In my recent response to Jason Ross' Leibniz Series, I mentioned that the way of displaying some of Leibniz achievements, in particular his method of the differential and integral calculus, was partly inconsistent. Now, I don't mind the incompleteness, because the topic is vast and certainly requires a whole series of sessions to really get into the topic. But inaccuracy is intolerable and there are a few other things that bother me about the way this is represented.
The fact that Jason started his Leibniz discourse in terms of examples with the comparison of Leibniz vis viva versus Descartes notion of mechanics could be kind of expected, because this work of his has been existing for a while and we know that Jason did these animations some time ago, which were easily taken out of the drawer to be represented here as a case. That's fair enough. But when in the 2nd installment of the series, Jason started to attempt to tackle the calculus, it was unfortunately done from the completely wrong end. I do not want to dive into the details here again as to why I think this is the case. Instead I am going to lay open my note's which I assembled for Jason as a feedback to his 2nd installment. These Notes, I did not yet send to Jason, because he was forestalling my sending the notes, by saying that he needed some time to study the material about Leibniz's quadrature, which I created over the last 3 years, more closely. Yet I think it is a good feedback also for others to read and it explains, why I think the calculus needs to be tackled differently in such a format as Jason's series on Leibniz.
Here are the Notes:
thanks for elaborating on the Calculus in your 2nd Installment on Leibniz. In your presentation you mentioned that Leibniz took further a problem, which Kepler could not resolve, namely to know what the change in motion is going to be at any given moment based on knowledge of the acting physical principle. While this is certainly so, I am yet still surprised that I have not encountered any work of his directly relating to Kepler's work other than shortly mentioning Kepler's "Barrel-Rule" in his treatise "On the arithmetic quadrature of the circle, ellipse and hyperbola of which a Corollary is the Trigonometry without tables". In your report you sound like you are suggesting that Leibniz was consciously working on his calculus in the continuation of Kepler's Legacy, but in fact according to Leibniz own works, at least those known to me, Kepler played little role in his progressing on the calculus, other than being one of many people before him having tried to conceptualize the notion of change as an effect of an acting physical principle, as expressed in his barrel rule. Unless you provide some prove, as to producing a written piece of work by Leibniz that clearly shows that he consciously rests some of his concepts on Kepler's footings, it would be a little far fetched to provide the impression that Leibniz consciously continued Kepler's work.
I must say that your chain of argument as to attempt to explain how the calculus came about is somewhat inconsistent and not rigorous. You started off with Kepler. Okay, we know that you are very familiar with his work, and the fact that physical principle is the cause of change like in motion certainly justifies that. Then you went on to Leibniz' difference series. According to his own words, Leibniz was working on these difference series already in his early youth, way before he was introduced to the arithmetic triangle by Huygens. His knowledge about this actually enabled him to react to Huygens challenge in the way he did. Interestingly and contrary to the way you displayed it, he was not initially using these difference series as a sole basis for the concept of what became the calculus, as you just put it, but rather as a means to prove that a sum of infinitely small sectorial triangles inscribed into a semi-circle is equal in area to a quadrilineum which is constructed by the boundaries of the abscissa the ordinates and the so-called RESECTING CURVE of the semi-circle. This prove was absolutely necessary, because it provides the fundamental geometric basis for building the areal relationships between various types of areas underneath and next to a continous curve, like any type of parabola for instance. Only once these areal relationships were worked out, and only after he has proven that the resecting figure principle works on all types of curves, like a circular curve, a cycloid curve, or all parabolas and hyperbolas for instance, Leibniz only then went on to use the difference series in relation to the arithmetic and the harmonic triangle which allowed him to turn the various geometric aerial PROPORTIONS, which according to Michelangelo Ricci relate to the powers of the very analytic expressions of the treated curves, into his notation of the calculus, which is still known and fully used today.
All this is to say that the creation of the calculus and its notation, derived from geometric relationships (which of course have their origin in physics!), and hence I have a huge problem when you jump in your presentation, straight from the difference series to the analytic expression of the parabola. You know, there was a long way from Apollonius Cone-Sections to the analytic expression Y=mXÂ²+n. As I already mentioned in my previous response to you, Leibniz was actually using Descartes "cartesian" coordinate system, which he got introduced to by Huygens in Paris and turned it into something useful, by means of his calculus among other things. He indeed turned it into something like a starting point for manifolds, which is certainly partly reason, why Riemann said the things he did about Leibniz.
In your straight jump to the analytic esxpression of the parabola, you completely left out as to how he came to the dy and the dx as infinitesimals. This is like trying to explain a principle with its effect, which is not a valid scientific method.
Furthermore, in your treatment of the analytic expression y+dy = (x+dx)Â² you are making the fundamental mistake of treating the infinitesimals dy and dx as finite quantities, which is the reason, why you are running into the contradictive last step, where you for no apparent reason have to remove one term dx on the right side of the expression without subtracting it on the other, in order to get to the commonly known 2x as a 1st derivative of XÂ². Now, one could resolve it by calculating the limits as the gentleman Marc Gordon suggested in one of the previous posts in this blog, but that would reduce the problem to one of quantitative convergence, which is not what the calculus is about. The calculus is about principle. And the coming abouts of the relationship y=xÂ² and its 1st derivative y' = dy/dx = 2x are best explained by the application of the Ricci-principle, in which it was proven that the powers of such an analytic expression relate to the proportions of the lengths between the tangents intersection with the abscissa and the ordinates intersection with the abscissa to the length between the apex origin point of the parabola and the intersection of the ordinate with the abscissa. This principle, dicovered and proven by Michelangelo Ricci in his treatise "Excercitatio edita de Maxima et Minima", is the key to Leibniz notation and methods of polinomial differentiation and integration in conjunction with the concept of an infinitesimal, which is nominally described as a infinitely small quantity, but in reality is an entity, which reduces the quantity to a remaining principle that created it in the first place and remains actually the same acting principle as in the finite domain. Now, in his treatise, which I mentioned above, Leibniz realized that the Ricci Principle was working for the paraboloids and for the hyperboloids, but with the exception of those hyperboloids which have equal dignities, meaning essentially the hyperbola. For that one he had to employ the logarithmic prove of Gregoire de St. Vincent.
Now before going on with trying to educate people about the creation of the calculus, I would strongly recommend studying Leibniz' works "On the artihmetic Quadrature of the Circle, Ellipse and Hyperbola" as well as his "History and Origin of the calculus". The latter one basically has the same content as the first one and was basically produced in defense of the attack by Newton and the british. As per your request I made some videos available, which are animated representations of the Proposition of Leibniz' Quadrature. Here are the links to the 1st eleven propositions:
I have a slight idea about what you guys want to achieve with this course and despite your presentation being somewhat inconsistent, I appreciate that you are trying hard to trigger a discussion, but this discussion will only be fruitful, if it is not entirely guided by purely political ambitions. Creativity expresses itself in various forms and denying one while supressing the other will not work.
However, there has not been any further correspondence, since Jason was letting me know that he is in need of some time to dig into the material, even though I was explicitely asking him to make this a bilateral work, because I am also interested in his thoughts. So from this I can currently only draw 2 possible conlcusions.
1.) The whole thing was just a flash in the pan, or
2.) The folks at the basement are still busy studying the material, in which case I would have expected some questions, feedback or bilateral sharing of working progress. But that doesn't seem to be the style of working of the Larouche Basement People.
One has to admit that there are other pressing topics going on in the world today, which easily can push the study of a principle like this far into the background, but I still think it would be a mistake, to make that a rule.