difficulties. They prove elaborately, by a reductio ad absurdum, II. PROJECTIVE GEOMETRY that the volumes cannot be unequal. This proof must be read in the Elements. We must, however, state that we have in the above It is difficult, at the outset, to characterize projective geometry not proved Euclid's Prop. 5, but only a special case of it. Euclid as compared with Euclidean. But a few examples will at least does not suppose that the bases of the two pyramids to be compared indicate the practical differences between the two. are equal, and hence he proves that the volumes are as the bases. In Euclid's Elements almost all propositions refer to the magni. The reasoning of the proof becomes clearer in the special case, from which the general one may be easily deduced. tude of lines, angles, areas or volumes, and therefore to measure$ 86. Prop. 6 extends the result to pyramids with polygonal ment. The statement that an angle is right, or that two straight bases. From these results follow again the rules at present given lines are parallel, refers to measurement. On the other hand, triangular prism having the same base and the same altitude. But the fact that a straight line does or does not cut a circle is inde a triangular prism is equal in volume to a parallelepiped which pendent of measurement, it being dependent only upon the has the same base and altitude. Hence it B is the base and k the mutual"position" of the line and the circle. This difference altitude, we have becomes clearer if we project any figure from one plane to another Volume of prism = BR, (see PROJECTION). By this the length of lines, the magnitude Volume of pyramid=fBk, of angles and areas, is altered, so that the projection, or shadow, statements which have to be taken in the sense that B means the of a square on a plane will not be a square; it will, however, number of square units in the base, h the number of units of length be some quadrilateral. Again, the projection of a circle will not in the altitude, or that B and h denote the numerical values of base be a circle, but some other curve more or less resembling a circle. and altitude. But one property may be stated at once-no straight line can cut $ 87. A method similar to that used in proving Prop. 5 leads to the following results relating to solids bounded by simple curved straight line can cut a circle in more than two points. There the projection of a circle in more than two points, because no surfaces Prop. 10. Every cone is the third part of a cylinder which has the are, then, some properties of figures which do not alter by same base, and is of an equal altinde with il. projection, whilst others do. To the latter belong nearly all Prop. 11. Cones or cylinders of the same altitude ere to one onolher properties relating to measurement, at least in the form in which as their bases Prop. 12. Similar cones or cylinders have to one another the triplicate they are generally given. The others are said to be projective ratio of that which the diameters of their bases have. properties, and their investigation forms the subject of projective Prop. 13. If a cylinder be cul by a plane parallel to its opposite geometry. plancs or bases, il divides the cylinder into two cylinders, one of which Different as are the kinds of properties investigated in the old is to the other as the axis of the first to the axis of the other; which may also be stated thus and the new sciences, the methods followed differ in a still Cylinders on the same base are proportional to their altitudes. greater degree. In Euclid each proposition stands by itself, Prop. 14. Çones or cylinders upon equal bases are to one another its connexion with others is never indicated; the leading ideas as their altitudes. contained in its proof are not stated; general principles do not Prop. 15. The bases and altitudes of equal cones or cylinders are reciprocally proportional, and if the bases and altitudes be reciprocally exist. In the modern methods, on the other hand, the greatest proportional, the concs or cylinders are equal to one another. importance is attached to the leading thoughts which pervade These theorems again lead to formulae in mensuration, if we the whole; and general principles, which bring whole groups of compare a cylinder with a prism having its base and altitude equal to theorems under one aspect, are given rather than separate promethod of exhaustion. We get, then, the result that their bases are positions. The whole tendency is towards generalization. equal, and have, if B denotes the numerical value of the base, and a straight line is considered as given in its entirety, extending h that of the altitude, both ways to infinity, while Euclid never admits anything but Volume of cylinder = Bhn finite quantities. The treatment of the infinite is in fact another Volume of cone jBk. fundamental difference between the two methods: Euclid avoids 9.88. The remaining propositions relate to circles and spheres. it; in modern geometry it is systematically introduced. of the sphere only one property is proved, viz. of the different modern methods of geometry, we shall treat Prop. 18. Spheres have to one another the triplicate ratio of that principally of the methods of projection and correspondence which which their diameters have. The mensuration of the sphere, like have proved to be the most powerful. These have become indethat of the circle, the cylinder and the cone, had not been settled pendent of Euclidean Geometry, especially through the Geometric in the time of Euclid. It was done by Archimedes. der Lage of V. Staudt and the Ausdehnungslehre of Grassmann. For the sake of brevity we shall presuppose a knowledge of BOOK XIII. Euclid's Elements, although we shall use only a few of his pro $ 89. The 13th and last book of Euclid's Elements is devoted to positions. the regular solids (see POLYHEDRON). It is shown that there are five of them, viz. : $ 1. Geometrtcal Elements. We consider space as filled with points, 1. The regular letrahedron, with 4 triangular faces and 4 vertices; lines and planes, and these we call the elements out of which our 2. The cube, with 8 vertices and 6 square faces; figures are to be formed, calling any combination of these elements a 3. The octahedron, with 6 vertices and 8 triangular faces; figure." 4. The dodecahedron, with 12 pentagonal laces, 3 at each of the By a line we mean a straight line in its entirety, extending both co vertices; ways to infinity; and by a plane, a plane surface, extending in all 5. The icosahedron, with 20 triangular faces, s at each of the directions to infinity. 12 vertices. We accept the three-dimensional space of experience the space It is shown how to inscribe these solids in a given sphere, and assumed by Euclid--which has for its properties (among others) how to determine the lengths of their edges. Through any two points in space one and only one line may be $90. The 13th book, and therefore the Elements, conclude withdrawn; the scholium, that no other regular solid exists besides the five Through any three points which are not in a line, one and only one ones enumerated." plane may be placed; The proof is very simple. Each face is a regular polygon, hence The intersection of two planes is a line; the angles of the faces at any vertex must be angles in equal regular A line which has two points in common with a plane lies in the polygons, must be together less than four right angles (XI. 21), and plane, hence the intersection of a line and a plane is a single pohit; and must be three or more in number. Each angle in a regular triangle Three planes which do not meet in a line have one single point in 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 results may be stated differently in the following formThese give the solid angles of the tetrahedron, the octahedron and I. A plane is determined A point is determined the icosahedron. The angle in a square (the regular quadrilateral) i. By three points which do 1. By three planes which do equals one right angle. Hence three will form a solid angle, that not lie in a line; of the cube, and four will not. The angle in the regular pentagon not pass through a line; 2. By two intersecting lines; equals f of a right angle. Hence three of them equal (i.e. less 2. By two intersecting lines, 3. By a line and a point 3. By a plane and a line than 4) right angles, and forin the solid angle of the dodecahedron. 'which does not lie in it. which does not lie in it Three regular polygons of six or more sides cannot form a solid II. A line is determinedangle. Therefore no other regular solids are possible. (O. H.) 1. By two poi its; 2. By two planes. common. B If we It will be observed that not only are planes determined by points, As immediate consequences we get the propositions: but also points by planes; that therefore the planes may be con Every line meets a plane in one point, or it lies in it; sidered as elements, like points; and also that in any one of the Every plane meets every other plane in a line; above statements we may interchange the words point and plane, Any two lines in the same plane meet. and we obtain again a correct statement, provided that these $ 5. Aggregates of Geometrical Elements:-We have called points, statements themselves are true. As they stand, we ought, in lines and planes the elements of geometrical figures. We also say several cases, to add if they are not parallel," or some such words, that an element of one kind contains one of the other it it lics in it parallel lines and planes being evidently lest altogether out of or passes through it. consideration. To correct this we have to reconsider the theory of All the elements of one kind which are contained in one or two parallels. elements of a different kind form aggregates which, have to be $ 2. Parallels. Point al Infinity --Let us take in a plane a line penumerated. They are the following: (fig. 1), a point S not in this line, and a line drawn through S. 1. Of one dimension. Then this line will meet 1. The row, or range, of points formed by all points in a line, the line p in a point A. If which is called its base. we turn the line q about S 2. The flat pencil formed by all the lines through a point in towards &. its point of a plane. Its base is the point in the plane. intersection with p. will 3. The axial pencil formed by all planes through a line move along P towards B, which is called its base or axis. 1. The field of points and lines that is, a plane with all its distance, until it is moved points and all its lines. out of our reach. 2. The pencil of lines and planes that is, a point in space q' turn q still farther, its con with all lines and all planes through it. The space of points-that is, all points in space. section has disappeared to IV. or four dimensions. The space of lines, or all lines in space, to the left. There is one intermediate position where g is parallel $6. Meaning of "Dimensions.”—The word dimension in the above to pthat is where it does not cut p., 'In every other position it needs explanation. If in a plane we take a row p and a pencil with cuts p in some finite point. If, on the other hand, we move the point centre , then through every point in p one line in the pencil will A to an infinite distance in p, then the line q which passes through pass, and every ray in Q will cut p in one point, so that we are A will be a line which does not cut p at any finite point. Thus we entitled to say a row contains as many points as a fat pencil lines, are led to say: Every line through s which joins it to any point and, we may add, as an axial pencil planes, because an axial pencil at an infinite distance in p is parallel to p. But by Euclid's 12th is cut by a plane in a flat pencil . axiom there is but one line parallel to p through S. The difficulty in The number of elements in the row, in the flat pencil, and in the which we are thus involved is due to the fact that we try to reason axial pencil is, of course, infinite and indefinite too, but the same in about infinity as if we, with our finite capabilities, could comprehend all. This number may be denoted by . Then a plane contains the infinite. To overcome this difficulty, we may say that all points o? points and as many lines. To see this, take a flat pencil in a at infinity in a line appear to us as one, and may be replaced by a plane. It contains lines, and each line contains co points, whilst single "ideal " point. each point in the plane lies on one of these lines. Similarly, in a We may therefore now give the following definitions and axiom : plane each line cuts a fixed line in a point. But this line is cut at Definition.--Lines which meet at infinity are called parallel. each point by co lines and contains es points ; hence there are o Axiom.--All points at an infinite distance in a line may be con lines in a plane. sidered as one single point. A pencil in space contains as many lines as a plane contains Definition.-This ideal point is called the point al infinity in the points and as many planes as a plane contains lines, for any plane line. cuts the pencil in a held of points and lincs. Hence a pencil conThe axiom is equivalent to Euclid's Axiom 12, for it follows from tains ? lines and op planes. The field and the pencil are of two either that through any point only one line may be drawn parallel dimensions. to a given line. To count the number of points in space we observe that each This point at infinity in a line is reached whether we move a point lies on some line in a pencil. But the pencil contains co? point in the one or in the opposite direction of a line to infinity. lines, and cach line s points; hence space contains and points. A line thus appears closed by this point, and we speak as if we Each plane cuts any fixed plane in a line. But a plane contains could move a point along the line from one position A to another oo? lines, and through each pass co planes; therefore space contains B in two ways, either through the point at inhnity or through finite planes. points only. Hence space contains as many planes as points, but it contains It must never be forgotten that this point at infinity is ideal; an infinite number of times more lines than points or planes. To in fact, the whole notion of "infinity" is only a mathematical count them, notice that every line cuts a fixed plane in one point. conception, and owes its introduction (as a method of research) to But co2 lines pass through each point, and there are co2 points in the the working generalizations which it permits. plane. Hence there are * lines in space. The space of points $ 3. Line and Plane at Infinity.--Having arrived at the notion of and planes is of three dimensions, but the space of lines is of four replacing all points at infinity in a line by one ideal point, there is no dimensions. difficulty in replacing all points at infinity in a plane by one ideal A field of points or lines contains an infinite number of rows and line. flat pencils; a pencil contains an infinite number of fat pencils To make this clear, let us suppose that a line p, which cuts two and of axial pencils; space contains a triple infinite number of fixed lines a and b in the points A and B; moves parallel to itself pencils and of fields, crows and axial pencils and co. Aat pencilsto a greater and greater distance, It will at last cut both a and or, in other words, each point is a centre of ? flat pencils. b at their points at infinity, so that a line which joins the two points $ 7. The above enumeration allows a classification of figures. at infinity in two intersecting lines lies altogether at infinity. Every Figures in a row consist of groups of points only, and figures in other line in the plane will meet it therefore at infinity, and thus it the flat or axial pencil consist of groups of lines or planes. In the contains all points at infinity in the plane. plane we may draw polygons; and in the pencil or in the point, All points al infinity in a plane lie in a line, which is called the line solid angles, and so on. at infinity in the plane. We may also distinguish the different measurements We have It follows that parallel planes must be considered as planes In the row, length of segment; having a common line at inhnity, for any other plane cuts them in In the flat pencil, angles; parallel lines which have a point at infinity in common. In the axial pencil, dihedral angles between two planes; If we next take two intersecting planes, then the point at infinity In the plane, areas; in their line of intersection lies in both planes, so that their lines In the pencil, solid angles; at infinity meet. Hence every line at infinity meets every other In the space of points or planes, volumes. line at infinity, and they are therefore all in one plane. Au points al infinity in space may be considered as lying in one SEGMENTS OF A LINE ideal plane, which is called the plane at infinity. $ 8. Any two points A and B in space determine on the line through 4. Parallelism.---We have now the following definitions: them a finite part, which may be considered as being described by Parallel lines are lines which meet at infinity; a point moving from A to B. This we shall denote by AB, and Parallel planes are planes which meet at infinity; distinguish it from BA, which is supposed as being described by : A line is parallel to a plane if it meets it at infinity. point moving from B to A, and hence in a direction or in a Theorems like this-Lines (or planes) which are parallel to a third opposite to AB. Such a finite line, which has a definite sense, we are parallel to each other-follow at once. shall call a " segment," so that AB and BA denote different segments, This view of parallels leads therefore to no contradiction of which are said to be equal in length but of opposite sense. The one Euclid's Elements. sense is often called positive and the other negative. X1.11 2a " sense sum A B sense. In introducing the word "sense" for direction in a line, we have from algebraic identities is very simple. For example, if a, b, 6 % the word direction reserved for direction of the line itself, so that be any lour qualititics, then different lines have different directions, unless they be parallel, b whilst in each line we have a positive and negative sense. Ta-b) (a-c)(x-a) 56-6) (6-a)(x-7)+ We may also say, with Clifford, that AB denotes the step " of going from A to B. 8 9. If we have three points A, B, C in a line (fig. 2), the step AB (-a)(c=b)(0-c) (-2)(x-7)(3-6) will bring us from A to B, and the step this may be proved, cumbrously, by multiplying up, or, simply, by BC from B to C. Hence both steps are decomposing the right hand member of the identity into partial equivalent to the one step AC. This is fractions. Now take a line ABCDX, and let AB = 2, ÁC=b, AD=C, expressed by saying that AC is the AX=x. Then obviously (0-6) = AB-AC=-BC, paying regard of AB and BC; in symbols to signs; (a-c)= AB-AD=DB, and so on. Substituting these AB+BC=AC, values in the identity we obtain the following relation connecting the segments formed by five points on a line :-- AB AX Conversely, if a metrical relation be given, its validity may be tested by reducing to an algebraic equation, which is an identity AB+BA=0, (1) if the relation be true. For example, if ABCDX be five collinear and similarly points, prove AD. AX, BD, BX, CD.CX (2) AB. AC + BC. BA +CA.CB which again is true for any three points in a line. We further write Clearing of fractions by multiplying throughout by AB. BC. Cá, we have to prove AB=-BA. -AD.AX.BC-BD.BX,CA-CD.CX.AB=AB.BC.CA. where - denotes negative sense. We can then, just as in algebra, change subtraction of segments for the segments in terms of a, b, c, x, we obtain on simplification Take A as origin and let AB=, AC=b, AD=C, AX =x. Substituting 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 a b-ab* =-ab' ta'b, an obvious identity, of this. The 'sense is, in fact, in every respect equivalent to the sign" of a number in algebra. An alternative method of testing a relation is illustrated in the 10. Of the many formulae which exist between points in a line following example:- If A, B, C, D, E, F be six collinear points, then 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 AE. AF BE. BF CE.CF DE . DF AB AC AD BC. BD.BA+CD.CA.CB+DA DB .DC=0. BC=BD+DC, CA =CD+DA, AB=AD+DB; or multiplying these by AD, BD, CD respectively, we get Clearing of fractions by multiplying throughout by AB.BC.CD.DA, and reducing to a common origin O (calling OA=a, OB = b, &c.), BC. AD=BD. AD+DC. AD=BD , AD-CD. AD an equation containing the second and lower powers of OA (=a), CA. BDCD.BD+DA. BD=CD, BD-AD. BD &c., is obtained. Calling OA=x, it is found that x=0, x=0, x=d AB. CD= AD. CD+DB.CD=AD.CD-BD.CD. are solutions. Hence the quadratic has three roots; consequently it is an identity. It will be seen that the sum of the right-hand sides vanishes, hence The relations connecting five points which we have instanced above that may be readily deduced from the six-point relation; the first by BC. AD+CA. BD+AB. CD=0 taking D at infinity, and the second by taking Fat infinity, and then for any four points on a line. making the obvious permutations of the points.) $11. 11 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 PROJECTION AND CROSS-Ratios sense of the two segments AC and CB. II C lies between A and B § 12. If we join a point A to a point $, then the point where the the ratio is positive, as AC and CB have the same sense. But it line SA cuts a fixed plane x is called the projection of A on the C lies without the segment AB, 1.c. il C divides AB externally, then plane , from S as centre of projection. If we have two planes + the ratio is negative and s' and a point ș, we may project every point A in to the B To see how the value of other plane. If A is the projection of A, then A is also the prothis ratio changes with ection of A', so that the relations are reciprocal. To every figure C, we will move Calong in 7 we get as its projection a corresponding figure in r'. the whole line (fig. 3), We shall determine such properties of figures as remain true for whilst A and B remain fixed. If C lies at the point A, then AC =0, the projection, and which are called projective properties. For this hence the ratio AC:CB vanishes. As C moves towards B, AC purpose it will be sufficient to consider at first only constructions in increases and CB decreases, so that our ratio increases. At the one plane. middle point M of AB it assumes the value +1, and then increases Let us suppose we have given in a plane two lines p and p' and a till it reaches an infinitely large value, when Carrives at B. On centre S (fig. 4); we way then project the points in p from 'S to p. passing beyond B the ratio becomes negative. If C is at P we have AC=AP=AB+BP, hence AC AB BP AB CB = PB +PB In the last expression the ratio AB:BP is positive, has its greatest value co when C coincides with B, and vanishes when BC becomes infinite. Hence, as C moves from B to the right to the point at p infinity, the ratio AC:CB varies from-to-i. If. on the other hand, C is to the left of A, say at Q. we have AC AB AC = AQ-AB+BQ-AB-QB, hence CB-QB-1 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 Cat A we get the ratio =0. 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 - !, reaches o at A, +1 at M, o at B, now changes sign to -60, and increases till at an infinite distance it reaches again the value-1. Il assumes therefore all possible values from -o lo+00, and each value only once, so that not only does every position of C determine a definite value of the ralio 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 Let A', B'... be the projections of A, B ..., the point at infinity in application of algebra to geometry. The genesis of such relations which we shall denote by I will be projected into a finite point Ο Α M P FIG. 3. BP-1. S FIG. 4. 1 1/(1-») (BD, CA) Di, as Au al Dil D ICO B Ic D l' in p'. viz. into the point where the parallel to p through S cuts 1. reducing to a common origin. There are therefore four equations Similarly, one point J in p will be projected into the point between three unknowns; hence if one cross-ratio be given, the at infinity in p'. This point J is of course the point where the remaining twenty-three are determinate. Moreover, two of the parallel top through $ cuts P. We thus see that every point in quantities 1,44 , v are positive, and the remaining one negative. is projected into a single point in p'. The following scheme shows the twenty-four cross-ratios expressed Fig. 5. shows that a segment AB will be projected into a segment | in terms of d, , v.] 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 (AB, CD) (AC, DB) (CA, BD) (N-1)/ SA'B'. Between three points in a line and their pro (DC, BA)) (DB, AC jections there exists therefore in general no relation. (AB, DC) (AD BC) But between four points a relation does exist. (BA, CD) $ 13. Let A, B, C, D be four points in P, A', B', ICD, BA) IA 1/(1-x) (BC, AD) (4-1)AM/64-1) C, D their projections in then the ratio of the two (DC, AB) (DA, CB) ratios AC:CB' and AD: DB into which C and D (AC, BD) (AD, CB divide the segment AB is equal to the corresponding (BD, ACÍ expression between A', B', C', D'. In symbols we have (BC, DA) v/(2-1) x/(1-1)(x-1) 1/v (DA, BC), CB: DB CB D'B' This is easily proved by aid of similar triangles. Through the points A and B on p draw parallels to p', which cut $ 17. If one of the points of which a cross-ratio is formed is the the projecting, rays in point at infinity in the line, the cross-ratio changes into a simple Inwide Boys Ca. Dz, B, and Ai, C, ratio. It is convenient to let the point at infinity occupy the last Tring indicated in place in the symbolic expression for the cross-ratio. Thus if I is a rity VISA fig. 6. The two triangles point at infinity, we have (AB, CI)=-AC/CB, because AI:IB=-1. ACC, and BCC, will be Every common ratio of three points in a line may thus be exweb de similar, as will also be pressed as a cross-ratio, by adding the point at infinity to the group wil ny the triangles ADD, and of points. HARMONIC RANGES $ 18. If the points have special positions, the cross-ratios may This result is of funda have such a value that, of the six different ones, two and two become mental importance. cqual. If the first two shall be equal, we get 1=1/, or \*= 1, The expression P AC/CB:AD/DB has been If we take 1 = +1, we have (AB, CD)= 1, or AC/CB = AD/DB; RHEU called by Chasles the that is, the points C and D coincide, provided that A and B are different. "anharmonic ratio of the Fig. 6. four points A, B, C, D." |-ADIDB. Hence c'and D divide AB internally and externally in the If we take 1=-1, so that (AB, CD)= -1, we have AC/CB = Professor Clifford pro cross-ratio." posed the shorter name of " We shall adopt the same ratio. latter. We have then the The four points are in this case said to be harmonic points, and FUNDAMENTAL THEOREM.—The cross-ratio of four points in a Cand D are said to be harmonic conjugates with regard to A and B. line is equal to the cross-ratio of their projections on any other line But we have also (CD, AB) = -1, so that A and B are harmonic which lies in the same plane with it. conjugates with regard to C and D. 14. Before we draw conclusions from this result, we must in The principal property of harmonic points is that their cross-ratio vestigate the meaning of a cross-ratio somewhat more fully. remains unaltered if we interchange the two points belonging to one If four points A, B, C, D are given, and we wish to form their pair, viz. cross-ratio, we have first to divide them into two groups of two, (AB, CD) = (AB, DC)=(BA, CD). the points in each group being taken in a definite order. Thus, For four harmonic points the six cross-ratios become equal two let A, B be the first, c, b the second pair, A and C being the first and two: 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 -1 Hence if we get four points whose cross-ratio is 2 or }, then they AC AD are harmonic, but not arranged so that conjugates are paired. If this is the case the cross-ratio=-I. This is easily remembered. In order to write it out, make first 9:19. If we equate any two of the above six values of the crossthe two lines for the fractions, and put above and below these ratios, we get either ^=1, 0, 0, ord=-1, 2, 1, or else X becomes a root of the equation /a-+1=0, that is, an imaginary cube root of A the letters A and B in their places, thus, B:48; and then fill -1. In this case the six values become three and three equal, so that only two different values remain. This case, though important up, crosswise, the first by C and the other by D. in the theory of cubic curves, is for our purposes of no interest, $15. If we take the points in a different order, the value of the whilst harmonic points are all-important. cross-ratio will change. We can do this in twenty-four different $ 20. From the definition of harmonic points, and by aid of $11, ways by forming all permutations of the letters. But of these the following properties are easily deduced. twenty-four cross-ratios groups of four are equal, so that there are If C and D are harmonic conjugates with regard to A and B, really only six different ones, and these six are reciprocals in pairs. then one of them lies in, the other without AB; it is impossible We have the following rules: to move from A to B without passing either through C or through 1. I in a cross-ratio the two groups be interchanged, its value D: the one blocks the finite way, the other the way through in remains unaltered, i.e. finity. This is expressed by saying A and B are "separated" by Cand D. (AB, CD) = (CD, AB) = (BA, DC) = (DC, BA). For every position of C there will be one and only one point II. If in a cross-ratio the two points belonging to one of the two D which is its harmonic conjugate with regard to any point pair groups be interchanged, the cross-ratio changes into its reciprocal, i.e. A, B. (AB.CD) = 1/(AB, DC)=1/(BA, CD) = 1/(CD, BA) = 1/(DC, AB). D does. But if A and B coincide, one of the points C or D, lying If A and B are different points, and if C coincides with A or B. From I. and II. we see that eight cross-ratios are associated with between them, coincides with them, and the other may be anywhere (AB, CD): in the line. It follows that, "it of four harmonic conjugates two 111. If in a cross-ratio the two middle letters be interchanged, coincide, then a third coincides wiih ihem, and the fourth may be any the cross-ratio a changes into its complement 1-a, i.e. (AB, CD)= point in the line.". I-(AC, BD). !1.C is the middle point between A and B, then D is the point at 1$ 16. II = (AB, CD), y=(AC, DB), v= (AD, BC), then , #, * infinity; for AC: CB=+!, hence AD: DB must be equal to - 1 and their reciprocals 1/4, 1/4, ily are the values of the total number The harmonic conjugale of the point al infinity in a line with regard of twenty-four cross-ratios. Moreover, d. Me v are connected by the lo two points A, B is the middle point of AB. relations This important property gives a first example how metric pro A +1/4=u+1/v=y+1/4 = -lur=1, perties are connected with projective ones. 15.21. Harmonic properties of the complete quadrilateral and quad. this proposition may be proved by substituting for , *, » and rangle. (AB, CD)=CB DB M FIG. 7. A figure formed by four lines in a plane is called a complete quadri Definition.-Four rays in a fat pencil and four plancs in an axial lateral, or, shorter, a four-side. The four sides meet in six points, pencil are said to be harmonic if their cross-ratio equals - 1, that is, named the “ vertices," which may be joined by three lines (other if they are cut by a line in four harmonic points. than the sides), named the " diagonals" or " harmonic lines." The If we understand by a "median line" of a triangle a line which diagonals enclose the harmonic triangle of the quadrilateral." In joins a vertex to the middle point of the opposite side, and by a fig. 7, A'B'C', B'AC, C'AB, CBA are the sides, A,A', B,B', C,C' '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 medion lines of a parallelogram form an harmonic pencil; and At a verlex of any triangle, the two sides, the median line, and the line parallel to the base form on harmonic pencil. Taking the parallelogram a rectangle, or the triangle isosceles, we get: Any two lines and the bisections of their ongles form on hormonic 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 conjugale rays, it is perpendicular lo its conjugale. This connects perpendicularity and bisection of angles with FIG. 8. projective properties. $24. We add a few theorems and problems which are easily proved the vertices, AA', BB', CC' the harmonic lines, and aBy the harmonic or solved by aid of harmonics. triangle of the quadrilateral. A figure formed by four coplanar An harmonic pencil is cut by a line parallel to one of its rays in points is named a complete quadrangle, or, shorter, a four-point: three equidistant points. The four points may be joined by six lines, named the sides," Through a given point to draw a line such that the segment which intersect in three other points, termed thu " diagonal or determined on it by a given angle is bisected at that point. harmonic points." The harmonic points are the vertices of the Having given two parallel lines, to bisect on either any given " harmonic triangle of the complete quadrangle." In fig. 8. AA: segment without using a pair of compasses. 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 through any given point in the plane a line parallel to the given line. Having given in a line a segment and its middle point, to draw of the quadrangle. To draw a line which joins a given point to the intersection of two The harmonic property of the complete quadrilateral is: Any given lines which meet off the drawing paper (by aid of $ 21). diagonal or harmonic line is harmonically divided by the other two; and of a complete quadrangle: The angle at any harmonic CORRESPONDENCE. HOMOGRAPHIC AND PERSPECTIVE RANGES point is divided harmonically by the joins to the other harmonic $ 25, Two rows, p and p', which are one the projection of the points. To prove the first theorem, we have to prove. (AA', By), other (as in fig. 5), stand in a definite relation to each other, char(BB', ya), (CC',Ba) are harmonic. Consider the cross-ratio (CC', aß). acterized by the following properties. Then projecting from A on BB' we have A(CC', aß) = A(B'B, ay). 1. To cach point in either corresponds one point in the other; that Projecting from A' on BB', A'(CC', aß) = A'(BB', ay); Hence is, those points are said to correspond which are projections of one (B'B, ay) = (BB', ay), i.e. the cross-ratio (BB', ay) equals that of its another. reciprocal; hence the range is harmonic. 2. The cross-ratio of any four points in one equals that of the correThe second theorem states that the pencils L(BANM),M(B'A,LN). sponding points in the other. N(BA, LM) are harmonic. Deferring the subject of harmonic pencils 3. The lines joining corresponding points all pass through the same to the next section, it will suffice to state here that any transversal point. intersects an harmonic pencil in an harmonic range. Consider the If we suppose corresponding points marked, and the rows brought pencil L(BA, NM), then it is sufficient to prove (BA', NM') is har into any other position, then the lines joining corresponding points monic. This follows from the previous theorem by considering A'B will no longer meet in a common point, and hence the third of the as a diagonal of the quadrilateral ALB M.) above properties will not hold any longer; but we have still a This property of the complete quadrilateral allows the solution correspondence between the points in the two rows possessing the first of the problem: two properties. Such a correspondence has been called a one-one To construct the harmonic conjugate D to a point C with regard lo two correspondence, whilst the two rows between which such correspondgiven poinls A and B. ence has been established are said to be projective or homographic. Through a draw any two lines, and through C one cutting the Two rows which are each the projection of the other are therefore former two in G and H. Join these points to B, cutting the former projective. We shall presently see, also, that any two projective two lines in E and F. The point D'where EF cuts AB will be the rows may always be placed in such a position that one appears as harmonic conjugate required. the projection of the other. If they are in such a position the rows This remarkable construction requires nothing but the drawing are said to be in perspective position, or simply to be in perspeclive. of lines, and is therefore independent of measurement. In a similar $26. The notion of a one-one correspondence between rows may manner the harmonic conjugate of the line VA for two lines VC, be extended to flat and axial pencils, viz. a flat pencil will be said VD is constructed with the aid of the property of the complete to be projective to a flat pencil is to each ray in the first corresponds quadrangle. one ray in the second, and if the cross-ratio of four rays in one equals $22. Harmonic Pencils.-The theory of cross-ratios may be ex- that of the corresponding rays in the second. tended from points in a row to lines in a flat pencil and to planes in Similarly an axial pencil may be projective to an axial pencil. an axial pencil. We have seen ($13) that if the lines which join four But a flat pencil may also be projective to an axial pencil, or either points A, B, C, D to any point S be cut by any other line in A', B', C', pencil may be projective to a row. The definition is the same in each D', then (AB, CD)=(A'B', C'D'). In other words, four lines in a case: there is a one-one correspondence between the elements, and Aat pencil are cut by every other líne in four points whose cross-ratio lour elements have the same cross-ratio as the corresponding ones. is constant. $ 27. There is also in each case a special position which is called Definition.-By the cross-ratio of four rays in a flat pencil is perspective, viz. meant the cross-ratio of the four points in which the rays are cut 1. Two projective rows are perspective if they lie in the same by any line. If a, b, , d be the lines, then this cross-ratio is denoted plane, and if the one row is a projection of the other. by (ab, cd). 2. Two profective flat pencils are perspective (1) if they lie in Definition.-By the cross-ratio of four planes in an axial pencil the same plane, and have a row as a common section; (2) 'if they is understood the cross-ratio of the four points in which any line lie in the same pencil fin space), and are both sections of the same cuts the planes, or, what is the same thing, the cross-ratio of the axial pencil; (3) if they are in space and have a row as common four rays in which any plane cuts the four planes. section, or are both sections of the same axial pencil, one of the In order that this definition may have a meaning, it has to be conditions involving the other. proved that all lines cut the pencil in points which have the same 3. Two projective axial pencils, if their axes meet, and if they cross-ratio. . This is seen at once for two intersecting lines, as their have a flat pencil as a common section. plane cuts the axial pencil in a flat pencil, which is itself cut by 4. A row and a projective flat pencil, if the row is a section of the the two lines. The cross-ratio of the four points on one line is pencil, each point lying in its corresponding line. therefore equal to that on the other, and equal to that of the four 5. A row and a projective axial pencil, if the row is a section of the rays in the flat pencil. pencil, each point lying in its corresponding line. If two non-intersecting lines p and q cut the lour planes in A, B, C, D and A’, B', C', D', draw a liner to meet both p and q, and of the other, cach ray lying in its corresponding plane. 6. Á flat and a projective axial pencil, if the former is a section let this line cut the planes in A', B’, C", D'. Then (AB, CD)= That in each case the correspondence established by the position (A'B', C'D'), for each is equal to (AB', Ć"D"). indicated is such as has been called projective follows at once from $ 23. We may now also extend the notion of harmonic elements, the definition. It is not so evident that the perspective position may viz. always be obtained. We shall show in $ 30 this for the first three |