Reading Newton’s Philosophiæ Naturalis Principia Mathematica is less challenging than listening to Vogon poetry. You just have to get your head working like a 17th Century genius who had just invented Calculus and who would have deep-fried his right arm in rancid skunk oil before he’d admit to using any of his rival Liebniz’ math notations or techniques.
Newton was essentially a geometer. These illustrations (from Book 1 of the Principia) will give you an idea of his style. He’d set himself a problem then solve it by constructing sometimes elaborate diagrams by which he could prove that certain components were equal or in strict proportion.
For instance, in the first diagram (Proposition II, Theorem II), we see an initial glimpse of his technique of successive approximation. He defines a sequence of triangles which as they proliferate get closer and closer to the curve he wants to characterize.
The lines and trig functions escalate in the second diagram (Prop XII, Problem VII), where he calculates the force on a body traveling along a hyperbola.
The third diagram is particularly relevant to the point I’ll finally get to when I get around to it. In Prop XLIV, Theorem XIV he demonstrates something weird. Suppose two objects A and B are orbiting around attractive center C, but B is moving twice as fast as A. If C exerts an additional force on B that is inversely dependent on the cube of the B-C distance, then A‘s orbit will be a perfect circle (yawn) but B‘s will be an ellipse that rotates around C, even though no external force pushes it laterally.
In modern-day math we’d write the additional force as F∼(1/rBC3), but Newton verbalized it as “in a triplicate ratio of their common altitudes inversely.” See what I mean about Vogon poetry?
Now, about that point I was going to get to. It’s C, in the center of that circle. If the force is proportional to 1/r3, what happens when r approaches zero? BLOOIE, the force becomes infinite.
In the previous post we used geometry to understand the optical singularity at the center of the Christmas ball. I said there that my modeling project showed me a deeper reason for a BLOOIE. That reason showed up partway through the calculation for the angle between the axis and the ring of reflected light. A certain ratio came out to be (1-x)/2x, where x is proportional to the distance between the LED and the ball’s center. Same problem: as the LED approaches the center, x approaches zero and BLOOIE. (No problem when x is one, because the ratio is 0/2 which is zero which is OK.)
Singularities happen when the formula for something goes to infinity.
Now, Newton recognized that his central-force (1/rn)-type equations covered gravity and magnetism and even the inward force on the rim of a rotating wheel. It’s surprising that he didn’t seem too worried about BLOOIE.
I think he had two excuses. First, he was limited by his graphical methodology. In most of his constructions, when a certain distance goes to zero there’s a general catastrophe — rectangles and triangles collapse to lines or even points, radii whirl aimlessly without a vertex to aim at… His lovely derivations devolve into meaninglessness. Further advances would depend on the algebraic approach to Calculus taken by the detested Liebniz.
Second (here’s the hook for this post’s title), Newton was looking outward, not inward. He was considering the orbits of planets and other sizable objects. r is always the distance between object centers. For sizable objects you don’t have to worry about r=0 because “center-to-center equals zero” never occurs. If the Moon (radius 1080 miles) were to drop down to touch the Earth (radius 3960 miles), their centers would still be 5000 miles apart. No BLOOIE.
Actually, there would be CRUMBLE instead of BLOOIE because a different physical model would apply — but that’s a tale for another post.
The moral of the story is this. Mathematical models don’t care about infinities, but Nature does. Any conditions where the math predicts an infinite value (for instance, where a denominator can become zero) are prime territory for new models that make better predictions.
~~ Rich Olcott