Lehmer, Derrick Norman - An Elementary Course in Synthetic Projective Geometry
L >>
Lehmer, Derrick Norman >> An Elementary Course in Synthetic Projective Geometry
Pages:
1 | 2 |
3 |
4 |
5 |
6 |
7 |
8
*14.* If to each line through a point in space we make correspond that
plane at right angles to it and passing through the same point, we see
that _the infinitude of planes through a point in space is of the second
order._
*15.* If to each plane through a point in space we make correspond the
line in which it intersects a given plane, we see that _the infinitude of
lines in a plane is of the second order._ This may also be seen by setting
up a one-to-one correspondence between the points on a plane and the lines
of that plane. Thus, take a point _S_ not in the plane. Join any point _M_
of the plane to _S_. Through _S_ draw a plane at right angles to _MS_.
This meets the given plane in a line _m_ which may be taken as
corresponding to the point _M_. Another very important method of setting
up a one-to-one correspondence between lines and points in a plane will be
given later, and many weighty consequences will be derived from it.
*16. Plane system and point system.* The plane, considered as made up of
the points and lines in it, is called a _plane system_ and is a
fundamental form of the second order. The point, considered as made up of
all the lines and planes passing through it, is called a _point system_
and is also a fundamental form of the second order.
*17.* If now we take three lines in space all lying in different planes,
and select _l_ points on the first, _m_ points on the second, and _n_
points on the third, then the total number of planes passing through one
of the selected points on each line will be _lmn_. It is reasonable,
therefore, to symbolize the totality of planes that are determined by the
[infinity] points on each of the three lines by [infinity]3, and to call
it an infinitude of the _third_ order. But it is easily seen that every
plane in space is included in this totality, so that _the totality of
planes in space is an infinitude of the third order._
*18.* Consider now the planes perpendicular to these three lines. Every
set of three planes so drawn will determine a point in space, and,
conversely, through every point in space may be drawn one and only one set
of three planes at right angles to the three given lines. It follows,
therefore, that _the totality of points in space is an infinitude of the
third order._
*19. Space system.* Space of three dimensions, considered as made up of
all its planes and points, is then a fundamental form of the _third_
order, which we shall call a _space system._
*20. Lines in space.* If we join the twofold infinity of points in one
plane with the twofold infinity of points in another plane, we get a
totality of lines of space which is of the fourth order of infinity. _The
totality of lines in space gives, then, a fundamental form of the fourth
order._
*21. Correspondence between points and numbers.* In the theory of
analytic geometry a one-to-one correspondence is assumed to exist between
points on a line and numbers. In order to justify this assumption a very
extended definition of number must be made use of. A one-to-one
correspondence is then set up between points in the plane and pairs of
numbers, and also between points in space and sets of three numbers. A
single constant will serve to define the position of a point on a line;
two, a point in the plane; three, a point in space; etc. In the same
theory a one-to-one correspondence is set up between loci in the plane and
equations in two variables; between surfaces in space and equations in
three variables; etc. The equation of a line in a plane involves two
constants, either of which may take an infinite number of values. From
this it follows that there is an infinity of lines in the plane which is
of the second order if the infinity of points on a line is assumed to be
of the first. In the same way a circle is determined by three conditions;
a sphere by four; etc. We might then expect to be able to set up a
one-to-one correspondence between circles in a plane and points, or planes
in space, or between spheres and lines in space. Such, indeed, is the
case, and it is often possible to infer theorems concerning spheres from
theorems concerning lines, and vice versa. It is possibilities such as
these that, give to the theory of one-to-one correspondence its great
importance for the mathematician. It must not be forgotten, however, that
we are considering only _continuous_ correspondences. It is perfectly
possible to set, up a one-to-one correspondence between the points of a
line and the points of a plane, or, indeed, between the points of a line
and the points of a space of any finite number of dimensions, if the
correspondence is not restricted to be continuous.
*22. Elements at infinity.* A final word is necessary in order to explain
a phrase which is in constant use in the study of projective geometry. We
have spoken of the "point at infinity" on a straight line--a fictitious
point only used to bridge over the exceptional case when we are setting up
a one-to-one correspondence between the points of a line and the lines
through a point. We speak of it as "a point" and not as "points," because
in the geometry studied by Euclid we assume only one line through a point
parallel to a given line. In the same sense we speak of all the points at
infinity in a plane as lying on a line, "the line at infinity," because
the straight line is the simplest locus we can imagine which has only one
point in common with any line in the plane. Likewise we speak of the
"plane at infinity," because that seems the most convenient way of
imagining the points at infinity in space. It must not be inferred that
these conceptions have any essential connection with physical facts, or
that other means of picturing to ourselves the infinitely distant
configurations are not possible. In other branches of mathematics, notably
in the theory of functions of a complex variable, quite different
assumptions are made and quite different conceptions of the elements at
infinity are used. As we can know nothing experimentally about such
things, we are at liberty to make any assumptions we please, so long as
they are consistent and serve some useful purpose.
PROBLEMS
1. Since there is a threefold infinity of points in space, there must be a
sixfold infinity of pairs of points in space. Each pair of points
determines a line. Why, then, is there not a sixfold infinity of lines in
space?
2. If there is a fourfold infinity of lines in space, why is it that there
is not a fourfold infinity of planes through a point, seeing that each
line in space determines a plane through that point?
3. Show that there is a fourfold infinity of circles in space that pass
through a fixed point. (Set up a one-to-one correspondence between the
axes of the circles and lines in space.)
4. Find the order of infinity of all the lines of space that cut across a
given line; across two given lines; across three given lines; across four
given lines.
5. Find the order of infinity of all the spheres in space that pass
through a given point; through two given points; through three given
points; through four given points.
6. Find the order of infinity of all the circles on a sphere; of all the
circles on a sphere that pass through a fixed point; through two fixed
points; through three fixed points; of all the circles in space; of all
the circles that cut across a given line.
7. Find the order of infinity of all lines tangent to a sphere; of all
planes tangent to a sphere; of lines and planes tangent to a sphere and
passing through a fixed point.
8. Set up a one-to-one correspondence between the series of numbers _1_,
_2_, _3_, _4_, ... and the series of even numbers _2_, _4_, _6_, _8_ ....
Are we justified in saying that there are just as many even numbers as
there are numbers altogether?
9. Is the axiom "The whole is greater than one of its parts" applicable to
infinite assemblages?
10. Make out a classified list of all the infinitudes of the first,
second, third, and fourth orders mentioned in this chapter.
CHAPTER II - RELATIONS BETWEEN FUNDAMENTAL FORMS IN ONE-TO-ONE
CORRESPONDENCE WITH EACH OTHER
*23. Seven fundamental forms.* In the preceding chapter we have called
attention to seven fundamental forms: the point-row, the pencil of rays,
the axial pencil, the plane system, the point system, the space system,
and the system of lines in space. These fundamental forms are the material
which we intend to use in building up a general theory which will be found
to include ordinary geometry as a special case. We shall be concerned, not
with measurement of angles and areas or line segments as in the study of
Euclid, but in combining and comparing these fundamental forms and in
"generating" new forms by means of them. In problems of construction we
shall make no use of measurement, either of angles or of segments, and
except in certain special applications of the general theory we shall not
find it necessary to require more of ourselves than the ability to draw
the line joining two points, or to find the point of intersections of two
lines, or the line of intersection of two planes, or, in general, the
common elements of two fundamental forms.
*24. Projective properties.* Our chief interest in this chapter will be
the discovery of relations between the elements of one form which hold
between the corresponding elements of any other form in one-to-one
correspondence with it. We have already called attention to the danger of
assuming that whatever relations hold between the elements of one
assemblage must also hold between the corresponding elements of any
assemblage in one-to-one correspondence with it. This false assumption is
the basis of the so-called "proof by analogy" so much in vogue among
speculative theorists. When it appears that certain relations existing
between the points of a given point-row do not necessitate the same
relations between the corresponding elements of another in one-to-one
correspondence with it, we should view with suspicion any application of
the "proof by analogy" in realms of thought where accurate judgments are
not so easily made. For example, if in a given point-row _u_ three points,
_A_, _B_, and _C_, are taken such that _B_ is the middle point of the
segment _AC_, it does not follow that the three points _A'_, _B'_, _C'_ in
a point-row perspective to _u_ will be so related. Relations between the
elements of any form which do go over unaltered to the corresponding
elements of a form projectively related to it are called _projective
relations._ Relations involving measurement of lines or of angles are not
projective.
*25. Desargues's theorem.* We consider first the following beautiful
theorem, due to Desargues and called by his name.
_If two triangles, __A__, __B__, __C__ and __A'__, __B'__, __C'__, are so
situated that the lines __AA'__, __BB'__, and __CC'__ all meet in a point,
then the pairs of sides __AB__ and __A'B'__, __BC__ and __B'C'__, __CA__
and __C'A'__ all meet on a straight line, and conversely._
[Figure 3]
FIG. 3
Let the lines _AA'_, _BB'_, and _CC'_ meet in the point _M_ (Fig. 3).
Conceive of the figure as in space, so that _M_ is the vertex of a
trihedral angle of which the given triangles are plane sections. The lines
_AB_ and _A'B'_ are in the same plane and must meet when produced, their
point of intersection being clearly a point in the plane of each triangle
and therefore in the line of intersection of these two planes. Call this
point _P_. By similar reasoning the point _Q_ of intersection of the lines
_BC_ and _B'C'_ must lie on this same line as well as the point _R_ of
intersection of _CA_ and _C'A'_. Therefore the points _P_, _Q_, and _R_
all lie on the same line _m_. If now we consider the figure a plane
figure, the points _P_, _Q_, and _R_ still all lie on a straight line,
which proves the theorem. The converse is established in the same manner.
*26. Fundamental theorem concerning two complete quadrangles.* This
theorem throws into our hands the following fundamental theorem concerning
two complete quadrangles, a _complete quadrangle_ being defined as the
figure obtained by joining any four given points by straight lines in the
six possible ways.
_Given two complete quadrangles, __K__, __L__, __M__, __N__ and __K'__,
__L'__, __M'__, __N'__, so related that __KL__, __K'L'__, __MN__, __M'N'__
all meet in a point __A__; __LM__, __L'M'__, __NK__, __N'K'__ all meet in
a __ point __Q__; and __LN__, __L'N'__ meet in a point __B__ on the line
__AC__; then the lines __KM__ and __K'M'__ also meet in a point __D__ on
the line __AC__._
[Figure 4]
FIG. 4
For, by the converse of the last theorem, _KK'_, _LL'_, and _NN'_ all meet
in a point _S_ (Fig. 4). Also _LL'_, _MM'_, and _NN'_ meet in a point, and
therefore in the same point _S_. Thus _KK'_, _LL'_, and _MM'_ meet in a
point, and so, by Desargues's theorem itself, _A_, _B_, and _D_ are on a
straight line.
*27. Importance of the theorem.* The importance of this theorem lies in
the fact that, _A_, _B_, and _C_ being given, an indefinite number of
quadrangles _K'_, _L'_, _M'_, _N'_ my be found such that _K'L'_ and _M'N'_
meet in _A_, _K'N'_ and _L'M'_ in _C_, with _L'N'_ passing through _B_.
Indeed, the lines _AK'_ and _AM'_ may be drawn arbitrarily through _A_,
and any line through _B_ may be used to determine _L'_ and _N'_. By
joining these two points to _C_ the points _K'_ and _M'_ are determined.
Then the line joining _K'_ and _M'_, found in this way, must pass through
the point _D_ already determined by the quadrangle _K_, _L_, _M_, _N_.
_The three points __A__, __B__, __C__, given in order, serve thus to
determine a fourth point __D__._
*28.* In a complete quadrangle the line joining any two points is called
the _opposite side_ to the line joining the other two points. The result
of the preceding paragraph may then be stated as follows:
Given three points, _A_, _B_, _C_, in a straight line, if a pair of
opposite sides of a complete quadrangle pass through _A_, and another pair
through _C_, and one of the remaining two sides goes through _B_, then the
other of the remaining two sides will go through a fixed point which does
not depend on the quadrangle employed.
*29. Four harmonic points.* Four points, _A_, _B_, _C_, _D_, related as
in the preceding theorem are called _four harmonic points_. The point _D_
is called the _fourth harmonic of __B__ with respect to __A__ and __C_.
Since _B_ and _D_ play exactly the same role in the above construction,
_B__ is also the fourth harmonic of __D__ with respect to __A__ and __C_.
_B_ and _D_ are called _harmonic conjugates with respect to __A__ and
__C_. We proceed to show that _A_ and _C_ are also harmonic conjugates
with respect to _B_ and _D_--that is, that it is possible to find a
quadrangle of which two opposite sides shall pass through _B_, two through
_D_, and of the remaining pair, one through _A_ and the other through _C_.
[Figure 5]
FIG. 5
Let _O_ be the intersection of _KM_ and _LN_ (Fig. 5). Join _O_ to _A_ and
_C_. The joining lines cut out on the sides of the quadrangle four points,
_P_, _Q_, _R_, _S_. Consider the quadrangle _P_, _K_, _Q_, _O_. One pair
of opposite sides passes through _A_, one through _C_, and one remaining
side through _D_; therefore the other remaining side must pass through
_B_. Similarly, _RS_ passes through _B_ and _PS_ and _QR_ pass through
_D_. The quadrangle _P_, _Q_, _R_, _S_ therefore has two opposite sides
through _B_, two through _D_, and the remaining pair through _A_ and _C_.
_A_ and _C_ are thus harmonic conjugates with respect to _B_ and _D_. We
may sum up the discussion, therefore, as follows:
*30.* If _A_ and _C_ are harmonic conjugates with respect to _B_ and _D_,
then _B_ and _D_ are harmonic conjugates with respect to _A_ and _C_.
*31. Importance of the notion.* The importance of the notion of four
harmonic points lies in the fact that it is a relation which is carried
over from four points in a point-row _u_ to the four points that
correspond to them in any point-row _u'_ perspective to _u_.
To prove this statement we construct a quadrangle _K_, _L_, _M_, _N_ such
that _KL_ and _MN_ pass through _A_, _KN_ and _LM_ through _C_, _LN_
through _B_, and _KM_ through _D_. Take now any point _S_ not in the plane
of the quadrangle and construct the planes determined by _S_ and all the
seven lines of the figure. Cut across this set of planes by another plane
not passing through _S_. This plane cuts out on the set of seven planes
another quadrangle which determines four new harmonic points, _A'_, _B'_,
_C'_, _D'_, on the lines joining _S_ to _A_, _B_, _C_, _D_. But _S_ may be
taken as any point, since the original quadrangle may be taken in any
plane through _A_, _B_, _C_, _D_; and, further, the points _A'_, _B'_,
_C'_, _D'_ are the intersection of _SA_, _SB_, _SC_, _SD_ by any line. We
have, then, the remarkable theorem:
*32.* _If any point is joined to four harmonic points, and the four lines
thus obtained are cut by any fifth, the four points of intersection are
again harmonic._
*33. Four harmonic lines.* We are now able to extend the notion of
harmonic elements to pencils of rays, and indeed to axial pencils. For if
we define _four harmonic rays_ as four rays which pass through a point and
which pass one through each of four harmonic points, we have the theorem
_Four harmonic lines are cut by any transversal in four harmonic points._
*34. Four harmonic planes.* We also define _four harmonic planes_ as four
planes through a line which pass one through each of four harmonic points,
and we may show that
_Four harmonic planes are cut by any plane not passing through their
common line in four harmonic lines, and also by any line in four harmonic
points._
For let the planes {~GREEK SMALL LETTER ALPHA~}, {~GREEK SMALL LETTER BETA~}, {~GREEK SMALL LETTER GAMMA~}, {~GREEK SMALL LETTER DELTA~}, which all pass through the line _g_, pass
also through the four harmonic points _A_, _B_, _C_, _D_, so that {~GREEK SMALL LETTER ALPHA~} passes
through _A_, etc. Then it is clear that any plane {~GREEK SMALL LETTER PI~} through _A_, _B_, _C_,
_D_ will cut out four harmonic lines from the four planes, for they are
lines through the intersection _P_ of _g_ with the plane {~GREEK SMALL LETTER PI~}, and they pass
through the given harmonic points _A_, _B_, _C_, _D_. Any other plane {~GREEK SMALL LETTER SIGMA~}
cuts _g_ in a point _S_ and cuts {~GREEK SMALL LETTER ALPHA~}, {~GREEK SMALL LETTER BETA~}, {~GREEK SMALL LETTER GAMMA~}, {~GREEK SMALL LETTER DELTA~} in four lines that meet {~GREEK SMALL LETTER PI~} in
four points _A'_, _B'_, _C'_, _D'_ lying on _PA_, _PB_, _PC_, and _PD_
respectively, and are thus four harmonic hues. Further, any ray cuts {~GREEK SMALL LETTER ALPHA~}, {~GREEK SMALL LETTER BETA~},
{~GREEK SMALL LETTER GAMMA~}, {~GREEK SMALL LETTER DELTA~} in four harmonic points, since any plane through the ray gives four
harmonic lines of intersection.
*35.* These results may be put together as follows:
_Given any two assemblages of points, rays, or planes, perspectively
related to each other, four harmonic elements of one must correspond to
four elements of the other which are likewise harmonic._
If, now, two forms are perspectively related to a third, any four harmonic
elements of one must correspond to four harmonic elements in the other. We
take this as our definition of projective correspondence, and say:
*36. Definition of projectivity.* _Two fundamental forms are protectively
related to each other when a one-to-one correspondence exists between the
elements of the two and when four harmonic elements of one correspond to
four harmonic elements of the other._
[Figure 6]
FIG. 6
*37. Correspondence between harmonic conjugates.* Given four harmonic
points, _A_, _B_, _C_, _D_; if we fix _A_ and _C_, then _B_ and _D_ vary
together in a way that should be thoroughly understood. To get a clear
conception of their relative motion we may fix the points _L_ and _M_ of
the quadrangle _K_, _L_, _M_, _N_ (Fig. 6). Then, as _B_ describes the
point-row _AC_, the point _N_ describes the point-row _AM_ perspective to
it. Projecting _N_ again from _C_, we get a point-row _K_ on _AL_
perspective to the point-row _N_ and thus projective to the point-row _B_.
Project the point-row _K_ from _M_ and we get a point-row _D_ on _AC_
again, which is projective to the point-row _B_. For every point _B_ we
have thus one and only one point _D_, and conversely. In other words, we
have set up a one-to-one correspondence between the points of a single
point-row, which is also a projective correspondence because four harmonic
points _B_ correspond to four harmonic points _D_. We may note also that
the correspondence is here characterized by a feature which does not
always appear in projective correspondences: namely, the same process that
carries one from _B_ to _D_ will carry one back from _D_ to _B_ again.
This special property will receive further study in the chapter on
Involution.
*38.* It is seen that as _B_ approaches _A_, _D_ also approaches _A_. As
_B_ moves from _A_ toward _C_, _D_ moves from _A_ in the opposite
direction, passing through the point at infinity on the line _AC_, and
returns on the other side to meet _B_ at _C_ again. In other words, as _B_
traverses _AC_, _D_ traverses the rest of the line from _A_ to _C_ through
infinity. In all positions of _B_, except at _A_ or _C_, _B_ and _D_ are
separated from each other by _A_ and _C_.
*39. Harmonic conjugate of the point at infinity.* It is natural to
inquire what position of _B_ corresponds to the infinitely distant
position of _D_. We have proved (§ 27) that the particular quadrangle _K_,
_L_, _M_, _N_ employed is of no consequence. We shall therefore avail
ourselves of one that lends itself most readily to the solution of the
problem. We choose the point _L_ so that the triangle _ALC_ is isosceles
(Fig. 7). Since _D_ is supposed to be at infinity, the line _KM_ is
parallel to _AC_. Therefore the triangles _KAC_ and _MAC_ are equal, and
the triangle _ANC_ is also isosceles. The triangles _CNL_ and _ANL_ are
therefore equal, and the line _LB_ bisects the angle _ALC_. _B_ is
therefore the middle point of _AC_, and we have the theorem
_The harmonic conjugate of the middle point of __AC__ is at infinity._
[Figure 7]
FIG. 7
*40. Projective theorems and metrical theorems. Linear construction.* This
theorem is the connecting link between the general protective theorems
which we have been considering so far and the metrical theorems of
ordinary geometry. Up to this point we have said nothing about
measurements, either of line segments or of angles. Desargues's theorem
and the theory of harmonic elements which depends on it have nothing to do
with magnitudes at all. Not until the notion of an infinitely distant
point is brought in is any mention made of distances or directions. We
have been able to make all of our constructions up to this point by means
of the straightedge, or ungraduated ruler. A construction made with such
an instrument we shall call a _linear_ construction. It requires merely
that we be able to draw the line joining two points or find the point of
intersection of two lines.
*41. Parallels and mid-points.* It might be thought that drawing a line
through a given point parallel to a given line was only a special case of
drawing a line joining two points. Indeed, it consists only in drawing a
line through the given point and through the "infinitely distant point" on
the given line. It must be remembered, however, that the expression
"infinitely distant point" must not be taken literally. When we say that
two parallel lines meet "at infinity," we really mean that they do not
meet at all, and the only reason for using the expression is to avoid
tedious statement of exceptions and restrictions to our theorems. We ought
therefore to consider the drawing of a line parallel to a given line as a
different accomplishment from the drawing of the line joining two given
points. It is a remarkable consequence of the last theorem that a parallel
to a given line and the mid-point of a given segment are equivalent data.
For the construction is reversible, and if we are given the middle point
of a given segment, we can construct _linearly_ a line parallel to that
segment. Thus, given that _B_ is the middle point of _AC_, we may draw any
two lines through _A_, and any line through _B_ cutting them in points _N_
and _L_. Join _N_ and _L_ to _C_ and get the points _K_ and _M_ on the two
lines through _A_. Then _KM_ is parallel to _AC_. _The bisection of a
given segment and the drawing of a line parallel to the segment are
equivalent data when linear construction is used._
Pages:
1 | 2 |
3 |
4 |
5 |
6 |
7 |
8