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E.J. Dijksterhuis included Heiberg's view in the 1987 "Archimedes" biography published by Princeton Press, The discussion begins with an Archimedes Lemma: In Quadrature of the Parabola Archimedes proves the following proposition on the sum of a geometrical progression with a common ratio of 1/4.
Given a series of magnitudes, each of which is equal to four times the order of the next, all of the magnitudes and one-third of the least added together will exceed the greatest by one-third.
Let the magnitudes A, B, C, D, E be given such that
A + B + C + D + E + 1/3E = (4/3)A
Dijksterhuis wrote out the 1/4 geometric infinite series:
4A/3 = A + A/4 + A/16 + A/64 + ...
an infinite series.
Heiberg published a finite Egyptian fraction series side of the discussion, as Dijksterhuis wrote as:
4A/3 = A + A/4 + A/12
that proved the accuracy of a finite 1/4 geometric series method that followed Eudoxus that used the same tradition.
planet math: Archimedes
Originally posted by AwakeinNM
Otherwise, I have no idea what you are talking about. I am right-brain dominant.
Originally posted by crazydaisy
I have no clue as to what you are
writing about, I probably wouldn't know if you explained it.
Sorry.
Originally posted by TrueBrit
However, OP has , fairly , refused to dumb down, and speak english,
Originally posted by TrueBrit
I will go crazy as if you had just shown me a new way to build mortice locks (Im a locksmith) .
Originally posted by bigfatfurrytexan
Damn. LOL, wanna tell me how better to forecast hotel occupancy in a hotel that hasn't even opened yet, in a down economy?
Originally posted by davidgrouchy
Just finished a preliminary analysis of the solar system.
Here are the Archimedes style formulas to get their orbital periods.
Mercury (9 +3/8)^2
Venus (14 +1/1)^2
Earth (19 +1/9)^2
Mars (26 +1/5)^2
Ceres (40 +1/1)^2
Jupiter (65 +4/5)^2
Saturn (104 +1/9)^2
Uranus (175 +1/2)^2
Neptune (245 +1/3)^2
out of 109,197.710156 total cummulative days for all orbital periods combined I get an error of 13.550595444444 days using the formulas above.
An error rate of 0.00012409230
Which compared to the Titus Bode law
an error rate of 0.88564294631
is a seven thousand fold improvement in accuracy.
/Grouch goes to make himself a sandwich
David Grouchy
Originally posted by wirehead
Furthermore.
Let's look at your math.
. . .
Using your value for Phi, the above expression simplifies to just:
E * 0.0808489
Originally posted by wirehead
So what, exactly, were you trying to show, again? That the earth's period, multiplied by an arbitrary number of your choosing, is close to the moon's period? I could have done the same thing in less steps and to greater accuracy... It doesn't prove anything physically.