Such as the area formula of a circle. Kepler calculated the area of a circle by dividing it into triangles. Therefore, the area of a circle is the sum of the areas of each triangle. A circle can be divided into four triangles with two diameters. However, the edges of these triangles cannot approach the curve correctly (excluding some spaces), so the calculated area is wrong.
To reduce this error, we can draw more diameters to create more short triangles. Although this reduces the error, it is still not zero. Therefore, we further divide the circle into more and more triangles until no space is excluded. However, in order to completely eliminate this error, we must divide it into infinite triangles. Because a straight line can be interpreted as a part of a great circle, we can say that the circle is composed of an infinite number of straight lines, which is approximated by the infinitesimal base of an infinite triangle.
People may notice that the order of triangles is reminiscent of China fans. All triangles have the same area. We can change the sector into a big right triangle by dispersing or stretching this area. Their perimeter has changed, but the whole area is still the same. The vertex of this right triangle is the center of the circle, its height is the length of the sector, that is, the radius of the circle, and its base is the circumference of the circle. The area is 1/2 times the bottom times the height, that is, 1/2 times r times 2πr, which equals π r 2. This is the correct answer, but the result is still wrong. These bases must be very small, so even if Kepler drew very, very small triangles, we know he can draw more.
When he stopped drawing triangles, he left a space. Although it is indeed a tiny space, it is still not zero. The curve is not completely approximate, and the calculation of the area of the circle is a bit wrong. Although this may make mathematicians feel uncomfortable, most people ignore these differences.
The calculus independently invented or discovered by Leibniz and Newton is also based on infinitesimal. This branch of mathematics is about curves and changes. For example, when we integrate a function, we are actually calculating the area under the curve it draws. However, just like calculating the area of a circle, we calculate it by approximating an infinitesimal rectangular curve. The thinner the rectangle, the smaller the error.
The area of a rectangle is its length, that is, the value of that point on the curve on the Y axis times its width, which is what we call the infinitesimal unit of dx. We calculate the area of each rectangle and add them to determine the area under the curve. This is very useful in physics. For example, the area under the object velocity curve gives the displacement value, but the result should not be wrong, right? Like the area of a circle?
After the appearance of calculus, this intractable problem puzzled mathematicians for two centuries until the concept of "limit" was perfected. In the research of Newton and Leibniz, limits were absolute, but they were modified and redefined in the early19th century. These new ideas are rigorous and consistent in mathematics. Although the limit makes mathematicians finally get rid of infinitesimal, what we haven't solved is infinity.