The quest of Science and Mathematics is to obtain predictive power, and what a powerful tool the statement above conveys. This equation has become so famous that the name of its author, who died roughly 2500 years ago, is still well known: Pythagoras. It was a momentous discovery and a secret that was at the heart of the secret Greek mathematics society. If one could measure two sides of any right triangle, one could then know, to an extremely high level of precision, what the length of the hypotenuse would be. This established relationship is so pervasive and well known that even carpenters (a group not well known for their grasp of mathematical concepts) use it to determine measurements when building a roof or other angular structure. In fact, it is more ubiquitous than it might appear.
Cartesian geometry, being a rectilinear system, does not have a rigorous system for the measurement of off-axis distances. Oh, sure, you can easily locate any point by a simple numerical callout, for example 5i + 3j + 2k (in Cartesian terms) locates the point reached by starting at the origin and going 5 units along the X axis, 3 along a line parallel to Y and then 2 in the Z direction (we’re staying in the positive quadrant for the sake of clarity in this example). But what is the overall length of the straight line distance from the origin to that point? Since there is no portable off-axis ‘yardstick’ this is a distance that must be calculated.
And this is the beauty of the Cartesian system. Since the axes are all defined to be perpendicular to each other, every measurement along an axis can be considered to be one of the sides of a right triangle, and in the immortal words of the Ozian Scarecrow;
“The sum of the squares of the length of the sides for any right triangle is equal to the square of the third”
So in the case listed above, we just apply Pythagoras’ theorem to three dimensions and we can prove that the distance from the origin to the point is:
x2 + y2 + z2 = 52 + 32 + 22 = 38 = length2 length = 6.164
With this powerful theorem, given our ‘a priori’ axes, we can know any distance in space! By combining the point equations, we can calculate relative distances, too. We won’t delve into the mathematics for an example of this at this point (or well, ever in this book, because the equations are too hard to type), but the point is that this methodology allows for the calculation of any relative distance between two points in Cartesian space.
This is a great system, and we use it every day for all sorts of scientific, mathematical and engineering applications. Unfortunately, there are a couple of problems that are not apparent on the surface.
The first has to do with the sign (positive or negative). Since this equation relies on the square root of a squared value for its answer, it has a ‘cleansing’ effect on the values. That is; since the square of any number (excepting ‘i’ of course) is positive due to the rules established to manage the multiplication values of our bipolar number system, and the square root of every positive number has a positive value, then every value calculated using this equation is naturally positive. But we have admitted the concept of negative length to our system, and the same value for the length of the distance would have been calculated if we had used the values -5, -3 and -2, which would clearly put the length in the negative octant.
How is one to know which value is correct? The answer is that you just have to know, and that the equation only calculates the relative value. But it is a little disingenuous for a system that gives physical, meaningful values to the concept of negative numbers to force the person doing the calculation to determine the geometric meaning of the answer. Because of the form and substance of the equation used to determine the overall length, the line segment generated could be in any one of the octants. It’s not very deterministic.
But there is a more insidious, pervasive problem here, and that is: it forces all of the equations of motion to be second order equations.
Let’s say that you wanted to go from point A to point B, both of which are totally off axis. The equation for determining the length is:
(xA – xB) 2 + (yA – yB) 2 + (zA – zB) 2 = AB2
Which, in the absence of actual values, can be expanded algebraically to be:
= [(xA – xB) * (xA – xB)] + [(yA – yB) * (yA – yB)] + [(zA – zB) * (zA – zB)] = AB2
= (xA2 – 2xAxB + xB2) + (yA2 – 2yAyB + yB2) + (zA2 – 2zAzB + zB2) = AB2
Pretty ugly isn’t it? And you still have to take the square root of the results of all of the sub-equations to get the answer. This is the sort of calculation that is behind every distance determination that utilizes Cartesian geometry. Why does this have to be so hard? Obviously, if one had a ruler, one could just go out and measure the length, and save oneself a whole lot of math. But if you are trying to calculate a velocity, which is a first order equation (distance/time), you’re stuck with this big, honkin’ set of quadratic equations in the middle.
The point of all of this is that the geometric system, because of the way that it is constructed and the way the expressions are put together, forces the analyst to utilize second order equations to calculate first order values, which introduces both complexity and uncertainty into the system where none is actually required.
What we really need is a geometry and measurement system that allows first order measurements, like velocity, to be calculated in first order terms. By relying on the three perpendicular axes and codifying direction in relative values, we have created a system that cannot do that.