SAND RECKONER by Archimedes Translated and edited by Henry Mendel Chapter Four

[1] With some of these supposed and others demonstrated, the proposed claim will be proved. For since it is supposed that the diameter of the poppy-seed is not less than a fortieth-part of an inch [lit. finger], it is clear that the sphere having an inch diameter is not larger than can contain six-ten-thousand and four-thousand poppy-seeds [6,4000]. For they are multiples of the sphere having its diameter a fortieth-part of an inch by the stated number. For it has been proved that spheres have the triplicate ratio to one another of their diameters. [2] It is clear that if a sphere having an inch diameter is filled with sand, the number of the sand would not be larger than ten-thousand-times six-ten-thousand and four-thousand [6,4000,0000]. But this number is 6 units of the second numbers and four-thousand myriads of the first numbers [62 4000,0001]. And so it is smaller than 10 units of the second numbers. But a sphere having a diameter of 100 inches is a multiple of a sphere having an inch diameter by 100 myriads, since the sphere have triplicate ratio to one another of their diameters. And so if there came to be a sphere of sand as large as a sphere having a 100 inch diameter, it is clear that the number of the sand will be smaller than the number which is the multiplication of ten units of the second numbers by 100 ten-thousand [102 * 1003 = 102 * 100,00001]. [3] But since ten units of the second numbers is the tenth number proportionally from the unit in the proportion of terms multiplying-by-ten, and one-hundred myriads is seventh from the unit in the same proportion, it is clear that the number which arises will be sixteenth from the unit of the numbers in the same proportion. For it was proved that it is distant from the unit by one less than the number of the sum [of the distances] which the numbers which multiplied one another are distant from the unit. But of these sixteen, eight are the first numbers with the unit of the so-called first numbers, while the eight after these are of the second numbers. And the last of them is a thousand myriads of the second numbers. And so it is obvious that the multitude of the sand having a magnitude equal to the sphere with a 100 inch diameter is smaller than a thousand myriads of the second numbers. 102 * 100,00001 = 10th + 7th - 1 = 16th number = 1000,00002 note: the n'th number is 10n-1 [4] Again a sphere having a diameter of ten-thousand inches is also a multiple of a sphere having an 100 inch diameter by 100 ten-thousand. And so if there came to be a sphere of sand as large as a sphere having a ten-thousand inch diameter, it is clear that the number of the sand will be smaller than the number which arises when one-thousand myriads of the second numbers is multiplied by 100 myriads. But since one-thousand myriads of the second numbers is the sixteenth number proportionally from the unit, while one-hundred myriads is seventh from the unit in the same proportion, it is clear that the number which arises will be twenty-second from the unit of the numbers in the same proportion. [5] But of these twenty-two, eight are the first numbers with the unit of the so-called first numbers, while the eight after these are of the so-called second numbers, and the remaining six are of the so-called third numbers, and the last of them is ten myriads of the third numbers. And so it is obvious that the multitude of the sand having a magnitude equal to the sphere with a ten-thousand inch diameter is smaller than 10 myriads of the third numbers. And since a sphere having a diameter of a stadium is smaller than a sphere having a diameter of ten-thousand inches, it is clear that the multitude of sand having a magnitude equal to the sphere having a stadium diameter is smaller than 10 myriads of the third number. 1000,00002 * 100,00001 = 16th + 7th - 1 = 22th number = 10,00003 [6] Again, a sphere having a diameter of 100 stadia is a multiple of a sphere having a stadium diameter by 100 myriads. And so if there came to be a sphere of sand as large as a sphere having diameter of 100 stadia, it is clear that the number of the sand will be smaller than the number which arises when 10 myriads of the third numbers is multiplied by 100 myriads. But since 10 myriads of the third numbers is the twenty-second number proportionally from the unit, while 100 myriads is seventh from the unit in the same proportion, it is clear that the number which arises will be twenty-eighth from the unit in the same proportion. [7] But of these twenty-eight, eight are the first numbers with the unit of the so-called first numbers, while the eight after these are of the so-called second numbers, while the eight after these are of the so-called third numbers, while the remaining four are of the so-called fourth numbers, and the last of them is one-thousand units of the fourth numbers. And so it is obvious that the multitude of the sand having a magnitude equal to the sphere with diameter of 100 stadia is smaller than one-thousand units of the fourth numbers. 10,00003 * 100,00001 = 22th + 7th - 1 = 28th number = 10004 [8] Again, a sphere having a diameter of ten-thousand stadia is a multiple of a sphere having a diameter of 100 stadia by 100 myriads. And so if there came to be a sphere of sand as large as a sphere having diameter of ten-thousand stadia, it is clear that the multitude of the sand will be smaller than the number which arises when one-thousand units of the fourth numbers is multiplied by 100 myriads. But since one-thousand units of the fourth numbers is the twenty-eighth number proportionally from the unit, while 100 myriads is seventh from the unit in the same proportion, it is clear that the number which arises will be thirty-fourth from the unit in the same propotion. [9] But of these twenty-eight, eight are the first numbers with the unit of the so-called first numbers, while the eight after these are of the second numbers, and the next eight after these are of the third numbers, and the eight after these are of the fourth numbers, while the remaining two are of the so-called fifth numbers, and the last of them is ten units of the fifth numbers. And so it is clear that the multitude of the sand having a magnitude equal to the sphere with diameter of ten-thousand stadia is smaller than 10 units of the fifth numbers. 10004 * 100,00001 = 28th + 7th - 1 = 34th number = 105 [10] Again, a sphere having a diameter of 100 myriads of stadia is a multiple of a sphere having a diameter of ten-thousand stadia by 100 myriads. And so if there came to be a sphere of sand as large as a sphere having diameter of 100 ten-thousand stadia, it is clear that the number of the sand will be smaller than the number which arises when ten units of the fifth numbers is multiplied by 100 myriads. And since ten units of the fifth numbers is the thirty-fourth number proportionally from the unit, while 100 myriads is seventh from the unit in the same proportion, it is clear that the number which arises will be fortieth from the unit in the same propotion. [11] But of these forty, eight are the first numbers with the unit of the so-called first numbers, while the next eight after these are of the second numbers, and the next eight after these are of the third numbers, while the eight after the third numbers are of the fourth numbers, while the eight after these are of the so-called fifth numbers, and the last of them is one-thousand myriads of the fifth numbers. And so it is obvious that the multitude of the sand having a magnitude equal to the sphere with diameter of 100 myriads of stadia is smaller than one-thousand myriads of the fifth numbers. 105 * 100,00001 = 34th + 7th - 1 = 40th number = 1000,00005 [12] But, a sphere having a diameter of ten-thousand myriads of stadia is a multiple of a sphere having a diameter of 100 myriads of stadia by 100 myriads. And so if there came to be a sphere of sand as large as a sphere having diameter of ten-thousand myriads of stadia, it is obvious that the multitude of the sand will be smaller than the number which arises when one-thousand myriads of the fifth numbers is multiplied by 100 myriads. And since one-thousand myriads of the fifth numbers is the fortieth number proportionally from the unit, while 100 myriads is seventh from the unit in the same proportion, it is clear that the number which arises will be forty-sixth from the unit. [13] But of these forty-six, eight are the first numbers with the unit of the so-called first numbers, while the eight after these are of the second numbers, and the next eight after these are of the third numbers, while the next eight after these are of the fourth numbers, and the eight after the fourth numbers are of the fifth numbers, while the remaining six are of the so-called sixth numbers, and the last of them is 10 myriads of the sixth numbers. And so it is obvious that the multitude of the sand having a magnitude equal to the sphere with diameter of ten-thousand myriads of stadia is smaller than 10 myriads of the sixth numbers. 1000,00005 * 100,00001 = 40th + 7th - 1 = 46th number = 10,00006 [14] But, a sphere having a diameter of 100 ten-thousand myriads of stadia is a multiple of a sphere having a diameter of ten-thousand myriads of stadia by 100 myriads. And so if there came to be a sphere of sand as large as a sphere having diameter of 100 ten-thousand myriads of stadia, it is obvious that the multitude of the sand will be smaller than the number which arises when 10 myriads of the sixth numbers is multiplied by 100 myriads. But since ten myriads of the sixth numbers is the forty-sixth number proportionally from the unit, while 100 myriads is seventh from the unit in the same proportion, it is clear that the number which arises will be fifty-second from the unit in the same proportion. [15] But of these fifty-two, forty-eight with the unit are the so-called first numbers as well as the second and third and fourth and fifth and sixth, while the remaining four are of the so-called seventh numbers, and the last of them is one-thousand units of the seventh numbers. And so it is obvious that the multitude of the sand having a magnitude equal to the sphere with diameter of 100 ten-thousand myriads of stadia is smaller than 1000 units of the seventh numbers. 10,00006 * 100,00001 = 46th + 7th - 1 = 52nd number = 10007 [16] And so since the diameter of the world was proved to be smaller than 100 ten-thousand myriads of stadia, it is clear that the multitude of sand having a magnitude equal to the world is less than 1000 units of the seventh numbers. And so it has been proved that the multitude of sand having a magnitude equal to the world so-called by most astronomers is less than 1000 units of the seventh numbers. But it will also be proved that the multitude of sand having a magnitude equal to a sphere as large as that which Aristarchus supposes the sphere of the fixed stars to be is smaller than 1000 myriads of the eighth numbers. [17] For since it is supposed that the earth has the same ratio to the world as described by us which the described world has to the sphere of the fixed stars which Aristarchus supposes, i.e., the diameters of the spheres have the same ratio to one another, but the diameter of the world has been proved to be smaller than ten-thousand-times the diameter of the earth, it is thus clear that the diameter of the sphere of the fixed stars is smaller than ten-thousand times the diameter of the world. [18] But since spheres have to one another the triplicate ratio of their diameters, it is obvious that the sphere of the fixed stars which Aristarchus supposes is smaller than the world multiplied by ten-thousand-times ten-thousand myriads. But it has been proved that the multitude of sand having a magnitude equal to the world is smaller than 1000 units of the seventh numbers. And so it is clear that if there came to be a sphere of sand as large as Aristarchus supposes the sphere of the fixed stars to be, the number of sand will be smaller than the number which arises when the thousand units are multiplied by ten-thousand-times ten-thousand myriads. [19] And since one-thousand units of the seventh numbers is the fifty-second number proportionally from the unit, while ten-thousand-times ten-thousand myriads is thirteenth from the unit in the same proportion, it is clear that the number which arises will be sixty-fourth from the unit in the same proportion. But this is the eighth of the eighth numbers, which would be one-thousand myriads of the eighth numbers. Thus, it is obvious that the multitude of sand having a magnitude equal to the sphere of the fixed stars which Aristarchus supposes is smaller than 1000 myriads of the eighth numbers. 10007 * (1,00001 * 1,00001 * 1,00001) = 10,00006 * (1,00001 * 1,00001 * 1,00001) = 52nd + ((5th + 5th - 1) + 5th - 1) - 1 = 52nd + 13th - 1 = 64th number = 1000,00008 [19] King Gelon, to the many who have not also had a share of mathematics I suppose that these will not appear readily believable, but to those who have partaken of them and have thought deeply about the distances and sizes of the earth and sun and moon and the whole world this will be believable on the basis of demonstration. Hence, I thought that it is not inappropriate for you too to contemplate these things.

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