Abstract: Mathis brings his comedy routine to Newton’s Principia, but can’t make it past page one before falling flat on his face.
This paper will analyze the following article and examine the authors claim that Newton’s first 8 Lemma from the Principia are false:
According to Mathis, Newton monitored the wrong angle in Lemma 6 of the Principia. However, based on this alone, Mathis asserts that the first 8 Lemma are false. Why this should invalidate the first 8 Lemma is never explained or even addressed in the article. In fact the entire article deals with Lemma 6, with only a brief mention of Lemma 7. So the following Mathis quote can’t possibly be true:
“Newton’s first 8 Lemmae from the Principia are shown to be false.” — Miles Mathis
Shown where? The best Mathis could ever hope to achieve in his article was to disprove Lemma 6; that’s the central topic of his article. But as this paper will show, Mathis simply misread Lemma 6; that’s right, the one and only Lemma he fully analyzed in his article he managed to misread. We could stop right here, and not continue any further. Mathis never disproved the first 8 Lemma; end of story. But let’s continue on and take a look at this supposed faulty Lemma 6 that has Mathis up in arms.
To set the stage we must first accurately assess Lemma 6 (Newton’s actual wording and intent), and then we will compare that to the unfounded claims made by Mathis. Fortunately, Lemma 6 is very simple; one single paragraph.
Book 1, Section 1, Lemma 6:
“If any arc ACB given in position should be subtended by the chord AB, and if at some point A in the middle of its continuous curvature, it should be touched by the straight line AD extended in either direction; then if points A and B should approach each other and coalesce; I assert that the angle BAD [generated] by the chord and tangent, would be diminished indefinitely and would ultimately vanish.” — Isaac Newton (See Figure 1)
AB is the chord
AD is the tangent line
ACB is the arc
BAD is the angle between the chord and the tangent line.
Don’t let the archaic language of Lemma 6 intimidate you; it’s deceptively simple. Lemma 6 essentially boils down to this. As point B approaches point A, along the arc, the angle between the chord and the tangent line will diminish. That’s really all there is to it. Think of it as an example of a diminishing angle.
So let’s put Lemma 6 to the test and see if it holds true. In Figure 2, point B has been moved along the arc so that it is now closer to point A.
Now, we compare the two Figures above. According to Lemma 6, we should find a diminished angle. Can you see that the angle between the chord and the tangent has gotten smaller? That’s the whole premise of Lemma 6; the angle is getting progressively smaller. The closer B gets to A, the smaller this angle becomes. That is the assertion made by Newton:
“I assert that the angle BAD [generated] by the chord and tangent, would be diminished indefinitely and would ultimately vanish.” — Isaac Newton
And finally, as point B moves closer and closer to point A, the angle between the chord and the tangent becomes infinitely small and eventually vanishes altogether when the two points coalesce (See Figure 3). This is what Newton claimed, and we have just seen that it is a valid claim.
So what’s the story? Why does Mathis insist that Newton monitored the wrong angle? Well, the answer is two-fold. First, Mathis has mixed together Lemma 6 and Lemma 8 into a jumbled, incoherent mess. And second, Mathis has misunderstood the method by which Newton specifies angles in the Principia.
Let’s begin with the mixed-up Lemma’s. Here is a quote from Mathis as he explains Lemma 6 to his readers:
“[I have added this paragraph after talks with many readers, who cannot visualize the manipulation here. It is very simple: you must slide the entire line RBD toward A, keeping it straight always. This was the visualization of Newton, and I have not changed it here. I am not changing his physical postulates, I am analyzing his geometry with greater rigor than even he achieved.]” — Miles Mathis
Huh? What? Slide the entire line RBD toward A? The only thing that moves in Lemma 6, is point B toward point A. Line RBD has nothing to do with Lemma 6. Line RBD applies to a different Lemma altogether; it applies to Lemma 8. But since Mathis never even mentions Lemma 8 in his article, he has confused one Lemma with another. Or as Mathis might phrase it himself: clueless with “greater rigor”.
Here is another example where Mathis has mixed together Lemma 6 and Lemma 8:
“In the Principia, Newton’s actual language in describing this problem (Lemma VI) is this: “if the points A and B approach one another. . .” Two things bear closer attention here. One, A cannot approach B without messing up the geometry. If we start moving the point A, we destroy our right triangle.” — Miles Mathis
Right triangle? What is he talking about? Lemma 6 doesn’t have any triangles. It is Lemma 8 that deals with triangles, not Lemma 6. Mathis has once again confused Lemma 6 with Lemma 8.
And further, A can indeed approach B “without messing up the geometry”. You can put A or B anywhere on the arc; they are just two points on a curve. Whenever the points are moved, you simply redraw the chord and tangent line. The geometry doesn’t get messed up, as Mathis claims. A can approach B, or B can approach A; the outcome is exactly the same.
As a last example, and by far the funniest and most ridiculous, Mathis claims that point B can never reach point A. This is not a joke, he even wrote a little 4 line proof (proving his confusion). Mathis actually believes that point B is prohibited from reaching point A:
“Of course this means that B never reaches A. If B actually reached A, then we would no longer have a triangle.” — Miles Mathis
By Mathis’ reasoning, point B cannot move along the arc and coalesce with point A. And why? It would mess up a nonexistent triangle. Is Mathis punch drunk? Did he even read Lemma 6? It’s as though he doesn’t understand any of this. His entire article is senseless blabbering about imaginary triangles. There are additional quotes that could be included here, but you get the idea. So let’s move on to angle specification.
Newton was very precise and methodic in the way he named his angles. Throughout the Principia, Newton specified angles by the path surrounding the angle; that is, you follow the path from points B to A to D, and it’s the angle enclosed by the path (See Figure 4). This is the convention Newton adopted for the Principia, and he uses this convention all throughout the book. Bear this in mind, as we now examine the crux of the Mathis article.
Finally, we come to the central theme (this is what prompted Mathis to write the article in the first place), his assertion that Newton monitored the wrong angle in Lemma 6 of the Principia:
“The first lemma in question here is Lemma VI, from Book I, section I (“Of the Motion of Bodies”). …Newton tells us that if we let B approach A, the angle BAD must ultimately vanish. In modern language, he is telling us that the angle goes to zero at the limit. This is false for this reason: If we let B approach A, we must monitor the angle ABD, not the angle BAD. As B approaches A, the angle ABD approaches becoming a right angle. When B actually reaches A, the angle ABD will be a right angle.” — Miles Mathis
Alright, so let’s have a look at Figure 1 again, and see if we can make any sense of what Mathis is saying.
Looking at Figure 1, it’s near impossible to know what Mathis even means by his statement above. When point B reaches point A the chord disappears altogether, leaving only the arc and the tangent line (See Figure 3). So where is this supposed right angle to which he refers? It doesn’t exist. His entire “angle ABD” argument is just meaningless nonsense (hogwash peddled by a pseudoscientist); Lemma 6 doesn’t have any right angles. In fact, using Newton’s convention above, angle ABD isn’t even an angle; it’s undefined.
So, did Newton make an easily spotted rookie error, and monitor the wrong angle in Lemma 6 of the Principia? Of course he didn’t; it never happened. And as we have just seen, this trumped-up wrong angle baloney was just a Mathis misread, coupled with an over-the-top exaggeration. Newton’s first 8 Lemma were never shown to be false. The only thing Mathis has shown is that his knowledge of Newton’s Principia is laughably thin, shallow, and feeble.
Note: This paper was last revised on 04/23/2013