1. Given a straight line and two points, one on each side of the same; on the given line one must determine a point in such a way that the difference of the distances to the two given points is a maximum.

2. On a given circle to find a point of such a nature that the sum of the squares of the distances from this point to two given points is a minimum.

The science program at the University of Christiania was new (established in 1851) and attracted good students in an enthusiastic environment. Guldberg particularly joined Peter Waage and Henrik Mohn. These three, together with the pharmacists Hans Hvoslef and Franz P Møller and Theodor Mohn, had a private "Physical-Chemical Society" in the years 1858-1860, with the purpose of "discussing matters of a physical-chemical nature". They met every Saturday and gave lectures to each other which were then discussed. The topics varied widely, from optics and astronomy to the factory production of aluminium; most of them lay in the border area between physics and chemistry. Guldberg contributed, among other things, a lecture on "mechanical heat theory" and several on how the melting point of metal alloys and other mixtures changes with the mixing ratio. The results of these last works were printed in the Proceedings of the Scientific Society of Christiania in 1860.

... three and four hours a week [of mathematics] were taught in the two upper forms. During the school's first few years, Ole Jacob Broch obviously led all the teaching of mathematics. When Broch left the school ... it was still in many ways [he] who ruled the teaching of mathematics. As professor of mathematics Broch not only taught future teachers ... [he] also determined the requirements for the 'examen artium'. Broch was known to be such a strict examiner and censor that many could not pass his requirements ...

Cato Guldberg was a mathematician and Peter Waage was a chemist. They were also brothers-in-law. Together they tried to find a way to express the "driving force" in chemical reactions mathematically. This was in the 1860s, at a time when it was thought that chemical forces acted when new substances were formed in a chemical reaction. Some explained reversibility, that is, the fact that chemical reactions can go both ways, as a balance between two opposing forces.

Building on the work of other researchers, Guldberg and Waage conducted about 300 experiments with reversible chemical reactions to explore the phenomenon further. Their strategy was to describe mathematically the forces acting in each of the opposing reactions, and then formulate a general law that said something about the relationship between them. They were also concerned with the conditions that affected the chemical forces acting in the reactions, such as temperature and solubility.

Guldberg and Waage eventually came to the conclusion that molecules of A and B react because they collide with sufficient force to cause them to react. How often they collide depends on the concentration of molecules A and B. Concentration was called "mass effect" by Guldberg and Waage. They called it mass effect because they argued that this was a factor that had to be taken into account in addition to affinity, the force that caused different substances to react with each other. In articles published in Norwegian, French and German in 1864, 1867 and 1879, they developed and refined their ideas towards what is today known as the law of mass action.

Wind direction and wind strength are determined by a balance between the pressure gradient, the Earth's rotation and the centrifugal forces in the vortex, and Guldberg set up equations for this balance. Using these equations, one can approximately calculate the wind from a map of the air pressure. He published the results in Polyteknisk Tidsskrift in 1872.

Since meteorological phenomena are highly complex, their mathematical study can only be approached by dealing with simple cases analogous to those found in nature. The equilibrium and movement of air constitute a still very underdeveloped branch of fluid mechanics, because there are too few observations for verifying numerical calculations. Encouraged by the excellent results obtained by Peslins, Reye, Colding, Ferrel, and Hann in this new application of analysis to meteorology, we have applied the principles of mechanics to atmospheric movements and have arrived at results that we believe are important for the development of meteorological science. First, we have found that one of the primary steps to ensure the success of meteorology is the establishment of meteorological stations at high altitudes: either on mountains or in balloons, and equipped, if possible, with recording instruments.

Horizontal winds or air currents at the Earth's surface are intimately linked to vertical currents; but the origin and movement of the latter depend not only on the physical state of the air at the Earth's surface, but also on the physical state of the air in the upper layers. Furthermore, wind speed and direction are both highly influenced by the Earth's surface, while their values ​​at a certain altitude would likely exhibit the regularity necessary to predict the course of meteorological phenomena.

We present here the first part of our studies, in which we have dealt with some simple cases of atmospheric mechanics. In treating atmospheric equilibrium, we introduced the virtual temperature, which is identical to the absolute temperature for dry air, but which depends on the amount of water vapor for humid air; the introduction of this quantity will simplify the formulas. We calculated the action of water vapor in the atmosphere, taking into account the variation of latent heat with temperature, which has been neglected in the usual formulas.

In studying horizontal currents under simple assumptions, we introduced friction along the Earth's surface, and we applied the theory to winds crossing the equator and vortices (C M Guldberg already, in 1872, developed part of this theory in the Norwegian Polytechnic Journal). The numerical calculations agree with the phenomena of nature within the limits that can be expected with the established hypotheses. It follows that the accurate observation of wind speed will be of great importance for meteorology.

Finally, we have also treated vertical currents in a special case to show the importance of meteorological observations at altitude.

We hope that the results derived from atmospheric mechanics will demonstrate the necessity of more extensive meteorological observations, especially in the tropics and in the upper layers of the atmosphere, and that the true path to progress in meteorology is founded on the development of atmospheric mechanics.