The ideals of and in other cases, when the imperfection seriously affects the result, this can be estimated and taken into account in our calculations. The ideals of geometrical theory have that perfection to geometry. A which we now refer. The scientific conceptions of the difficulty ex- point, the straight line, the plane, the curved surface, and plained. the regular solid set forth things of a finer quality than any which present themselves to the senses. The ordinary definitions of some of these ideals have been the occasion of perplexity both to metaphysicians and to those mathematicians who have critically examined their own conceptions. In particular, the point, the line, and the surface, as described in geometry, are impossible entities. The existence of that which has neither length, breadth, nor thickness, but position only, or of that which has length, position, and direction, but no width and no thickness, or of that which has length and breadth but no thickness or depth, is inconceivable. Thus, apparently, geometry sets out by asking us to accept absurd conceptions. The difficulty here presented cannot properly be ascribed to the imaginary perfection of the entities considered. There is nothing impossible or absurd in imaginary perfection. The difficulty originates in connection with the peculiar scientific use for which the ideals of geometry are intended, and which they serve. Yet, as it could have arisen only where such ideals were employed, it may be considered in the present connection. A solution of it is offered in the two following statements: with attri Geometry First, strictly speaking, geometrical science is not con concerned cerned with any independent entities which can be called butes rather points, lines, and surfaces, but only with those inherent than with parts of solid bodies which these names may indicate, or bodies. rather to speak more strictly still with the characteristic attributes of these parts. A surface, as its name signifies, is properly the boundary of a solid body; a line is the edge at which one surface meets with another; a point is the termination of some sharp projection of the solid; the first of these is considered only with reference to its superficial extent; the second with reference only to its length and course; and the third with reference to its position only. Even the solid body itself, though possessing an independent or substantial existence, is thought of only so far as it has shape and size, so that, in truth, the shape and size of the solid, rather than the solid itself, are considered. For in geometry solidity means simply space-filling extension. -- This fact that the proper objects of geometrical thought are not independent entities, but attributes of solid bodies or of their inherent parts, helps to explain the character of geometrical definitions. Though no surface can exist without solidity, we can think of its breadth without thinking of the solidity beneath it; though no line can exist save as a slender solid strip, we can think of its length without thinking of the solidity accompanying that; and though no point can exist save as the terminal part of a line or sharpened body, we can think of its position, or of the position of the centre of it, without thinking of its solidity. Therefore, in a science which concerns itself with surfaces, lines, and points only that it may consider their characteristic attributes, it is natural that these entities should be spoken of as if they possessed these attributes alone, although, as we have said, these attributes cannot exist, nor even really be conceived to exist, in separation from each other and from solidity. Geometry uses auxili ary concep tions. This mode of speech will be further justified by the second statement which we have to make. This is that ideal conceptions of lines, points, and surfaces, as separate entities, are used by us as supports of geometrical thought. The mind dislikes to conceive of mere attributes, even though these may be the proper subjects of its consideration; so, instead of attributes simply, it conceives of objects as having them. In this way one's conceptions are made more to resemble fact. But in the combinations of thought it is needful that each attribute, or each system of attributes, should be allowed its own proper value and effect; therefore we fashion for ourselves objects in which all other attributes than those specially given to them exist in the lowest conceivable degree. In short, we imagine entities which have no appreciable force or value, except in those particulars with which we have characterized them. Hence geometrical ideals are things more perfect for the purposes of thought than any that can be made or found. But they are not absurdities. The point occupies space, though it is infinitesimally small; the line has width and thickness, but it is of the utmost conceivable attenuation, and is without the slightest roughness or irregularity; the superficies is a film of indescribable thinness, and absolutely continuous; while the solid is bounded by such surfaces, and is free from all interstices, so as fully to fill the space assigned to it. These conceptions involve no absurdity; they are consistent with the necessary laws of being. But the size of the point, the width of the line, the thickness of the surface, are so insignificant that they can be disregarded in reasoning. And the solid, being of perfect density, is such that it is measured exactly by the space it occupies. When, therefore, the geometrician says that the point has position only, the line length only, and the surface breadth only, and identifies the solid with the full possible content of a given space, we are to understand that these ideals are such as may simply represent certain attributes, and such that by means of them we reason, more easily than we otherwise could, regarding the position, length, superficial extent, and solid contents of material objects. The forma The manner in which men of genius form hypotheses tion and use and scientific theories is essentially the same with that in of scientific which we form suppositions to account for facts which hypotheses. interest us. The phenomenon to be explained is attentively studied, and is compared with similar phenomena whose causes are known. Thereupon a cause is conjectured similar to some known cause or causes, but differing from it or them in some way to account for the peculiarities of the case in hand. But often an hypothesis when made is found unsatisfactory. Deductions from it conflict with some of the observed facts, or with facts not previously considered. Then that conjecture is abandoned for another, constructed in a similar way, but either wholly or partially different. Another process of trial takes place with this hypothesis; and so the work goes on till either hope of discovery is given up, or an hypothesis is framed which satisfactorily explains the facts. Then, if the cause assigned by this supposition be found really to exist and operate, or if, in any other way, we can prove that no other cause can possibly produce the results to be accounted for, the hypothesis becomes a doctrine fully received and confidently held. Such has been the history of almost all important theories. The second The use of philosophic invention, in which we suppose use of hy- things to exist for the purpose of deducing from them impotheses. aginary consequences, is next in importance to that which aims at the explanation of facts and the discovery of causes. Indeed, the formation of hypotheses or conjectures would be comparatively ineffectual toward the ascertainment of truth if these could not be tested by a deductive process. This is done when one combines the hypothesis to be tested with some known fact or principle, and then marks the legitimate inference. For he can now inquire whether this inference agrees with the various facts known to him which relate to the subject in hand, or with such facts as he can discover, or with the results of his experiment, that is, with such facts as he can create. If there be agreement, the hypothesis is confirmed; if there be conflict with fact, it is overthrown. Thus suppositional inference is a test of hypothesis. But it has uses more immediately its own; because the full significance of any scientific truth cannot be understood unless we combine it with one supposition and another, so as to perceive its different possible bearings. For example, the importance of solar light and heat cannot well be estimated, unless we should suppose them suddenly to cease to illuminate and warm the earth, and should consider what midnight darkness and frigid death would then enwrap all beings that are living now. Useful in- A yet more notable use of imagination, in connection vention. with a deductive process, is exhibited in useful contrivance. Such was the invention of the air-pump, by Otto Guericke; of the thermometer, by Sanctorius; of the reflecting telescope, by Gregory; of the safety-lamp, by Sir Humphry Davy; of logarithms, by Napier; and of the Calculus, by Sir Isaac Newton. The steam-engine, the cottongin, the electric telegraph, the telephone, the daguerreotype; and machines for carding, spinning, weaving, knitting, sewing; for type-setting and printing, for mowing, reaping, threshing; and many others employed in modern civilization, are the products of that invention of which we now speak. For invention, in the narrower sense, indicates only one species of philosophical imagination or invention, and signifies the work of discovering methods by which laws and instrumentalities already known may be made to serve useful ends. This work is similar to that of discovering the causes and conditions of phenomena, but it is more completely dependent on the constructive power of the imagination. That conjecture which uses hypotheses for the purpose of discovering antecedents starts out from the perception or assumption of facts; but this invention, which aims to realize an end through the use of means, has only a possibility in view. Moreover, causes may often be found by simple inquiry and search, without the aid of supposition; but mental combination alone can afford us any hope of the production of a new agency. Sometimes the discovery of a useful adaptation may appear to result from chance; but it seldom or never results from chance alone. Ordinarily, the inventor must try many combinations, one after another, without producing the effect hoped for. But if the end be a possible one, his work makes progress. Every new attempt reduces the likelihood of failure in the next, and increases the probability of success. But, generally, some uncertainty still remains; so that in most instances the end seems attained or suggested, at last, by some fortunate circumstance, and has the appearance of being found rather than achieved. Hence it is that the term "invention," which originally signified only discovery, has come to be applied to the laborious process of contrivance, and especially to the contrivance of useful instrumentalities. Imaginative That exercise of the philosophic imagination which furillustration. nishes illustrations of truth may be passed without extended discussion. It is a fact that a principle is sometimes better stated and understood by means of suppositions and similitudes than it can be by means of direct statement, or even by describing any actual example of its operation. The right illustration of truth is a work of less difficulty than the formation of wise hypotheses, or the invention of useful applications. Yet it involves care and skill. An illustration which does not truly present the point to be considered, only confuses the mind; and an illustration which sets forth with equal or greater prominence some other point also, may be the cause of positive error. CHAPTER XLVII. THE RATIONAL FACULTY. 1. THAT power of thought which manifests itself prominently as the controlling element in the rational or discursive phase of intellect, is commonly known as reason. The common Most logical and metaphysical writers define this definition of faculty as that by which the mind forms general nothe rational tions and uses these notions in inference and in other faculty. operations pertaining to the perception of truth. This definition does not appear to be correct. On the one hand, general notions are employed in operations which belong to the perceptive and reproductive faculties; and, on the other, certain exercises of the reason do not involve general notions. The cognitions of acquired perception, which are common to man and the brutes, and are not exercises of reason, involve the instinctive use of rules of inference, which rules are of the nature of general notions. In short, several operations which are often described as belonging to the rational faculty exclusively, occur in mental phases which are contrasted with reason. And the doctrine that every exercise of reason involves the use of general thought cannot be sustained. It is now commonly admitted that trains of geometrical ratiocination can, and often do, take place from the simple inspection and consideration of diagrams, and without the intervention of universal principles. Yet such reasonings are among the purest products of the rational faculty. Locke says that reason is" that faculty whereby definition. man is supposed to be distinguished from beasts, and wherein it is evident that he much surpasses them." To make this definition explicit and satisfactory, we must say "that faculty of perception and judgment;" for man surpasses the brutes in imagination as well as in reason. Locke's As Locke's " Essay" was directed to the consideration of the understanding, the limitation we have suggested was doubtless in his mind. Indeed, this is evident; for he goes on to describe reason as the faculty which first distinctly ascertains the grounds for belief or knowledge, and which then applies them so as to obtain either certainty or probable conviction. Kant's em the term Other authors such as Kant, Coleridge, and ployment of Morell - give the name "reason" to a faculty which they distinguish from the understanding, or reasoning power, and by means of which we immediately possess ourselves of the necessary elements or eternal principles of truth. We can discover no good ground to believe that we have any such independent faculty, and therefore shall not dwell on this meaning of the term. Nor need we discuss those teachings which make reason something impersonal, separate from the soul, and communicated to it, a revelation of the Absolute Intelligence! Philosophers should leave such language to orators and poets. power, but a dowment of Reason is not An exact definition of the rational faculty can be a single obtained only by a careful scrutiny of that conception peculiar en- of reason which those employ who use the term withmental abil- out making it the expression of any philosophical theory. An examination of this usage, together with a consideration of the mental facts immediately related to it, will lead to the following results: : ity. In the first place, reason is not a single power, but rather a collection of powers which operate in conjunction with each other. Both thought and belief, together with attention, association, analysis, synthesis, abstraction, conception, generalization, spe |