Mathematical Physics. We have thus named the parts of Physics in which we bring together observation and experiment with mathematical calculations and where we apply these calculations to the phenomena of nature. We have already seen with the word Application the abuses which can happen when calculus is applied in Physics; we will add the following thoughts.

It is easy to see that the different subjects of Physics cannot have Geometry applied to them equally well. If the observations that serve as the basis for calculations are in small numbers, if they are simple and obvious, then a geometer can then draw the greatest advantage from them, deducing that physical knowledge which is most capable of satisfying the intellect. Less perfect observations are often used to lead him in his research and to give his discoveries a new degree of certainty. Sometimes, mathematical reasoning can inform and enlighten him, however, if the experiment discloses nothing, the final analysis will remain confused. Finally, if the matters that he proposes to understand do not allow any progress toward a solution by calculation, he should then proceed based on the simple facts of the observations. If one’s method leads to false starts, then vague and unclear reasoning should certainly not take the place of rigorous proofs.

This is principally the method that one must follow in relation to these phenomena, when reasoning is not able to help, when we cannot see the link, or when we see it only very imperfectly as in the case of the phenomena of magnets, electricity, and an infinity of others, etc. See Experimental.

The physico-mathematical sciences are also in great numbers as there are branches in mixed Mathematics . See Mathematics and the explanation of the Map of the System of Human Knowledge in the first volume of this work and following the Preliminary Discourse.

We can therefore list the numbers of physico-mathematical sciences as follows: Mechanics, Statics, Hydrostatics, Hydrodynamics or Hydraulics, Optics, Catoptrics, Dioptrics, Aerometrics, Music, Acoustics, etc. Concerning Acoustics, which we have promised to mention here, see the article Fundamental, where we have already fulfilled our promise; see also concerning Optics, the article Vision, and Fluid concerning Hydrodynamics.

One of the most brilliant and most useful of the physico-mathematical sciences is Astronomical physics, see Astronomy. What is meant by Astronomical physics is not the chimera of vortices, but the explanation of astronomical phenomena using the admirable theory of gravity. See Gravitation, Attraction, Newtonianism. If Astronomy is one of the sciences which provides the greatest honor to the human intellect, Newtonian astronomical physics is one of those which provides the most to modern Philosophy. The research of the causes of celestial phenomena in which today there is so much progress is not empty guessing but is rather at the limits of the magnitude of its object and the difficulty in grasping it. This research must still contribute to the speedy advances in Astronomy itself. We cannot flatter ourselves by thinking that we have found the true causes of planetary movement because we are not yet able to assign the proper calculations to causes and effects and thereby demonstrate that they are in accordance with what our observations seem to have demonstrated. Since the combination of these effects is still considerable there is still a great deal to discover; consequently once we are aware of the principle, the geometrical conclusions that we will be able to deduce will allow for us quickly to grasp and predict even those phenomena which have been hidden and are hard to capture, and perhaps may need an extended period of time to be known, unraveled and determined through observation alone.

Amongst the different assumptions that we can imagine in attempting to explain an effect, the only ones worthy of our attention are those which by their nature provide us with unmistakable ways to assure that they are correct. The gravitational system is considered to be one of them and for that reason alone merits the attention of Philosophers. Here there is no need to fear the abuses of calculus and geometry to which physicists have all too often succumbed in the defense of or to fight hypotheses. Since planets are supposed to move either in emptiness or at least in a non-resistant medium, and the forces by which they act upon each other are also known, it is now a purely mathematical problem to determine the phenomena which issue from this. We have the rare advantage of being able to determine [the movement of the planets], for once and for all, using Newton’s system and this should happen as soon as possible. One can also hope that all questions in Physics can all be incontestably decided in a similar fashion. For it will not be possible to ascertain the veracity of gravitation except after having been assured through precise calculations that it responds precisely to the phenomena; otherwise the Newtonian hypothesis would have little value over that of vortices by which many circumstances concerning planetary motion are explained. However, in such an incomplete and so to say underhanded way, that if the motion was anything but what it is, one might continue to explain it in the same way; sometimes as good, sometimes even better. The gravitational system does not permit us any such sort of illusion; one such point is when the observations would undermine the calculation and would bring the whole structure crashing down and would relegate the Newtonian system into the class of so many others that imagination has given birth and that analysis has destroyed. However the agreement that has been noted between celestial phenomena and those calculations based on the gravitational system is that a consensus is proved true more and more every day and appears to have fully won over the Philosophers in favor of this system. See the quoted articles.

In regards to the other physico-mathematical sciences, consult the articles for each.