!). MECHANICAL METHODS OF COMPUTATION 149improvements consist of scaling in such a way that the logarithm of 1 is 0 and the
logarithm of 10 is 1. which is the basic idea of what we now call common logarithms.
These further improvements to the theory of logarithms were made by Henry Briggs
(1561-1630), who was in contact with Napier for the last two years of Napier's life
and wrote a commentary on the appendix to Napier's treatise. As a consequence,
logarithms to base 10 came to be known as Briggsian logarithms.
9.2. Hardware: slide rules and calculating machines. The fact that log-
arithms change multiplication into addition and that addition can be performed
mechanically by sliding one ruler along another led to the development of rulers
with the numbers arranged in proportion to their logarithms (slide rules). When
one such scale is slid along a second, the numbers pair up in proportion to the
distance slid, so that if 1 is opposite 5, then 3 will be opposite 15. Multiplication
and division are then just as easy to do as addition and subtraction would be. The
process is the same for both multiplication and division, as it was in the Egyptian
graphical system, which was also based on proportion. Napier himself designed
a system of rods for this purpose. A variant of this linear system was a system
of sliding circles. Such a circular slide rule was described in a pamphlet entitled
Grammelogia written in 1630 by Richard Delamain (1600-1644), a mathematics
teacher living in London. Delamain urged the use of this device on the grounds
that it made it easy to compute compound interest. Two years later the English
clergyman William Oughtred (1574-1660) produced a similar description of a more
complex device. Oughtred's circles of proportion, as he called them, gave sines
and tangents of angles in various ranges on eight different circles. Because of their
portability, slide rules remained the calculating machine of choice for engineers for
350 years, and improvements were still being made in them in the 1950s. Different
types of slide rule even came to have different degrees of prestige, according to the
number of different scales incorporated into them.
Portions of the C, D, and CI scales of a slide rule. Adjacent
numbers on the C and D scales are in proportion, so that 1 : 1.23 ::
1.3 : 1.599 :: 1.9 : 2.337. Thus, the position shown here illustrates
the multiplication 1.23 1.3 = 1.599, the division 1.722-M.4 = 1.23,
and many other computations. Some visual error is inevitable.
The CI (inverted) scale gives the reciprocals of the numbers on
the C scale, so that division can be performed as multiplication,
only using the CI scale instead of the C scale. Decimal points
have to be provided by the user.
Slide rule calculations are floating-point numbers with limited accuracy and
necessary round-off error. When computing with integers, we often need an exact
answer. To achieve that result, adding machines and other digital devices have
been developed over the centuries. An early design for such a device with a series
of interlocking wheels can be found in the notebooks of Leonardo da Vinci (1452-
1519). Similar machines were designed by Blaise Pascal (1623 1662) and Gottfried
Wilhelm Leibniz (1646-1716). Pascal's machine was a simple adding machine that