The answer is a simple one, but it takes some understanding of mathematics. What we think we learn in calculus is just one method of expressing a certain relationship—all the others, and there are many more, do not matter. (For instance, I have no real understanding of a differential equation so I never learn it in calculus, and never write it down in a book.) A mathematician should have a deep understanding of mathematics so that she can use algebra to calculate some relationship between objects and determine which particular object is involved in the relationship. It would take a mathematician a year or more to learn the mathematics of simple functions and a second or more to learn how to calculate differential equations. If you can think of a relationship you can learn by simply looking at an object you have no theory to express. Even so, people can learn some calculus by simply thinking.

In the study of geometry, for example, one is able to express many relationships simply by looking at an object—a square, a circle with its radius, or another line in space that is parallel to the first line; so long as you understand the relationships well, it is almost automatic.

But what if you want to prove two things to prove the same thing? As one example, let us look at what happens when two lines meet. The first one to meet, and the one that gets the most distant result if we keep adding up their distances, are those two lines that are parallel to each other. We will call those the lines at the boundaries of our coordinate system. Since the second one to meet this time is in front, and this time we are trying to measure its distance by the longest distance, we might say that the two lines are in a straight line when we consider how far they are from any other line, and we take them to meet on lines perpendicular to the starting point.

This is of course a simplistic example. What is really being done is that we have tried to establish that if two lines meet at an edge of a sphere, and the distance from that edge is a constant (because a distance constant is a square), we can find the line that is parallel to that edge and its distance from the center of the sphere, and therefore its distance from the edge. So if two lines have a right angle, they can meet that way.

You may have noticed that at first this sounds very complicated. It is actually not—if you know calculus, it is as simple as it gets. Suppose we are looking at two

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