The Sand Reckoner

• Archimedes,The Sand-Reckoner (Arenarius), ch. 1 (sects. 1-20) ©
• translated by Henry Mendell (Cal. State U., L.A.)
• Source https://web.calstatela.edu/faculty/hmendel/Ancient%20Mathematics/Archimedes/SandReckoner/Ch.1/Ch1.html

CHAPTER 1

1. Some people believe, King Gelon, that the number of sand is infinite in multitude. I mean not only of the sand in Syracuse and the rest of Sicily, but also of the sand in the whole inhabited land as well as the uninhabited. There are some who do not suppose that it is infinite, and yet that there is no number that has been named which is so large as to exceed its multitude.
2. It is clear that if those who hold this opinion should conceive of a volume composed of the sand as large as would be the volume of the earth when all the seas in it and hollows of the earth were filled up in height equal to the highest mountains, they would not know, many times over, any number that can be expressed exceeding the number of it.
3. I will attempt to prove to you through geometrical demonstrations, which you will follow, that some of the numbers named by us and published in the writings addressed to Zeuxippus exceed not only the number of sand having a magnitude equal to the earth filled up, just as we said, but also the number of the sand having magnitude equal to the world.
4. You grasp that the world is called by most astronomers the sphere whose center is the center of the earth and whose line from the center is equal to the straight-line between the center of the sun and the center of the earth, since you have heard these things in the proofs written by the astronomers. But Aristarchus of Samos produced writings of certain hypotheses in which it follows from the suppositions that the world is many times what is now claimed.
5. For he supposes that the fixed stars and the sun remain motionless, while the earth revolves about the sun on the circumference of a circle which is placed on the middle road, but that the sphere of the fixed stars, which is placed about the same center as the sun, is so large in magnitude that the circle on which he supposes the earth to revolve has the sort of proportion to the distance of the fixed stars that the center of the sphere has to the surface.
6. This is trivially impossible, since the center of the sphere has no magnitude. One must suppose that it doesn't have any ratio either to the surface of the sphere. We must understand this such that Aristarchus means this: since we suppose the earth is just like the center of the world, the ratio which the earth has to the world described by us is the same as the ratio that the sphere on which the circle is on which he supposes the earth to revolve has to the sphere of the fixed stars. For he applies the demonstrations of the phainomena[sic] to what is supposed here, and the magnitude of the sphere on which he makes the earth move appears above all to be supposed equal to the world described by us.
7. In fact we say that even if a sphere of sand were to become as large in magnitude as Aristarchus supposes the sphere of the fixed stars to be, we will also prove that some of the initial numbers having an expression exceed in multitude the number of sand having a magnitude equal to the mentioned sphere, when the following are supposed.
8. First that the perimeter of the earth is about 300,000 stadia and not larger, although some have attempted to demonstrate it , as you too follow them, as being about 30,000. But since I am exceeding this and posit the magnitude of the earth as ten-times what was believed by the earlier astronomers, I suppose the perimeter of it to be about 300,000 and not larger.
   • After this, I suppose that the diameter of the earth is larger than the diameter of the moon and that the diameter of the sun is larger than the diameter of the earth, and assume the same things in like manner as most of the earlier astronomers.
9. After these, I suppose that the diameter of the sun is about thirty-times the diameter of the moon and not larger, although of earlier astronomers Eudoxus declared about nine-times, Pheidias, my father, about twelve-times, and Aristarchus has attempted to prove that the diameter of the sun is more than eighteen-times the diameter of the moon and smaller than twenty-times. But I will also exceed this amount, so that what is proposed be indisputably proved, and suppose that the diameter of the sun is about thirty-times the diameter of the moon and not larger.
10. In addition to these, I suppose that the diameter of the sun is larger than the side of the chiliagon inscribed in the largest circle of those in the world. I suppose this given that Aristarchus has found the sun appears about one seven hundred and twentieth of the circle of the zodia, but having examined it in the following manner I attempted with instruments to get the angle into which the sun fits and which has its vertex at the eye.
11. And so it is not easy to get precision since neither the eye nor the hands nor the instruments through which we must get it are trustworthy at declaring precision. For the present it is not timely to lengthen our discussion about these things, especially since these sorts of things have been explained many times. For the demonstration of the proposed claim, it suffices for me to get an angle which is no larger than the angle into which the sun fits and which has its vertex at the eye, and again to get another angle which is not smaller than the angle into which the sun fits and which has its vertex at the eye.
12. And so a long ruler was placed on a straight footing lying in a place from where the sun would be seen rising, and a small bored cylinder was placed on the ruler upright straightaway after the rising of the sun, and then when it was on the horizon and could be looked at right on, the ruler was turned around into the sun, and the eye was positioned at the end of the ruler. Lying between the sun and the eye, the cylinder blocked the view of the sun. And so with the cylinder separated from the eye, the cylinder was positioned in the spot where a little bit of the sun would begin to appear on each side of the cylinder.
13. And so if it had been the case that the eye sees from one point, with straight-lines drawn from the end of the ruler in the place where the eye was positioned and tangent to the cylinder, then the angle enclosed by the lines drawn would have been smaller than the angle into which the sun fits and which has its vertex at the eye, since a bit of the sun was glimpsed on each side of the cylinder. Since eyes do not see from one point, but from a magnitude, a round magnitude was taken not smaller than an eye, and with the magnitude placed on the end of the ruler in the place where the eye was positioned, after straight lines tangent to the magnitude and the cylinder were drawn, the angle enclosed by the drawn lines was thus smaller than the angle into which the sun fits and which has its vertex at the eye.
14. The magnitude no smaller than the eye is found in this way. Two narrow cylinders equal to one another in thickness are taken, one white, one not, and they are placed before the eye, with the white set apart from it and the non-white as near as possible to the eye, so that it is even touching the face. And so, if the cylinders taken are narrower than the eye, the cylinder nearby will be encompassed by the eye and the white will be seen by it, and if they are much narrower, the whole will be seen, while if they are not much narrower, certain parts of the white will be seen on each side of the cylinder near the eye. But when these cylinders are somehow taken suitably for their thickness, one of them blocks the view of the other and not a larger place. Now such a magnitude which is the thickness of the cylinders which produce this effect is above all, I suppose, not smaller than the eye.
15. The angle which is not smaller than the angle into which the sun fits and which has its vertex at the eye was taken in this way. With the cylinder positioned on the ruler away from the eye in such a way that the cylinder blocked the view of the whole sun and with straight-lines drawn from the end of the ruler in the place where the eye was positioned and tangent to the cylinder, (diagram 3) the angle enclosed by the straight-lines drawn becomes no smaller than the angle into which the sun fits and which has its vertex at the eye.
16. When a right angle is measured out by angles taken in this way, with the right angle divided into 164 parts the angle at the point comes to be smaller than one of these parts, and, with the right angle divided into 200 parts, the smaller angle comes to be larger than one of these parts. And so it is clear that the angle into which the sun fits and which has its vertex at the eye is also smaller than, with the right angle divided into 164, one of these parts, but larger than, with the right angle divided into 200, one of these parts.
17. When we put our trust in these things, the diameter of the sun is proved to be larger than the side of the chiliagon inscribed in the largest circle of those in the world.
    1. For let there be conceived a plane extended through the center of the sun and the center of the earth and through the eye, with the sun a little above the horizon.
    2. Let the extended plane cut the world along circle ABG, the earth along circle DEZ, the sun along circle SH, and let there be a center of the earth, Q, a center of the sun, K, and let there be an eye, D.
    3. And let straight-lines tangent to circle SH be drawn from D, namely DL, DX, with tangents at N and T, (diagram 3) but from Q, QM, QO, with tangent at C and R, and let QM, QO cut circle ABG at A and B
    4. In fact QK is larger than DK, since the sun is supposed to be above the horizon.
    5. Thus the angle enclosed by DL, DX is larger than the angle enclosed by QM, QO.
    6. But the angle enclosed by DL DX is larger than a two hundredth part of a right angle and smaller than, with the right angle divided into 164 parts, one of these parts. For it is equal to the angle into which the sun fits and which has its vertex at the eye.
    7. Thus the angle enclosed by QM, QO is smaller than, with the right angle divided into 164, one of these parts, (diagram 8) but straight-line AB is smaller than the line subtending one segment of the circumference of circle ABG divided into 656.
    8. But the perimeter of the mentioned polygon to the line from the center of circle ABG has a smaller ratio than 44 to 7 since the perimeter of every polygon inscribed in a circle to the line from the center has a smaller ratio than 44 to 7.
    9. For you know what was proved by us, that the circumference of every circle is larger than three-times the diameter by smaller than a seventh part, but the perimeter of the inscribed polygon is smaller than this.
    10. And so BA to QK has a ratio smaller than 11 to 1148.
    11. Thus BA is smaller than a hundredth part of QK.
    12. But the diameter of circle SH is equal to BA, since its half FA is also equal to KR.
    13. For, since QK, QA are equal, perpendiculars from their end-points are joined under the same angle.
    14. And so it is clear that the diameter of circle SH is smaller than a hundredth part of QK.
    15. And diameter EQU is smaller than diameter SH, since circle DEZ is smaller than circle SH. Therefore, both QU, KS are smaller than a hundredth part of QK.
    16. Thus QK to US has a ratio smaller than 100 to 99. [US = QK – (KS + QU, while a < b/n  b-a : b > n-1 : n  b : (b-a) < n : (n-1))]
    17. And since QK is not smaller than QR, but SU is smaller than DT, therefore QR to DT would have a ratio smaller than 100 to 99. [a � c & b < d  a : b > c : d]
    18. Since, given that QKR, DKT are right-angled triangles, sides KR, KT are equal, while QR, DT are unequal with QR larger, the angle enclosed by DT, DK to the angle enclosed by QR, QK has a ratio larger than QK to DK, but smaller than QR to DT.
    19. For if in two right-angled triangles one pair of sides about the right angle are equal and the others are unequal, the larger of the angles at the unequal sides to the smaller has a ratio larger than the larger of the lines subtending the right angle to the smaller, but smaller than the larger of the lines at the right angle to the smaller.
    20. Thus the angle enclosed by DL, DX to the angle enclosed by QO, QM has a ratio smaller than QR to DT, which has a ratio smaller than 100 to 99. Thus too the angle enclosed by DL, DX to the angle enclosed by QM, QO has a ratio smaller than 100 to 99.
    21. And since the angle enclosed by DL, DX is larger than a two hundredth part of a right angle, the angle enclosed by QM, QO would be larger than 99 of these parts of the right angle divided into 2,0000. [since LDX > /200 & LDX : MQO < 100 : 99] Thus it is larger than, with the right angle divided into 200 and 3, one of these parts. [since /203 < 99*/2,0000] Therefore, BA is larger than the line subtending one segment and dividing the circle into 812 [= 4 * 203]. But the diameter of the sun is equal to AB. And so it is clear that the diameter of the sun is larger than the side of the chiliagon.

CHAPTER 2

1. Given these as supposed, the following will also be proved, namely the diameter of the world is less than ten-thousand-times the diameter of the earth, and furthermore the diameter of the world is less than 100 ten-thousand-times ten-thousand stadia, For since it is supposed that the diameter of the sun is not larger than thirty-times the diameter of the moon, but that the diameter of the earth is larger than the diameter of the moon, it is clear that the diameter of the sun is less than thirty-times the diameter of the earth. Again, since the diameter of the sun was proved to be larger than the side of a chiliagon inscribed in the greatest circle of those on the world, it is obvious that the perimeter of the mentioned chiliagon is smaller than one-thousand-times the diameter of the sun. But the diameter of the sun is smaller than thirty-times the diameter of the earth. Thus the perimeter of the chiliagon is smaller than three-ten-thousand-times [3,0000] the diameter of the earth.
2. And so since the perimeter of the chiliagon is smaller than thirty-thousand-times the diameter of the earth, it is larger than three-times the diameter of the world. For it has, in fact, been proved that because the diameter of every circle is smaller than a third part of the perimeter of every polygon which is equilateral and having more angles than the hexagon inscribed in the circle. The diameter of the world would then be smaller than ten-thousand-times the diameter of the earth. And so the diameter of the world has been proved smaller than ten-thousand-times the diameter of the earth,
3. while it is clear from this that the diameter of the world is smaller than 100 ten-thousand-times ten-thousand stadia. For since it is supposed that the perimeter of the earth is not larger than three-hundred ten-thousand (300,0000), but the perimeter of the earth is larger than three-times the diameter due to the fact that the circular-arc of every circle is larger than three times the diameter of every circle, it is clear that the diameter of the earth is smaller than 100 ten-thousand stadia (100,0000). And so since the diameter of the world is smaller than ten-thousand-times the diameter of the earth, it is clear that the diameter of the world is smaller than 100 ten-thousand-times ten-thousand stadia.

4. 

   The image is just an illustration, illustrating the definition of 1 grain of sand

   And so I suppose these concerning the magnitudes and distances, while concerning the sand these. If there is a magnitude composed from the sand not larger than a poppy-seed, the number of it is not larger than ten-thousand, while the diameter of the poppy-seed is not larger than a fortieth-part of an inch, I suppose this after examining it in this way. Poppy-seeds were placed on a smooth ruler in a straight line, lying one by one and touching one another. 25 poppy-seeds took up more place than an inch length. And so, having put the diameter of the poppy-seed as smaller, I suppose it to be about a fortieth-part of an inch and no smaller, since I wish also through these to prove indisputably what's proposed.

CHAPTER 3

Well, in chapter 3, Archimedes has started discussing googology, and finally he was able to make the biggest googology he ever made and maybe the oldest googologism in the world.

1. And so these are what I suppose, but I consider it useful for the way of expressing numbers to be stated, so that even those others who haven't come across the book written to Zeuxippus are not at a loss since nothing has yet been said about it in this book.
2. In fact the names of the numbers up to the name of ten-thousand happen to have been provided to us, and beyond the name of ten-thousand we ascertain a number of ten-thousands of units when we say, "even up to ten-thousand myriads." And so let the presently stated numbers up to ten-thousand myriads be called by us first numbers, and let ten-thousand myriads of the first numbers be called a unit of the second numbers, and let units of the second numbers be counted, i.e. from the units decads (10's), hekatontads (100's), chiliads (1000's), myriads (1,0000's) up to ten-thousand myriads. Again let ten-thousand myriads of the second numbers be called a unit of the third numbers, and let units of the third numbers be counted and from the units decads, hekatontads, chiliads, and myriads up to ten-thousand myriads.
3. In the same manner let ten-thousand myriads of the third numbers also be called a unit of the fourth numbers, and let ten-thousand myriads of the fourth numbers be called a unit of the fifth numbers, and repeatedly proceeding in this way let the numbers have names up to the ten-thousand myriads of the ten-thousand ten-thousandth numbers. $(ex:100,000,000^{100,000,000}=10^{800.000.000})$And so numbers are adequately ascertained up to so many, but it is also possible to proceed further.
4. For let the presently described numbers be called of-the-first-period. Let the last number of the first period be called a unit of the second period of the first numbers. Again, let ten-thousand myriads of the first numbers of the second period be called a unit of the second numbers of the second period. Similarly, let the last of these be called a unit of third numbers of the second period, and repeatedly proceeding in this way the numbers of the second period have names up to ten-thousand myriads of the ten-thousand ten-thousandth numbers. Again let the last number of the second period also be called a unit of the first numbers of the third period, and given that let them repeatedly proceed in this way up to ten-thousand myriads of the ten-thousand-ten-thousandth numbers of the ten-thousand-ten-thousandth period. $(ex:100,000,000^{100,000,000^{100,000,000}}=10^{10^{10^{8.90}}})$
5. Given these numbers as expressed in this way, if numbers are set out in continuous proportion from a unit, with a decad next to the unit, then the first eight numbers including the unit will be among what we called the first numbers, while the next eight after them will be among what we called second numbers, and the others in the same way as these will belong to numbers called with the same name by the distance [counted out by] an octad of numbers from the first octad of numbers.
   1. And so in the first octad of numbers, the eighth number is a thousand ten-thousand, while in the second octad, the first, since it is ten-times what preceeds it, will be ten-thousand myriads. But this is the unit of the second numbers. The eighth of the second octad is a thousand myriads of the second numbers. Again the first of the third octad, since it is ten-times what preceeds it, will be ten-thousand myriads of the second numbers. But this is a unit of the third numbers. It is obvious that there will also be many, many octads, as was said.
6. This too is usefully ascertained. If when numbers are proportional from the unit, some of the numbers from the same proportion multiply one another, the number which arises will be from the same proportion and will be distant from the larger of the numbers which multiplied one another as much as the smaller of the numbers which multiplied one another is distant proportionally from the unit, but it will be 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.
7. For let there be some numbers in proportion from a unit, A, B, C, D, E, F, G, H, I, J, K, and let A be a unit, and let D be multiplied by H, and let the number that arises be X. Let K be taken from the proportion as being as distant from H as D is from unit A. We must prove that X is equal to K. And so since, given that the numbers are proportional, D from A and K from H have equal distances, D to A has the same ratio as K to H. But D is a multiple of A by D. Therefore, K is also a multiple of H by D. Thus K is equal to X.
8. And so it is clear that the number which arises is from the proportion and is distant from the larger of the numbers which multiplied one another as much as the smaller is from the unit. But it is obvious that it also is distant from the unit by one less than the number of the sum [of the distances] which D, H are distant from the unit. For A, B, C, D, E, F, G, H are as many as H is distant from the unit, while I, J, K are fewer by one than the distance that D is from the unit. For with H they are as many.

CHAPTER 4

This chapter is about calculating the amount of sand in the limited heliocentric universe using mathematical calculations that are difficult for ordinary people to understand.

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 [6,400]. 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, 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
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.
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.
   1. 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.
   1. 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.
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.
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.
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.
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.
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.
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.
20. King Gelon, to the many who have not also had a share of mathematics I suppose that g 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 hand 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.

ARCHIMEDES' WRITINGS ABOUT THE SAND RECKONER ARE FINISHED HERESee below to see my comments and writings about this,

The Two Notation

IMPORTANT NOTE: THIS CHAPTER WAS BORN FROM MY MISCONCEPTION ABOUT THE SAND RECKONER, THIS IS NOT AN ORIGINAL ARCHIMEDES NOTATION, THIS IS NOT TRUE, I HAVE CONFIRMED IT IN THE NEXT CHAPTER,

NOTATION O_n and Q_n I MADE IT 100% NOT ARCHIMEDES, IF I, TAMATAK SOCKPUPPET SAID ARCHIMEDES MADE THIS, I WAS REALLY WRONG, BECAUSE I JUST REALIZED A FEW DAYS LATER SORRY IF I MADE YOU WRONG, THIS WAS TAMATAK SOCKPUPPET FAULT!

,

this number seems to be the large number that was known at that time because numbers that are bigger than this might be defined as grains of sand on the beach or stars in the sky because they don't know how big they are and it came to pass that this myriad is a first order number and how about second order, second order is 10^{4 x 2} or 10^{4 x (2^1))

$O_{2}=(10^{4})\times (10^{4})=10^{8}$

This number was considered large at that time because there was no scientific notation or exponentiation. If there was exponentiation, maybe people would compete to make large numbers bigger than this, but because in the past there was only multiplication and addition, the numbers were small.

then as Archimedes said, we have to be able to multiply the myriad and myriad to create the second order, Archimedes here has made a googological notation even though the growth is still slow, why? This happened because before there was no such thing as hyperoperation such as exponentiation and tetration, so Archimedes only used addition, (Note It was confirmed in the next chapter that this O_n notation came from my misconception of the actual Archimedes notation)

$O_{3}=O_{2}\times {O_{2}}=1\underbrace {0^{8}+10^{8}+10^{8}\times 10^{8}+10^{8}+10^{8}+...10^{8}} _{10^{8}}$

as said in the following quote

In the same manner let ten-thousand myriads of the third numbers also be called a unit of the fourth numbers, and let ten-thousand myriads of the fourth numbers be called a unit of the fifth numbers, and repeatedly proceeding in this way let the numbers have names up to the ten-thousand myriads of the ten-thousand ten-thousandth numbers. And so numbers are adequately ascertained up to so many, but it is also possible to proceed further.

  —Psammites 3:3

THESE TWO NOTATIONS WERE BORN FROM MY MISCONCEPTIONS ABOUT SAND RECKONER:

In this case I have arleady used recursion to create large numbers even though he only used multiplication

In the end we will understand the formula for this primitive notation of addition as follows

$O_{n}=O_{n-1}\times {O_{n}-1}=\underbrace {O_{n-1}+O_{n-1}+O_{n-1}+O_{n-1}+...} _{O_{n}-1}whereO_{1}=10,000$

where O_1 = 10,000 (myriad)

then there are those who are confused about the 4th order or 5th order, maybe there are those who are confused too, to understand this I will ask a simple question

what is the number of googol x googol

This is a simple question but many people are confused. Maybe people who haven't studied power law will answer that the number is Googolplex or Googolplexian, but in fact the number is smaller than that, much smaller than that. This is power law.

One of the ten exponent rules put forward by Archimedes in Sand Reckoner is that if the numbers 10^a and 10^b are multiplied then the result is 10^(a+b), the number is very small and above it only uses addition.

If we apply this to the problem I created previously the results may be very surprising to lay people who have not studied exponents

$10^{100}\times 10^{100}=10^{100+100}=100^{200}$

Look at the results, it might be a bit surprising for people who are still laymen, for those who don't believe it, try checking the results for yourself on wolfram alpha

This applies not only to Googol but also to other exponents, for example 10^{8} * 10^{8} as follows

This is a definite formula that is easier to understand than the previous formula, it is quite easy

$10^{8}=\times 10^{8}=10^{8+8}=10^{8\times 2}=10^{16}$

If there are two numbers, how will it happen, we can make a list as follows, this is for a period of up to 10 periods

1. O_1 = 10^4
2. O_2 = (10^4)*(10^4) = 10^(4+4) = 10^(4*2) = 10^16
3. O_3 = (10^16)*(10^16) = 10^(16+16) = 10^(16*2) = 10^32
4. O_4 = (10^32)*(10^32) = 10^(32+32) = 10^(32*2) = 10^61
5. O_5 = (10^64)*(10^64) = 10^(64+64) = 10^(64*2) = 10^128

now we have seen a very orderly pattern. Maybe you often see this sequence of numbers everywhere, especially in computing, in RAM specifications, right? For those who don't know, THIS IS THE POWER OF 2

FINALLY we have found a formula that is truly certain to determine what this number is. In fact, this is a definite formula without using formulas that are confusing to see.

$Q_{n}=10^{(2^{n})}$

what if instead of using addition and multiplication, we would use exponentiation and tetration, this makes the notation I made very fast, I named this notation Q(n) why Q because the letter Q is similar to the letter O, later there will also be the omega and omicron notation,

$Q_{n}={Q_{n-1}}\underbrace {Q_{n-1}^{Q_{n-1}^{Q_{n-1}^{Q_{n-1}^{Q_{n-1}^{Q_{n-1}...}}}}}} =Q_{n-1}\uparrow \uparrow Q_{n-1}whereQ(1)is10,000$

for the rule Q(1) is the same as 10,000 while Q(0) is "super root of 10,000 I can't say because it's small and I can't count, and the number n must be a positive integer

"Kuklinasio" this number may be included in the tetration class or up arrow notation level, I don't know how big this number is

or if I want I will create a number, namely Q(10.000) which I will name "Kuklinasio" (KOOK LEE NA SEE YO), maybe this number is in the tetration level or uparrow notation level, I don't know,

for example, for those who don't understand, Q(1) is 10,000 or thousand, seen from Tamatak' order notation, then Q(2) is how much, Q(2) is

$Q_{2}=\underbrace {10.000^{10.000^{10.000^{10.000^{10.000^{10.000^{10.000^{...}}}}}}}} _{10.000}$

then Q(3) is even bigger, Q(3) is Q(2) which is tetrased and repeated for Q(2) times so Q(3) = Q(2)↑↑Q(2), while we know that Q(2) is an unimaginable number and 100% has exceeded googolplex even all numbers in class 5 and below

now what about "Kuklinasio" kuklinasio is Q(10.000), you know

for the number Q(2) I will give the name "Kuklinasiol" why because this comes from the name grahal and graham number even though the name kuklinasio (not be confused with kuklinasiol without L) does not come from any journal or video, this is just an extension for the first Archimedes notation

then understand Q3, Q3 itself is Q2 which is tetralated with the number itself so Q3 = Q2↑↑Q2, this number can no longer be imagined because Q2 is already very large, exceeding the googolplex and class 5 numbers then Q3 is Q2 which is tetrased, if Q2 is named Kuklinasiol then Q3 is named Kuklinatriol

$Q_{3}=(10,000\uparrow \uparrow 10,000)\uparrow \uparrow (10,000\uparrow \uparrow 10,000)\underbrace {Q_{2}^{Q_{2}^{Q_{2}^{Q_{2}^{Q_{2}^{Q_{2}^{Q_{2}^{Q_{2}^{...}}}}}}}}} _{Q_{2}(10,000\uparrow \uparrow 10,000)}$

In short, the number Q3 is (10,000↑↑10,000)↑↑(10,000↑↑10,000), this number may look big but learn from the power law, maybe a number that looks big like this is actually still small, so maybe this number is still in class. tetration, at least has exceeded class 5

Then we can continue to Q4 and so on until Q(10,000)

$Q_{4}=(((10^{4})\uparrow \uparrow (10^{4}))\uparrow \uparrow ((10^{4})\uparrow \uparrow (10^{4})))\uparrow \uparrow (((10^{4})\uparrow \uparrow (10^{4}))\uparrow \uparrow ((10^{4})\uparrow \uparrow (10^{4})))$

look at the brackets to find out how big this number is of course this is smaller than pentation because pentation can also be written a↑↑a↑↑a↑↑a↑↑a↑↑a↑↑a↑↑a b times whereas this looks like it's mostly brackets and may still be in the tetration class, growth is quite slow

in the end and it came to pass that I will make a regiment for numbers that come from the notation Q(n) this is a small notation and the growth is quite slow and all these numbers up to 100 cannot go beyond the uparrow level because of the rules but it is possible that Q (10^10^100) is at the beginning of the uparrow notation level

KUKLINASIOL REGIMENT
around tetrational class

Myriad (Kuklimonol) Q1

Kuklinasiol (Q2)

Kuklinatriol (Q3)

Kuklitetrol (Q4)

Kuklipentol (Q5)

Kuklihexiol (Q6)

Kukliheptol (Q7)

Kuklioctol (Q8)

Kukliennol (Q9)

Kuklidekol (Q10)

Kuklihektol (Q100)

Kuklichillol (Q1000)

Kuklinasio (Kuklimyriol) (Q10000)

a misunderstanding

Then now I will go back from sequence Q(n) to Archimedes' current writings to see the final formulation of Archimedes' order notation because this makes me confused about this.

Now indeed I was wrong that O_n = 10^(2^(n)+1) is a wrong formulation because I haven't confirmed it, that is my own representation not by Archimedes and by other people let's just say that the order notation is not artificial Archimedes, that's what I, Tamatak sockpuppet made

The O_n notation that I showed earlier came from my understanding of the actual Archimedes notation, in the end I redefined it and even continued to the Q(n) notation,

remember that starting from 10^4 to 10^8 is a multiplication between powers, but what Archimedes means here is the first order (don't be confused withorder sequence)

remember that I defined sand reckoner with the wrong sequence, for my own O_n notation the definition is different from what Archimedes intended, the short definition of the O_n sequence is

$O_{n}=10^{2^{{n}+1}}$

where the number n is all positive or negative real numbers 0 and O_1 is equal to 10,000 (the number n can be any real number)

for example if O_2 is equal to $10^{2^{{2}+1}}=10^{8}$

and if n is a large enough number such as 100 it can be determined by

$10^{2^{{100}+1}}=10^{10^{30}}$

Of course this number grows faster than the one in the sand reckoner, because if you increase the number to 10,000 then the number is bigger than Googolplex, even believe it or not, it is already bigger. τῶν ἀριθμῶν μυρίαι μυριάδες, so I can conclude 100% that the O_n notation is different with sandreckoner notation and everything above is my misunderstanding except for the Q(n) notation which has not been created before

$10^{2^{{10,000}+1}}=10^{10^{3010}}>10^{10^{100}}$

Apart from integers, this O_n notation can also be applied to irrational numbers or fractions, I take the irrational number e as an example.

$O_{e}=10^{2^{{e}+1}}=1.45134941442156151736784628770092755486391443744934090484069\times 10^{13}$

what if a negative number is used? If a negative number is used, what will happen? I made an example of -5

$O_{-5}=10^{2^{{-5}+1}}={\sqrt[{16}]{10}}$