Calculus and the Third Dimension

The hidden realm.

This article is part of the mathematics sequence of the Tom Swift Academy.

While the advent of differential and integral calculus made classical physics as we know it possible, many problems would not be solvable until the Cartesian system was extended to the third dimension. This was accomplished by Newton and Leibniz in the most rudimentary form. However, a full development of this branch of mathematics would have to wait until a new era.

The eighteenth century saw the development of calculus from the mathematical underpinning of theoretical physics, to a tool fundamental to navigation and military logistics. The field was analyzed with a new mathematical rigor, leading to the analysis of infinite geometric series and the advent of many new applications.

The great ideas of the Enlightenment were often advanced not in universities, but in free and open discussion, in what were known as salons, often organized by aristocratic women. One of this was organized by the Madame Sofie de Grouchy, later to be the wife of the Marquis de Condorcet. In our age of increasing instability and academic decay, this system of inquiry may very well again prove to be meritorious. It is also in this age that calculus was integrated into the mathematical curriculum used to educate engineers. This Newtonian synthesis was especially advantageous in military engineering, and therefore was critical to the advances of the Age of Discovery.

Thus were many unique characteristics of three-dimensional calculus developed within the salons and coffeehouses of the Enlightenment. One of the primary developers of this form of calculus was Leonhard Euler. Leonhard Euler was a true polymath, responsible for many different fields, including the calculus of variations. Perhaps his most rigorous contribution to calculus was his extension of its laws to exponential and trigonometric functions.

The Marquis de Condorcet is an unusual figure indeed. He was one of the leading figures of the French Enlightenment, sympathizing greatly with the rationalistic and anti-clerical view of progress first espoused by Voltaire. In his view, mankind was to advance through ten ages of increasing societal elevation, the last being a state of true equality. I shall leave it to the perfect hindsight of our twenty-first century readers to determine whether this turned out to be truth, or merely a pleasant delusion. Perhaps this affinity for the new and rational led him to employ the first mathematical symbol that did not appear first in an ancient language, that of the partial derivative.