*181. Poncelet and Cauchy.* The efforts of Poncelet to compel the acceptance of this principle independent of analysis resulted in a bitter and perhaps fruitless controversy between him and the great analyst Cauchy. In his review of Poncelet's great work on the projective properties of figures(18) Cauchy says, "In his preliminary discourse the author insists once more on the necessity of admitting into geometry what he calls the 'principle of continuity.' We have already discussed that principle ... and we have found that that principle is, properly speaking, only a strong induction, which cannot be indiscriminately applied to all sorts of questions in geometry, nor even in analysis. The reasons which we have given as the basis of our opinion are not affected by the considerations which the author has developed in his Traite des Proprietes Projectives des Figures." Although this principle is constantly made use of at the present day in all sorts of investigations, careful geometricians are in agreement with Cauchy in this matter, and use it only as a convenient working tool for purposes of exploration. The one-to-one correspondence between geometric forms and algebraic analysis is subject to many and important exceptions. The field of analysis is much more general than the field of geometry, and while there may be a clear notion in analysis to, correspond to every notion in geometry, the opposite is not true. Thus, in analysis we can deal with four cooerdinates as well as with three, but the existence of a space of four dimensions to correspond to it does not therefore follow. When the geometer speaks of the two real or imaginary intersections of a straight line with a conic, he is really speaking the language of algebra. Apart from the algebra involved, it is the height of absurdity to try to distinguish between the two points in which a line fails to meet a conic!



*182. The work of Poncelet.* But Poncelet's right to the title "The Father of Modern Geometry" does not stand or fall with the principle of contingent relations. In spite of the fact that he considered this principle the most important of all his discoveries, his reputation rests on more solid foundations. He was the first to study figures in homology, which is, in effect, the collineation described in 175, where corresponding points lie on straight lines through a fixed point. He was the first to give, by means of the theory of poles and polars, a transformation by which an element is transformed into another of a different sort. Point-to-point transformations will sometimes generalize a theorem, but the transformation discovered by Poncelet may throw a theorem into one of an entirely different aspect. The principle of duality, first stated in definite form by Gergonne,(19) the editor of the mathematical journal in which Poncelet published his researches, was based by Poncelet on his theory of poles and polars. He also put into definite form the notions of the infinitely distant elements in space as all lying on a plane at infinity.



*183. The debt which analytic geometry owes to synthetic geometry.* The reaction of pure geometry on analytic geometry is clearly seen in the development of the notion of the class of a curve, which is the number of tangents that may be drawn from a point in a plane to a given curve lying in that plane. If a point moves along a conic, it is easy to show—and the student is recommended to furnish the proof—that the polar line with respect to a conic remains tangent to another conic. This may be expressed by the statement that the conic is of the second order and also of the second class. It might be thought that if a point moved along a cubic curve, its polar line with respect to a conic would remain tangent to another cubic curve. This is not the case, however, and the investigations of Poncelet and others to determine the class of a given curve were afterward completed by Pluecker. The notion of geometrical transformation led also to the very important developments in the theory of invariants, which, geometrically, are the elements and configurations which are not affected by the transformation. The anharmonic ratio of four points is such an invariant, since it remains unaltered under all projective transformations.



*184. Steiner and his work.* In the work of Poncelet and his contemporaries, Chasles, Brianchon, Hachette, Dupin, Gergonne, and others, the anharmonic ratio enjoyed a fundamental role. It is made also the basis of the great work of Steiner,(20) who was the first to treat of the conic, not as the projection of a circle, but as the locus of intersection of corresponding rays of two projective pencils. Steiner not only related to each other, in one-to-one correspondence, point-rows and pencils and all the other fundamental forms, but he set into correspondence even curves and surfaces of higher degrees. This new and fertile conception gave him an easy and direct route into the most abstract and difficult regions of pure geometry. Much of his work was given without any indication of the methods by which he had arrived at it, and many of his results have only recently been verified.



*185. Von Staudt and his work.* To complete the theory of geometry as we have it to-day it only remained to free it from its dependence on the semimetrical basis of the anharmonic ratio. This work was accomplished by Von Staudt,(21) who applied himself to the restatement of the theory of geometry in a form independent of analytic and metrical notions. The method which has been used in Chapter II to develop the notion of four harmonic points by means of the complete quadrilateral is due to Von Staudt. His work is characterized by a most remarkable generality, in that he is able to discuss real and imaginary forms with equal ease. Thus he assumes a one-to-one correspondence between the points and lines of a plane, and defines a conic as the locus of points which lie on their corresponding lines, and a pencil of rays of the second order as the system of lines which pass through their corresponding points. The point-row and pencil of the second order may be real or imaginary, but his theorems still apply. An illustration of a correspondence of this sort, where the conic is imaginary, is given in 15 of the first chapter. In defining conjugate imaginary points on a line, Von Staudt made use of an involution of points having no double points. His methods, while elegant and powerful, are hardly adapted to an elementary course, but Reye(22) and others have done much toward simplifying his presentation.



*186. Recent developments.* It would be only confusing to the student to attempt to trace here the later developments of the science of protective geometry. It is concerned for the most part with curves and surfaces of a higher degree than the second. Purely synthetic methods have been used with marked success in the study of the straight line in space. The struggle between analysis and pure geometry has long since come to an end. Each has its distinct advantages, and the mathematician who cultivates one at the expense of the other will never attain the results that he would attain if both methods were equally ready to his hand. Pure geometry has to its credit some of the finest discoveries in mathematics, and need not apologize for having been born. The day of its usefulness has not passed with the invention of abridged notation and of short methods in analysis. While we may be certain that any geometrical problem may always be stated in analytic form, it does not follow that that statement will be simple or easily interpreted. For many mathematicians the geometric intuitions are weak, and for such the method will have little attraction. On the other hand, there will always be those for whom the subject will have a peculiar glamor—who will follow with delight the curious and unexpected relations between the forms of space. There is a corresponding pleasure, doubtless, for the analyst in tracing the marvelous connections between the various fields in which he wanders, and it is as absurd to shut one's eyes to the beauties in one as it is to ignore those in the other. "Let us cultivate geometry, then," says Darboux,(23) "without wishing in all points to equal it to its rival. Besides, if we were tempted to neglect it, it would not be long in finding in the applications of mathematics, as once it has already done, the means of renewing its life and of developing itself anew. It is like the Giant Antaeus, who renewed, his strength by touching the earth."



INDEX

(The numbers refer to the paragraphs)

Abel (1802-1829), 179

Analogy, 24

Analytic geometry, 21, 118, 119, 120, 146, 176, 180

Anharmonic ratio, 46, 161, 184, 185

Apollonius (second half of third century B.C.), 70

Archimedes (287-212 B.C.), 176

Aristotle (384-322 B.C.), 169

Asymptotes, 111, 113, 114, 115, 116, 117, 118, 148

Axes of a conic, 148

Axial pencil, 7, 8, 23, 50, 54

Axis of perspectivity, 8, 47

Bacon (1561-1626), 162

Bisection, 41, 109

Brianchon (1785-1864), 84, 85, 86, 88, 89, 90, 95, 105, 113, 174, 184

Calculus, 176

Carnot (1796-1832), 179

Cauchy (1789-1857), 179, 181

Cavalieri (1598-1647), 162

Center of a conic, 107, 112, 148

Center of involution, 141, 142

Center of perspectivity, 8

Central conic, 120

Chasles (1793-1880), 168, 179, 180, 184

Circle, 21, 73, 80, 145, 146, 147

Circular involution, 147, 149, 150, 151

Circular points, 146

Class of a curve, 183

Classification of conics, 110

Collineation, 175

Concentric pencils, 50

Cone of the second order, 59

Conic, 73, 81

Conjugate diameters, 114, 148

Conjugate normal, 151

Conjugate points and lines, 100, 109, 138, 139, 140

Constants in an equation, 21

Contingent relations, 180, 181

Continuity, 180, 181

Continuous correspondence, 9, 10, 21, 49

Corresponding elements, 64

Counting, 1, 4

Cross ratio, 46

Darboux, 176, 186

De Beaugrand, 170

Degenerate pencil of rays of the second order, 58, 93

Degenerate point-row of the second order, 56, 78

De la Hire (1640-1718), 168, 171, 175

Desargues (1593-1662), 25, 26, 40, 121, 125, 162, 163, 164, 165, 166, 167, 168, 169, 170, 171, 174, 175

Descartes (1596-1650), 162, 170, 171, 174, 176

Descriptive geometry, 179

Diameter, 107

Directrix, 157, 158, 159, 160

Double correspondence, 128, 130

Double points of an involution, 124

Double rays of an involution, 133, 134

Duality, 94, 104, 161, 180, 182

Dupin (1784-1873), 174, 184

Eccentricity of conic, 159

Ellipse, 110, 111, 162

Equation of conic, 118, 119, 120

Euclid (ca. 300 B.C.), 6, 22, 104

Euler (1707-1783), 166

Fermat (1601-1665), 162, 171

Foci of a conic, 152, 153, 154, 155, 156, 157, 158, 159, 160, 161, 162

Fourier (1768-1830), 179

Fourth harmonic, 29

Fundamental form, 7, 16, 23, 36, 47, 60, 184

Galileo (1564-1642), 162, 169, 170, 176

Gauss (1777-1855), 179

Gergonne (1771-1859), 182, 184

Greek geometry, 161

Hachette (1769-1834), 179, 184

Harmonic conjugates, 29, 30, 39

Harmonic elements, 86, 49, 91, 163, 185

Harmonic lines, 33, 34, 35, 66, 67

Harmonic planes, 34, 35

Harmonic points, 29, 31, 32, 33, 34, 35, 36, 43, 71, 161

Harmonic tangents to a conic, 91, 92

Harvey (1578-1657), 169

Homology, 180, 182

Huygens (1629-1695), 162

Hyperbola, 110, 111, 113, 114, 115, 116, 117, 118, 162

Imaginary elements, 146, 180, 181, 182, 185

Infinitely distant elements, 6, 9, 22, 39, 40, 41, 104, 107, 110

Infinity, 4, 5, 10, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 41

Involution, 37, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 161, 163, 170

Kepler (1571-1630), 162

Lagrange (1736-1813), 176, 179

Laplace (1749-1827), 179

Legendre (1752-1833), 179

Leibniz (1646-1716), 173

Linear construction, 40, 41, 42

Maclaurin (1698-1746), 177, 178

Measurements, 23, 40, 41, 104

Mersenne (1588-1648), 168, 171

Metrical theorems, 40, 104, 106, 107, 141

Middle point, 39, 41

Moebius (1790-1868), 179

Monge (1746-1818), 179, 180

Napier (1550-1617), 162

Newton (1642-1727), 177

Numbers, 4, 21, 43

Numerical computations, 43, 44, 46

One-to-one correspondence, 1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 24, 36, 87, 43, 60, 104, 106, 184

Opposite sides of a hexagon, 70

Opposite sides of a quadrilateral, 28, 29

Order of a form, 7, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21

Pappus (fourth century A.D.), 161

Parabola, 110, 111, 112, 119, 162

Parallel lines, 39, 41, 162

Pascal (1623-1662), 69, 70, 74, 75, 76, 77, 78, 95, 105, 125, 162, 169, 171, 172, 173

Pencil of planes of the second order, 59

Pencil of rays, 6, 7, 8, 23; of the second order, 57, 60, 79, 81

Perspective position, 6, 8, 35, 37, 51, 53, 71

Plane system, 16, 23

Planes on space, 17

Point of contact, 87, 88, 89, 90

Point system, 16, 23

Point-row, 6, 7, 8, 9, 23; of the second order, 55, 60, 61, 66, 67, 72

Points in space, 18

Pole and polar, 98, 99, 100, 101, 138, 164, 166

Poncelet (1788-1867), 177, 179, 180, 181, 182, 183, 184

Principal axis of a conic, 157

Projection, 161

Protective axial pencils, 59

Projective correspondence, 9, 35, 36, 37, 47, 71, 92, 104

Projective pencils, 53, 64, 68

Projective point-rows, 51, 79

Projective properties, 24

Projective theorems, 40, 104

Quadrangle, 26, 27, 28, 29

Quadric cone, 59

Quadrilateral, 88, 95, 96

Roberval (1602-1675), 168

Ruler construction, 40

Scheiner, 169

Self-corresponding elements, 47, 48, 49, 50, 51

Self-dual, 105

Self-polar triangle, 102

Separation of elements in involution, 148

Separation of harmonic conjugates, 38

Sequence of points, 49

Sign of segment, 44, 45

Similarity, 106

Skew lines, 12

Space system, 19, 23

Sphere, 21

Steiner (1796-1863), 129, 130, 131, 177, 179, 184

Steiner's construction, 129, 130, 131

Superposed point-rows, 47, 48, 49

Surfaces of the second degree, 166

System of lines in space, 20, 23

Systems of conics, 125

Tangent line, 61, 80, 81, 87, 88, 89, 90, 91, 92

Tycho Brahe (1546-1601), 162

Verner, 161

Vertex of conic, 157, 159

Von Staudt (1798-1867), 179, 185

Wallis (1616-1703), 162



FOOTNOTES

1 The more general notion of anharmonic ratio, which includes the harmonic ratio as a special case, was also known to the ancients. While we have not found it necessary to make use of the anharmonic ratio in building up our theory, it is so frequently met with in treatises on geometry that some account of it should be given.

Consider any four points, A, B, C, D, on a line, and join them to any point S not on that line. Then the triangles ASB, GSD, ASD, CSB, having all the same altitude, are to each other as their bases. Also, since the area of any triangle is one half the product of any two of its sides by the sine of the angle included between them, we have

[formula]

Now the fraction on the right would be unchanged if instead of the points A, B, C, D we should take any other four points A', B', C', D' lying on any other line cutting across SA, SB, SC, SD. In other words, the fraction on the left is unaltered in value if the points A, B, C, D are replaced by any other four points perspective to them. Again, the fraction on the left is unchanged if some other point were taken instead of S. In other words, the fraction on the right is unaltered if we replace the four lines SA, SB, SC, SD by any other four lines perspective to them. The fraction on the left is called the anharmonic ratio of the four points A, B, C, D; the fraction on the right is called the anharmonic ratio of the four lines SA, SB, SC, SD. The anharmonic ratio of four points is sometimes written (ABCD), so that

[formula]

If we take the points in different order, the value of the anharmonic ratio will not necessarily remain the same. The twenty-four different ways of writing them will, however, give not more than six different values for the anharmonic ratio, for by writing out the fractions which define them we can find that (ABCD) = (BADC) = (CDAB) = (DCBA). If we write (ABCD) = a, it is not difficult to show that the six values are

[formula]

The proof of this we leave to the student.

If A, B, C, D are four harmonic points (see Fig. 6, p. *22), and a quadrilateral KLMN is constructed such that KL and MN pass through A, KN and LM through C, LN through B, and KM through D, then, projecting A, B, C, D from L upon KM, we have (ABCD) = (KOMD), where O is the intersection of KM with LN. But, projecting again the points K, O, M, D from N back upon the line AB, we have (KOMD) = (CBAD). From this we have

(ABCD) = (CBAD),

or

[formula]

whence a = 0 or a = 2. But it is easy to see that a = 0 implies that two of the four points coincide. For four harmonic points, therefore, the six values of the anharmonic ratio reduce to three, namely, 2, [formula], and -1. Incidentally we see that if an interchange of any two points in an anharmonic ratio does not change its value, then the four points are harmonic.

[Figure 49]

FIG. 49

Many theorems of projective geometry are succinctly stated in terms of anharmonic ratios. Thus, the anharmonic ratio of any four elements of a form is equal to the anharmonic ratio of the corresponding four elements in any form projectively related to it. The anharmonic ratio of the lines joining any four fixed points on a conic to a variable fifthpoint on the conic is constant. The locus of points from which four points in a plane are seen along four rays of constant anharmonic ratio is a conic through the four points. We leave these theorems for the student, who may also justify the following solution of the problem: Given three points and a certain anharmonic ratio, to find a fourth point which shall have with the given three the given anharmonic ratio. Let A, B, D be the three given points (Fig. 49). On any convenient line through A take two points B' and D' such that AB'/AD' is equal to the given anharmonic ratio. Join BB' and DD' and let the two lines meet in S. Draw through S a parallel to AB'. This line will meet AB in the required point C.

2 Pappus, Mathematicae Collectiones, vii, 129.

3 J. Verneri, Libellus super vigintiduobus elementis conicis, etc. 1522.

4 Kepler, Ad Vitellionem paralipomena quibus astronomiae pars optica traditur. 1604.

5 Desargues, Bruillon-project d'une atteinte aux evenements des rencontres d'un cone avec un plan. 1639. Edited and analyzed by Poudra, 1864.

6 The term 'pole' was first introduced, in the sense in which we have used it, in 1810, by a French mathematician named Servois (Gergonne, Annales des Matheematiques, I, 337), and the corresponding term 'polar' by the editor, Gergonne, of this same journal three years later.

7 Euler, Introductio in analysin infinitorum, Appendix, cap. V. 1748.

8 OEuvres de Desargues, t. II, 132.

9 OEuvres de Desargues, t. II, 370.

10 OEuvres de Descartes, t. II, 499.

11 OEuvres de Pascal, par Brunsehvig et Boutroux, t. I, 252.

12 Chasles, Histoire de la Geometrie, 70.

13 OEuvres de Desargues, t. I, 231.

14 See Ball, History of Mathematics, French edition, t. II, 233.

15 Newton, Principia, lib. i, lemma XXI.

16 Maclaurin, Philosophical Transactions of the Royal Society of London, 1735.

17 Monge, Geometrie Descriptive. 1800.

18 Poncelet, Traite des Proprietes Projectives des Figures. 1822. (See p. 357, Vol. II, of the edition of 1866.)

19 Gergonne, Annales de Mathematiques, XVI, 209. 1826.

20 Steiner, Systematische Ehtwickelung der Abhaengigkeit geometrischer Gestalten von einander. 1832.

21 Von Staudt, Geometrie der Lage. 1847.

22 Reye, Geometrie der Lage. Translated by Holgate, 1897.

23 Ball, loc. cit. p. 261.

THE END