But to obtain an intrinsic representation of an equation of more than three variables, without introducing auxiliary variables, it is necessary to make use of some other principle of representation.
The first principle of this kind to appear was that of collinear points, which was introduced by M. d'Ocagne in 1884, and has proved to be so usable that, for equations with four variables, such as are found in practical work, it is employed almost uni versally. The method was derived from a particular application of Chasles's principle of duality, together with a special system of tangential co-ordinates (called parallels) ; it substituted for the concurrent lines of the older anamorphosic diagrams the collinear points ; and where in the older method it was only possible to represent a simply infinite system of lines, one could diagram matize a doubly infinite system of points. This explains why, with concurrent lines, intrinsic representation was limited to three variables, and why it is now possible to include as many as six variables.
As a typical example of an equation which it was not possible to represent by means of concurrent lines, which can easily be treated by means of collinear points, there may be mentioned quadrinomial equations, or more particularly, a complete cubic equation, which we meet with in certain problems of hydraulics and resistance of materials ; also the fundamental formulas of spherical trigonometry (which leads to the nomographic solu tion of all problems in astronomy of position and in navigation) ; d'Ocagne has also shown that all such problems can be resolved by means of a special unique nomogram. (Comptes Rendus de l'Academie des Sciences, vol. 138, p. 70, 1904 ; and Bulletin de la Societe Mathematique de France, vol. 32, p. 196, 1904.) In 1891, under the name Nomography, then used for the first time, M. d'Ocagne worked out a synthesis of all the methods of graphic representation proposed up to that date, taking the vari ous points of difference and welding them into a unity of concep tion. Since then he has continued to search for means of extending these methods into various fields of practice.
In considering a number of planes superimposed upon each other, each with its given geometrical elements numbered or not (called constants), among which is established a certain graphic liaison, he has been able to extend these methods and make them as general as possible with a given number of variables. To de
velop this idea, let us call a geometrical element (point or line) defined by the parameter z,. Now, the aggregate of elements E, constitutes the system (El), and each element is determined by the corresponding value of z,. • If such systems as (E1), (E2), . . . , (E.), fixed or subject to displacement, one with respect to another, co-exist in a plane, and if they are placed in such a way as to realize between the elements a given graphic liaison, this liaison will be expressed by an ana lytical relation of the corresponding values of the variables, . . . , z. such as f (z, z2 . . = 0. This aggregate of num bered systems, completed at the moment when graphic liaison is realized between them, constitutes a nomogram of the equation. Such a nomogram permits, by a simple reading (which guides the graphic liaison in question), the determination of the values of any one of the variables entering into the equation considered, when the values of the other n— I are given.
After these explanations it is easy to perceive what the object is in the more general problems of nomography; one might formu late it thus : one equation being given, to construct a nomogram by means of systems (E1), (E2) . . . (En) as quickly and easily as possible (i.e., by means consisting exclusively, if possible, of points, straight lines, or circles) ; a graphic liaison would be estab lished at once between them.
To do this, it is necessary at the outset by a kind of inverse process, to study the types of nomograms conceived a priori as the most simple for obtaining the canonical forms of the corre sponding equations; for recognizing the conditions necessary for an equation to be reduced to one of these forms; and when this reduction is possible, for deciding the simplest and most direct means of constructing the corresponding systems. It was the need for presenting the nature of mathematical problems that gave rise to nomography, problems into the details of which we need not enter here.