A tangent line only intersects a curve at one point.
It takes two points to determine a straight line.
However, there is only one tangent line through a point.
These three sentences are completely fine when viewed separately, but when connected together, Ella feels logically confused. Countless straight lines can be drawn through a point, but the tangent line passing through this point is unique. Isn't this true?
Some questions?
After some thinking, Ella figured out the reason: the properties of each individual in infinity are different, and there is only one with the property of 'tangent'.
In other words, the tangent line is the only one among infinity.
But the problem is that without two points, people cannot make a definite straight line. And if we use the elimination method to eliminate all the infinite possibilities and work backwards to find the only one among them, it sounds like only gods can do it.
.
"I know the tangent is there, and I can understand all its properties, but I can't make it?"
This reminded Ella of the entire natural world - people can understand the properties of water, the properties of air, and the properties of soil, but people cannot create water, create air, or create soil.
This sense of powerlessness made Ella begin to feel that the teachings of the Abrahamic Church were correct - God has arranged everything in the world, and humans can only watch and cannot imitate.
"No, if you admit this, you will never be able to learn Pythagorean magic!"
Ella patted her cheek. Although she was in a spiritual state now, this action itself could cheer her up.
"Ella, don't be afraid of infinity!" she encouraged herself. "Since two points can determine a straight line, then just find two points!"
If we really find two points on a straight line, the line we draw will be a secant line instead of a tangent line. But Ella knows that as the two points continue to approach, the secant line we draw will continue to be close to the tangent line.
With previous experience in finding the area of a curve, Ella quickly thought of a similar trick - making the distance between the two points infinitely close, but not equal to zero. This number must be small enough, otherwise the line will be drawn.
There is a deviation from the tangent line; at the same time, it must not be equal to zero, otherwise there will be only one point left and the straight line cannot be determined.
Ella expressed these two points like this: (x, y), (xdx, ydy), Among them, dx and dy represent the numbers that are infinitely close to zero but not equal to zero. Just calculate the tangent line through these two points
slope, you can find this tangent line.
After putting these numbers into the curve of y=x2, the formula is very simple, unlike the infinite series that is involved in finding the area of the curve that adds up to infinity.
Ella tried changing the function to y=x3, y=x4, y=x5, but the difficulty of calculation did not change much.
"This is too simple!"
Ella shouted happily. She wanted to share this discovery with others and wanted to kill fifty cows to celebrate this discovery!
However, no one paid attention to Ella.
The members of the Ancient Abrahamic Church had re-translated "The Technique of Chariots to the Sky" from memory. For several days, like Jochebed, they buried their heads deeply between their knees and chanted it over and over again.
Protective mantras, trying to understand the miracles of God.
This scene seemed familiar to Ella - when she was in Constantiniya, whenever she tried to explain her new discoveries about celestial bodies to people, they would always lower their heads in panic and use vague words to look towards the gods in fear.
prayer.
Whenever this happens, Ella will feel that she is an alien, the ugly duckling among a group of white swans.
She lowered her head in disinterest. Looking at the manuscript in front of her, she suddenly felt a little doubtful - did these things have any value? This seemed to be just a trick of playing with numbers. The clown on the stage could also use his tricks to amuse people.
Make everyone happy, but this trick can't even do that.
Someone came in front of her, but she had no intention of looking up. No matter how many people gathered around her, all she felt was boundless loneliness, just like every time she observed the starry sky.
A drop of water - or something like water dripped on the empty manuscript paper. The person who came to her looked at the things on her manuscript paper carefully and said: "It's amazing."
"Do you understand? It's interesting, right?"
Ella suddenly became excited. But when she raised her head, she saw a terrifying sight - the face of the spirit in front of her was covered with red corpse spots, there were diatoms stuck to her hair, and there were incisions in her nose and nails.
It was full of mud and sand, and the skin all over the body shriveled as if it had been soaked in water for too long.
Ella screamed in fright and hurriedly moved back a few steps, but the spirit pursued him relentlessly. He tilted his head, staring at the manuscript paper in Ella's hand with extremely empty eyes, and mouthed
Li repeated two words over and over: "Incomprehensible, amazing; incomprehensible, amazing; incomprehensible, amazing..."
"You...who are you?"
The spirit mechanically moved its gaze from the manuscript paper to Ella. The crazy rereading stopped, he smiled strangely, and suddenly raised a hand to grab Ella's face.
Ella was so frightened that she let out another scream. Through the transparent spirit body, she saw that the members of the Abrahamic Ancient Church still kept their heads buried in their knees, and actually screamed completely at her twice.
Indifferent.
Ella smiled miserably: "Ah... So that's it... I'm completely... just a superfluous person."
The spirit caught her. In an instant, the tragic memories from that spirit poured into Ella's heart——
"Sibsos, look what has been proved?"
"Sibsos, you unleashed the devil!"
"You shouldn't have entered this realm! Sibersos!"
——Transparent sea water, illusory sunshine.
There were also ugly fish swimming in front of her eyes.
The biting seawater stimulated every nerve in her. She felt suffocated, but could not struggle, because her hands and feet were tied to death with ropes. She felt frightened, but could not scream, because the seawater had filled her esophagus and stomach.
A group of mathematicians looked at her without mercy on the boat. The leader was Pythagoras. His voice penetrated through the sea water and penetrated every bone in Ella's body: "Sink to the bottom of the sea... along with
That pathetic proof of yours."
In 500 BC, Sibersus discovered irrational numbers. It was the first time that humans understood infinity mathematically.
Then, he was thrown into the sea by members of the Pythagoreans.
She could do nothing but let her body fall into the abyss of the sea, watching helplessly as the sky got further and further away from her.
The ship full of mathematicians continued to float on the sea, like a carefree fish.
…
"Come back quickly, that spirit body has been blown up by your memory. I'm not telling you that the amount of your memories... is really unbearable."
"But thanks to this, I know what you're doing."
Gottfried's voice penetrated the water and dragged Ella.
"This is a great job that has never been done before. Come on, I will help you complete it."
