By the sixteenth
century, European navies had conquered the seas and were expanding their
empires outside the continent, especially in the New World. Navigational
and cartographic
problems were solved by appeal to astronomical techniques and required
accurate calculations (to ensure that one's ship would cross the ocean
and not wind up 500 miles off course, or that one's map would correctly
determine the extent of the king's realm!). For instance, a typical
astronomical calculation involves determining the sides or angles in a
spherical triangle.

This standard problem from spherical
trigonometry concerns triangle *ABC* on the surface of a sphere (representing
the triangle formed by three bodies on the celestial sphere). The
sides opposite points *A*, *B*, *C* are labeled *a*,
*b*,
*c*,
and the angles at *A*, *B*, *C* are
a,
b,
g.
Since the sides of the triangles are arcs of circles on the sphere centered
at the center of the sphere, they, like the angles a,
b,
g,
are measured in degrees. The trigonometric relationships amongst
the elements of the triangle include equations like

cos*a* = cos*b* cos*c*
+ sin*b* sin*c* cos a and
sin*b* sin a
= sin*a* sin b.

Solving for the missing values required
repeated multiplications and divisions of sine and cosine values, and this
just for a single triangle. Preparation of ephemeral
tables (which tell when and where certain objects will be in the sky)
demanded reams of tedious calculations like this. Mathematicians,
both professional and amateur, would give considerable attention to the
problem of speeding up the time required to perform these types of calculations.

Christopher
Clavius (1538 - 1612), a Jesuit priest and scholar who became famous
for leading the papal commission under Gregory XIII to reform the
calendar in 1582, devised a clever method for speeding calculations called
**prosthaphaeresis**
(from the Greek "prosth-" = "adding" + "-aphaeresis" = "subtracting").
This method was employed to great effect by Tycho
Brahe (1546 - 1601), the Danish astronomer who, upon witnessing a new
star in the sky in November of 1572 (now known as Tycho's supernova), built
an observatory called Uraniborg in 1575 on an island in Copenhagen harbor
from which he made voluminous and painstaking astronomical observations
for over 20 years. Prosthaphaeresis makes use of the trigonometric
identity

cos *a* cos *b* = [cos(*a*
+ *b*) + cos(*a* - *b*)]/2

called the *prosthaphaeretic rule*;
it allowed a calculator to perform a multiplication of two numbers (viewed
as cosine values) by averaging the values of the cosines of the sum and
difference of these angles. In other words, a multiplication was
replaced by a table lookup to find the angles *a* and *b* whose
cosines are the two factors, an addition and subtraction of these angles,
table lookups of the cosines of these two new angles, and an averaging
of the resulting values. Once a calculator became adept at this method,
he would find a substantial time savings when computing many multiplications.

For instance,
to multiply 147809 by 6756 by standard methods requires the partial multiplications
of the first factor by each of the digits of the second, then the addition
of these partial products is necessary. By prosthaphaeresis, we interpret
the two factors as cosines of angles and search for .147809 and .6756 in
a cosine table. (The cosine of an angle is the ratio of the side
adjacent to the angle and the hypotenuse of a right triangle that contains
that angle, whence it must be less than 1; artificially introducing a decimal
point before the leading digit of a number allows us to interpret it as
a cosine and only affects the outcome of the multiplication in the placement
of the decimal point, which can be altered afterward.) We find from
trigonometric tables (or our calculator!) that cos(81.5º) = .147809
and cos(47.5º) = .675590 = .6756. By prosthaphaeresis,

(.147809)(.6756) = cos(81.5º) cos(47.5º) = [cos(129º) + cos(34º)]/2 = [-.62932 + .82904]/2 = .09986

from which we deduce that (147809)(6756)
= 998600000. While the exact value of the product is 998597604, note
that our answer is correct to the number of decimal places used in the
calculation (5).

Despite the
advance that prosthaphaeresis provided, it never became a popular method
for quick calculation, for it was soon eclipsed by the invention of **logarithms**
by the Scotsman John
Napier (1550 - 1617). Napier was a wealthy landowner (he held
the title of Baron of Merchiston) with business interests in the new British
colonies of America. He was also an amateur astronomer and mathematician.
His discovery derived from the observation that to multiply any two powers
of a fixed number *r*, say
*r ^{m}* and

Napier published his ideas in a pair of treatises,

Briggs'

Logarithms soon became an important tool for calculation, expecially when tools like the slide rule were manufactured. A slide rule is essentially a pair of rulers, one marked on a standard scale of 1 to 10 (the logarithms) and another marked on a geometric scale, showing powers of 10 (the numbers one wishes to calculate with), that slide against each other. Logarithms for absolute numbers are read right off the rule and calculations are quickly dispatched. Precision slide rules were manufactured and sold for centuries, and were used extensively right up to the 1960s, when they were finally displaced by the hand-held electronic calculator.

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last modified 10/30/02

Copyright (c) 2000. Daniel E. Otero