Most of what went on in the ancient times is speculation, so bare with me for a minute. It's speculated that the peoples of the ancient times (before 2000 B.C.) discovered relationships, such as the bigger the object, the heavier it is. We speculate that they could have found the relationship between diameter and circumference.
The of course weren't able to eloquently write it out as we have today. The Greek letter, π, until the 18th century. The relationship was discovered before then, but the latin phrase, Quantitas, in quam cum multiplicetur diameter, proveniet circumferential, was what was used.
By 2000 B.C., ancient Babylonians knew that pi was between 3 1/7 and 3 1/8. Nobody knows how they knew this. One guess what by drawing a circle in the sand, and using a rope as the diameter. From there, they could notice that the diameter went around the circle 3 and somewhere between 1/7 and 1/8 of a time.
In I Kings vii. 23 and 2 Chronicles iv. 2 from the old testament of the bible, "Also, he made a molten sea of ten cubits from brim to brim, round in compass, and five cubits the height there of; and a line of thirty cubits did compass it round about." This is stating that pi is equal to 3.
Last but not least, another speculation is that they used rearrangement of pieces. See: Top
Early Egyptians and Greeks
The ancient Babylonians knew that the perimeter of a hexagon is equal to six times the radius of a circumscribed circle. This gives us the ratio of 6r/C, where r is the radius and C is the circumference of the circumscribed circle. Using the Babylonian sexagesimal system and the definition π = C/2r, we get 3/π = 57/60 + 36/(60*60). Which then gives us π= 3 1/8 = 3.125.
Between 2000 and 1800 B.C. there was an ancient Egyptian named Ahmes that discovered the value of pi to be approximately 4*(8/9)*(8/9). He used a circular field with a diameter of 9 units, and said it was equal to a square field with a side of 8 units. π(9/2)*(9/2) = 8*8. Which then using a little bit of algebra, we get π = 4*(8/9)*(8/9).
Ahmes used the relation of a circle, square and octagon to arrive at that weird number also. The circle was inscribed in a square, with an irregular octagon being formed by trisecting the square with side length 9 units,
and cutting off the corners of the square. This means that the area of the octagon is 63 square units.
Ahmes then states that 63 is approximately 64 which is then 8*8. With the circle being about the same size as the octagon, he then states that the area of the circle is then about 64 square units. This then leads to that funky value of 4*(8/9)*(8/9). Anaxagoras was in jail for impiety and there he tried to figure out how to "square the circle".
Squaring the circle is being able to find a square that has exactly the same area as a circle.
Another Greek philosopher, named Antiphon thought that a circle could be squared. If you started with an inscribed square and then doubled it's sides, it will eventually become a circle so you will have "squared the circle." Hippias of Elis was the first person to define a curve outside of a segment of a circle. The curve that Hippias defined was called a quadratrix
because it could be used to trisect an angle, and could be used to square the circle, with the mere use of compasses and straightedges. This did not follow the rules set forth by Greek geometry, which will be explained later. Let the straight segment AB move uniformly from the indicated position until the coincides with CD. The curve traced by the intersection (L) of the two segments during this motion is Hippias's quadratrix.
Using only a compass and straightedge, one could construct this curve by dividing the angle AOP into 2^n parts and the segment AO into the same number of parts, with n being arbitrarily large. The proof was given by Pappus (late 3rd century A.D.)
Let OL=ρ and OA=r. From the definition of a quadratrix, we have AL'/(π/(2-θ))=const=2r/π and since AL' = r-ρ sin(θ) we obtain (1)2r= (π ρ(sin(θ)))/θ). For θ->0 we have sin(θ)/θ -> 1 and ρ -> OQ; also 2r=AD so that (1) yields AD:OQ = π and we have the geometrical construction for the circle ratio π
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Euclid did not do much in regards to helping discover pi. The main reason he is mentioned is because he is the one that set up mathematical rigor in proofs. Because of how he had written the elements, it disproved people's ways of figuring out how to square the circle.
The three rules that he had to follow in order to “square the circle” were 1. square the circle, 2. use a straightedge and compass only, and 3. do it in a finite number of steps.
Hippias had used an infinite number of steps and he had used a French curve , which could not be constructed by using a straightedge and compass only.
The five axioms that Euclid used were 1. a straight line may be drawn from any point to any other point 2. A finite straight line may be extended continuously in a straight line, 3. A circle may be described with any center and any radius,
4. all right angles are equal to one another, and 5. given a line and a point not on that line, there is not more than one line which can be drawn through that point parallel to the original line.
Euclid was able to square a rectangle, but was never able to square the circle. It has been proven that one cannot square the circle, following the 3 rules allowed.
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Archimedes
If for whatever reason the applet is not working, please use Archimedes applet
If it could not be guessed, by the applet, Archimedes found an upper and lower bound by using inscribed and circumscribed polygons. The only difference between Archimedes and the applet is that Archimedes did the calculations by hand and he started with a hexagon and doubled the amount of sides the polygon had, up to a 96-gon. Top
One of Euler's contributions to pi was that he made another, faster series to calculate the value of pi. While it was William Jones who was the first person to use the Greek letter, π, as the ratio between the circumference and diameter of a circle, it was Euler who popularized it. (Posamier and Lehmann) Top
Digit Hunters and Computers
Between the time of Archimedes and Newton, there were many people trying to solve for what pi actually is. It wasn't until the time of Euler that they started to realize that there was a possibility of pi not being an exact number. These people were known as the digit hunters. There were a number of different people that I care not too much to mention by name that came to calculate pi to 200+ decimal places without the use of computers.
Once as computers came about, the number of digit hunters went down, but the number of digits that have been calculated, have sky rocketed. Top
There are books now that talk about getting rid of pi, and using tau instead. Tau would just be 2π. This movement is being called the " Tau Manifesto . . Pi is based upon the circumference of a circle being divided by the diameter of the circle. What the pushers of getting rid of pi are saying is that it makes more sense for the “circle constant”
to be the number produced by dividing the circumference of a circle by its radius.They bring to light that the radian measure for 1/4 of a circle would be τ/4 instead of π/2 which tends to confuse people as they learn about the unit circle. They say that we would not have to burn down all of the textbooks and rewrite proofs to match this new constant, but they could coincide. It just makes more sense to have τ instead of π.
Should we keep pi presentation. Top
