SOLID, SURFACE, LINE, l'OINT. (Geometry.) We have thought it best to bring together the remarks which it is necessary to make upon these fundamental terms of geometry. According to Euclid, a point has no dimensions ; a line, length only ; a surface, length and breadth : a solid, length, breadth, and thickness. No one has the least doubt about each of these terms representing a clear and distinct notion already in the mind ; in spite of this, however, the propriety of the definitions has been made matter of much discussion.
Space being distinctly conceived, parts of space become perfectly intelligible. Hence arises the notion of a boundary separating one part of space from the rest. That a material object, a desk or an ink stand, occupies a certain portion of space, separated by a boundary from all that is external, needs no explanation : this boundary is called surface, and possesses none of the solidity either of the desk or ink stand, or of the external space. Surface itself, when distinctly under stood, is capable of division into parts, and the boundary which separates two parts of a surface has none of the surface, either on one side or the ether : it therefore presents length only to the imagination. Again, length itself is capable of division into parts : the boundaries do not possess any portion of length, either on one side or the other : they are only partition marks or points. Euclid reverses the order of our explanation, requiring first the conception of a point, then of a line, then of a surface, then of a solid.
That when we think of a point, we deny length, breadth, and thickness ; that when we think of a line, it is length without breadth that we figure to ourselves ; that in the same manner the surface of our thoughts possesses no thickness whatever—are, to us at least, real truths. We cannot, for instance, imagine what Dr. Beddoes meant when be said (` Obs. on Demonstrative Evidence,' p. 33), "Draw your lines as narrow as you conveniently can, your diagrams will be the clearer ; but you cannot, and you need not, conceive length without breadth." Why are diagrams the clearer, the narrower the lines of which they consist I Diagrams have no clearness in themselves ; the comprehension of them is in the mind of the observer. If diagrams having (so called) lines of one-bundredtb of an inch in breadth be clearer than others of five-hundredths of an inch, it is because the former approach nearer than the latter to a true representation of that which is in the mind, or of that which the mind desires to see per trayed. If the smaller the breadth the better the diagram in the clearness which it gives to the mind, it must be because the mind would have no breadth at all.
It matters nothing that the point, line, and surface are mechanical impossibilities ; that no point or line, if they actually existed, could reflect light to show them ; and that no surface could continue to exist for any perceptible time, even supposing it to have one moment of existence. Neither does it signify whether the ideas be necessary, or acquired from the senses ; the question in geometry is, Have you got them f not, How did they come ? There may be danger that some students should need at first to be frequently reminded of the abstract limits of which the conceptions must be made permanent, lest they should accustom themselves to rest in the imperfect approaches to these conceptions which are realised in their diagrams ; but it is always found that a moment's recollection will produce a satisfactory answer to any question upon this point.
There is, it is true, one circumstance in which the pupil may acquire a permanently false notion of the object of geometry. if an instructor should require what is called a very well-drawn figure in every case, with very thin lines and very small points, be may perhaps succeed in giving the learner some idea that geometry consists in that approach to accuracy which constitutes practical excellence in the applications of the science. No idea can be more false : let the good line be ex amined under a microscope, and it is seen to be a solid mound of black lead or ink, as the case may be. Hence it is perhaps desirable that the demonstrations should be frequently conducted with what are called ill-drawn figures, in order that no reliance may be placed on the diagram, further than as serving to remind the student of the ideal con ception which is the real object of his demonstration. This of course is recommended without prejudice to his learning the accurate use of the ruler and compasses for another distinct purpose, namely, the intention of producing avowedly approximate practical results.
It is to be noted that these definitions, so called, are in Euclid more than definitions. They appeal to conceptions supposed to exist, in words which are considered sufficient not to give, but to recall, the neces sary ideas. This they actually do, to the satisfaction of the learner, who would never dream of their containing anything dubious, if it were not for the ill-advised interference of the psychologist. Whatever of pleasure or profit there may be in the subsequent union of the sciences, there is, we doubt that the young geometer should not be required to examine the foundations of his notions of space : he cannot do this with effect until he has seen what these notions are by the light of their geometrical consequences.