adding to or subtracting from both members magnitude in the same proportion with that between the original members themselves, no alteration will be made in the proportion, which will be as much increased by the addition of magnitude to the antecedent, as it is diminished by the addition of proportional quantity to the consequent, and vice versa. When the proportion between two dimensions or their relation in respect of magnitude is the same with that between two other dimensions, the four are elliptically said to be proportional, or to be in proportion, meaning that they are in the same or in equal proportion.

From the foregoing view of the connection be tween magnitude and proportion, all the propositions of the fifth book of Euclid are immediate consequences. Thus, the multiplication of both members of a proportion, or a ratio, as it is called in Euclid, is the continued addition to each of magnitude in the same proportion with the original, and the division of both members by the same quantity being the converse of the foregoing operation, must give rise to parts in the same proportion with the wholes from whence they were derived.

Again, if A be to B as C to D, and C be substituted for B in the first ratio, the like alteration must be made in the second, in order to preserve the identity of the two proportions; that is to say,

the consequent in the second ratio must also be altered in the proportion of B to C, or the antecedent in the opposite proportion of C to B. Therefore, if C be substituted for B in the first ratio, the identity of the two relations will be preserved by substituting B for C in the opposite member of the second ratio, or if A be to B as C to D, A will be to C as B to D.

The notion of magnitude is transferred from linear to superficial extension by reference to the comparative extent of motion for which scope is given by surfaces of different size, without going twice over the same ground. The magnitude of a surface is apprehended by passing the finger over the entire surface; in which operation the size of the bodily organ must evidently be taken into account, inasmuch as the motion of which a mathematical point is capable without going twice over the same ground, is infinite in extent as well in a small surface as in a large one. In reasoning, therefore, concerning the proportion of superficial magnitudes, the surfaces compared must be considered as ultimately composed of lines of finite thickness, however small. We may thus conceive rectangles as composed of a series of straight lines, each equal and parallel to the base, and all of the same thickness. Rectangles, therefore, of the same height, will be composed of the same number of parallel lines, each equal to the base of the

rectangle to which it belongs; and the magnitude of the rectangles, which is measured by the total length of line into which each rectangle can be divided, will be proportional to the length of their bases.

We have indicated in the preceding pages the chief deficiencies to be supplied and alterations to be made in the premises of Euclid. One or two considerable variations will also be found in the body of the science. In the first place, the problems are omitted as being wholly unessential to the demonstration of the theorems with which they are connected. The student has credit in the postulates of Euclid for the possession of a ruler and a pair of compasses, and whenever the proof of a proposition requires any addition to the figures mentioned in the statement of the proposition, a problem is thought necessary in order to show him the means of describing the additional figure with those implements. The only object which can be gained by such a proceeding, is either to teach the student to describe the figure with exactitude, or to show the inherent possibility of the construction. With respect to the former purpose, it must be borne in mind, that the figure by which the proof is commonly accompanied is not itself the subject of reasoning, but merely an illustration in order to aid the student in conceiving the species to which the reasoning relates, and to enable the

geometer to speak with clearness of its separate parts. So long as it serves this purpose, it matters little how rude the illustration may be. On the other hand, the possibility of a construction properly framed will need no extraneous proof. The student of geometry must obviously have credit for the conception of the figures which form the subject of the science; and to that effect he must be acquainted with the elementary materials of form, with the attributes and relations by which they are to be moulded into definite systems. Now, the impossibility of a certain construction, or its incapacity of actual existence, must arise from some essential incongruity in the conditions on which the construction is based, and the same objection would equally be fatal to the distinct conception of the system. All our ideas being ultimately derived from experience, whatever can be distinctly conceived is inherently capable of exhibition in actual existence. The student, therefore, will carry in his own mind the only proof he requires of the possibility of any construction which he can distinctly imagine, nor will his conviction in such possibility be in any way increased by a problem shewing him a particular means of mechanical execution. Thus the capacity of division into parts being an essential attribute of every kind of magnitude, the student will be capable of conceiving the division of any magnitude, as an

angle, into two equal parts, or in any other proportion, and will require no further proof of the possibility of such an operation when called for in the course of demonstration. The geometer will, of course, have to take care that he proposes such constructions only as the student is able to carry out, an end which will effectually be secured so long as the position of each fresh element to be used in construction be defined by a simple relation to portions already fixed in the system.

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The principle of arrangement adopted in the following system, has been to place the propositions in an order in which each admits of being proved by direct reasoning, wholly avoiding the employment of ex absurdo demonstration. For that purpose, it has been found necessary to take some propositions relating to circles before all the geometry of triangles has been exhausted; and thus, somewhat to disturb the symmetry of Euclid's arrangement. It is not to be supposed that the conclusion from direct, is of greater cogency than that from ex absurdo reasoning. But, in the one case, the reasoning shews that the subject of the proposition is in the demonstrated predicament; in the other, that it must be so, leaving a craving in the mind to know how the necessity directly arises out of the essential nature of the figure. The two modes of proof are, in fact, examples of what are called synthetical and analytical reason