Page:Encyclopædia Britannica, Ninth Edition, v. 15.djvu/661

Rh M A T M A T 629 boundaries of the surface be plane or capable of being turned on the lathe, the desired form is best approximated to by working wood or plaster blocks. Plaster is not easily worked on the lathe, but a plane surface is readily got by rubbing, and if not too dry it may be cut with the knife. A surface which cannot be conveniently approximated to by the above method may be built up of strips cut to pattern, which are then filled in with some plastic material. To accomplish this, a system of sections either parallel or having a single axis is laid through the region to be filled up ; the bounding lines of these sec tions are calculated or obtained by geometrical construction. Strips of plate zinc are then cut to the required form and fixed securely by cross-pieces or soldered if necessary. Between the interstices of this zinc scaffolding some plastic material is filled in, such as embossing wax or damp clay ; and thus the form of the surface is rendered. The substance known in trade as plastilin is especially suitable for use in this way, as it retains its plastic property a long time. The finishing touches are given to the surface by means of a sculptor s style. From the clay model a plaster cast is formed and vell dried ; and its imperfections are removed by means of plaster- files and other instruments familiar to those who work in plaster. Lines which are to be shown on the model are drawn through points already marked on the original clay model, and engraved with fine files. A galvanoplastic copy of such a plaster model, not too deeply deposited, shows the surface even better than the plaster itself. MATHEMATICS. Any conception which is definitely and completely determined by means of a finite number of specifications, say by assigning a finite number of elements, is a mathematical conception. Mathematics has for its function to develop the consequences involved in the definition of a group of mathematical conceptions. Interdependence and mutual logical consistency among the members of the group are postulated, otherwise the group would either have to be treated as several distinct groups, or would lie beyond the sphere of mathematics. As an example of a mathematical conception we may take &quot;a triangle&quot;; regarded without reference to its position in space, this is determined when three elements are specified, say its three sides ; or we may take a &quot; colour sensation,&quot; which, on Young s theory, is determined when the amounts of the three fundamental colour sensations that enter into it are stated. As an example of a non -mathematical con ception we may take &quot; a man,&quot; &quot; a mineral,&quot; &quot; iron,&quot; no one of which admits of being so determined by a finite number of specifications that all its properties can be truly said to be deducible from the definition. A mathematical conception is, from its very nature, abstract; indeed its abstractness is usually of a higher order than the abstractness of the logician. Thus, for instance, we may neglect the other attributes of a body and consider merely its form ; we thus reach the abstract idea of &quot; form.&quot; But the form of an irregular fragment of stone does not admit of being finitely specified, and is therefore not susceptible of mathematical treatment. If, however, we have a carefully squared cubical block of granite to deal with, for most practical purposes its form is specified by stating that it is a cube, and assigning one element, viz., an edge of the abstract mathematical cube by which we replace it. This example illustrates at once the limits of mathematical reasoning and the nature of the bearing of mathematics on practice. A variety of words have been used to denote the dependence of a mathematical conception upon its elements. It is frequently said, for instance, that the conception is a &quot; function &quot; of its elements. One word has recently come into use which is very convenient, inasmuch as it draws attention at once to the fundamental idea involved in mathematical conception and to the prime object of mathematical contemplation, viz., &quot;manifoldness.&quot; Number is involved in the notion of a manifoldness both directly, as any one can see, and also indirectly in a manner which the mind untrained to mathematical thinking does not so readily understand. Take on the one hand the case of a triangle considered without reference to its position but merely as composed of three limited straight lines, it may be completely determined in various ways by assigning three elements. A triangle may therefore be called a triple discrete manifoldness. A plane quadrilateral considered in the same way (being fully determined when four sides and a diagonal are known) is a quintuple discrete manifold- ness ; and a plane polygon of n sides a (2n - 3)-ple discrete manifoldness. Consider on the other hand the assemblage of points on a given straight line, they are infinite in number yet so related that any one of them is singled out by assigning its distance from an arbitrarily chosen fixed point on the line. Such an assemblage is called a onefold continuous manifoldness, or simply a onefold manifoldness ; another example of the same kind is the totality of instants in a period of time. The assemblage of points on a surface is a twofold manifoldness ; the assemblage of points in tridimensional space is a threefold manifoldness ; the values of a continous function of n arguments an ??-fold manifoldness. It should be observed that the distinction between discrete and continuous manifoldness is not of necessity inherent in the conception. For one purpose we may treat a conception as a discrete manifoldness, for another as a continuous manifoldness. Thus we have seen that an unlimited straight* line may be treated as a onefold con tinuous manifoldness ; but, if we regard it as a whole, and with reference to the fact that its position in space is determined by four data, it becomes a quadruple discrete manifoldness. The primary, although not the only, operation in the treatment of a discrete manifoldness is numbering or counting ; hence arises the pure mathematical science of number, comprehending (abstract) Arithmetic and its higher branch commonly called the Theory of Numbers. Without entering into a discussion of the definitions and axioms of the science of number, it will be sufficient here to remark that all numerical operations are reducible to three fundamental laws commonly called the commutative, associative, and distributive laws. The four fundamental processes, or four species, as they are sometimes called, two of which, addition and multiplication, are direct, and two, subtraction and division, inverse, are solely defined by and derive their meaning from the three laws of operation just mentioned. A careful consideration of the methods in vogue for dealing with continuous manifoldness shows that they reduce themselves to two, which may be called the synoptic method and the analytic method. In the synoptic method we deduce the properties of a manifoldness by contemplat ing it as a whole, aiding our understanding, when it is necessary to do so, by a diagram, a model, or any other concrete device more or less refined according to circum stances. In the analytic method we fix our attention upon the individual elements of the manifoldness, usually defining each element by a definite number of specifications the variation of which leads us from element to element of the given manifoldness. We examine the properties of an element in the most general manner, and from them we predicate the properties of the manifoldness as a whole. The best and most familiar examples of the synoptic treatment of manifoldness are the different varieties of pure geometry. Among these we may mention the apagogic geometry of the Greeks, which starts with a collection of definitions and axioms, enunciates and proves proposition after proposition with great attention to strict logical form and with continual reference to the grounds of inference, but pays little attention to the ordering of theorems with a view to mutual illustration, and carefully suppresses all traces of the method by which the propositions were or might have been discovered. It is true that the Creeks