Opening page to Rudolf Clausius' 1875 Mathematical Introduction chapter.
In famous publications, “Mathematical Introduction: on Mechanical Work, on Energy, and on the Treatment of Non-Integrable Differential Equations” is the opening twenty-page chapter to German physicist Rudolf Clausius’ 1875 textbook The Mechanical Theory of Heat in which a number of key thermodynamics principles are introduced, namely: work (or mechanical work), the Cartesian system, energy of a body, and the criterion for an exact differential. The famous opening paragraph reads as follows: [1]

“Every force tends to give motion to the body on which it acts; but it may be prevented from doing so by other opposing forces, so that equilibrium results, and the body remains at rest. In this case the force performs no work. But as soon as the body moves under the influence of the force, work is performed.”

The mathematical introduction originated in the 1858 article “On the Treatment of Differential Equations which are Not Directly Integrable” published in Dingler’s Polytechnisches Journal, which was then expanded on as the introduction section to the first (1865) and second (1875) editions of The Mechanical Theory of Heat. [2] A good part of the derivation seems to be based on French physicist Gustave Coriolis’ 1829 textbook Calculation of the Effect of Machines, although Coriolis is not explicitly mentioned. The classic 1875 edition of the mathematical introduction is divided into nine section, as discussed below

I.1 Definition and Measurement of Mechanical Work
In this section, Clausius introduces the basics of French physicist Gustave Coriolis’ 1829 principle of the transmission of work (although he dosen't site Coriolis), in that whenever a body moves under the influence of a force, work is performed, and that this movement can be quantified as a product of the line of motion of the particle and the component of the force in the direction of motion.

I.2 Mathematical Determination of the Word Done by the Variable Components of the Force
In this section, Clausius introduces the basic differential work equation:

dW = S ds (1)
where S is a component of the force (or FS in modern notation) action on a material point through a differential length of space ds. He then explains that, to facilitate further calculations, it is expedient to resolve both the direction of movement and the direction of the force into a Cartesian coordinate system, whereby the differential of work, in the two-dimensional case, becomes a function of force in the x- and y-directions:

dW = X dx + Y dy (3)

I.3 Integration of the Differential Equation for Word Done
Clausius here introduces the reader to the elusive importance of the “complete differential”, whereby to be a complete differential, the functions X and Y of the right hand side of any given two or more variable expression, such as the Cartesian expression for work, must satisfy the condition for an exact differential:

$\frac{dX}{dy} = \frac{dY}{dx} \,$ (4)

If the the condition of equation (4) is satisfied then, according to Clausius, the expression on the right of equation (3) becomes immediately integrable. This section is very important in that it sets the definition of the state variable.

I.4 Geometrical Interpretation of the Foregoing Results, and Observations on Partial differential Coefficients

I.5 Extension of the Above to Three Dimensions

I.6 On the Ergal

I.7 General Extension of the Foregoing

I.8 Relation between Work and Vis Viva