The Great Calculation According to the Indians, of Maximus Planudes - The Digits of the System, I

"The So-Called Great Calculation According to the Indians"

of the monk Maximos Planoudes.

A Translation

Since numbers continue without bound, but knowledge of the boundless is not possible, the more eminent of the astronomers invented certain signs and a method relating to them, so that the representation of those numbers they needed might be more easily and more clearly apprehended at a glance.  There are only nine signs required which are these: 1 2 3 4 5 6 7 8 9.¹  They also use a certain other sign which they call a cipher, which, according to the Indians, signifies 'nothing'. These nine signs are themselves of Indian origin and the cipher is written as 0.

When each of these 9 signs² stands alone by itself and in the first place beginning from the right-hand side, the symbol 1 indicates one, 2 indicates two, 3 three, 4 four, 5 five, 6 six, 7 seven, 8 eight and 9 nine. If, however, it is in the second place, then the symbol 1 indicates ten, 2 twenty, 3 thirty and so on. In the third position 1 indicates a hundred,³ 2 two hundred, 3 three hundred and so on. The pattern continues for the remaining places.

Thus, in the first position the signs are to be regarded as units,⁴ which beginning from one proceed to nine. (Two, three, four, up to nine will be reckoned as monadic numbers, since they are all bounded by ten, neither reaching nor exceeding it.) Hence any sign which occurs in the first position will be regarded as monadic, and in the second position as decadic, that is, between ten and ninety, and in the third position as hecatontadic, that is, between a hundred and nine hundred. So also a sign in the fourth position is regarded as a multiple of a thousand and in the fifth as myriads⁵ and in the sixth as tens of myriads, the seventh as hundreds of myriads, the eighth as thousands of myriads, and in the ninth as myriads of myriads. If one were to proceed even beyond this point, the tenth position counts as tens of myriads of myriads⁶ and the eleventh as hundreds of myriads of myriads, the twelfth as thousands of myriads of myriads and the thirteenth as myriads of myriads of myriads. Indeed one could proceed even further.

Now to clearly illustrate what I have said by an example, suppose the given number is 8136274592 which occupies ten of these places.  We⁷ begin from the right hand side, as has been said, and so the sign 2 in the first place indicates the number two, which is a monadic number. In the second place, the sign 9, is ninety, which is a decadic number, consisting in fact only of tens, just as the two before it, being monadic, consisted only of units. The sign 5 in the third place is five hundred, which is a hecatontadic number. The number 4, in the fourth place, is four thousand, which is a chiliadic number,⁸ and 7 in the fifth represents seven myriads and is a myriadic number. The sign 2 in the sixth place is twenty myriads; it is a decakismyriadic number. 6 in the seventh place is six hundred myriads, a hecatontakismyriadic number. The sign 3 in the eighth is three thousand myriads, a chiliontakismyriadic number. The sign 1 in the ninth is a myriad of myriads, and is a myriontakismyriadic number. The sign 8 in the tenth place is eighty myriads of myriads, which is a decakismyriontakismyriadic number. The given number is thus read in its entirety as eighty myriads of myriads and a myriad of myriads and three thousand six hundred and twenty seven myriads, four thousand five hundred and ninety two.