Do We Really Need Calculus?

I once took a class titled "Calculus-Based Physics". I was still learning calculus, and was more adept with spatial mathematics like geometry and trigonometry. I could not help noticing that just about anything that can be solved with calculus can also be solved without calculus. We live in a spatial universe, and the graphing used in calculus is just another way of solving spatial problems.

I find that an under-appreciated gem of basic physics is the Inverse Square Law. The Inverse Square Law states that an object that is twice as far away will appear as one-quarter the size or, if two radio antennae are broadcasting with equal strength, the signal from one twice as far away will have one-quarter the strength of the one that is closer.

If we look at a building some distance away, for example, the result is an isoceles triangle (one with two equal angles) with the observer at the point of the triangle and the width of the building forming the base of the triangle. This could also be expressed as a right triangle (one with a right angle) with the height of the building as the vertical axis of the triangle. The Inverse Square Law applies in that, if the building were twice as far away from the observer it would appear to the observer as having only a quarter it's former width, or height.

The reason for the Inverse Square Law is that the circumference of a circle is pi (3.1415927 is as many decimal places as I have it memorized) times the diameter of the circle, and the diameter is twice the radius. This means that if we double the radius, which represents the distance to the object, there are now four times the original radius in the diameter of the circle that the object lies on, with the observer at the center.

So, why can't we make use of the Inverse Square Law when dealing with anything that forms a triangle? It does not necessarily have to involve an actual spatial triangle, this law of physics can be applied to anything that forms a triangle in it's pattern of events. This opens up a whole world of possibilities.

Actually, anything which changes at a steady rate forms a triangle in pattern. Picture a right triangle, or a cone. If some entity begins at zero, and proceeds at a steady rate to some maximum, it can easily be expressed as a triangle. Let's replace the triangle formed by the observer looking at the building with the beginning at zero replacing the position at the point of the observer, and the maximum replacing the building.

Now, let's have a look at fractions. I find that fractions represent the way reality really operates. We count in tens, and so we prefer decimal expression. But that is an artificial numbering system and use of decimal tends to make patterns in numbers less apparent than if we used fractions.

Using a simple example of the Inverse Square Law, we can see that triangles have a very useful relationship with the squares of fractions.

Suppose that we have a right angle between two lines. The vertical line has a length of four units, and the horizontal line a length of six units. Let's draw a line from the end of the horizontal line to the top of the vertical line to form a right triangle. The triangle would have an area of twelve square units, since the triangle is half of what a square involving the two lines would be and such a square would have an area of 4 x 6 = 24 square units.

Next, let's consider the half of the horizontal line from the furthest point and moving toward the vertical line. This is the narrowest half of the triangle along the horizontal line. The vertical dimension of the triangle would be zero at the beginning point, and two units at the halfway point of the horizontal line of the triangle. This is because the vertical dimension reaches it's maximum of four units, and we have gone halfway there from the opposite point of the triangle.

The area of this narrow half of the triangle, along the horizontal axis line, would be half of six because 6 = 2 x 3. Thus, the area of the narrow horizontal half of the triangle would be three square units.

Do you see the Inverse Square Law at work? Starting at the narrow end of the triangle, we proceeded halfway toward the wide end of the triangle. In doing so, we passed one quarter of the area of the triangle because the total area of the triangle is twelve square units and the narrow half of the triangle, along the horizontal axis line, has a volume of three square units.

This means that we can do all manner of measurements of anything forming a triangle using the squares of fractions. If the narrow half of a triangle (or cone) contains one quarter of it's area or volume, it must mean that the widest half of the triangle contains 3/4 of it's area or volume. Likewise, the narrowest 1/3 of a triangle contains 1/9 of it's total area or volume. The widest 1/9 of the triangle contains 1/3 of it's total area or volume, and so on.

Now, let's move on. No one says that this very useful Inverse Square Law has to be limited to actual spatial applications. It must also apply to anything that forms a triangle in pattern, even if it does not involve an actual triangle in space. When you think about it, anything that proceeds between zero, or a minimum, and a maximum forms a triangle pattern if it were displayed on a graph.

An object in motion with a steady acceleration or deceleration forms a triangle, with the minimum at the point of the triangle and the maximum at it's widest part. Of course, if the minimum is other than zero, all we have to do is to add a rectangle beneath the triangle so that the width of the rectangle represents the value of the minimum. The most common use of calculus is to measure change, and change proceeds between a maximum and a minimum.

A falling object forms a definite triangle. The acceleration of falling due to gravity is the well-known 32 feet per second squared ( I won't convert this to metric because it is easier to express in feet). This means that if an object is dropped, it will go into the first second of fall with a velocity of zero feet per second and end the first second with a velocity of 32 feet per second, with the increase coming at a steady rate. This means that the average velocity of the object, in it's first second of fall, will be 16 feet per second. So, it will fall 16 feet in the first second.

The object enters it's second second of fall with a velocity of 32 feet per second, and ends the second with a velocity of 64 feet per second. This means that it's average velocity throughout the second second of fall was 48 feet per second. So, it fell 48 feet in the second second of fall.

In two seconds, the object has fallen 64 feet. The 16 feet that it fell in the first of the two seconds is 1/4 of 64. Can you see the triangle that is formed in this pattern, and the applicability of the Inverse Square Law?

(By the way, this 16 feet would be a very useful unit of vertical measurement because of how it relates to the velocity of falling objects. I named this unit a "grav", for gravity, and described it's use in the posting on this blog, "The Way Things Work", and in the book "The Patterns Of New Ideas").

A rising ballistic object forms a triangle in reverse to that of a falling object. Throw a ball into the air and it will form one triangle on the way up, by starting at a maximum vertical velocity and proceeding to zero as a result of the action of gravity, and then another triangle on the way down as it's velocity starts from zero, at the maximum altitude, and proceeds to a maximum.

Something like a ball rolling across the ground, with a steady deceleration, also forms a neat triangle that can readily be measured with this method.

What about a dam holding back a body of water? The pressure of the water against the dam also forms a triangle. The water pressure starts at zero at the surface of the water, and proceeds steadily to a maximum at the bottom of the water.

Anything spreading steadily along a circular front, such as an oil spill, forms the base of a cone in pattern that we can easily measure using this method. When half of the time from the beginning of the spill, if it was from an area of zero, to now had elapsed, the area covered by the spill was 1/4 of what it is now. We can also measure withdrawal at a steady rate in the same way.

Possibly the most useful application of the Inverse Square Law and the squares of fractions involves the total earnings of money which earns interest, with the interest rolled back in. This also forms a triangle, increasing at a steady rate between between minimum and maximum.

To find the sum total of any calculation, how much distance has been covered in the case of velocity, or how much money has been earned in the case of interest, just form a triangle and find the area under the triangle.

So far, we have seen how calculations can be done on anything involving change at a steady rate by using the Inverse Square Law of fundamental physics and basic fractions, with no need whatsoever to use calculus. But now, let's get a little bit more complicated.

It is easy enough to do measurements involving constant change, such as acceleration. But what if the rate of change is itself changing? For example, a graph of velocity will appear as a straight horizontal line for constant, unchanging velocity and a slanted line for constant change in velocity (acceleration or deceleration). But if the rate of acceleration was also changing, a graph of the velocity would show as a curve. The area under the curve would represent the total distance travelled. The trouble is that we do need calculus to find the area under a curve.

But a simple curve is the synthesis of two straight lines, with one of the lines changing in length, which forms the two axes of a triangle. There may be constant acceleration, or change of some kind, which would be expressed as a straight slanted line on a graph, which could be the hypotenuse of a right triangle. But there may be a change in the rate of acceleration, or a change in the change in the rate of acceleration. There may even be a change in the change of the change in the rate of change or acceleration.

To dispense with calculus, all we need to do is to arrive at triangles on our graph so that we can easily find the total distance travelled (or money earned, etc.) using ordinary geometry. No matter how complex the curve, we can find this by simply using multiple triangles and then adding their values to get a total. Of course, we would subtract the value that we get from the area under a triangle if it represented a negative value, such as deceleration, instead of positive acceleration.

Suppose that we wanted to find the total distance travelled by a moving object over a given period of time. But, the velocity of the object was contantly changing.

We would start with one triangle representing the acceleration of the object at the beginning. It would be graphed as a rectangle if it were a contant velocity, without acceleration.

If the object began to accelerate at a given point in time, we would start another triangle beginning at an axis representing the point in time at which the acceleration began.

If that acceleration rate was changing, rather than acting at a contant rate, we would set up another triangle representing that change, and continuing between the appropriate points in time represented by the common vertical axes of the triangles.

If there was a change in that rate, we would set up yet another triangle to represent it. If there was deceleration, we could set that up with the common time axis at the top, instead of the bottom of the graph, and subtract that from our final total rather than adding it.

Isn't this easier, and more enjoyable, than using calculus?