Newton’s Lemma

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:

The Un-Unified Field, Chapter 2:
A Disproof of Newton’s Fundamental Lemmae

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)

lemma 6 fig 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.

lemma 6 fig 2

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.

lemma 6 fig 3

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.

lemma 6 f 4

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.

lemma 6 fig 1

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.

lemma 6 fig 3

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.

Isaac Newton: Principia
Book I, Section I, Lemma I thru XI
The Key to Newton’s Dynamics
Analytical View of Newton’s Principia

Note: This paper was last revised on 04/23/2013


33 thoughts on “Newton’s Lemma

  1. It is so sad to see how despite the obvious and easily verifiable means, you continue to milead readers and continue to insult the intelligence of an average mind.

    Lets start with the alleged quote of Mathis: “Newton’s first 8 Lemmae from the Principia are shown to be false.” — Miles Mathis . IT IS NOT true to be his cite, this text CAN NOT BE FOUND in his paper you are linking his quote to. If you have any other wording to it than I have from the same link in your post, I encourage you to forward it to me, I left my email as required, by my post. Lemmae 6&7 are mentioned, and since Lemma 7 builds on Lemma 6 and Mathis claims that Lemma 6 was not correctly postulated, logically, Lemma 7 is wrongly postulated. That’s all to it, what you are implying is just incorrect and misleading.

    The diagram you posted as to prove Mathis wrong , is again – MISLEADING. Not to be manipulated afterwards, I am here refering to your own diagram, which you use to demonstrate Mathis’ being confused. If you click on your own link to Newton’s Principia, where Lemma 6 is introduced, his (Newton’s) diagram DOES show R (marked as “r” in his diagram), that would represent circle’s center. For interpretation of logic, we’d refer to it as point R, i.e. when describing a line going from point R, through point B to point D. As well, it actually shows drawn line , making it “BRD” line, as descriptive form. Whereas in your post, the point or marking “r” (R) as circle’s center simply disapeared. As well, the “BRD” in your diagram disapeared But you don’t stop there, making MAthis look as if he was making things up from thin air, when he’s describing the line i.e. RBD. The same line that exists in both his and Newton’s diagram, by the way, that is the same line only your diagram is missing.

    So, honestly, it was repulsive to read your post, I will not comment on your logic of argument unless you provoke me. It is already insulting enough to see you quoting non-existing quotes and misleading those, who are not rigorous enough to do the basic homework od cross-checking. Especially physicists by their title, for every single word they say or post.

    Shame on you.

  2. Please read Newton:

    The lemma VI which Mathis misreads is actually the 2nd edition proof, a compromise proof after Huygens told the editor that the lemma was so trivial that no proof was required. Or something so. My latin is not very good.

    Huygens says:

    “Lemma 6 probatur per suppositionem ipsius lemmatis. Tota ejus demonstratio ita legi poterit expunctis inutilibus quibusdam. Nam punctis AB coeuntibus, nullaque adeo ipsius AB parte jacente intra curvam, manifestum est quod haec recta AB vel coincidet cum tangente AD cujus nulla etiam pars jacet intra curvam, vel ducitur inter tangentem et curvam. Sed casus posterior est contra naturam curvarum quae unicam in puncto A tangentem admittit ergo.”

    Note that both the original lemma 6 and lemma 7 do heavy use of the finite segments AB and AD. If you ignore its use, you are misreading Newton too!

    • Lemma 6 is so simple, and so straightforward, that it’s hard to believe that anyone (even a half-wit like Mathis) could misread it. But nevertheless, Mathis did manage to misread Lemma 6; and then, he foolishly included the mistake in his first book: The Un-Unified Fairy Tale.

      A diminishing angle is by far one of Newton’s simplest concepts in the Principia. And as you have pointed out above, Huygens considered it “trivial”. But then, Huygens was a brilliant, learned man; whereas Mathis, on the other hand, falls squarely into the imbecile category.

  3. Lee is correct. Miles Mathis is also correct. Steve never addresses Miles argument that angle ABD approaches 90 degrees. Read what Miles writes: “If we let B approach A, we must monitor the angle ABD”
    Why does Miles say this? Because for ABD to be a triangle it has to have 3 angles! The only way that BAD diminishes all the way to zero is if B is on the line tangent to the circle at A. But since B is by definition said to be on the curve, it cannot possibly be on the tangent line! This is Miles whole point, one that is never addressed in the original post.

    • First off, there aren’t any triangles in Lemma 6. That would be Lemma 8. Stop reading the drivel that Mathis is spouting, and actually read what Newton wrote; it’s quoted in the paper above, scroll up and read it. Newton’s description of Lemma 6 takes priority; not the incoherent ramblings of a half-wit ballet dancer obsessed with imaginary triangles (that is, when he’s not obsessed with the CIA). Again, Lemma 6 doesn’t have any triangles; not even one.

      Second, the whole point of Lemma 6 is to monitor the angle between the chord and the tangent line (angle BAD). If you don’t do this, and instead monitor some other angle (as Mathis contends), then how would you even know whether the angle between the chord and the tangent had diminished or not? Lemma 6 requires that you monitor angle BAD. It is Mathis who is monitoring the wrong angle, not Newton.

      And finally, angle ABD doesn’t even exist until you get to Lemma 8; it’s undefined. Look at Figure 1 above. Angle ABD is nonexistent; you can’t monitor an angle that doesn’t exist.

      Mathis doesn’t have the slightest idea what he is talking about; he’s in way over his head, and has made himself look foolish and ridiculous; something he does frequently.

  4. ”This is another disastrous outcome of Newton’s lemma VI from the Principia, where he found the arc approaching the tangent at the limit.” — Miles Mathis [Lev Landau Fudges The Orbital Math] 5/6/2013

    To his credit, Mathis has finally figured out how to spell the word “lemma”. Unfortunately, he is still hopelessly confused. Lemma 6 has nothing to do with the arc approaching the tangent at the limit; that would be Lemma 7. Will Mathis ever get it right?

  5. I don’t understand the following words:
    “…but always keeping point B at 90 degrees with the line D-B”.

    Question: How can a point be perpendicular (at 90deg.) to a line ?

    • Oh, you are referring to the comment “Lee” made below.

      Yeah, it doesn’t make any sense. It sounds like something Mathis might say; which means it’s nothing but hogwash. And in all likelihood, Lee probably is Mathis.

  6. Ok, so Miles is wrong about lemma 6. But what about lemma 7? Miles claims the chord, arc, and tangent are not equal at the limit. Is he wrong about that too?

    • Yes, Mathis got Lemma 7 wrong too. When point B is very close to point A (see Figure 3 above), the chord, arc, and tangent become indistinguishable; that is, across that infintesimal gap between A and B, the three lines are essentially all sitting atop of one another.

      As I said in the paper above, Mathis doesn’t understand any of this; he’s clueless.

      • Ah, I get it now, it makes perfect sense. Newton didn’t give an explanation with Lemma 7. And Miles drones on and on about a different angle where the tangent is longer than the chord – brother, did he take a wrong turn. Thanks, I finally understand it. Newton was right all along.

        • zack, sounds like there’s some hope for you. At least you tried to cite an example of MandM being right. Notice in his Lemma 6 article he does the bait and switch thing of ”it’s not right, because even though it’s obvious, you need to be monitoring this irrelevant thing over here to do the Miles-certified calculation.”

          He strikes me as someone who had some potential at some point, followed by a series of mental breakdowns. Still haven’t found any confirmation of his PSAT story.

  7. Really like this blog. Most of what’s on the internet about Miles Mathis is just ad hominem attacks. This blog has some real substance. Miles sounds so convincing and seems to make perfect sense. Glad to see someone showing where the mistakes are. Keep it up, hope there will be more.

    • There is nothing wrong with ad hominem attacks per se. Those who have claimed that Mathis is full of crap, have been vindicated. His articles are filled with contradictions and errors. To call Mathis a crackpot isn’t really an ad hominem attack at all; it’s a fact. Mathis *is* a crackpot. Anyone willing to take a close look at his work will quickly discover that it is nothing but error-ridden nonsense; page after page of worthless rubbish.

  8. Don’t you often get the impression that too many people are schooled in philosophy, and not enough in physics? Philosophy has always been the easy option because – by their lights – one can never be wrong (unless one mixes up the character defects of past philosophers; gee, which one spoke to horses?). Unfortunately, their standard bumper-sticker bag-of-tricks (e.g. ‘one cannot prove a negative’) impresses the ‘intelligent layman’ and lifts from the latter the obligation to study anything more difficult..

    • “Don’t you often get the impression that too many people are schooled in philosophy, and not enough in physics?”Crackpotwatch

      Mathis falls squarely into that camp; he’s a philosopher, painter, and ballet dancer without any math or science education at all (Bachelor’s Degree in Liberal Arts). He does, however, spend a great deal of time pretending to be a mathematician and physicist.

      Myron Evans, on the other hand, supposedly has a PhD in science.

      So who’s the bigger crackpot of the two; Miles Mathis or Myron Evans? Well, it’s a tough call; both men can easily redline a Crank-O-Meter. But which one is the crankiest of the two? Let’s examine the evidence.

      Both claim to have unified gravity and electromagnetism; so that has to be a draw. Now Evans is famous for having invented a perpetual motion machine; while Mathis is famous for having discovered that pi equals 4. So which one is the more outlandish claim? It’s really just too damn close to call. At present, Mathis and Evans must agree to share a dual seated throne, as the undisputed Kings of Crackpottery.

      • Well, Evans hasn’t really invented anything himself but he does use his madcap theory to ‘rubber-stamp’ the inventions of others (e.g. John Searl, John Bedini, Aureliano Horta, Bessler [!]). The worst thing about Evans is that people ln the future will see him on the same list as Faraday, Heaviside, Herschel, etc., and believe that he was their equal. How tragic.

    • Miles Mathis is precisely as poor and untrained a philosopher as he is a physicist. Contrary to popular misconception, proper philosophy isn’t something that a person can practice without an education; the rigorous discipline of logic, which underlies mathematical proof, is a good example. I’d say the homeless guy on the corner “philosophizing” is every bit a crank in that field as Mathis is when he tries to write about physics.

      • We have no problem with the ‘good’ parts of philosophy, such as pure logic and the propositional calculus, but have no patience with the antics of its stars – such as the infamous exchange between Russell and Wittgenstein. The latter might have done some good if he had stuck to engineering.

  9. I’m glad you did this. By analyzing this claim in language even more plain than Mathis’ “straight talk,” anyone can see how wrong he is — even as he attempts to dazzle us with esoteric talk of “lemmae.” It’s completely insane if Mathis thinks he’s actually exposed flaws in Einstein’s field equations, when he can’t even get something this simple right. Then there’s his other insanities about the coriolis force, why the sky is blue, and 532nm light not being green (a worldwide conspiracy by laser-pointer manufacturers, it seems), etc. By the way, did Miles ever get his paper on plate tectonics published in that Australian journal? Oddly enough, his site still has his announcement from August that it was being published that month. Hmmm.

    • Immensely enjoy your Mathis-Watch on You do an outstanding job of lampooning the new Leonardo. Your rebuttal to “The Coriolis Effect Deconstructed” was exceptionally good.

      Predictably, just like everything else, Mathis attributes the Coriolis Effect to his fictitious, make-believe charge field. My favorite Mathis quote from that article is this one:

      “…the charge field causes almost everything.”

      There it is, his entire schtick reduced to a single sentence. Well that, with a pratfall and rimshot.

      The Coriolis Effect (Michael Norris)

  10. My problem with Miles Mathis isn’t that he’s a voice in the wilderness heard by no one. Thanks to the magic of the internet and his print-on-demand books, he’s actually reached a few poor souls and convinced them he’s right. (I’m not one of those who thinks every Mathis supporter is just Mathis writing under a new nickname.) In doing so, he has convinced these few people that physics is a conspiracy, and that you have to be in on the conspiracy to be in the game, in which case you will make money from it. This is dangerous — like a late-night TV charlatan who convinces infomercial viewers that modern medicine is a fraud and a conspiracy, and all you have to do is take his miracle elixir pills to cure yourself of cancer, diabetes, heart disease, you name it. (Thank God for the FDA!)

    Mathis and his “feed the web kitty” donation buttons are selling a miracle elixir to fools who want to believe they can understand high-level physics with little or no effort. If folks like him actually caught on with the masses, mere conspiracy theory would devolve into paranoia and superstition, basically ushering in new anti-science dark age. This is why Miles Pantload Mathis is a bad bad man.

    • “I’m not one of those who thinks every Mathis supporter is just Mathis writing under a new nickname.” — Jack Sprite

      You may well be right about that. Can anyone really know with certainty, who is who on the internet? I become suspicious whenever someone can cite Mathis chapter and verse, and wholeheartedly believe every word of it. Anyone foolish enough to be hoodwinked by Mathis, doesn’t have the wherewithal to memorize 3,000 plus pages of navel-gazing nonsense.

      But hey, there could be a handful of true-believers out there who are convinced that Mathis is a real scientist. As P. T. Barnum discovered, “there’s a sucker born every minute”.

  11. You could save yourself a lot of time and trouble by just listing everything Mathis has been right about. So far he hasn’t been right about anything. A blank page would do it. Why even bother? Everyone knows he’s a crackpot.

    • It’s a valid point. Very few people are even aware of Mathis, and those who are have already written him off as a crank. So why bother? As a rule, I normally never publicly laugh at anyone’s stupidity. But since Mathis is a misinformed braggart and windbag who takes himself much too seriously, he is well deserving of ridicule; which I am more than happy to heap upon him.

  12. The Newton lemma is missing the finesse of the Mathis insight in regards to a circle.
    Geometry was Newton’s ace. Mathis points out geometry doesn’t allow for a solution
    because: it is the problem.

    • Lemma 6 demonstrates a diminishing angle; that’s all it does. Mathis merely misread it. The new Leonardo can’t navigate his way through even the simplest of Newton’s concepts.

    • If the whole static vs. dynamic thing is an issue, then imagine it not as though the point is moving along the circle and the angle diminishing. One could also just phrase it as a comparison between two independent, static, circular geometries, let’s call them S and S’.
      “If the arc between A’ and B’ is smaller than the arc between A and B, then the angle B’A’D’ is smaller than the angle BAD.”
      Nothing to do with time or movement, or anything Mathis might claim.

      • It doesn’t really matter whether Lemma 6 is static or dynamic; it holds true in both cases. Mathis routinely drags time and his calculus papers into the mix, primarily to cloud and confuse the issue. But in the case of Lemma 6, that tired and stale harangue won’t rescue him; he’s simply flat-out wrong.

        What is truly funny about all this is that it’s Chapter 2 of his book; a book he is currently selling on He can’t correct the problem; it’s already been published. He could, however, publish a revised edition of the book; but since it only sold three copies, why even bother? Besides, cranks never correct previous foul-ups; they just spin new crackpot theories.

        This isn’t information buried deep in the book; it’s Chapter 2 for crying out loud. How could Dr. Tahir Yaqoob have missed this obvious and glaring error? The man has a PhD in physics for Pete’s sake. Did Dr. Yaqoob actually read the book before writing its introduction?

        I have e-mailed Dr. Yaqoob and invited him to read the above paper and comment publicly on this forum. Unfortunately, he has refused my request; instead, clinging to a desperate hope that his entire association with the matter will simply fade into obscurity. But sadly for the good doctor, that’s just wishful thinking; whistling past the graveyard, so to speak. It’s too late to walk it back.

        Note: Dr. Tahir Yaqoob can be contacted via the following e-mail address:

      • Mathis is making the point: when dealing with a circle in motion in a gravity field the radius, tangent and the arc are equal at 1/8 of the circle and not at the instant angle BAD diminishes as stated in the lemma. Newton’s lemma applies to “static” flat geometry.
        You have to read his paper to fully understand his view

  13. Read Lemma 6; it’s in the article above. Do you see any mention of the radius being equal to the tangent line? That’s something Mathis made up; that isn’t found anywhere in Lemma 6. Mathis doesn’t have the slightest idea what he is talking about; he never has.

  14. In the Mathis diagram A-D is equal to the length of the radius. From D he draws a line to the center of the circle and says something like – Where this line intersects the circle represents gravitational acceleration. This point corresponds to B.

    If so: gravitational acceleration is always perpendicular to the circle meaning B is 90 degrees with line D-B so that moving point B towards point A along an arc really means diminishing the circle but always keeping point B at 90 degrees with the line D-B, gravitational acceleration never disappears until the circle is no more, therefore the tangent will always be the hypotenuse in the triangle ABD.
    The tangent can therefore never equal the arc and angle BAD can never diminish before ABD.

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