*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 Nürnberg
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 mass 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 pass 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 space.* As an
illustration of his remarkable powers of generalization, we may note that
Desargues extended the notion of poles and polars to space 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
space 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 space analogous 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 compasses,
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 friendship 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 manuscript 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 fashion
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 assure you I have nowhere found anything similar to
what you describe. Go, my son, and tranquilize yourself; be assured that
what you take for spots on the sun are the faults of your glasses, 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 glass 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 analytic 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 associates 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, entitled "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 __μ__ the
intersection of __MV__ and __SK__; and if through two of the four points
__A__, __K__, __μ__, __V__, which are not in the same straight line with
__M__ and __S__, such as __K__ and __V__, we pass the circumference 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 résumé of a more
extended treatise on conics which, owing partly to Pascal’s extreme youth,
partly to the difficulty of publishing 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 analysis
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 analysis should come a species of contempt for the methods of the
ancients, so that in his preface to his "Méchanique Analytique," 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 analysis
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.
*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 analysis 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, Möbius, 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 space of three dimensions
by means of space of two dimensions. The method commonly used consists in
projecting the space 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 Mézières, where Monge was a draftsman and pupil, viewed the
results with distrust. Monge afterward became professor of mathematics at
Mézières 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, homology, continuity, contingent relations.* Analytic
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, homology, 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 analysts. Their dream was the building up of a system of geometry
which should be independent of analysis. 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 analyst 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 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 Traité des Propriétés
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 coördinates 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 Plücker. 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 rôle. 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
Möbius (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 événements des
rencontres d’un cône 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 Mathéématiques_, 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 Œuvres de Desargues, t. II, 132.
9 Œuvres de Desargues, t. II, 370.
10 Œuvres de Descartes, t. II, 499.
11 Œuvres de Pascal, par Brunsehvig et Boutroux, t. I, 252.
12 Chasles, Histoire de la Géométrie, 70.
13 Œuvres 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, Géométrie Descriptive. 1800.
18 Poncelet, Traité des Propriétés Projectives des Figures. 1822. (See
p. 357, Vol. II, of the edition of 1866.)
19 Gergonne, _Annales de Mathématiques, XVI, 209. 1826._
20 Steiner, Systematische Ehtwickelung der Abhängigkeit 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.