- MECHANICAL METHODS OF COMPUTATION 147
The problem here is to find the fourth proportional x, satisfying r : a = b : x,
where r is the radius of the circle and a and b are two sines (half-chords) in the
circle. We can see immediately that ÷ = ab/r, but as Pitiscus says, the idea is to
avoid the multiplication and division, since in the trigonometric tables of the time
a and b might easily have seven or eight digits each.
The key to prosthaphaeresis is the well-known formula
sin(a + â)+ sin[a - â)
smacos/3 =.
This formula is applied as follows: If you have to multiply two large numbers, regard
one of them as the sine of an angle, the other as the cosine of a second angle. (Since
Pitiscus had only tables of sines, he had to use the complement of the angle having
the second number as a sine.) Add the angles and take the sine of their sum to
obtain the first term; then subtract the angles and take the sine of their difference
to obtain a second term. Finally, divide the sum of the two terms by 2 to obtain
the product. To take a very simple example, suppose that we wish to multiply
155 by 36. A table of trigonometric functions shows that sin 8° 55' = 0.15500 and
cos68° 54' = 0.36000. Hence
36 .155 = 105 sin77°49- + sin(-59°59') = 97748 - 86588 =
2 2
In general, some significant figures will be lost in this kind of multiplication.
For large numbers this procedure saves labor, since multiplying even two seven-digit
numbers would tax the patience of most modern people. A further advantage is
that prosthaphaeresis is less error-prone than multiplication. Its advantages were
known to the Danish astronomer Tycho Brahe (1546-1601), who used it in the
astronomical computations connected with the extremely precise observations he
made at his observatory during the latter part of the sixteenth century.
Logarithms. The problem of simplifying laborious multiplications, divisions, root
extractions, and the like, was being attacked at the same time in another part
of the world and from another point of view. The connection between geometric
and arithmetic proportion had been noticed earlier by Chuquet, but the practical
application of this fact had never been worked out. The Scottish laird John Napier,
Baron of Murchiston (1550-1617), tried to clarify this connection and apply it. His
work consisted of two parts, a theoretical part based on a continuous geometric
model, and a computational part, involving a discrete (tabular) approximation of
the continuous model. The computational part was published in 1614. However,
Napier hesitated to publish his explanation of the theoretical foundation. Only
in 1619, two years after his death, did his son publish the theoretical work under
the title Mirifici logarithmorum canonis descriptio (A Description of the Marvelous
Rule of Logarithms). The word logarithm means ratio number, and it was from the
concept of ratios (geometric progressions) that Napier proceeded.
To explain his ideas Napier used the concept of moving points. He imagined
one point Ñ moving along a straight line from a point Ô toward a point S with
decreasing velocity such that the ratio of the distances from the point Ñ to S at
two different times depends only on the difference in the times. (Actually, he called
the line ending at S a sine and imagined it shrinking from its initial size TS, which
he called the radius.) A second point is imagined as moving along a second line
at a constant velocity equal to that with which the first point began. These two
motions can be clarified by considering Fig. 4.