An Elementary Course in Synthetic Projective Geometry
by Lehmer, Derrick Norman
Edition 1, (November 4, 2005)
PREFACE
The following course is intended to give, in as simple a way as possible,
the essentials of synthetic projective geometry. While, in the main, the
theory is developed along the well-beaten track laid out by the great
masters of the subject, it is believed that there has been a slight
smoothing of the road in some places. Especially will this be observed in
the chapter on Involution. The author has never felt satisfied with the
usual treatment of that subject by means of circles and anharmonic ratios.
A purely projective notion ought not to be based on metrical foundations.
Metrical developments should be made there, as elsewhere in the theory, by
the introduction of infinitely distant elements.
The author has departed from the century-old custom of writing in parallel
columns each theorem and its dual. He has not found that it conduces to
sharpness of vision to try to focus his eyes on two things at once. Those
who prefer the usual method of procedure can, of course, develop the two
sets of theorems side by side; the author has not found this the better
plan in actual teaching.
As regards nomenclature, the author has followed the lead of the earlier
writers in English, and has called the system of lines in a plane which
all pass through a point a _pencil of rays_ instead of a _bundle of rays_,
as later writers seem inclined to do. For a point considered as made up of
all the lines and planes through it he has ventured to use the term _point
system_, as being the natural dualization of the usual term _plane
system_. He has also rejected the term _foci of an involution_, and has
not used the customary terms for classifying involutions—_hyperbolic
involution_, _elliptic involution_ and _parabolic involution_. He has
found that all these terms are very confusing to the student, who
inevitably tries to connect them in some way with the conic sections.
Enough examples have been provided to give the student a clear grasp of
the theory. Many are of sufficient generality to serve as a basis for
individual investigation on the part of the student. Thus, the third
example at the end of the first chapter will be found to be very fruitful
in interesting results. A correspondence is there indicated between lines
in space and circles through a fixed point in space. If the student will
trace a few of the consequences of that correspondence, and determine what
configurations of circles correspond to intersecting lines, to lines in a
plane, to lines of a plane pencil, to lines cutting three skew lines,
etc., he will have acquired no little practice in picturing to himself
figures in space.
The writer has not followed the usual practice of inserting historical
notes at the foot of the page, and has tried instead, in the last chapter,
to give a consecutive account of the history of pure geometry, or, at
least, of as much of it as the student will be able to appreciate who has
mastered the course as given in the preceding chapters. One is not apt to
get a very wide view of the history of a subject by reading a hundred
biographical footnotes, arranged in no sort of sequence. The writer,
moreover, feels that the proper time to learn the history of a subject is
after the student has some general ideas of the subject itself.
The course is not intended to furnish an illustration of how a subject may
be developed, from the smallest possible number of fundamental
assumptions. The author is aware of the importance of work of this sort,
but he does not believe it is possible at the present time to write a book
along such lines which shall be of much use for elementary students. For
the purposes of this course the student should have a thorough grounding
in ordinary elementary geometry so far as to include the study of the
circle and of similar triangles. No solid geometry is needed beyond the
little used in the proof of Desargues’ theorem (25), and, except in
certain metrical developments of the general theory, there will be no call
for a knowledge of trigonometry or analytical geometry. Naturally the
student who is equipped with these subjects as well as with the calculus
will be a little more mature, and may be expected to follow the course all
the more easily. The author has had no difficulty, however, in presenting
it to students in the freshman class at the University of California.
The subject of synthetic projective geometry is, in the opinion of the
writer, destined shortly to force its way down into the secondary schools;
and if this little book helps to accelerate the movement, he will feel
amply repaid for the task of working the materials into a form available
for such schools as well as for the lower classes in the university.
The material for the course has been drawn from many sources. The author
is chiefly indebted to the classical works of Reye, Cremona, Steiner,
Poncelet, and Von Staudt. Acknowledgments and thanks are also due to
Professor Walter C. Eells, of the U.S. Naval Academy at Annapolis, for his
searching examination and keen criticism of the manuscript; also to
Professor Herbert Ellsworth Slaught, of The University of Chicago, for his
many valuable suggestions, and to Professor B. M. Woods and Dr. H. N.
Wright, of the University of California, who have tried out the methods of
presentation, in their own classes.
D. N. LEHMER
BERKELEY, CALIFORNIA
CONTENTS
Preface
Contents
CHAPTER I - ONE-TO-ONE CORRESPONDENCE
1. Definition of one-to-one correspondence
2. Consequences of one-to-one correspondence
3. Applications in mathematics
4. One-to-one correspondence and enumeration
5. Correspondence between a part and the whole
6. Infinitely distant point
7. Axial pencil; fundamental forms
8. Perspective position
9. Projective relation
10. Infinity-to-one correspondence
11. Infinitudes of different orders
12. Points in a plane
13. Lines through a point
14. Planes through a point
15. Lines in a plane
16. Plane system and point system
17. Planes in space
18. Points of space
19. Space system
20. Lines in space
21. Correspondence between points and numbers
22. Elements at infinity
PROBLEMS
CHAPTER II - RELATIONS BETWEEN FUNDAMENTAL FORMS IN ONE-TO-ONE
CORRESPONDENCE WITH EACH OTHER
23. Seven fundamental forms
24. Projective properties
25. Desargues’s theorem
26. Fundamental theorem concerning two complete quadrangles
27. Importance of the theorem
28. Restatement of the theorem
29. Four harmonic points
30. Harmonic conjugates
31. Importance of the notion of four harmonic points
32. Projective invariance of four harmonic points
33. Four harmonic lines
34. Four harmonic planes
35. Summary of results
36. Definition of projectivity
37. Correspondence between harmonic conjugates
38. Separation of harmonic conjugates
39. Harmonic conjugate of the point at infinity
40. Projective theorems and metrical theorems. Linear construction
41. Parallels and mid-points
42. Division of segment into equal parts
43. Numerical relations
44. Algebraic formula connecting four harmonic points
45. Further formulae
46. Anharmonic ratio
PROBLEMS
CHAPTER III - COMBINATION OF TWO PROJECTIVELY RELATED FUNDAMENTAL FORMS
47. Superposed fundamental forms. Self-corresponding elements
48. Special case
49. Fundamental theorem. Postulate of continuity
50. Extension of theorem to pencils of rays and planes
51. Projective point-rows having a self-corresponding point in common
52. Point-rows in perspective position
53. Pencils in perspective position
54. Axial pencils in perspective position
55. Point-row of the second order
56. Degeneration of locus
57. Pencils of rays of the second order
58. Degenerate case
59. Cone of the second order
PROBLEMS
CHAPTER IV - POINT-ROWS OF THE SECOND ORDER
60. Point-row of the second order defined
61. Tangent line
62. Determination of the locus
63. Restatement of the problem
64. Solution of the fundamental problem
65. Different constructions for the figure
66. Lines joining four points of the locus to a fifth
67. Restatement of the theorem
68. Further important theorem
69. Pascal’s theorem
70. Permutation of points in Pascal’s theorem
71. Harmonic points on a point-row of the second order
72. Determination of the locus
73. Circles and conics as point-rows of the second order
74. Conic through five points
75. Tangent to a conic
76. Inscribed quadrangle
77. Inscribed triangle
78. Degenerate conic
PROBLEMS
CHAPTER V - PENCILS OF RAYS OF THE SECOND ORDER
79. Pencil of rays of the second order defined
80. Tangents to a circle
81. Tangents to a conic
82. Generating point-rows lines of the system
83. Determination of the pencil
84. Brianchon’s theorem
85. Permutations of lines in Brianchon’s theorem
86. Construction of the penvil by Brianchon’s theorem
87. Point of contact of a tangent to a conic
88. Circumscribed quadrilateral
89. Circumscribed triangle
90. Use of Brianchon’s theorem
91. Harmonic tangents
92. Projectivity and perspectivity
93. Degenerate case
94. Law of duality
PROBLEMS
CHAPTER VI - POLES AND POLARS
95. Inscribed and circumscribed quadrilaterals
96. Definition of the polar line of a point
97. Further defining properties
98. Definition of the pole of a line
99. Fundamental theorem of poles and polars
100. Conjugate points and lines
101. Construction of the polar line of a given point
102. Self-polar triangle
103. Pole and polar projectively related
104. Duality
105. Self-dual theorems
106. Other correspondences
PROBLEMS
CHAPTER VII - METRICAL PROPERTIES OF THE CONIC SECTIONS
107. Diameters. Center
108. Various theorems
109. Conjugate diameters
110. Classification of conics
111. Asymptotes
112. Various theorems
113. Theorems concerning asymptotes
114. Asymptotes and conjugate diameters
115. Segments cut off on a chord by hyperbola and its asymptotes
116. Application of the theorem
117. Triangle formed by the two asymptotes and a tangent
118. Equation of hyperbola referred to the asymptotes
119. Equation of parabola
120. Equation of central conics referred to conjugate diameters
PROBLEMS
CHAPTER VIII - INVOLUTION
121. Fundamental theorem
122. Linear construction
123. Definition of involution of points on a line
124. Double-points in an involution
125. Desargues’s theorem concerning conics through four points
126. Degenerate conics of the system
127. Conics through four points touching a given line
128. Double correspondence
129. Steiner’s construction
130. Application of Steiner’s construction to double correspondence
131. Involution of points on a point-row of the second order.
132. Involution of rays
133. Double rays
134. Conic through a fixed point touching four lines
135. Double correspondence
136. Pencils of rays of the second order in involution
137. Theorem concerning pencils of the second order in involution
138. Involution of rays determined by a conic
139. Statement of theorem
140. Dual of the theorem
PROBLEMS
CHAPTER IX - METRICAL PROPERTIES OF INVOLUTIONS
141. Introduction of infinite point; center of involution
142. Fundamental metrical theorem
143. Existence of double points
144. Existence of double rays
145. Construction of an involution by means of circles
146. Circular points
147. Pairs in an involution of rays which are at right angles. Circular
involution
148. Axes of conics
149. Points at which the involution determined by a conic is circular
150. Properties of such a point
151. Position of such a point
152. Discovery of the foci of the conic
153. The circle and the parabola
154. Focal properties of conics
155. Case of the parabola
156. Parabolic reflector
157. Directrix. Principal axis. Vertex
158. Another definition of a conic
159. Eccentricity
160. Sum or difference of focal distances
PROBLEMS
CHAPTER X - ON THE HISTORY OF SYNTHETIC PROJECTIVE GEOMETRY
161. Ancient results
162. Unifying principles
163. Desargues
164. Poles and polars
165. Desargues’s theorem concerning conics through four points
166. Extension of the theory of poles and polars to space
167. Desargues’s method of describing a conic
168. Reception of Desargues’s work
169. Conservatism in Desargues’s time
170. Desargues’s style of writing
171. Lack of appreciation of Desargues
172. Pascal and his theorem
173. Pascal’s essay
174. Pascal’s originality
175. De la Hire and his work
176. Descartes and his influence
177. Newton and Maclaurin
178. Maclaurin’s construction
179. Descriptive geometry and the second revival
180. Duality, homology, continuity, contingent relations
181. Poncelet and Cauchy
182. The work of Poncelet
183. The debt which analytic geometry owes to synthetic geometry
184. Steiner and his work
185. Von Staudt and his work
186. Recent developments
INDEX
CHAPTER I - ONE-TO-ONE CORRESPONDENCE
*1. Definition of one-to-one correspondence.* Given any two sets of
individuals, if it is possible to set up such a correspondence between the
two sets that to any individual in one set corresponds one and only one
individual in the other, then the two sets are said to be in _one-to-one
correspondence_ with each other. This notion, simple as it is, is of
fundamental importance in all branches of science. The process of counting
is nothing but a setting up of a one-to-one correspondence between the
objects to be counted and certain words, ’one,’ ’two,’ ’three,’ etc., in
the mind. Many savage peoples have discovered no better method of counting
than by setting up a one-to-one correspondence between the objects to be
counted and their fingers. The scientist who busies himself with naming
and classifying the objects of nature is only setting up a one-to-one
correspondence between the objects and certain words which serve, not as a
means of counting the objects, but of listing them in a convenient way.
Thus he may be able to marshal and array his material in such a way as to
bring to light relations that may exist between the objects themselves.
Indeed, the whole notion of language springs from this idea of one-to-one
correspondence.
*2. Consequences of one-to-one correspondence.* The most useful and
interesting problem that may arise in connection with any one-to-one
correspondence is to determine just what relations existing between the
individuals of one assemblage may be carried over to another assemblage in
one-to-one correspondence with it. It is a favorite error to assume that
whatever holds for one set must also hold for the other. Magicians are apt
to assign magic properties to many of the words and symbols which they are
in the habit of using, and scientists are constantly confusing objective
things with the subjective formulas for them. After the physicist has set
up correspondences between physical facts and mathematical formulas, the
"interpretation" of these formulas is his most important and difficult
task.
*3.* In mathematics, effort is constantly being made to set up one-to-one
correspondences between simple notions and more complicated ones, or
between the well-explored fields of research and fields less known. Thus,
by means of the mechanism employed in analytic geometry, algebraic
theorems are made to yield geometric ones, and vice versa. In geometry we
get at the properties of the conic sections by means of the properties of
the straight line, and cubic surfaces are studied by means of the plane.
[Figure 1]
FIG. 1
[Figure 2]
FIG. 2
*4. One-to-one correspondence and enumeration.* If a one-to-one
correspondence has been set up between the objects of one set and the
objects of another set, then the inference may usually be drawn that they
have the same number of elements. If, however, there is an infinite number
of individuals in each of the two sets, the notion of counting is
necessarily ruled out. It may be possible, nevertheless, to set up a
one-to-one correspondence between the elements of two sets even when the
number is infinite. Thus, it is easy to set up such a correspondence
between the points of a line an inch long and the points of a line two
inches long. For let the lines (Fig. 1) be _AB_ and _A’B’_. Join _AA’_ and
_BB’_, and let these joining lines meet in _S_. For every point _C_ on
_AB_ a point _C’_ may be found on _A’B’_ by joining _C_ to _S_ and noting
the point _C’_ where _CS_ meets _A’B’_. Similarly, a point _C_ may be
found on _AB_ for any point _C’_ on _A’B’_. The correspondence is clearly
one-to-one, but it would be absurd to infer from this that there were just
as many points on _AB_ as on _A’B’_. In fact, it would be just as
reasonable to infer that there were twice as many points on _A’B’_ as on
_AB_. For if we bend _A’B’_ into a circle with center at _S_ (Fig. 2), we
see that for every point _C_ on _AB_ there are two points on _A’B’_. Thus
it is seen that the notion of one-to-one correspondence is more extensive
than the notion of counting, and includes the notion of counting only when
applied to finite assemblages.
*5. Correspondence between a part and the whole of an infinite
assemblage.* In the discussion of the last paragraph the remarkable fact
was brought to light that it is sometimes possible to set the elements of
an assemblage into one-to-one correspondence with a part of those
elements. A moment’s reflection will convince one that this is never
possible when there is a finite number of elements in the
assemblage.—Indeed, we may take this property as our definition of an
infinite assemblage, and say that an infinite assemblage is one that may
be put into one-to-one correspondence with part of itself. This has the
advantage of being a positive definition, as opposed to the usual negative
definition of an infinite assemblage as one that cannot be counted.
*6. Infinitely distant point.* We have illustrated above a simple method
of setting the points of two lines into one-to-one correspondence. The
same illustration will serve also to show how it is possible to set the
points on a line into one-to-one correspondence with the lines through a
point. Thus, for any point _C_ on the line _AB_ there is a line _SC_
through _S_. We must assume the line _AB_ extended indefinitely in both
directions, however, if we are to have a point on it for every line
through _S_; and even with this extension there is one line through _S_,
according to Euclid’s postulate, which does not meet the line _AB_ and
which therefore has no point on _AB_ to correspond to it. In order to
smooth out this discrepancy we are accustomed to assume the existence of
an _infinitely distant_ point on the line _AB_ and to assign this point
as the corresponding point of the exceptional line of _S_. With this
understanding, then, we may say that we have set the lines through a point
and the points on a line into one-to-one correspondence. This
correspondence is of such fundamental importance in the study of
projective geometry that a special name is given to it. Calling the
totality of points on a line a _point-row_, and the totality of lines
through a point a _pencil of rays_, we say that the point-row and the
pencil related as above are in _perspective position_, or that they are
_perspectively related_.
*7. Axial pencil; fundamental forms.* A similar correspondence may be set
up between the points on a line and the planes through another line which
does not meet the first. Such a system of planes is called an _axial
pencil_, and the three assemblages—the point-row, the pencil of rays, and
the axial pencil—are called _fundamental forms_. The fact that they may
all be set into one-to-one correspondence with each other is expressed by
saying that they are of the same order. It is usual also to speak of them
as of the first order. We shall see presently that there are other
assemblages which cannot be put into this sort of one-to-one
correspondence with the points on a line, and that they will very
reasonably be said to be of a higher order.
*8. Perspective position.* We have said that a point-row and a pencil of
rays are in perspective position if each ray of the pencil goes through
the point of the point-row which corresponds to it. Two pencils of rays
are also said to be in perspective position if corresponding rays meet on
a straight line which is called the axis of perspectivity. Also, two
point-rows are said to be in perspective position if corresponding points
lie on straight lines through a point which is called the center of
perspectivity. A point-row and an axial pencil are in perspective position
if each plane of the pencil goes through the point on the point-row which
corresponds to it, and an axial pencil and a pencil of rays are in
perspective position if each ray lies in the plane which corresponds to
it; and, finally, two axial pencils are perspectively related if
corresponding planes meet in a plane.
*9. Projective relation.* It is easy to imagine a more general
correspondence between the points of two point-rows than the one just
described. If we take two perspective pencils, _A_ and _S_, then a
point-row _a_ perspective to _A_ will be in one-to-one correspondence with
a point-row _b_ perspective to _B_, but corresponding points will not, in
general, lie on lines which all pass through a point. Two such point-rows
are said to be _projectively related_, or simply projective to each other.
Similarly, two pencils of rays, or of planes, are projectively related to
each other if they are perspective to two perspective point-rows. This
idea will be generalized later on. It is important to note that between
the elements of two projective fundamental forms there is a one-to-one
correspondence, and also that this correspondence is in general
_continuous_; that is, by taking two elements of one form sufficiently
close to each other, the two corresponding elements in the other form may
be made to approach each other arbitrarily close. In the case of
point-rows this continuity is subject to exception in the neighborhood of
the point "at infinity."
*10. Infinity-to-one correspondence.* It might be inferred that any
infinite assemblage could be put into one-to-one correspondence with any
other. Such is not the case, however, if the correspondence is to be
continuous, between the points on a line and the points on a plane.
Consider two lines which lie in different planes, and take _m_ points on
one and _n_ points on the other. The number of lines joining the _m_
points of one to the _n_ points jof the other is clearly _mn_. If we
symbolize the totality of points on a line by [infinity], then a
reasonable symbol for the totality of lines drawn to cut two lines would
be [infinity]2. Clearly, for every point on one line there are [infinity]
lines cutting across the other, so that the correspondence might be called
[infinity]-to-one. Thus the assemblage of lines cutting across two lines
is of higher order than the assemblage of points on a line; and as we have
called the point-row an assemblage of the first order, the system of lines
cutting across two lines ought to be called of the second order.
*11. Infinitudes of different orders.* Now it is easy to set up a
one-to-one correspondence between the points in a plane and the system of
lines cutting across two lines which lie in different planes. In fact,
each line of the system of lines meets the plane in one point, and each
point in the plane determines one and only one line cutting across the two
given lines—namely, the line of intersection of the two planes determined
by the given point with each of the given lines. The assemblage of points
in the plane is thus of the same order as that of the lines cutting across
two lines which lie in different planes, and ought therefore to be spoken
of as of the second order. We express all these results as follows:
*12.* If the infinitude of points on a line is taken as the infinitude of
the first order, then the infinitude of lines in a pencil of rays and the
infinitude of planes in an axial pencil are also of the first order, while
the infinitude of lines cutting across two "skew" lines, as well as the
infinitude of points in a plane, are of the second order.
*13.* If we join each of the points of a plane to a point not in that
plane, we set up a one-to-one correspondence between the points in a plane
and the lines through a point in space. _Thus the infinitude of lines
through a point in space is of the second order._
*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.