However, without entering into detail, one can take account, more or less, of the manner by which one forms rationally the fundamental types of nomograms. To do this it is first necessary to understand what, being immediately discernible, constitutes a graphic liaison. The only relation of position between two geo metrical elements that one can judge at a glance with sufficient exactness is a contact and, more specifically, a contact between line and point ; in this connection the parallelism of two straight lines (which one can also judge with some precision) may also be considered, for it is in effect the contact of one of the straight lines with the point at infinity of the other.
The notion of the graphic contact being thus understood, we come easily to the conviction that every graphic liaison intervening in a nomogram consists simply in the simultaneity of many con tacts between numbered elements or constants. Such is the postu late that is to be found at the base of nomography. While we must occasionally consider intervening contacts, confined in practice almost always to contacts between points and lines (parallelism included), the contacts between lines are so infrequent that we may neglect them in this sketch.
It is proper to recall here, as we have noted above, that we can only consider, on a plane, those systems of lines which depend upon a single parameter; on the other hand, it is possible to con sider a doubly infinite system of points which has been formed by the mutual intersections of lines taken in two simply infinite systems. The aggregate of these two systems forms what is known as a reseau (net). From this it follows that the maximum number of variables directly intervening in the same contact is, at most, three ; two being affected by the point, the third by the line enter ing into the contact. But there must be three contacts, from one plane to the other, to fix the position of a moving plane in refer ence to a fixed plane. The position being thus fixed, it is then possible to establish a contact between elements belonging to each of the planes; we may then secure an intrinsic representation for equations having as many as 12 variables. This principle was communicated to the Academie des Sciences de Paris by M. d'Ocagne in 1893 (Comptes Rendus, vol. I p. 216 and 277). If one of four intervening contacts is reduced to a parallelism of the axes OX and O'X' of the two planes, the orientation of the moving plane remains constant in relation to the fixed plane, and can be treated by a type of nomogram which permits intrinsic represen tation of as many as nine variables. This problem was made the subject of a profound study by Margoulis under the title L'abaque a transparent oriente.
If we consider a case where n moving planes are fixed simul taneously on a fixed plane, the fixation of each of them is made by three contacts, and it can be seen that we thus arrive at the most general type of nomogram permitting the intrinsic representation of equations up to 3 (3n+I) variables.
Nomography reached its full development in the comprehensive Traite of M. d'Ocagne, published in 1899, and crowned with the Prix Poncelet (in 1902) by the Academie des Sciences. Since the
appearance of this work, later theoretical developments have been made by Soreau, Farid Boulad, Clark, Gercevanoff, Margoulis and others, and the applications have multiplied to an extraordinary degree. Among these applications, the most numerous are those derived from the principle of the collinear points, relating more over to all kinds of technical fields. The library of the Ecole des Ponts et Chausees in Paris has a considerable collection of mate rial, brought together for most part by M. d'Ocagne. Next to the above method, the orientation of transparents, due to Margoulis, seems to be the most fruitful. Indeed, he has opened up a large number of interesting applications relating to hydrodynamics, aerodynamics, aviation, etc.
of M. d'Ocagne: Nomographie, Les calculs usuels affectues au moyen des abaques (Paris, 1891), Trazte de nomo graphie (Paris, 1899 ; 2nd ed., 1921) , Exposé synthetique des principes fondamentaux de la nomographie (Paris, 5903), Calcul graphique et nomographie (Paris, 1908; 2nd ed. 1914; 3rd ed. 1924), Esquisse d'en semble de la nomographie (Paris, 1925).
Works of other authors: Rev. P. de Beaurepaire, Graphs and Abacuses (Madras, 1907) ; F. Boulad-Bey, al Nomografia (Arabian; Cairo, 1908) ; S. Brodetsky, A first course in Nomography (Leeds, 1920) ; A. Esnouf, Nomography in a Nutshell (Port Louis, Mauritius; 1926) ; N. Gercevanoff, Les principes du calcul nomographique (St. Petersburg, 1906) ; R. K. Heylet, Nomography or the graphic repre sentation of formulae (Woolwich, 1913) ; Hews and Seward, Design of diagram for Engineers. Formule and the theory of the Nomography (New York, 1926) ; J. Jonesco, Nomografia (Rumanian; Bucharest, 1900) ; B. M. Kanorski, Die Grundlagen der Nomographie (Berlin, 1923) ; F. Krauss, Die Nomographie oder Fluchtlinienkunst (Berlin, 1923) ; 0. Laemann, Die Herstellung gegeichneter Rechentafeln (Ber lin, 1923) ; W. Laska, Wyklady nomografii (Polish, Lemberg, 1905) ; J. Lipka, Graphical and mechanical computation (New York, 1921) ; P. Luckey, Einfiihrung in die Nomographia (Berlin, 1918) ; K. Ogura, Calcul graphique et .Nomographie (Japanese, Tokyo, 1923) ; J. Peddle, The construction of graphical charts (London, 1910, 2nd eel. 1919) ; G. Pesci, Cenni di Nomografia (Livarno, 1901) ; G. Ricci, La Nomo grafia (Rome, 1901) ; C. Runge, Graphical Methods (New York, 1912) ; F. Schilling, Ueber die Nomographie von M. d'Ocugne (Leipzig, 1900) ; H. Schwerdt, Lehrbuch der Nomographie (Berlin, 1924) ; R. Seco de la Garza, Nomografia (Madrid, 1910), Nomogramas del Ingeniero (Mad rid, 1907 ; French trans. 1912) ; R. Sareau, Contribution a la theorie et aux applications de la nomographie (Paris, 1901), Nomographie, Theorie des abanes (Paris, 1920) ; R. C. Strachan, "Nomographic solu tions for formulas of various types," Proceedings of Amer. Soc. of Civ. Eng., vol. 4o (1914) P. and Transactions, vol. 78, pp. I., 408 ; T. Tanimura, Elements de Nomographie (Japanese, Tokyo, 1928) ; F. Ulkowski, 0 Nomografii (Polish, Lemberg, 1905) ; F. J. Vaes, Nomographie (Dutch, Rotterdam, 1901) ; P. Werkmeister, Das Ent werten von graphische Rechentaphs (Berlin, 1923). (M. d'O.)