Sect. II. Principles conducting to a more refined analysis. Art. 25. The investigation of art. 17, conducted by asssuming, as known, the figure which is the object of inquiry. &c. must also depend on AB, and admit of an exact appreciation by the rules which contain the mutual dependence existing between the lines of rectilinear figures. These rules, we again repeat, are to be the objects of our further research, and will be established when we are prepared to treat in a more methodical manner the principles of abstract geometry; but in point of fact these rules are a subordinate part of the question we are considering; and supposing them established, a question immediately suggests itself, how, from a knowledge of the lines just mentioned, are we to determine the form of the section 1 m n p? A very obvious reflection will enable us to answer this question. It will be perceived, that since we assumed the section to be a circle, the length of each of the lines, ol, om, on, &c. must be the same, these lines being radii of that section. Taking, therefore, the known altitudes il, 2 m, 3 n, &c. from the calculated lengths of O i, o 2, o 3, &c., we must observe whether the remainders are equal, and should this appear to be the case, the truth of the hypothesis, or, in other words, the circular form of the section, is established. Much, indeed, would yet be wanting to prove the several steps of the process with geometrical accuracy, but enough has been accomplished to exhibit the powers of geometry, and to demonstrate the immediate dependence of the practical branches upon those which are ab stract. 26. A single observation, however, remains, before we conclude the illustration that has been afforded by the Chap. I. Classification of the varieties of form. Art. 26. The relations of form and magnitude reduced to relations of number. preceding problem, of the successive stages through which geometry may be supposed to have passed. Of the two principles mentioned in art. 25, the second has yet to be explained. Both, we have remarked, are founded on the independence of form and magnitude; and both ought to be regarded as only particular methods of expressing that truth. There is, however, another view that can be taken of the second principle; and as this method of considering the subject possesses many advantages, we shall for the future adopt it. The forms of rectilinear figures depend, we have seen, upon the lengths of their sides, and the angles at which the latter are mutually inclined. Let us recall the principles upon which these two species of quantity are measured. The lengths of lines must have reference to some other line that is taken as a standard; and the quantities of angles are measured by their relation to the whole plane space about a point. The ratios, then, of either of these quantities to the standards that measure them, will be just expressions of the quantities themselves. But as the ratios in question are numbers, it appears that every relation of form and magnitude-or, according to what we have hitherto seen-every relation of the form and magnitude of rectilinear figures-will be reduced to a relation of numbers. The rules which guided our researches concerning the properties of number, will thus become subservient to the present inquiry; and every improvement that re Sect. II. Principles conducting to a more refined analysis. Art. 26. The relations of form and magnitude reduced to relations of number. warded the diligence bestowed upon the former, will be a step in the acquisition of the latter science. 27. With these remarks we shall close the preliminary investigation that has occupied our attention in this division of the work: but, prior to entering on a more methodical inquiry, it will not be amiss to state briefly the elements to which we have ultimately traced the relations of form and magnitude. The elements peculiar to geometry may be considered as among the simplest ideas we possess. They are few in number; and neglecting certain notions about continuity, may perhaps be reduced to three-place, direction and distance. The first steps beyond these simple ideas have an immediate reference to them, teaching to designate by signs the places spoken of, and to obtain proper measures of their directions and distances. For science, which has grown out of art by refining and extending its processes, and rendering the operative part of them purely mental, has been unable to dispense with many conceptions having their origin in the ruder operations of the measurer of land, or in the labours of the artizan. The pegs and stakes used by the former to mark the most prominent places in his fields, probably gave birth to mathematical points, symbols used for the same purpose in the mental measurements of the geometrician. The size of the points is neglected in either case; it is omitted by the surveyor because the nature of his work does not make account of such small quantities; by the mathematician, for an analogous reason-the sensible L Chap. I. Classification of the varieties of form. points he employs are regarded as symbols of position, that ought not to be considered as existences or magnitudes. In this way we pass by an easy transition, from an acquaintance with external objects, to the ideas of place, direction and quantity. We reason on the parts of space, which are every where alike, by making use of something adventitious to them-of objects, course and rude in the first infancy of science, and gradually diminishing in size as it becomes necessary to indicate with increasing accuracy the place of which they are the signs this process may be continued without limit, until, neglecting the magnitudes of the symbols, and depriving them, as far as possible, of physical properties, we are led to the abstract notion of mathematical points, signs that are not regarded as having a real existence. The comparison of two points, the only idea we have of direction, will again lead to a similar abstractionthe straight line, or the symbol of space that has every where the same direction. And following this process, and passing from the direction of points to the distance that separates them, we obtain the notion of quantity, and its correlative-mea surement. The distance of the points is the straight line which joins them. And we estimate the quantity of the line by a continued superposition of its own parts, or the parts of some other line similar to it-a process also borrowed from the operations of art, and forming the pri mary notion involved in every species of measurement; operations which have their origin in the same humble source: whether the surveyor carelessly stretches his line over the asperities of the ground; or the philosophic Sect. II. Principles conducting to a more refined analysis. artist exhausts all the refinements of ingenuity in adjusting the microscopic divisions of his standard unit; or, lastly, the geometrician conceives the superposition of his imaginary lines, and proves the possibility of their coincidence, the leading idea is in all these cases the same: yet how wide is the step made by the latter in substituting the possibility of performing an operation for the operation itself; and thus replacing the limited testimony of the senses by evidence purely mental! And, as, from the comparison of points situated in one direction we obtained the notion of distance, so, by the comparison of two directions we form the idea of an angle; an idea altogether distinct from the former, but, like it, capable of measurement by the superposition of its own parts. Symmetry might be classed with the elements of position but as it seems a combination of the ideas already enumerated with that of order, which is common to our conceptions of time, space and number; there would perhaps be as much impropriety in ranking it with the elements peculiar to geometry, as in referring equality to the same source. A similar observation will apply to our ideas of continuity and infinity, both of which seem to involve the conceptions of motion and time; they must be classed among those obscure ideas that all minds perceive with difficulty, and yet that perhaps all perceive alike. The reader may be surprised to find in a science having such pretensions to accuracy, the foundation of much that is essential referred to ideas concerning which no distinct notion can be formed. But it may often happen that, whilst the primary idea is obscure, many of its relatives are distinct; and enter so extensively into our reasoning, |