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 equation of the first degree satisfied by x and y. No point not on the line satisfies the same equation; for the line from C to any point off the line would make with CR some angle S different from a, and the point in question would satisfy an equation y —k = tan li(x —h), which is inconsistent with the above equation.
The equation of the line may also be written y=mx+b, where m=tan a, and b=k—h tan a. Here b is the value obtained for y from the equation when o is put for x, i.e. it is the numerical measure, with proper sign, of OB, the intercept made by the line on the axis of y, measured from the origin. For different straight lines, m and b may have any constant values we like.
Now the general equation of the first degree Ax+By+C =0 may
be written y= x—B, unless B =o, in which case it represents a
line parallel to the axis of y; and —A/B, —C/B are values which can be given to m and b, so that every equation of the first degree represents a straight line. It is important to notice that the general equation, which in appearance contains three constants A, B, C, in effect depends on two only, the ratios of two of them to the third. In virtue of this last remark, we see that two distinct conditions suffice to determine a straight line. For instance, it is easy from the above to see that
a+v=t
is the equation of a straight line determined by the two conditions that it makes intercepts OA, OB on the two axes, of which a and b are the numerical measures with proper signs: note that in fig. 50 a is negative. Again,
_Y2—yt
yyt 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 condition that (x2, y2) lies on the line.
In this paragraph the coordinates have been assumed rectangular. Had they been oblique, the doctrine of similar triangles would have given the same results, except that in the forms of equation y—k = m(x—h), y=mx+b, we should not have had m=tan a.
9. The Circle.—It is easy to write down the equation of a given circle. Let (h, k) be its given centre C, and p the numerical measure of its given radius. Take P (x, y) any point on its circumference, and construct the triangle CRP, in fig. 50 as above. The fact that this is rightangled tells us that
CR2+RP2=CP2,,
and this at once gives the equation
(xh)2+(y k)2=p2.
A point not upon the circumference of the particular circle is at some distance from (h, k) different from p, and satisfies an equation inconsistent with this one; which accordingly represents the circumference, or, as we say, the circle.
The equation is of the form
x2+y2+2Ax+2By+C =0.
Conversely every equation of this form represents a circle: we have only to take —A, —B, A2+B2—C for h, k, p2 respectively, to obtain its centre and radius. But this statement must appear too unrestricted. Ought we not to require A2+B2—C to be positive? Certainly, if by circle we are only to mean the visible round circumference of the geometrical definition. Yet, analytically, we contemplate altogether imaginary circles, for which p2 is negative, and circles, for which p=o, with all their reality condensed into their centres. Even when p2 is positive, so that a visible round circumference exists, we do not regard this as constituting the whole of the circle. Giving to x any value whatever in (x —h)2+ (y—k)2 =p2, we obtain two values of y, real, coincident or imaginary, each of which goes with the abscissa x as the ordinate of a point, real or imaginary, on what is represented by the equation of the circle.
The doctrine of the imaginary on a circle, and in geometry generally, is of purely algebraical inception; but it has been, in its entirety accepted by modern pure geometers, and signal success has attended the efforts of those who, like K. G. C. von Staudt, have striven to base its conclusions on principles not at all algebraical in form, though of course cognate to those adopted in introducing the imaginary into algebra.
A circle with its centre at the origin has an equation x2+y2=p2. In oblique coordinates the general equation of a circle is x2+2xy cos w+y2+2Ax+2By+C =o.
to. The conic sections are the next simplest loci; and it will be seen later that they are the loci represented by equations of the second degree. Circles are particular cases of conic sections; and
Fin. 5o.
they have just been seen to have for their equations a particular class of equations of the second degree. Another particular class of such equations is that included in the form (Ax+By+C) (A'x+ B'y+C') =o, which represents two straight lines, because the product on the left vanishes if, and only if, one of the two factors does, i.e. if, and only if, (x, y) lies on one or other of two straight lines. The condition that axe+2hxy+bye+2gx+2fy+c=o, which is often written (a, b, c, f, g, h) (x, y, 1)2 = o, takes this form is abc+2fgh—af2—bg2—ch2=o. Note that the two lines may, in particular cases, be parallel or coincident.
Any equation like Fi(x, y) F2(x, y) . . . F„(x, y) =o, of which the lefthand side breaks up into factors, represents all—the loci separately represented by F,(x, y) =o, F2(x, y) =o, . . . F„(x, y) =o. In particular an equation of degree n which is free from x represents n straight lines parallel to the axis of x, and one of degree n which is homogeneous in x and y, i.e. one which upon division by x' becomes an equation in the ratio y/n, represents n straight lines through the origin.
Curves represented by equations of the third degree are called cubic curves. The general equation of this degree will be written (*) (x, y, i)3=o.
II. Descriptive Geometry.—A geometrical proposition is either descriptive or metrical: in the former case the statement of it is independent of the idea of magnitude (length, inclination, &c.), and in the latter it has reference to this idea. The method of coordinates seems to be by its inception essentially metrical. Yet in dealing by this method with descriptive propositions we are eminently free from metrical considerations, because of our power to
use general equations, and to avoid all assumption that measurements implied are any particular measurements.
12. It is worth while to illustrate this by the instance of the wellknown theorem of the radical centre of three circles. The theorem is that, given any three circles A, B, C (fig'. 51), the common chords aa', $j3', yy' of the three pairs of circles meet in a point.
The geometrical proof is metrical throughout:
section of aa', 0#', and joining
this with y', suppose that 7'O does not pass through y, but that it meets the circles A, B in two distinct points 72 , 71 respectively. We have then the known metrical property of intersecting chords of a circle; viz. in circle C, where aa', 00', are chords meeting at a point 0,
Oa.Oa' =00.00',
where, as well as in what immediately follows, Oa, &c., denote, of course, lengths or distances.
Similarly in circle A,
00.00'=072.07',
and in circle B,
Oa.Oa' = 071.07'.
Consequently O71.Oy'=072.07', that is, 0y,=072, or the points 71 and 72 coincide; that is, they each coincide with y.
We contrast this with the analytical method:
Here it only requires to be known that an equation Ax+By+C =o represents a line, and an equation x2+y2+Ax+By+C =o represents a circle. A, B, C have, in the two cases respectively, metrical significations; but these we are not concerned with. Using S to denote the function x2+y2+Ax+By+C, the equation of a circle is S=o. Let the equation of any other circle be S', =x2+y2+A'x+ B'y+C'=o; the equation S—S'=o is a linear equation (S—S' is in fact=(A—A')x+(B—B') y+C—C'), and it thus represents a line; this equation is satisfied by the coordinates of each of the points of intersection of the two circles (for at each of these points S=o and S'=o, therefore also S—S'=o); hence the equation S—S'=o is that of the line joining the two points of intersection of the two circles, or say it is the equation of the common chord of the two circles. Considering then a third circle S",=x2+y2+A"x+B"y+C"=o, the equations of the common chords are S—S'=o, S—S"=o, S'—S"=o (each of these a linear equation) ; at the intersection of the first and second of these lines S=S' and S=S", therefore also S'=S", or the equation of the third line is satisfied by the coordinates of the point in question; that is, the three chords intersect in a point 0, the coordinates of which are determined by the equations S=S'=S". `
It further appears that if the two circles S =o, S' =o donot intersect in any real points, they must be regarded as intersecting in two imaginary points, such that the line joining them is the real line represented by the equation S—S'=o; or that two circles, whether their intersections be real or imaginary, have always a real common chord (or radical axis), and that for any three circles the common chords intersect in a point (of course real) which is the radical centre. And by this very theorem, given two circles with imaginary intersections, we can, by drawing circles which meet each of them in real points, construct the radical axis of the firstmentioned two circles.
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 writing for (x, y) the values (xo, yo), then the value of kisk=So=S'o. The equation in fact is SS's —SoS'=o; and starting from this equation we at once verify it a posteriori; the equation is a linear equation satisfied by the values of (x, y) which make S=o, S'=o; and satisfied also by the values (xo, yo); and it is thus the equation of the line in question.
If, as before, S = o, S'=o represent circles, then (k being arbitrary) S—kS'=o is the equation of any circle passing through the two points of intersection of the two circles; and to make this pass through a given point (xo, yo) we have again k=So=S'o• In the particular case k =I, the circle becomes the common chord (more accurately it becomes the common chord together with the line infinity; see § 23 below).
If S denote the general quadric function,
S =axe+2hxy+by2+2fy+2gx+e,
then the equation S=o represents a conic; assuming this, then, if S'=o represents another conic, the equation S—kS'=o represents any conic through the four points of intersection of the two conics.
14. The object still being to illustrate the mode of working with coordinates for descriptive purposes, we consider the theorem of the polar of a point in regard to a circle. Given a circle and a point 0 (fig. 52), we draw through 0 any two lines meeting the circle in the points A, A' and B, B' respectively, and then taking Q as the intersection of the lines AB' and A'B, the theorem is that the locus of the point Q is a right line depending only upon 0 and the circle, but independent of the particular lines OAA' and OBB'.
Taking 0 as the origin, and for the axes any two lines through 0 at right angles to each other, the equation of the circle will be x2+y2+2Ax+2By+C =o;
and if the equation of the line OAA' is taken to be y=mx, then the points A, A' are found as the intersections of the straight line with the circle; or to determine x we have
x2 (1 +m2) +2 x (A+ B m) +C = o.
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 divide the first of these equations by x,x4, and the second by x2x3,)SSand then add, we obtain S)S] 1(
x m ~x3+x4) —m' (x,+x2) —y ( x3+x4— (x,+x 1J 2/ +2m' 2m =0,
or, what ris the same thing, (73, \x,+xsl2/ (y—m'x)— x4/ 0—mx)+2m'—2m=o,
which by what precedes is the equation of a line through the point Q.
Substituting herein for i3+?x4 their foregoing values, the x, x2 x
equation becomes
— (A+Bm) (y—m'x)+(A+Bm') (y —mx) +C(m' —m) =o; that is,
(m—m') (Ax+By+C)=o;
or finally it is Ax+By+C=o, showing that the point Q lies in a line the position of which is independent of the particular lines OAA', OBB' used in the construction.. It is proper to notice that there is no correspondence to each other of the points A, A' and B, B'; the grouping might as well have been A, A' and B', B; and it thence appears that the line Ax+By+C=o just obtained is in fact the line joining the point Q with the point R which is the intersection of AB and A'B'.
15. In § 8 it has been seen that two conditions determine the equation of a straight line, because in Ax+By+C=o one of the coefficients may be divided out, leaving only two parameters to be determined. Similarly five conditions instead of six determine an equation of the second degree (a, b, c, f, g, h) (x, y, 1)2=0, and nine instead of ten determine a cubic (*) (x, y, 1)3=o. It thus appears that a cubic can be made to pass through 9 given points, and that the cubic so passing through 9 given points is completely determined. There is, however, a remarkable exception. Considering two given cubic curves S=o, S'=o, these intersect in 9 points, and through these 9 points we have the whole series of cubics SkS'=o, where k is an arbitrary constant: k may be determined so that the cubic shall pass through a given tenth point (k=So=S'o, if the coordinates are (xo, yo), and So, S'o denote the corresponding values of S, S'). The resulting curve SS'oS'So=o may be regarded as the cubic determined by the conditions of passing through 8 of the 9 points and through the given point (xo, yo); and from the equation it thence appears that the curve passes through the remaining one of the 9 points. In other words, we thus have the theorem, any cubic curve which passes through 8 of the 9 intersections of two given cubic curves passes through the 9th intersection.
The applications of this theorem are very numerous; for instance, we derive from it Pascal's theorem of the inscribed hexagon. Consider a hexagon inscribed in a conic. The three alternate sides constitute a cubic, and the other three alternate sides another cubic. The cubics intersect in 9 points, being the 6 vertices of the hexagon, and the 3 Pascalian points, or intersections of the pairs of opposite sides of the hexagon. Drawing a line through two of the Pascalian points, the conic and this line constitute a cubic passing through 8 of the 9 points of intersection, and it therefore passes through the remaining point of intersection—that is, the third Pascalian point; and since obviously this does not lie on the conic, it must lie on the line—that is, we have the theorem that the three Pascalian points (or points of intersection of the pairs of opposite sides) lie on a line.
16. Metrical Theory resumed. Projections and Perpendiculars.—It is a metrical fact of fundamental importance, already used in § 8, that, if a finite line PQ be projected on any other line 00' by perpendiculars PP', QQ' to 00', the length of the projection P'Q' is equal to that of PQ multiplied by the cosine of the acute angle between the two lines. Also the algebraical sum of the projections of the sides of any closed polygon upon any line is zero, because as a point goes round the polygon, from any vertex A to A again, the point which is its projection on the line passes from A' the projection of A to A' again, i.e. traverses equal distances along the line in positive and negative senses. If we consider the polygon as consisting of two broken lines, each extending from the same initial to the same terminal point, the sum of the projections of the lines which compose the one is equal, in sign and magnitude, to the sum of the projections of the lines composing the other. Observe that the projection on a line of a length perpendicular to the line is zero.
Let us hence find the equation of a straight line such that the perpendicular .OD on it from the origin is of length p taken as positive, and is inclined to the axis of x at an angle xOD = a, measured counterclockwise from Ox. 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 sin a= p.
Observe that cos a and sin a here are the sin a and cos a, or the sin a and cos a of § 8 according to circumstances.
We can write down an expression for the perpendicular distance from this line of any point (x', y') which does not lie upon it. If the parallel through (x', y') to the line meet OD in E, we have x' cos a+ y' sin a=OE, and the perpendicular distance required is ODOE, i.e. px' cos ay' sin a; it is the perpendicular distance taken positively or negatively according as (x', y') lies on the same side of the line as the origin or not.
The general equation Ax+By+C =o may be given the form x cos a+y sin ap =o by dividing it by s/ (A2+B2). Thus (Ax'+ By'+C) =J (A2+B2) is in absolute value the perpendicular distance of (x', y') from the line Ax+By+C =0. Remember, however, that there is an essential ambiguity of sign attached to a square root. The expression found gives the distance taken positively when (x', y') is on the origin side of the line, if the sign of C is given to J (A2+B2)•
17. Transformation of Coordinates.—We often need to adopt new axes of reference in place of old ones; and the above principle of projections readily expresses the old coordinates of any point in terms of the new.
Suppose, for instance, that we want to take for new origin the point 0' of old coordinates OA=h, AO'=k, and for new axes of X and Y lines through 0' obtained by rotating parallels to the old axes of x and y through an angle 0 counterclockwise. Construct (fig. 53) the old and new co
ordinates of any point P. Expressing 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 0Y sin 0, and
y=k+X cos(4Zr0)+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 change of reference from one set of rectangular axes to another. It might have been required to take O'X, O'Y' for the positive directions of the new axes, so that the change of directions of the axes could not be effected by rotation. We must then write Y for Y in the above.
Were the new axes oblique, making angles a, $ respectively with the old axis of x, and so inclined at the angle ,8a, the same method would give the formulae
x=h+X cos a+Y cos 13, y=k+X sin a+Y sin 9.
18. The Conic Sections.—The conics, as they are now called, were at first defined as curves of intersection of planes and a cone; but Apollonius substituted a definition free from reference to space of three dimensions. This, in effect, is that a conic is the locus of a point the distance of which from a given point, called the focus, has a given ratio to its distance from a given line, called the directrix (see CONIC SECTION). 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 ap=o for the directrix. The absolute values ofJ{(xh)2+(yk)21 andpxcosay sin a are to have the ratio e : 1; and this gives
(xh)2+(yk)2 = e2(px cos ay 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 special case when the given equation represents two straight lines, as in § to, but there is speciality: if the two lines intersect, the intersection and either bisector of the angle between them are a focus and directrix; if they are united in one line, any point on the line and a perpendicular to it through the point are: if they are parallel, the case is a limiting one in which e and h2+k2 have become infinite while e2(h2+k2) remains finite. In the case (§ 9) of an equation such as represents a circle there is another instance of proceeding to a limit: e has to become o, while ep remains finite: moreover a is indeterminate. The centre of a circle is its focus, and its directrix has gone to infinity, having no special direction. This last fact illustrates the necessity, which is also forced on plane geometry by threedimensional considerations, of treating all points at infinity in a plane as lying on a single straight line.
Sometimes, in reducing an equation to the above focus and directrix form, we find for h, k, e, p, tan a, or some of them, only imaginary values, as quadratic equations have to be solved; and we have in fact to contemplate the existence of entirely imaginary conics. For instance, no real values of x and y satisfy x2+2y2+3 =o. Even when the locus represented is real, we obtain, as a rule, four sets of values of h, k, e, p, of which two sets are imaginary; a real conic has, besides two real foci and corresponding directrrces, two others that are imaginary.
In oblique as well as rectangular coordinates equations of the second degree represent conics.
19. The three Species of Conics.—A real conic, which does not degenerate into straight lines, is called an ellipse, parabola or hyperbola according as e<, =, or >I. To trace the three forms it is best so to choose the axes of reference as to simplify their equations.
In the case of a parabola, let 2c be the distance between the given focus and directrix, and take axes referred to which these are the point (c, o) and the line x = c. The equation becomes (x c)2+y2 = (x+c)2, i.e. y2=4cx.
In the other cases, take a such that a(e e1) is the distance of focus from directrix, and so choose axes that these are (ae, o) and x=ae1, thus getting the equation(x ae)2+y2 = e2(xae1)2,i.e. (1e2)x2+y2 = a2(1e2). When e<1, i.e. in the case of an ellipse, this may be written x2/a2+y2/b2 =1, where b2 = a2(i e2) ; and when e> I, i.e. in the case of an hyperbola, x2/a2y2/b2=I, where b2=a2(e2I).
The axes thus chosen for the ellipse and hyperbola are called the principal axes.
In figs. 54, 55, 56 in order, conics of the three species, thus referred, are depicted.
The oblique straight lines in fig. 56 are the asymptotes x/a = y/b of the hyperbola, lines to which the curve tends with unlimited
closeness as it goes to infinity. The hyperbola would have an equation of the form xy=c if referred to its asymptotes as axes, the coordinates being then oblique, unless a =b, in which case the hyperbola is called rectangular. An ellipse has two imaginary asymptotes. In particular a circle x2+y2=a2, a particular ellipse, has for asymptotes the imaginary lines x= =ys/1. These run from the centre to the socalled circular points at infinity.
f2o. Tangents and Curvature.—Let (x , y') and (x'+h, y'+k) be two neighbouring points P, P' on a curve. The equation of the line on which both lie is h(yy')=k(xx'). Now keep P fixed, and let P' move towards coincidence with it along the curve. The connecting line will tend towards a limiting position, to which it can never attain as long as P and P' are distinct. The line which occupies this limiting position is the tangent at P. Now if we subtract the equation of the curve, with (x', y') for the coordinates in it, from the like equation in (x'+h, y'+k), we obtain a relation in h and k, which will, as a rule, be of the form o=Ah+Bk+ terms of higher degrees in h and k, where A, B and the other coefficients involve x' and y'. This gives k/h=A/B+ terms which tend to vanish as h and k do, so that A : B is the limiting value tended to by k : h. Hence the equation of the tangent is B (y y') +A(x  x') = o.
The normal at (x', y') is the line through it at right angles to the tangent, and its equation is A(yy')B(xx')=o.
In the case of the conic (a, b, c, f, g, h) (x, y, 1)2=o we find that A/B = (ax'+hy'+g)/(hx'+by'+f).
We can obtain the coordinates of Q, the intersection of the normals QP, QP' at (x', y') and (x'+h, y'+k), and then, using the limiting value of k : h, deduce those of its limiting position as P' moves up to P. This is the centre of curvature of the curve at P (x', y'), and is so called because it is the centre of the circle of closest contact with the curve at that point. That it is so follows from the facts that the closest circle is the limit tended to by the circle which touches the curve at P and passes through P', and that the arc from P to P' of this circle lies between the circles of centre Q and radii QP, QP', which circles tend, not to different limits as P' moves up to P, but to one. The distance from P to the centre of curvature is the radius of curvature.
21. Differential Plane Geometry.—The language and notation of the differential calculus are very useful in the study of tangents and curvature. Denoting by (, n) the current coordinates, we find, as above, that the tangent at a point (x, y) Of a curve is ry= (Ex)dy/dx, where dy/dx is found from the equation of the curve. If this be f(x, y) =o the tangent is (Ix) (af/ax)+(,ly) (af/ay) =o. If p and (a, /3) are the radius and centre of curvature at (x, y), we find that q(ax)p(I+p2),q(Ry)=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 aX and /3y, 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 pole) 0, and the inclination xOP of 0Pto a fixed line Ox, determine the point: r, the numerical measure of OF, the radius vector, and 0, the circular measure of xOP, the inclination, are called polar coordinates of P. The formulae x= r cos 0, y =r sin 0 connect Cartesian and polar coordinates, and make transition from either system to the other easy. In polar coordinates the equations of a circle through 0, and of a conic with 0 as focus, take the simple forms r=2a cos (0a), rl1e cos (Oa)} =1. The use of polar coordinates is very convenient in discussing curves which have properties of symmetry akin to that of a regular polygon, such curves for instance as r=a cos mO, with m integral, and also the curves called spirals, which have equations giving r as functions of B itself, and not merely of sin 0 and cos B. In the geometry of motion under central forces the advantage of working with polar coordinates is great.
23. Trilinear and Areal Coordinates.—Consider a fixed triangle ABC, and regard its sides as produced without limit. Denote, as in trigonometry, by a, b, c the positive numbers of units of a chosen scale contained in the lengths BC, CA, AB, by A, B, C the angles, and by 0 the area, of the triangle. We might, as in § 6, take CA, CB as axes of x and y, inclined at an angle C. Any point P (x, y) in the plane is at perpendicular distances y sin C and x sin C from CA and CB. Call these S and a respectively. The signs of 13 and a are those of y and x, i.e. /3 is positive or negative according as P lies on the same side of CA as B does or the opposite, and similarly for a. An equation in (x, y) of any degree may, upon replacing in it x and y by a cosec C and /3 cosec C, be written as one of the same degree in (a, 13). Now let y be the perpendicular distance of P from the third side AB, taken as positive or negative as P is on the C side of AB or not. The geometry of the figure tells us that as+b/3+cy=2A. By means of this relation in a, 13, y we can give an equation considered countless other forms, involving two or all of a, /3, y. 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 present in the equation by a power of (aa+bfl+c7)/2L1 just sufficient to raise them, in each case, to the highest degree.
We call (a, /3, y) trilinear coordinates, and an equation in them the trilinear equation of the locus represented. Trilinear equations are, as a rule, dealt within their homogeneous forms. An advantage thus gained is that we need not mean by (a, 13, y) the actual measures of the perpendicular distances, but any properly signed numbers which have the same ratio two and two as these distances.
In place of a, 0, y it is lawful to use, as coordinates specifying the position of a point in the plane of a triangle of reference ABC, any given multiples of these. For instance, we may use x=as/2LS, y = b/3/2o, z =cy/20, the properly signed ratios of the triangular areas PBC, PCA, PAB to the triangular area ABC. These are called the areal coordinates of P. In areal coordinates the relation which enables us to make any equation homogeneous takes the simple form x+y+z=l; and, as before, we need mean by x, y, z, in a homogeneous equation, only signed numbers in the right ratios.
Straight lines and conics are represented in trilinear and in areal, because in Cartesian, coordinates by equations of the first and second degrees respectively, and these degrees are preserved when the equations are made homogeneous. What must be said about points infinitely far off in order to make universal the statement, to which there is no exception as long as finite distances alone are considered, that every homogeneous equation of the first degree represents a straight line? Let the point of areal coordinates (x', y', z') move infinitely far off, and mean by x, y, z finite quantities in the ratios which x', y', z' tend to assume as they become infinite. The relation x'+y'+z'=I gives that the limiting state of things tended to is expressed by x+y+z=o. 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 px cos 0y 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+n22mn cos A2nl cos B2lm 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 accord with a convention that oppositeness of sign implies distinction between one side of the line and the other. Three applications of the result above give
t/1=20/{al, bm, en) =n/m=I'/n;
and we thus have the important fact that tx'+ny'+iz' is the perpendicular distance between a point of areal coordinates (x'y'z') and a line on which the perpendiculars from A, B, C are t, n, respectively. We have also that fx+ny+3"z=o is the areal equation of the line on which the perpendiculars are t', n, 1; and, by equating the two expressions for the perpendiculars from (x', y', z') on the line, that in all cases {at, bn, q}2=4A2.
25. Linecoordinates. Duality.—A quite different order of ideas may be followed in applying analysis to geometry. The notion of a straight line specified may precede that of a point, and points may be dealt with as the intersections of lines. The specification of a line may be by means of coordinates, and that of a point by an equation, satisfied by the coordinates of lines which pass through it. Systems of linecoordinates will here be only briefly considered. Every such system is allied to some system of pointcoordinates; and space will be saved by giving prominence to this fact, and not recommencing ab initio.
Suppose that any particular system of pointcoordinates, in which lx+my+nz—o may represent any straight line, is before us: notice that not only are trilinear and areal coordinates such systems, but Cartesian coordinates also, since we may write x/z, y/z for the Cartesian x, y, and multiply through by z. 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 linecoordinates 1, m, n.
Linecoordinates are also called tangential coordinates. A curve is the envelope of lines which touch it, as well as the locus of points which lie on it. A homogeneous equation of degree above the first in 1, m, n is a relation connecting the coordinates of every line which touches some curve, and represents that curve, regarded as an envelope. For instance, the condition that the line of coordinates (1, m, n), i.e. the line of which the allied pointcoordinate equation is lx+my+nz=o, may touch a conic (a, b, c, f, g, h)(x, y, z)2=o, is readily found to be of the form (A, B, C, F, G, H) (l, m, n)2=o, i.e. to be of the second degree in the linecoordinates. It is not hard to show that the general equation of the second degree in 1, m, n thus represents a conic ; but the degenerate conics of linecoordinates are not linepairs, as in pointcoordinates, but pointpairs.
The degree of the pointcoordinate equation of a curve is the order of the curve, the number of points in which it cuts a straight line. That of the linecoordinate equation is its class, the number of tangents to it from a point. The order and class of a curve are generally different when either exceeds two.
26. The system of linecoordinates allied to the areal system of pointcoordinates has special interest.
The 1, m, n of this system are the perpendiculars t, n, of § 24; and x'i;+y'n+z'l'=o is the equation of the point of areal coordinates (x', y', z'), i.e. is a relation which the perpendiculars from the vertices of the triangle of reference on every line through the point, but no other line, satisfy. Notice that a nonhomogeneous equation of the first degree in t, n, i• does not, as a homogeneous one does, represent a point, but a circle. In fact x't+y'n+z`l=R expresses the constancy of the perpendicular distance of the fixed point x'f{y'n+ z'a=o from the variable line (t, n, g), i.e. the fact that (g, n, O touches a circle with the fixed point for centre. The relation in any t, n, iwhich enables us to make an equation homogeneous is not linear, as in pointcoordinates, but quadratic, viz. it is the relation {at, bn, c~}2=402 of § 24. Accordingly the homogeneous equation of the above circle is
402(x't+y'n+z'i')2=R2tat, bn, cI"}2.
Every circle has an equation of this form in the present system of linecoordinates. Notice that the equation of any circle is satisfied by those coordinates of lines which satisfy both x'I+y'n+z'i=o, the equation of its centre, and tat, bn, c;}2 =o. This last equation, of which the lefthand side satisfies the condition for breaking up into two factors, represents the two imaginary circular points at infinity, through which all circles and their asymptotes pass.
There is strict duality in descriptive geometry between pointlinelocus and linepointenvelope theorems. But in metrical geometry duality is encumbered by the fact that there is in a plane one special line only, associated with distance, while of special points, associated with direction, there are two: moreover the line is real, and the points both imaginary.
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 paper taken parallel to the plane zOx, the points of a solid figure being projected on that plane by parallels to some chosen line through 0 in the positive octant. Sometimes it is clearer to delineate, as in fig. 58, by projection parallel to that line in the octant which is equally inclined to Ox, Oy, Oz upon a plane of the paper perpendicular to it. It is possible by parallel projection to delineate equal scales along Ox, Oy, Oz by scales having any ratios we like along lines in a plane having any mutual inclinations we like.
For the delineation of a surface of simple form it frequently suffices to delineate the sections by the coordinate planes; and, in particular, when the surface has symmetry about each coordinate plane, to delineate the
quartersections belonging to a single octant. Thus fig. 59
conveniently represents an octant of the wave surface, which cuts each coordinate plane in a circle and an ellipse. Or we may delineate a series of contour lines, i.e. sections by planes parallel to xOy, or some other chosen plane; of course other sections may be
indicated too for greater clearness. For the delineation of a curve a good, method y
is to represent, as FIG. 59.
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 part indeed of the subject which can be even referred to in the present article.
In the case of a surface we have between the coordinates (x, y, z) a single, or say a onefold relation, which can be represented by a single relation f(x, y, z) =o; or we may consider the coordinates expressed each of them as a given function of two variable parameters p, q; the form z=f(x, y) is a particular case of each of these modes of representation; in other words, we have in the first mode f(x, y, z)=z f(x, y), and in the second mode x=p, y=q for the expression of two of the coordinates in terms of the parameters.
x'
z'
these are expressions for the current coordinates in terms of a parameter p, which is in fact the distance from the fixed point
(a, b, c).
It is easy to see that, if the coordinates (x, y, z) are connected by any two linear equations, these equations can always be brought into the foregoing form, and hence that the two linear equations represent a line.
Secondly, taking for greater simplicity the point Q to be coincident with the origin, and a', p', y', p to be constant, then p is the perpendicular distance of a plane from the origin, and a', /3', 7' are the cosineinclinations of this distance to the axes (a'2+0'2+72=1). P is any point in this plane, and taking its coordinates to be (x, y, z) then (E, n, r) are = (x, y, z), and the foregoing equation p= a E+$'nF
becomes
a'x+/3'y+7'z=p,
which is the equation of the plane in question.
If, more generally, Q is not coincident with the origin, then, taking its coordinates to be (a, b, c), and writing pi instead of p, the equation is
In the case of a curve we have between the coordinates (x, y, z) a twofold relation: two equations f(x, y, z) =0, '(x, y, z) =o give such a relation; i.e. the curve is here considered as the intersection of two surfaces (but the curve is not always the complete intersection of two surfaces, and there are hence difficulties); or, again, the coordinates may be given each of them as a function of a single variable parameter. 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 nonmetrical character of the method of coordinates when used for the proof of a descriptive proposition, apply also to solid geometry; and they might be illustrated in like manner by the instance of the theorem of the radical centre of four spheres. The proof is obtained from the consideration that S and S' being each of them a function of the form x2+y2+z2+ ax+by+cz+d, the difference S—S' is a mere linear function of the coordinates, and consequently that S—S'=o is the equation of the plane containing the circle of intersection of the two spheres S=0 and S'=o.
30. Metrical Theory.—The foundation in solid geometry of the
metrical theory is in fact the beforementioned theorem that if a
finite right line PQ be projected upon any other line 00' by lines perpendicular to 00', then the length of the projection P'Q' is equal to the length of PQ into the cosine of its inclination to P'Q'—or
T (in the form in which it is now convenient to state the theorem) the perpendicular distance P'Q' of two parallel planes is equal to the inclined distance PQ into the cosine of the inclination. The principle of § i6, that the algebraical sum of the projections of the sides of any closed polygon on any line is zero, or that the two sets of sides of the polygon which connect a vertex A and a vertex B have the same sum of projections on the line, in sign and magnitude, as we pass from A to B, is applicable when the sides do not all lie in one plane.
31. Consider the skew quadrilateral QMNP,
the sides QM, MN, NP being respectively
parallel to the three rectangular axes Ox,
Oy, Oz; let the lengths of these sides be
E, n, and that of the side QP be =p; and let the cosines of the inclinations (or say the cosineinclinations) of p to the three axes be a, 13, y; then projecting successively on the three sides and on QP we have
E, n, 3' =Pa. pR, py,
and P = aE+O*1+'Yi',
whence p2=E2}n2+12, which is the relation between a distance p and its projections E, n, t. upon three rectangular axes. And from the same equations we obtain a2+02+72= i, which is a relation connecting the cosineinclinations of a line to three rectangular axes.
Suppose we have through Q any other line QT, and let the cosineinclinations of this to the axes be a', S', y', and S be its cosineinclination to QP; also let p be the length of the projection of QP upon QT; then projecting on QT we have
P=a't+R'n+ti r=Ps.
And in the last equation substituting for i;, n, their values pa, pt3, p7 we find
S =ace +AT +77',
which is an expression for the mutual cosineinclination of two lines, the cosineinclinations of which to the axes are a, 13, y and a', 13', 7' respectively. 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 cosineinclinations (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 coordinates 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 cosineinclinations 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, $, 7, instead of being equal, will only. be proportional, to the cosineinclinations.
Using the' last equation, and writing
x, y, z=a+ap, b+Rp, c+7P,a'(x—a)+$'(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 tii,t2+f2=p2 becomes
(x—a)2+(y—b)2+(z—c)2 =d2,
which is the equation of the sphere, coordinates of the centre = (a,b,c), and radius =d.
A quadric equation wherein the terms of the second order are x2+y2+z2, viz. an equation
x2+y2+z2+Ax+By+Cz+D =o,
can always, it is clear, he brought into the foregoing form; and it thus appears that this is the equation of a sphere, coordinates of the centre 4A, 4C,andsquared radius =.ay(A2+B2+C2)—D.
33. 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 cover a surface, and the equation of the surface is obtained by eliminating the parameter between the two equations.
If the lines all pass through a given point, then the surface is a cone; and, in particular, if the lines are all parallel to a,given line, then the surface is a cylinder.
Beginning with this last case, suppose the lines are parallel to the line x=mz, y=nz, the equations of a line of the system are x=mz+a, y=nz+b,—where a, b are supposed to be functions of the variable parameter, or, what is the same thing, there is between them a relation f(a, b) =o: we have a=x—mz, b=y—nz, and the result of the elimination of the parameter therefore is f(x—mz, y—nz)=o, which is thus the general equation of the cylinder the generating lines whereof are parallel to the line x=mz, y=nz. The equation of the section by the plane z=o is f(x, y) =o, and conversely if the cylinder be determined by means of its curve of intersection with the plane z=o, then, taking the equation of this curve to be f(x, y) =o, the equation of the cylinder is f(x—mz, y—nz) =0. Thus, if the curve of intersection be the circle (x—a)2+(y—S)2=y2, we have (x—mz—a)2+(y—nz—/3)2=y2 as the equation of an oblique cylinder on this base, and thus also (x—a)2+(y—ti')2='Y2 as the equation of the right cylinder.
If the lines all pass through a given point (a, b, c), then the equations of a line are x—a=a(z—c), y—b=,B(z—c), where a, S are functions of the variable parameter, or, what is the same thing, there exists between them an equation f(a, 0) =0; the elimination
of the parameter gives,' therefore, f (Z=d, Z_b) =o; and this
equation, or, what is the same thing, any homogeneous equation f(x—a, y—b, z—c)=o, or, taking f to be a rational and integral function of the order n, say (*) (x—a, y—b, z—c),=o, is the general equation of the cone having the point (a, b, c) for its vertex. Taking the vertex to be at the origin, the equation is (*) (x, y, z)"=o; and, in particular, (*) (x, y, z)2=0 is the equation of a cone of the second order, or quadricone, having the origin for its vertex.
34. In the general case of a singly infinite system of lines, the locus is a ruled surface (or regulus). Now, when a line is changing its position in space, it may be looked upon as in a state of turning about some point in itself, while that point is, as a rule, in a state of moving out of the plane in which the turning takes place. If instantaneously it is only in a state of turning, it is usual, though not strictly accurate, to say that it intersects its consecutive position. A regulus such that consecutive lines on it do not intersect, in this sense, is called a skew surface, or scroll; one on which they do is called a developable surface or terse.
Suppose, for instance, that the equations of a line (depending on
the variable parameter 0) are ..+Ys0 c (I +b) ¢ c 6—Z)
'
then, eliminating B, we have a2 i = 1—y, or say a2 + b —?= I, the equation of a quadric surface, afterwards called the hyperboloid of one sheet; this surface is consequently a scroll. It is to be re
marked that we have upon the surface a second singly infinite series of lines; the equations of a line of this second system (depending on the variable parameter 0) are
a+c—~(I b)' a c(I+b)
It is easily shown that any line of the one system intersects every line of the other system.
Considering any curve (of double curvature) whatever, the tangent lines of the curve form a singly infinite system of lines, each line intersecting the consecutive line of the system,—that is, they form a developable, or torse; the curve and torse are thus inseparably connected together, forming a single geometrical figure. An osculating plane of the curve (see § 38 below) is a tangent plane of the torse all along a generating line.
35. Transformation of Coordinates.—There is no difficulty in changing the origin, and it is for brevity assumed that the origin remains unaltered. We have, then, two sets of rectangular axes, Ox, Oy, Oz, and Oxi, Oyl, Ozi, the mutual cosineinclinations being
shown by the diagram x y z
xi a 0 7
yl A' 7'
zi a" 0" y"
that is, a, fi, y are the cosineinclinations of Oxi to Ox, Oy, Oz; a', 13', y' those of Oyl, &c.
And this diagram gives also the linear expressions of the coordinates (x1, y', zi) or (x, y, z) of either set in terms of those of the other set; we thus have
xl=a x+R y+y z, x=axi+ayi+a"zi,
yi=a'x+l;'y+y'z, y=Qxi+0'yi+0"zi,
z1=a"x+ry+y"z, z=yx1 +7'yi+7"zi,
which are obtained by projection, as above explained. Each of these equations is, in fact, nothing else than the beforementioned equation p=a't+,',t+y'i', adapted to the problem in hand.
But we have to consider the relations between the nine coefficients. By what precedes, or by the consideration that we must have identically x2+y2+z2=x12+y22+z12, it appears that these satisfy the relations
a2 +02 .+72 =1, aa1+aa'2 +a"2 =1,
a 2 +$'2 +7'21 =1, Y +N'2 +Y "2 =1,
+7"2 =1, 72+y'2 +7"2 =1,
a'a"+,a'$'+7'y"=0, 07 +I0'y'+0"7"=0,
a"a +{4"0 +y"y =0, ya+7'a'+y"a"=0,
aa' +0 +77' =0, =0, either set of six equations being implied in the other set.
It follows that the square of the determinant
la , , I
a", 9", y"
is= I ; and hence that the determinant itself is= t I. The distinction of the two cases is an important one: if the determinant is = +1, then the axes Oxi, Oyl, Ozi are such that they can by a rotation about 0 be brought to coincide with Ox, Oy, Oz respectively; if it is=—1, then they cannot. But in the latter case, by measuring x1, yl, z1 in the opposite directions we change the signs of all the coefficients and so make the determinant to be =+1; hence the former case need alone be considered, and it is accordingly assumed that the determinant is = +1. This being so, it is found that we have the equality a=fi'y"—$"y', and eight like ones, obtained from this by cyclical interchanges of the letters a, 13, 7, and of unaccented, singly and doubly accented letters.
36. The nine cosineinclinations above are, as has been seen, connected by six equations. It ought then to be possible to express them all in terms of three parameters. An elegant means of doing this has been given by Rodrigues, who has shown that the tabular express
+(I +X2+1i2+v2),
the meaning being that the coefficients in the transformation are fractions, with numerators expressed as in the table, and the common denominator.
37. Tke Species of Quadric Surfaces. Surfaces represented by equations of the second degree are called quadric surfaces. Quadric surfaces are either proper or special. The special ones arise when the coefficients in the general equation are limited to satisfy certain special equations; they. comprise (I) planepairs, 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++=', ellipsoid.
x2 y2 z2
(4) 5,1+y—Ti =1, hyperboloid of one sheet.
x2 y2 zz
(5) a+L22=—I,_hyperboloid of two sheets.
It is at once seen that these are distinct surfaces; and the equations also show very readily the general form and mode of generation of the several surfaces.
In the elliptic paraboloid (fig. 61) the sections by the planes of zx and zy are the parabolas
. x2 y2
z=2a,z gib,
having the common axes Oz; and the section by any plane z=y parallel to that of xy is the ellipse
x2 y2
y 2a+2b'
so that the surface is generated by Fm. 6i.
a variable ellipse moving parallel to itself along the parabolas as directrices.
In the hyperbolic paraboloid (figs. 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 saddle.
In the ellipsoid (fig. 64) the sections by the planes of zx, zy, and xy are each of them an ellipse, and the section by any parallel plane is also an ellipse. The surface may be considered as generated by
an ellipse moving parallel to itself along two ellipses as directrices.
x y z
x, I+X2—u2—v2 2(X z—v) 2(vX+1+)
y: 2(X1++v) I ._x2+112—v2 2(F+v+X)
z, 2(vT—µ) 2(µv+X) I —X2'µ2
l'0
In the hyperboloid of one sheet (fig. 65), the sections by the planes of zx, zy are the hyperbolas
x2_cz=1, —c2=1,
having a common conjugate axis zOz'; the section by the plane of x, y, and that by any parallel plane, is an ellipse; and the surface may be considered as generated by a variable ellipse moving parallel to itself along the two hyperbolas as directrices. If we imagine two equal and parallel circular disks, their points connected by strings of equal lengths, so that these are the generators of a right circular cylinder, and if we turn one of the disks about its centre through an angle in its plane, the strings in their new positions will be one system of generators of a hyperboloid of one sheet, for which a=b; and if we turn it through the same angle in the opposite direction,
a
we get in like manner the generators of the other system; there will be the same general configuration when a+b. The hyperbolic paraboloid is also covered by two systems of rectilinear generators as a method like that used in § 34 establishes without difficulty. 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
c2a2 =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'(ny)/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'(tx)[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 size and position. The limit tended to is the sphere of closest contact with the curve at x, y, z); its centre and radius are called the centre and radius of spherical curvature. It cuts the osculating plane in a circle, called the circle of absolute curvature; and the centre and radius of this circle are the,centre and radius of absolute curvature. The centre ofabsolute curvature is the limiting position of the point where the principal normal at (x, y, z) is cut by the normal plane at a neighbouring point, as that point moves up to (x, y, z).
39. Differential Geometry of Surfaces.—Let (x, y, z) be any chosen point on a surface f(x, y, z) =o. As a second point of the surface moves up to (x, y, z), its connector with (x, y, z) tends to a limiting position, a tangent line to the surface at (x, y, z). All these tangent lines at (x, y, z), obtained by approaching (x, y, z) from different directions on a surface, lie in one plane
axx)+ay(+1—y)+az(i—z)=0.
This plane is called the tangent plane at (x, y, z). 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 node or double point at (x, y, z). Two of the tangent lines touch this curve at the node. They are called the " chief tangents " (Haupttangenten) at (x, y, z) ; they have closer contact with the surface than any other tangents.
In the case of a quadric surface the curve of intersection of a tangent and the surface is of the second order and has a node, it must therefore consist of two straight lines. Consequently a quadric surface is covered by two sets of straight lines, a pair through every point on it; these are imaginary for the ellipsoid, hyperboloid of two sheets, and elliptic paraboloid.
A surface of any order is covered by two singly infinite systems of curves, a pair through every point, the tangents to which are all chief tangents at their respective points of contact. These are called chieftangent curves; on a quadric surface they are the above straight lines.
40. The tangents at a point of a surface which bisect the angles between the chief tangents are called the principal tangents at the point. They are at right angles, and together with the normal constitute a convenient set of rectangular axes to which to refer the surface when its properties near the point are under discussion. At a special point which is such that the chief tangents there run to the circular points at infinity in the tangent plane, the principal tangents are indeterminate; such a special point is called an umbilic of the surface.
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 chieftangents 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 chieftangents are imaginary; in the latter case a and b have opposite signs, the paraboloid is hyperbolic, and the chieftangents 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 (Meunier's theorem) that the radius of curvature of this inclined section of the surface is ap cos ch.
Arthur Cayley in the 9th edition of this work. Of early and im
portant recent publications on analytical geometry, special mention
II
is to be made of R. Descartes, Geometrie (Leyden, 1637) ; John Wallis, Tractatus de sectionibus conicis nova methodo expositis (1655, Opera mathematica, i., Oxford, 1695); de l'Hospital, Traite analytique des sections conigues (Paris, 1720) ; Leonhard Euler, Introductio in analysin infinitorum, ii. (Lausanne, 1748) ; Gaspard Monge, " Application d'algebre a la geometrie " (Journ. Ecole Polytech., 1801) ; Julius Plucker, Analytischgeometrische Entwickelungen, 3 Bde. (Essen, 1828—1831) ; System der analytischen Geometric (Berlin, 1835); G. Salmon, A Treatise on Conic Sections (Dublin, 1848; 6th ed., London, 1879) ; Ch. Briot and J. Bouquet, Lecons de geometric analytique (Paris, 1851; 16th ed., 1897) ; M. Chasles, Trait: de geometrie superieure (Paris, 1852); Wilhelm Fiedler, Analytische Geometric 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 Geometric (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. ClebschF. Lindemann, Vorlesungen caber Geometrie, Bd. i. (Leipzig, 1876, 2te Aufl., 1891) ; R. Baltser, Analytische Geometric (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); SalmonFiedler, 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 Encyklopadie der mathematischen Wissenschaften, Bd. in. 1, 2.
(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 element. Just as ordinary geometry deals primarily with points and systems of points, this theory deals in the first instanceare 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 Its, n2, ns, 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 2nlnin3n4. 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. 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 whichever 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 nulplane of that point, and those lying in a fixed plane Linear pass through a point called the nulpoint or pole of the complex. plane. If the nulplane of A pass through B, then the
nulplane of B will pass through A; the nulplanes 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. 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 nulplane of any point 0, 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 0 to meet them is therefore a ray of the complex; similarly, by choosing another four we can find ancther ray through O: these rays lie in the nulplane, 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 nulpoint 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 nulplane of any point on it is perpendicular to it. This is called the central axis; if d be the shortest distance, a the angle between it and a ray of the complex, then d tan 0 = 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 nulmoment
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 x,y1z,w1 and x2y2z2w2 are two points on the line, it is easily verified that the six determinants of the array
1 x1y1z1w1 x2y2z2w2 are in the same ratios for all pointpairs 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 1, X, m, ,u, n, v where l=x,w2—x2w1, X=yiz2—y2z1, &c. There is the further advantage that if a1bicid, and a2b2c2d2 be two planes through the line, the six determinants
alblc,d,
a2b2c2d2
are in the same ratios as the foregoing, so that except as regards a
factor of proportionality we have x=b,c2—b2c,, l=cld2—c2d,, &c.
The identical relation la+mo+nv=o reduces the number of inde
pendent 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
la'+l'X+tny'+m'µ+nv'+n'v =o
where the accented letters refer to the second line. If the coordinates are Cartesian and 1, 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 triplyinfinite,•adoublyinfinite and a singlyinfinite number of lines; they
722
for the system form a linear complex of which the given line is the central axis and .the quotient G/R is the pitch. Any motion of a rigid body can be reduced to a screw motion about a certain line, i.e. to an angular velocity w about that line combined with a linear velocity u along the line. The plane drawn through any point perpendicular to the direction of its motion is its nulplane with respect to a linear complex having this line for central axis, and the quotient u/co for pitch (cf. Sir R. S. Ball, Theory of Screws).
The following are some properties of a configuration of two linear complexes:
The lines common to the twocomplexes also belong to an infinite number of linear complexes, of which two reduce to single straight lines. These two lines are conjugate lines with respect to each of the complexes, but they may coincide, and then some simple modifications are required. The locus of the central axis of this system of complexes is a surface of the third degree called the cylindroid, which plays a leading part in the theory of screws as developed synthetically by Ball. Since a linear complex has an invariant of the second degree in its coefficients, it follows that two linear complexes have a lineolinear invariant. This invariant is fundamental : if the complexes be both straight lines, its vanishing is the condition of their intersection as given above; if only one of them be a straight line, its vanishing is the condition that this line should belong to the other complex. When it vanishes for any two complexes they are said to be in involution or ¢polar; the nulpoints P, Q of any plane then divide harmonically the points in which the plane meets the common conjugate lines, and each complex is its own reciprocal with respect to the other. As regards a configuration of these linear complexes, the common lines from one system of generators of a quadric, and the doubly infinite system of complexes containing the common lines, include an infinite number of straight lines which form the other system of generators of the same quadric.
If the equation of a linear complex is AI+Bm+Cn+DX+E,i+ Fv=o, then for a line not belonging to the complex we may regard
General the expression on the lefthand side as a multiple of the
line co moment of the line with respect to the complex, the word ordinates. moment being used in the statical sense; and we infer
that when the coordinates are replaced by linear functions of themselves the new coordinates are multiples of the moments of the line with respect to six fixed complexes. The essential features of this coordinate system are the same as those of the original one, viz. there are six coordinates connected by a quadratic equation, but this relation has in general a different form. 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 tetrahedron.
The system of coordinates referred to six complexes mutually in involution was introduced by Felix Klein, and in many cases is more useful than that derived directly from point coordinates; e.g. in the discussion of quadratic complexes: by means of it Klein has developed an analogy between line geometry and the geometry of spheres as treated by G. Darboux and others. In fact, in that geometry a point is represented by five coordinates, connected by a relation of the same type as the one just mentioned when the five fundamental spheres are mutually at right angles and the equation of a sphere is of the first degree. Extending this to four dimensions of space, we obtain an exact analogue of line geometry, in which (i.) a point corresponds to a line; (ii.) a linear complex to a hypersphere; (iii.) two linear complexes in involution to two orthogonal hyperspheres; (iv.) a linear complex and 'two conjugate lines to a hypersphere and two inverse points. Many results may be obtained by this principle, and more still are suggested by trying to extend the properties of circles to spheres in three and four dimensions. Thus the elementary theorem, that, given four lines, the circles circumscribed to the four triangles formed by them are concurrent, may be extended to six. hyperplanes in four dimensions; and then we can derive a result in line geometry by translating the inverse of this theorem. Again, just as there is an infinite number of spheres touching a surface at a given point, two of them having contact of a closer nature, so there is an infinite number of linear complexes touching a nonlinear complex at a given line, and three of these have contact of a closer nature (cf. Klein, Math. Ann. v.).
Sophus Lie has pointed out a different analogy with sphere geometry. 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'—r2de.=o, and it is easy to see that two spheres touch, if
[LINE
the polar form 2aatFabb1+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 light by admitting the ideas of space of four and five dimensions.
Thus, regarding the coordinates of a line as homogeneous coordinates in five dimensions, we may say that line geometry is equivalent to geometry on a quadric surface in five dimensions. A linear complex is represented by a hyperplane section; and if two such complexes are in involution, the corresponding hyperplanes are conjugate with respect to the fundamental quadric. By projecting this quadric stereographically into space of four dimensions we obtain Klein's analogy. In the same way geometry in a linear complex is equivalent to geometry on a quadric in four dimensions; when two lines intersect the representative points are on the same generator of this quadric. Stereographic projection, therefore, converts a curve in a linear complex, i.e. one whose tangents all belong to the complex, into one whose tangents intersect a fixed conic: when this conic is the imaginary circle at infinity the curve is what Lie calls a minimal curve. Curves in a linear complex have been extensively studied. The osculating plane at any point of such a curve is the nulplane 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, NonLinear 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(ni)', 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 complexcones of all points on 1 touch a and the complexcurves of all planes through t touch 1 at A. It follows that 1 is a double line of the complexcone of A, and a double tangent of the complexcurve 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 hypersurface 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 account of the lines of a quadratic complex that meet a given line.
The cones whose vertices are on the given line all pass through eight fixed points and envelop a surface of the fourth degree; the conics whose planes contain the given line all lie on a surface of the fourth class and touch eight fixed planes. It is easy to see by elementary geometry that these two surfaces are identical. Further, the given line contains four singular points Al, A2, A3, A4, and the planes into which their cones degenerate are the eight common tangent planes mentioned above; similarly, there are four singular planes, al, a2, a3, a4, through the line, and the eight points into which their conics degenerate are the eight common points above. The locus of the pole of the line with respect to all the conics in planes through it is a straight line called the polar line of the given one; and through this line passes the polar plane of the given line with respect to each of the cones. The name polar is applied in the ordinary analytical sense; any line has an infinite number of polar complexes with respect to the given complex, for the equation of the latter can be written in an infinite number of ways; one of these polars is a straight line, and is the polar line already introduced. 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 cross ratio as that of the range in which the Plucker surface of li cuts 1. Applying this to the case in which 11 is the polar of 1, we find that the cross ratios of (A1, A2, A3, A4) and (ai, a2, a3, a4) are equal. The identity of the locus of the A's with the envelope of the a's follows at once; moreover, a line meets the singular surface in four points having the same cross ratio as that of the four tangent planes drawn through the line to touch the surface. The Plucker surface has eight nodes, eight singular tangent planes, and is a double line. The relation between a line and its polar line is not a reciprocal one with respect to the complex; but W. Stahl has pointed out that the relation is reciprocal as far as the singular surface is concerned.
To facilitate the discussion of the general quadratic complex we Quadratk introduce Klein's canonical form. We have, in fact, to complxes. deal with two quadratic equations in six variables; and by
suitable linear transformations these can be reduced to the form
a1x12 +a2x22+a3x32±a4x42+asx52+a6x62 = o
x12 + x22+ x32+ x42+ x52+ x52=o
subject to certain exceptions, which will be mentioned later.
Taking the first equation to be that of the complex, we remark that both equations are unaltered by changing the sign of any coordinate; the geometrical meaning of this is, that the quadratic complex is its own reciprocal with respect to each of the six fundamenial complexes, for changing the sign of a coordinate is equivalent to taking the polar of a line with respect to the corresponding fundamental complex. It is easy to establish the existence of six systems of bitangent linear complexes, for the complex llxi 12x2+13x3+14x4+15x5+lex6=o is a bitangent when
11=0, and 122 + 132 + 142 + 152 + 152 =0, a2ai a3a1 a4a1 a5a2 a5a1
and its lines of contact are conjugate lines with respect to the first fundamental complex. We therefore infer the existence of six systems of bitangent lines of the complex, of which the first is given by
xi=o, x22 x32 +x42 + x52 + zr2 =o.
a2–ai a3–ai a4–al a5–a1 a5–a1
Each of these lines is a bitangent of the singular surface, which is therefore completely determined as.being the focal surface of the (2, 2) congruence above. 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 hyperelliptic functions (cf. R. W. H. T. Hudson, Kammer's Quartic, 1905).
As might be expected from the magnitude of a form in six variables, the number of projectivally distinct varieties of quadratic complexes Classiflca is very great; and in fact Adolf Weiler, by whom the cl a of question was first systematically studied on lines indicated quadratic by Klein, enumerated no fewer than fortynine different complexes, types. But the principle of the classification is so im
portant, and withal so simple, that we give a brief sketch which indicates its essential features.
We have practically to study the intersection of two quadrics F and F' in six variables, and to classify the different cases arising we make use of the results of Karl Weierstrass on the equivalence conditions of two pairs of quadratics. As far as at present required, they are as follows: Suppose that the factorized form of the determinantal equation Disct (F+XF') =o is
(A–a)'1F'2F'2...(~_ll)~i F°2+ia f
where the root a occurs si}s2+s3 . . . times in the determinant, s2+s3 . . . times in every first minor, s3+ . . . times in every second minor, and so on; the meaning of each exponent is then perfectly definite. Every factor of the type (X. a)' is called an elementartheil (elementary divisor) of the determinant, and the condition of equivalence of two pairs of quadratics is simply that their determinants have the same elementary divisors. We write the pair of forms symbolically thus [(sis2 ...), (t1t2 ...), . . .], letters in the inner brackets referring to the same factor. Returning now to the two quadratics representing the complex, the sum of the exponents will be six, and two complexes are put in the same class if they have the same symbolical expression; i. e. the actual values of the roots of the determinantal equation need not be the same for both, but their manner of occurrence, as far as here indicated, must be identical in the two. 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 ill]; the complex of tangents to a quadric is [(iii) (iii)] and that of lines meeting a conic is [(222)]. Full information will be found in Weiler's memoir, Math.
Ann. vol. vii.
The detailed study of each variety of complex opens up a vast subject; we only mention two special cases, the harmonic complex and the tetrahedral complex.
The harmonic complex, first studied by Battaglini, is generated in an infinite number of ways by the lines cutting two quadrics harmonically. Taking the most general case, and referring the quadrics to their common selfconjugate tetrahedron, we can find its equation in a simple form, and verify that this complex really depends only on seventeen constants, so that it is not the most general quadratic complex. It belongs to the general type in so far as it is discussed above, but the roots of the determinant are in involution. The singular surface is the " tetrahedroid " discussed by Cayley. As a particular case, from a metrical point of view, we have L. F. Painvin's complex generated by the lines of intersection of perpendicular tangent planes of a quadric, the singular surface now being Fresnel's wave surface. 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 family of confocal quadrics, and for many investigations it is convenient to deal with this complex referred to the principal axes. For example, Lie has developed the theory of curves in a Reye complex (i.e. curves whose tangents belong to the complex) as solutions of a differential equation of the form (b–c)xdydz+(c–a)ydzdx+(a–b)zdxdy=o, and we can simplify this equation by a logarithmic transformation. Many theorems connecting complexes with differential equations have been given by Lie and his school. A line complex, in fact, corresponds to a Mongian equation having oo 3 line integrals.
As the coordinates of a line belonging to a congruence are functions of two independent parameters, the theory of congruences is analogous to that of surfaces, and we may regard it as a fundamental Congrainquiry to find the simplest form of surface into which en4es
a given congruence can be transformed. Most of those
whose properties have been extensively discussed can be represented on a plane by a birational transformation. But in addition to the difficulties of the theory of algebraic surfaces, a subject still in its infancy, the theory of congruences has other difficulties in that a congruence is seldom completely represented, even by two equations.
A fundamental theorem) is that the lines of a congruence are in general bitangents of a surface; in fact, since the condition of intersection of two consecutive straight lines is ldX+dmdu+dndv=o, a line 1 of the congruence meets two adjacent lines, say 11 and 12. Suppose 1, 11 lie in the plane pencil (Alai) and 1, 12 in the plane pencil (A2a2), then the locus of the A's is the same as the envelope of the a's, but a2 is the tangent plane at Al and a1 at A2. This surface is called the focal surface of the congruence, and to it all the lines I are bitangent. The distinctive property of the points A is that two of the congruence lines through them coincide, and in like manner the planes a each contain two coincident lines. The focal surface consists of two sheets, but one or both may degenerate into curves;
thus, for example, the normals to a surface are bitangents of the surface of centres, and in the case of Dupin's cyclide this surface degenerates into two conics.
In the discussion of congruences it soon becomes necessary to introduce another number r, called the rank, which expresses the number of plane pencils each of which contains an arbitrary line and two lines of the congruence. The order of the focal surface is 2m(nI)2r, and its class is m(mI)2r. Our knowledge of congruences is almost exclusively confined to those in which either m or n does not exceed two. We give a brief account of those of the second order without singular lines, those of order unity not being especially interesting. A congruence generally has singular points through which an infinite number of lines pass; a singular point is said to be of order r when the lines through it lie on a cone of the rth degree. By means of formulae connecting the number of singular points and their orders with the class m of quadratic congruence Kummer proved that the class cannot exceed seven. The focal surface is of degree four and class 2m; this kind of quartic surface has been extensively studied by Kummer, Cayley, Rohn and others. The varieties (2, 2), (2, 3), (2, 4), (2, 5) all belong to at least one Reye complex; and so also does the most important class of (2, 6) congruences which includes all the above as special cases. The congruence (2, 2) belongs to a linear complex and forty different Reye complexes; as above remarked, the singular surface is Kammer's sixteennodal quartic, and the same surface is focal for six different congruences of this variety. The theory of (2, 2) congruences is completely analogous to that of the surfaces called cyclides in three dimensions. Further particulars regarding quadratic congruences will be found in Kummer's memoir of 1866, and the second volume of Sturm's treatise. The properties of quadratic congruences having singular lines, i.e. degenerate focal surfaces, are not so interesting as those of the above class; they have been discussed by Kummer, Sturm and others.
Since a ruled surface contains only l elements, this theory is
practically the same as that of curves. If a linear complex contains
Ruled more than n generators of a ruled surface of the nth degree,
surfaces. it contains all the generators, hence for n=2 there are
three linearly independent complexes, containing all the generators, and this it a wellknown property of quadric surfaces. In ruled cubics the generators all meet two lines which may or may not coincide; these two cases correspond to the two main classes of cubics discussed by Cayley and Cremona. As regards ruled quartics, the generators must lie in one and may lie in two linear complexes. The first class is equivalent to a quartic in four dimensions and is always rational, but the latter class has to be subdivided into the elliptic and the rational, just like twisted quartic curves. A quintic skew may not lie in a linear complex, and then it is unicursal, while of sextics we have two classes not in a linear complex, viz. the elliptic variety, having thirtysix places where a linear complex contains six consecutive generators, and the rational, having six such places.
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 spacecurve 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 = (lull+mlm2 I nln2 — rir2)/{ (l12 I miln12 — 1'12)
(122+m22+n22 — r22)14 (2)
These planes only have a real angle of inclination if they possess a line of intersection within the actual space, i.e. if they Intersect. Planes which do not intersect possess a shortest distance along a line which is perpendicular to both of them. If this shortest distance is 012, we have
cosh (012/y) = (1112+mim2 1 nin2 — rlr2)/{(42Tm12+ni2 — r12)
(122 m22+ n22 — r22)I (3)
Thus in the case of the two planes one and only one of the two, 012 and an, is real. The same considerations hold for coplanar straight lines (see VII. Axioms of Geometry). Let 0 (fig. 67) be the point (o, o, o, I), OX the line y=o,
z=o, OY the line z=o, x=o, and OZ the line x=o, y=o. These are the coordinate axes and are at right angles to each other. 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 coordinates 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(xABC). 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. NONEUCLIDEAN GEOMETRY
The various metrical geometries are concerned with the properties of the various types of congruencegroups, which are defined in the study of the axioms of geometry and of their immediate consequences. But this point of view of the subject is the outcome of recent research, and historically the subject has'a different origin. NonEuclidean geometry arose from the discussion, extending from the Greek period to the present day, of the various assumptions which are implicit in the traditional Euclidean system of geometry. In the course of these investigations it became evident that metrical geometries, each internally consistent but inconsistent in many respects with each other and with the Euclidean system, could be developed. A short historical sketch will explain this origin of the subject, and describe the famous and interesting progress of thought on the subject. But previously a description of the chief characteristic properties of elliptic and of hyperbolic geometries will be given, assuming the standpoint arrived at below under VII. Axioms of Geometry.
First assume the equation to the absolute (cf. loc. cit.) to be w2x2y2z2=o. 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 parallelism.'
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+dw)' . . (9).
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 limitsurfaces. Thus (9) denotes a limitsurface, if d2—a'—b2—c'=o. Two limitsurfaces only differ in position. Thus the two limitsurfaces which touch the plane YOZ at 0, but have their concavities turned in opposite directions, have as their equations
w2x2y2z2=(wtx)2.
The geodesic geometry of a sphere is elliptic, that of a surface of equal distance is hyperbolic, and that of a limitsurface 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 die, the other is Try—die. Every straight line returns into itself, forming a closed series. Thus there are two segments between any two points, together forming the whole line which contains them; one distance is associated with one segment, and the other distance with the other segment. The complete length of every straight line is ley.
The angle between the two planes llx+mly+nlz+rlw=o and 12x +m2y +n2z + r2w = o is
cos 812=(1112+mim2+ninz+r,r,)/{(ll' hm12 Fni2 } r12) (122I+22+n22+r22)}+ (II).
The polar plane with respect to the absolute of the point (xi, yi, zl, wi) is the real plane xix+yly+ziz+wiw=o, and the pole of the plane lix+m,y+niz+riw=o is the point (li, m1, ni, ri). Thus (from equations lo and ii) it follows that the angle between the polar planes of the points (x1,...) and (x2,...) is die/y, and that the distance between the poles of the planes (li,...) and (12,...) is y0i2. Thus there is complete reciprocity between points and planes in respect to all properties. This complete reign of the principle of duality is one of the great beauties of this geometry. The theory of lines and planes at right angles is simply the theory of conjugate elements with respect to the absolute. A tetrahedron selfconjugate with respect to the absolute has all its intersecting elements (edges and planes) at right angles. 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. Whitehead, Universal Algebra, Bk. vi. (Cambridge, 1898).by the plane p, but P and R are not separated by p, nor are Q and R.
Let A, B, C be any three noncollinear points, then four triangles are defined by these points. 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, Tryc, 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 watch is laid on the plane OXY, face upwards with its centre at 0, and the line 12 to 6 (as marked on dial) along the line YOY. 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, 1lacy—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. Clifford. The two lines are called Clifford's right and left parallels to 1 through the point. This property of parallelism is reciprocal, so that if m is a left parallel to 1, then l is a left parallel to m. Note also that two parallel lines 1 and m are not coplanar. Many of those properties of Euclidean parallels, which do not hold for Lobatchewsky's parallels in hyperbolic geometry, do hold for Clifford's parallels in elliptic geometry. The geodesic geometry of spheres is elliptic, the geodesic geometry of surfaces of equal distance from lines (cylinders) is Euclidean, and surfaces of revolution can be found' of which the geodesic geometry is hyperbolic. But it is to be noticed that the connectivity of these surfaces is different to that of a Euclidean plane. For instance there are only 002 congruence transformations of the cylindrical surfaces of equal distance into themselves, instead of the 00 3 for the ordinary plane. It would obviously be possible to state " axioms " which these geodesics satisfy, and thus to define independently, and not as loci, quasispaces of these peculiar types. The existence of such Euclidean quasigeometries was first pointed out by Clifford.'
In both elliptic and hyperbolic geometry the spherical geometry, i.e. the relations between the angles formed by lines and planes passing through the same point, is the same as the " spherical trigonometry " in Euclidean geometry. The constant y, which appears in the formulae both of hyperbolic and elliptic geometry, does not by its variation produce different types of geometry. There is only one type of elliptic geometry and one type of hyperbolic geometry; and the magnitude of the constant y in each case simply depends upon the magnitude of the arbitrary unit of length in comparison with the natural unit of length
' Cf. A. N. Whitehead, loc. cit.
'Cf. A. N. Whitehead, " The Geodesic Geometry of Surfaces in nonEuclidean Space," Proc. Lond. Math. Soc. vol. xxix.
' Cf. Klein, " Zur nichtEuklidischen Geometrie," Math. Annal. vol. xxxvii.
and
which each particular instance of either geometry presents. equal. The first hypothesis is that these are both right angles; The existence of a natural unit of length is a peculiarity common
both to hyperbolic and elliptic geometries, and differentiates them from Euclidean geometry. It is the reason for the failure of the theory of similarity in them. If y is very large, that is, if the natural unit is very large compared to the arbitrary unit, and if the lengths involved in the figures considered are not large compared to the arbitrary unit, then both the elliptic and hyperbolic geometries approximate to the Euclidean. For from formulae (4) and (5) and also from (12) and (13) we find, after retaining only the lowest powers of small quantities, as the formulae for any triangle ABC,
a/ sin A=b/ sin B=c/ sin C,
and
a'=b2+c2—2bc cos A,
with two similar equations. Thus the geometries of small figures are in both types Euclidean.
History.—" In pulcherrimo Geometriae corpore," wrote Sir Henry Savile in 1621, " duo sunt naevi, duae labes nec quod Theory of sciam plures, in quibus eluendis et emaculendis cum parallels veterum turn recentiorum . . . vigilavit industria."
before These two blemishes are the theory of parallels and
Gauss. the theory of proportion. The " industry of the
moderns," in both respects, has given rise to important branches of mathematics, while at the same time showing that Euclid is in these respects more free from blemish than had been previously credible. It was from endeavours to improve the theory of parallels that nonEuclidean geometry arose; and though it has now acquired a far wider scope, its historical origin remains instructive and interesting. Euclid's " axiom of parallels " appears as Postulate V. to the first book of his Elements, and is stated thus, " And that, if a straight line falling on two straight lines make the angles, internal and on the same side, less than two right angles, the two straight lines, being produced indefinitely, meet on the side on which are the angles less than two right angles." The original Greek is Kai Eiv as no ei'Oefas eb0tia i,u ri rrovra TQS Evros Kai hri T a abra µEprl yowias no bpBwv EXaQQOVas srotO, EK/3aXAou&as Tas %o evOeias Ea' a1retpov ovµaiarewv, E4' & µEpf doiv al Ti;'.'v
OpO&av EXOo roves.
To Euclid's successors this axiom had signally failed to appear selfevident, 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 discovery of equivalent axioms. Christoph Clavius in 1574 deduced the axiom from the assumption that a line whose points are all equidistant from a straight line is itself straight. John Wallis in 1663 showed that the postulate follows from the possibility of similar triangles on different scales. Girolamo Saccheri (1933) showed that it is sufficient to have a single triangle, the sum of whose angles is two right angles. Other equivalent forms may be obtained, but none shows any essential superiority to Euclid's. Indeed plausibility, which is chiefly aimed at, becomes a positive demerit where it conceals a real assumption.
A new method, which, though it failed to lead to the desired
goal, proved in the end immensely fruitful, was invented by
Saccheri Saccheri, in a work entitled Euclides ab omni naevo
vindicatus (Milan, 1733). If the postulate of parallels
is involved in Euclid's other assumptions, contradictions must
emerge when it is denied while the others are maintained. This
led Saccheri to attempt a reductio ad absurdum, in which he
mistakenly believed himself to have succeeded. What is interest
ing, however, is not his fallacious conclusion, but the non
Euclidean results which he obtains in the process. 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 nonEuclidean geometry only because he does not perceive the possible truth of his nonEuclidean hypotheses.
Some advance is made by Johann Heinrich Lambert in his Theorie der Parallellinien (written 1766; posthumously published 1786). Though he still believed in the necessary Lambert truth of Euclidean geometry, he confessed that, in
all his attempted proofs, something remained undemonstrated. He deals with the same three hypotheses as Saccheri, showing that the second holds on a sphere, while the third would hold on a sphere of purely imaginary radius. The second hypothesis he succeeds in condemning, since, like all who preceded Bernhard Riemann, he is unable to conceive of the straight line as finite and closed. But the third hypothesis, which is the same as Lobatchewsky's, is not even professedly refuted.'
NonEuclidean geometry proper begins with Karl Friedrich Gauss. The advance which he made was rather philosophical than mathematical: it was he (probably) who first Three recognized that the postulate of parallels is possibly periods of false, and should be empirically tested by measuring nonthe angles of large triangles. The history of non Luctfaeea Euclidean geometry has been aptly divided by Felix geometry. Klein into three very distinct periods. The first—which contains only Gauss, Lobatchewsky and Bolyai—is characterized by its synthetic method and by its close relation to Euclid. . The attempt at indirect proof of the disputed postulate would seem to have been the source of these three men's discoveries; but when the postulate had been denied, they found that the results, so far from showing contradictions, were just as selfconsistent as Euclid. They inferred that the postulate, if true at all, can only be proved by observations and measurements. Only one kind of nonEuclidean space is known to them, namely, that which is now called hyperbolic. The second period is analytical, and is characterized by a close relation to the theory of surfaces. It begins with Riemann's inaugural dissertation, which regards space as a particular case of a manifold; but the characteristic standpoint of the period is chiefly emphasized by Eugenio Beltrami. The conception of measure of curvature is extended by Riemann from surfaces to spaces, and a new kind of space, finite but unbounded (corresponding to the second hypothesis of Saccheri and Lambert), is shown to be possible. As opposed to the second period, which is purely metrical, the third period is essentially projective in its method. It begins with Arthur Cayley, who showed that metrical properties are projective properties relative to a certain fundamental quadric, and that different geometries arise according as this quadric is real, imaginary or degenerate. Klein, to whom the development of Cayley's work is due, showed further that there are two forms of Riemann's space, called by him the elliptic and the spherical. Finally, it has been shown by Sophus Lie, that if figures are to be freely movable throughout all space in oc 6 ways, no other threedimensional spaces than the above four are possible.
Gauss published nothing on the theory of parallels, and it was not generally known until after his death that he had interested himself in that theory from a very early pane& date. In 1799 he announces that Euclidean geometry
would follow from the assumption that a triangle can be drawn greater than any given triangle. Though unwilling to assume this, we find him in 1804 still hoping to prove the postulate of parallels. In 1830 he announces his conviction that geometry is not an a priori science; in the following year he explains that nonEuclidean geometry is free from contradictions, and that, in this system, the angles of a triangle diminish without limit when all the sides are increased. He also gives for the
On the theory of parallels before Lobatchewsky, see Stackel and Engel, Theorie der Parallellinien von Euklid bis auf Gauss (Leipzig, 1895). The foregoing remarks are based upon the materials collected in this work.
GEOMETRY 727
The works of Lobatchewsky and Bolyai, though known and valued by Gauss, remained obscure and ineffective until,in 1866, they were translated into French by J. Hotiel. But Riemann. at this time Riemann's dissertation, Ober die Hypothesen,
welche der Geometrie zu Grunde liegen,' was already about to be published. In this work Riemann, without any knowledge of his predecessors in the same field, inaugurated a far more profound discussion, based on a far more general standpoint; and by its publication in 1867 the attention of mathematicians and philosophers was at last secured. (The dissertation dates from 1854, but owing to changes which Riemann wished to make in it, it remained unpublished until after his death.)
Riemann's work contains two fundamental conceptions, that of a manifold and that of the measure of curvature of a continuous manifold possessed of what he calls flatness in the smallest parts. By means of these conceptions space is made to appear
at the end of agradual series of more and more specialized Definitinontconceptions. Conceptions of magnitude, he explains, ffama fold. are only possible where we have a general conception
capable of determination in variousways. The manifold consists of all these various determinations, each of which is an element of the manifold. The passage from one element to another may be discrete or continuous; the manifold is called discrete or continuous accordingly. Where it is discrete two portions of it can be compared, as to magnitude, by counting; where continuous, by measurement. But measurement demands superposition, and consequently some magnitude independent of its place in the manifold. In passing, in a continuous manifold, from one element to another in a determinate way, we pass through a series of intermediate terms, which form a onedimensional manifold. If this whole manifold be similarly caused to pass over into another, each of its elements passes through a onedimensional manifold, and thus on the whole a twodimensional manifold is generated. In this way we can proceed to n dimensions. Conversely, a manifold of n dimensions can be analysed into one of one dimension and one of (n—1) dimensions. By repetitions of this process the position of an element may be at last determined by n magnitudes. We may here stop to observe that the above conception of a manifold is akin to that due to Hermann Grassmann in the first edition (1847) of his Ausdehnungslehre.'
Both concepts have been elaborated and superseded by the modern procedure in respect to the axioms of geometry, and by the conception of abstract geometry involved therein. Measure of Riemann proceeds to specialize the manifold by con curvature.
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 confine ourselves to lines along which the ratios dx,: dx2:... : a;xk alter continuously. Let us also assume that the element of length, ds, is unchanged (to the first order) when all its points undergo the same infinitesimal motion. Then if all the increments dx be altered in the same ratio, ds is also altered in this ratio. Hence ds is a homogeneous function of the first degree of the increments dx. Moreover, ds must be unchanged when all the dx change sign. The simplest possible case is, therefore, that in which ds is the square root of a quadratic function of the dx. This case includes space, and is alone considered in what follows. It is called the case of flatness in the smallest parts. 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 onedimensional series, of shortest lines, any one of which is defined
4 Abhandlungen d. Konigl. Ges. d. Wiss. zu Gottin en, Bd. xiii. ; Ges. math. Werke, pp. 254269; 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 formula ak(e*lk—e' lb), where k is a constant depending upon the nature of the space. In 1832, in reply to the receipt of Bolyai's Appendix, he gives an elegant proof that the amount by which the sum of the angles of a triangle falls short of two right angles is proportional to the area of the triangle. From these and a few other remarks it appears that Gauss possessed the foundations of hyperbolic geometry, which he was probably the first to regard as perhaps true. It is not known with certainty whether he influenced Lobatchewsky and Bolyai, but the evidence we possess is against such a view.'
The first to publish a nonEuclidean geometry was Nicholas Lobatchewsky, professor of mathematics in the new university
of Kazan.2 In the place of the disputed postulate
Lobat
he puts the following: " All straight lines which, in
chewsky.
a plane, radiate from a given point, can, with respect to any other straight line in the same plane, be divided into two classes, the intersecting and the nonintersecting. The boundary line of the one and the other class is called parallel to the given line." It follows that there are two parallels to the given line through any point, each meeting the line at infinity, like a Euclidean parallel. (Hence a line has two distinct points at infinity, and not one only as in ordinary geometry.) The two parallels to a line through a point make equal acute angles with the perpendicular to the line through the .point. If p be the length of the perpendicular, either of these angles is denoted by II(p). The determination of II(p) is the chief problem (cf. equation (6) above); it appears finally that, with a suitable choice of the unit of length,
tan a A(p) =e—".
Before obtaining this result it is shown that spherical trigonometry is unchanged, and that the normals to a circle or a sphere still pass through its centre. 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 limitline or limitsurface. The geometry on such a surface is shown to be Euclidean, limitlines 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 solution of triangles. He points out that his formulae result from those of spherical trigonometry by substituting ia, ib, ic, for the sides a, b, c.
John Bolyai, a Hungarian, obtained results closely corresponding to those of Lobatchewsky. These he published in an appendix eotyat to a work by his father, entitled Appendix Scientiam
spatii absolute veram exhibens: a veritate aut falsitate Axiomatis XI. Euclidei (a priori haud unquam decidenda) independentem: adjecta ad casum falsilatis, quadratura circuli geometrica.3 This work was published in 1831, but its conception dates from 1823. It reveals a profounder appreciation of the importance of the new ideas, but otherwise differs little from Lobatchewsky's. Both men point out that Euclidean geometry as a limiting case of their own more general system, that the geometry of very small spaces is always approximately Euclidean, that no a priori grounds exist for a decision, and that observation can only give an approximate answer. Bolyai gives also, as his title indicates, a geometrical construction, in hyperbolic space, for the quadrature of the circle, and shows that the area of the greatest possible triangle, which has all its sides parallel and all its angles zero, is 7rr2,where i is what we should now call the spaceconstant.
'See Stackel and Engel, op. cit., and "Gauss, die beiden Bolyai, and die nichtEuklidische Geometrie," Math. Annalen, Bd. xlix.; also Engel's translation of Lobatchewsky (Leipzig, 1898), pp. 378 ff..;
' Lobatchewsky's works on the subject are the following:—" On the Foundations of Geometry," Kazan" Messenger, 1829—183o; •" New Foundations of Geometry, with a complete Theory of Parallels," Proceedings of the University of Kazan", 1835 (both in Russian, but translated into German by Engel, Leipzig, 1898); " G6ometrie imaginaire," Crelle's Journal, 1837; Theorie der Parallellinien (Berlin, 184o; 2nd ed., 1887; translated by Halsted, Austin, Texas, 1891). His results appear to have been set forth in a paper (now lost) which he read at Kazan in 1826.
Translated by Halsted (Austin, Texas, 4th ed., 1896).
distance of u, v from the origin, we have, for a geodesic through the origin,
by Xdx+p5x, where the parameter X : p may have any value. This pencil generates a twodimensional series of points, which may be regarded as a surface, and for which we may apply Gauss's formula for the measure of curvature at any point. Thus at every point of our manifold there is a measure of curvature corresponding to every such pencil; but all these can be found when n.n—1/2 of them are known. If figures are to be freely movable, it is necessary and sufficient that the measure of curvature should be the same for all points and all directions at each point. Where this is the case, it a be the measure of curvature, the linear element can be put into the form
ds = (Edx2) /(I +lane').
If a be positive, space is finite, though still unbounded, and every straight line is closed—a possibility first recognized by Riemann. It is pointed out that, since the possible values of a form a continuous series, observations cannot prove that our space is strictly Euclidean. It is also regarded as possible that, in the infinitesimal, the measure of curvature of our space should be variable.
There are four points in which this profound and epochmaking work is open to criticism or development—(1) the idea of a manifold requires more precise determination; (2) the introduction of coordinates is entirely unexplained and the requisite presuppositions are unanalysed; (3) the assumption' that ds is the square root of a quadratic function of dxi, dx2, . . . is arbitrary; (4) the idea of superposition, or congruence, is not adequately analysed. The modern solution of these difficulties is properly considered in connexion with the general subject of the axioms of geometry.
The publication of Riemann's dissertation was closely followed by two works of Hermann von Helmholtz,' again undertaken Ne/mholtz. in ignorance of the work of predecessors. In these a
proof is attempted that ds must be a rational integral quadratic function of the increments of the coordinates. This proof has since been shown by Lie to stand in need of correction (see VII. Axioms of Geometry). Helmholtz's remaining works on the subject' are of almost exclusively philosophical interest. We shall return to them later.
The only other writer of importance in the second period is Eugenio Beltrami, by whom Riemann's work was brought into
connexion with that of Lobatchewsky and Bolyai. BekraAi!. As he gave, by an elegant method, a convenient Euclidean interpretation of hyperbolic plane geometry, his results will be stated at some length.' The Saggio shows that Lobatchewsky's plane geometry holds in Euclidean geometry on surfaces of constant negative curvature, straight lines being replaced by geodesics. 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/R2 is the measure of curvature of our surface (note that k=y as used above). The angle between two geodesics u=const., v=const. is 8, where
cos e=uv/J ((a2u2)(a2—v2)}, sing=aw/J ((a2—u2)(a2—v2)}.
Thus u=o is orthogonal to all geodesics v=const., and vice versa.
In order that sin B may be real, w2 must be positive; thus geo
desics have no real intersection when the corresponding straight
lines intersect outside the circle u2+v2=a2. When they intersect on
this circle, 0 =o. Thus Lobatchewsky's parallels are represented
by straight lines intersecting on the circle. Again, transforming
to polar coordinates u =r cos p, v =r sin p, and calling p the geodesic
Wiss. Abh. vol. ii. pp. 6io, 628 (1866, 2868).
2 Mind, O.S., vols. i. and iii.; Vortrage and Reden, vol. ii. pp. 1, 256.
His papers are " Saggio di interpretazione delta geometria nonEuclidea," Giornale di matematiche, vol. vi. (1868) ; ' Teoria fondamentale degli spazii di curvatura costante," Annali di matematica, vol. ii. (1868_1869). Both were translated into French by J. Hoiiel, Annales scientifiques de l'Ecole Normale superieure, vol. vi. (1869). (a2 —uuo —vvo)2 = cosh2(p/R)wow2 = C2w2 (say). This equation remains real when p is a pure imaginary, and remains finite when wo=o, provided p becomes infinite in such a way that wo cosh (p/R) remains finite. In the latter case the equation represents a limitline. In the former case, by giving different values to C, we obtain concentric circles with the imaginary centre uovo. One of these, obtained by putting C=o, is the straight line a2—uuo—via =o. 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= }(a2x2—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/a2r2)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 auxiliary plane by a triangle inscribed in the limiting circle. The angle of parallelism is also easily obtained. The perpendicular to v=o at a distance b from the origin is u=a tanh (S/R), and the parallel to this through the origin is a=v sinh (S/R). Hence II (S), the angle which this parallel makes with v =o, is given by
tan 110) . sinh (S/R) =1, or tan aII(S) =eslR
which is Lobatchewsky's formula. We also obtain easily for the area of a triangle the formula R2(w—A—B—C).
Beltrami's treatment connects two curves which, in the earlier treatment, had no connexion. These are limitlines 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, clan ..dzn). But —i of this quotient is defined by Riemann as the measure of curvature.' Hence the measure of curvature is —i/R2, i.e. is constant and negative. The properties of parallels, triangles, &c., are as in the Saggio. It is also shown that the analogues of limit surfaces have zero curvature; and that spheres of radius p have constant positive curvature i/R2 sinh2 (p/R), so that spherical geometry may be regarded as contained in the pseudospherical (as Beltrami calls Lobatchewsky's system).
The Saggio, as we saw, gives a Euclidean interpretation confined to two dimensions. But a consideration of the auxiliary Transition plane suggests a different interpretation, which may be to the extended to any number of dimensions. If, instead
projective of referring to the pseudosphere, we merely define method.
distance and angle, in the Euclidean plane, as those
functions of the coordinates which gave us distance and angle on the pseudosphere, we find that the geometry of our plane has become Lobatchewsky's. All the points of the limiting circle are now at infinity, and points beyond it are imaginary. 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 nonEuclidean 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 stage of thought which finds its expression in the modern treatment of the axioms of geometry.
The projective method leads to a discrimination, first made by Klein,3 of two varieties of Riemann's space; Klein calls The two these elliptic and spherical. They are also called the kinds of polar and antipodal forms of elliptic space. The latter
elliptic names will here be used. 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 antipodes. According to the definition of geometry adopted in section VII. (Axioms of Geometry), the antipodal form is not to be termed " geometry," since any pair of coplanar straight lines intersect each other in two points. It may be called a " quasigeometry." Similarly in the antipodal form two diameters always determine a plane, but two points on a sphere do not determine a great circle when they are antipodes, and two great circles always intersect in two points. Again, a plane does not form a boundary among lines through a point: we can pass from any one such line to any other without passing through the plane. But a great circle does divide the surface of a sphere. So, in the polar form, a complete straight line does not divide a plane, and a planedoes not divide space, and does not, like a Euclidean plane, have two sides.' But, in the antipodal form, a plane is, in these respects, like a Euclidean plane.
It is explained in section VII. in what sense the metrical
geometry of the material world can be considered to be deter
minate and not a matter of arbitrary choice. The scientific ' Beltrami shows also that this definition agrees with that of Gauss. ' " Sur la theorie des foyers," Nouv. Ann. vol. xii.
Math. Annalen, iv. vi., 18711872.
' For an investigation of these and similar properties, see Whitehead, Universal Algebra (Cambridge, 1898), bk. vi. ch. ii. The polar form was independently discovered by Simon Newcomb in 1877.question as to the best available evidence concerning the nature of this geometry is one beset with difficulties of a peculiar kind. We are obstructed by the fact that all existing physical science assumes the Euclidean hypothesis. This hypothesis has been involved in all actual measurements of large distances, and in all the laws of astronomy And physics. The principle of simplicity would therefore lead us, in general, where an observation conflicted with one or more of those laws, to ascribe this anomaly, not to the falsity of Euclidean geometry, but to the falsity of the laws in question. This applies especially to astronomy. On the earth our means of measurement are many and direct, and so long as no great accuracy is sought they involve few scientific laws. Thus we acquire, from such direct measurements, a very high degree of probability that the spaceconstant, if not infinite, is yet large as compared with terrestrial distances. But astronomical distances and triangles can only be measured by means of the received laws of astronomy and optics, all of which have been established by assuming the truth of the Euclidean hypothesis. It therefore remains possible (until a detailed proof of the contrary is forthcoming) that a large but finite spaceconstant, with different laws of astronomy and optics, would have equally explained the phenomena. We cannot, therefore, accept the measurements of stellar paraliaxes, &c., as conclusive evidence that the spaceconstant 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 parallax for very distant stars, or a negative parallax for any star, could not be accepted as conclusive evidence that our geometry is nonEuclidean, unless it were shown—and this seems scarcely possible—that no modification of astronomy or optics could account for the phenomenon. Thus although we may admit a probability that the spaceconstant is large in comparison with stellar distances, a conclusive proof or disproof seems scarcely possible.
Finally, it is of interest to note that, though it is theoretically possible to prove, by scientific methods, that our geometry is nonEuclidean, it is wholly impossible to prove by such methods that it is accurately Euclidean. For the unavoidable errors of observation must always leave a slight margin in our measurements. A triangle might be found whose angles were certainly greater, or certainly less, than two right angles; but to prove them exactly equal to two right angles must always be beyond our powers. If, therefore, any man cherishes a hope of proving the exact truth of Euclid, such a hope must be based, not upon scientific, but upon philosophical considerations.
For Lobatchewsky's writings, cf. Urkunden zur Geschichte der nichteuklidischen Geometrie, i., Nikolaj Iwanowitsch Lobatschefsky, by F. Engel and P. Stackel (Leipzig, 1898). 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 Budapest; the second volume contains the appendix. Cf. also J. Frischauf, Elemente der absoluten, Geometrie (Leipzig, 1876) ; M. L. Gerard, Sur la geometric nonEuclidienne (thesis for doctorate) (Paris, 1892); de Tilly, Essai sur les principes fondamentales de la geometrie et de la mecanique (Bordeaux, 1879) ; Sir R. S. Ball, " On the Theory of Content," Trans. Roy. Irish Acad. vol. xxix. (1889) ; F. Lindemann, " Mechanik bei projectiver Maasbestimmung," Math. Annal. vol. vii. ; W. K. Clifford, " Preliminary Sketch of Biquaternions," Proc. 'of Land. Math. Soc. (1873), and Coll. Works; A. Buchheim, " On the Theory of Screws in Elliptic Space," Proc. Land. Math. Soc. vols. xv., xvi., xvii. ; H. Cox, " On the Application of Quaternions and Grassmann's Algebra to different Kinds of Uniform Space," Trans. Carob. Phil. Soc. (1882) ; M. Dehn, " Die Legendarischen Satze fiber die Winkelsumme im Dreieck," Math. Ann. vol. 53 (1900), and " Ober den Rauminhalt," Math. Annal. vol. 55 (1902).
For expositions of the whole subject, cf. F. Klein, NichtEuklidische Geometrie (Gottingen, 1893)`; R. Bonola, La Geometria nonEuclidea (Bologna, 1906); P. Barbarin, La Geometrie nonEuclidienne (Paris, 1902) ; W. Killing, Die nichtEuklidischen Raumformen in analytischer Behandlung (Leipzig, 1885). The lastnamed work also deals with geometry of more than three dimensions; in this connexion cf. also G. Veronese, Fondamenti di geometria a piiii dimensioni ed a pill specie
di unitd rettilinee . (Padua, 1891, German translation, Leipzig, 1894) ; G. Fontene, L'Hyperespace d (n—i) dimensions (Paris, 1892) ; and A. N. Whitehead, loc. cit. Cf. also E. Study, " TJber nichtEuklidische and Liniengeometrie," Jahr. d. Deutsch. Math. Ver. vol. xv. (1906) ; W. Burnside, " On the Kinematics of nonEuclidean Space," Proc. Lond. Math. Soc. vol. xxvi. (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, Index operum ad geometriam absolutam spectantium .
(1902, and Leipzig, 1903). (B. A. W. R.; A. N. W.)
Until the discovery of the nonEuclidean 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. NonEuclidean 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 Newton), space has an existence, in some sense whatever it may be, independent of the bodies which it contains. The bodies occupy space, and it is not intrinsically unmeaning to say that any definite body occupies this part of space, and not that part of space, without reference to other bodies occupying space. According to the relational theory of space, of which the chief exponent was Leibnitz,l space is nothing but a certain assemblage of the relations between the various particular bodies in space. The idea of space with no bodies in it is absurd. Accordingly there can be no meaning in saying that a body is here and not there, apart from a reference to the other bodies in the universe. Thus, on this theory, absolute motion is intrinsically unmeaning. 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 adoption of the relational theory by modern men of science. But no decisive argument for either view has at present been elaborated.2 Kant's view of space as being a form of perception at first sight appears to cut across this controversy. But he, saturated as he was with the spirit of the Newtonian physics, must (at least in both editions of the Critique) be classed with the upholders of the absolute theory. The form of perception has a type of existence proper to itself independently of the particular bodies which it contains. For example he writes: 3 " Space does not represent any quality of objects by themselves, or objects in their relation to one another, i.e. space does not represent any determination which is inherent in the objects themselves, and would remain, even if all subjective conditions of intuition were removed."
The second controversy is that between the view that the axioms applicable to space are known only from experience, Aaloms. and the' view that in some sense these axioms are
given a priori. Both these views, thus broadly stated, are capable of various subtle modifications, and a discussion of them would merge into a general treatise on epistemology. The cruder forms of the a priori view have beers made quite untenable by the modern mathematical discoveries. Geometers now profess ignorance in many respects of the exact axioms which apply to existent space, and it seems unlikely that a profound study of the question should thus obliterate a priori' intuitions.
Another question irrelevant to this article, but with some relevance to the above controversy, is that of the derivation
For an analysis of Leibnitz's ideas on space, cf. B. Russell, The Philosophy of Leibnitz, chs. viii.x.
2 Cf. Hon. Bertrand Russell, " Is Position in Time and Space Absolute or Relative?" Mind, n.s. vol. 10 (1901), and A. N. Whitehead, " Mathematical Concepts of the Material World," Phil. Trans.
(1906), p. 205.
3 Cf. Critique of Pure Reason, 1st section; " Of Space," conclusion A, .Max Mailer's translation.of our perception of existent space from our various types of sensation. This is a question for psychology.'
Definition of Abstract Geometry.—Existent space is the subject matter of only one of the applications of the modern science of abstract geometry, viewed as a branch of pure mathematics. Geometry has been defined a as " the study of series of two or more dimensions." It has also been defined' as " the science of cross classification." These definitions are founded upon the actual practice of mathematicians in respect to their use of the term " Geometry." Either of them brings out the fact that geometry is not a science with a determinate subject matter. It is concerned with any subject matter to which the formal axioms may apply. Geometry is not peculiar in this respect. All branches of pure mathematics deal merely with types of relations. Thus the fundamental ideas of geometry (e.g. those of points and of straight lines) are not ideas of determinate entities, but of any entities for which the axioms are true. And a set of formal geometrical axioms cannot in themselves be true or false; since they are not determinate propositions, in that they do not refer to a determinate subject matter. The axioms are propositional functions? When a set of axioms is given, we can ask (I) whether they are consistent, (2) whether their " existence theorem " is proved, (3) whether they are independent. Axioms are consistent when the contradictory of any axiom cannot be deduced from the remaining axioms. Their existence theorem is the proof that they are true when the fundamental ideas are considered as denoting some determinate subject matter, so that the axioms are developed into determinate propositions. It follows from the logical law of contradiction that the proof of the existence theorem proves also the consistency of the axioms. 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 independence of a given axiom is proved by establishing the consistency of the remaining axioms of the set, together with the contradictory of the given axiom. The enumeration of the axioms is simply the enumeration of the hypotheses' (with respect to the undetermined subject matter) of which some at least occur in each of the subsequent propositions.
Any science is called a " geometry " if it investigates the theory of the classification of a set of entities (the points) into classes (the straight lines), such that (I) there is one and only one class which contains any given pair of the entities, and (a) every such class contains more than two members. In the two geometries, important from their relevance to existent space, axioms which secure an order of the points on any line also occur. These geometries will be called " Projective Geometry " and " Descriptive Geometry." In projective geometry any two straight lines in a plane intersect, and the straight lines are closed series which return into themselves, like the circumference of a circle. In descriptive geometry two straight lines in a plane do not necessarily intersect, and a straight line is an open series without beginning or end. Ordinary Euclidean geometry is a descriptive geometry; it becomes a projective geometry when the socalled " points at infinity " are added.
Projective Geometry.
Projective geometry may be developed from two undefined fundamental ideas, namely, that of a " point " and that of a " straight line." These undetermined ideas take different specific meanings for the various specific subject matters to which projective geometry can be applied. The number of the axioms is always to some extent arbitrary, being dependent upon the verbal forms of statement which are adopted. They will
' Cf. Ernst Mach, Erkenntniss and Irrtum (Leipzig) ; the relevant chapters are translated by T. J. McCormack, Space and Geometry (London, 1906); also A. Meinong, fiber die Stellung der Gegenstandstheorie im System der Wissenschaften (Leipzig, 1907).
a Cf. Rusself, Principles of Mathematics, § 352 (Cambridge, 1903). 'Cf. A. N. Whitehead, The Axioms of Projective Geometry, § 3 (Cambridge, 1906).
7 Cf. Russell, Princ. of Math., ch. i.
8 Cf. Russell, loc. cit., and G. Frege, " eber die Grundlagen der Geometrie," Jahresber. der Deutsch. Math. Ver. (1906).
all the points.
5. If A, B, C are noncollinear points, and A' is on the straight line BC, and B' is on the straight line CA, then the straight lines AA' and BB' possess a point in common.
Definition.—If A, B, C are any three noncollinear points, the plane ABC is the class of points lying on the straight lines joining A with the various points on the straight line BC.
6. 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. (ABCD) symbolizes the following conjoint statements: (i) that the points A, B, C, D are collinear, and (2) that a quadrilateral can be found with one pair of opposite sides intersecting at A, with the other pair intersecting at C, and with its diagonals passing through B and D respectively. Then B and D are said to be " harmonic conjugates " with respect to A and C.
8. Harm. (ABCD) implies that B and D are distinct points. In the above axioms 4 secures at least two dimensions; axiom 5 is the fundamental axiom of the plane, axiom 6 secures at least three dimensions, and axiom 7 secures at most three dimensions. From axioms 15 it can be proved that any two distinct points in a straight line determine that line, that any three noncollinear points in a plane determine that plane, that the straight line containing any two points in a plane lies wholly in that plane, and that any two straight lines in a plane intersect. From axioms 16 Desargues's wellknown theorem on triangles in perspective can be proved.
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 15, 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 15, and such that Desargues's theorem does not hold.
It follows from axioms 15 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. Farm,' who has produced a three dimensional geometry of fifteen points, i.e. a method of cross classification of fifteen entities, in which each straight line contains three points, and each plane contains seven straight lines. In this geometry axiom 8 does not hold. Also from axioms 16 and 8 it follows that Harm. (ABCD) implies Harm. (BCDA).
Definitions.—When two plane figures can be derived from one another by a single projection, they are said to be in perspective. When two plane figures can be derived one from the other by a finite series of perspective relations between intermediate figures, they
' This formulation—though not in respect to number—is in all essentials that of M. Pieri, cf. " I principii della Geometria di Posizione." Accad. R. di Torino (1898) ; also cf. Whitehead, loc. cit.
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. Moulton, " A Simple nonDesarguesian Plane Geometry," Trans. Amer. Math. Soc., vol. iii. (1902).
' Cf. " Suit postulati fondamentali della geometria projettiva," Giorn: di matematica, vol. xxx. (1891); also of Pieri, loc. cit., and Whitehead., loc. cit.are said to be projectively related. Any property of a plane figure which necessarily also belongs to any projectively related figure, is called a projective property.
The following theorem, known from its importance as " the fundamental theorem of projective geometry," cannot be proved 4 from axioms 18. The enunciation is: " A projective correspondence between the points on two straight lines is completely determined when the correspondents of three distinct points on one line are determined on the other." This theorem is equivalent' (assuming axioms 18) to another theorem, known as Pappus's Theorem, namely: " If 1 and 1' are two distinct coplanar lines, and A, B, C are three distinct points on 1, and A', B', C' are three distinct points on 1', then the three points of intersection of AA' and B'C, of A'B and CC', of BB' and C'A, are collinear." This theorem is obviously Pascal's wellknown theorem respecting a hexagon inscribed in a conic, for the special case when the conic has degenerated into the two lines 1 and 1'. Another theorem also equivalent (assuming axioms 18) to the fundamental theorem is the following:' If the three collinear pairs of points, A and A', B and B', C and C', are such that the three pairs of opposite sides of a complete quadrangle pass respectively through them, i.e. one pair through A and A' respectively, and so on, and if also the three sides of the quadrangle which pass through A, B, and C, are concurrent in one of the corners of the quadrangle, then another quadrangle can be found with the same relation to the three pairs of points, except that its three sides which pass through A, B, and C, are not concurrent.
Thus, if we choose to take any one of these three theorems as an axiom, all the theorems of projective geometry which do not require ordinal or metrical ideas for their enunciation can be proved. Also a conic can be defined as the locus of the points found by the usual construction, based upon Pascal's theorem, for points on the conic through five given points. But it is unnecessary to assume here any one of the suggested axioms; for the fundamental theorem can be deduced from the axioms of order together with axioms 18.
Axioms of Order.—It is possible to define (cf. Pieri, loc. cit.) the property upon which the order of points on a straight line depends. 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 rest of the points on the line. This definition is elucidated by noticing that with our ordinary geometrical ideas, if B and X are any two points between A and C, then the two pairs of points, A and C, B and X, define an involution with real double points, namely, the Y and Y' of the above definition. The property of belonging to a segment ABC is projective, since the harmonic relation is projective.
The first three axioms of order (cf. Pieri, loc. cit.) are:
9. If A, B, C are three distinct collinear points, the supplementary segment ABC is contained within the segment BCA.
so. If A, B, C are three distinct collinear points, the common part of the segments BCA and CAB is contained in the supplementary segment ABC.
11. If A, B, C are three distinct collinear points, and D lies in the segment ABC, then the segment ADC is contained within the segment ABC.
From these axioms all the usual properties of a closed order follow. It will be noticed that, if A, B, C are any three collinear points, C is necessarily traversed in passing from A to B by one route along the line, and is not traversed in passing from A to B along the other route. Thus there is no meaning, as referred to closed straight lines, in the simple statement that C lies between A and B. But there may be a relation of separation between two pairs of collinear points, such as A and C, and B and D. The couple B and D is said to separate A and C, if
4 Cf. Hilbert, loc. cit.; for a fuller exposition of Hilbert's proof cf. K. T. 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 112 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 oneone correspondence. The arbitrary elements in the
establishment of this relation are the points on the line associated with o, 1 and co.
This association 3 is most easily effected by considering a
class of projective relations of the line with itself, called by F. Schur (loc. cit.) prospectivities.
Let l (fig. 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 oneone 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 " numerationsystem (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 associate points on OU, OV, OW with numbers according to the numeration systems (OE1U), (OE2V), and (OE3W). Let the planes VWP, WUP, UVP cut OU, OV, OW in L, M, N respectively; and let x, y, z be the numbers associated with L, M, N respectively. o Then P is the point (x, y, z). Also
homogeneous coordinates can be introduced as before, thus avoiding the infinitieson 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 twotermed serial relation, as considered in the logic of relations, ranging the points between which it holds into a linear open order. Thus the relations are the straight lines, and the terms between which they hold are the points. But a combination of these two points of view yields 5 the simplest statement of all. Descriptive geometry is then conceived as the investigation of an undefined fundamental relation between three terms (points); and when the relation holds between three points A, B, C, the points are said to be " in the [linear] order ABC."
0. Veblen's axioms and definitions, slightly modified, are as follows:
1. If the points A, B, C are in the order ABC, they are in the order CBA.
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 BCD and CEA, then a point F exists in the order AFB, and such that
D, E, F are collinear.
Definition.—If A, B, C are three
noncollinear points, the plane ABC
is the class of points which lie on any
one of the lines joining any two of the
n points belonging to the boundary of
formed by the segments BC, CA and AB. The interior of the triangle ABC is formed by the points in segments such as PQ, where P and Q are points respectively on two of the segments BC, CA, AB.
8. There exists a plane ABC, which does not contain all the points.
Definition.—If A, B, C, D are four noncoplanar points, the space ABCD is the class of points which lie on any of the lines containing two points on the surface of the tetrahedron ABCD, the surface being formed by the interiors of the triangles ABC, BCD, DCA, DAB.
9. There exists a space ABCD which contains all the points.
'Cf. loc. cit.
2 Cf. I Principii di geometria (Turin, 1889) and " Sui fondamenti della geometria," Rivista di mat. vol. iv. (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 19 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 halflines ll and 12. Then it can be proved that the set of halflines, emanating from A and intersecting h (such as m), are bounded by two halflines, of which ABC is one. Let r be the other. Then it can be proved that r does not intersect 11. Similarly for the halfline, such as n, intersecting 12. Let s be its bounding halfline. Then two cases are possible. (I) The halflines 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
halflines 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 nonsecant lines which do not intersect 1. The two boundary nonsecant 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 startingpoint from which Lobatchewsky constructed the first explicit coherent theory of nonEuclidean 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 (1ro) 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, " Sulla introduzione degli enti improprii in geometria projettive," Giorn. di mat. vol. xxxviii. (1900) ; and Whitehead, Axioms of Descriptive Geometry (Cambridge, 1907).
A
If the planes intersect, the ideal line is termed proper, otherwise it is improper. It can be proved that any two planes, with which any two of the ideal points are both coherent, will serve as the guiding planes used in the definition. The ideal planes are defined as in projective geometry, and all the other definitions (for segments, order, &c.) of projective geometry are applied to the ideal elements. If an ideal plane contains some proper ideal points, it is called proper, otherwise it is improper. Every ideal plane contains some improper ideal points.
It can now be proved that all the axioms of projective geometry hold of the ideal elements as thus obtained; and also that the order of the ideal points as obtained by the projective method agrees with the order of the proper ideal points as obtained from that of the associated points of the descriptive geometry. 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 justification for the ordinary use of the " points at infinity " in the ordinaryEuclidean geometry; the reasoning has been transferred from the original descriptive space to the associated projective space of ideal elements; and with the Euclidean parallel axiom the improper ideal elements reduce to the ideal points on a single improper ideal plane, namely, the plane at infinity.'
Congruence and Measurement.—The property of physical space which is expressed by the term " measurability " has now to be considered. This property has often been considered as essential to the very idea of space. For example, Kant writes,2 " Space is represented as an infinite given quantity." This quantitative aspect of space arises from the measurability of distances, of angles, of surfaces and of volumes. These four types of quantity depend upon the two first among them as fundamental. The measurability of space is essentially connected with the idea of congruence, of which the simplest examples are to be found in the proofs of equality by the method of superposition, as used in elementary plane geometry. The mere concepts of " part " and of " whole " must of necessity be inadequate as the foundation of measurement, since we require the comparison as to quantity of regions of space which have no portions in common. The idea of congruence, as exemplified by the method of superposition in geometrical reasoning, appears to be founded upon that of the " rigid body," which moves from one position to another with its internal spatial relations unchanged. But unless there is a previous concept of the metrical relations between the parts of the body, there can be no basis from which to deduce that they are unchanged.
It would therefore appear as if the idea of the congruence, or metrical equality, of two portions of space (as empirically suggested by the motion of rigid bodies) must be considered as a fundamental idea incapable of definition in terms of those geometrical concepts which have already been enumerated. 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 coordinates have already been introduced. The displacement of a rigid body is simply a mode of defining to the senses a oneone 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." Sect. r.
2 Cf. loc. cit.
' Cf. Ober die Grundlagen der Geometrie (Leipzig, Ber., 189o) ; and Theorie der Transformationsgruppen (Leipzig, 1893), vol.cessive displacements of a rigid body from position A to position B, and from position B to position C, are the same in effect as one displacement from A to C. But this is the characteristic "group" property. Thus the transformations of space into itself defined by displacements of rigid bodies form a group.
Call this group of transformations a congruencegroup. Now according to Lie a congruencegroup is defined by the following characteristics:
r. A congruencegroup is a finite continuous group of oneone transformations, containing the identical transformation.
2. It is a subgroup 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 nonhomogeneous coordinates 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+wax+cy(ux+vy+wz), (A)
dz/dt =w —w2x+w,y+cz(ux+vy+wz).
In the case considered in theorem r (Q), with the proper choice of coordinates the three equations defining, the general infinitesimal transformation are:
dx/dt =u —way+w2z,
dy/dt=v+wax, (B)
(h/dt = w —w2x lwl y.
In this case the latent plane is the plane for which at least one of x, y, z are infinite, that is, the plane o.x+o.y+o.z+a=o; and the latent conic is the conic in which the cone x2+y2+z2=o intersects the latent plane.
It follows from theorems r and 2 that there is not one unique
congruencegroup, but an indefinite number of them. There is one congruencegroup corresponding to each closed real quadric, one to each imaginary quadric with a real equation, and one to each imaginary conic in a real plane and with a real equation. The quadric thus associated with each congruencegroup is called the absolute for that group, and in the degenerate case of r (3) the absolute is the latent plane together with the latent imaginary conic. If the absolute is real, the congruencegroup is hyperbolic; if imaginary, it is elliptic; if the absolute is a plane and imaginary conic, the group is parabolic. Metrical geometry is simply the theory of the properties of some particular congruencegroup selected for study.
The definition of distance is connected with the corresponding congruencegroup 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) log{A,P,A2P2}+ log{A1PsA,Pa} =log{A,P1A2Pa}, and
(2), if the points Al, As, P,, P2 are transformed into A',, A's, P',, P's by any transformation of the congruencegroup, (a) {A1P,A2Ps} _ {A',P',A'sP's}, since the transformation is projective, and 0) A',. are on the absolute since As and As are on it. Thus if we define
AXIOMS]
the distance PIP2 to be ;k log {A1P1A2P2}, where A, and A2 are the points in which the line P,P2 cuts the absolute, and k is some constant, the two characteristic properties of distance, namely, (i) the addition of consecutive lengths on a straight line, and (2) the invariability of distances during a transformation of the congruencegroup, are satisfied. .This is the wellknown CayleyKlein projective definition 1 of distance, which was elaborated in view of the addition property alone, previously to Lie's discovery of the theory of congruencegroups. For a hyperbolic group when Pi and P2 are in the region enclosed by the absolute, log{AiPiA2P2} is real, and therefore k must be real. For an elliptic group Al and A2 are conjugate imaginaries, and log {A1P2A2P2} is a pure imaginary, and k is chosen to be Kit, where K is real and =I/ —.
Similarly the angle between two planes, pi and p2, is defined to be (1/2i) log (tipitlp2), where ti and t2 are tangent planes to the absolute through the line plp2. The planes tl and 12 are imaginary for an elliptic group, and also for an hyperbolic group when the planes pi and p2 intersect at points within the region enclosed by the absolute. The development of the consequences of these metrical definitions is the subjct of nonEuclidean geometry.
The definitions for the parabolic case can be arrived at as limits of those obtained in either of the other two cases by making k ultimately to vanish. It is also obvious that, if Pi and P2 be the points (xi, yi, zi) and (x2, y2, z2), it follows from equations (B) above that {(xi—x2)2+(yi—y2)2+(zi—z2)2}i is unaltered by a congruence transformation and also satisfies the addition property for collinear distances. 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 pipe to the two points where the plane pipz cuts the imaginary conic. These points are in fact the
circular points at infinity " on the plane. The development of the consequences of these definitions for the parabolic case gives the ordinary Euclidean metrical geometry.
Thus the only metrical geometry for the whole of projective
space is of the elliptic type. But the actual measurerelations (though not their general properties) differ according to the elliptic congruencegroup selected for study. In a descriptive space a congruencegroup should possess the four characteristics of such a group throughout the whole of the space. Then form the associated ideal projective space. The associated congruencegroup for this ideal space must satisfy the four conditions throughout the region of the proper ideal points. Thus the boundary of this region is the absolute. Accordingly there can be no metrical geometry for the whole of a descriptive space unless its boundary (in the associated ideal space) is a closed quadric or a plane. If the boundary is a closed quadric, there is one possible congruencegroup of the hyperbolic type. If the boundary is a plane (the plane at infinity), the possible congruencegroups are parabolic; and there is a congruencegroup 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 congruencegroups 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 congruencegroup.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 Sixth Memoir on Quantics," Trans. Roy. Soc., i859, and Coll. Papers, vol. ii. ; and F. 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. Poincare, La Science et 1'hypothese, ch. iii.735
we have, in fact, presented to our senses a definite set of transformations forming a congruencegroup, resulting in a set of measure relations which are in no respect arbitrary. Accordingly our scientific laws are to be stated relevantly to that particular congruencegroup. Thus the investigation of the type (elliptic, hyperbolic or parabolic) of this special congruencegroup is a perfectly definite problem, to be decided by experiment. The consideration of experiments adapted to this object requires some development of nonEuclidean geometry (see section VI., NonEuclidean Geometry). But if the doctrine means that, assuming some sort of objective reality for the material universe, beings can be imagined, to whom either all congruencegroups are equally important,or some other congruencegroup is specially important, the doctrine appears to be an immediate deduction from the mathematical facts. Assuming a definite congruencegroup, the investigation of surfaces (or threedimensional loci in space of four dimensions) with geodesic geometries of the form of metrical geometries of other types of congruencegroups forms an important chapter of nonEuclidean geometry. Arising from this investigation there is a widelyspread fallacy, which has found its way into many philosophic writings, namely, that the possibility of the geometry of existent threedimensional space being other than Euclidean depends on the physical existence of Euclidean space of four or more dimensions. The foregoing exposition shows the baselessness of this idea.
The numberless works which have been written to suggest equivalent alternatives to Euclid's parallel axioms may be neglected as being of trivial importance, though many of them are marvels of geometric ingenuity.
The second stream of thought confined itself within the circle of ideas of Euclidean geometry. Its origin was mainly due to a
GEOPONICIGEORGE, SAINT
rearing of cattle, and the breeding of fishes. He was the first to systematize what had been written on the subject, and supplemented the labours of others by practical experience gained during his travels. In the Augustan age Julius Hyginus wrote on farming and beekeeping, Sabinus Tiro on horticulture, and during the early empire Julius Graecinus and Julius Atticus on the culture of vines, and Cornelius Celsus (best known for his De medicina) on farming. The chief work of the kind, however, is that of Lucius Junius Moderatus Columella (q.v.). About the middle of the 2nd century the two Quintilii, natives of Troja, wrote on the subject in Greek. It is remarkable that Columella's work exercised less influence in Rome and Italy than in southern Gaul and Spain, where agriculture became one of the principal subjects of instruction in the superior educational establishments that were springing up in those countries. One result of this was the preparation of manuals of a popular kind for use in the schools. In the 3rd century Gargilius Martialis of Mauretania compiled a Geoponica in which medical botany and the veterinary art were included. The De re rustica of Palladius (4t1l century), in fourteen books, which is almost entirely borrowed from Columella, is greatly inferior in style and knowledge of the subject, It is a kind of farmer's calendar, in which the different rural occupations are arranged in order of the months. The fourteenth book (on forestry) is written in elegiacs (85 distichs). The whole of Palladius and considerable fragments of Martialis are extant.
The best edition of the Scriptores rei rusticae is by J. G. Schneider (17441797), and the whole subject is exhaustively treated by A. Magerstedt, Bilder aus der romischen Landwirtschaft (1858—1863) ; see also TeuffelSchwabe, Hist. of Roman Literature, 54; C. F. Bahr in Ersch and Gruber's Allgemeine Encyklopadie.
End of Article: CRP 

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