An Elementary Course in Synthetic Projective Geometry Part 18

*177. Newton and Maclaurin.* But in hastening on to the epoch of Poncelet and Steiner we should not omit to mention the work of Newton and Maclaurin. Although their results were obtained by a.n.a.lysis for the most part, nevertheless they have given us theorems which fall naturally into the domain of synthetic projective geometry. Thus Newton's "organic method"(15) of generating conic sections is closely related to the method which we have made use of in Chapter III. It is as follows: _If two angles, __AOS__ and __AO'S__, of given magnitudes turn about their respective vertices, __O__ and __O'__, in such a way that the point of intersection, __S__, of one pair of arms always lies on a straight line, the point of intersection, __A__, of the other pair of arms will describe a conic._ The proof of this is left to the student.

*178.* Another method of generating a conic is due to Maclaurin.(16) The construction, which we also leave for the student to justify, is as follows: _If a triangle __C'PQ__ move in such a way that its sides, __PQ__, __QC'__, and __C'P__, turn __ around three fixed points, __R__, __A__, __B__, respectively, while two of its vertices, __P__, __Q__, slide along two fixed lines, __CB'__ and __CA'__, respectively, then the remaining vertex will describe a conic._

*179. Descriptive geometry and the second revival.* The second revival of pure geometry was again to take place at a time of great intellectual activity. The period at the close of the eighteenth and the beginning of the nineteenth century is adorned with a glorious list of mighty names, among which are Gauss, Lagrange, Legendre, Laplace, Monge, Carnot, Poncelet, Cauchy, Fourier, Steiner, Von Staudt, Mobius, Abel, and many others. The renaissance may be said to date from the invention by Monge(17) of the theory of _descriptive geometry_. Descriptive geometry is concerned with the representation of figures in s.p.a.ce of three dimensions by means of s.p.a.ce of two dimensions. The method commonly used consists in projecting the s.p.a.ce figure on two planes (a vertical and a horizontal plane being most convenient), the projections being made most simply for metrical purposes from infinity in directions perpendicular to the two planes of projection. These two planes are then made to coincide by revolving the horizontal into the vertical about their common line. Such is the method of descriptive geometry which in the hands of Monge acquired wonderful generality and elegance. Problems concerning fortifications were worked so quickly by this method that the commandant at the military school at Mezieres, where Monge was a draftsman and pupil, viewed the results with distrust. Monge afterward became professor of mathematics at Mezieres and gathered around him a group of students destined to have a share in the advancement of pure geometry. Among these were Hachette, Brianchon, Dupin, Chasles, Poncelet, and many others.

*180. Duality, h.o.m.ology, continuity, contingent relations.* a.n.a.lytic geometry had left little to do in the way of discovery of new material, and the mathematical world was ready for the construction of the edifice.

The activities of the group of men that followed Monge were directed toward this end, and we now begin to hear of the great unifying notions of duality, h.o.m.ology, continuity, contingent relations, and the like. The devotees of pure geometry were beginning to feel the need of a basis for their science which should be at once as general and as rigorous as that of the a.n.a.lysts. Their dream was the building up of a system of geometry which should be independent of a.n.a.lysis. Monge, and after him Poncelet, spent much thought on the so-called "principle of continuity," afterwards discussed by Chasles under the name of the "principle of contingent relations." To get a clear idea of this principle, consider a theorem in geometry in the proof of which certain auxiliary elements are employed.

These elements do not appear in the statement of the theorem, and the theorem might possibly be proved without them. In drawing the figure for the proof of the theorem, however, some of these elements may not appear, or, as the a.n.a.lyst would say, they become imaginary. "No matter," says the principle of contingent relations, "the theorem is true, and the proof is valid whether the elements used in the proof are real or imaginary."

*181. Poncelet and Cauchy.* The efforts of Poncelet to compel the acceptance of this principle independent of a.n.a.lysis resulted in a bitter and perhaps fruitless controversy between him and the great a.n.a.lyst 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 a.n.a.lysis. 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 a.n.a.lysis is subject to many and important exceptions. The field of a.n.a.lysis is much more general than the field of geometry, and while there may be a clear notion in a.n.a.lysis to, correspond to every notion in geometry, the opposite is not true. Thus, in a.n.a.lysis we can deal with four coordinates as well as with three, but the existence of a s.p.a.ce 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 t.i.tle "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 h.o.m.ology_, 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 s.p.a.ce as all lying on a plane at infinity.

*183. The debt which a.n.a.lytic geometry owes to synthetic geometry.* The reaction of pure geometry on a.n.a.lytic geometry is clearly seen in the development of the notion of the _cla.s.s_ 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 cla.s.s. 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 cla.s.s of a given curve were afterward completed by Plucker. 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 a.n.a.lytic 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 a.s.sumes 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 pa.s.s through their corresponding points. The point-row and pencil of the second order may be real or imaginary, but his theorems still apply. An ill.u.s.tration 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 s.p.a.ce. The struggle between a.n.a.lysis 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 pa.s.sed with the invention of abridged notation and of short methods in a.n.a.lysis.

While we may be certain that any geometrical problem may always be stated in a.n.a.lytic 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 s.p.a.ce. There is a corresponding pleasure, doubtless, for the a.n.a.lyst 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."

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 alt.i.tude, 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_ pa.s.s 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_.