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
cosa = cosb cosc + sinb sinc cos a and sinb sin a = sina 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
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 rm and rn, one need only add the exponents to obtain rm+n. Further, if r were chosen to be a number only slightly larger than 1, and one could calculate many of the powers of r, storing them in a list to obtain a table of absolute numbers, as he called them, that increase slowly from 1 to, say, 10, then any two of these numbers could be multiplied by adding the corresponding exponents, which Napier called artificial numbers or logarithms, which he tabulated in an adjoining list, and then find the corresponding absolute value in the first list as the desired product. If the list of absolute numbers could be filled with values sufficiently close together, then one could effectively multiply any two numbers by this method (after appropriately shifting decimal places, as we did in the example above).
Napier published his ideas in a pair of treatises, Mirifici logarithmorum canonis descriptio (A Description of theWonderful Table of Logarithms) in 1614 and Mirifici logarithmorum canonis constructio (The Construction of theWonderful Table of Logarithms) in 1619. He enlisted the aid of Henry Briggs (1561 - 1630), a mathematics professor at Gresham College in London, who traveled to Scotland in the summers of 1615 and 1616 to consult with Napier. In the course of this collaboration, the two men were able to considerably simplify the process of associating logarithms to numbers. Briggs took over the work upon Napier's death in 1617, preparing the edition of the Constructio in 1619 and later writing a work of his own, Arithmetica Logarithmica in 1624. Here, Briggs published a table of logarithms for the numbers 1 to 20,000 and from 90,000 to 100,000; the second edition of the work, published in 1628 (from which our selection comes), completed the table for the intermediate values 20,000 to 90,000.
Briggs' common logarithms were based on powers of 10: the logarithm of a number corresponded to the power of 10 that gave the number. So, for example, log 1 = 0, log 10 = 1, and log 3.16228 = .5 since 100 = 1, 101 = 10 and 100.5 = 3.16228 is the square root of 10. This made multiplications even easier to compute than with prosthaphaeresis: recalling our example from earlier, to multiply 147809 by 6756, we find in the logarithm tables that log 147809 = 5.16970 and log 6756 = 3.82969; adding logarithms gives 8.99939, and searching the tables again finds that log 998596409 = 8.9939, whence (147809)(6756) = 998596409, which is accurate to the first six decimal places. Once again, a multiplication is performed without any multiplying; instead, the table is searched three times and one addition takes place.
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