05 01 Interpolation

05.
01
History of Interpolation
After reading this chapter, you should be able to:

1. Know the history of Interpolation and its current uses by the HNMI.
History
Sir Edmund Whittaker, a professor of Numerical Mathematics at the University of
Edinburgh from 1913 to 1923, observed the most common form of interpolation occurs
when we seek data from a table which does not have the exact values we want. Throughout
history, interpolation has been used in one form or another for just about every purpose under
the sun.
Speaking of the sun, some of the first surviving evidence of the use of interpolation came
from ancient Babylon and Greece. Around 300 BC, they were using not only linear, but also
more complex forms of interpolation to predict the positions of the sun, moon, and the
planets they knew of. Farmers, timing the planting of their crops, were the primary users of
these predictions. Also in Greece sometime around 150 BC, Hipparchus of Rhodes used
linear interpolation to construct a chord function, which is similar to a sinusoidal function,
to compute positions of celestial bodies.
Farther east, Chinese evidence of interpolation dates back to around 600 AD. Liu
Zhuo used the equivalent of second order Gregory-Newton interpolation to construct an
Imperial Standard Calendar. In 625 AD, Indian astronomer and mathematician
Brahmagupta introduced a method for second order interpolation of the sine function and,
later on, a method for interpolation of unequal-interval data.
Many similar land-based purposes were found for interpolation over the ages, but
ocean navigation was found to be one of the most important applications for centuries.
Tables of special function values were constructed using numerical methods, and seafarers
used certain ones to determine latitude and longitude values. The French government started
production on an extensive set of such tables when the metric system was introduced.
Ideally, one would want mathematicians to construct a large set of tables due to their
proficiency at the subject. However, the primary source of work on the project ended up
being hairdressers who had lost their gaudy-wigged customers to the guillotine.
The unfortunate truth about special function tables is that most of them were
plagiarized. Since the computers, the workers who carried out and recorded the
calculations, were prone to making many errors during the creation of these daunting tables,
05.05.1
05.05.2
Chapter 05.05
plagiarism only propagated more errors. Charles Babbage tried to solve this problem with
the invention of his difference engine, a mechanical computer programmed by the use of
punch cards. On the side, Babbage also tried inventing a system that would choose winning
horse race numbers, hoping to raise extra money. Although he was not short of funds, his
life ran short and never saw the completion of the invention. Over a century and a quarter
later, as we plunge into the nano-technology era, Babbage is now considered the grandfather
of modern computing.
During the Great Depression, one final burst of manual table-making found its way
into the United States. The Works Progress Administration began the Mathematical Tables
Project shortly before World War II.
As with the French project, the desired
mathematician workers ended up being unskilledthis time to the point that negative
numbers were puzzling. The solution: black pencils for positive numbers and red ones for
negative numbers. Having each calculation in this project iterated twice (each by a different
person), and extensive proof reading carried out, these tables were possibly the most
accurate ever produced. Many of them were collected in a book by Milton Abromowitz and
Irene Stegun, which is still in worldwide use today. With computers (not the people type,
either), tables are no longer manually constructed, but the Australian Government produces
life tables which describe mortality rates. Relevant to the life insurance industry and the
study of demography, these tables are extended using modern interpolation methods. No
matter how advanced or extensive, interpolation will always be needed to find values in
modern tables due to their nature. Since they arent continuous functions, there will be
infinitely many missing values.
Two of the methods of interpolation taught at the HNMI are credited to Newton and
Lagrange. Newton began his work on the subject in 1675, which laid the foundation of
classical interpolation theory. In 1795, Lagrange published the interpolation formula now
known under his name, despite the fact that Waring had already produced the same formula
sixteen years earlier.
Bibliography
Kahaner, David, Cleve Moler, and Stephen Nash.
Englewood Cliffs, NJ: Prentice Hall, 1989.
Numerical Methods and Software.
Meijering, Erik. A Chronology of Interpolation: From Ancient Astronomy to Modern

Signal and Image Processing. Proceedings of the IEEE. vol. 90, no. 3, pp. 319-42. March
2002.
Mills, Terry. Historical Notes. Join the Dots and See the World. La Trobe University,
Bendigo, Australia. http://www.bendigo.latrobe.edu.au/rahdo/research/worner96.htm.
05.05.3
INTERPOLATION
Topic
Summary
Textbook notes on the history of Interpolation and its current uses in
the HNMI.
Major
All Majors of Engineering
Authors
Autar Kaw, Michael Keteltas
Last Revised December 23, 2009
Web Site
http://numericalmethods.eng.usf.edu

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05.

After reading this chapter, you should be able to:

Numerical Methods and Software.

Meijering, Erik. A Chronology of Interpolation: From Ancient Astronomy to Modern

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