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 rôle 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 α, β, γ, δ, which all pass through the line _g_, pass
also through the four harmonic points _A_, _B_, _C_, _D_, so that α passes
through _A_, etc. Then it is clear that any plane π 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 π, and they pass
through the given harmonic points _A_, _B_, _C_, _D_. Any other plane σ
cuts _g_ in a point _S_ and cuts α, β, γ, δ in four lines that meet π 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 α, β,
γ, δ 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._
*42.* It is not difficult to give a linear construction for the problem
to divide a given segment into _n_ equal parts, given only a parallel to
the segment. This is simple enough when _n_ is a power of _2_. For any
other number, such as _29_, divide any segment on the line parallel to
_AC_ into _32_ equal parts, by a repetition of the process just described.
Take _29_ of these, and join the first to _A_ and the last to _C_. Let
these joining lines meet in _S_. Join _S_ to all the other points. Other
problems, of a similar sort, are given at the end of the chapter.
*43. Numerical relations.* Since three points, given in order, are
sufficient to determine a fourth, as explained above, it ought to be
possible to reproduce the process numerically in view of the one-to-one
correspondence which exists between points on a line and numbers; a
correspondence which, to be sure, we have not established here, but which
is discussed in any treatise on the theory of point sets. We proceed to
discover what relation between four numbers corresponds to the harmonic
relation between four points.
[Figure 8]
FIG. 8
*44.* Let _A_, _B_, _C_, _D_ be four harmonic points (Fig. 8), and let
_SA_, _SB_, _SC_, _SD_ be four harmonic lines. Assume a line drawn through
_B_ parallel to _SD_, meeting _SA_ in _A’_ and _SC_ in _C’_. Then _A’_,
_B’_, _C’_, and the infinitely distant point on _A’C’_ are four harmonic
points, and therefore _B_ is the middle point of the segment _A’C’_. Then,
since the triangle _DAS_ is similar to the triangle _BAA’_, we may write
the proportion
_AB : AD = BA’ : SD._
Also, from the similar triangles _DSC_ and _BCC’_, we have
_CD : CB = SD : B’C._
From these two proportions we have, remembering that _BA’ = BC’_,
[formula]
the minus sign being given to the ratio on account of the fact that _A_
and _C_ are always separated from _B_ and _D_, so that one or three of the
segments _AB_, _CD_, _AD_, _CB_ must be negative.
*45.* Writing the last equation in the form
_CB : AB = -CD : AD,_
and using the fundamental relation connecting three points on a line,
_PR + RQ = PQ,_
which holds for all positions of the three points if account be taken of
the sign of the segments, the last proportion may be written
_(CB - BA) : AB = -(CA - DA) : AD,_
or
_(AB - AC) : AB = (AC - AD) : AD;_
so that _AB_, _AC_, and _AD_ are three quantities in hamonic progression,
since the difference between the first and second is to the first as the
difference between the second and third is to the third. Also, from this
last proportion comes the familiar relation
[formula]
which is convenient for the computation of the distance _AD_ when _AB_ and
_AC_ are given numerically.
*46. Anharmonic ratio.* The corresponding relations between the
trigonometric functions of the angles determined by four harmonic lines
are not difficult to obtain, but as we shall not need them in building up
the theory of projective geometry, we will not discuss them here. Students
who have a slight acquaintance with trigonometry may read in a later
chapter (§ 161) a development of the theory of a more general relation,
called the _anharmonic ratio_, or _cross ratio_, which connects any four
points on a line.
PROBLEMS
*1*. Draw through a given point a line which shall pass through the
inaccessible point of intersection of two given lines. The following
construction may be made to depend upon Desargues’s theorem: Through the
given point _P_ draw any two rays cutting the two lines in the points
_AB’_ and _A’B_, _A_, _B_, lying on one of the given lines and _A’_, _B’_,
on the other. Join _AA’_ and _BB’_, and find their point of intersection
_S_. Through _S_ draw any other ray, cutting the given lines in _CC’_.
Join _BC’_ and _B’C_, and obtain their point of intersection _Q_. _PQ_ is
the desired line. Justify this construction.
*2.* To draw through a given point _P_ a line which shall meet two given
lines in points _A_ and _B_, equally distant from _P_. Justify the
following construction: Join _P_ to the point _S_ of intersection of the
two given lines. Construct the fourth harmonic of _PS_ with respect to the
two given lines. Draw through _P_ a line parallel to this line. This is
the required line.
*3.* Given a parallelogram in the same plane with a given segment _AC_,
to construct linearly the middle point of _AC_.
*4.* Given four harmonic lines, of which one pair are at right angles to
each other, show that the other pair make equal angles with them. This is
a theorem of which frequent use will be made.
*5.* Given the middle point of a line segment, to draw a line parallel to
the segment and passing through a given point.
*6.* A line is drawn cutting the sides of a triangle _ABC_ in the points
_A’_, _B’_, _C’_ the point _A’_ lying on the side _BC_, etc. The harmonic
conjugate of _A’_ with respect to _B_ and _C_ is then constructed and
called _A"_. Similarly, _B"_ and _C"_ are constructed. Show that _A"B"C"_
lie on a straight line. Find other sets of three points on a line in the
figure. Find also sets of three lines through a point.