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Four Areal Views

This project presents the third of four vignettes regarding the measure of area at different times in history. Chapter One related that the Greeks, through quadrature, attempted to use the square as a basis of comparison for all areas. In Chapter Two, we saw that Islamic mathematicians formalized algebraic procedures involving quadratic equations and brought them into agreement with geometric theorems. The work of al-Khwarizmi became especially influential in the 12th century awakening of Europe which eventually led to the Renaissance. It is a recurring theme in math history, that when a people rediscovered Euclid and Archimedes, they mounted full attacks on, and solved many of the outstanding questions of mathematics. Seventeenth century inquiries into the areas in and around conic sections were extensions of the accomplishments of Archimedes, and their resolutions led to the most dramatic advances in understanding the mechanics of the universe.

Chapter 3. Saint-Vincent Squares the Hyperbola

Introduction

As in the rest of this project, we will use a broad broom and sweep a wide swath in covering the events which contributed to such major advances as the Universal Law of Gravitation, the invention of calculus, new computational methods, and, taking the long view, to the exploration of space begun in the 1950s. As you will see, Saint-Vincent had a very minor role, but he is generally credited with being the first to discover the nature of the areas under the hyperbolic curve.

It is very tempting for this author to give you all the details of every discovery along with the proofs. Of course, to do that, we would start the project over again, or at least return to Eudoxus, whose method of exhaustion was the key inspiration for the calculus. Archimedes made brilliant use of the method in work on the circle and the parabola. Isaac Newton applied his own laws of motion, and the calculus, to the elliptical orbits of the planets, and thus solved the mechanics of the universe. It would be no small volume which told these stories in full, but I will attempt to give you samplings of each.

We will place our major focus on the areas associated with a hyperbola. Many mathematicians contributed, and many strands of mathematics are connected with the problem.

As mentioned in an earlier chapter, Archimedes was slain in 212 B.C. He had made amazing advances in both practical and theoretical problems. His solutions cut across the modern disciplines of engineering, physics, and mathematics. We know his works from the translations of Arab and Latin scholars, and we know from inference that many of his works are lost. His solution to the area measure of a circle was geometric, and yet he established very accurate numerical upper and lower bounds for it, and by association, for pi. He accomplished the quadrature of the parabolic segment in two ways, one geometric, and the other using the quasi-physical concept of equilibrium, regarding areas as masses which are transformed and brought into balance with more familiar areas [1]. For a diagram and explanation of the geometric solution, the reader can see Appendix 1.

In the first month of the year 1600 A.D. the young Kepler went to work for the astronomer Tycho Brahe. In two years time Tycho was dead, and volumes of observational data came into Kepler's hands. Kepler managed to induce from the data three laws governing the orbits of the known planets: 1) that the orbits were ellipses; 2) that the areas swept by the focal rays in different portions of an orbit were proportional to the times spent in those portions; and 3) that the square of the period for each body was proportional to the cube of the semi-major axis of the orbit. Kepler was right and he cites Archimedes as the inspiration, but also by his own admission, he drove himself crazy in failing to find the physical causes which determined the mathematics [2].

At about this time, or perhaps a little before, John Napier began work on a new system of computing. He was aware that astronomers of the time were typically observing all night and forced to calculate the fruits of the observations all the next day. They had endless products and ratios to figure, and those required eight-place accuracy in order to be useful. Needless to say, they were a weary bunch. Napier went public with logarithms in 1614, and then received a visit from Henry Briggs, a professor of geometry from London, who persuaded the Scotsman to use a base of ten, rather than 0.9999999 for the log tables [3]. The astronomers' multiplications were reduced to additions, and the divisions to subtractions, all thanks to logarithms; however, it is as a function that the logarithm was to supply a missing link in the quadrature of conic curves: no one had squared any segment related to a hyperbola.

Mathematical events then happened in rapid succession. In 1637 Rene DesCartes published a new system of charting functions using coordinate pairs. The concept was so novel that Isaac Newton, at first reading, did not understand it [4]. In 1640, Pierre de Fermat discovered the formula for the area under the curve y = x^n. The formula has (n + 1) as a divisor, and so it doesn't work, of course, for n = -1; consequently, y = 1/x, remained the unsquared, "square" hyperbola [5]. I call it square because its asymptotes are parallel to the coordinate axes, and "unsquared" because no one had yet accomplished a quadrature, or so it was thought. In a 1654 series of letters over a problem in gaming, Fermat and Blaise Pascal established a basis for the science of probability. To say that Newton eventually mastered Cartesian geometry is an understatement. He made it the basis for his Calculus, the essential analytical tool of mathematics. In 1665, his new Calculus, combined with his mastery of "old" geometry, confirmed Kepler's laws and explained the physical causes. See Appendix 2 for a glimpse of Newton's explanation of the "equal areas," Kepler's second law of orbits. But now we have gotten a bit ahead of the story.

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