difference between which is small, to have been found, which, when substituted for x, give results with different signs; then a root of the equation lies between them. To determine it, let us suppose the root to be a + A; then f{a + A) = 0; but by Art. 116, equation (21), /(a + A) =f(a)+f'(a + $h)h, ■ A - - f{a) ■ "~ f'(a + 6h)' and neglecting Oh when added to a, since A is small, we have in which case however /'(a) ought to be large in comparison of f(a), otherwise the result is inconsistent with our supposition of A being small. And here I must conclude this inquiry. I have investigated the theory of algebraical expressions so far as the principles of Infinitesimal Calculus cast light on them; and I have incidentally given one or two theorems which enable us to determine approximate values of the roots. A systematic inquiry into the discovery of roots of equations would take me through the theory of quadratic, cubic, and biquadratic equations in their most general form; and would carry me on to the great discovery of Abel, that the roots of an equation of a degree higher than the fourth cannot generally be expressed in an algebraical form; and to make the subject complete I should also have to discuss the theory of simultaneous equations, and of elimination; and I should in the course of such discussion enter on the subject of determinants. All this is manifestly beyond the scope of the present work. To the student however who desires informatiou on the profound investigations of the ablest mathematicians in these subjects, I would recommend the study of (1) Cours d'Algebre Superieure, par J. A. Serret, 2me edit., Paris, 1854; (2) some memoirs in the works of Abel, edited by Holinboe, Christiania, 1839; (3) the memoirs of Evariste Galois in Liouville's Journal. He will also in M. Serret's treatise find references to the different sources whence he may derive information on these and kindred subjects. DIFFERENTIAL CALCULUS. PART II. CHAPTER IX. ON GEOMETRY. Section 1.—On the adjustment of the Principles of Geometry and Infinitesimal Calculus. 185.] It will by this time have become tolerably plain to the attentive reader, that the characteristic property of Number, which is the foundation of Infinitesimal Calculus, is that of continuous and infinitesimal growth; and that Differentiation is the mathematical expression of the Law of Continuity. Now our object in the following pages is to apply the propositions which have been proved' above to questions of pure geometry; and therefore it is necessary so to modify or enlarge the principles of that science, as to adjust them to those of Infinitesimal Calculus. As it is not however our intention to write a treatise on the principles or difficulties of Elementary Geometry, we shall rather enuntiate axioms and definitions, and state results, than prove propositions, leaving the last to be effected by our applications; neither shall we discuss the methods by which we have arrived at them, in the belief that a rational understanding of the first Chapter of the present Treatise is sufficient to explain them. In accordance with illustrations therein given, we have introduced the ideas of motion and of limits; motion perhaps as having to do with the generation of geometrical quantities, but chiefly as involving the property of infinitesimal divisibility, Price, Vol. i. Q q which is necessary to a due conception of the latter property of limits; motion however, as we have introduced it, does not encroach on the subject of mechanics, wherein we treat of motion as the effect of certain causes, and discuss its circumstances, as, for example, the particular law of force which produces it, the velocity with which the moving material changes position, which necessarily involves time, and so on: but in what follows we consider motion as a simple act, a primary conception as a quality of matter; and if it tends, as it does, to give clearness to our first geometrical conceptions, it is nothing but a servile adherence to an inferior, though customary method, which would hinder us from introducing it. It is conceived that all geometrical quantity, whether linear, superficial, or spatial, is from its very nature capable of increase or decrease to an infinite extent. A line may be very long, nay of an infinite length, or very short; space may be very small, such as, so to speak, it would require a microscope of almost infinite power to render visible, or it may be very large. Whenever such quantities vary, they vary in accordance with the law of continuity; they cannot pass from one magnitude to another without passing through all intermediate magnitudes; they grow larger and larger, or less and less. This capability of increase or decrease is involved in our idea of geometrical quantity; it is necessary to its completion; and if it is omitted, our notions fall short of the properties of the subject-matter of the science. There are however limits within which this variation is included; the superior limit of geometrical magnitude of the concrete kind called space is infinite space: so of superficial and linear magnitudes, the superior limits are respectively infinite superficies, and a line of infinite length. The inferior limit of all these is the same, the geometrical zero, a point. 186.] Of the definitions of geometrical quantities founded on such notions, the following are useful for our present object. I. A point is the inferior limit of geometrical space. II. A sphere is the locus of a point of space, which is always at the same distance from a given point. III. A plane is the surface of a sphere, the radius of which is infinitely great. IV. A circle is the locus of a point, which is always at the same distance from a given point, all the points being in one plane. V. A straight line is the arc of a circle, the radius of which is infinitely'grcat. VI. A triangle is a plane figure contained by three straight lines meeting one another, two and two. VII. And if the triangle is isosceles, the sides of that triangle, having a finite base and the vertex at an infinite distance, are parallel straight'lines. As this is not intended to be an accurate treatise on the principles of geometry, many words are used which have not been defined, as line, locus, &c.; these however are to be taken in their ordinary significations: and it is to be observed, with respect to these definitions and conceptions, that the surfaces, lines, &c. they refer to, are only approximations to the accurate ones. But they are such approximations as may differ from the real ones by quantities as small as we please; and as these small quantities may be infinitesimals, such that it would require an infinity of them to make a finite quantity, and as we do not take an infinite number of them, these differences must, in conformity with what has been said in the first Chapter, be neglected, and our definitions are rigorously exact. Having defined a plane, as we have done, to be the limiting spherical surface when the radius becomes infinitely great, it follows that the extreme positive side of the plane, when continued, runs into the extreme negative side; that is, having traced the plane as far as we can on the positive side, we meet it again on the negative; and although the surface appears to be discontinuous, it is not in reality so: the positive side being continued into the negative, and the apparent discontinuity arising from the defect in our power of apprehending and symbolizing such quantities. Thus then, if we have any continuous curve traced on the plane, and the curve runs off to the extreme positive side of the plane, we ought not to consider it to stop or to have points of discontinuity, but we must consider the branches of it to be continued, and must look for them on the negative side of the plane. We may borrow from the figure of the earth, and our mode of determining position on its surface, an illustration of what is here intended. We measure, say from the meridian of Greenwich, degrees along the equator to 180D east longitude; and then, instead of proceeding further on and measuring in the same direction, we measure backwards, and reckon degrees of longitude west: and what would be 181° east longitude becomes 179° west. If then east corresponds to the positive direction, west does to the negative. It is .worth remarking how exactly our notion of a plane coincides with the definition which I have given. We speak of the surface of water as a plane; whereas it is a portion of the surface of a sphere, whose radius is very large compared with the area we take, say, 4000 miles compared with a few inches. So again as to our conception of a straight line. A straight line being a particular case of a circle, is a continuous line; it does not terminate at positive infinity or at negative infinity; but the two branches of the line are connected with one another, running, if we may so speak, round the circle of which the radius is infinity, and joining together. If then we take any given point on the circle as the origin, the distance to the opposite extremity of the diameter of the circle is positive infinity, and we do not measure or follow the line further in this direction, but considering the line to be continued beyond that point, we meet it on the opposite side, and measure it backwards. There is no point of discontinuity in the line: the line proceeds in the same direction; it has been positive infinity; the pole or extremity of the diameter of the circle has been passed, and then the line becomes negative infinity. The illustration above given from the figure of the earth aptly illustrates our meaning in this case. Considering any meridian to be the very large circle, and taking any place on it to be the origin, the "antipodes" to it becomes either positive infinity or negative infinity, according as we measure in the positive or negative direction; the sign of the quantity changes immediately after the pole has been passed, and what was positive infinity becomes negative infinity. Therefore in this point of view infinity is not a quantity incapable of increase, for the line may be continued round and round the meridianal circle as often as we please; there is no limit to the quantity: the limit is to our powers of symbolizing such quantities. It is worth observing too, that the definition given of parallel |