630 MATHEMATICS were in possession of a method, called by them analysis, which had for its object the discovery of geometrical truth. But this consisted merely in taking any proposition suspected to be true and tracing its consequences until one was reached which either contradicted a known proposition or else was true and capable of leading by a direct process of reasoning (synthesis) to the proposition in question. In this we have no trace of the systematic development of geometric truth, and the method was apparently regarded by the ancients themselves as imperfect, for it makes no figure in such of their systematic treatises as have reached us. In somewhat sharp contrast with the Grecian geometry, but still essentially synoptic in method, stand the different varieties of modern geometry, which aims at greater generality in its definitions, pays less explicit attention to logical form, but arranges geometrical propositions as much as possible in the natural order of development or dis covery, and above all makes extensive use of the principle of continuity. As examples of the modern geometry may be cited the descriptive geometry (Geometric Descriptive, Darstellende Geometrie) of Monge ; the projective geometry (Geometric Projective, Geometrie der Lage) of Poncelet, Steiner, and Von Staudt ; and the geometry of transforma tion in general, of which projective geometry is but a parti cular case. There is one other highly interesting form of modern geometry, which, although analytic in some of its developments, and often exhibited in close alliance with other analytical methods, is nevertheless synoptic as to its fundamental principle, viz., arithmic geometry (Abzahlende Geometrie) or theory of characteristics, which originated in the characteristic equations of Pliicker, and was developed into a powerful special method by Chasles and others. See GEOMETRY and CURVE. Geometry, however, is not the only field for the synoptic treatment of manifoldness. This is obvious if we reflect that any magnitude whatever may be represented by a line; so that any function of not more than two elements may be represented by a geometrical construction and treated by any method applicable in geometry. Since the famous dissertation of Riemann, On the Hypotheses that form the Basis of Geometry, mathematicians have been familiar with the fact that the methods of geometry suitably generalized can be applied to the treatment of an re-fold manifoldness; and in point of fact the synoptic treatment of manifoldness under the name of ?i-dimensional geometry has been usefully employed by Cayley and others as an adjunct to the analytic method. The fundamental characteristic of the analytical treatment of an re-fold manifoldness is the specification of an element by means of n continuously varying quantities or variables (see MEASUREMENT). For dealing with continuous as distinguished from discrete quantity we have the special analytical method of the INFINITESIMAL CALCULUS (q.v.), built upon the notion of a limit, with its various branches, viz., the differential calculus, the integral calculus includ ing differential equations, the calculus of functions, and the theory of functions in general (see FUNCTION). But, whether we make use of the algorithm of the infinitesimal calculus or not, we find upon examination that all ana lytical operations with continuous quantity fall under the three laws of commutation, association, and distribution, so that they are fundamentally identical with the opera tions with discrete quantity ; the difference so far as there is any consists simply in the greater generality of the operand. The same fact may be looked at instructively in another light. Whether we consider analytical processes in concrete applications or look at them abstractly, we are equally led to the notion of a unit, by the multiplication or subdivision of which all the other quantities that enter into our calculus are derived. The exigencies of continuity are met by allowing that the multiplication or subdivision of the unit can be carried on to an unlimited extent ; but in any case where analytical formula; have to be reduced to arithmetical calculation (in which of course only a finite number of figures or arithmetical symbols can be used) the subdivision (or multiplication) of the unit actually stops short at a certain point ; in other words, all our methods are, in practice at least, discrete. Here therefore we have the meeting point of discrete and continuous quantity, and on this ground alone we might infer the fundamental identity of their laws of operation. The abstract science of quantity which we have just seen to be the essential part of the analytic treatment of manifoldness receives the name of ALGEBRA (q.v.). It was found very early in the history of that science that the full development of which it is capable could not be attained without great extension of the idea of quantity. This necessity first arose in connexion with the inverse operations, such as subtraction, the extraction of roots, and the numerical solution of algebraical equations (see EQUA TION), of which root extraction is merely a particular case. In this way arose essentially negative quantities, and the so-called impossible or imaginary quantities. The former may be said to depend on a new abstract unit - 1, and the latter upon new units V^i. The numbers having 1 for abstract unit are usually classed as real numbers, and in that case we may regard the ordinary imaginaries of algebra as depending on the new unit + V- 1, or i, defined by the equation i 2 + 1 = 0. But the extension was soon carried farther by the classical researches of Hamilton and Grassmann. 1 The theory of sets and the QUATERNIONS (q.v.) of the former and the Ausdehnungslehre of the latter opened up a boundless field for algebra, and led to a total revolution in our ideas of quantity. In view of the great extension thus effected in the meaning of quantity, it becomes an interesting if somewhat difficult undertaking to define the word. The following maybe taken as a provisional definition: Quantity is that which is operated with according to fixed mutually consistent laivs. Both operator and operand must derive their mean ing from the laws of operation. In the case of ordinary algebra these are the three laws already indicated, in the algebra of quaternions the same save the law of commuta tion for multiplication and division, and so on. It may be questioned whether this definition is sufficient, and it may be objected that it is vague ; but the reader will do well to reflect that any definition must include the linear algebras of Peirce, the algebras of logic, and others that may be easily imagined, although they have not yet been developed. This general definition of quantity enables us to see how operators may be treated as quantities, and thus to understand the rationale of the so-called symbolical methods. In combining operations, it is often observed that the combinations of operators fall under a few simple laws, in some cases in fact under the three laws of ordinary algebra ; these operators are therefore quantities according to the general definition, and can be treated as such. From the historical development of the analytic method there is little danger of the error arising that its application is peculiar to any special kind of manifoldness. As examples of its use in deducing the properties of tridimen- sional space we may cite the Cartesian geometry, the Geometrie de Position of Carnot, and the line geometry of Pliicker (see GEOMETRY). Its use in the various branches of applied mathematics, of which geometry is merely one of the simplest, is far more common than that of the - 1 In this connexion should be mentioned the great services of De Morgan, whose bold speculations on the fundamental principles of mathematical science have perhaps met with less than their due share
of appreciation.