The Great Calculation According to the Indians, of Maximus Planudes - Outline of the Work

In the first section of the Great Calculation, Planudes tries to explain the numeral system and its symbols in terms of Greek arithmetic, which used monadic numbers (units), decadic numbers (tens), and so on. He also introduces the symbol for zero (the cipher), which was generally absent from Greek arithmetic. The algorithm for addition is identical (as we would expect) to our modern algorithm, and involves the notion of carry. He also mentions the old check, known as casting out nines, to verify (the reasonableness of) the answer. For subtraction he gives the two standard algorithms: the borrow and pay back algorithm and the algorithm often referred to in modern primary school books as the trading method. These are explained in great and often unnecessary detail with general descriptions of the methods followed by worked examples. The examples chosen by Planudes are not always the best - on several occasions in the work a poor choice of digits leads to confusing ambiguity in the written explanations.

Planudes' first method of long multiplication is rather different from that used by the modern school child. He uses a kind of chiastic, or 'cross' multiplication, which involves sums of products, combined with carry. For example, to find 24×35 he works out 4×5 = 20 and carries 2. Then he finds 4×3 + 2×5 = 22 and adds on the 2 to obtain 24. He again carries 2 and works out 2×3 which becomes 8 with the carry, so the answer is 840. For larger numbers, the process is similar. For example, to find 432 ×264, we have to find the products 2×4, 3×4 + 2×6, 4×4 + 3×6 + 2×2, 4×6 + 3×2 and finally 4×2, remembering to record the units and carry the tens as we go.

Planudes then gives a second method which involves writing and then erasing numbers. The explanation of this method is complicated and made worse by the fact that the erasure of symbols makes it very difficult to follow his examples. His explanation of long division is particularly difficult to follow.

The third section introduces the sexagesimal system using signs of the zodiac (that is, multiples of 30˚ which are read modulo 12), and degrees, minutes and seconds, which are all base 60. The four operations are repeated using this new system.

Finally, there is a long section on finding square roots of non-squares to various degrees of accuracy, based essentially on the formula \[\sqrt{a^2 +\epsilon}\approx a+\frac{\epsilon}{2a}.\] The latter parts of this section are particularly obscure and hard to make sense of.

Planudes' algorithm for square roots is very similar to the 'modern one' which goes as follows:

Take, for example, 235. Divide the digits from right to left into pairs, thus 2 | 35. Find the root of the largest square less than the first number, i.e. 1, and subtract its square from the number, i.e. 2 - 1² = 1. Record the root and carry down the next number 35, so we have \[\begin{array} {l l} & {\phantom{22}}1 \\ & \sqrt{2|35} \\& {\phantom{22}}1 \\ \hline 2 & {\phantom{22}}135 \\ \end {array}\]

Also, double the square root 1 and write it on the left beside the number 135. We now find the largest digit x such that 2x (i.e. 20+x) times x is less than 135. The digit is 5, since 25×5 = 125 < 135. We write the 5 on top and subtract 135 - 125 = 10.

\[\begin{array} {l l} & {\phantom{22}}1\,5 \\ & \sqrt{2|35} \\& {\phantom{22}}1 \\ \hline 25 & {\phantom{22}}1\,35 \\ & {\phantom{22}}1\,25 \\ \hline & {\phantom{111}}10 \\ \end {array}\]

Then the square root of \(235\) is \(15\frac{10}{30} = 15\frac{1}{3},\) noting that the denominator is twice the root \(15.\) In longer examples, Planudes makes the description more complicated by not properly using place value.

What follows are a series of excerpts from my (fairly literal) translation from Greek into English of the whole work. These excerpts consist of: the introductory passage on the numerals themselves and the section on addition; the first of the two algorithms for subtraction; the first algorithm for multiplication; the introduction to the sexigesimal system and an example of addition in that system; and finally the beginning of the section on square roots.  This should be sufficient to give the general flavour of the work. The full translation can be downloaded here (pdf file).

I used the edition of Gerhardt (1865) for the translation, making corrections and additions as necessary. Such alterations are noted when they occur.

There is also an old and very literal German translation by Wäschke (1878) listed in the References.