MATHEMATICS AND COMPUTATION I

The beginning of mathematics was primitive man's discovery of counting; adding one and one to make two is pictured on this stamp. Upon seeing two birds, an Egyptian makes the cerebral leap to count them on his fingers. (Detail) This stamp is the first in a set of ten issued by Nicaragua in 1970 which features important mathematical formulas that changed the face of the earth. Besides showing the law, equation, or formula, the name of its originator, and an application, the reverse of each stamp is printed with a brief paragraph in Spanish explaining the significance of the formula and its far-reaching applications in modern life. Presumably the user can ponder this educational message while licking the stamp; whether the recipient would appreciate it or be aware of it is another matter. The illustrations contain a wealth of interesting detail, and the sci-philatelic sleuth can enjoy identifying the many clues and their relationship to the original formula.
The Inca civilization of the 15th century, centered in Peru, did not have a written language, but kept records (and count) of items on an array of colored, knotted cords called quipu. Color, type of knot, spacing and placement of the cords were all meaningful and part of a code that was used to send information on production, census, resources, and taxes owed or collected to distant parts of the empire. Runners carried the quipi which were then decoded at their destination.

A well-known theorem in geometry is named after Pythagoras, who flourished in the 6th century BC, and was a teacher in Samos, Babylon, and Egypt: the sum of the squares of the sides of a right triangle equals the square of the hypotenuse. Actually, the so-called Pythagorean triples were known already in Babylonian times. The striking design of the Greek stamp is a visual representation of the theorem. (Detail)

	Archimedes of Syracuse, the great mathematician of the 3rd century BC, considered one of his most important accomplishments to be the proof of the relationship between the volumes of a sphere and a circumscribed cylinder: the spherical volume is 2/3 that of the cylinder. He asked that a sphere and cylinder be placed on his tomb, where they were seen by Cicero in 75 BC. This Italian stamp commemorates the World Mathematical Year 2000 with just such a sphere and cylinder. For some of Archimedes' many other accomplishments seeArchimedes .
	Tsu Ch'ung Chi (430-501) was a Chinese mathematician and astronomer. His approximation of pi was 355/113, which is correct to six decimal places. In astronomy, he arrived at the precise time of the solstice by measuring the sun's shadow at noon on days around the solstice.
	Muhammed ibn Musa al-Khwarizmi was a Persian mathematician of the 9th century whose name comes down to us in the word "algorithm." The translation of his arithmetic treatises into Latin in the 12th century brought the advantages of the Indo-Arabic method of positional counting, including the use of zero, to western Europe, where counting tables based on the abacus and Latin numerals were still used. The title of his work Al-jabr wa'l muqabalah has become our word "algebra." (Detail)
	Luca Pacioli (1445-1517) (Detail) was an Italian monk and mathematician who in 1494 published an encyclopedic compendium of all that was known in the field of mathematics at the time: Summa de arithmetica, geometria,proportioni et proportionalita. This Italian stamp commemorates the 500th anniversary of this event with a painting based undoubtedly on an original by Jacopo de Barbari of Pacioli surrounded by mathematical figures and artifacts shown here. His work contained many chapters on double entry bookkeeping based on an unpublished manuscript of Benedetto Cotrugli, and elsewhere he drew on the work of many earlier mathematicians, including Fibonacci. His last great work was Divina proportione, (divine proportion, a term coined by Pacioli) or the golden section, known to Greek mathematicians and philosophers. This work was illustrated by his friend Leonardo da Vinci.
	Adam Riese (1489-1559) was the most famous and influential German arithmetician of the 16th century, and author of many popular commercial arithmetic books which made use of Indo-Arabic numerals instead of counters.(Detail)

Pedro Nunes (1502-1578) was the foremost mathematician and cosmographer of Portugal during the Age of Discoveries. He wrote extensively on Cosmography, Spherical Geometry, Astronomic Navigation, and Algebra. Three stamps issued by Portugal in 2002 reproduce two of the most important discoveries of Nunes: the loxodromic curve and the nonius. Loxodromic curves appear on the first stamp from the left, both over a globe and on space, and cover part of the second stamp, on which appears a typical face of the epoch. It is not a portrait of Nunes, since no drawing or painting of his face survives. The stamp on the right reproduces an instrument now in the Museum for the History of Science in Florence of nonius scales, as well as a stylized drawing of these scales. Loxodromic curves, also called rhumb lines, are spirals that converge to the poles. They are lines that maintain a fixed angle with the meridians. A ship following a fixed compass direction travels along a loxodromic. Nunes discovered the loxodromic lines and advocated the drawing of maps in which loxodromic spirals would appear as straight lines. This led to the celebrated Mercator projection, constructed along these recommendations. Nonius scales were invented by Nunes in order to allow a more precise reading of the height of stars on a quadrant. The set of concentric scales provides a higher precision than the individual precision of any particular scale. The device was used and perfected at the time by Tycho Brahe, Jacob Kurtz, Christopher Clavius and others. Following these various improvements, the French Pierre Vernier constructed in 1630 a device with one sliding scale over a fixed one, which proved to be a most practical solution. For some centuries, this device was called nonius. During the 19th century, many countries, most notably France, started to call it vernier.(1)

	John Napier's (1550-1617) invention of logarithms immensely facilitated mathematical computations before the age of computers. Lengthy multiplications and divisions of large numbers were replaced by the addition or subtraction of their logarithms, which were calculated and conveniently available in tables of great accuracy. The Nicaraguan stamp shows astronomical equipment and the Big Dipper.(Detail) Astronomical calculations benefitted enormously from the use of accurate logarithmic tables. Napier's "bones," or sliding scales, later became the slide rule, an essential tool for engineers and scientists until replaced by the hand-held calculator of the 1970's. On the Romanian stamps of 1957 publicizing a Congress of Engineers and Technicians, calipers and slide rule represent quintessential tools for measurement and calculation.
	Johannes Kepler (1571-1630) was a German mathematician who is remembered for his three laws of planetary motion, derived empirically from Tycho Brahe's data and observations. The laws describe the solar system as having the sun at the focus of elliptic planetary orbits. In his writings on conic sections he introduced the word focus into mathematical language. The second stamp, issued for the International Congress of Mathematicians in Tokyo in 2000, shows an origami construction of the stella octangula, a composed polyhedron first discovered in by Kepler. It is made up of two interpentrating tetrahedra and its vertices are the vertices of a cube. The stella octangula is also called a stellated tetrahedron, and is the only stellation of the octahedron (s.a. Astronomy and Cosmology I)
	Pierre de Fermat (1601-1665), the French mathematician, left mathematical posterity with a tantalizing problem, known as Fermat's Last Theorem, which states that the Pythagorean equation a² + b² = c² has no solution for integer values of a, b, and c for any power higher than 2. He professed to have a proof of this theorem but did not have space to write it down in the margin of the text he was then annotating. Not until 1994 was a satisfactory proof published, by Andrew Wiles, who has the distinction to see his name emblazoned on the Czech Republic stamp, while Fermat's enigmatic smile on the French stamp at left reflects perhaps the amusement he would have felt at the feverish attempts of mathematicians to prove him either right or wrong over the last four centuries.

(1) Thanks to Professor Nuno Crato of Lisbon for this detailed commentary.

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