Although I would bet that archimedes spent alot of time getting the most accurate number possible, even though he didn't have a calculator.
Is there someone who knows how Archimedes did when he got the approximation 355/113? I mean, calculating pi with float values can be done, but he didn't have a calculator, did he calculate pi as a decimal number first, with a fixed number of decimals, and then made it a fraction, or did he keep it a fraction all the time? What formula did he use?
Although I would bet that archimedes spent alot of time getting the most accurate number possible, even though he didn't have a calculator.
The concept of the continued fraction originated from Leonardo Pisano Fibonacci from Medieval Ages.
It is a fraction of the form,
And they can get longer.
Sometimes even infinite.
There is a way to get a continued fraction for any number.
starts out (the fraction is infinite),
Now if you terminate this fraction you get,
Are all the convergents.
The beauty about this fractions is that they are the best possible fractions with that size. There can be no better. Also the rate at which they reach decimal accurate is excellent.
There are several problems however, the ideas of continued fractions did not exist at the time of Archimedes (but who knows he is considered to be the Greatest mathemation). Also, I do not think that fraction belongs to Archimedes.
I think how it goes it that Archimedes was able to show geometrically that is a close estimate. I do not think that belongs to him.
used ratios or ordinary fractions to express the values he found.
What he did was to calculate the perimeter of inscribed and circumscribed
regular polygons for a circle of a given diameter , and as knew that
the circumference of the circle was , he had for any given number of
sides of the polygon trapped this between the perimeter of the inscribed and
Now he just did the calculation for as many sides as he needed for his
estimates (96 seems to be the maximum number of sides he used). Which
Note that A. did not just estimate but an interval that
contains the value of .
The question is about what Archimedes did, not what the ImPerfectHacker
A valuable lesson here for when you have to do your exams is to
answer the question asked, not the question you would have asked.
(unless there is an obvious mistake in the question, when you
should do whatever your school advises - the advice in my day
was to point out the error then if the question was impossible move
on and you would be given full marks, otherwise answer what was asked).
As CB mentioned, polygons inscribed in a circle was Archimedes' attack.
Start with a square inscrobed in a circle and begin doubling the number of sides to obtain a octagon then a 16-gon, then a 32-gon. Get the picture.
Archimedes allegedly went to a 96-gon.
We know the perimeter of these polygons will tend to a limit of twice . The lengths of the sides of the polygons can be derived by Pythagoras. Let s and t be the sides of two successive polygons, one having twice the number of sides as the other.
From the diagram:
AC=s and AB=t. B is the midpt of arc AC. Square the binomial in a:
Use b to sub into the parentheses in the above:
Also, a gives:
Multiply both sides by 2:
we get a formula for the one side of the (n+1)st polygon in terms of that of the nth polygon:
But, is the side of the inscribed square in the diagram.
So, we get:
In general we have:
, with n radicals.
Since the perimeter of the nth polygon is
an approaximation to PI is
We can write this as:
I once researched this because I found it interesting.
Hope it helps.
he produced not a point estimate for , but an interval estimate.
Which makes his thinking modern and completely ahead of his time (about 2000 years).
(P.S. A family joke based on my Mum's translation of the meaning of is name is
that "Archimedes" translates into English as "Big Head")
seige of Syracuse:
Marcellus now moved with his whole army to Syracuse, and, camping near the wall, proceeded to attack the city both by land and by sea. The land forces were conducted by Appius: Marcellus, with sixty galleys, each with five rows of oars, furnished with all sorts of arms and missiles, and a huge bridge of planks laid upon eight ships chained together, upon which was carried the engine to cast stones and darts, assaulted the walls, relying on the abundance and magnificence of his preparations, and on his own previous glory; all which, however, were, it would seem, but trifles for Archimedes and his machines.
These machines he had designed and contrived, not as matters of any importance, but as mere amusements in geometry; in compliance with King Hiero's desire and request, some little time before, that he should reduce to practice some part of his admirable speculations in science, and by accommodating the theoretical truth to sensation and ordinary use, bring it more within the appreciation of people in general. Eudoxus and Archytas had been the originators of this far-famed and highly prized art of mechanics, which they employed as an elegant illustration of geometrical truths, and as a means of sustaining experimentally, to the satisfaction of the senses, conclusions too intricate for proof by words and diagrams. As, for example, to solve the problem, so often required in constructing geometrical figures, given the two extreme, to find the two mean lines of a proportion, both these mathematicians had recourse to the aid of instruments, adapting to their purpose certain curves and sections of lines. (The 'mesolabes or mesalabium, was the name by which this instrument was commonly known.) But what with Plato's indignation at it, and his invectives against it as the mere corruption and annihilation of the one good of geometry,--which was thus shamefully turning its back upon the unembodied objects of pure intelligence to recur to sensation, and to ask help (not to be obtained without haste subservience and depravation) from matter; so it was that mechanics came to be separated from geometry, and, being repudiated and neglected by philosophers, took its place as a military art. Archimedes, however, in writing to King Hiero, whose friend and near relation he was, had stated, that given the force, any weight might be moved, and even boasted, we are told, relying on the strength of demonstration, that if there were another earth, by going into it he could remove this. Hiero being struck with amazement at this, and entreating him to make good this problem by actual experiment, and show some great weight moved by a small engine, he fixed accordingly upon a ship of burden out of the king's arsenal, which could not be drawn out of the dock without great labor and many men; and, loading her with many passengers and a full freight, sitting himself the while far off, with no great endeavor, but only holding the head of the pulley in his hand and drawing the cord by degrees, he drew the ship in a straight line, as smoothly and evenly as if she had been in the sea. The king, astonished at this, and convinced of the power of the art, prevailed upon Archimedes to make him engines accommodated to all the purposes, offensive and defensive, of a siege. These the king himself never made use of, because he spent almost all his life in a profound quiet, and the highest influence. But the apparatus was, in a most opportune time, ready at hand for the Syracusans, and with it also the engineer himself.
When, therefore, the Romans assaulted the walls in two places at once, fear and consternation stupefied the Syracusans, believing that nothing was able to resist that violence and those forces. But when Archimedes began to ply his engines, he at once shot against the land forces all sorts of missile weapons, and immense masses of stone that came down with incredible noise and violence, against which no man could stand; for they knocked down those upon whom they fell, in heaps, breaking all their ranks and files. In the mean time huge poles thrust out from the walls over the ships, sunk some by the great weights which they let down from on high upon them; others they lifted up into the air by an iron hand or beak like a crane's beak, and, when they had drawn them up by the prow, and set them on end upon the poop, they plunged them to the bottom of the sea; or else the ships, drawn by engines within, and whirled about, were dashed against steep rocks that stood jutting out under the walls, with great destruction of the soldiers that were aboard them. A ship was frequently lifted up to a great height in the air (a dreadful thing to behold), and was rolled to and fro, and kept swinging, until the mariners were all thrown out, when at length it was dashed against the rocks, or let fall. In the meantime, Marcellus himself brought up his engine upon the bridge of ships, which was called "Sambuca," from some resemblance it had to an instrument of music, but while it was as yet approaching the wall, there was discharged at it a piece of rock of ten talents' weight, then a second and a third, which, striking upon it with immense force and with a noise like thunder, broke all its foundations to pieces, shook out all its fastenings, and completely dislodged it from the bridge. So Marcellus, doubtful what counsel to pursue, drew off his ships to a safer distance, and sounded a retreat to his forces on land. They then took a resolution of coming up under the walls, if it were possible, in the night; thinking that as Archimedes used ropes stretched at length in playing his engines, the soldiers would now be under the shot, and the darts would, for want of sufficient distance to throw them, fly over their heads without effect. But he, it appeared, had long before framed for such occasion engines accommodated to any distance, and shorter weapons; and had made numerous small openings in the walls, through which, with engines of a shorter range, unexpected blows were inflicted on the assailants. Thus, when they who thought to deceive the defenders came close up to the walls, instantly a shower of darts and other missile weapons was again cast upon them. And when stones came tumbling down perpendicularly upon their heads, and, as it were, the whole wall shot out arrows at them, they retired. And now, again, as they were going off, arrows and darts of a longer range inflicted a great slaughter among them, and their ships were driven one against another; while they themselves were not able to retaliate in any way; for Archimedes had fixed most of his engines immediately under the wall. The Romans, seeing that infinite mischiefs overwhelmed them from no visible means, began to think they were fighting with the gods.
Yet Marcellus escaped unhurt, and, deriding his own artificers and engineers, exclaimed "What! Must we give up fighting with this geometrical Briareus, who plays pitch and toss with our ships, and, with the multitude of darts which he showers at a single moment upon us, really outdoes the hundred-handed giants of mythology?" The rest of the Syracusans were but the body of Archimedes' designs, one soul moving and governing all; for, laying aside all other arms, with his alone they infested the Romans, and protected themselves. In fine, when such terror had seized upon the Romans, that, if they did but see a little rope or a piece of wood from the wall, they instantly cried out, "There it is again! Archimedes is about to let fly another engine at us," and turned their backs and fled, Marcellus desisted from conflicts and assaults, putting all his hope in a long siege.
I like to make a comment here.
I once made the most simple proof for the area of circle. Look hier.
Now, I am not satisfied with it. In fact, I am never satisfied with these "it is almost like..." arugments. That I keep on seeing in my engineering class. But yesterday I had a revealation, when I was thinking about this post. I finally realized what Archimedes done to make it a proof.
Say, that the following is true,
But when I multiply by a large number, say, 100000
This approximation fails,
My point is this.... when we divide a circle into many polygons we multiply the area of that small triangle by the number of triangles (which is a huge number) so how do we know that the appoximation is valid? How do we know that they do not start to diverge?
This will answer why my approach cannot be considered legit is because I did not show that. Arcimedes did. He first insribed the circle by polygonals and then he outscribed it also. And he was able to show that the two approximations converged to the same number! (Which is the squeeze principle from Calculus). Thus he had no need to fear that it will fail to approximate eventually. How wonderful his approach.
I agree with you, PerfectHacker, small errors can become huge if we treat them wrong. That's what I realized too when i was thinking about trying to make a computer program that uses this root method to calculate pi.
In the end I'll get a very small number, that I multiply with a very large number. And I get that that very small number by subtracting a number that is almost two from two. It's impossible for me to save the almost-two-number as 2 minus a very small number, cause I get the number itself by taking the square root out of a value that is almost four. Hence I may eventually loose that very small number if I'm not using enough with decimals. Hence, using this method, even though the equation becomes more and more acurate the more times I iterate, if I'm not using numbers with really many decimals when I'm calculating, I can expect my result to get worse and worse the more times I iterate, after a certain number of iterations. Or is it any way to walk around this problem?
So, I really admire Archimedes if this was the method he used. Especially since he hadn't any calculator. But in other hand, was his approximations that good?
And sorry if my English is poor, I am from Sweden. It can be hard sometimes to find the right words writing this kind of text.