difficulties. They prove elaborately, by a reductio ad absurdum, that the volumes cannot be unequal. This proof must be read in the Elements. We must, however, state that we have in the above not proved Euclid's Prop. 5, but only a special case of it. Euclid does not suppose that the bases of the two pyramids to be compared are equal, and hence he proves that the volumes are as the base. The reasoning of the proof becomes clearer in the special case, from which the general one may be easily deduced.

§ 86. Prop. 6 extends the result to pyramids with polygonal bases. From these results follow again the rules at present given for the mensuration of solids, viz. a pyramid is the third part of a triangular prism having the same base and the same altitude. But a triangular prism is equal in volume to a parallelepiped which has the same base and altitude. Hence if B is the base and h the altitude, we have

$87. A method similar to that used in proving Prop. 5 leads to the following results relating to solids bounded by simple curved Prop. 10. Every cone is the third part of a cylinder which has the same base, and is of an equal altitude with it.

as their bases.

Prop. 11. Cones or cylinders of the same altitude are to one another Prop. 12. Similar cones or cylinders have to one another the triplicate ratio of that which the diameters of their bases have.

Prop. 13. If a cylinder be cut by a plane parallel to its opposite planes or bases, it divides the cylinder into two cylinders, one of which is to the other as the axis of the first to the axis of the other; which may also be stated thus:

Cylinders on the same base are proportional to their altitudes. Prop. 14. Cones or cylinders upon equal bases are to one another as their altitudes. Prop. 15. The bases and altitudes of equal cones or cylinders are

reciprocally proportional, and if the bases and altitudes be reciprocally proportional, the cones or cylinders are equal to one another.

These theorems again lead to formulae in mensuration, if we compare a cylinder with a prism having its base and altitude equal to the base and altitude of the cylinder. This may be done by the method of exhaustion. We get, then, the result that their bases are equal, and have, if B denotes the numerical value of the base, and h that of the altitude,

§89. The 13th and last book of Euclid's Elements is devoted to the regular solids (see POLYHEDRON). It is shown that there are five of them, viz.:

1. The regular tetrahedron, with 4 triangular faces and 4 vertices; 2. The cube, with 8 vertices and 6 square faces;

3. The octahedron, with 6 vertices and 8 triangular faces;

4. The dodecahedron, with 12 pentagonal faces, 3 at each of the eo vertices;

5. The icosahedron, with 20 triangular faces, 5 at each of the 12 vertices.

It is shown how to inscribe these solids in a given sphere, and how to determine the lengths of their edges.

§ 90. The 13th book, and therefore the Elements, conclude with the scholium, that no other regular solid exists besides the five ones enumerated."

The proof is very simple. Each face is a regular polygon, hence the angles of the faces at any vertex must be angles in equal regular polygons, must be together less than four right angles (XI. 21), and must be three or more in number. Each angle in a regular triangle equals two-thirds of one right angle. Hence it is possible to form a solid angle with three, four or five regular triangles or faces. These give the solid angles of the tetrahedron, the octahedron and the icosahedron. The angle in a square (the regular quadrilateral) equals one right angle. Hence three will form a solid angle, that of the cube, and four will not. The angle in the regular pentagon equals of a right angle. Hence three of them equal (i.e. less than 4) right angles, and form the solid angle of the dodecahedron. Three regular polygons of six or more sides cannot form a solid angle. Therefore no other regular solids are possible. (O. H.)

II. PROJECTIVE GEOMETRY

It is difficult, at the outset, to characterize projective geometry as compared with Euclidean. But a few examples will at least indicate the practical differences between the two.

In Euclid's Elements almost all propositions refer to the magnitude of lines, angles, areas or volumes, and therefore to measurement. The statement that an angle is right, or that two straight lines are parallel, refers to measurement. On the other hand, the fact that a straight line does or does not cut a circle is independent of measurement, it being dependent only upon the mutual "position" of the line and the circle. This difference becomes clearer if we project any figure from one plane to another (see PROJECTION). By this the length of lines, the magnitude of angles and areas, is altered, so that the projection, or shadow, of a square on a plane will not be a square; it will, however, be some quadrilateral. Again, the projection of a circle will not be a circle, but some other curve more or less resembling a circle. But one property may be stated at once-no straight line can cut straight line can cut a circle in more than two points. There the projection of a circle in more than two points, because no are, then, some properties of figures which do not alter by projection, whilst others do. To the latter belong nearly all properties relating to measurement, at least in the form in which they are generally given. The others are said to be projective properties, and their investigation forms the subject of projective geometry.

Different as are the kinds of properties investigated in the old and the new sciences, the methods followed differ in a still greater degree. In Euclid each proposition stands by itself; its connexion with others is never indicated; the leading ideas contained in its proof are not stated; general principles do not exist. In the modern methods, on the other hand, the greatest importance is attached to the leading thoughts which pervade the whole; and general principles, which bring whole groups of theorems under one aspect, are given rather than separate propositions. The whole tendency is towards generalization. A straight line is considered as given in its entirety, extending both ways to infinity, while Euclid never admits anything but finite quantities. The treatment of the infinite is in fact another fundamental difference between the two methods: Euclid avoids

it; in modern geometry it is systematically introduced.

Of the different modern methods of geometry, we shall treat principally of the methods of projection and correspondence which have proved to be the most powerful. These have become independent of Euclidean Geometry, especially through the Geometrie der Lage of V. Staudt and the Ausdehnungslehre of Grassmann.

For the sake of brevity we shall presuppose a knowledge of Euclid's Elements, although we shall use only a few of his pro positions.

§1. Geometrical Elements. We consider space as filled with points, lines and planes, and these we call the elements out of which our figures are to be formed, calling any combination of these elements a figure."

By a line we mean a straight line in its entirety, extending both ways to infinity; and by a plane, a plane surface, extending in all directions to infinity.

We accept the three-dimensional space of experience-the space assumed by Euclid-which has for its properties (among others):Through any two points in space one and only one line may be drawn;

Through any three points which are not in a line, one and only one plane may be placed;

The intersection of two planes is a line;

A line which has two points in common with a plane lies in the plane, hence the intersection of a line and a plane is a single point; and Three planes which do not meet in a line have one single point in

common.

It will be observed that not only are planes determined by points, but also points by planes; that therefore the planes may be considered as clements, like points; and also that in any one of the above statements we may interchange the words point and plane, and we obtain again a correct statement, provided that these statements themselves are true. As they stand, we ought, in several cases, to add "if they are not parallel," or some such words, parallel lines and planes being evidently left altogether out of consideration. To correct this we have to reconsider the theory of parallels.

As immediate consequences we get the propositions:Every line meets a plane in one point, or it lies in it; Every plane meets every other plane in a line;

Any two lines in the same plane meet.

A

B

FIG. 1.

Then this line 9 will meet the line p in a point A. If we turn the line q about S towards q', its point of intersection with will move along towards B, passing, on continued turning, to a greater and greater distance, until it is moved out of our reach. If we q'turn q still farther, its continuation will meet p, but now at the other side of A. The point of intersection has disappeared to the right and reappeared to the left. There is one intermediate position where q is parallel to that is where it does not cut p. In every other position it cuts p in some finite point. If, on the other hand, we move the point A to an infinite distance in p, then the line q which passes through A will be a line which does not cut p at any finite point. Thus we are led to say: Every line through S which joins it to any point at an infinite distance in p is parallel to p. But by Euclid's 12th axiom there is but one line parallel to p through S. The difficulty in which we are thus involved is due to the fact that we try to reason about infinity as if we, with our finite capabilities, could comprehend the infinite. To overcome this difficulty, we may say that all points at infinity in a line appear to us as one, and may be replaced by a single "ideal" point.

I. The row, or range, of points formed by all points in a line, which is called its base.

2. The flat pencil formed by all the lines through a point in a plane. Its base is the point in the plane.

3. The axial pencil formed by all planes through a line which is called its base or axis.

The space of points-that is, all points in space. The space of planes-that is, all planes in space. IV. Of four dimensions.

The space of lines, or all lines in space.

86. Meaning of "Dimensions."-The word dimension in the above needs explanation. If in a plane we take a row p and a pencil with centre Q, then through every point in p one line in the pencil will pass, and every ray in Q will cut p in one point, so that we are entitled to say a row contains as many points as a flat pencil lines, and, we may add, as an axial pencil planes, because an axial pencil is cut by a plane in a flat pencil.

The number of elements in the row, in the flat pencil, and in the axial pencil is, of course, infinite and indefinite too, but the same in all. This number may be denoted by ∞. Then a plane contains 2 points and as many lines. To see this, take a flat pencil in a plane. It contains lines, and each line contains points, whilst each point in the plane lies on one of these lines. Similarly, in a each point by o lines and contains ∞ points; hence there are ∞ lines in a plane.

Definition. This ideal point is called the point at infinity in the

line.

The axiom is equivalent to Euclid's Axiom 12, for it follows from either that through any point only one line may be drawn parallel to a given line.

This point at infinity in a line is reached whether we move a point in the one or in the opposite direction of a line to infinity. A line thus appears closed by this point, and we speak as if we could move a point along the line from one position A to another B in two ways, either through the point at infinity or through finite points only.

It must never be forgotten that this point at infinity is ideal; in fact, the whole notion of "infinity is only a mathematical conception, and owes its introduction (as a method of research) to the working generalizations which it permits.

3. Line and Plane at Infinity-Having arrived at the notion of replacing all points at infinity in a line by one ideal point, there is no difficulty in replacing all points at infinity in a plane by one ideal

line.

To make this clear, let us suppose that a line p, which cuts two fixed lines a and b in the points A and B, moves parallel to itself to a greater and greater distance, It will at last cut both a and b at their points at infinity, so that a line which joins the two points at infinity in two intersecting lines lies altogether at infinity. Every other line in the plane will meet it therefore at infinity, and thus it contains all points at infinity in the plane.

All points at infinity in a plane lie in a line, which is called the line at infinity in the plane.

It follows that parallel planes must be considered as planes having a common line at infinity, for any other plane cuts them in parallel lines which have a point at infinity in common.

If we next take two intersecting planes, then the point at infinity in their line of intersection lies in both planes, so that their lines at infinity meet. Hence every line at infinity meets every other line at infinity, and they are therefore all in one plane.

All points at infinity in space may be considered as lying in one ideal plane, which is called the plane at infinity.

§ 4. Parallelism.-We have now the following definitions:-
Parallel lines are lines which meet at infinity;
Parallel planes are planes which meet at infinity;
A line is parallel to a plane if it meets it at infinity.

Theorems like this-Lines (or planes) which are parallel to a third are parallel to each other-follow at once.

This view of parallels leads therefore to no contradiction of Euclid's Elements.

A pencil in space contains as many lines as a plane contains points and as many planes as a plane contains lines, for any plane cuts the pencil in a field of points and lines. Hence a pencil contains lines and planes. The field and the pencil are of two dimensions.

To count the number of points in space we observe that each point lies on some line in a pencil. But the pencil contains ∞ lines, and cach line points; hence space contains points. Each plane cuts any fixed plane in a line. But a plane contains ∞ lines, and through each pass ∞ planes; therefore space contains planes.

Hence space contains as many planes as points, but it contains an infinite number of times more lines than points or planes. To count them, notice that every line cuts a fixed plane in one point. But lines pass through each point, and there are 2 points in the plane. Hence there are lines in space. The space of points and planes is of three dimensions, but the space of lines is of four dimensions.

A field of points or lines contains an infinite number of rows and flat pencils; a pencil contains an infinite number of flat pencils and of axial pencils; space contains a triple infinite number of pencils and of fields, rows and axial pencils and ∞ flat pencilsor, in other words, each point is a centre of 2 flat pencils. 87. The above enumeration allows a classification of figures. Figures in a row consist of groups of points only, and figures in the flat or axial pencil consist of groups of lines or planes. In the plane we may draw polygons; and in the pencil or in the point, solid angles, and so on.

We may also distinguish the different measurements We have→ In the row, length of segment;

In the flat pencil, angles;

In the axial pencil, dihedral angles between two planes;

In the plane, areas;

In the pencil, solid angles;

In the space of points or planes, volumes.

SEGMENTS OF A LINE

"sense

§ 8. Any two points A and B in space determine on the line through them a finite part, which may be considered as being described by a point moving from A to B. This we shall denote by AB, and distinguish it from BA, which is supposed as being described by a point moving from B to A, and hence in a direction or in a opposite to AB. Such a finite line, which has a definite sense, we shall call a " segment," so that AB and BA denote different segments, The one which are said to be equal in length but of opposite sense. sense is often called positive and the other negative.

2a

a

b

B

9. If we have three points A, B, C in a line (fig. 2), the step AB will bring us from A to B, and the step BC from B to C. Hence both steps are equivalent to the one step AC. This is expressed by saying that AC is the sum "of AB and BC; in symbolsAB+BC=AC,

A

B

where denotes negative sense.

(1)

(2)

We can then, just as in algebra, change subtraction of segments into addition by changing the sense, so that AB-CB is the same as AB+(-CB) or AB+BC. A figure will at once show the truth of this. The sense is, in fact, in every respect equivalent to the "sign" of a number in algebra.

10. Of the many formulae which exist between points in a line we shall have to use only one more, which connects the segments between any four points A, B, C, D in a line. We have

BC BD+DC, CA=CD+DA, AB=AD+DB;

or multiplying these by AD, BD, CD respectively, we get

BC. AD BD. AD+DC. AD BD. AD-CD. AD
CA. BD CD. BD+DA. BD=CD, BD-AD, BD
AB. CD=AD. CD+DB. CD=AD. CD-BD. CD.

It will be seen that the sum of the right-hand sides vanishes, hence
that
BC. AD+CA. BD+AB. CD=0

for any four points on a line.

(3)

§ 11. If C is any point in the line AB, then we say that C divides the segment AB in the ratio AC/CB, account being taken of the sense of the two segments AC and CB. If C lies between A and B the ratio is positive, as AC and CB have the same sense. But if C lies without the segment AB, i.c. if C divides AB externally, then the ratio is negative. To see how the value of

QA

M

B

P

this ratio changes with FIG. 3. C, we will move C along the whole line (fig. 3), whilst A and B remain fixed. If C lies at the point A, then AC =0, hence the ratio AC:CB vanishes. As C moves towards B, AC increases and CB decreases, so that our ratio increases. At the middle point M of AB it assumes the value +1, and then increases till it reaches an infinitely large value, when C arrives at B. On passing beyond B the ratio becomes negative. If C is at P we have AC AP AB+BP, hence

Here AB <QB, hence the ratio AB:QB is positive and always less than one, so that the whole is negative and <1. If C is at the point at infinity it is-1, and then increases as C moves to the right, till for C at A we get the ratio = o. Hence

"As C moves along the line from an infinite distance to the left to an infinite distance at the right, the ratio always increases; it starts with the value-1, reaches o at A, +1 at M, at B, now changes sign too, and increases till at an infinite distance it reaches again the value-1. It assumes therefore all possible values from - ∞ to +∞, and each value only once, so that not only does every position of C determine a definite value of the ratio AC:CB, but also, conversely, to every positive or negative value of this ratio belongs one single point in the line AB.

[Relations between segments of lines are interesting as showing an application of algebra to geometry. The genesis of such relations

(a−b) (a−c) (x (x − a)+(b−c) (b − a) (x−b) +

(c − a) (c − b ) ( x − c ) = (x − a) (x − b ) (x −c); this may be proved, cumbrously, by multiplying up, or, simply, by decomposing the right-hand member of the identity into partial fractions. Now take a line ABCDX, and let AB-a, ÁCb, ÁD=c, AX=x. Then obviously (a-b) AB-AC-BC, paying regard to signs; (a-c)=AB-AD=DB, and so on. Substituting these values in the identity we obtain the following relation connecting the segments formed by five points on a line :-AB AC

AD

AX BC. BD. BX+CD. CB. CX+DB. DC. DX BX. CX. DX Conversely, if a metrical relation be given, its validity may be tested by reducing to an algebraic equation, which is an identity if the relation be true. For example, if ABCDX be five collinear points, prove

+ BC. BA+

= 1.

AD. AX, BD. BX, CD. CX AB.AC CA.CB Clearing of fractions by multiplying throughout by AB. BC. Cá, we have to prove -AD.AX. BC-BD.BX.CA-CD.CX.AB=AB. BC.CA. Take A as origin and let AB = a, AC=b, AD=c, AX=x. Substituting for the segments in terms of a, b, c, x, we obtain on simplification a2b-ab2=-ab2+ab, an obvious identity.

An alternative method of testing a relation is illustrated in the following example:-If A, B, C, D, E, F be six collinear points,

AE. AF BE. BF AB AC AD BC. BD.BA+CD.CA.CB DA DB. DC =0. Clearing of fractions by multiplying throughout by AB. BC.CD.DA, and reducing to a common origin O (calling OA=a, OB=b, &c.), an equation containing the second and lower powers of OA (=a), &c., is obtained. Calling OA=x, it is found that x=b, x=c, x=d are solutions. Hence the quadratic has three roots; consequently it is an identity.

The relations connecting five points which we have instanced above may be readily deduced from the six-point relation; the first by taking D at infinity, and the second by taking F at infinity, and then making the obvious permutations of the points.]

PROJECTION AND CROSS-RATIOS

§ 12. If we join a point A to a point S, then the point where the line SA cuts a fixed plane is called the projection of A on the plane from S as centre of projection. If we have two planes and and a point S, we may project every point A in to the other plane. If A' is the projection of A, then A is also the projection of A', so that the relations are reciprocal. To every figure in we get as its projection a corresponding figure in '.

We shall determine such properties of figures as remain true for the projection, and which are called projective properties. For this purpose it will be, sufficient to consider at first only constructions in one plane.

Let us suppose we have given in a plane two lines p and p' and a centre S (fig. 4); we may then project the points in p from S to p'.

Let A', B'... be the projections of A, B.. ..., the point at infinity in p which we shall denote by I will be projected into a finite point

I' in p', viz. into the point where the parallel to through S cuts Similarly one point J in p will be projected into the point at infinity in p'. This point J is of course the point where the parallel to p through S cuts p. We thus see that every point in p is projected into a single point in p'.

Fig. 5 shows that a segment AB will be projected into a segment A'B' which is not equal to it, at least not as a rule; and also that the ratio AC: CB is not equal to the ratio A'C': C'B' formed by the projections. These ratios will become equal only if and p' are parallel, for in this case the triangle SAB is similar to the triangle SA'B'. Between three points in a line and their projections there exists therefore in general no relation. But between four points a relation does exist.

813. Let A, B, C, D be four points in p, A', B', C', D' their projections in p', then the ratio of the two ratios AC:CB and AD: DB into which C and D divide the segment AB is equal to the corresponding expression between A', B', C', D'. In symbols we have AC AD_A'C' A'D' CB DB CB D'B''

Α'

(AB, CD)) (BA, DC) CD, AB) DC, BA))

(AC, DB)

(BD, CA)

λ

I/(1-x)

(CA, BD)

1/p (v-1)/v

(DB, AC)

(AD, BC)

(BC, AD)

(CB, DA)

(DA, CB)

(AD, CB)

(AB, DC) (BA, CD) (CD, BA) (DC, AB)) (AC, BD) (BD, AC) (CA, DB) (DB, CA).

posed the shorter name of "cross-ratio." We shall adopt the latter. We have then the

FUNDAMENTAL THEOREM.-The cross-ratio of four points in a line is equal to the cross-ratio of their projections on any other line which lies in the same plane with it.

§ 14. Before we draw conclusions from this result, we must investigate the meaning of a cross-ratio somewhat more fully.

If four points A, B, C, D are given, and we wish to form their cross-ratio, we have first to divide them into two groups of two, the points in each group being taken in a definite order. Thus, let A, B be the first, C, D the second pair, A and C being the first points in each pair. The cross-ratio is then the ratio AC: CB divided by AD: DB. This will be denoted by (AB, CD), so that

This is easily remembered. In order to write it out, make first the two lines for the fractions, and put above and below these A A the letters A and B in their places, thus, :; and then fill up, crosswise, the first by C and the other by D.

15. If we take the points in a different order, the value of the cross-ratio will change. We can do this in twenty-four different ways by forming all permutations of the letters. But of these twenty-four cross-ratios groups of four are equal, so that there are really only six different ones, and these six are reciprocals in pairs. We have the following rules:

I. If in a cross-ratio the two groups be interchanged, its value remains unaltered, i.e.

(AB, CD) = (CD, AB) = (BA, DC) = (DC, BA).

11. If in a cross-ratio the two points belonging to one of the two groups be interchanged, the cross-ratio changes into its reciprocal, i.e. (AB, CD) = 1/(AB, DC) = 1/(BA, CD) = 1/(CD, BA) = 1/(DC, AB). From I. and II. we see that eight cross-ratios are associated with (AB, CD).

III. If in a cross-ratio the two middle letters be interchanged, the cross-ratio a changes into its complement 1-a, i.e. (AB, CD) I-(AC, BD).

(-1)/1

11

17. If one of the points of which a cross-ratio is formed is the point at infinity in the line, the cross-ratio changes into a simple ratio. It is convenient to let the point at infinity occupy the last place in the symbolic expression for the cross-ratio. Thus if I is a point at infinity, we have (AB, CI)=-AC/CB, because AI: IB = -1. Every common ratio of three points in a line may thus be expressed as a cross-ratio, by adding the point at infinity to the group of points.

HARMONIC Ranges

18. If the points have special positions, the cross-ratios may have such a value that, of the six different ones, two and two become equal. If the first two shall be equal, we get A=1/A, or A2=1,

If we take +1, we have (AB, CD) = =1, or AC/CB=AD/DB; that is, the points C and D coincide, provided that A and B are different.

If we take A-1, so that (AB, CD)=-1, we have AC/CB= -AD/DB. Hence C and D divide AB internally and externally in the

same ratio.

The four points are in this case said to be harmonic points, and C and D are said to be harmonic conjugates with regard to A and B. conjugates with regard to C and D. But we have also (CD, AB)=-1, so that A and B are harmonic

The principal property of harmonic points is that their cross-ratio remains unaltered if we interchange the two points belonging to one pair, viz.

(AB, CD) = (AB, DC) = (BA, CD). For four harmonic points the six cross-ratios become equal two and two:

I

X=-1,1-λ=2,-1,-1,-1,-2

Hence if we get four points whose cross-ratio is 2 or, then they are harmonic, but not arranged so that conjugates are paired. If this is the case the cross-ratio=1.

§19. If we equate any two of the above six values of the crossratios, we get either A1, o, ∞, or λ=-1, 2, 4, or else à becomes a root of the equation X2-x+1=o, that is, an imaginary cube root of -1. In this case the six values become three and three equal, so that only two different values remain. This case, though important in the theory of cubic curves, is for our purposes of no interest, whilst harmonic points are all-important.

§ 20. From the definition of harmonic points, and by aid of § 11, the following properties are easily deduced.

If C and D are harmonic conjugates with regard to A and B, then one of them lies in, the other without AB; it is impossible to move from A to B without passing either through C or through D; the one blocks the finite way, the other the way through infinity. This is expressed by saying A and B are "separated by

For every position of C there will be one and only one point D which is its harmonic conjugate with regard to any point pair A, B.

D does. But if A and B coincide, one of the points C or D, lying If A and B are different points, and if C coincides with A or B, between them, coincides with them, and the other may be anywhere in the line. It follows that, "if of four harmonic conjugates two coincide, then a third coincides with them, and the fourth may be any point in the line."

If C is the middle point between A and B, then D is the point at infinity; for AC: CB=+1, hence AD: DB must be equal to -1 The harmonic conjugate of the point at infinity in a line with regard to two points A, B is the middle point of AB.

This important property gives a first example how metric properties are connected with projective ones.

[§ 21. Harmonic properties of the complete quadrilateral and quad.

A figure formed by four lines in a plane is called a complete quadrilateral, or, shorter, a four-side. The four sides meet in six points, named the "vertices," which may be joined by three lines (other than the sides), named the "diagonals" or " harmonic lines." The diagonals enclose the harmonic triangle of the quadrilateral." In fig. 7, A'B'C', B'AC, C'AB, CBA are the sides, A‚A', B‚B′, C,C'

the vertices, AA', BB', CC' the harmonic lines, and aßy the harmonic triangle of the quadrilateral. A figure formed by four coplanar points is named a complete quadrangle, or, shorter, a four-point. The four points may be joined by six lines, named the "sides," which intersect in three other points, termed the " diagonal or harmonic points." The harmonic points are the vertices of the "harmonic triangle of the complete quadrangle." In fig. 8, AA', BB' are the points, AA', BB', 'A'B', B’A, AB, BA' are the sides, L, M, N are the diagonal points, and LMN is the harmonic triangle of the quadrangle.

The harmonic property of the complete quadrilateral is: Any diagonal or harmonic line is harmonically divided by the other two; and of a complete quadrangle: The angle at any harmonic point is divided harmonically by the joins to the other harmonic points. To prove the first theorem, we have to prove (AA', By), (BB', ya), (CC', Ba) are harmonic. Consider the cross-ratio (CC', as). Then projecting from A on BB' we have A(CC', aß)=A(B'B, ar). Projecting from A' on BB', A'(CC', aß)=A'(BB', ay). Hence (B'B, ay)=(BB', ay), i.e. the cross-ratio (BB', ay) equals that of its reciprocal; hence the range is harmonic.

The second theorem states that the pencils L(BA, NM),M(B’A,LN), | N(BA, LM) are harmonic. Deferring the subject of harmonic pencils to the next section, it will suffice to state here that any transversal intersects an harmonic pencil in an harmonic range. Consider the pencil L(BA, NM), then it is sufficient to prove (BA', NM') is harmonic. This follows from the previous theorem by considering A'B as a diagonal of the quadrilateral ALB'M.]

This property of the complete quadrilateral allows the solution of the problem:

To construct the harmonic conjugate D to a point C with regard to two given points A and B. Through A draw any two lines, and through C one cutting the former two in G and H. Join these points to B, cutting the former two lines in E and F. The point D'where EF cuts AB will be the harmonic conjugate required.

This remarkable construction requires nothing but the drawing of lines, and is therefore independent of measurement. In a similar manner the harmonic conjugate of the line VA for two lines VC, VD is constructed with the aid of the property of the complete quadrangle.

$22. Harmonic Pencils.-The theory of cross-ratios may be extended from points in a row to lines in a flat pencil and to planes in an axial pencil. We have seen (§ 13) that if the lines which join four points A, B, C, D to any point S be cut by any other line in A', B', C', D', then (AB, CD) = (A'B', C'D'). In other words, four lines in a flat pencil are cut by every other line in four points whose cross-ratio is constant.

Definition. By the cross-ratio of four rays in a flat pencil is meant the cross-ratio of the four points in which the rays are cut by any line. If a, b, c, d be the lines, then this cross-ratio is denoted by (ab, cd).

Definition. By the cross-ratio of four planes in an axial pencil is understood the cross-ratio of the four points in which any line cuts the planes, or, what is the same thing, the cross-ratio of the four rays in which any plane cuts the four planes.

In order that this definition may have a meaning, it has to be proved that all lines cut the pencil in points which have the same cross-ratio. This is seen at once for two intersecting lines, as their plane cuts the axial pencil in a flat pencil, which is itself cut by the two lines. The cross-ratio of the four points on one line is therefore equal to that on the other, and equal to that of the four rays in the flat pencil.

If two non-intersecting lines p and q cut the four planes in A, B, C, D and A', B', C, D, draw a liner to meet both p and q, and let this line cut the planes in A", B, C, D. Then (AB, CD) = (A'B', C'D'), for each is equal to (AB, Ċ'D').

§ 23. We may now also extend the notion of harmonic elements,

viz.

Definition.-Four rays in a flat pencil and four planes in an axial pencil are said to be harmonic if their cross-ratio equals -1, that is, if they are cut by a line in four harmonic points.

If we understand by a "median line" of a triangle a line which joins a vertex to the middle point of the opposite side, and by a "median line" of a parallelogram a line joining middle points of opposite sides, we get as special cases of the last theorem:

The diagonals and median lines of a parallelogram form an harmonic pencil; and

At a vertex of any triangle, the two sides, the median line, and the line parallel to the base form an harmonic pencil.

Taking the parallelogram a rectangle, or the triangle isosceles, we get:

Any two lines and the bisections of their angles form an harmonic pencil. Or:

In an harmonic pencil, if two conjugate rays are perpendicular, then the other two are equally inclined to them; and, conversely, if one ray bisects the angle between conjugate rays, it is perpendicular to its conjugate.

This connects perpendicularity and bisection of angles with projective properties.

824. We add a few theorems and problems which are easily proved or solved by aid of harmonics.

An harmonic pencil is cut by a line parallel to one of its rays in three equidistant points.

Through a given point to draw a line such that the segment determined on it by a given angle is bisected at that point. Having given two parallel lines, to bisect on either any given segment without using a pair of compasses.

Having given in a line a segment and its middle point, to draw through any given point in the plane a line parallel to the given line. To draw a line which joins a given point to the intersection of two given lines which meet off the drawing paper (by aid of § 21).

CORRESPONDENCE. HOMOGRAPHIC and Perspective Ranges $25. Two rows, p and p', which are one the projection of the other (as in fig. 5), stand in a definite relation to each other, characterized by the following properties.

1. To each point in either corresponds one point in the other; that is, those points are said to correspond which are projections of one another. 2. The cross-ratio of any four points in one equals that of the corresponding points in the other. 3. The lines joining corresponding points all pass through the same point.

If we suppose corresponding points marked, and the rows brought into any other position, then the lines joining corresponding points will no longer meet in a common point, and hence the third of the above properties will not hold any longer; but we have still a correspondence between the points in the two rows possessing the first two properties. Such a correspondence has been called a one-one correspondence, whilst the two rows between which such correspondence has been established are said to be projective or homographic. Two rows which are each the projection of the other are therefore projective. We shall presently see, also, that any two projective rows may always be placed in such a position that one appears as the projection of the other. If they are in such a position the rows are said to be in perspective position, or simply to be in perspective.

§ 26. The notion of a one-one correspondence between rows may be extended to flat and axial pencils, viz. a flat pencil will be said to be projective to a flat pencil if to each ray in the first corresponds one ray in the second, and if the cross-ratio of four rays in one equals that of the corresponding rays in the second.

Similarly an axial pencil may be projective to an axial pencil. But a flat pencil may also be projective to an axial pencil, or either pencil may be projective to a row. The definition is the same in each case: there is a one-one correspondence between the elements, and four elements have the same cross-ratio as the corresponding ones. § 27. There is also in each case a special position which is called perspective, viz.

1. Two projective rows are perspective if they lie in the same plane, and if the one row is a projection of the other.

2. Two projective flat pencils are perspective (1) if they lie in the same plane, and have a row as a common section; (2) if they lie in the same pencil (in space), and are both sections of the same axial pencil; (3) if they are in space and have a row as common section, or are both sections of the same axial pencil, one of the conditions involving the other.

3. Two projective axial pencils, if their axes meet, and if they have a flat pencil as a common section.

4. A row and a projective flat pencil, if the row is a section of the pencil, each point lying in its corresponding line.

5. A row and a projective axial pencil, if the row is a section of the pencil, each point lying in its corresponding line.

6. A flat and a projective axial pencil, if the former is a section of the other, each ray lying in its corresponding plane.

That in each case the correspondence established by the position indicated is such as has been called projective follows at once from the definition. It is not so evident that the perspective position may always be obtained. We shall show in § 30 this for the first three