The History of Mathematics: A Brief Course

148 6. CALCULATION

S

ï a b c d

FIGURE 4. Geometric basis of logarithms.

The first point sets out from Ô at the same time and with the same speed
with which the second point sets out from o. The first point, however slows down,
while the second point continues to move at constant speed. The figure shows the
locations reached at various times by the two points: When the first point is at A,
the second is at o; when the first point is at B, the second is at 6; and so on. The
point moving with decreasing velocity requires a certain amount of time to move
from Ô to A, the same amount of time to move from A to B, from  to C, and
from C to D. Consequently, TS : AS = AS : BS = BS : CS = CS : DS.
The first point will never reach 5, since it keeps slowing down, and its velocity
at S would be zero. The second point will travel indefinitely far, given enough
time. Because the points are in correspondence, the division relation that exists
between two positions in the first case is mirrored by a subtractive relation in the
corresponding positions in the second case. Thus, this diagram essentially changes
division into subtraction and multiplication into addition. The top scale in Fig. 4
resembles a slide rule, and this resemblance is not accidental: a slide rule is merely
an analog computer that incorporates a table of logarithms.
Napier's definition of the logarithm can be stated in the modern notation of
functions by writing log(i45) = oa, log(Z?S) = ob, and so on; in other words,
the logarithm increases as the "sine" decreases. These considerations contain the
essential idea of logarithms. The quantity Napier defined is not the logarithm as
we know it today. If points T, A, and Ñ correspond to points ï, a, and p, then

op = oa logfc

where k = AS/TS.

Arithmetical implementation of the geometric model. The geometric model just

discussed is theoretically perfect, but of course one cannot put the points on a line

into a table of numbers. It is necessary to construct the table from a finite set of

points; and these points, when converted into numbers, must be rounded off. Napier

was very careful to analyze the maximum errors that could arise in constructing

such a table. In terms of Fig. 4, he showed that oa, which is the logarithm of AS,

satisfies

TA<oa<TA(l + -).

(These inequalities are simple to prove, since the point describing oa has a velocity

larger than the velocity of the point describing Ô A but less than TS/AS times the

velocity of that point.) Thus, the tabular value for the logarithm of AS can be

taken as the average of the two extremes, that is, TA[l + (TA/2AS)], and the

relative error will be very small when Ô A is small.

Napier's death at the age of 67 prevented him from making some improve-

ments in his system, which are sketched in an appendix to his treatise. These

Ô A Â C D