ORIGINAL BUFFON NEEDLE PROBLEM

Georges Buffon (1707 – 1788) initially presented his Needle Problem to the Paris Academy of Sciences in 1733. In 1734, it was first published in a minor journal intended for worthy work of non Academy members. It was published again in a minor journal in 1776. Its first major popular publication was in 1777, in a supplement (4) to Buffon's Histoire Naturelle.

The original Needle introduced geometric probability as it delivered the first random proof of pi. The geometric nature came from proving relative 1/4 pi, relative to the field's diameter, as both 1/4 of a circle (that is: 1/4 C) and, simultaneously, the universal random average (relative 1/4 pi) relative to the diameter.

The original Needle proved 1/4 pi to be a percentage of the diameter: .78539.... . Therefore: the random value of a diameter: "1.".

In delivering the universal random average, the original Needle also introduced the methodology of serial random measurements. Since WWII, this has generally been known as: "Monte Carlo methodology."

Buffon originally used a baguette (French loaf) on a plank floor. He suggested a needle on a checkerboard would be more convenient and the name understandably stuck.

The original Needle asks: with a series of equidistant parallel lines, if two players want an even chance a randomly dropped needle will cross a line, how long must the needle be?

“Je suppose que dans une chambre, dont le parquet est simplement divise par des joints paralleles, on jette en l’air une baguette, & que l’un des joucurs parie que la baguette ne croisera aucune des paralleles du parquet, & que l’autre au contraire parie que la baguette croisera quelques-unes de ces paralleles; on demande le sort de ces deux joucurs.”

Buffon concluded that the needle must be approximately 3/4 of the diameter distance (shortest possible distance) between the lines. More precisely, that percentage: .78539.... . That is: relative 1/4 pi, relative to the diameter.

“...que la longueur de la baguette doit saire a peu-pres les trios quarts de la distance dees joints du parquet.”

Buffon also identified that percentage as one fourth of the circle described between and just touching two adjacent lines. That is: 1/4 C.

“...quart de la circonference du cercle don’t la longue de la baguette est le diameter...”

A unique and fascinating feature of the original Needle is that it readily and deductively demonstrates that, in the first instance, every series of random measurements --of anything-- automatically turns the circle or field or game or object (of any shape) into a dimensional game of relative pi in rotation. More specifically, into a series random measurements tending to average 1/4 pi each.

The original Needle proves a field or circle or game’s diameter has a random mathematical value of: “1.”. Therefore, deductively, the radius of the diameter has a random mathematical value: .50 .

It is here that the original Needle makes its random point. It deductively and inferentially proves the relative cross diameter (the dimension of relative “width”) has a random mathematical value of relative pi in rotation ...which therefore, for the reasons given herein, deductively and inferentially values the relative cross radius as: relative 1/2 pi!

The original Needle coincidentally (since it appeared a year before Boskovic's finesse) supports Boskovic’s methodology of a geometric finesse for predicting the orbits of comets. Since the original Needle’s deductions and inferences invite the finesse, it is quite possible the original Needle inspired Boskovic. This is discussed in the history section.

The heart and soul of Laplace’s work was based on the random quadrature he usurped from the original Needle. More accurately, it now appears he may have been quietly handed the Needle by Buffon and Condorcet in 1770. It appears that Laplace quietly used the Needle’s point (without mention or reference of the Needle or Buffon) in 1772, when he announced he had “discovered” that the “second degree of every equation lies in quadrature.” That, of course, is the point of the original Needle.

“Amongst the minor discoveries of Laplace in pure mathematics I may mention his discussion (simultaneously with Vandermonde) of the general theory of determinants in 1772; his proof that every equation of an even degree must have at least one real quadratic factor….” (Ball, Rouse. A Short History of Mathematics. Macmillan, 1908. p.419).

When Buffon unexpectedly published the Needle, in both 1776 and 1777, in the middle of the Laplace/Boskovic “debate” over “action at a distance,” the geometric truth surely started its inexorable atomic fizz. There for the taking was a .08333 flat bet advantage (or .16666 depending on the "game") over the very traditional random theory of quadrature that Laplace was advocating …and that would appear two centuries later as the same advantage in and of the Quantum sciences.

The flat bet advantage is the randomly measured geometric difference between a circle and diameter. In the Boskovic/Laplace debate, relative to the mean inclination of comets, relative to the totality of the field (360 possible degrees) it is also the difference between 45 degrees and 60 degrees, factored by two possible directions. That is: 60 – 45 = 15. Next: 15 /360 = .08333…. . Next: 2 (.08333) = .16666).

It is the difference between one of four algebraic poles on a circle or orbit, each with a .25 algebraic possibility …and one of three geometric poles on the circle or orbit’s diameter, each with a geometric probability of .33333, factored by two directions.

That is: 2 (.33333 - .25) = .16666 .

The same random advantage is found on a semi circle as discussed within.

As is the subject of this book, an arc of 60 degrees is not only .16666 of a circle, but is the geometric probability, over three random measurements, of a randomly measured field’s relative pi angle pole. Therein is the flat bet advantage: .16666 .

In short, the original Needle structures the algebra of a randomly measured circle as four Cardinal poles of 90 degrees each. To define the circle …just multiply a Cardinal pole by 4. This is the algebra on which every random table game is based

In short, when the original Needle is extended with “action at a distance,” the algebra of a randomly measured circle is geometrically defined as six relative pi angle poles of 60 degrees each. Each pole is .16666 of circle or "game." To define a circle (just a random statement of algebra) after using "action at a distance …just algebraically multiply the randomly found pi angle pole by 6.

As discussed within, geometrically, the use of "action at a distance" only makes mathematical sense when it is an extension of the original Needle. This is due to the requirement of "action at a distance" that the methodology of serial random measurements be used ("Monte Carlo" methodology). Since the original Needle automatically introduced the methodology when it introduced geometric probability ...and since the original Needle proved that geometric probability to be a statement of relative 1/4 pi ...and since the original Needle was the first random proof of pi ..."action at a distance" is best understood in terms of pi and as an extension of the original Needle.

This appears to shatter traditional random theory.

Did Laplace see it?

Of course! Most probably, it was after it was given to him by Buffon (see History). Today, the Needle often bears his name and is frequently referenced as the Buffon/Laplace Needle Problem.

In 1812, Laplace published the Needle under his own name without crediting or mentioning Buffon.

“Imaginons un plan divise par des lignes paralleles, equidistantes de la quantitie a; concevons de plus un cylindre tres-etroit don’t 2r soit la longueur, supposee eqale ou moindre que a. On demande la probabilite qu’en le projetant, il rencontrera une divisions du plan.” (Theorie Analytique des Probabilities, Simon Laplace, p. 569).
["Let us imagine a plane divided by equally spaced parallel lines of the distance a; and a cylinder of the length 2r and suppose the cylinder to be equal or less that the distance a. Give the probability of a toss touching a line of the plane."]

Buffon’s original take on the Needle left a clear inference that the value of a randomly measured field’s radius is: .50 .

When Laplace changed the Needle’s ("cylinder’s") length from .78539 (that is: the universal random average) of the field’s diameter to the complete length of the field’s diameter (or any other length) he eliminated the universal random average.

In his analysis, Laplace fundamentally changed the original Needle while discrediting its original length. He also, without giving reason or discussion, arbitrarily and casually mentioned that his calculated result (without mention of “pi angles” or the Needle or Boskovic or “action at a distance” or any other reason) must be multiplied by 16 ...?!  

The only reason to multiply the result of Laplace’s calculations of the Needle by “16” is if the randomly found arc on the Needle’s circle of 360 degrees is a relative geometric probability of 60 degrees (that is: 1/6 of a circle) without going through three random measurements and the finesse of “action at a distance.” This seemingly impossible mathematical phenomenon is the result of Laplace’s disingenuous convolutions (see History).

Simply as a matter of quadrature and perspective, the structure of the original Needle’s “game” can also be perceived as delivering an angle of 45 degrees. This is the straight line from South to West. It isn’t the Needle’s length. It is just a way of measuring with quadrature without going through pi. That is, without going through a series of random measurements to find the original Needle’s arc of relative 1/4 pi (aka: 1/4 C).  

The only way to randomly turn a straight line of 45 degrees into a straight line of 60 degrees is to use three random measurements and “action at distance.” This is the sine of the inclination relative to the diameter. It also, when “action at a distance” is used, delivers a random arc of 60 degrees on the circle, relative to a game or field’s diameter.

Sixty degrees of arc is 1/6 of a circle. However, when found randomly with “action at a distance,” its relativity to a diameter comes with a unique directional factor. Like a circle, the random measurements of a diameter are also subject to the possibility of two directions but with a difference.

Two possible algebraic directions on a diameter come with a geometric certainty of one direction.

Two possible algebraic directions on a circle come without geometry or geometric certainty. They are just two algebraic possibilities.

With “action at a distance,” all four directional factors come into play, but not equally. The randomness of gaming relative to a diameter is different from the randomness of gaming relative to a circle.

When two possibilities are factored into the Needle’s quadrant arc of 90 degrees (or into a gaming quadrant) they also contain the two possible directions transferred from the diameter, through 1/2 pi, with a unit of measure of relative 1/4 pi. This reduces a Quadrant to 45 degrees of inclination relative to a diameter base or, relative to the COR ...to 22.5 degrees of inclination.

When 22.5 degrees is made relative to a circle of 360 degrees (the “game”) the result is .08333 . This result (Laplace’s result ((and also the advantage of the Quantum Sciences))) must be multiplied by 16 to order to define the circle (or game or field or orbit or particle or randomly measured circumference of anything). That is: 16 (22.5) = 360. This completes the field of 360 degrees, to which the result of a random measurement is relative. However, it is only relative as an algebraic statement relative to the circle being measured. Since a circle (or "game") is already only a statement of algebra ...this renders it meaninglessly relative to the circle and quadrature Laplace was promoting and the geometry he was looking for.  

That is what Laplace necessarily did to obtain the 60 degrees he needed to be able to multiply 22.5 degrees by 16 to obtain the results in degrees (or 60 degrees by 6 if the results are expressed as percentages). He simply changed the Needle’s length.  

The problem for Laplace and his take on the Needle is that he couldn’t use “action at distance” over three random measurements to get the result of 60 degrees that is delivered by the randomness of “action at a distance” and the original Needle. That would have exposed the sham of both his acceptance/usurpation of the original Needle in 1770, and of his fraudulent "discovery" of quadrature (the Needle’s quadrature) in 1772, and his use of it to attack Boskovic’s finesse methodology of three random measurements in 1776.

To protect his plagiarism of the Needle, Laplace had to start with 60 degrees instead of the original Needle’s 45 degrees (straight line connecting the ends of a 90 degree arc which is the original Needle’s length) because otherwise, in 1812, he would have been right back where he was in 1776 …where he was proved wrong in the Boskovic debate. Laplace’s solution was to arbitrarily lengthen the Needle. This intentionally lost the unique geometry of the original Needle’s random length. It made the original Needle’s length geometrically meaningless and no more than equally algebraic with any or all other lengths.  

Without the original Needle and its insistence on relative 1/4 pi, Laplace could use the original Needle’s paradox of quadrature and twist it into an appearance of randomness without going through “action at a distance” and pi....  

...All Laplace had to do was make his "cylindre" longer than the original Needle’s random length.

Laplace made his "cylindre" the entire length of the diameter distance between the lines. Laplace claimed there were fewer “errors” when using his length. By that he meant his longer length crossed a line more often than a shorter length (such as --by implication-- Buffon’s length of Needle). Therefore, according to Laplace, pi could be calculated faster.

Instead of recognizing the original Needle’s relative valuation being relative to a field’s complete diameter (which is random “1.” and to which everything random is relative) Laplace left the muddled inference that the value of a randomly measured field’s radius is “1.” ...from which it may be algebraically deduced that the complete diameter therefore has a random value of: "2." ...and therefore a circle is deductively valued as: "2pi."

History has since interpreted Laplace's disingenuous comment to mean that Buffon made an “error” that Laplace somehow “corrected.”

Ever since, the original Needle has been considered as no more that a quaint way of calculating pi. It is now sometimes called the "Buffon/Laplace Needle Problem" or the "Laplace/Buffon Needle Problem" or even the "Laplace Needle Problem."

In point of fact, there is no error in the original Needle and no one has ever found or identified one. The "error" was entirely Laplace’s.

Laplace did not make an error of algebraic calculation. Rather, it appears a matter of malice in which Laplace was concealing the original Needle's random geometric truth of pi and quadrature and his usurpation of it. Laplace's alteration of the original Needle is perhaps the greatest disaster in the history of commerce and science and education, especially including mathematics. It even derailed Albert Einstein's quest for the grail!

Laplace's change made the random geometry of relative 1/4 pi algebraically equally to any other length of Needle or cylindre. In doing so, he dismissed and distained the unique geometric values which the original Needle attached to the random gravitational nature of relative 1/4 pi. That lost the unique fundamental geometric values of “1.” and pi that are only found with the random circumstances of the original Needle.

History, including Einstein, followed Laplace's lead. "One complete revolution corresponds to the angle 2pi in the absolute angular measure customary in physics...." (Einstein, Albert, tr. Lawson. Relativity. Crown, 1961. p.125).

The original random values of "1." and pi ...are restored and explored here. As previously noted, even after it was altered, the power of the Needle is such that when physicists built the first atomic reactor they had to randomly toss nails on a grid floor to determine the critical geometric probability of random particle collision.

The formula for the gravity bet is the formula for the original Needle, extended with the geometric finesse, factored by two directions. This delivers the random flat bet advantage: .16666 !

In the world of science and education, the original Needle has never fully recovered since its loss in 1795, and again in 1812. The original Needle never had a chance to develop beyond its original introduction.

Laplace’s conduct and cover up was surely the most colossal fraud in history. It has been the basis of science ever since, including the stock market and the insurance and gaming industries!

It may be fairly said that any deep understanding of randomness and geometric probability and "action at a distance," must start with the original Needle.