### Conic Sections All-Stars, and Mr. Napier

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.

The circle and the parabola are two of the conic curves, that is, intersections between a plane and a double-ended cone, and Archimedes had solutions for the associated areas of each. In the eighteen hundred years after Archimedes, the areas were largely ignored, but at the beginning of the seventeenth century, Kepler made a breakthrough discovery about the solar system, and the areas within ellipses became suddenly significant.

In the first month of the year 1600 A.D. the young Kepler went to work for the Danish astronomer Tycho Brahe. In two years time Tycho was dead, and volumes of observational data came into Kepler's hands. He managed to induce from the data three laws governing the orbits of the known planets, the first two being: 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. 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.

At about this time, Scotsman 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. Astronomers were a weary bunch.

To say that Napier was an eccentric individual might be understating the case. He wrote religious tracts railing against the Roman Pope. He envisioned great destructive machines of war, including tanks and submarines. He even drugged his neighbor's pigeons who were always decimating his supply of grain.

Napier went public with logarithms in 1614. 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 (area calculation) of conic curves: no one had calculated any area related to a hyperbola. For example, the area under a portion of the hyperbolic curve y = 1/x, shown below, was unknown.

In 1665, Sir Isaac Newton’s new Calculus, combined with his mastery of "old" geometry, confirmed Kepler's laws and explained the physical causes. He concluded that the only possible paths of a body with mass acted on by gravity are: 1) the parabola – a failed, captured orbit; 2) the ellipse – or its ideal, the circle; or 3) the hyperbola, the path of a flyby, used to great effect in modern space exploration. A capsule or probe passes a planet, has its path altered but is not captured, and receives a gravity boost toward its next destination.

Gregoire de Saint Vincent is credited with relating the hyperbolic segments to logarithms, and thus Mr. Napier's numbers became a function with an application, and a useful one at that. In the figure above, if the right hand boundaries increased by a factor, the area increases would all be equal. For instance, if the sequence of righthand limits were proportional to10, 100, 1000, ..., then the cumulative areas would be proportional to the base ten logarithms 1, 2, 3, ....

A great hole in the early calculus of areas was patched, and since then we've merely flown by Pluto to points beyond.

Further readings: Maor, e: The Story of a Number;

Mead, Four Areal Views, Chapter 3.   http://www.unclebobpuzzles.com/Permasite/UB&AC/chapter3part1.htm