Sect. I. Most obvious principles of classification. plained, and the second, or the comparison so frequently alluded to in the preceding paragraphs, will be understood from the following simple example: Let 52 and 53 be two rectilinear plane figures, whose corresponding sides and angles are equal. It is required, by a strict comparison of these figures, to prove their identity; and to show that if right lines be drawn from the extremity of the first to the extremity of the last side-these lines, which cause the figure to enclose a space, will also be equal. Fig. 52. Fig. 53. To prove the first part of the proposition, it is only necessary that we should sUPPOSE the figure A'B'C'D'E' placed upon the figure ABCDE, or, if the nature of the surface whereon the figures are drawn renders this supposition em L B barrassing that we should suppose an exact model, or copy, of fig. 53 carried from that figure and placed on 52. These two figures may, then, be so adjusted that A' shall fall upon A, and the line A'B' take the direction AB. But when this is effected it involves a necessary GONSEQUENCE: the point B', necessarily, falls upon B; since if it did not, the sides A'B' and AB could not be of the same length, a supposition which is contrary to the hypothesis. This consequence, again, produces another, the points B' and B coinciding, and the angles A'B'C' being equal by hypothesis, the sides B'C' and BC "must" coincide in direction; and, as B'C' and BC are equal, the point C' must fall on C. Chap. I. Classification of the varieties of form. Continuing this reasoning, it is evident, both that D' must fall on D and E' upon E: and the two figures, thus coinciding throughout, are equal in every respect. 16. The second part of the proposition more particularly deserves attention, as asserting the equality of lines that are not assumed to be equal, and which, indeed, are not mentioned in the hypothesis. Such a result will give rise to important consequences, and offers an example of a relation among certain lines necessarily involving a relation among others. In the case before us it is the equality in the lengths and positions of the sides AB, BC, CD, DE, fig. 52, with the corresponding sides A'B', B'C', C'D' and D'E', fig. 53, that is asserted to involve an equality between EA and E'A'. That it does so we have now abundant proof; since, on placing one of the figures, or an exact copy of one of the figures, upon the other, it has been shown that, A' and E' can be made to coincide with A and E; and consequently that an exact coincidence can take place between the lines AE and A'E'. Figures such as these, that is, whose extremities coincide, are called polygons or closed figures; and act a very conspicuous part in geometrical investigations. Any one side of such figures has been shown to have a necessary relation to the other sides; so that angles and sides taken at pleasure, cannot always be so placed as to form a closed figure. The reader will by this time, it is hoped, fully comprehend the nature and object of the very important science whose principles we are preparing to investigate. Sect. I. Most obvious principles of classification. Art. 16. A relation among certain lines may necessarily involve a relation among others. He will recollect the successive illustrations of the following results. First. That by a process of arrangement conducted according to the obvious analogies which figures bear to each other, the complex may be reduced to others more simple, until the forms of an extensive class are made to depend on the places of their more prominent points. Secondly. That a principle of classification, not less important, distinguishes form from magnitude. And, Finally, That a method of comparison may be employed which, substituting the "mental" for the "actual" superposition of figures, furnishes demonstrations not dependent on the magnitude of the diagram; and applicable therefore to all cases where the form investigated is the same. F Chap. I. Classification of the varieties of form. SECTION II. PRINCIPLES CONDUCTING TO A MORE REFINED ANALYSIS. Example of a geometrical investigation-reflections on the preceding example-branches into which geometry is divided-advantages peculiar to abstract geometry- principles on which geometrical investigations should be conducted-the comparison of figures reduced to the relation of points in space—the magnitude of all the parts of a figure deduced from its form and the magnitude of one part--the investigation of article 17 made to depend on a single lineal measurement—the investigation of article 17 conducted by assuming as known the figure which is the object of inquiry-relations of form and magnitude reduced to the relations of number-recapitulation—elements peculiar to the sub ject. 17. The recapitulation that closed the preceding section contains the chief ideas that have resulted from our inquiries, and little as they seem removed from those which a moment's reflection would have suggested, they, nevertheless, constitute facts imperfectly developed in Sec. II. Principles conducting to a more refined analysis. the earlier works on geometry, and that are capable of leading to the most important consequences. By their assistance we are enabled to penetrate some of the most hidden secrets of nature; and to answer questions apparently far removed beyond the reach of human inquiry. How difficult it appears, for example, to estimate the magnitude of the earth!-persons wholly ignorant of geometry are unable to divine the first steps of the inquiry, and look on its success as an ingenious fictionand yet the solution of this great problem follows immediately from the elementary principles we have developed. There are two ways in which the form and magnitude of the earth may be determined: one proceeds on a theory respecting the heavenly bodies; but the other is altogether geographical, and measures the earth as we would measure a work of art-a colossal statue, or a ship, or any other object of doubtful figure, and dimensions greatly exceeding our own. Confining ourselves to this last method, and supposing, for the present, the object to be a work of art; let us consider in what manner we must proceed in order to determine its magnitude and form. Our knowledge of the forms of objects is best acquired and recorded by means of pictures, and these pictures are most conveniently examined when all the parts of them lie in one plane. The intersections of planes with the surface investigated are figures of this kind: and when the planes are parallel and placed at equal distances, they afford, we have seen, art. a series of pictures that express in a very compendious way, the figure of the object. |