In this section of the project well explore the work of Archimedes entitled the “Sand-Reckoner.” My goal is to present the reasons why Archimedes spent his time on this problem, and I also want to show some of the Geometric techniques and relations that Archimedes used in proving his hypothesis. The goal of the “Sand-Reckoner” is to show that there are finitely many grains of sand contained in the universe. Before I proceed, I need to define what I mean when I use the Words finite and infinite. Finite is a term used in mathematics that describes any number that one can eventually count too and infinite is the term used far a number that no matter how long you count or whatever method you use to count, you can never reach it.

The numerical values that were used in this problem were used in this problem were exaggerated quite a bit, but this really does not matter because the result was not meant to be an exact calculation as you will see soon. Many of the values used by Archimedes are presented in a form of “x” is less than “y” but “x” is greater than “z” were x, y and z are all numbers or in mathematical form y > x > z. “There are some, king Gelon, who think that the number of the sand is infinite in multitude; and I mean by the sand not only that which exists about Syracuse and the rest of Sicily but also that which is found in every region whether inhabited or uninhabited. Again there are some who, without regarding it as infinite, yet think of no number has been named which is great enough to exceed its multitude” (1, p. 221). In this opening passage of the “Sand-Reckoner” you can see Archimedes has two goals. One of which is to prove to king Gelon and others that the number of grains of sand are indeed finite. The other goal, which is arguably the main goal of the work, was to represent very large numbers; because the Greeks used letters in the alphabet to describe numbers and the larger the number gets the greater the difficulty in representing the number.

The first goal of Archimedes was to show that the grains of sand are finite. To begin, Archimedes made five assumptions,

“1) The perimeter of the earth is about 3,000,000 stadia and not greater.” (1, p. 222)

“2) The diameter of the earth is greater than the diameter of the moon, and the diameter of the sun is greater than the diameter of the earth.” (1, p. 223)

“3) The diameter of the sun is about 30 times the diameter of the moon and not greater.” (1, p. 223)

“4) The diameter of the sun is greater than the side of the chiliagon inscribed in the greatest circle in the (sphere of the) universe.” (1, p. 223)

“5) Suppose a quantity of sand taken not greater than a poppy-seed, and suppose that it contains not more than 10,000 grains.

Next suppose the diameter of the poppy-seed to be not less than 1/40^th of a finger-breadth” (1, p.227).

I feel like I need to cover some definitions before I go any further. The radius of a circle is the distance from the center to the perimeter of a circle. The diameter is twice the radius. The chiliagon is a geometric object called a polygon, which has 1,000 sides with equal length.

To begin to form a geometric argument, Archimedes needed to find out the angle of the sun with respect to his position on the earth. “To determine the first ration Archimedes made an experiment which consisted in sliding a disc along a beam until its distance from his eye was just such as to obscure the sun: the angle subtended by the sun” (2, p. 273). “The result of the experiment was to show that the angle subtended by the diameter of the sun was less than 1/164^th part, and greater than 1/200^th part, of a right angle” (1, p. 224).

 Archimedes' discs slides along the bar until it just obscures the disc of the sun or moon. The angle subtended by the disc to the eye is then equal to that of the sun or moon. Since the latter are nearly equal, the diameters are in the ratio of distances. (2, Figure 4).

These numbers look kind of funny to our way of seeing numbers, so I will show these numbers in a decimal form. So the angle subtended by the diameter of the sun was less than 0.549 degrees, and greater than 0.450 degrees ( 0.549 > (angle subtended) > 0.450). What I mean by subtended angel, is the angle formed by the central line with either diagonal line.

I need to define what Archimedes refers to as the universe. Archimedes refers to the universe as the sphere that has the center of the earth as the center of the universe, and the distance from the center of the earth to the center of the sun is the radius of the universe. The sphere of fixed stars is all of the surrounding stars which is so much larger than the sphere of the universe.

This is my interpritation of Archimedes' description.

Archimedes shows by many different geometric methods which I am not able to show in this format, so I will just go over some of the major achievements that were obtained by simple geometry.

Let Ds=(the diameter of the sun), De=(the diameter of the earth), Dm=(the diameter of the moon), and Du=(the diameter of the universe)

This is my interpretation of the universe again, but this time with the radi and diameters labeled.

This is my version of the geometric relations that Archimedes used for his proof. (1, P. 225)

Showing a few single gemetric figures obtained by the previous figure.

 The diameter of the universe is less than 10,000 times the diameter of the Earth, and the diameter of the universe is less than 10,000,000,000 stadia, (Du < 10,000 x De and Du < 10,000,000,000 stadia). By similar methods, he was able to approximate the diameter of the sphere of fixed stars.

By assumption 5, a poppy seed contains no more than 10,000 grains of sand and the diameter of a poppy-seed is no less than 1/40th of a finger-breadth. Archimedes was then found the number of finger-breadths in a stadia. After finding this relationship, it was easy for him to calculate the number of sand grains contained in the sphere of the universe and the fixed stars. The only problem left is that Archimedes could not represent his results in the number system at the time (1, P. 227-32). 

Now I will discuss the number system Archimedes developed for this problem. Do not worry about understanding it completely because it is quite different in it’s representation than what we commonly use as numbers. If you notice it is an early form of scientific notation for numbers. A word of caution is that the way the numbers are represented are arranged so that we can understand how they read, because Greek numbers are quite different than our numbers.

“I. We have traditional names for numbers up to a myriad (10,000); we can therefore express numbers up to a myriad myriad (100,000,000). Let these numbers be called numbers of the first order. Suppose the 100,000,000 to be the unit of the second order, and le the second order consist of the numbers from that unit up to (100,000,000)^2”… and so on, until we reach the 100,000,000^th order of numbers ending with (1000,000,000)^(100,000,000), which we call P” (1, p. 227-228).

“II. Suppose the numbers from 1 to P just describe the first period. Let P be the unit of the first order of the second period and this consist of the numbers from P up to 100,000,000P. Let the last number be the unit of second order of the second period end with (100,000,000)^2 *P” (1, p. 228).

“III. Taking P^2 as the unit of first order  in the third period, we proceed in the same way till we reach the 100,000,000^th order of the third period ending with P^3” (1, p.228).

IV. Proceeds in the same way until we end up with P^P which is (100,000,000)^ (100,000,000) (1, p. 228).

I bet you have questions about what all of this means. The most important thing about this notation that I want for the reader to understand is that by using this system you can represent very large numbers with the existing numbers used in Greek writing.

In combination with all of this data Archimedes was able to calculate an approximation of the number of grains it would take to fill the universe and the sphere of fixed stars. The number of grains that Archimedes calculated to exist in the sphere of fixed stars is 10^63 (1, p. 232).

This is my interpretation of the sphere of fixed stars based on what Archimedes' theory is based off of.

Interpretation

The "Sand-Reckoner" was a work with two motives. One motive was to show that the grains of sand is not infinite, and also to create a number system to represent very large numbers which was at that time not possible. Archimedes showed the grains of sand are finite as follows, if the grains of sand are infinite then this implies the universe is infinite. Since there are infinite grains of sand the infinite universe is full of sand and nothing else. Since we know that the whole universe is not composed of sand alone we can see that there cannot be infinite grains of sand. In presenting this argument mathematically Archimedes created the “P-system,” as described earlier, to show the number of grains of sand to fill the known universe.  

Sources

(1   1.)   Archimedes, The Works of Archimedes: Edited in Modern Notation With Introductory Chapters. T. L. Heath, Cambridge: Cambridge University Press, 1897.

       Notes: This work was essential to my project because it presented an English translation to the original work that was done by Archimedes. It contained notes and diagrams which were also essential to my project.

(2    2.)   Brown, G. Burniston, Why Do Archimedes And Eddington Both Get 10^79 For The Total Number Of Particles in the Universe?. Cambridge: Cambridge University Press. Philosophy, Vol. 15, No. 59 (Jul., 1940), pp. 269-284.

      Notes: This article was essential to my project because it presented the procedure in which Archimedes was able to obtain the angle of the sun with respect to the earth. This procedure was unclear in the original text.