Three carriages lined up in a row, driving quietly through the countryside of the Duchy of Valois.
More than twenty members of the Ancient Church of Abraham were seated on these three carriages. "Chariot Climbing Technique" was also cut into three parts, held by the people on the three carriages respectively. On the one hand, this is to speed up the deciphering of cipher text, and on the other hand, it is to prevent someone from trying to monopolize the "Chariot Ascension Technique".
According to Ella, it has been a full month since they were magically teleported to Île-de-France. With the joint efforts of more than twenty members of the Ancient Abrahamic Church, the deciphering work of "Chariot Techniques for Ascending to Heaven" is nearing completion.
But in terms of space, they were still circling around the Île-de-France - although they were not caught by the Earl of Île-de-France's people, they did not get close to Stade at all. If you draw the route they took on a map, you will find that rather than escaping from Île-de-France, they are more like conducting a carpet search around Île-de-France.
No one came forward to explain this strange route. Because the planner of this weird route, Ella Cornelius Scipio, was doing math problems from beginning to end.
In one month, Ella's mathematics research has made great progress.
Using the inspiration from the spider's dream, Ella drew two horizontal and vertical number axes at ninety-degree angles on the white paper.
Ella named the system formed by these two number axes coordinate axes.
In this way, any point on the white paper can be represented by the coordinates of a number. Any geometric figure is composed of infinite points. In other words, using this coordinate axis, any geometric figure can be converted into numbers.
——Geometry and numbers are unified on this basis.
Ella felt that she had taken a big step towards the Pythagorean concept of "everything is number", and she couldn't help but want to tell Gottfried this discovery several times.
However, Gottfried, who was working on cracking the "Chariot Climbing Technique", had no intention of listening to what Ella was chattering about. Ella ran over several times, but Gottfried ignored her.
Later, whenever Ella appeared in front of Gottfried's carriage, the people next to him would shoo her away like flies without Gottfried even saying a word.
Habiba seemed to have seen a good opportunity to make money. He came over to Ella with a smile and said: "Miss, my apprentice has a bad temper and only listens to me. If you really want to learn mathematics, give me five nomis." Ma, I guarantee that he will teach you honestly. If he doesn't want to, I can tie him up and throw him into your room."
But even Habiba added in the end: "However, it will have to wait until we have decrypted the "Chariot Ascension Technique"."
According to this group of members of the Ancient Abrahamic Church, the Technique of Chariot Ascension to Heaven records the method of measuring the infinite gods. By learning it, you can understand the nature of the Supreme God and gain a power far beyond any kind of protection.
In order to get rid of the predicament of being chased by the apostles as soon as possible, they worked day and night to decipher the "Chariot Ascension Technique", with an average of only three hours of sleep per person per day. After this month, they have reached their limit and have no time to worry about math problems.
Ella could only retreat to the corner of the carriage angrily, and continued to write and draw on the paper alone. In retaliation, when someone asked her why she took this route, she always said perfunctorily: "Wait until I finish this question."
During this time, she expressed all common geometric figures using functional expressions based on coordinate axes. Then, the problem comes back to that parabola.
A parabola is a curve. Experience tells Ella that whenever the problem is related to a curve, the difficulty will suddenly become greater.
Through the coordinate axes, Ella can already describe various curves with numbers. In order to give herself some confidence, she first chose the simplest parabola: y=x2 for research.
She made a straight line y=1, which intersects the parabola at point a. In this way, the three lines of parabola, straight line and x-axis form an irregular geometric figure.
Ella wants to find the area of this irregular shape.
She found points on the parabola and made two lines perpendicular to the x-axis and y-axis, thus dividing the irregular figure into rectangles. The combined area of these rectangles is obviously greater than the area of the irregular figure. However, the finer you divide these rectangles, the closer their area will be to the irregular shape.
Ella assumes that n rectangles are divided from the origin of the coordinate axis to the straight line y=1, then the width of each rectangle is 1/n. And because the functional formula of the parabola is y=x2, then the height of the first rectangle is (1/n) 2, and the height of the second rectangle is (2/n) 2...
Then, the sum of the areas of all rectangles is:
s=1/n×(1/n)2+1/n×(2/n)2+……+1/n×(n/n)2
This is an infinite series. However, Gottfried had taught Ella the formula for the sum of squares of infinite polynomials. After using this formula to simplify the infinite series, she obtained an extremely simple formula:
s=1/3+1/（2n）+1/（6n2）
The larger n is, the closer the area sum of the rectangle is to the irregular shape. Then when n is infinite, the sum of the areas of the rectangles, s, will be equal to the area of the irregular figure. At this time, 1/(2n) and 1/(6n2) are infinitesimally small and can be completely discarded.
So the area of this irregular figure is obvious: s=1/3.
——Infinitely large and infinitely small
Ella whispered the two concepts that had just appeared. The concept of infinity appeared in mathematical operations, which made her feel somewhat uncomfortable.
She shook her head to put the discomfort behind her, and then changed the functional formula from y=x2 to y=x3
Although it was only a slight change, it immediately became several times more difficult to find the area. This time, Ella wrote two full pages, but failed to simplify the formula as before.
“Why is it that when it comes to curves, there is always infinity!”
Ella dropped her pen, held her head and cried.
Infinity is an abyss that is difficult for all mathematicians to cross.
Parabolas and circles are just the simplest curves, just a small branch sticking out from the edge of the infinite abyss. Ella grabbed the twig. But when she continued to look down, what she saw was an even more terrifying abyss - using coordinate axes and functional expressions, she found many complex curves that Archimedes could not describe at all.
She discovered them, but couldn't control them at all. This seems to be a warning from God: People, do what you should do!
Infinite, this is a forbidden area that humans must not enter.
"Pythagorean magic is too difficult to learn!"
Ella yelled again.
"Keep your voice down!"
The people of the Abrahamic Orthodox Church cast dissatisfied looks at Ella, who was so scared that Ella hurriedly covered her mouth.
