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153. The circle and the parabola. The above discussion
applies only to the central conics, apart from
the circle. In the circle the two foci fall together at the
center. In the case of the parabola, that part of the
investigation which proves the existence of two foci on
one of the axes will not hold, as we have but one
axis. It is seen, however, that as P moves to infinity,
carrying the line q with it, q becomes the line at infinity,
which for the parabola is a tangent line. Its pole
Q is thus at infinity and also the point P', so that P
and P' fall together at infinity, and therefore one focus
of the parabola is at infinity. There must therefore be
another, so that
A parabola has one and only one focus in the finite
part of the plane.
154. Focal properties of conics. We proceed to develop
some theorems which will exhibit the importance
of these points in the theory of the conic section.
Draw a tangent to the conic, and also the normal
at the point of contact P. These
two lines are clearly conjugate
normals. The two points T and
N, therefore, where they meet the
axis which contains the foci, are
corresponding points in the involution
considered above, and are
therefore harmonic conjugates with respect to the foci
(Fig. 44); and if we join them to the point P, we
shall obtain four harmonic lines. But two of them
are at right angles to each other, and so the others
make equal angles with them (Problem 4, Chapter II).
Therefore
The lines joining a point on the conic to the foci make
equal angles with the tangent.
It follows that rays from a source of light at one
focus are reflected by an ellipse to the other.
155. In the case of the parabola, where one of the
foci must be considered to be at infinity in the direction
of the diameter, we have
A diameter makes the same
angle with the tangent at its
extremity as that tangent does
with the line from its point of
contact to the focus (Fig. 45).
156. This last theorem is the basis for the construction
of the parabolic reflector. A ray of light from the
focus is reflected from such a reflector in a direction
parallel to the axis of the reflector.
157. Directrix. Principal axis. Vertex. The polar of
the focus with respect to the conic is called the directrix.
The axis which contains the foci is called the principal
axis, and the intersection of the axis with the curve is
called the vertex of the curve. The directrix is at right
angles to the principal axis. In a parabola the vertex
is equally distant from the focus and the directrix,
these three points and the point at infinity on the axis
being four harmonic points. In the ellipse the vertex is
nearer to the focus than it is to the directrix, for the
same reason, and in the hyperbola it is farther from
the focus than it is from the directrix.
158. Another definition of a conic. Let P be any point
on the directrix through which a line is drawn meeting
the conic in the points A and B (Fig. 46). Let the tangents
at A and B meet in T, and call the focus F. Then
TF and PF are conjugate lines, and as they pass through
a focus they must be at right angles to each other. Let
TF meet AB in C. Then P, A, C, B are four harmonic
points. Project these four points parallel to TF upon
the directrix, and we then get
the four harmonic points P,
M, Q, N. Since, now, TFP is
a right angle, the angles MFQ
and NFQ are equal, as well
as the angles AFC and BFC.
Therefore the triangles MAF
and NFB are similar, and
FA : FM = FB : BN. Dropping
perpendiculars AA and BB'
upon the directrix, this becomes
FA : AA' = FB : BB'. We
have thus the property often taken as the definition
of a conic:
The ratio of the distances from a point on the conic to
the focus and the directrix is constant.
159. Eccentricity. By taking the point at the vertex
of the conic, we note that this ratio is less than unity
for the ellipse, greater than unity for the hyperbola,
and equal to unity for the parabola. This ratio is called the
eccentricity.
160. Sum or difference of focal
distances. The ellipse and the
hyperbola have two foci and
two directrices. The eccentricity, of course, is the same
for one focus as for the other, since the curve is symmetrical
with respect to both. If the distances from
a point on a conic to the two foci are r and r', and
the distances from the same point to the corresponding
directrices are d and d'
(Fig. 47), we have r : d = r' : d';
(r ± r') : (d ± d'). In the
ellipse (d + d') is constant,
being the distance between
the directrices. In the hyperbola
this distance is (d - d').
It follows (Fig. 48) that
In the ellipse the sum of the
focal distances of any point
on the curve is constant, and
in the hyperbola the difference between the focal distances
is constant.
PROBLEMS
1. Construct the axis of a parabola, given four tangents.
2. Given two conjugate lines at right angles to each
other, and let them meet the axis which has no foci on it
in the points A and B. The circle on AB as diameter will
pass through the foci of the conic.
3. Given the axes of a conic in position, and also a
tangent with its point of contact, to construct the foci and
determine the length of the axes.
4. Given the tangent at the vertex of a parabola, and
two other tangents, to find the focus.
5. The locus of the center of a circle touching two given
circles is a conic with the centers of the given circles for
its foci.
6. Given the axis of a parabola and a tangent, with its
point of contact, to find the focus.
7. The locus of the center of a circle which touches a
given line and a given circle consists of two parabolas.
8. Let F and F' be the foci of an ellipse, and P any
point on it. Produce PF to G, making PG equal to PF'.
Find the locus of G.
9. If the points G of a circle be folded over upon a
point F, the creases will all be tangent to a conic. If F is
within the circle, the conic will be an ellipse; if F is without
the circle, the conic will be a hyperbola.
10. If the points G in the last example be taken on a
straight line, the locus is a parabola.
11. Find the foci and the length of the principal axis of
the conics in problems 9 and 10.
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 ratio1
the involution of six points cut out by any transversal
which intersects the sides of a complete quadrilateral
as studied by Pappus2;
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 Verner3
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 Kepler4 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 Descartes10 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. Leibniz12 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 Monge17 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 figures18
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
Reye22
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."
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