Characteristic of his time, Pierre-Simon Laplace (1749-1827) published a comprehensive review of the scientific work done by his predecessors with an overlay of his new mathematical interpretation. Titled Celestial Mechanics, the five-volume work replaced the old-style geometric methodology with Laplace’s unique style of calculus, suggesting a broad range of new subject matter – everything from stability of planetary orbits to a study of probability, applicable to the European gaming table.
For centuries, the language of mathematics had been geometry, the study of shapes. Shortly before the time of Laplace, Gottfried Leibniz and Isaac Newton, more or less simultaneously and independently, had introduced into scientific discourse a new perspective: calculus, the language of change. As geometry is to space, calculus is to time.
Differential calculus takes as its subject matter rates of change and slopes of curves, while integral calculus is concerned with accumulation of quantities and the areas shown graphically that are under or between curves. An infinite sequence or infinite series is seen in either discipline to converge at the region of a definite limit.
Laplace, in this context, formulated the equation named for him, the Laplace theorem and the Laplacian differential operator. The Laplace equation is an elliptic partial differential equation. Its solution gives rise to harmonic functions, which play a central role in electrodynamics, describing gravitational and fluid potentials as well.
The Laplace transform is similar to the Fourier transform, discussed in a previous article. But rather than expressing a function as a superposition of sine waves, it expresses a function as a superposition of moments. Both the Fourier and Laplace transforms permit the translation of the time domain to the frequency domain. In an oscilloscope, rather than by means of difficult computation, this transformation can be achieved by pressing a button.
The Laplace transform is a powerful tool that, in changing from the time domain to the frequency domain, shifts the terms of the inquiry from differential to algebraic equations.
Laplace had a wide-ranging intellect. He investigated diverse topics other than celestial mechanics, electrostatics and electromagnetism. He became interested at various times in probability theory, politics and theology. He was among early thinkers who propounded the idea of a black hole. With no observational evidence, he envisioned a body in space that would have such immense density and gravity that no light could escape, so that it became invisible to the outside world.
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