NOMOGRAPHY, a comparatively recent development in applied geometry, analogous to descriptive geometry or to graph ical statics. Its object is the general study of the representation, by means of diagrams call nomograms, of mathematical laws (Gr. nomos, a law) which are expressed analytically by means of equa tions with any number of variables. The principle involved in rep resentations of this kind was first applied, in the case of a law involving only two variables, by Descartes when, by means of co-ordinates, he showed that every equation in two variables cor responds to a plane curve.
In constructing such a graph we may neglect all measurements, simply drawing on squared paper, divided, say, into millimetres and having two graduated axes, OX and OY . A simple reading is then sufficient to give the different values of the variables as thus connected. This method of representation is now perfectly famil iar even to those who are not mathematicians.
Thus, to represent the law, f (z, =0, trace a curve whose cartesian equation is, f(x, y) =0, on squared paper where and In the case of a law involving three variables, the solution that first suggests itself is to give to any arbitrary value and then to proceed as in the example above. If, however, we give to a series of sufficiently contig uous values, and for each of these construct the lines to represent the laws of and then it suffices to write beside each line the value of which served in its construction; for then the lines and parallel to the axes of the diagram, and the line pass through the same point if the values of a,, and sat isfy the given equation. Margetts applied such an idea in his Longitude and Horary Tables (published in London in 1791). It was also put into systematic use by Pouchet and published in his Arithmetique lineaire (an appendix to his Metrologie terrestre [Rouen, 1797]). Since that time the idea has gained wide usage.
It should be observed that it is arbitrary to take z„ and and that we can as well replace the axes by two simply infinite systems of lines, the one determined by the values of z,, and the other by the values of At the outset, one will be tempted to see in this only a useless complication, since representation by means of rectangular axes in a plane does not lend itself to equations in three variables. However, there are many instances where, in re
placing the rectangular axes by other systems of lines made to correspond to and we obtain straight lines also for which formerly had to be traced out point by point on the ordinary squared paper. Hence arises a greater rapidity, simplicity and precision of the new construction.
The underlying idea of such a transformation was formulated in 1842 by Lalanne, under the name, the principle of anamorphosis. This transformation was generalized to a high degree in 1884 by Massau. M. d'Ocagne also helped to generalize this transforma tion when recognizing the ease of drawing circles, he developed a theory of representation of equations by three systems of circles (S.52 in the first edition of his Traite de Nomographie, 1899).
By means of concurrent lines it is impossible to represent equa tions of more than three variables, since it is not possible to represent on a plane any system of lines of more than two para meters. Nevertheless, it was possible to use this method for cer tain equations in more than three variables if, by introducing auxiliary variables, one could replace the given equation by a series of equations in three variables. A particular example of this device was proposed by Lallemand, in 1885, in the form of binary scales which he introduced in his abaques hexagonaux (an ingeniously combined variation, in view of certain practical ends, of Lalanne's abaques with their three pencils of parallel lines). In view of the nature of a nomographic diagram, Lalanne applied to it the name abacus (a(ctE), and the term has remained.