The way that they usually teach calculus is HORRIBLE. They start off with the fundamental theorem of calculus and all of this difficult math without relating it to anything which makes it very difficult to understand. I was a couple of weeks into my first calculus class (many many years ago) and started relating it to geometry and then everything clicked, and after that it was simple. Once you understand what you are doing, calculus is easy to understand.

My son is taking calculus this year in high school. I didn’t know if they had come up with a better way of teaching it in the past 30+ years or so since I took my first class but apparently they were doing the same old crap. I explained to him that the derivative is just figuring out the slope of a curve and an integral is just the area under the curve, and once he got the basic concepts he was like wow, this actually is pretty simple.

It’s not just all about geometry though. Calculus shows up all over the place in basic physics. Acceleration, velocity, and position all have an integral/differential relationship, for example. And in electronics, if you have a simple coil of wire, the current and voltage also have an integral/differential type of relationship (v = L di/dt). And a capacitor is similar (i = C dv/dt). So simple electronics 101 type stuff dives right into calculus. When you get into more advanced things like feedback control, PID loops are a very common type of control. PID is Proportional Integral and Differential. Basically, in the feedback loop you measure the proportional error, then you add in an integral of the error to correct for any long-term errors in your control loop, and add in some differential error to make the system respond better to quick changes.

So yeah, the point is, it’s not just geometry. But if you relate it to geometry that makes it very easy to understand (IMHO).

As for why it took so long to develop calculus, the answer is that for a lot of stuff you don’t need it. You don’t need calculus to build pyramids. In fact, you can actually tell that they used very simple geometry to build the pyramids because the ratio of the length of the sides to the heights includes a factor of pi in it. This makes ancient astronaut theorists go off on strange ideas about aliens and advanced mathematics, but all it really means is that they took a wheel and attached a stick to it to mark off distances along the ground (google “measuring wheel” if you can’t picture what I mean here). If your wheel is one cubit in diameter, and you put a mark on one point on that wheel, then all you do is pace off the distance and count how many times that mark comes around. In other words, if you run your little wheel along the ground and you count ten ticks as the mark goes by, then the distance you traveled will end up being 10 x pi cubits. It’s just simple geometry. No advanced mathematics or aliens required.

Simple geometry only gets you so far, though. When you try to describe things that depend on the rate of change of other things, calculus makes your life a lot easier. The ancient Greeks actually came up with a lot of the concepts, but it wasn’t all put together into the modern form that we use until the 17th century. Newton and Leibniz get credit for inventing modern calculus, but they built their work on the foundations laid by previous mathematicians. Both Newton and Leibniz were trying to solve problems that couldn’t be adequately expressed using classical geometry type mathematics. Modern calculus allowed them to solve their problems.