146 6. CALCULATION9. Mechanical methods of computation
Any study of the history of calculation must take account of the variety of com-
puting hardware that people have invented and the software algorithms that are
developed from time to time. In ancient China the software (decimal place-value
system) was so good that the hardware (counting rods, counting boards, and aba-
cus) worked with it very smoothly. The Greek and Roman system of writing num-
bers, however, was not a good representation of the decimal system, and the abacus
was probably an essential tool of computation. When the graphical methods associ-
ated with Hindu-Arabic numerals were introduced into Europe, they were thought
to be superior to the abacus.
9.1. Software: prosthaphaeresis and logarithms. The graphic arithmetic that
had vanquished the counting board a few centuries earlier still had certain labori-
ous aspects connected with multiplication and division, which mathematicians kept
trying to simplify. Consider, for example, the two three-digit numbers 476 and 835.
To add these numbers we must perform three simple additions, plus two more that
result from "carrying," a total of eight simple additions. In general, at most 3n - 1
simple additions with ç — 1 carryings will be required to add two ra-digit numbers.
Similarly, subtracting these numbers will require at most two borrowings, with con-
sequent modification of the digits borrowed from, and three simple subtractions.
For an n-digit number that is at most ç simple subtractions and ç — 1 borrowings.
On the other hand, to multiply two three-digit numbers will require nine simple
multiplications followed by addition of the partial products, which will involve
up to 10 more simple additions if carrying is involved. Thus we are looking at
considerably more labor, with a number of additions and multiplications on the
order of 2n^2 if the two numbers each have ç digits. Not only is a greater amount of
time and effort needed, the procedure is obviously more error-prone. On the other
hand, in a practical application in which we are multiplying, say, two seven-digit
numbers (which would involve more than 100 simple multiplications and additions),
we seldom need all 14 or 15 digits of the result. If we could improve the speed of
rhe operation at the expense of some precision, the trade-off would be worthwhile.
Prosthapharesis. The increased accuracy of astronomical instruments, among other
applications, led to a need to multiply numbers having a large number of digits. As
just pointed out, the amount of labor involved in multiplying two numbers increases
as the product of the numbers of digits, while the labor of adding increases according
to the number of digits in the smaller number. Thus, multiplying two 15-digit
numbers requires over 200 one-digit multiplications, while adding the two numbers
requires only 15 such operations (not including carrying). It was the large number
of digits in the table entries that caused the problem in the first place, but the key
to the solution turned out to be in the structural properties of sines and cosines.
The process was called prosthaphaeresis, from two Greek prefixes pros-, meaning
toward, and apo-, meaning from., together with the root verb haird, meaning / seize
or / take. Together these parts mean simply addition and subtraction.
There are hints of this process in several sixteenth-century works, but we shall
quote just one example. In his Trigonometria, first published in Heidelberg in
1595, the theologian and mathematician Bartholomeus Pitiscus (1561-1613) posed
the following problem: to solve the proportion in which the first term is the radius,
while the second and third terms are sines, avoiding multiplication and division.