On Newton, Raphson, and successive linear approximation

I seem to be in a bad mood.

My hero, our hero, Isaac Newton, writes in Latin, so now I have to learn Latin, so I start taking notes with Google Translate from the first page of De Aequationes With Infinitely Many Terms, and immediately he has a mistake, page 1, example 5 is wrong by a factor 2/3. Really, Isaac? Okay, so he's human. No, it's me that's human, I copied the problem inserting a mystery 2/3 into it where there wasn't actually one there.

Now I get serious and try to find Newton Raphson, but it's not in there, it's in Method of Fluxions. That would be 150 pages of unpublished medieval Latin, which I wouldn't understand if I could find it, and I don't see it, and at this point don't trust him or myself anyway, so I find a translation: it turns out John Colton, quite the suck-up, translated it into English in 1734, and gave Comments, more than doubling the book length.

So I grind through all of his 150 pages of Newton trying to find Newton's method, and I don't find it. Second pass, the closest I get is this, p 47, in Problem IV, To Draw Tangents to Curves:

Make the Terms of the proposed Equation equal to nothing: multiply by the proper number of the Dimensions of the Ordinate, and put the result in the numerator: Then multiply the Terms of the same Equation by the proper number of the Dimensions of the Absciss, and put the Product divided by the Absciss, in the Denominator of the value of BT. Then take BT towards A, if its value be affirmative, but the contrary way if that Value be negative.

If that sounds like Newton Raphson to you, you are a better mathematician than I.

Then I read Colton's explanation of this "Newton"'s method (p187), which gets closer:

"His (Newton's) Method may be easily apprehended from this one Instance, as it is contain'd in his Diagram, and the Explanation of it..."

Not so, since Newton's tangent diagram has a non-perpendicular line as what we would call the y-axis, or what would be from (x,0) to (x,y), so translation to normal coordinates does not apply, and the explanation thereof is what I quoted above, inscrutable.

"Yet for farther Illustration I shall venture to give a short rationale of it...."

What he means is, Newton neither explained, nor evidently understood enough to be clear about, this method, but his drawing of a tangent to a curve, plus Newton's talking around it, produced greater clarity in the mind of the translator than evidently in Newton's, and so it was Colton who came up with this much clearer rewrite of, really, Raphson's idea. Newton might have used it in his examples, but he did not explain it so that someone could understand. Whereas Colton did, both for Newton and later for Raphson, as follows.

"When a Numeral Equation is propos'd to be resolved, he takes as near an Approximation to the Root as can be readily and conveniently obtain'd.

"If this be greater or less than the Root, the Excess or Defect, indifferently call'd the Supplement, may be represented by \(p\), and the assumed Approximation, together with this Supplement, are to be substituted in the given Equation instead of the Root. By this means, (expunging what will be superfluous,) a Supplemental Equation will be form'd, whose Root is now \(p\), which will consist of the Powers of the assumed Approximation orderly descending, involved with the Powers of the Supplement regularly ascending [which inappropriately limits the discussion to polynomial functions], on both which accounts the Terms will be continually decreasing, in a decuple [10:1] ratio or faster, if the assumed Approximation be suppos'd to be at least ten times greater than the Supplement [not actually so]. Therefore to find a new Approximation, which shall nearly exhaust the Supplement \(p\), it will be sufficient to retain only the two first Terms of this Equation, and to seek the Value of \(p\) from the resulting simple Equation. Therefore to find a new Approximation, which shall nearly exhaust the Supplement \(p\), it will be sufficient to retain only the two first Terms of this Equation, and to seek the Value of \(p\) from the resulting simple Equation...

"This new Approximation, together with a new Supplement \(q\), must be substituted instead of \(p\) in this last supplemental Equation, in order to form a second, whose Root will be \(q\). And the same things may be observed of this second supplemental Equation as of the first; and its Root, or an Approximation to it, may be discovered after the same manner. And thus the Root of the given Equation may be prosecuted as far as we please, but finding new Supplemental Equations, the Root of every one of which will be a correction to the preceding Supplement..."

Then Colton goes on to describe Raphson's method, which does the same but instead of incrementing to \(p\)'s and \(q\)'s it increments the root \(x\) by the same amounts.

In Colton's case, and Raphson's, but not exactly clearly in Newton's case (though Newton does teach successive approximation intensively elsewhere in the Method of Fluxions), they have the idea of successive approximation, by evaluating some polynomial or multivariate curve, then solving it for only the linear part of it. If the formula is \(ax^3 + bx^2 + cx + d = 0\) they would insert the approximation, I'll call it \(\hat x + \Delta\), in for \(x\), where the unknown \(\Delta\) will measure how far off the final target we are. Multiply out your formula, then delete the \(x^3\)'s, \(x^2\)'s, etc., and solve for the \(\Delta\)'s value using the remaining, linear part of the equation.

So, yes, they do have some idea of using linear approximations to re-estimate something, but they mention no zero-crossing, no derivatives, no fluxions, no sense that following the tangent down to a zero at the x axis is what will give you a sense of a better Approximation, in fact nothing about Newton-Raphson as I learned it is given in Newton, or apparently Raphson, or even the translator Colton.

So I'm going to call it the method of Successive Linear Approximation for the zero of a function, which is what it is.

Until I find myself in a better mood.

There is a whole thesis on this topic.

	Your thoughts?
Name+:
Email*:	(will not be shared or abused)
Comment:
	Feedback is welcome.