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LINE] is to be made of R. Descartes, Géométrie (Leyden, 1637); John Wallis, Tractatus de sectionibus conicis nova methodo expositis (1655, Opera mathematica, i., Oxford, 1695); de l’Hospital, Traité analytique des sections coniques (Paris, 1720); Leonhard Euler, Introductio in analysin infinitorum, ii. (Lausanne, 1748); Gaspard Monge, “Application d’algèbre à la géométrie” (Journ. École Polytech., 1801); Julius Plücker, Analytisch-geometrische Entwickelungen, 3 Bde. (Essen, 1828–1831); System der analytischen Geometrie (Berlin, 1835); G. Salmon, A Treatise on Conic Sections (Dublin, 1848; 6th ed., London, 1879); Ch. Briot and J. Bouquet, Leçons de géométrie analytique (Paris, 1851; 16th ed., 1897); M. Chasles, Traité de géométrie supérieure (Paris, 1852); Wilhelm Fiedler, Analytische Geometrie der Kegelschnitte nach G. Salmon frei bearbeitet (Leipzig, 5te Aufl., 1887–1888); N. M. Ferrers, An Elementary Treatise on Trilinear Coordinates (London, 1861); Otto Hesse, Vorlesungen aus der analytischen Geometrie (Leipzig, 1865, 1881); W. A. Whitworth, Trilinear Coordinates and other Methods of Modern Analytical Geometry (Cambridge, 1866); J. Booth, A Treatise on Some New Geometrical Methods (London, i., 1873; ii., 1877); A. Clebsch-F. Lindemann, Vorlesungen über Geometrie, Bd. i. (Leipzig, 1876, 2te Aufl., 1891); R. Baltser, Analytische Geometrie (Leipzig, 1882); Charlotte A. Scott, Modern Methods of Analytical Geometry (London, 1894); G. Salmon, A Treatise on the Analytical Geometry of three Dimensions (Dublin, 1862; 4th ed., 1882); Salmon-Fiedler, Analytische Geometrie des Raumes (Leipzig, 1863; 4te Aufl., 1898); P. Frost, Solid Geometry (London, 3rd ed., 1886; 1st ed., Frost and J. Wolstenholme). See also E. Pascal, Repertorio di matematiche ''superiori, II. Geometria'' (Milan, 1900), and articles now appearing in the Encyklopädie der mathematischen Wissenschaften, Bd. iii. 1, 2.

V. Line geometry is the name applied to those geometrical investigations in which the straight line replaces the point as element. Just as ordinary geometry deals primarily with points and systems of points, this theory deals in the first instance with straight lines and systems of straight lines. In two dimensions there is no necessity for a special line geometry, inasmuch as the straight line and the point are interchangeable by the principle of duality; but in three dimensions the straight line is its own reciprocal, and for the better discussion of systems of lines we require some new apparatus, e.g., a system of coordinates applicable to straight lines rather than to points. The essential features of the subject are most easily elucidated by analytical methods: we shall therefore begin with the notion of line coordinates, and in order to emphasize the merits of the system of coordinates ultimately adopted, we first notice a system without these advantages, but often useful in special investigations.

In ordinary Cartesian coordinates the two equations of a straight line may be reduced to the form y = rx + s, z = tx + u, and r, s, t, u may be regarded as the four coordinates of the line. These coordinates lack symmetry: moreover, in changing from one base of reference to another the transformation is not linear, so that the degree of an equation is deprived of real significance. For purposes of the general theory we employ homogeneous coordinates; if x1y1z1w1 and x2y2z2w2 are two points on the line, it is easily verified that the six determinants of the array are in the same ratios for all point-pairs on the line, and further, that when the point coordinates undergo a linear transformation so also do these six determinants. We therefore adopt these six determinants for the coordinates of the line, and express them by the symbols l,, m, , n, where l = x1w2 − x2w1,  = y1z2 − y2z1, &c. There is the further advantage that if a1b1c1d1 and a2b2c2d2 be two planes through the line, the six determinants are in the same ratios as the foregoing, so that except as regards a factor of proportionality we have = b1c2 − b2c1, l = c1d2 − c2d1, &c. The identical relation l + m + n = 0 reduces the number of independent constants in the six coordinates to four, for we are only concerned with their mutual ratios; and the quadratic character of this relation marks an essential difference between point geometry and line geometry. The condition of intersection of two lines is

l′ + l′ + m′ + m′ + n′ + n′ = 0

where the accented letters refer to the second line. If the coordinates are Cartesian and l, m, n are direction cosines, the quantity on the left is the mutual moment of the two lines.

Since a line depends on four constants, there are three distinct types of configurations arising in line geometry—those containing a triply-infinite, a doubly-infinite and a singly-infinite number of lines; they are called Complexes, Congruences, and Ruled Surfaces or Skews respectively. A Complex is thus a system of lines satisfying one condition—that is, the coordinates are connected by a single relation; and the degree of the complex is the degree of this equation supposing it to be algebraic. The lines of a complex of the nth degree which pass through any point lie on a cone of the nth degree, those which lie in any plane envelop a curve of the nth class and there are n lines of the complex in any plane pencil; the last statement combines the former two, for it shows that the cone is of the nth degree and the curve is of the nth class. To find the lines common to four complexes of degrees n1, n2, n3, n4, we have to solve five equations, viz. the four complex equations together with the quadratic equation connecting the line coordinates, therefore the number of common lines is 2n1n2n3n4. As an example of complexes we have the lines meeting a twisted curve of the nth degree, which form a complex of the nth degree.

A Congruence is the set of lines satisfying two conditions: thus a finite number m of the lines pass through any point, and a finite number n lie in any plane; these numbers are called the degree and class respectively, and the congruence is symbolically written (m, n).

The simplest example of a congruence is the system of lines constituted by all those that pass through m points and those that lie in n planes; through any other point there pass m of these lines, and in any other plane there lie n, therefore the congruence is of degree m and class n. It has been shown by G. H. Halphen that the number of lines common to two congruences is mm′ + nn′, which may be verified by taking one of them to be of this simple type. The lines meeting two fixed lines form the general (1, 1) congruence; and the chords of a twisted cubic form the general type of a (1, 3) congruence; Halphen’s result shows that two twisted cubics have in general ten common chords. As regards the analytical treatment, the difficulty is of the same nature as that arising in the theory of curves in space, for a congruence is not in general the complete intersection of two complexes.

A Ruled Surface, Regulus or Skew is a configuration of lines which satisfy three conditions, and therefore depend on only one parameter. Such lines all lie on a surface, for we cannot draw one through an arbitrary point; only one line passes through a point of the surface; the simplest example, that of a quadric surface, is really two skews on the same surface.

The degree of a ruled surface qua line geometry is the number of its generating lines contained in a linear complex. Now the number which meets a given line is the degree of the surface qua point geometry, and as the lines meeting a given line form a particular case of linear complex, it follows that the degree is the same from whichever point of view we regard it. The lines common to three complexes of degrees, n1n2n3, form a ruled surface of degree 2n1n2n3; but not every ruled surface is the complete intersection of three complexes.

In the case of a complex of the first degree (or linear complex) the lines through a fixed point lie in a plane called the polar plane or nul-plane of that point, and those lying in a fixed plane pass through a point called the nul-point or pole of the plane. If the nul-plane of A pass through B, then the nul-plane of B will pass through A; the nul-planes of all points on one line l1 pass through another line l2. The relation between l1 and l2 is reciprocal; any line of the complex that meets one will also meet the other, and every line meeting both belongs to the complex. They are called conjugate or polar lines with respect to the complex. On these principles can be founded a theory of reciprocation with respect to a linear complex.

This may be aptly illustrated by an elegant example due to A. Voss. Since a twisted cubic can be made to satisfy twelve conditions, it might be supposed that a finite number could be drawn to touch four given lines, but this is not the case. For, suppose one such can be drawn, then its reciprocal with respect to any linear complex containing the four lines is a curve of the third class, i.e. another twisted cubic, touching the same four lines, which are unaltered in the process of reciprocation; as there is an infinite number of complexes containing the four lines, there is an infinite number of cubics touching the four lines, and the problem is poristic.

The following are some geometrical constructions relating to the unique linear complex that can be drawn to contain five arbitrary lines:

To construct the nul-plane of any point O, we observe that the two lines which meet any four of the given five are conjugate lines of the complex, and the line drawn through O to meet them is therefore a ray of the complex; similarly, by choosing another four we can find another ray through O: these rays lie in the nul-plane, and there is clearly a result involved that the five lines so obtained all lie in one plane. A reciprocal construction will enable us to find the nul-point of any plane. Proceeding now to the metrical properties and the statical and dynamical applications, we remark that there is just one line such that the nul-plane of any point on it is perpendicular to it. This is called the central axis; if d be the shortest distance, the angle between it and a ray of the complex, then d tan = p, where p is a constant called the pitch or parameter. Any system of forces can be reduced to a force R along a certain line, and a couple G perpendicular to that line; the lines of nul-moment