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Mathematics is the only instructional material that can be presented in an entirely undogmatic way.
In The Mathematical Intelligencer, v. 5, no. 2, 1983.
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.
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
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.
Then the square root of 235 is 15[10/30] = 15[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.