Lehmer, Derrick Norman - An Elementary Course in Synthetic Projective Geometry
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Lehmer, Derrick Norman >> An Elementary Course in Synthetic Projective Geometry
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[Figure 46]
FIG. 46
*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._
[Figure 47]
FIG. 47
*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_.
[Figure 48]
FIG. 48
*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 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 Nuernberg
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 __{~GREEK SMALL LETTER MU~}__ the
intersection of __MV__ and __SK__; and if through two of the four points
__A__, __K__, __{~GREEK SMALL LETTER MU~}__, __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 resume 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 "Mechanique 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.
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