While the previous book of Cauchy I discussed remained mostly on the level of pure mathematics, the very title of this book, Leçons sur les applications de calcul infinitésimal, promises to deal with applied mathematics. Of course, even applying can happen at different levels, and Cauchy is here dealing not with, say, application of mathematics to other sciences, but with application of one part of mathematics within another part mathematics, more precisely, in geometry.
What is applied in geometry is infinitesimal calculus, which consists of differential and integral calculus, two methods which in a sense are counterparts to one another. Yet, in a sense this is not enough, since Cauchy is actually using a variety of mathematical tools. For instance, the book begins with a long introductory section on trigonometric functions. These functions express various relations between lines and angles and can thus be used in simplifying the formulas dealt in calculus.
Another example of mathematical tools used by Cauchy is provided by polar coordinates. Unlike the regular xy-coordinates, polar coordinates express all positions through a distance between the position and the origin and the angle that the line expressing the distance forms with one of the axes. As Cauchy notes, some geometric shapes are easier to express with the polar coordinates. This is particularly true of spirals. Spirals circle in a regular fashion around a centre, which we can think as located at the origin. While the distance between the centre and a point in spiral grows, the direction of the point from the centre changes in a regular fashion.
Differential and integral calculus are still the primary methods used in the book. Originally both methods have been justified through the idea of infinitesimals or infinitely small quantities - hence, their common name, that is, infinitesimal calculus. In my previous discussion of Cauchy, I noted that by an infinitesimal calculus he meant actually a variable which was thought to be diminishing into nothing. Here, he returns to the more relaxed notion of true infinitesimal quantities, perhaps because such deep theoretical questions need not be addressed in a more applied context.
A good example of Cauchy’s tendency to ignore the theoretical questions is his rather free use of the notion of different levels of infinitesimals. It is undoubtedly difficult to understand how one infinitely small quantity can be of a different level from another infinitely small quantity, that is, larger or smaller than it. Cauchy’s theoretical account of infinitesimals is rather enlightening. Different levels of infinitesimals could be defined through different rates at which quantities approach nothing. For example, if a quantity is approaching zero, its square will also approach it, except that the square will approach the goal in a quicker fashion than the original quantity.
Of the two methods, integral calculus seems simpler in the sense that it has less areas of application - and indeed, this part of Cauchy’s book is just a fraction compared to the part dealing with differentiation. Integration in its original sense was thought to consist of dividing something (curve, area or body) into infinitely many infinitely small parts and then, as it were, adding them up. In more modern terms, integration means finding more and more fine grained divisions of curve, area and body and sums of these divisions, and examining whether these sums approach some definite limit. The result of integration is then, simply, the magnitude of the curve, area or body. In fact, the result is at worst only an assumption that the curve, area or body will have some magnitude, and for actually defining this magnitude, information on other relations between the geometrical entities is often needed.
Basic use of differential calculus, on the other hand, is to find out relations between differentials, or again in terms Cauchy used in the previous book, to find out whether relations between variable, diminishing quantities approach some definite limit. An obvious application for this is the relation between variable x- and y-coordinates of points of a curve. One begins by looking at cords connecting ever closer points of the curve and the relationship of their coordinates, which determines the direction or inclination of the cord. The limit of this approach is then a direction or inclination of a tangent moving through this one point of the curve - or, one might even say, the inclination of the curve at that point. That is, in case the tangent even exists.
The last sentence is important. Cauchy already understands that differentiation is not a universal method, but fails at some points. The curve might have a sudden change in its direction, or it might loop back and touch itself, making it impossible to say what is its inclination at this point. It is an important change in self-understanding of mathematicians to accept mathematics as not interested merely of general rules, but also of particular exceptions to these rules.
Inclination of a curve is only one of its characteristics. Indeed, a curve and its tangent have the same inclination at one point and still look quite different around that point. One would like to say that the line has less of a curve than the curve, but it needs a more precise definition to do this. Cauchy asks us to think of a circle - the bigger the circle the more the circle looks like a straight line from a given point and less curved it seems. Hence, since the circle grows with its ray, Cauchy defines the curvature of this circle as inverse of its ray.
Now, while inclinations are represented by tangents of a curve, curvature of a curve can be explained by circles touching the curve - or more precisely, the inverse of their ray - which are known as osculating circles. This is still not a proper definition of curvature, since it is not clear what circle touching the curve one is to take as the osculating circle.
Still, we can use the circle as a clue for finding out what curvature is. Picture a tangent moving through the circle, always remaining a tangent and changing its inclination as it advances through an arc of the circle. Taking smaller and smaller arcs around a point, the relation of the inclination of the tangent to length of the arc might approach a certain constant, which happens to be the inverse of the ray of the circle or its curvature.
This notion of curvature is at once applicable to other curves, since they also involve the variables of the arc and the inclination. True, the concept of curvature does not work with all curves, at least not in all points. In particular, it requires the curve in question to be twice differentiable.
Cauchy’s book deals with many other important concepts. Worth mentioning is what happens when we move from a setting with two coordinates to a case with three conditions. I won’t go into details, but note merely that then mere curvature defined in the sense above is not enough, because this first curvature deals only with a two-dimensional issue. We also need then to take into account another quantity defining the curvedness of the figure when looking at the third dimension - this curvedness Cauchy calls second curvature.
