An Elementary Course in Synthetic Projective Geometry Part 17

12. In problem 10 a correspondence is set up between straight lines and parabolas. As there is a fourfold infinity of parabolas in the plane, and only a twofold infinity of straight lines, there must be some restriction on the parabolas obtained by this method. Find and explain this restriction.

13. State and explain the similar problem for problem 9.

14. The last four problems are a study of the consequences of the following transformation: A point _O_ is fixed in the plane. Then to any point _P_ is made to correspond the line _p_ at right angles to _OP_ and bisecting it. In this correspondence, what happens to _p_ when _P_ moves along a straight line? What corresponds to the theorem that two lines have only one point in common? What to the theorem that the angle sum of a triangle is two right angles? Etc.

CHAPTER X - ON THE HISTORY OF SYNTHETIC PROJECTIVE GEOMETRY

*161. Ancient results.* The theory of synthetic projective geometry as we have built it up in this course is less than a century old. This is not to say that many of the theorems and principles involved were not discovered much earlier, but isolated theorems do not make a theory, any more than a pile of bricks makes a building. The materials for our building have been contributed by many different workmen from the days of Euclid down to the present time. Thus, the notion of four harmonic points was familiar to the ancients, who considered it from the metrical point of view as the division of a line internally and externally in the same ratio(1) the involution of six points cut out by any transversal which intersects the sides of a complete quadrilateral as studied by Pappus(2); but these notions were not made the foundation for any general theory. Taken by themselves, they are of small consequence; it is their relation to other theorems and sets of theorems that gives them their importance. The ancients were doubtless familiar with the theorem, _Two lines determine a point, and two points determine a line_, but they had no glimpse of the wonderful law of duality, of which this theorem is a simple example. The principle of projection, by which many properties of the conic sections may be inferred from corresponding properties of the circle which forms the base of the cone from which they are cut-a principle so natural to modern mathematicians-seems not to have occurred to the Greeks. The ellipse, the hyperbola, and the parabola were to them entirely different curves, to be treated separately with methods appropriate to each. Thus the focus of the ellipse was discovered some five hundred years before the focus of the parabola! It was not till 1522 that Verner(3) of Nurnberg undertook to demonstrate the properties of the conic sections by means of the circle.

*162. Unifying principles.* In the early years of the seventeenth century-that wonderful epoch in the history of the world which produced a Galileo, a Kepler, a Tycho Brahe, a Descartes, a Desargues, a Pascal, a Cavalieri, a Wallis, a Fermat, a Huygens, a Bacon, a Napier, and a goodly array of lesser lights, to say nothing of a Rembrandt or of a Shakespeare-there began to appear certain unifying principles connecting the great ma.s.s of material dug out by the ancients. Thus, in 1604 the great astronomer Kepler(4) introduced the notion that parallel lines should be considered as meeting at an infinite distance, and that a parabola is at once the limiting case of an ellipse and of a hyperbola. He also attributes to the parabola a "blind focus" (_caecus focus_) at infinity on the axis.

*163. Desargues.* In 1639 Desargues,(5) an architect of Lyons, published a little treatise on the conic sections, in which appears the theorem upon which we have founded the theory of four harmonic points (-- 25).

Desargues, however, does not make use of it for that purpose. Four harmonic points are for him a special case of six points in involution when two of the three pairs coincide giving double points. His development of the theory of involution is also different from the purely geometric one which we have adopted, and is based on the theorem (-- 142) that the product of the distances of two conjugate points from the center is constant. He also proves the projective character of an involution of points by showing that when six lines pa.s.s through a point and through six points in involution, then any transversal must meet them in six points which are also in involution.

*164. Poles and polars.* In this little treatise is also contained the theory of poles and polars. The polar line is called a _traversal_.(6) The harmonic properties of poles and polars are given, but Desargues seems not to have arrived at the metrical properties which result when the infinite elements of the plane are introduced. Thus he says, "When the _traversal_ is at an infinite distance, all is unimaginable."

*165. Desargues's theorem concerning conics through four points.* We find in this little book the beautiful theorem concerning a quadrilateral inscribed in a conic section, which is given by his name in -- 138. The theorem is not given in terms of a system of conics through four points, for Desargues had no conception of any such system. He states the theorem, in effect, as follows: _Given a simple quadrilateral inscribed in a conic section, every transversal meets the conic and the four sides of the quadrilateral in six points which are in involution._

*166. Extension of the theory of poles and polars to s.p.a.ce.* As an ill.u.s.tration of his remarkable powers of generalization, we may note that Desargues extended the notion of poles and polars to s.p.a.ce of three dimensions for the sphere and for certain other surfaces of the second degree. This is a matter which has not been touched on in this book, but the notion is not difficult to grasp. If we draw through any point _P_ in s.p.a.ce a line to cut a sphere in two points, _A_ and _S_, and then construct the fourth harmonic of _P_ with respect to _A_ and _B_, the locus of this fourth harmonic, for various lines through _P_, is a plane called the _polar plane_ of _P_ with respect to the sphere. With this definition and theorem one can easily find dual relations between points and planes in s.p.a.ce a.n.a.logous to those between points and lines in a plane. Desargues closes his discussion of this matter with the remark, "Similar properties may be found for those other solids which are related to the sphere in the same way that the conic section is to the circle." It should not be inferred from this remark, however, that he was acquainted with all the different varieties of surfaces of the second order. The ancients were well acquainted with the surfaces obtained by revolving an ellipse or a parabola about an axis. Even the hyperboloid of two sheets, obtained by revolving the hyperbola about its major axis, was known to them, but probably not the hyperboloid of one sheet, which results from revolving a hyperbola about the other axis. All the other solids of the second degree were probably unknown until their discovery by Euler.(7)

*167.* Desargues had no conception of the conic section of the locus of intersection of corresponding rays of two projective pencils of rays. He seems to have tried to describe the curve by means of a pair of compa.s.ses, moving one leg back and forth along a straight line instead of holding it fixed as in drawing a circle. He does not attempt to define the law of the movement necessary to obtain a conic by this means.

*168. Reception of Desargues's work.* Strange to say, Desargues's immortal work was heaped with the most violent abuse and held up to ridicule and scorn! "Incredible errors! Enormous mistakes and falsities!

Really it is impossible for anyone who is familiar with the science concerning which he wishes to retail his thoughts, to keep from laughing!"

Such were the comments of reviewers and critics. Nor were his detractors altogether ignorant and uninstructed men. In spite of the devotion of his pupils and in spite of the admiration and friends.h.i.+p of men like Descartes, Fermat, Mersenne, and Roberval, his book disappeared so completely that two centuries after the date of its publication, when the French geometer Chasles wrote his history of geometry, there was no means of estimating the value of the work done by Desargues. Six years later, however, in 1845, Chasles found a ma.n.u.script copy of the "Bruillon-project," made by Desargues's pupil, De la Hire.

*169. Conservatism in Desargues's time.* It is not necessary to suppose that this effacement of Desargues's work for two centuries was due to the savage attacks of his critics. All this was in accordance with the fas.h.i.+on of the time, and no man escaped bitter denunciation who attempted to improve on the methods of the ancients. Those were days when men refused to believe that a heavy body falls at the same rate as a lighter one, even when Galileo made them see it with their own eyes at the foot of the tower of Pisa. Could they not turn to the exact page and line of Aristotle which declared that the heavier body must fall the faster! "I have read Aristotle's writings from end to end, many times," wrote a Jesuit provincial to the mathematician and astronomer, Christoph Scheiner, at Ingolstadt, whose telescope seemed to reveal certain mysterious spots on the sun, "and I can a.s.sure you I have nowhere found anything similar to what you describe. Go, my son, and tranquilize yourself; be a.s.sured that what you take for spots on the sun are the faults of your gla.s.ses, or of your eyes." The dead hand of Aristotle barred the advance in every department of research. Physicians would have nothing to do with Harvey's discoveries about the circulation of the blood. "Nature is accused of tolerating a vacuum!" exclaimed a priest when Pascal began his experiments on the Puy-de-Dome to show that the column of mercury in a gla.s.s tube varied in height with the pressure of the atmosphere.

*170. Desargues's style of writing.* Nevertheless, authority counted for less at this time in Paris than it did in Italy, and the tragedy enacted in Rome when Galileo was forced to deny his inmost convictions at the bidding of a brutal Inquisition could not have been staged in France.

Moreover, in the little company of scientists of which Desargues was a member the utmost liberty of thought and expression was maintained. One very good reason for the disappearance of the work of Desargues is to be found in his style of writing. He failed to heed the very good advice given him in a letter from his warm admirer Descartes.(8) "You may have two designs, both very good and very laudable, but which do not require the same method of procedure: The one is to write for the learned, and show them some new properties of the conic sections which they do not already know; and the other is to write for the curious unlearned, and to do it so that this matter which until now has been understood by only a very few, and which is nevertheless very useful for perspective, for painting, architecture, etc., shall become common and easy to all who wish to study them in your book. If you have the first idea, then it seems to me that it is necessary to avoid using new terms; for the learned are already accustomed to using those of Apollonius, and will not readily change them for others, though better, and thus yours will serve only to render your demonstrations more difficult, and to turn away your readers from your book. If you have the second plan in mind, it is certain that your terms, which are French, and conceived with spirit and grace, will be better received by persons not preoccupied with those of the ancients....

But, if you have that intention, you should make of it a great volume; explain it all so fully and so distinctly that those gentlemen who cannot study without yawning; who cannot distress their imaginations enough to grasp a proposition in geometry, nor turn the leaves of a book to look at the letters in a figure, shall find nothing in your discourse more difficult to understand than the description of an enchanted palace in a fairy story." The point of these remarks is apparent when we note that Desargues introduced some seventy new terms in his little book, of which only one, _involution_, has survived. Curiously enough, this is the one term singled out for the sharpest criticism and ridicule by his reviewer, De Beaugrand.(9) That Descartes knew the character of Desargues's audience better than he did is also evidenced by the fact that De Beaugrand exhausted his patience in reading the first ten pages of the book.

*171. Lack of appreciation of Desargues.* Desargues's methods, entirely different from the a.n.a.lytic methods just then being developed by Descartes and Fermat, seem to have been little understood. "Between you and me,"

wrote Descartes(10) to Mersenne, "I can hardly form an idea of what he may have written concerning conics." Desargues seems to have boasted that he owed nothing to any man, and that all his results had come from his own mind. His favorite pupil, De la Hire, did not realize the extraordinary simplicity and generality of his work. It is a remarkable fact that the only one of all his a.s.sociates to understand and appreciate the methods of Desargues should be a lad of sixteen years!

*172. Pascal and his theorem.* One does not have to believe all the marvelous stories of Pascal's admiring sisters to credit him with wonderful precocity. We have the fact that in 1640, when he was sixteen years old, he published a little placard, or poster, ent.i.tled "Essay pour les conique,"(11) in which his great theorem appears for the first time.

His manner of putting it may be a little puzzling to one who has only seen it in the form given in this book, and it may be worth while for the student to compare the two methods of stating it. It is given as follows: _"If in the plane of __M__, __S__, __Q__ we draw through __M__ the two lines __MK__ and __MV__, and through the point __S__ the two lines __SK__ and __SV__, and let __K__ be the intersection of __MK__ and __SK__; __V__ the intersection of __MV__ and __SV__; __A__ the intersection of __MA__ and __SA__ (__A__ is the intersection of __SV__ and __MK__), and ____ the intersection of __MV__ and __SK__; and if through two of the four points __A__, __K__, ____, __V__, which are not in the same straight line with __M__ and __S__, such as __K__ and __V__, we pa.s.s the circ.u.mference of a circle cutting the lines __MV__, __MP__, __SV__, __SK__ in the points __O__, __P__, __Q__, __N__; I say that the lines __MS__, __NO__, __PQ__ are of the same order."_ (By "lines of the same order" Pascal means lines which meet in the same point or are parallel.) By projecting the figure thus described upon another plane he is able to state his theorem for the case where the circle is replaced by any conic section.

*173.* It must be understood that the "Essay" was only a resume of a more extended treatise on conics which, owing partly to Pascal's extreme youth, partly to the difficulty of publis.h.i.+ng scientific works in those days, and also to his later morbid interest in religious matters, was never published. Leibniz(12) examined a copy of the complete work, and has reported that the great theorem on the mystic hexagram was made the basis of the whole theory, and that Pascal had deduced some four hundred corollaries from it. This would indicate that here was a man able to take the unconnected materials of projective geometry and shape them into some such symmetrical edifice as we have to-day. Unfortunately for science, Pascal's early death prevented the further development of the subject at his hands.

*174.* In the "Essay" Pascal gives full credit to Desargues, saying of one of the other propositions, "We prove this property also, the original discoverer of which is M. Desargues, of Lyons, one of the greatest minds of this age ... and I wish to acknowledge that I owe to him the little which I have discovered." This acknowledgment led Descartes to believe that Pascal's theorem should also be credited to Desargues. But in the scientific club which the young Pascal attended in company with his father, who was also a scientist of some reputation, the theorem went by the name of 'la Pascalia,' and Descartes's remarks do not seem to have been taken seriously, which indeed is not to be wondered at, seeing that he was in the habit of giving scant credit to the work of other scientific investigators than himself.

*175. De la Hire and his work.* De la Hire added little to the development of the subject, but he did put into print much of what Desargues had already worked out, not fully realizing, perhaps, how much was his own and how much he owed to his teacher. Writing in 1679, he says,(13) "I have just read for the first time M. Desargues's little treatise, and have made a copy of it in order to have a more perfect knowledge of it." It was this copy that saved the work of his master from oblivion. De la Hire should be credited, among other things, with the invention of a method by which figures in the plane may be transformed into others of the same order. His method is extremely interesting, and will serve as an exercise for the student in synthetic projective geometry. It is as follows: _Draw two parallel lines, __a__ and __b__, and select a point __P__ in their plane. Through any point __M__ of the plane draw a line meeting __a__ in __A__ and __b__ in __B__. Draw a line through __B__ parallel to __AP__, and let it meet __MP__ in the point __M'__. It may be shown that the point __M'__ thus obtained does not depend at all on the particular ray __MAB__ used in determining it, so that we have set up a one-to-one correspondence between the points __M__ and __M'__ in the plane._ The student may show that as _M_ describes a point-row, _M'_ describes a point-row projective to it. As _M_ describes a conic, _M'_ describes another conic. This sort of correspondence is called a _collineation_. It will be found that the points on the line _b_ transform into themselves, as does also the single point _P_. Points on the line _a_ transform into points on the line at infinity. The student should remove the metrical features of the construction and take, instead of two parallel lines _a_ and _b_, any two lines which may meet in a finite part of the plane. The collineation is a special one in that the general one has an invariant triangle instead of an invariant point and line.

*176. Descartes and his influence.* The history of synthetic projective geometry has little to do with the work of the great philosopher Descartes, except in an indirect way. The method of algebraic a.n.a.lysis invented by him, and the differential and integral calculus which developed from it, attracted all the interest of the mathematical world for nearly two centuries after Desargues, and synthetic geometry received scant attention during the rest of the seventeenth century and for the greater part of the eighteenth century. It is difficult for moderns to conceive of the richness and variety of the problems which confronted the first workers in the calculus. To come into the possession of a method which would solve almost automatically problems which had baffled the keenest minds of antiquity; to be able to derive in a few moments results which an Archimedes had toiled long and patiently to reach or a Galileo had determined experimentally; such was the happy experience of mathematicians for a century and a half after Descartes, and it is not to be wondered at that along with this enthusiastic pursuit of new theorems in a.n.a.lysis should come a species of contempt for the methods of the ancients, so that in his preface to his "Mechanique a.n.a.lytique," published in 1788, Lagrange boasts, "One will find no figures in this work." But at the close of the eighteenth century the field opened up to research by the invention of the calculus began to appear so thoroughly explored that new methods and new objects of investigation began to attract attention.

Lagrange himself, in his later years, turned in weariness from a.n.a.lysis and mechanics, and applied himself to chemistry, physics, and philosophical speculations. "This state of mind," says Darboux,(14) "we find almost always at certain moments in the lives of the greatest scholars." At any rate, after lying fallow for almost two centuries, the field of pure geometry was attacked with almost religious enthusiasm.