin on a word of math》

The four basic operation is addition, subtraction, multiplication, and division.

《The translation which doesn’t fit nicely somehow》

The author said that he could understand how the people took pains when translating then, but the translation on 数学 from math struck him as incongruous, for the field which the math handles hasn’t been limited numbers alone.

The number is a conception which shows us an order or an amount on something. It started from natural numbers like 1.2.3..., but it’s expanded its area like decimal...

【Math used to be music or astronomy?】

《The translation which doesn’t fit nicely somehow》

...but it’s expanded its area like decimal, a fraction, and irrational number and it means real number and the whole imaginary numbers at present. As the number is a conception which is abstracted, so it doesn’t have any unit.

On the other hand, quantity is an objection for measure like length, area, volume, angle, weight, time and speed. The quantity has an unit basically. In geometry when asking the value on length of a side or on area of a figure, it means that we measure the quantity exactly.

【Math used to be music or astronomy?】

《The translation which doesn’t fit somehow》

It’s sure that not only the number but the quantity have been in the center on math from the time of Ancient Greece. Then should we translate it into a learning on number and quantity? The author said it’s not enough yet.

When a function which some amounts cause other amounts appears in 17th century, lots of mathematicians began to take in interest for studying change between them closely.

When saying y is the function for x, it means that y is the number which is fixed with x. The author says when imaging...

【Math used to be music or astronomy?】

《The translation which doesn’t fit nicely somehow》

The author said if imaging that putting into a value of x on an instrument, we can get other value of y. It’s easy to understand.

As the x is changed into the y with the instrument of the function, the conception on function was born, as a result, lots of the people, including mathematicians and other ones who aren’t always interested in math started pay attention to the change.

After that, a big field of mathematical analysis of which center is differential and integral calculus has been established.

【Math used to be music or astronomy?】

《The translation which doesn’t fit nicely somehow》

Then the area which math handles has expanded so much that it covers natural science, so it is no exaggeration to say that dealing with the change has established the present position on the math.

Euclid made rules on nature which the space has as axiom for the first time in his book, the Elements of Euclid, in the time of Ancient Greece.

For example, the shortest distance between two points is a straight line, or two straight lines parallel each other don’t cross forever.

It’s so natural that it...

【Math used to be music or astronomy】

《The translation doesn’t fit nicely somehow》

It’s so natural that it seems that we don’t have to say, but mathematicians can define other special space which doesn’t have the nature like that, so thinking of the space which doesn’t belong to the Elements of Euclid led to a new math and new science.

I’m wondering what kind of space it is.

《Area where math handles is by far larger》

Considering those things, the translation of 数学 from math isn’t enough, the author said.

When being a student in high school, he used to have some doubts why the word of...

【Math used to be music or astronomy】

《An area where math deals with is by far larger》

When being a student in high school, the author used to have some doubts why 数学 was adopted.

Instead of numbers, some letters have been used in equations. It’s algebra, 代数学 He knew it, so 代数学 was shorten and it changed into 数学 he thought like that, but it means that function, geometry and probability aren’t included. He seemed to be worried about it.

Actually the math deals with the number, quantity, space, change and construction, and math has been put to practical use at the wide variety.

There is a...

【Math used to be music and astronomy】

《The area where math deals with is by far larger》

When there is a question that what math is, if trying to answer, it is hard not only mathematically but philosophically at present.

There is a field of learning that philosophical study on an object or a way for math has been done. We call it mathematical philosophy.

《Arithmetic, music, geometry and astronomy》

In that meaning, the origin of the word on math is something which they should learn in Greek, I’m interested in it, the author said like that.

Something which they should learn in the..

【Math used to be music or astronomy】

《Arithmetic, music, geometry and astronomy》

Something which they should learn in the Ancient Greece is arithmetic which is quiet number, music which is active number, geometry which is quiet quantity, and astronomy which is active quantity.

Platoon used to open an academia for himself, and when fixing its curriculum, he thought the ones whose age from 16 to 17 should be trained so as to learn philosophical questions and answers, so he prepared for those four subjects.

As he was influenced over by Pythagoras and his sects so greatly that he made much...

【Math used to be music or astronomy】

《Arithmetic, music, geometry and astronomy》

Pythagoras and his sects had such a tremendous impact on Platoon that Platoon made much of those four subjects. Why were they divided into four?

At first Pythagoras divided into two, number and quantity, and after that separated into something quiet and something active for each of two.

Quiet means itself and active is the one which changes according to its relation with other.

An ability on algebra which they learn number itself, which means something quiet, is the base on everything, so it’s one of...

【Math used to be music or astronomy】

《Arithmetic, music, geometry and astronomy》

...so it’s one of things which they should learn, none of us raise an object against it.

Pythagoras and his disciples accomplished remarkable outcome in the field of geometry where they learn on figures which means something quantitative, at the same time established a logical way on thinking.

Then learning geometry means learning a logical way of thinking of something then, if trying to learn philosophy, geometry was indispensable. It was natural for them then.

《Mystery on numbers and music in the...》

【Math used to be music or astronomy】

《Something mysterious on numbers and music in celestial sphere》

Pythagoras and his disciples were aware that something mysterious was hiding in beautiful consonance, and expounded everything was numbers. They enlightened that the cosmos was made of harmony which was caused a relation between numbers.

Their enlightenment made the people in the Ancient Greece think the foundamental principle on the cosmos was ムジカ and its harmony was ハルモニア, ムジカ means music, and ハルモニア harmony in English.

To tell the truth, music had been thought as rather something...

【Math used to be music or astronomy】

《Something mysterious on numbers and music in celestial sphere》

In fact music had been thought as something symbolic for for order and harmony rather than an entertainment until the Middle Ages. As a result, they learned music so as to grasp the principle in cosmos. In other words, music was indispensable for them to understand the language from the God.

Besides, Pythagoras created an idea of a music on celestial sphere.

Then they thought all the stars were fixed to a gigantic spherical surface of which center was the earth, and the rotation of the...

【Math used to be music or astronomy】

《Something mysterious on numbers and music in celestial sphere》

...and its rotation on the spherical surface made all the stars move, they thought like that then and it’s Ptolemaic theory 天動説.

In the viewpoint of the Ptolemaic theory, a movement on planets was so mysterious, and when trying to express it with a relation between figures which means active quantity, they need to contrive a complicated geometry on spherical surface.

Even if it looked very complicated, a heavenly body revolved on a harmonious orbit, so the cosmos should be filled with...

【Math used to be music or astronomy】

《Something mysterious on numbers and music on celestial sphere》

...so the cosmos should be filled with beautiful music on the celestial sphere which we the humankind couldn’t listen to, Pythagoras and his disciples thought like that.

Platoon set arithmetic, music, geometry, and astronomy as required subject in the Academia where he established for himself, but the one who had a strong spirit for being a leader in the next era should learn with its own free will, so it shouldn’t be forced to do from anyone, Platoon thought like that.

The four required...

【Math used to be music or astronomy】

《Something mysterious on numbers and music on celestial sphere》

Platoon set the required subjects like arithmetic, music, geometry and astronomy, and it was called マテマータ in the Ancient Greek and it was an origin of word on mathematics, and the マテマータ learned to be called liberal arts in English later. It means several technical skills which should be acquired with free will.

IT and AI have tried to change the industrial structure in a large scale at present, so some people seem to call it the fourth industrial revolution, so math is indispensable which...

【Math used to be music and astronomy】

《To liberal arts》

...so math is indispensable which we should learn. It’s returned to its original meaning.

It seems that math used to tend to be learned by the ones who adopted science course or others who were good at it alone when learning math or making use of it as if they had hidden from others people. What we call オタク in Japanese slang.

But from now on the ones who adopted the humanities and others who were not good at math won’t be able to avoid math. The author continued.

But I don’t think I want math to be the one which is forced to do ...

【Math used to be music or astronomy】

《To liberal arts》

I don’t think I want to math to be the one which is forced to learn from others. When learning everyone can find something fun from math, and I want math to be the liberal arts which we’re ready to tackle. I’ve believed in that math has something tolerable.

I’m sure the author has loved math.

【Welcome to a museum on curves】

《Nature is a scene which creates curves》

Hideki Yukawa who accepted a Nobel prize for physics as the Japanese for the first time used to say that nature creates curves, and we the humankind does straight lines.

【Welcome to a museum on curves】

《Nature is a scene which creates curves》

If looking around us, it is certain that almost all of the things of the straight line is something artificial like a pen, an end of a desk, or an outline on electric appliance, but needless to say, there is a thing which looks straight in the natural world like a cedar 杉, for its origin of the word is a straight tree, according to the author, though I’ve never heard of it until now.

But strictly speaking, the cedar isn’t the straight line. Both a stone, a flower, a mountain, and a cloud are made of complicated...

【Welcome to a museum on curves】

《Nature is a scene which creates curves》

Both a stone, a flower, a mountain and a cloud are made with complicated combination by curves, but in spite of it, except for a circle, we the humankind have never paid attention to other curves for a long time.

After Pythagoras, it has been dots, street lines and circles which geometry treated, except for a single exception, Aplllonios, who played an active role in the era of the Ancient Greece.

He’d studied a cut end of a cone when it was cut not through its vertex minutely, and named three kinds of curves which...

【Welcome to a museum on curves】

《Nature is a scene which creates curves》

...and named three kinds of curves which appeared as the cut end of the cone then. The one is ellipsis, and the other parabole and the last was hyperbole. Each of them is the origin of word, ellipse 楕円, parabola, and hyperbola in English.

Why did Apollonios call like that?

There is an illustration on the cone when it was cut, then the cone was illustrated as an isosceles triangle 二等辺三角形, and the illustration was the one when we look at the each of cut end from right at our side.

When cutting the isosceles triangle...

【Welcome to a museum on curves】

《Nature is a scene which creates curves》

Roughly speaking,when the isosceles triangle was cut, three other isosceles triangles appear, according to the cut end. As to the three isosceles triangles, when comparing the angle of the cut end with the one of the base angle, there are three situations.

First is both of the angles are the same, and the second is the angle of the cut end is smaller than the one on the base angle, and the third one is the cut end is larger. Then we can distinguish difference on each curve.

When the angle on the cut end was small...

【Welcome to a museum on curves】

《Nature is a scene which creates a curve》

When the cut end was smaller than the base angle, its curve was an ellipse, the cut end was the same, its curve was a parabola, and it was larger, a hyperbola 双曲線.

When the cut end was parallel to the base, a circle appears in the cut end, but the circle has been thought as a kind of ellipse in math, the author said like that.

《Equations on curves》

Collecting, a circle, an ellipse, a parabola, and hyperbola together, and we call them quadratic curves, for equations for each of curves are expressed with quadratic...

【Welcome to a museum on curves】

《Equations on curves》

...for the equations for each of curves are expressed with quadratic equations of x squared or y squared.

The author expressed equations on curves, but we can do because Descartes introduced coordinates 座標, a coordinate axis, and variables into math in the 17th century, so we can express the curves with the equations.

The coordinates means a single point on a plain or in the space are expressed with a pair of numbers or each of combination. The single point on the plain is expressed with a pair of numbers（2.1）and the one in the space...

【Welcome to a museum on curves】

《Equations on curves》

...and the single point in the space like （2.4.3）is expressed with three combinations on numbers.

The single point and the coordinates are dealt with one to one, and there is a straight line on its base, and we call it the coordinate axis.

We’ve learned the way of expressing the point on the plain or in the space using the coordinates in junior high school or in high school, so we find it natural at present.

However every kind of point in the plain or in the space is able to be expressed with a pair of numbers or there combinations...

【Welcome to a museum on curves】

《Equations on curves》

However every kind of point on the plain or in the space is able to be expressed with a pair of numbers or three combinations on numbers was a novel and epoch-making then.

Descartes gave some letters each role which contained some values, and called the values variables 変数.

For example, x and y are variables, let’s suppose there was an equation, x ＋ y = 2, then there are some coordinates on equation of x ＋ y = 2 like the next.

（x,y）＝（1,1）,（x,y）＝ （2、0）,（x, y）＝ （0, 2）

When connecting each of three points, there is a straight line on...

【Welcome to a museum on curves】

《Equations on curves》

When connecting each point on the coordinates, there was a straight line, so Descartes thought the equation of x ＋ y ＝ 2 is the straight line expressing the numerical formula.

At last we the humankind succeeded in connecting a figure with an equation. It was so epoch-making that it was said it was an evolution on the history of math.

Descartes has not only made us express curves which have already been existed in the world but made us create characteristic curves on mathematically or physically from numerical formulas.

【Welcome to a museum on curves】

《Gaudiy and Sagrada Familia》

Apollonios discovered curves on cone which appears as cut end of cone and about 1700 years passed after it, so it learned to be expressed with a numerical formula, and one of them is a parabola, and it turns out that it corresponds to an orbit of throwing something.

While as to the Pythagorean theorem, if using a circle, it seems to be easy to understand it, though I have little knowledge on it, if trying to demonstrate a theorem on Fermat, curves on ellipse is deeply related to it.

What is the theorem on Fermat?

When some ..

【Welcome to a museum on curves】

《Gaudiy and Sagrada Familia》

When some number is bigger than 3, natural numbers of x, y, and z which meet an equation of x cubed ＋ y cubed = z cubed don’t exist.

Though it took no less than 350 years so as to demonstrate the theorem on Fermat, the great feat was accomplished because numerical formulas and curves were connected.

I’ve wanted to express it some day, but to my sorrow I find it impossible with my own ability on math at present.

The author said he was going to show us two curves and each of equation, which we’ve never learned in junior high...

【Welcome to a museum on curves】

《Gaudiy and Sagrada Familia》

...which never appears in junior high school nor in high school.

『Catenary』

When holding both ends of a string or chain of which density is fixed, there is a curve, and we call it a catenary. The density means a unit on weight for a single cube. Catenary is 懸垂曲線 in Japanese.

When holding both ends of the chain, there is a curve made by the chain, and we’d thought it was a parabola for a long time, but Christian Huygens in Holland demonstrated it wasn’t the parabola but the catenary.

We can see lots of catenaries in both...

《Gaudiy and Sagrada Familia》

『Catenary』

We can see lots of catenaries in artificial things created by us the humankind or natural thing which were created by nature like a power transmission line, a suspension bridge, and a spiderweb.

When the catenary is turned upside down it prove to be stable physically, so when holding both ends of the chain, there is the curve, and the curve is turned upside down, it’s stable.

As a result a design of shape is arch that the catenary is turned upside down has been adopted for lots of buildings, especially Antonio Gaudiy who is a typical architect...

《Gaudiy and Sagrada Familia》

『Catenary』

...especially Antonio Gaudiy who is a typical architect from Spain adopted catenary for lots of his works and it has been well known.

Gaudiy thought a beautiful shape has a stable structure, so we have to learn the structure from nature, so his design wasn’t decided by a desk plan but actual experiments.

When designing curves on buildings, he used numberless weights made of sandbag and string, and created curves of catenary, and he thought the curves of catenary were the most natural and strongest structure against vertical weighting.

He had ...

《Gaudiy and Sagrada Familia》

『Catenary』

Gaudy had a absolute confidence on design of the structural strength, so when workmen feared for accumulating tremendous stone like arch, he got rid of scaffoldings for himself, and demonstrated the design was safe.

An experimental instrument which Gaudy used for the design was called Fujikura, and we can look at it in archives beside the Sagrada Familia.

The equation is expressed in the book, but it’s too complicated to express here. It’s regrettable.

《A roller coaster and spirals on Euler》

『Clothoid』

It was in 1895 when Flip Flap appeared in...

《A roller coaster and curves on Euler》

『Clothoid』

It was in 1895 when Flip Flap appeared at suburbs of New York in America for the first time. Flip Flap was a roller coaster which rotates vertically.

A crowd who loved something new gathered and after the roller coaster started to drive with passengers, some of them suffered from whiplash injury むちうち症, and others were hurt on the neck, so lots of them were damaged.

Why was it caused?

Because the shape on the rail which rotated a single time vertically was almost a circle. If connecting the part of the circle with the part of the...

《A roller coaster and curves on Euler》

『Clothoid』

If connecting the part of the circle with the part of the straight line, passengers suffered from violent burden in the place where the rail changed from the curve to the straight line.

Instead of the circle, a curve called Clothoid was adopted for the roller coaster which rotated vertically in order to prevent the dangerous situation form happening.

Leo hard Euler in Germany studied the curves minutely, so it’s also called spirals by Euler. Clothoid started from a straight line, and the more it go ahead, the sharper the way of curving.

《A roller coaster and curves on Euler》

『Clothoid』

If comparing the Clothoid with driving a car, when driving at a fixed speed and turning the steering wheel with the fixed ratio, curves are drawn by the car. We call the curve the Clothoid.

If we have to drive form a straight line to curve composed of a circle and to the straight line again, the driver has to cut the steering wheel in a hurry both at first and the last of the curve.

Unless the driver decreases its speed so less, not only driving the car is hard to do but physical burden to the passengers is serious.

On the other hand...