CRP tells us that, with the signs appropriate to their directions attached to CR and RP, RP=CR tan a, i.e. MP—DC e. (OM —OD) tan a, and this gives that y—k=tan a (x —h), an See also:

• EQUATION (from Lat. aequatio, aequare, to equalize)

Again, _Y2—yt y-yt -x2—xl(x -xl), i.e. (yi — y2)x — (xi —x2) y+xiy2 —= 0, represents the line determined by the data that it passes through two given points (xi, yl) and (x2, y2). To prove this find m in the equation y—y1=m(x—xi) of a line through (x,, from the See also:

• CONDITION (Lat. condicio, from condicere, to agree upon, arrange; not connected with conditio, from condere, conditum, to put together)

The equation is of the See also:

• FORM (Lat. forma)

In oblique coordinates the general equation of a circle is x2+2xy See also:

• COS
• COS, or STANKO (Ital. Stanchio, Turk. Islan-keui, by corruption from Etc rav KW)

. . F„(x, y) =o. In particular an equation of degree n which is See also:

• FREE

51), the See also:

• COMMON

13. The prin'biple employed in showing that the equation of the common chord of two circles is S—S' =o is one of very extensive application, and some more illustrations of it may be given. Suppose S=o, S'=o are lines (that is, let S, S now denote linear functions Ax+By+C, A'x+B'y+C'), then S — kS' =o (k an arbitrary constant) is the equation of any line passing through the point of intersection of the two given lines. Such a line may be made to pass through any given point, say the point (xo, yo) ; i.e. if So, S'o are what S, S' respectively become on See also:

• WRITING
• WRITING (the verbal noun of " to write," O. Eng. writan, to inscribe)

If (x,, y,) are the coordinates of A, and (x2, y2) of A', then the roots of this equation are x1, x2, whence easily 1 i A+Bm x,+x2 -2 C And similarly, if the equation of the line OBB' is taken to be y=m'x, and the coordinates of B, B' to be (x3, y3) and (x4, y4) respectively, then I I + m r x3+x4=—2 C, We have then by § 8 x(Y, —y4) —Y(x,—x4)+x,Y4—x47, =0, x(Y2 — y3) +Y (X2 —x3) +x2Y3 — x3Y2 =0, as the equations of the lines AB' and A'B respectively. Reducing by means of the relations y,—mx,=o, y2—mx2=o, y3—m'x2=o, y4—m'x4=o, the two equations become x(mx, —m'x4) —Y(x, —x4)+(m'-m)x,x4 =o, x (mx2 — m'x3) —Y (x2 —x3) + (n2' —m)x2x3 = o, and if we See also:

• DIVIDE

The applications of this theorem are very numerous; for instance, we derive from it See also:

• PASCAL, BLAISE (1623-1662)
• PASCAL, JACQUELINE (1625-T661)

Take any point P (x, y) on the line, and construct OM and MP as in fig. 48. The sum of the projections of OM and MP on OD is OD itself; and this gives the equation of the line x cos a+y See also:

• SIN
• SIN (O. Eng. syn: a common Teutonic word, cf. Dutch zonde, Ger. Siinde)

Construct (fig. 53) the old and new co- ordinates of any point P. Ex-pressing that the projections, first on the old axis of x and secondly on the old axis of y, of OP are equal to the sums of the projections, on those axes respectively, of the parts of the broken line 00'M'P, we obtain: x=h+X cos 0+Y cos (O+ier) = h+X cos 0-Y sin 0, and y=k+X cos(4Zr-0)+Y cos 0= k+X sin 0+Y cos 0. Be careful to observe that these FIG. 53. formulae do not apply to every conceivable See also:

• CHANGE (derived through the Fr. from the Late Lat. cambium, cambiare, to barter; the ultimate derivation is probably from the root which appears in the Gr. «a urmu', to bend)

If e : I is the ratio, e is called the eccentricity. The distances are considered signless. Take (h, k) for the focus, and x cos a+y sin a-p=o for the directrix. The absolute values ofJ{(x-h)2+(y-k)21 andp-xcosay sin a are to have the ratio e : 1; and this gives (x-h)2+(y-k)2 = e2(p-x cos a-y sin a)2 as the general equation, in rectangular coordinates, of a conic. It is of the second degree, and is the general equation of tlf9"t degree. If, in fact, we multiply it by an unknown a, we can, by solving six simultaneous equations in the six unknowns T, h, k, e, p, a, so choose values for these as to make the coefficients in the equation equal to those in any equation of the second degree which may be given. There is no failure of this statement in the See also:

• SPECIAL

Even when the locus represented is real, we obtain, as a See also:

• RULE

54, 55, 56 in See also:

• ORDER
• ORDER (through Fr. ordre, for earlier ordene, from Lat. ordo, ordinis, rank, service, arrangement; the ultimate source is generally taken to be the root seen in Lat. oriri, rise, arise, begin; cf. " origin ")
• ORDER, HOLY

The line which occupies this limiting position is the tangent at P. Now if we sub-See also:

• TRACT (from Lat. tractare, to treat of a matter, through Provencal tractat and Ital. trattato)

Denoting by (, n) the current coordinates, we find, as above, that the tangent at a point (x, y) Of a curve is r-y= (E-x)dy/dx, where dy/dx is found from the equation of the curve. If this be f(x, y) =o the tangent is (I--x) (af/ax)+(,l-y) (af/ay) =o. If p and (a, /3) are the radius and centre of curvature at (x, y), we find that q(a-x)-p(I+p2),q(R-y)=I+p2, g2p2=(1+p2)2, where p, q denote dy/dx, d2y/dx2 respectively. (See INFINITESIMAL CALCULUS.) In any given case we can, at all events in theory, eliminate x, y between the above equations for a-X and /3-y, and the equation of the curve. The resulting equation in (a, R) represents the locus of the centre of curvature. This is the evolute of the curve. 22. Polar Coordinates.—In plane geometry the distance of an; point P from a fixed origin (or See also:

• POLE (FAMILY)
• POLE, REGINALD (1500-1558)
• POLE, RICHARD DE LA (d. 1525)
• POLE, WILLIAM (1814—1900)

23. Trilinear and Areal Coordinates.—Consider a fixed triangle ABC, and regard its sides as produced without limit. Denote, as in See also:

• TRIGONOMETRY (from Gr. rpiywvov, a triangle, /ATpov, measure)

In particular we may make it homogeneous in a, /3, -y: to do this we have only to multiply the terms of every degree less than the highest See also:

• PRESENT

This particular equation of the first degree is satisfied by no point at a finite distance; but we see the propriety of saying that it has to be taken as satisfied by all the points conceived of as actually at infinity. Accordingly the special property of these points is expressed by saying that they lie on a special straight line, of which the areal equation is x+y+z =o. In trilinear coordinates this line at infinity has for equation as+b/3+ c7=0. On the one special line at infinity parallel lines are treated as meeting. There are on it two special (imaginary) points, the circular points at infinity of § 19, through which all circles pass in the same sense. In fact if S=0 be one circle, in areal coordinates, S+(x+y+z) (lx+my+nz) =o may, by proper choice of 1, m, n, be made any other; since the added terms are once lx+my+nz, and have the generality of any expression like a'x+b'y+c' in Cartesian coordinates. Now these two circles intersect in the two points where either meets x+y+z=0 as well as in two points on the radical axis lx+my+nz =o. 24. Let us consider the perpendicular distance of a point (a', ,e', y') from a line la+m1+ny. We can take rectangular axes of Cartesian coordinates (for clearness as to equalities of angle it is best to choose an origin inside ABC), and refer to them, by putting expressions p-x cos 0-y sin 0, &c., for a &c.; we can then apply §.16 to get the perpendicular distance; and finally revert to the trilinear notation. The result is to find that the required distance is (la'+m13'+ny)/1l, m, n}, where {l, m, n}2=12+m2+n2-2mn cos A-2nl cos B-2lm Cos C. In areal coordinates the perpendicular distance from (x', y', z') II to lx+my+nz=o is 20(lx'+my'+nz')/{al, bm, cn}.

In both cases the coordinates are of course actual values. Now let t, n, 1- be the perpendiculars on the line from the vertices A, B, C, i.e. the points (I, o, o), (o, I, o), (o, o, I), with signs in See also:

• ACCORD (from Fr. accorder, to agree)

The line is exactly assigned if 1, m, n, or their mutual ratios, are known. Call (1, m, n) the coordinates of the line. Now keep x, y, z constant, and let the coordinates of the line vary, but always so as to satisfy the equation. This equation, which we now write xl+ym+zn=o, is satisfied by the coordinates of every line through a certain fixed point, and by those of no other line; it is the equation of that point in the line-coordinates 1, m, n. Line-coordinates are also called tangential coordinates. A curve is the envelope of lines which See also:

• TOUCH (derived through Fr. toucher from a common Teutonic and Indo-Germanic root, cf. " tug," " tuck," O. H. Ger. zucchen, to twitch or draw)

26. The system of line-coordinates allied to the areal system of point-coordinates has special See also:

• INTEREST

II. Solid Analytical Geometry. 27. Any point in space may be specified by three coordinates. We consider three fixed planes of reference, and generally, as in allthat follows, three which are at right angles two and two. They intersect, two and two, in lines x'Ox, y'Oy, z'Oz, called the axes of x, y, z respectively, and divide all space into eight parts called octants. If from any point P in space we draw PN parallel to zOz' to meet the plane xOy in N, and then from N draw NM parallel z N FIG. 57. FIG. 58. to yOy' to meet x'Ox in M, the coordinates (x, y, z) of P are the numerical measures of OM, MN, NP; in the case of rectangular coordinates these are the perpendicular distances of P from the three planes of reference. The sign of each coordinate is positive or negative as P lies on one side or the other of the corresponding plane.

In the octant delineated the signs are taken all positive. In fig. 57 the delineation is on a plane of the See also:

• PAPER
• PAPER (Fr. papier, from Lat. papyrus)

above, a series of points P thereof, each accompanied by its ordinate PN, which serves to refer it to the plane of xy. The employment of stereographic projection is also interesting. 28. In plane geometry, reckoning the line as a curve of the, first order, we have only the point and the curve. In solid geometry, reckoning a line as a curve of the first order, and the plane as a surface of the first order, we have the point, the curve and the surface; but the increase of complexity is far greater than would hence at first sight appear. In plane geometry a curve is considered in connexion with lines (its tangents) ; but in solid geometry the curve is considered in connexion with lines and planes (its tangents and osculating planes), and the surface also in connexion with lines and planes (its tangent lines and tangent planes) ; there are surfaces arising out of the line—cones, skew surfaces, developables, doubly and triply infinite systems of lines, and whole classes of theories which have nothing analogous to them in plane geometry: it is thus 'a very small See also:

• PART

The form y=4(x), z=,y(x), where two of the coordinates are given in terms of the third, is a particular case of each of these modes of representation. 29. The remarks under plane geometry as to descriptive and metrical propositions, and as to the non-metrical See also:

• CHARACTER (Gr. xapareri7p, from xap&crew, to scratch)

We have of course a2+$ +72 = i and a'2+13'2+y2 =I; and hence also 1_ S2 = (a2+132+Y2) (a2+r +y 2) — (aft/ +RR'+yy')2, =(Or'—0'7)2+(ya —y'a)2+(ap'—a'/9)2; so that the sine of the inclination can only be expressed as a square root. These formulae are the foundation of spherical trigonometry. 32. Straight Lines, Planes and Spheres.—The foregoing formulae give at once the equations of these loci. For first, taking Q to be a fixed point, coordinates (a, b, c), and the cosine-inclinations (a, 13, y) to be constant, then P will be a point in the line through Q in the direction thus determined; or, taking (x, y, z) for its coordinates, these will be the current co-ordinates of a point in the line. The values of E, n, 1' then are x—a, y—b, z—c, and we thus have xx=a —y=b z=c , which (omitting the last equation, = p) are the equations of the line through the point (a, b, c), the cosine-inclinations to the axes being a, $, y, and these quantities being connected by the relation a2+132+y2 = i. This equation may be omitted, and then a, $, 7, instead of being equal, will only. be proportional, to the cosine-inclinations. Using the' last equation, and writing x, y, z=a+ap, b+Rp, c+7P,a'(x—a)+$'(y—b)+7'(z —c) =pi; and we thence have Pi =P — (aa'+b$'+cy'), which is an expression for the perpendicular distance of the point (a, b, c) from the plane in question. It is obvious that any linear equation Ax+By+Cz+D =o between the coordinates can always be brought into the foregoing form, and hence that such an equation represents a plane. Thirdly, supposing Q to be a fixed point, coordinates (a, b, c), and the distance QP, =p, to be constant, say this is =d, then, as before, the values of E, n, g- are x—a, y—b, z—c, and the equation ti-i-,t2+f-2=p2 becomes (x—a)2+(y—b)2+(z—c)2 =d2, which is the equation of the See also:

• SPHERE (Gr. vclsa?pa, a ball or globe)

Cylinders, Cones, ruled Surfaces.—If the two equations of a straight line involve a parameter to which any value may be given, we have a singly infinite system of lines. They See also:

• COVER (from the Fr. convert, from couvrir, to cover, Lat. cooperire)

A regulus such that consecutive lines on it do not intersect, in this sense, is called a skew surface, or See also:

• SCROLL

It follows that the square of the See also:

• DETERMINANT

The special ones arise when the coefficients in the general equation are limited to satisfy certain special equations; they. comprise (I) plane-pairs, including in particular one plane twice repeated, and (2) cones, including in particular cylinders; there is but one form of cone, but cylinders may be elliptic, parabolic or hyperbolic. A discussion of the general equation of the second degree shows that the proper quadric surfaces are of five kinds, represented respectively, when referred to the most convenient axes of reference, by equations of the five types (a and b positive) : x2 y2 (I) z=2a+2b' elliptic paraboloid. 2 (2) z=212b' hyperbolic paraboloid. s 2 (3) a2++=', See also:

• ELLIPSOID

62 and 63) the sections by the planes of zx, zy are the parabolas z=2a,z = having the opposite axes Oz, Oz', and the section by a plane z =7 parallel to that of xy is the hyperbola -y =06;—;;, which has its transverse axis parallel to Ox or Oy according as y is positive or negative. The surface is thus z generated by a variable hyperbola moving parallel to itself along the parabolas as directrices. The form is best seen from fig. 63, which represents the sec- tions by planes parallel to the plane of xy, or say the contour lines; the continuous lines are the sections above the plane of xy, and the dotted lines the sections below this plane. The form is, in fact, that of a See also:

• SADDLE (a word common to Teutonic languages, cf. Ger. Sattel, Dut. zadel, also in Russ. siedlo and Lat. sella, for sedla; it is not derived directly from Lat. sedile, which means a chair, but all the words are to be referred to the root sad-, which gives

The figures should be studied to see how they can lie. In the hyperboloid of two sheets (fig. 66) the sections by the planes of zx and zy are the hyperbolas g x2 Z2 2 c2-a2 =I, c'—.,_=I ' having a common transverse axis along z'Oz; the section by any plane z= ty parallel to that of xy is the ellipse x2 2 " +=y—I, a' b2 c' provided y2>c2, and the surface, consisting of two distinct portions or sheets, may be considered as generated by a variable ellipse moving parallel to itself along the hyperbolas as directrices. 38. Differential Geometry of Curves.—For convenience consider the coordinates (x, y, z) of a point on a curve in space to be given as functions of a variable parameter 0, which may in particular be one of themselves. Use the notation x', x" for dx/de, d2x/del, and similarly as to y and z. Only a few formulae will be given. Call the current coordinates (Z, i, ~). The tangent at (x, y, z) is the line tended to as a limit by the connector of (x, y, z) and a neighbouring point of the curve when the latter moves up to the former: its equations are (!:—x)/x'(n-y)/y'=(;•-z)/z'. The osculating plane at (x, y, z) is the plane tended to as a limit by that through (x, y, z) and two neighbouring points of the curve as these, remaining distinct, both move up to (x, y, z) : its one equation is ( —x)(y'z"—y"z')+(7—y) (z'x"—z"x')+ (I'—z)(x'y" —x"y') =o. The normal plane is the plane through (x, y, z) at right angles to the tangent line, i.e. the plane x'(t-x)-[-y'(a—y)+z'(r—z) =o. It cuts the osculating plane in a line called the principal normal.

Every line through (x, y, z) in the normal plane is a normal. The normal perpendicular to the osculating plane is called the binormal. A tangent, principal normal, and binormal are a convenient set of rectangular axes to use as those of reference,+when the nature of a' curve near a point on it is to be discussed. Through (x, y, z) and three neighbouring points, all on the curve, passes a single sphere ; and as the three points all move up to (x, y, z) continuing distinct, the sphere tends to a limiting See also:

• SIZE

One line through (x, y, z) is at right angles to the tangent plane. This is the normal (—x)/Lx =(0—y)/ay=(i'—z)/az The tangent plane is cut by the surface in a curve, real or imaginary, with a See also:

• NODE (Lat. nodus, a loop)

There are two singly infinite systems of curves on a surface, a pair cutting one another at right angles through every point upon it, all tangents to which are principal tangents of the surface at their respective points of contact. These are called lines of curvature, because of a property next to be mentioned. As a point Q moves in an arbitrary direction on a surface from coincidence with a chosen point P, the normal at it, as a rule, at once fails to meet the normal at P; but, if it takes the direction of a line of curvature through P, this is instantaneously not the case. We have thus on the normal two centres of curvature, and the distances of these from the point on the surface are the two principal radii of curvature of the surface at that point; these are also the radii of curvature of the sections of the surface by planes through the normal and the two principal tangents respectively; or say they are the radii of curvature of the normal sections through the two principal tangents respectively. Take at the point the axis of z in the direction of the normal, and those of x and y in the directions of the principal tangents respectively, then, if the radii of curvature be a, b (the signs being such that the coordinates of the two centres of curvature are z=a and z=b respectively), the surface has in the neighbourhood of the point the form of the paraboloid 2 y2 z_2a+ 2b' x and the chief-tangents are determined by the equation o=as+2b. The two centres of curvature may be on the same side of the point or on opposite sides; in the former case a and b have the same sign, the paraboloid is elliptic, and the chief-tangents are imaginary; in the latter case a and b have opposite signs, the paraboloid is hyperbolic, and the chief-tangents are real. The normal sections of the surface and the paraboloid by the same plane have the same radius of curvature; and it thence readily follows that the radius of curvature of a normal section of the surface by a plane inclined at an angle 0 to that of zx is given by the equation 1 cos20 sine = + p a b The section in question is that by a plane through the normal and a line in the tangent plane inclined at an angle 0 to the principal tangent along the axis of x. To complete the theory, consider the section by a plane having the same trace upon the tangent plane, but inclined to the normal at an angle 0; then it is shown without difficulty (See also:

• MEUNIER, CONSTANTIN (1831-1905)

Ecole Polytech., 1801) ; See also:

• JULIUS

Clebsch-F. Lindemann, Vorlesungen caber Geometrie, Bd. i. (Leipzig, 1876, 2te Aufl., 1891) ; R. Baltser, Analytische Geometric (Leipzig, 1882) ; See also:

• CHARLOTTE

(E. B. EL.) V. LINE GEOMETRY Line geometry is the name applied to those geometrical investigations in which the straight line replaces the point as See also:

• ELEMENT (Lat. elementum)

H. Haiphen 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 (I, 1) congruence; and the chords of a twisted cubic form the general type of a (I, 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 which-ever point of view we regard it. The lines common to three com- Elexes of degrees, nin2ns, form a ruled surface of degree 2nln2ns; 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 Linear pass through a point called the nul-point or pole of the complex. 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 11 pass through another line 12. The relation between 11 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. See also:

• VOSS, JOHANN HEINRICH (1751–1826)
• VOSS, RICHARD (1851– )

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 co-ordinates 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 x,y1z,w1 and x2y2z2w2 are two points on the line, it is easily verified that the six determinants of the See also:

• ARRAY (from the O. Fr. areyer, Med. Lat. arredare, to get ready)

See also:

• SIR

By suitable choice of the six fundamental complexes, as they may be called, this connecting relation may be brought into other simple forms of which we mention two: (i.) When the six are mutually in involution it can be reduced to x12+x22+xa2+x42+xi2+x62=o; (ii.) When the first four are in involution and the other two are the lines common to the first four it is xi2+x22+xs'+x42—2xsxs=o. These generalized coordinates might be explained without reference to actual magnitude, just as homogeneous point coordinates can be; the essential remark is that the equation of any coordinate to zero represents a linear complex, a point of view which includes our original system, for the equation of a coordinate to zero represents all the lines meeting an edge of the fundamental See also:

• TETRAHEDRON (Gr. riepa-, four, Ebpa, face or base)

Suppose, in fact, that the equation of a sphere of radius ris x2+y'+s2+2ax+2by+2ca+d =o, so that 0=a'+b2+c2—d; then introducing the quantity e to make this equation homogeneous, we may regard the sphere as given by the six coordinates a, b, c, d, e, r connected by the equation a'+ b2+4'—r2-de.=o, and it is easy to see that two spheres touch, if [LINE the polar form 2aat-Fabb1+2cc1—2rr1—del—die vanishes. Comparing this with the equation x12+x22+xa2+x4'—2xsxs=o given above, it appears that this sphere geometry and line geometry are identical, for we may write a=xi, b=x2, c=x2, r=x4S—i, d=xi, e = 2x6; but it is to be noticed that a sphere is really replaced by two lines whose coordinates only differ in the sign of x4, so that they are polar lines with respect to the complex x4=o, Two spheres which touch correspond to two lines which intersect, or more accurately to two pairs of lines (p, p') and(q, q'), of which the pairs (p, q) and (p', q') both intersect. By this means the problem of describing a sphere to touch four given spheres is reduced to that of drawing a pair of lines (t, t') (of which t intersects one line of the four pairs (pp'), (qq'), (re'), (ss'), and t' intersects the remaining four). We may, however, ignore the accented letters in translating theorems, for a configuration of lines and its polar with respect to a linear complex have the same projective properties. In Lie's transformation a linear complex corresponds to the totality of spheres cutting a given sphere at a given angle. A most remarkable result is that lines of curvature in the sphere geometry become asymptotic lines in the line geometry. Some of the principles of line geometry may be brought into clearer See also:

• LIGHT

Curves in a linear complex have been extensively studied. The osculating plane at any point of such a curve is the nul-plane of the point with respect to the complex, and points of superosculation always coincide in pairs at the points of contact of stationary tangents. When a point of such a curve is given, the osculating plane is determined, hence all the curves through a given point with the same tangent have the same torsion. The lines through a given point that belong to a complex of the nth degree lie on a cone of the nth degree: if this cone has a double line the point is said to be a singular point. Similarly, Non-Linear a plane is said to be singular when the envelope of the ~m- lines in it has a double tangent. It is very remarkable plexes. that the same surface is the locus of the singular points and the envelope of the singular planes: this surface is called the singular surface, and both its degree and class are in general 2n(n-i)', which is equal to four for the quadratic complex. The singular lines of a complex F=o are the lines common to F and the complex SF SF SF SF SF SF Sl SX +Sm Su ' Sn Sv =fi' As already mentioned, at each line 1 of a complex there is an infinite number of tangent linear complexes, and they all contain the lines adjacent to 1. If now l be a singular line, these complexes all reduce to straight lines which form a plane pencil containing the line I. Suppose the vertex of the pencil is A, its plane a, and one of its lines then 1' being a complex line near 1, meets E, or more accurately the mutual moment of 1', and is of the second order of small quantities. If P be a point on 1, a line through P quite near l in the plane a will meet b and is therefore a line of the complex; hence the complex-cones of all points on 1 touch a and the complex-curves of all planes through t touch 1 at A. It follows that 1 is a double line of the complex-cone of A, and a double tangent of the complex-curve of G. Conversely, a double line of a cone or curve is a singular line, and a singular line clearly touches the curves of all planes through it in the same point.

Suppose now that the consecutive line 1' is also a singular line, A' being the allied singular point, a' the singular plane and i;' any line of the pencil (A', a') so that E' is a tangent line at 1' to the complex: the mutual moments of the pairs 1', E and 1, f are each of the second order; hence the plane a' meets the lines 1 and f' in two points very near A. This being true for all singular planes, near a the point of contact of a with its envelope is in A, i.e. the locus of singular points is the same as the envelope of singular planes. Further, when a line touches a complex it touches the singular surface, for it belongs to a plane pencil like (Aa), and thus in Klein's analogy the analogue of a focus of a hyper-surface being a bitangent line of the complex is also a bitangent line of the singular surface. The theory of cosingular complexes is thus brought into line with that of confocal surfaces in four dimensions, and guided by these principles the existence of cosingular quadratic complexes can easily be established, the analysis required being almost the same as that invented for confocal, cyclides by Darboux and others. Of cosingular complexes of higher degree nothing is known. Following J. Plucker, we give an See also:

• ACCOUNT
• ACCOUNT (through O. Fr. acont, Late Lat. comptum, cornputare, to calculate)

The surface on which lie all the conics through a line l is called the Plucker surface of that line: from the known properties of (2, 2) correspondences it can be shown that the Plucker surface of 1 cuts 11 in a range of the same See also:

• CROSS

It is thence easy to verify that the two complexes Eax2 = o and Ebx2 = o are cosingular if b,= a,X +11/ara +P. The singular surface of the general quadratic complex is the famous quartic, with sixteen nodes and sixteen singular tangent planes, first discovered by E. E. Kummer. We cannot give a full account of its properties here, but we deduce at once from the above that its bitangents break up into six (2, 2) congruences, and the six linear complexes containing these are mutually in involution. The nodes of the singular surface are points whose complex cones are coincident planes, and the complex conic in a singular tangent plane consists of two coincident points. This configuration of sixteen points and planes has many interesting properties; thus each plane contains six points which lie on a conic, while through each point there pass six planes which touch a quadric cone. In many respects the Kummer quartic plays a part in three dimensions analogous to the general quartic curve in two; it further gives a natural representation of certain relations between hyper-elliptic functions (cf. R. W. H. T.

See also:

• HUDSON
• HUDSON, GEORGE (180o-1871)
• HUDSON, HENRY
• HUDSON, JOHN (1662–1719)

The enumeration of all possible cases is thus reduced to a simple question in combinatorial analysis, and the actual study of any particular case is much facilitated by a useful rule of Klein's for writing down in a simple form two quadratics belonging to a given class—one of which, of course, represents the equation connecting line coordinates, and the other the equation of the complex. The general complex is naturally [iii See also:

• ILL

The tetrahedral or Reye complex is the simplest and best known of proper quadratic complexes. It is generated by the lines which cut the faces of a tetrahedron in a constant cross ratio, and therefore by those subtending the same cross ratio at the four vertices. The singular surface is made up of the faces or the vertices of the fundamental tetrahedron, and each edge of this tetrahedron is a double line of the complex. The complex was first discussed by K. T. Reye as the assemblage of lines joining corresponding points in a homographic transformation of space, and this point of view leads to many important and elegant properties. A (metrically) particular case of great interest is the complex generated by the normals to a See also:

• FAMILY

But in addition to the difficulties of the theory of algebraic surfaces, a subject still in its See also:

• INFANCY

The focal surface is of degree four and class 2m; this See also:

• KIND (O. E. ge-cynde, from the same root as is seen in " kin," supra)

The general theory of skews in two linear complexes is identical with that of curves on a quadric in three dimensions and is known. But for skews lying in only one linear complex there are difficulties; the curve now lies in four dimensions, and we represent it in three by stereographic projection as a curve meeting a given plane in n points on a conic. To find the maximum deficiency for a given degree would probably be difficult, but as far as degree eight the space-curve theory of Halphen and Nother can be translated into line geometry at once. When the skew does not lie in a linear complex at all the theory is more difficult still, and the general theory clearly cannot advance until further progress is made in the study of twisted curves. The only points to which the metrical geometry applies are those within the region enclosed by the quadric; the other points are " improper ideal points." The angle (012) between two planes, 11x+mly+nlz+r1w=o and l2x+m2y+n2z+r2w = o, is given by cos 012 = (See also:

• LULL (or LvLLY), RAIMON, or RAYMOND (c. 1235-1315)

Let P be any point, and let p be the distance OP, 0 the angle POZ, and 4) the angle between the planes ZOX and ZOP. Then the co-ordinates of P can be taken to be sinh (p/y) sin 0 cos sinh (ply) sin sin 4), sinh (p/y) cos0, cosh (ply). If ABC is a triangle, and the sides and angles are named according to the usual convention,we have sinh (a/y)/sinA=sinh (b/y)/sin B=sinh (c/y)/sin C, (4) and also cosh (a/y) =cosh (b/y) cosh (c/y)—sinh (b/y) sinh (c/y) cos A, (5) with two similar equations. The sum of the three angles of a triangle is always less than two right angles. The area of the triangle ABC is a2(x-A-B-C). If the base BC of a triangle is kept fixed and the vertex A moves in the fixed plane ABC so that the area ABC is constant, then the locus of A is a line of equal distance from BC. This locus is not a straight line. The whole theory of similarity is inapplicable; two triangles are either congruent, or their angles are not equal two by two. Thus the elements of a triangle are determined when its three angles are given. By keeping A and B and the line BC fixed, but by making C move off to infinity along BC, the lines BC and AC become parallel, and the sides a and b become infinite. Hence from equation (5) above, it follows that two parallel lines (cf. Section VII.

Axioms of FIG. 68. Geometry) must be considered as making a zero angle with each other. Also if B be a right angle, from the equation (5), remembering that, in the limit, cosh (a/y)/cosh (by) =cosh (a/y)/sinh (b/y) = I, VI. NON-EUCLIDEAN GEOMETRY The various metrical geometries are concerned with the properties of the various types of congruence-See also:

• GROUPS
• GROUPS, THEORY

The absolute is then real, and the geometry is hyberbolic. The distance (du) between the two points (xi, yi, z1, w1) and (x2, y2, 2)(d given cosh (W1 W2 — x1x2 — ylyz — 21zg)/{(w12 — x12 — yl2 — zit) (w22 — x22 — y22 — Z22) 1i (I) we have cos A=tanh (c/2y) . . . (6). The angle A is called by N. I. Lobatchewsky the " angle of parallel-ism.' The whole theory of lines and planes at right angles to each other is simply the theory of conjugate elements with respect to the absolute, where ideal lines and planes are introduced. Thus if 1 and 1' be any two conjugate lines with respect to the absolute (of which one of the two must be improper, say 1'), then any plane through 1' and containing proper points is perpendicular to 1. Also if p is any plane containing proper points, and P is its pole, which is necessarily improper, then the Ines through P are the normals to P. The equation of the sphere, centre (xi, y1, z1, w1) and radius p, is (w12 —x12—y,2 —z12) (w2—x'—y'—z') cosh'(/)/y) _ (wiw—xix—yiy—ziz)2 (7)• The equation of the surface of equal distance (a) from the plane lx+my+nz+rw=o is (12+m2+n2—r') (w2—x2—y2—z2) sinh2(v/y) (rw+lx+my+nz)2 (8). A surface of equal distance is a sphere whose centre is improper; and both types of surface are included in the family k'(w2—x2—y2—z')= (ax+by+cz+See also:

• DW(i+at)X587

But this family also includes a third type of surfaces, which can be looked on either as the limits of spheres whose centres have approached the absolute, or as the limits of surfaces of equal distance whose central planes haye approached a position tangential to the absolute. These surfaces are called limit-surfaces. Thus (9) denotes a limit-surface, if d2—a'—b2—c'=o. Two limit-surfaces only differ in position. Thus the two limit-surfaces which touch the plane YOZ at 0, but have their concavities turned in opposite directions, have as their equations w2-x2-y2-z2=(wtx)2. The geodesic geometry of a sphere is elliptic, that of a surface of equal distance is hyperbolic, and that of a limit-surface is parabolic (i.e. Euclidean). The equation of the surface (cylinder) of equal distance (S) from the line OX is (w' —x') tanh'(S/y) —y' —z2 = o. This is not a ruled surface. Hence in this geometry it is not possible for two straight lines to be at a constant distance from each other. Secondly, let the equation of the absolute be x2+y'+z2+ w' = o. The absolute is now imaginary and the geometry is elliptic.

The distance (dl2) between the two points (xi, yi, w1) and (x2, y2, Z2, W2) is given by cos (d12/y) (xIx2+yly2+zlz2+wiw2)/{(x12+y12+zl2+wl2) (x221 I y22+z22+w22)}i (10). Thus there are two distances between the points, and if one is See also:

• DIE
• DIE (Fr. de, from Lat. datum, given)

If 1 and 1' are two conjugate lines, the Vanes through one are the planes perpendicular to the other. If is the pole of the plane p, the lines through P are the normals to the plane p. The distance from P to p is 17ry. Thus every sphere is also a surface of equal distance from the polar of its centre, and , conversely. A plane does not divide space; for the line joining any two points P and Q only cuts the plane once, in L say, then it is 'always possible to go from P to Q by the segment of the line PQ which does not contain L. But P and Q may be said to be separated by a plane p, if the point in which PQ cuts p lies on the shortest segment between P and Q. With this sense of " separation," it is possible ' to find three points P, Q, R such that P and Q are separated 'Cf. A. N. See also:

• WHITEHEAD, WILLIAM (1715-1785)

Thus if a, b, c and A, B, C are the elements of any one triangle, then the four triangles have as their elements: (i) a, b, c, A, B, C. (2) a, Try—b, Try--c, A, 7r—B, 7r—C. (3) icy—a, b, Try—c, 7r—A, B, 7r—C. (4) Try—a, Try — b, c, 7r—A, 7r—B, C. The formulae connecting the elements are sin A/sin (a/7) =sin B/sin (b/y) =sin C/sin (c/y), . (I2) cos (a/y) =cos (bjy) cos (c/y) +sin (b/y) sin (c/y) cos A, (13) with two similar equations. Two cases arise, namely (I.) according as one of the four triangles has as its sides the shortest segments between the angular points, or (II.) according as this is not the case. When case I. holds there is said to be a " principal triangle."' If all the figures considered lie within a sphere of radius 47ry only case I. can hold, and the principal triangle is the triangle wholly within this sphere, also the peculiarities in respect to the separation of points by a plane cannot then arise. The sum of the three angles of a triangle ABC is always greater than two right angles, and the area of the triangle is 7'(A+B+C—7r). Thus as in hyperbolic geometry the theory of similarity does not hold, and the elements of a triangle are determined when its three angles are given. The coordinates of a point can be written in the form sin (ply) sin 8 cos 0, sin (ply) sine sin ¢, sin (ply) cos 0, cos (p/-y), where p, 0 and q have the same meanings as in the corresponding formulae in -hV{perbolic geometry. Again, suppose a See also:

• WATCH (in O. Eng. wcecce, a keeping guard or watching, from wacian, to guard, watch, wacan, to wake)

Let the watch be continually pushed along the plane along the line OX, that is, in the direction 9 to 3. Then the line XOX being of finite length, the watch will return to 0, but at its first return it will be found to be face downwards on the other side of the plane, with the line 12 to 6 reversed in direction along the line YOY. This peculiarity was first pointed out by Felix Klein. The theory of parallels as it exists in hyperbolic 'space has no application in elliptic geometry. But another property of Euclidean parallel lines holds in elliptic geometry, and by the use of it parallel lines are defined. For the equation of the surface (cylinder) of equal distance (S) from the line XOX is (x'+w') tan '(S/y) — (y2+z') = o. This is also the surface of equal distance, 1-lacy—S, from the line conjugate to XOX. Now from the form of the above equation this is a ruled surface, and through every point of it two generators pass. But these generators are lines of equal distance from XOX. Thus throughout every point of space two lines can be drawn which are lines of equal distance from a given line 1. This property was discovered by W. K.

See also:

• CLIFFORD
• CLIFFORD, JOHN (1836- )
• CLIFFORD, WILLIAM KINGDON (1845-1879)

A. N. Whitehead, loc. cit. 'Cf. A. N. Whitehead, " The Geodesic Geometry of Surfaces in non-Euclidean Space," Proc. Lond. Math. See also:

• SOC

Annal. vol. See also:

• XXXVII

To Euclid's successors this axiom had signally failed to appear self-evident, and had failed equally to appear indemonstrable. Without the use of the postulate its converse is proved in Euclid's 28th proposition, and it was hoped that by further efforts the postulate itself could be also proved. The first step consisted in the See also:

• DISCOVERY

Saccheri 'distinguishes three hypotheses (corresponding to what are now known as Euclidean or parabolic, elliptic and hyperbolic geo- metry), and proves that some one of the three must be univer sally true. His three hypotheses are thus obtained: equal perpendiculars AC, BD are drawn from a straight line AB, and CD are joined. It is shown that the angles ACD, BDC are the second, that they are both obtuse; and the third, that they are both acute. Many of the results afterwards obtained by Lobatchewsky and Bolyai are here developed. Saccheri fails to be the founder of non-Euclidean geometry only because he does not perceive the possible truth of his non-Euclidean hypo-theses. Some advance is made by Johann Heinrich See also:

• LAMBERT [alias NICHOLSON], JOHN (d. 1538)
• LAMBERT, DANIEL (1770-1809)
• LAMBERT, FRANCIS (c 1486-153o)
• LAMBERT, JOHANN HEINRICH (1728–1777)
• LAMBERT, JOHN (1619-1694)

Klein into three very distinct periods. The first—which contains only Gauss, Lobatchewsky and Bolyai—is characterized by its synthetic method and by its See also:

• CLOSE
• CLOSE (from Lat. clausum, shut)
• CLOSE, MAXWELL HENRY (1822-1903)

Gauss published nothing on the theory of parallels, and it was not generally known until after his See also:

• DEATH

Conceptions of magnitude, he explains, ffama See also:

• FOLD

siderations as to measurement. If measurement is to be possible, some magnitude, we saw, must be independent of position; let us consider manifolds in which lengths of lines are such magnitudes, so that every line is measurable by every other. The coordinates of a point being x2, x2, . . . x,,, let us con-See also:

• FINE

Its further discussion depends upon the measure of curvature, the second of Riemann's fundamental conceptions. This conception, derived from the theory of surfaces, is applied as follows. Any one of the shortest lines which issue from a given point(say the origin) is completely determined by the initial ratios of the dx. Two such lines, defined by dx and bx say, determine a pencil, or one-dimensional series, of shortest lines, any one of which is defined 4 Abhandlungen d. Konigl. Ges. d. Wiss. zu Gott-in en, Bd. xiii. ; Ges. math. Werke, pp. 254-269; translated by Clifford, Collected Mathematical Papers. 5 Cf. Gesamm. math. and phys.

Werke, vol. i. (Leipzig, 1894). circumference of a circle of radius r the See also:

• FORMULA
• FORMULA (Lat. diminutive of forma, shape, pattern, &c., especially used of rules of judicial procedure)

When the radius of the circle or sphere becomes infinite all these normals become parallel, but the circle or sphere does not become a straight line or plane. It becomes what Lobatchewsky calls a limit-line or limit-surface. The geometry on such a surface is shown to be Euclidean, limit-lines replacing Euclidean straight lines. (It is, in fact, a surface of zero measure of curvature.) By the help of these propositions Lobatchewsky obtains the above value of 11(p), and thence the See also:

• SOLUTION (from Lat. solvere, to loosen, dissolve)

Annalen, Bd. xlix.; also Engel's See also:

• TRANSLATION

There are four points in which this profound and See also:

• EPOCH (Gr. E7roxij, holding in suspense, a pause, from E1rxav, to hold up, to stop)

Such surfaces are capable of a conformal representation on a plane, by which geodesics are represented by straight lines. Hence if we take, as coordinates on the surface, the Cartesian coordinates of corresponding points on the plane, the geodesics must have linear equations. Hence it follows that ds2 = R2w — '( (a2 —v2)du2 +2uvdudv + (a2—u') dal where w2=a2—u2—v2, and -1/See also:

• R2(rt-s2)

1, 256. His papers are " Saggio di interpretazione See also:

• DELTA (from the shape of the Gr. letter A, delta, originally used of the mouth of the Nile)

Hence the others are each throughout at a constant distance from this line. (It may be shown that all motions in a hyperbolic plane consist, in a general sense, of rotations; but three types must be distinguished according as the centre is real, imaginary or at infinity. All points describe, accordingly, one of the three types of circles.) The above Euclidean interpretation fails for three or more dimensions. In the Teoria fondamentale, accordingly, where n dimensions are considered, Beltrami treats hyperbolic space in a purely analytical spirit. The paper shows that Lobatchewsky's space of any number of dimensions has, in Riemann's sense, a constant negative measure of curvature. Beltrami starts with the formula (analogous to that of the Saggio) ds2 = Rex 2(dx2+dxi2+dx22+ ... +dxa2) where x2+xi2-•-x22-... +xn2=a2. He shows that geodesics are represented by linear equations between xi, x2, ..., x,,, and that the geodesic distance p between two points x and x' is given by a2 —xi—x2x'2 —... —x,,x'a cosh RR= }(a2-x2—x2—... —x2) (a'—x 1 2 —x 2 2—.. —x,?) }lt2 i 2 n (a formula practically identical with Cayley's, though obtained by a very different method).

In order to show that the measure of curvature is constant, we make the substitutions xi=rai, x2=r)2...xa=rXa, where MX2=1. Hence ds2 =(Radr/a2-r2)2+R2ridA2/(a2—r2). where dA2=Zda2. Also calling p the geodesic distance from the origin, we have cosh (p/R) =II (aa r2j , sinh (p/R) = d (a2~-r2) Hence ds2=dp2+(R sinh (p/R))2dA'. Putting zi=pXi, z2=pA2, ••.zn=pXa, we obtain ds2=Zdz2+. (R sink R) 2—1 E(zidzk—zkdzi)2. Hence when p is small, we have approximately ds2dz2 { 32E(zidzk—zkdzi)2 . (I). Considering a surface element through the origin, we may choose our axes so that, for this element, z2=za=...=za=0. Thus ds2=dzl+ dz2+3R2(zidz2—z2dzi)2 . (2). Now the area of the triangle whose vertices are (o, o), (zi, z2), (dzi, dz2) is i(zt, dz2—z2dzi).

Hence the quotient when the terms of the fourth order in (2) are divided by the square of this triangle is dp=Radr/(a2—r2), p=1RlogQ—, r = a tanh R. Thus points on the surface corresponding to points in the plane on the limiting circle r=a, are all at an infinite distance from the origin. Again, considering r constant, the arc of a geodesic circle subtending an angle p at the origin is o=Rrti/J (a2—r2) =pR sink (p/R), whence the circumference of a circle of radius p is 2,rR sinh (p/R). Again, if a be the angle between any two geodesics V —v=m(U—u), V —v=n(U—u), then tan a=a(n—m)w/ 1(1+tan)a2—(v—mu) (v —nu)]. Thus a is imaginary when u, v is outside the limiting circle, and is zero when, and only when, u, v is on the limiting circle. All these results agree with those of Lobatchewsky and Bolyai. The maximum triangle, whose angles are all zero, is represented in the See also:

• AUXILIARY (from Lat. auxilium, help)

These are limit-lines and curves of constant distance from a straight line. Both may be regarded as circles, the first having an infinite, the second an imaginary radius. The equation to a circle of radius p and centre uovo is 4/3R2; hence, returning to general axes, the same is the quotient when the terms of the fourth order in (i) are divided by the square of the triangle whose vertices are (o, o,. ..o), (al, z2, z3,...Zn), (dzl, dz2, See also:

• CLAN (Gaelic clann, O. Ir. cland, connected with Lat. planta, shoot or scion, the ancient Gaelic or Goidelic substituting k for p)

If we give our circle an imaginary radius the geometry on the plane becomes elliptic. Replacing the circle by a sphere, we obtain an analogous representation for three dimensions. Instead of a circle or sphere we may take any conic or quadric. With this definition, if the fundamental quadric be lxx=o, and if Exy be the polar form of lax, the distance p between x and x' is given by the projective formula cos (p/k) =Z,z'/ l zx. x'z'l L That this formula is projective is rendered evident by observing that a 2ip/k is the anharmonic ratio of the range consisting of the two points and the intersections of the line joining them with the fundamental quadric. With this we are brought to the third or projective period. The method of this period is due to Cayley; its application to previous non-Euclidean geometry is due to Klein. The projective method contains a generalization of discoveries already made byLaguerre2 in 1853 as regards Euclidean geometry. The arbitrariness of this procedure of deriving metrical geometry from the properties of conics is removed by Lie's theory of congruence. We then arrive at the See also:

• STAGE (Fr. 6/age; from Lat. stare, to stand)

The difference is strictly space, analogous to that between the diameters and the points of a sphere. In the polar form two straight lines in a plane always intersect in one and only one point; in the antipodal form they intersect always in two points, which are See also:

• ANTIPODES (Gr. avri, opposed to, and 7r6 es, feet)

Math. Annalen, iv. vi., 1871-1872. ' For an investigation of these and similar properties, see See also:

• WHITE
• WHITE, ANDREW DICKSON (1832– )
• WHITE, GILBERT (1720–1793)
• WHITE, HENRY KIRKE (1785-1806)
• WHITE, HUGH LAWSON (1773-1840)
• WHITE, JOSEPH BLANCO (1775-1841)
• WHITE, RICHARD GRANT (1822-1885)
• WHITE, ROBERT (1645-1704)
• WHITE, SIR GEORGE STUART (1835– )
• WHITE, SIR THOMAS (1492-1567)
• WHITE, SIR WILLIAM ARTHUR (1824--1891)
• WHITE, SIR WILLIAM HENRY (1845– )
• WHITE, THOMAS (1628-1698)
• WHITE, THOMAS (c. 1550-1624)

We cannot, therefore, accept the measurements of stellar paraliaxes, &c., as conclusive evidence that the space-constant is large as compared with stellar distances. For the present, on grounds of simplicity, we may rightly adopt this view; but it must remain possible that, in view of some hitherto undiscovered discrepancy, a slight correction of the sort suggested might prove the simplest alternative. But conversely, a finite See also:

• PARALLAX (Gr. aapa?X6, alternately)

For John Bolyai's Appendix, cf. Absolute Geometrie nach Johann Bolyai, by J. Frischauf (Leipzig, 1872), and also the new edition of his father's large work, Tentamen . . ., published by the Mathematical Society of See also:

• BUDAPEST

(1889) ; F. Lindemann, " Mechanik bei projectiver Maasbestimmung," Math. Annal. vol. vii. ; W. K. Clifford, " Preliminary Sketch of Biquaternions," Proc. 'of See also:

• LAND

Land. Math. Soc. vols. xv., xvi., xvii. ; H. See also:

• COX, DAVID (1783-1859)
• COX, JACOB DOLSON (1828-1900)
• COX, KENYON (1856– )
• COX, RICHARD (I3oo ?–1581)
• COX, SAMUEL (1826–1893)
• COX, SAMUEL HANSON (1793-1880)
• COX, SIR GEORGE WILLIAM (1827-1902)

Annal. vol. 55 (1902). For expositions of the whole subject, cf. F. Klein, Nicht-Euklidische Geometrie (See also:

• GOTTINGEN

N. Whitehead, loc. cit. Cf. also E. Study, " TJber nicht-Euklidische and Liniengeometrie," Jahr. d. See also:

• DEUTSCH, IMMANUEL OSCAR MENAHEM (1829-1873)

(1894). A bibliography on the subject up to 1878 has been published by G. B. Halsted, Amer. Journ. of Math. vols. i. and ii.; and one up to 1900 by R. Bonola, See also:

• INDEX

W.) Until the discovery of the non-Euclidean geometries (Lobatchewsky, 1826 and 1829; J. Bolyai, 1832; B. Riemann, 1854), Theories geometry was universally considered as being exotspace. clusively the science of existent space. (See section VI. Non-Euclidean Geometry.) In respect to the science, as thus conceived, two controversies may be noticed. First, there is the controversy respecting the absolute and relational theories of space. According to the absolute theory, which is the traditional view (held explicitly by See also:

• NEWTON
• NEWTON, ALFRED (1829–1907)
• NEWTON, JOHN (1725-1807)
• NEWTON, JOHN (1823-1895)
• NEWTON, SIR CHARLES THOMAS (1816-1894)
• NEWTON, SIR ISAAC (1642-1727)

It is admitted on all hands that in practice only relative motion is directly measurable. Newton, however, maintains in the Principia (scholium to the 8th definition) that it is indirectly measurable by means of the effects of " centrifugal force " as it occurs in the phenomena of rotation. This irrelevance of absolute motion (if there be such a thing) to science has led to the general See also:

• ADOPTION (Lat. adoptio, for adoptalio, from adoptare, to choose for oneself)

See also:

• RUSSELL (FAMILY)
• RUSSELL, ISRAEL COOK (1852- )
• RUSSELL, JOHN (1745-1806)
• RUSSELL, JOHN (d. 1494)
• RUSSELL, JOHN RUSSELL, 1ST EARL (1792-1878)
• RUSSELL, JOHN SCOTT (1808–1882)
• RUSSELL, LORD WILLIAM (1639–1683)
• RUSSELL, SIR WILLIAM HOWARD
• RUSSELL, THOMAS (1762-1788)
• RUSSELL, WILLIAM CLARK (1844– )

This is a question for See also:

• PSYCHOLOGY
• PSYCHOLOGY (>Guxrl, the mind or soul, and X yos, theory)

This is the only method of proof of consistency. The axioms of a set are independent of each other when no axiom can be deduced from the remaining axioms of the set. The See also:

• INDEPENDENCE
• INDEPENDENCE, DECLARATION OF
• INDEPENDENCE, WAR OF

They will ' Cf. See also:

• ERNST, HEINRICH WILHELM (1814–1865)

Russell, Princ. of Math., ch. i. 8 Cf. Russell, loc. cit., and G. Frege, " See also:

• EBER, PAUL (1511-1569)

There is at least one plane which does not contain all the points. 7. There exists a plane a, and a point A not incident in a, such that any point lies in some straight line which contains both A and a point in a. Definition.—Harm. (See also:

• ABCD

The enunciation of this theorem is as follows: If ABC and A'B'C' are two coplanar triangles such that the lines AA', BB', CC' are concurrent, then the three points of intersection of BC and B'C' of CA and C'A', and of AB and A'B' are collinear; and conversely if the three points of intersection are collinear, the three lines are concurrent. The proof which can be applied is the usual projective proof by which a third triangle A"B C" is constructed not coplanar with the other two, but in perspective with each of them. It has been proved 2 that Desargues's theorem cannot be deduced from axioms 1-5, that is, if the geometry be confined to two dimensions. All the proofs proceed by the method of producing a specification of " points " and " straight lines " which satisfies axioms 1-5, and such that Desargues's theorem does not hold. It follows from axioms 1-5 that Harm. (ABCD) implies Harm. (ADCB) and Harm. (CBAD), and that, if A, B, C be any three distinct collinear points, there exists at least one point D such that Harm. (ABCD). But it requires Desargues's theorem, and hence axiom 6, to prove that Harm. (ABCD) and Harm. (ABCD') imply the identity of D and D'.

The necessity for axiom 8 has been proved by G. See also:

• FARM

2 Cf. G. Peano, " Sui fondamenti della Geometria," p. 73, Rivista di matematica, vol. iv. (1894), and D. Hilbert, Grundlagen der Geometric (Leipzig, 1899) ; and R. F. See also:

• MOULTON, LOUISE CHANDLER (1835-1908)

' Cf. " Suit postulati fondamentali della geometria projettiva," Giorn: di matematica, vol. See also:

• XXX

But to secure that this property does in fact range the points in a serial order, some axioms are required. A straight line is to be a closed series; thus, when the points are in order, it requires two points on the line to divide it into two distinct complementary segments, which do not overlap, and together form the whole line. Accordingly the problem of the definition of order reduces itself to the definition of these two segments formed by any two points on the line; and the axioms are stated relatively to these segments. Definition.—If A, B, C are three collinear points, the points on the segment ABC are defined toa be those points such as X, for which there exist two points Y and Y' with the property that Harm. (AYCY') and Harm. (BYXY') both hold. The supplementary segment ABC is defined to be the See also:

• REST (O. Eng. rest, reste, bed, cognate with other Teutonic forms, e.g. Ger. Rast, Riiste, rest, and probably Gothic Rasta, league, i.e. resting or stopping place)

so. If A, B, C are three distinct collinear points, the common part of the segments BCA and See also:

• CAB (shortened about 1825 from the Fr. cabriolet, derived from cabriole, implying a bounding motion)

Vahlen, A bstrakte Geometric (Leipzig, 1905), also Whitehead, loc. cit. i Cf. H. Wiener, Jahresber. der Deutsch. Math. Ver. vol. i. (189o) ; and F. Schur, " Ober den Fundamentalsatz der projectiven Geometrie," Math. Ann. vol. li. (1899). e Cf. Hilbert, loc. cit., and Whitehead, loc. cit.

be presented' here as twelve in number, eight being "axioms of classification," and four being " axioms of order." Axioms of Classification.—The eight axioms of classification are as follows: r. Points form a class of entities with at least two members. 2. Any straight line is a class of points containing at least' three members. 3. Any two distinct points lie in one and only one straight line. 4. There is at least one straight line which does not contain the four points are collinear and D lies in the segment complementary to the segment ABC. The property of the separation of pairs of points by pairs of points is projective. Also it can be proved that Harm. (ABCD) implies that B and D separate A and C. Definitions.—A series of entities arranged in a serial order, open or closed, is said to be compact, if the series contains no immediately consecutive entities, so that in traversing the series from any one entity to any other entity it is necessary to pass through entities distinct from either.

It was the merit of R. Dedekind and of G. Cantor explicitly to formulate another fundamental property of series. The Dedekind property' as applied to an open series can be defined thus: An open series possesses the Dedekind property, if, however, it be divided into two mutually exclusive classes u and v, which (1) contain between them the whole series, and (2) are such that every member of u precedes in the serial order every member of v, there is always a member of the series, belonging to one of the two, u or v, which precedes every member of v (other than itself if it belong to v), and also succeeds every member of u (other than itself if it belong to u). Accordingly in an open series with the Dedekind property there is always a member of the series marking the junction of two classes such as u and v. An open series is continuous if it is compact and possesses the Dedekind property. A closed series can always be transformed into an open series by taking any arbitrary member as the first term and by taking one of the two ways round as the ascending order of the series. Thus the definitions of compactness and of the Dedekind property can be at once transferred to a closed series. 12. The last axiom of order is that there exists at least one straight line for which the point order possesses the Dedekind property. It follows from axioms 1-12 by projection that the Dedekind property is true for all lines. Again the harmonic system ABC, where A, B, C are collinear points, is defined2 thus: take the harmonic conjugates A', B', C' of each point with respect to the other two, again take the harmonic conjugates of each of the six points A, B, C, A', B', C' with respect to each pair of the remaining five, and proceed in this way by an unending series of steps.

The set of points thus obtained is called the harmonic system ABC. It can be proved that a harmonic system is compact, and that every segment of the line containing it possesses members of it. Furthermore, it is easy to prove that the fundamental theorem holds for harmonic systems, in the sense that, if A, B, C are three points on a line 1, and A', B', C' are three points on a line 1', and if by any two distinct series of projections A, B, C are projected into A', B', C', then any point of the harmonic system ABC corresponds to the same point of the harmonic system A'B'C' according to both the projective relations which are thus established between 1 and 1'. It now follows immediately that the fundamental theorem must hold for all the points on the lines 1 and 1', since (as has been pointed out) harmonic systems are " everywhere dense " on their containing lines. Thus the fundamental theorem follows from the axioms of order. A system of numerical coordinates can now be introduced, possessing the property that linear equations represent planes and straight lines. The outline of the argument by which this remarkable problem (in that " distance " is as yet undefined) is solved, will now be given. It is first proved that the points on any line can in a certain way be definitely associated with all the positive and negative real numbers, so as to form with them a one-one correspondence. The arbitrary elements in the See also:

• ESTABLISHMENT (0. Fr. establissement, Fr. etablissement, late Norm. Fr. establishement, from O. Fr. establir, Fr. etablir, Lat. stabilire, to make stable)

69) be the given line, m and n any two lines intersecting at U on 1, S and S' two points on n. Then a projective relation between l and itself is formed by projecting 1 from S on to m, and then by projecting m from S' back on to 1. All such projective 1 Cf. Dedekind, Stetigkeit and irrationale Zahlen (1872). 2 Cf. v. Staudt, Geometrie der Loge (1847). 'Cf. Pasch, Vorlesungen uber neuere Geometric (Leipzig, 1882), a classic work; also Fiedler, Die darstellende Geometrie (1st ed., 1871, 3rd ed., 1888) ; Clebsch, Vorlesungen fiber Geometric, vol. iii.; Hilbert, loc. cit. ; F. Schur, Math. Ann. Bd. lv.

(1902) ; Vahlen, loc. cit. ; Whitehead, loc. cit.relations, however m, n, S and S' be varied, are called " prospectivities," and U is the double point of the prospectivity. If a point 0 on 1 is related to A by a prospectivity, then all prospectivities, which (r) have the same double point U, and (2) relate 0 to A, give the same correspondent (Q, in figure) to any point P on the line 1; in fact they are all the same prospectivity, however m, n, S, and S' may have been varied subject to these conditions. Such aOAU) ectivity will be denoted by The sum of two prospectivities, written (OAU2)-l-(OBU'), is defined FIG. 6o. to be that transformation of the line 1 into itself which is obtained by first applying the prospectivity (OAU2) and then applying the prospectivity (OBU'). Such a transformation, when the two summands have the same double point, is itself a prospectivity with that double point. With this definition of addition it can be proved that prospectivities with the same double point satisfy all the axioms of magnitude. Accordingly they can be associated in a one-one correspondence with the positive and negative real numbers. Let E (fig. 70) be any point on 1, distinct from 0 and U. Then the prospectivity (OEU2) is associated with unity, the prospectivity (OOUi') is associated with zero, and (ODU2) with co . The prospectivities of the type (OPU'), where P is any point on the segment OEU, correspond to the positive numbers; also if P' is the harmonic conjugate of P with respect to O and U, the prospectivity (OP'U2) is associated with the corresponding negative number.

(The subjoined figure explains this relation of the positive and negative prospectivities.) Then any point P on 1 is associated with the same number as is the prospectivity (OPU2). It can.be proved that the order of the numbers in algebraic order of magnitude agrees with the order on the line of the associated points. Let the numbers, assigned according to the preceding specification, be said to be associated with the points according to the " numeration-system (OEU)." The introduction of a coordinate system for a plane is now managed as follows: Take any triangle OUV in the plane, and on the lines OU and OV establish the numeration systems (OEIU) and (OE2V), where El and E2 are arbitrarily chosen. Then (cf. fig. 71) if M and N are associated with the numbers x and y according to these systems, the coordinates of P are x and y. It then follows that the equation of a straight line is of the form ax+by+c=o. Both coordinates of any point on the line UV are infinite. This can be avoided by introducing homogeneous coordinates X, Y, Z, where x=X/Z, and y=Y/Z, and Z =o is the equation of UV. The procedure for three dimensions is similar. Let OUVW (fig. 72) be any tetrahedron, and See also:

• ASSOCIATE (Lat. associates, from ad, to, and sociare to join)

Also homogeneous coordinates can be introduced as before, thus avoiding the infinities-on the plane UVW. o The cross ratio of a range of four collinear points can now be defined FIG. 72. as a number characteristic of that range. Let the coordinates of any point Pr of the range PI P2 P3 P4 be ara+µ,+a' a.b+µrb' arc+µrc' Xr+µr 1'r+µr ' X,+µr ' and let (N,µ,) be written for X,µ,—X,µr• Then the cross ratio (PI P2 P3 P4} is defined to be the number (1,Iµ2)(4µ4);(AI (X31"2). The equality of the cross ratios of the ranges (PI P2 P3 F4) and (Q, Q2 Q3 Q4) is proved to be the necessary and sufficient condition for their mutual projectivity. The cross ratios of all harmonic ranges are then easily seen to be all equal to – 1, by comparing with the range (OEIUE'I) on the axis of x. Thus all the ordinary propositions of geometry in which distance and angular measure do not enter otherwise than in cross ratios can now be enunciated and proved. Accordingly the greater part of the analytical theory of conics and quadrics belongs to geometry w (r=1, 2, 3, 4) at this stage The theory of distance will be considered after the principles of descriptive geometry have been developed. Descriptive Geometry. Descriptive geometry is essentially the science of multiple order for open series. The first satisfactory system of axioms was given by M. Pasch.' An improved version is due to G.

Peano.2 Both these authors treat the idea of the class of points constituting the segment lying between two points as an undefined fundamental idea. Thus in fact there are in this system two fundamental ideas, namely, of points and of segments. It is then easy enough to define the prolongations of the segments, so as to form the complete straight lines. D. Hilbert's3 formulation of the axioms is in this respect practically based on the same fundamental ideas. His work is justly famous for some of the mathematical investigations contained in it, but his exposition of the axioms is distinctly inferior to that of Peano. Descriptive geometry can also be considered ' as the science of a class of relations, each relation being a two-termed serial relation, as considered in the See also:

• LOGIC (Xoy1K7, sc. rixvrt, the art of reasoning)

2. If the points A, B, C are in the order ABC, they are not in the order BCA. 3. If the points A, B, C are in the order ABC, A is distinct from C. 4. if A and B are any two distinct points, there exists a point C such that A, B, C are in the order ABC. Definition.—The line AB (A B) consists of A and B, and of all points X in one of the possible orders, ABX, AXB, XAB. The points X in the order AXB constitute the segment AB. 5. If points C and D (CD) lie on the line AB, then A lies on the line CD. 6. There exist three distinct points A, B, C not in any of the orders ABC, BCA, CAB. 7.

If three distinct points A, B, C (fig. 73) do not lie on the same line, and D and E are two distinct points in the orders See also:

• BCD

(1894). ° Cf. loc. cit. ' Cf. Vailati, Rivista di mat. vol. iv. and Russell, loc. cit. § 376. 6 Cf. O. Veblen, " On the Projective Axioms of Geometry," Trans. Amer. Math. Soc. vol. iii. (1902).

ro. The Dedekind property holds for the order of the points on any straight line. It follows from axioms 1-9 that the points on any straight line are arranged in an open serial order. Also all the ordinary theorems respecting a point dividing a straight line into two parts, a straight line dividing a plane into two parts, and a plane dividing space into two parts, follow. Again, in any plane a consider a line 1 and a point A (fig. 74). Let any point B divide 1 into two See also:

• HALF

Then two cases are possible. (I) The half-lines r and s are collinear, and together form one complete line. In this case, there is one and only one line (viz. r+s) through A and lying in a which does not intersect 1. This is the Euclidean case, and the assumption that this case holds is the _ Euclidean parallel axiom. But (2) the c half-lines r and s may not be collinear. FIG. 74. In this case there will be an infinite number of lines, such as k for instance, containing A and lying in a, which do not intersect 1. Then the lines through A in a are divided into two classes by reference to 1, namely, the secant lines which intersect 1, and the non-secant lines which do not intersect 1. The two boundary non-secant lines, of which r and s are respectively halves, may be called the two parallels to 1 through A. The perception of the possibility of case 2 constituted the starting-point from which Lobatchewsky constructed the first explicit coherent theory of non-Euclidean geometry, and thus created a revolution in the philosophy of the subject. For many centuries the speculations of mathematicians on the foundations of geometry were almost confined to hopeless attempts to prove the ` parallel axiom " without the introduction of some equivalent axiom .° Associated Projective and Descriptive Spaces.—A region of a projective space, such that one, and only one, of the two supplementary segments between any pair of points within it lies entirely within it, satisfies the above axioms (1-ro) of descriptive geometry, where the points of the region are the descriptive points, and the portions of straight lines within the region are the descriptive lines.

If the excluded part of the original projective space is a single plane, the Euclidean parallel axiom also holds, otherwise it does not hold for the descriptive space of the limited region. Again, conversely, starting from an original descriptive space an associated projective space can be constructed by means of the concept of ideal points? These are also called projective points, where it is understood that the simple points are the points of the original descriptive space. An ideal point is the class of straight lines which is composed of two coplanar lines a and b, together with the lines of intersection of all pairs of intersecting planes which respectively contain a and b, together with the lines of intersection with the plane ab of all planes containing any one of the lines (other than a or b) already specified as belonging to the ideal point. It is evident that, if the two original lines a and b intersect, the corresponding ideal point is nothing else than the whole class of lines which are concurrent at the point ab. But the essence of the definition is that an ideal point has an existence when the lines a and b do not intersect, so long as they are coplanar. An ideal point is termed "proper, if the lines composing it intersect; otherwise it is improper. A theorem essential to the whole theory is the following: if any two of the three lines a, b, c are coplanar, but the three lines are not all coplanar, and similarly for the lines a, b, d, then c and d are coplanar. It follows that any two lines belonging to an ideal point can be used as the pair of guiding lines in the definition. An ideal point is said to be coherent with a plane, if any of the lines composing it lie in the plane. An ideal line is the class of ideal points each of which is coherent with two given planes. 6 Cf.

P. Stdckel and F. Engel, Die Theorie der Parallellinien von Euklid bis auf Gauss (Leipzig, 1895). Cf Pasch, loc. cit., and R. Bonola, " See also:

• SULLA, LUCIUS CORNELIUS (138–78 B.C.)

Thus a projective space has been constructed out of the ideal elements, and the proper ideal elements correspond element by element with the associated descriptive elements. Thus the proper ideal elements form a region in the projective space within which the descriptive axioms hold. Accordingly, by substituting ideal elements, a descriptive space can always be considered as a region within a projective space. This is the See also:

• JUSTIFICATION

This was in effect the point of view of Pasch .3 It has, however, been proved by Sophus Lie' that congruence is capable of definition without recourse to a new fundamental idea. This he does by means of his theory of finite continuous groups (see GROUPS, THEORY OF), of which the definition is possible in terms of our established geometrical ideas, remembering that co-ordinates have already been introduced. The displacement of a rigid body is simply a mode of defining to the senses a one-one transformation of all space into itself. For at any point of space a particle may be conceived to be placed, and to be rigidly connected with the rigid body; and thus there is a definite correspondence of any point of space with the new point occupied by the associated particle after displacement. Again two suc- ' The original idea- (confined to this particular case) of ideal points is due to von Staudt (loc. cit.). 2 Cf. Critique, " Trans. Aesth." See also:

• SECT

Thus the transformations of space into itself defined by displacements of rigid bodies form a group. Call this group of transformations a congruence-group. Now according to Lie a congruence-group is defined by the following characteristics: r. A congruence-group is a finite continuous group of one-one transformations, containing the identical transformation. 2. It is a sub-group of the general projective group, i.e. of the group of which any transformation converts planes into planes, and straight lines into straight lines. 3. An infinitesimal transformation can always be found satisfying the condition that, at least throughout a certain enclosed region, any definite line and any definite point on the line are latent, i.e. correspond to themselves. 4. No infinitesimal transformation of the group exists, such that, at least in the region for which (3) holds, a straight line, a point on it, and a plane through it, shall all be latent. The property enunciated by conditions (3) and (4), taken together, is named by Lie " Free mobility in the infinitesimal." Lie proves the following theorems for a projective space: r. If the above four conditions are only satisfied by a group throughout part of projective space, this part either (a) must be the region enclpsed by a real closed quadric, or (13) must be the whole of the projective space with the exception of a single plane.

In case (a) the corresponding congruence group is the continuous group for which the enclosing quadric is latent; and in case (9) an imaginary conic (with a real equation) lying in the latent plane is also latent, and the congruence group is the continuous group for which the plane and conic are latent. 2. If the above four conditions are satisfied by a group throughout the whole of projective space, the congruence group is the continuous group for which some imaginary quadric (with a real equation) is latent. By a proper choice of non-homogeneous co-ordinates the equation of any quadrics of the types considered, either in theorem 1(a), or in theorem 2, can be written in the form r+c(x2+y2+z2) =o, where c is negative for a real closed quadric, and positive for an imaginary quadric. Then the general infinitesimal transformation is defined by the three equations: dx/dt = u —way+w2z+cx(ux+vy+wz), dy/dt = v —w,z+See also:

• WAX

The definition of distance is connected with the corresponding congruence-group by two considerations in respect to a range of five , points (A,, As, P,, Ps, Ps), of which A, and As are on the absolute. Let {AiP,A2P2} stand for the cross ratio (as defined above) of the range (A,P,A,Ps), with a similar notation for the other ranges. Then (r) See also:

• LOG (a word of uncertain etymological origin,possibly onomatopoeic; the New English Dictionary rejects the derivation from Norwegian lag, a fallen tree)

Also the previous definition of an angle can be adapted to this case, by making t1 and t2 to be the tangent planes through the line pi 1'2 to the imaginary conic. Similarly if pi and p2 are intersecting lines, the same definition of an angle holds, where ti and 12 are now the lines from the point See also:

• PIPE

If the boundary is a plane (the plane at infinity), the possible congruence-groups are parabolic; and there is a congruence-group corresponding to each imaginary conic in this plane, together with a Euclidean metrical geometry corresponding to each such group. Owing to these alternative possibilities, it would appear to be more accurate to say that systems of quantities can he found in a space, rather than that space is a quantity. Lie has also deduced 2 the same results with respect to congruence-groups from another set of defining properties, which explicitly assume the existence of a quantitative relation (the distance) between any two points, which is invariant for any transformation of the congruence-group.3 The above results, in respect to congruence and metrical geometry, considered in relation to existent space, have led to the doctrine' that it is intrinsically unmeaning to ask which system of metrical geometry is true of the physical world. Any one of these systems can be applied, and in an indefinite number of ways. The only question before us is one of convenience in respect to simplicity of statement of the physical laws. This point of view seems to neglect the consideration that science is to be relevant to the definite perceiving minds of men; and, that (neglecting the ambiguity introduced by the invariable slight inexactness of observation which is not relevant to this special doctrine) 1 Cf. A. Cayley, " A See also:

• SIXTH

Klein, Math. Ann. vol. iv., 1871. 2 Cf. loc. cit. 2 For similar deductions from a third set of axioms, suggested in essence by Peano, Riv. mat. vol. iv. loc. cit. cf. Whitehead, Desc. Geom. loc. cit.. ' Cf. H. See also:

• POINCARE, RAYMOND (186o— )

But if the doctrine means that, assuming some sort of See also:

• OBJECTIVE, or OBJECT GLASS

One result of this was the preparation of manuals of a popular kind for use in the See also:

• SCHOOLS