A Blogger Joining The Convo

I have read your recent blog Newton and your response Leibniz and through plenty of analysis about the both of you I have come to a concensus.

You both are great men in possesion with a new analysis before either of you made it known to the world. If priority of publication determined priority of discovery Leibniz would have completely gained his cause; but this is not sufficient on the present occasion. The inventor may have long kept the secret to himself; he may have allowed some hints to escaope him on which another may have seized. Therefore I had to trace it to the sources of discovery.

In the whole of this business there are three pieces that are truly decisive; first the letter you, Newton wrote to Oldenburg dated October 24th 1676 which was communicated to Leibniz the year following, the second the reply, Leibniz, that you returned to Oldenburg in response to the first letter, and finally teh scholia to Newton's Principa published towards the end of 1688.

From the analysis of these three peices it is clear that if Newton first invented the method of fluxions, as is pretended to be proved in his letter, Leibniz equally invented it on his part, without borrowing anything from his rival.

You two are great men that by the strength of your genius' arrived at the same discovery through differnt paths. It would be wonderful for you two to realize this and meet up at a pub and hare a beer once and talk about both of your discoveries. Another breakthrough just might happen!

-Charles Bossut

Fluxions and Integral Calculus

Using my terminology a variable quanitity, x, depending on time is called a fluent; and its rate of change with time is said to be fluxions of the fluent. I also chose to use the notation o to represent infinitely small quantities.

Also through my discoveries in calculus i generalized the bionomial theorum for expanding expressions in the form (1+a)^n, with n being a positive integer, to the case where n is a fractional exponent, positive or negative, with the result being an infinte binomial series rather than a polynomial.

"No great discovery was ever made without a bold guess"

-Newton

Post Script

Gifted by nature with superior intellect, I was born at the time of Wren, Wallis, Barrow and others, I had already rendered the mathematical sciences flourished in England, and was able to recieve lessons from Barrow himself. I was much more priviaged than said discoverers who leech off of the ideas of my own and try to take credit for my discoveries. I have been said to have "laid the foundations of the grand theories at the age of twenty-five" and i believe i deserve to be credited with the full discovery.

Notation

With my notation in my discoveries of calculus I can provide suitable symbolism that allow the geometric arguments of my predecessors to be translated into operation.

lx^2=x^3l3

The symbol l allows me to represent the sum of infintely small rectangles. It is the script form of s, the initial letter in summa.

Differentials:

The notation dy/dx allows mathematicians to treat it as a quotient of differentials[infinitely small increments of a variable]. With this notation I am able to solve for the derivative of Descartes folium equation...

"Nothing is more important than to see the sources of invention which are
more interesting than the inventions themselves."

- Leibniz

Folium of Descartes

As seen it forms a loop in the first qudrant of the cartesian plane [which I also designed] with a double point at the origin and an asymptote.

This algebraic curve is represented by the equation:

x^3+y^3-3axy=0

I discovered this form of calculus when challenging Fermat to find the tangent line of a curve at an arbitrary point since Fermat has just discovered a method for finding tangent lines.

Although Fermat was able to solve the problem easily, i was able to determine that the slope of the tangent line can be found easily using implicit differentiation.

"I think therefore I am"

- Rene Descartes

The Method of Mechanical Theorums

Using the law of levers it is possible to determine the areas of the figures from the known centre of mass of the other figures.

The points A and B are on the curve. The line AC is parallel to the axis of the parabola. The line BC is tangent to the parabola.

My first proposition states:
The area of the triangle ABC is exactly three times the area bounded by the parabola and the secant line AB.

Proof: Let D be the midpoint of AC. The point D is the fulcrum of a lever, which is the line JB. The points J and B are at equal distances from the fulcrum. As Archimedes had shown, the center of gravity of the interior of the triangle is at a point I on the "lever" so located that DI:DB = 1:3. Therefore, it suffices to show that if the whole weight of the interior of the triangle rests at I, and the whole weight of the section of the parabola at J, the lever is in equilibrium. If the whole weight of the triangle rests at I, it exerts the same torque on the lever as if the infinitely small weight of every cross-section EH parallel to the axis of the parabola rests at the point G where it intersects the lever. Therefore, it suffices to show that if the weight of that cross-section rests at G and the weight of the cross-section EF of the section of the parabola rests at J, then the lever is in equilibrium. In other words, it suffices to show that EF:GD = EH:JD. That is equivalent to EF:DG = EH:DB. And that is equivalent to EF:EH = AE:AB. But that is just the equation of the parabola.

"Give me a fulcrum and I will move the world!"

- Archimedes