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GEOMETRY OF FOUR DIMENSIONS
THE MACMILLAN COMPANY
NEW YORK - BOSTON CHICAGO DALLAS
ATLANTA SAN FKANUSCO
MACMILLAN & CO., LIMITED
LONDON BOMBAY CALCUTTA
MELBOURNE
THE MACMILLAN CO. OF CANADA, LTD.
TORONTO
GEOMETRY
OF FOUR DIMENSIONS
BY
HENRY PARKER MANNING, PH.D.
ASSOCIATE PROFESSOR OF PURE MATHEMATICS
IN BROWN UNIVERSITY
Weto gork
THE MACMILLAN COMPANY
1914
All rights reserved
COPYRIGHT, 1914,
BY THE MACMILLAN COMPANY.
Set up and electrotyped. Published September, 1914.
J. S. Gushing Co. Berwick & Smith Co.
Norwood, Mass., U.S.A.
PREFACE
THE object and plan of this book are explained in the
Introduction (page 16). I had hoped to give some account
of the recent literature, but this would have delayed work
that has already taken several years. I have prepared a
list of technical terms as found in a few of the more
familiar writings, very incomplete, and, I fear, not with-
out errors. The list may be of service, however, to those
who wish to consult the authors referred to ; it will also
indicate something of the confusion that exists in a subject
whose nomenclature has not become fixed. It has been
necessary for me to introduce a considerable number of
terms, but most of these have been formed in accordance
with simple or well-established principles, and no attempt
has been made to distinguish them from the terms that
have already been used.
I am indebted to the kindness of Mr. George A. Plimp-
ton of New York for an opportunity to examine his copy
of Rudolph's Coss referred to on page 2. I am also under
many obligations to Mrs. Walter C. Bronson of Providence,
to Mr. Albert A. Bennett, Instructor at Princeton Univer-
sity, and to my colleagues, Professors R. C. Archibald
and R. G. D. Richardson, from all of whom I have re-
ceived valuable criticisms and suggestions. Many of the
references in the first four pages were found by Professor
Archibald ; several of these are not given in the leading
bibliographies, and the reference to Ozanam I have not
seen anywhere.
HENRY P. MANNING.
PROVIDENCE,
July, 1914.
CONTENTS
PAGE
INTRODUCTION i
CHAPTER I
THE FOUNDATIONS OF FOUR-DIMENSIONAL GEOMETRY
I.
II.
III.
POINTS AND LINES
TRIANGLES
PLANES ........
23
. 29
IV.
V.
VI.
VII.
VIII.
IX.
CONVEX POLYGONS
TETRAHEDRONS
HYPERPLANES
CONVEX PYRAMIDS AND PENTAHEDROIDS
SPACE OF FOUR DIMENSIONS
HYPERPYRAMIDS AND HYPERCONES
. 40
. 45
. 50
55
. 59
. . 63
CHAPTER II
PERPENDICULARITY AND SIMPLE ANGLES
INTRODUCTORY 73
I. LINES PERPENDICULAR TO A HYPERPLANE ... 74
II. ABSOLUTELY PERPENDICULAR PLANES .... 80
III. SIMPLY PERPENDICULAR PLANES 85
IV. PERPENDICULAR PLANES AND HYPERPLANES . . 90
V. HYPERPLANE ANGLES 95
vii
viii CONTENTS
CHAPTER III
ANGLES OF TWO PLANES AND ANGLES OF HIGHER
ORDER
PAGE
I. THE COMMON PERPENDICULAR OF Two LINES . . 105
II. POINT GEOMETRY 112
III. THE ANGLES OF Two PLANES 114
IV. POLYHEDROIDAL ANGLES 126
V. PLANO-POLYHEDRAL ANGLES 133
CHAPTER IV
SYMMETRY, ORDER, AND MOTION
I. ROTATION AND TRANSLATION 141
II. SYMMETRY 146
III. ORDER 153
IV. MOTION IN GENERAL 167
V. RECTANGULAR SYSTEMS 179
VI. ISOCLINE PLANES 180
CHAPTER V
HYPERPYRAMIDS, HYPERCONES, AND THE HYPER-
SPHERE
I. PENTAHEDROIDS AND HYPERPYRAMIDS .... 199
II. HYPERCONES AND DOUBLE CONES 204
III. THE HYPERSPHERE 207
CHAPTER VI
EUCLIDEAN GEOMETRY. FIGURES WITH PARALLEL
ELEMENTS
THE AXIOM OF PARALLELS 221
I. PARALLELS 221
II. THE "HYPERPLANE AT INFINITY" .... 230
CONTENTS ix
PAGE
III. HYPERPRJSMS .... ... 235
IV. DOUBLE PRISMS 241
V. HYPERCYLINDERS 253
VI. PRISM CYLINDERS AND DOUBLE CYLINDERS . . 256
CHAPTER VII
MEASUREMENT OF VOLUME AND HYPERVOLUME IN
HYPERSPACE
I. VOLUME 265
II. HYPERVOLUME 270
CHAPTER VIII
THE REGULAR POLYHEDROIDS
I. THE FOUR SIMPLER REGULAR POLYHEDROIDS . . 289
II. THE POLYHEDROID FORMULA 300
III. RECIPROCAL POLYHEDKOIDS AND RECIPROCAL NETS
OF POLYHEDROIDS 303
IV. CONSTRUCTION OF THK REGULAR 6oo-HEDROin AND
THE REGULAR 1 20-11 EDKOID 317
TECHNICAL TERMS 327
INDEX 335
INTRODUCTION
THE geometry of more than three dimensions is entirely
a modern branch of mathematics, going no farther back
than the first part of the nineteenth century. There are,
however, some early references to the number of dimensions
of space.
In the first book of the Heaven of Aristotle (384-322
B.C.) are these sentences: "The line has magnitude in one
way, the plane in two ways, and the solid in three ways,
and beyond these there is no other magnitude because the
three are all," and " There is no transfer into another kind,
like the transfer from length to area and from area to a
solid."* Simplicius (sixth century, A.D.) in his Commen-
taries says, "The admirable Ptolemy in his book On Dis-
tance well proved that there are not more than three dis-
tances, because of the necessity that distances should be
defined, and that the distances defined should be taken
along perpendicular lines, and because it is possible to take
only three lines that are mutually perpendicular, two by
which the plane is defined and a third measuring depth ;
so that if there were any other distance after the third it
would be entirely without measure and without definition.
Thus Aristotle seemed to conclude from induction that
there is no transfer into another magnitude, but Ptolemy
proved it." f
* Aristoteles, De Caelo, ed. Prantl, Leipzig, 1881, 268a, 7 and 30.
t Simplicii in Aristotdis De Caelo Commentaria, cd. Heiberg, Berlin, 1894, 7 ff >
33- Ptolemy lived about 150 A.D. The book on distance, irepi Smtrrdo-ew, is lost, and
with it Ptolemy's " proof " except so far as it may be reproduced in the above quo-
tation from Simplicius.
2 INTRODUCTION
There is also in the early history of algebra a use of terms
analogous to those derived from the plane and solid geom-
etry, but applicable only to geometry of more dimensions.
With the Greeks, and then in general with the mathe-
maticians that came after them, a number was thought
of as a line (of definite length), the product of two numbers
as a rectangle or plane , and the product of three numbers
as a parallelepiped or solid; or, if the numbers were equal,
the product of two was a square and of three a cube. When
they began to study algebra, other terms were required
for the higher powers, and so in Diophantus (third century)
we find square-square, square-cube, and cube-cube* In
later times there was a variation in the use of these terms.
Thus the square-cube came to mean the square of the cube,
or sixth power, while with Diophantus it means the square
times the cube, or fifth power. This change required the
introduction of new terms for powers of prime orders, and,
in particular, for the fifth power, which was finally called
a sur solid .^ The geometrical conception of equations
and the geometrical forms of their solutions J hindered
* Cantor, Vorlesungen uber Geschichte der Uathematik, vol. I, 3d ed , Leipzig,
1907, p. 470.
f In the edition of Rudolph's Coss (algebra) revised by Stifel (Konigsberg, 1553,
described by David Eugene Smith in Kara Artlhmetica, Boston, 1908, p 258) Sur-
solidum denotes the fifth power, Bsursohdum the seventh power, and so on (Part I,
chap 5, fol 63) Paciuolo (about 1445-1514) in his Swnma de Anthmetica Geo~
metria Proportiom el Proportionalita, printed in 1494, uses the terms primo relato and
sec undo relato (Cantor, Vorlesungen, etc , vol. If, 2d ed , 1000, p. 317) On the other
hand, Vieta (1540-1603) follows Diophantus. He expresses all the powers above
the third by compounds of quadrato and cubo, cubo-cubo-cubus being the ninth power
(Francisci Vieta opera malhematica, Leyden, 1646, p. 3 and elsewhere). The term
sursolid occurs several times in the geometry of Descartes (1596-1650). It is to be
noted, however, that a product with Descartes always means a line of definite
length derived from given lengths by proportions. Problems which lead to equations
of the fifth or sixth degrees require for their geometrical solution curves "one degree
more complicated than conies." Conies were called by the Greeks solid loci, and
these more complicated curves were called by Descartes sursolid loci (La Gtomttrie.
See pp. 20 and 29 of the edition published by Hermann, Paris, 1886).
I Such solutions are given in the second and sixth books of Euclid's Elements.
^See Heath's edition, Cambridge University Press, 1908, vol. I, p. 383.
EARLY REFERENCES TO DIMENSIONS 3
the progress of algebra with the ancients. Higher equations
than the third were avoided as unreal,* and when the study
of higher equations forced itself upon mathematicians, it
meant an impossible extension of geometrical notions,
which met with many protests, and only in later times
gave way to a purely numerical conception of the nature
of algebraic quantities. Thus Stifel (1486 ?-is67), in the
Algebra of Rudolph already referred to (footnote, pre-
ceding page), speaks of "going beyond the cube just as
if there were more than three dimensions," "which is,"
he adds, "against nature." f John Wallis (1616-1703)
in his Algebra objects to the "ungeometrical" names
given to the higher powers. He calls one of them a
"Monster in Nature, less possible than a Chimaera or
Centaure." He says: "Length, Breadth and Thickness,
take up the whole of Space. Nor can Fansie imagine how
there should be a Fourth Local Dimension beyond these
Three. "J Ozanam (1640-1717), after speaking of the
product of two letters as a rectangle and the product of
three as a rectangular parallelepiped, says that a product
of more than three letters will be a magnitude of "as many
dimensions as there are letters, but it will only be imaginary
because in nature we do not know of any quantity which
has more than three dimensions."
Again, we find in the writings of some philosophers ref-
erences to a space of four dimensions. Thus Henry More
(1614-1687), an English philosopher, in a book published
in 1671, says that spirits have four dimensions,^ and Kant
(1724-1804) refers in several places to the number of
dimensions of space. ||
* Matthicssen, Grundzugc der antikcn und modernen Algebra, 2d ed., Leipzig,
1896, pp. 544 and 921. f Part I, chap, i, fol. Q recto. J London, 1685, p. 126.
Dictionairc mathematiquc, Amsterdam, 1691, p. 62.
T Enchiridion metaphysicum, Pt. I, chap. 28, 7, p. 384.
|( For example, he says in the Critique of Pure RMSJ.I, ''For if the intuition
4 INTRODUCTION
Finally, there is a suggestion made by certain writers
that mechanics can be considered a geometry of four
dimensions with time as the fourth dimension (see below,
p. n). This idea is usually credited to Lagrange (1736-
1813), who advanced it in his Theorie des fonctions analy-
tiques, first published in 1797. * It is expressed, however, in
an article on " Dimension" published in 17 54 by d'Alembert
(1717-1783) in the Encyclopedic edited by Diderot and
himself. D'Alembert attributes the suggestion to "un
homme d'esprit de ma connaissance."f
These are the only ways in which we have found our
subject referred to before 1827.
In the period beginning with 1827 we may distinguish
those writings which deal with the higher synthetic geom-
etry from those whose point of view is that of analysis.
In synthetic geometry our attention is confined at first
chiefly to the case of four dimensions, while in analysis
we are ready for n variables by the time we have con-
sidered two and three.
So far as we know, the first contribution to the synthetic
geometry of four dimensions is made by Mobius, who points
out that symmetrical figures could be made to coincide if
there were a space of four dimensions.! In 1846 Cayley
were a concept gained a posteriori . . we should not be able to say any more than
that, so far as hitherto observed, no space has yet been found having more than three
dimensions" (translation by F Max Muller, 2d ed revised, Macmillan, 1905, p. 19).
C. H. Hinton finds in four-dimensional space illustration and interpretation of
the ideas of Plato, Aristotle, and other Greek philosophers (sec Fourth Dimension
London, 1904, chap. iv).
*p. 223; (Euvres, vol. IX, Paris, 1881, p. 337.
t See paper by R. C. Archibald, "Time as a Fourth Dimension," Bulletin of the
American Mathematical Society, vol. 20, 1914, pp. 409-412.
t He states very clearly the analogy with symmetrical figures in a plane and
symmetrical groups of points on a line. Reasoning from this analogy, he says that
the coincidence of two symmetrical figures in space would require that we should
be able to let one of them make a rotation in space of four dimensions Then he
adds, "Da aber ein soldier Raum nicht gedacht werden kann, so ist auch die Coin-
BEGINNINGS OF THIS GEOMETRY 5
makes use of geometry of four dimensions to investigate
certain configurations of points, suggesting a method that
is systematically developed by Veronese.* Cayley had al-
ready published a paper with the title " Chapters in the An-
alytical Geometry of (ri) Dimensions/' f but as this paper
contains no actual reference to such a geometry, we may
think of the paper of 1846 as the beginning of his published
writings on this subject. Some of the most interesting
examples of the direct study of these geometries were given
by Sylvester. In 1851, in a paper on homogeneous func-
tions,! he discusses tangent and polar forms in ^-dimensional
geometry; in 1859, in some lectures on partitions, he
makes an application of hyperspace ; and in 1863, in a
memoir "On the Centre of Gravity of a Truncated Tri-
angular Pyramid,"^ he takes up the corresponding
figures in four and n dimensions and proves his theorems
for all of these figures, using analytic methods to some
extent, but appealing freely to synthetic conceptions. Clif-
ford also, about this time, makes a very interesting applica-
tion of the higher geometry to a problem in probability. ||
cidenz in diesem Falle unmoglich " (Der barycentrische Calcul, Leipzig, 1827,
140, p. 184).
*"Sur quelques th6oremes de la geometric dc position," Crclle's Journal, vol.
31, pp 213-226 (in particular, pp. 217-218); Collected Mathematical Papers,
Cambridge, vol. I, 1889, No. 50. See also Veronese, Fondamcnti, etc. (the full title
is given below on p. 9), p. 690 of the German translation, and Veronese's memoir
(mentioned on p. 8). In introducing this method of reasoning, Cayley says:
"On peut en effet, sans recourir a aucune notion metaphysique a 1'egard de la pos-
sibilite de Tespace a quatre dimensions, raisonner comme suit (tout cela pourra
aussi e"tre traduit facilement en langue purement analytique)." . . .
t Cambridge Mathematical Journal, vol. 4, 1844; Math. Papers, vol. I, No. n.
J Cambridge and Dublin Mathematical Journal, vol. 6, p. i ; Collected Mathe-
matical Papers, Cambridge, vol. I, 1904, No 30.
Outlines of these lectures are published in the Proceedings of the London Math-
ematical Society, vol. 28, 1896, p. 33; Mathematical Papers, vol. II, 1908, No. 26.
^[ Philosophical Magazine, fourth series, vol. 26, Sept., 1863, pp. 167-183 ; Math-
ematical Papers t vol. II, No. 65.
\\Educational Times, Jan., 1866; Mathematical Reprints, vol. 6, pp. 83-87;
Mathematical Papers, Macmillan, 1882, p. 601.
6 INTRODUCTION
Quite independently of this beginning of its synthetic
development, we find a notion of a higher geometry spring-
ing out of the applications of analysis. Certain geometri-
cal problems lead to equations which can be expressed
with any number of variables as well as with two or
three. Thus, in 1833, Green reduces the problem of the
attraction of ellipsoids to analysis, and then solves it for
any number of variables, saying, "It is no longer confined
as it were to the three dimensions of space.' 1 * Other
writers make the same kind of generalization, though not
always pointing out so directly its geometrical significance, f
It was but a step farther to apply the language of geometry
to all the forms and processes of algebra and analysis.
This principle is clearly announced by Cauchy in 1847,
in a memoir on analytical loci, where he says, "We shall
call a set of n variables an analytical point, an equation
or system of equations an analytical locus/' etc.f
The most important paper of this period is that of
Riemann, "On the Hypotheses which Lie at the Founda-
tions of Geometry." In this paper Riemann builds
up the notion of multiply-extended manifolds and their
measure-relations. He discusses the nature of the line-
element ds when the manifold is expressed by means of n
variables. When ds is equal to the square root of the sum
* Mathematical Papers of George Green, edited by N. M. Ferrers, Macmillan, 1871,
p. 188.
t C. G. J. Jacobi, " De binis quibuslibet functionibus homogeneis," etc., Crelle's
Journal, vol. 12, 1834, p. i; Cayley, two papers published in the Cambridge Ma-
thematical Journal, vol. 3, 1841 ; Mathematical Papers, vol. I, Nos. 2 and 3 ; Schlafli,
"Ueber das Minimum des Integrals /(V</*i 2 + dx<? + . . . -M#n 2 )," etc., Crette's
Journal, vol. 43, 1852, pp. 23-36 ; "On the Integral ! n dx dy . . . dz," etc., Quarterly
Journal, vois. 2 and 3, 1858-1860.
% "Memoir sur les lieux analytiques," Comptes Rendus, vol. 24, p. 885.
"Ueber die Hypothesen, welche der Geometric zu Grunde liegen," presented
to the philosophical faculty at Gdttingen in 1854, but not published till 1866;
Gesammelte Werke, Leipzig, 1892, No. xiii, pp. 272-287; translated by Clifford
in Nature, vol. 8, 1873, pp, 14 and 36; Mathematical Papers, No. 9, pp. 55-69.
RIEMANN, GRASSMANN 7
of the squares of the quantities dx, as in the ordinary plane
and space, the manifold is fiat. In general there is a devia-
tion from flatness, or curvature; and the simplest cases
are those in which the curvature is constant. Riemann
points out that space may be unbounded without being
infinite that, in fact, it cannot be infinite if it has a con-
stant positive curvature differing at all from zero. We
therefore attribute to Riemann the Elliptic Non- Euclidean
Geometry, which from this time on takes its place beside
that other discovered by Bolyai and Lobachevsky. His
paper has a bearing on our subject in two ways: in the
first place, his manifold of n dimensions is a space of n
dimensions, and geometrical conceptions are clearly before
the mind throughout the discussion ; and then the notion
of a curvature of space suggests at once a space of four
dimensions in which the curved three-dimensional space
may lie. Soon after, it was shown by Beltrami that the
planimetry of Lobachevsky could be represented upon real
surfaces of constant negative curvature just as the Elliptic
Two-dimensional Geometry is represented upon the sphere,
and the way was fully opened for the study of spaces of
constant curvature and of curvature in general.*
Another work that has an important influence on recent
developments of hypergeometry, especially in its applica-
tion to physical theories, is the Ausdehnungslehre of Grass-
mann, first published in 1844, though little noticed at the
* Beltrami, " Saggio di interpretazione della geometria non-euclidea," Giornale di
matematiche, vol. 6, 1868; Opere, Milan, vol. I, 1902, pp. 374-405.
Another memoir by Beltrami, "Teoria fondamentale degli spazii di curvatura
cost ante," Annali di matematica pur a ed applicata, Ser. 2, vol. 2, 1868-1869;
Opere, vol. I, pp. 406-429, develops and explains much in Riemann's paper that is
difficult to understand. There are French translations of both memoirs by Houel,
Anndes Sdentifiques de VEcole Normale Superieure, vol. 6, 1869.
Beltrami considers the representations of the three-dimensional geometries upon
curved spaces as only analytic, while the representations of the two-dimensional
geometries upon surfaces of constant curvature are real. See Opere, vol. I, p. 396
and p. 427.
8 INTRODUCTION
time. His theory of extensive magnitudes is a vector
analysis, and the applications which he makes to plane
geometry and to geometry of three dimensions can be made
in the same way to geometry of any number of dimensions.
The number of memoirs and books relating to geometry
of four or more dimensions has increased enormously in
recent years. We can mention only a few. In 1870,
Cayley published his " Memoir on Abstract Geometry/'
in which he lays down the general principles of w-dimen-
sional geometry.* Another important contribution to the
science was an unfinished paper "On the Classification of
Loci" by Clifford. f An important paper by Nother on
birational transformations was published in iSyo.J
Other papers were published by Halphen in 1873 and by
Jordan in 1875, the latter giving a methodical generaliza-
tion of metrical geometry by means of Cartesian coordi-
nates. Perhaps the most important of all was a memoir
by Veronese published in 1882,^ in which he takes up a
study of the properties of configurations, the quadratic
in any number of variables, the characteristics of curves,
correspondence of spaces, etc. : he employed synthetic,
not analytic methods, and inaugurated a purely synthetic
method of studying these geometries. Veronese's Fonda-
menti di geometria contains an elementary synthetic treat-
ment of the geometry of four dimensions and the geometry
of n dimensions; and the Mehrdimensionale Geometric of
* Philosophical Transactions, vol. 160 ; Mathematical Papers, vol. VI, 1893, No 413.
\ Philosophical Transactions, vol. 169, 1878; Mathematical Papers, No. 33, pp.
305-331-
t "ZurTheoriedes eindeutigen Entsprechens algebraischcr Gebildevon beliebig
vielen Dimensionen," Mathematische Annalen, vol. 2, pp. 293-316.
Halphen, "Recherches de geometric a n dimensions," Bulletin de la Soci&U
Mathematique de France, vol. 2, pp. 34-52; Jordan, "Essai sur la ge'ome'trie a n
dimensions," id. vol. 3, pp. 103-174.
If "Behandlung der projectivischen Verhaltnisse der Raume von verschiedenen
Dimensionen durch das Princip des Projicirens und Schneidens," Mathematische
Annalen, vol. 19, pp. 161-234.
SLOW RECOGNITION 9
Schoute, employing a variety of methods, makes these
subjects very clear and interesting. * A bibliography with
nearly six hundred titles, up to 1907, is to be found in
Loria's // passato ed il presente delle principali teorie geo-
metriche. f The latest bibliography is that of Sommerville, J
which contains 1832 references on n dimensions up to
1911: about one-third of these are Italian, one-third
German, and the rest mostly French, English, and Dutch.
We see that the geometries of more than three dimen-
sions were slow in gaining recognition. The general
notion that geometry is concerned only with objective
external space made the existence of any kind of geometry
seem to depend upon the existence of the same kind of
space. Consequently some of our leading mathematicians
hesitated to use the higher geometry,^ although the work-
* Veronese, Fondamenti di geomclria a piu dimensioni ed a piu spczie di unita
rcttilinee esposti in forma elementare, Padua, 1891 ; German translation by Schepp,
Grundziige der Geometric von mehreren Dimensionem, etc , Leipzig, 1894. Schoute,
Mehrdimensionale Geometric, Sammlung Schubert, XXXV and XXXVI, Leipzig,
1902 and 1905. Another elementary treatment of the subject is by Jouffret,
Geometric a quatre dimensions, Paris, 1903
t 3d ed., Turin, 1907.
t Bibliography of Non-Euclidean Geometry, Including the Theory of Parallels t
the Foundations of Geometry, and Space of n Dimensions, University of St. Andrews,
Scotland, 1911.
There is now a considerable popular interest in the four-dimensional geometry,
because of the many curious things about it, and because of attempts which have
been made to explain certain mysterious phenomena by means of it. This interest
has produced numerous articles and books written to describe the fourth dimension
in a non-mathematical way. In 1908 a prize of #500 was offered through the
Scientific American for the best non-mathematical essay on the fourth dimension.
Two hundred and forty-five essays were submitted in this competition. Some of
these have been published in a book, whose Introduction, by the present writer,
gives quite a full discussion of the various questions connected with the subject
(The Fourth Dimension Simply Explained, Munn and Company, New York, 1910).
II Thus Darboux, in a memoir presented in 1869 at the Academy of Sciences and
published in 1873, speaks of a lacune in geometry of space as compared with plane
geometry, for certain plane curves can be studied with advantage as projections from
space, but "Comme on n'a pas d'espace a quatre dimensions, les me"thodes de pro-
jection ne s'ctendent pas a la ge'eme'trie dc 1'espace" (Sur itne classe remarquabk de
courbes et de surfaces algebriques, Paris, p. 164). Even in 1903, in his Report at
10 INTRODUCTION
ing out of its details presented comparatively little diffi-
culty to them. This objection has led some writers to
emphasize those applications of four-dimensional geometry
that can be made in three-dimensional space, interpreting
it as a geometry four-dimensional in some other element
than the point just as we have interpretations of the
non-Euclidean geometries, which cannot, however, take
the place of their ordinary interpretation.* As long ago as
1846 it was pointed out by Pliicker that four variables
the Congress at St. Louis, he says, "Une seule objection pouvait fctre faite . . .
Tabsence de toute base reele, de tout substratum," etc. (Bulletin des sciences mathe-
matiques, ser. 2, vol. 28, p. 261, Congress of Arts and Sciences , edited by H. J. Rogers,
Houghton, Mifflin and Co., Boston, vol. I, 1005, p. 557). But Darboux himself has
made important contributions to the geometry of n dimensions : see, for example, his
Leqons sur les systemes orthogonaux, 2d ed., Paris, 1910; in particular, Bk. I, chap. 6,
and Bk. II, chap. i.
Poincare", in speaking of the representation of two complex variables in space of
four dimensions, says, "On est expose* a rebuter la plupart des lecteurs et de plus
on ne possede que 1'avantage d'un langage commode, mais incapable de parler aux
sens." Acia Mathematica, vol. o, 1886-1887, p. 324.
On the other hand, we have the following from Sylvester : "There are many who
regard the alleged notion of a generalized space as only a disguised form of algebraic
formulization ; but the same might be said with equal truth of our notion of infinity,
or of impossible lines, or lines making a zero angle in geometry, the utility of dealing
with which no one will be found to dispute. Dr. Salmon in his extension of Chasles*
theory of characteristics to surfaces, Mr. Clifford in a question of probability, and
myself in my theory of partitions, and also in my paper on barycentric projection,
have all felt and given evidence of the practical utility of handling space of four
dimensions as if it were conceivable space" ("A Plea for the Mathematician,"
Nature, vol. i, 1869, p. 237 ; Mathematical Papers, vol. II, p. 716).
A statement of Cayley's has been given in a previous footnote (p. 5). For
other expressions of his views we may refer to the first paragraph of the "Memoir
on Abstract Geometry " mentioned above, and to a statement quoted by Forsyth
in his "Biographical Notice," Cayley's Mathematical Papers, vol. VIII, 1895, p. xxxv.
As to the existence of a higher space, Gauss also is said to have considered it
a possibility (W. Sartorius von Waltershausen, "Gauss zum Gedachtniss," Gauss
Werke, Gottingen, vol. VIII, 1000, p. 267).
Segre, referring to the first of the two remarks that we have quoted from Darboux,
says, "Maintenant nous faisons usage de 1'espace a quatre dimensions sans nous
pre*occuper de la question de son existence, que nous regardons comme une question
tout-a-fait secondaire, et personne ne pense qu'on vienne ainsi a perdre de la rigeur."
Mathematische Annalen, vol. 24, 1884, p. 318.
*See Emory McClintock, "On the Non-Euclidean Geometry," Bulletin of the
New York Mathematical Society, vol. 2, 1892, pp. 21-33.
EXTENT AND VARIETY OF APPLICATIONS II
can be regarded as the coordinates of a line in space.*
Another four-dimensional geometry that has been sug-
gested is that of spheres. f
But this higher geometry is now recognized as an indis-
pensable part of mathematics, intimately related to many
other branches, and with direct applications in mathe-
matical physics. The most important application for
the mathematician is the application as analytic geometry
to algebra and analysis : it furnishes concise terms and
expressions, and by its concrete conceptions enables him
to grasp the meanings of complicated formulae and in-
tricate relations. This is true of all the geometries as well
as the geometry of four dimensions. The latter is of special
use in connection with two complex variables, both in the
study of one as a function of the other, and when it is
desired to study functions of both considered as inde-
pendent variables. J Another very important applica-
tion of geometry of four dimensions is that mentioned by
d'Alembert, making time the fourth dimension: within
a few years this idea has been developed very fully, and
has been found to furnish the simplest statement of the
new physical principle of relativity.
* System der Geometric des Raumes, Diisseldorf, p. 322.
t See article by Professor Keyser, " A Sensuous Representation of Paths that
Lead from the Inside to the Outside of a Sphere in Space of Four Dimensions/'
Bulletin of the American Mathematical Society, vol. 18, ign, pp. I&-22.
I See reference given on the preceding page to Poincare"'s memoir in the Acta
Mathematica; also Kwietnewski, Ueber Flachen des vierdimensionalen Raumes,
deren sdmtliche Tangentialebenen untereinander gleichwinklig sind, und ihre Beziehung
zu den ebenen Kurven, Zurich, 1902.
The theory has been developed somewhat as follows : If time is represented by
a coordinate t measured on an axis perpendicular to the hyperplane of the space-
axes, the /-axis itself or any parallel line will represent a stationary point, and uni-
form motion will be represented by lines oblique to the /-axis, forming an angle with
the /-axis which depends on the rate of the motion. A certain velocity (the velocity
of light) is taken as the greatest possible velocity and the same for all systems of
measurement. The lines through the origin, or through any point, representing
this velocity are the elements of a conical hypersurface. All lines not parallel to
12 INTRODUCTION
With these various applications have been developed
many methods of studying the higher geometries, besides
the ordinary synthetic and analytic methods. We now
have the synthetic and analytic projective geometries,
including the projective theories of measurement ; we have
the theories of transformations and transformation groups ;
the geometry of algebraic curves and algebraic functions;
the geometry associated with the representation of two
complex variables; differential geometry and the trans-
formation of differential expressions; analysis situs, enu-
merative geometry, kinematics, and descriptive geometry ;
the extensive magnitudes of Grassmann and different kinds
of vector geometry; the application of quaternions to
four dimensions; and the very recent application of four-
dimensional vector analysis to the principles of relativity.*
these elements are divided into two classes the lines of one class, less inclined to
the /-axis, represent possible motions, while the lines of the other class can represent
only imaginary motions. The system may be regarded as a non- Euclidean geometry
in which the conical hypersurface plays the part of absolute for angles, while dis-
tances along lines of the two classes are independent and cannot be compared. Now
a point moving uniformly may be regarded as stationary, and the points which are
really stationary as moving uniformly in the opposite space-direction. This change
of view is represented by a transformation of coordinates, the new /-axis being the
line representing the given uniform motion. In this theory the angles of planes
play an important part, and line and plane vectors are freely used.
This application of four-dimensional geometry was developed by Minkowski. For
further elaboration sec article by E. B Wilson and G. N. Lewis, "The Space-time
Manifold of Relativity. The Non-Euclidean Geometry of Mechanics and Electro-
magnetics," Proceedings of the American Academy of Arts and Sciences, vol. 48,
No. n, Nov., 1912.
*On the projective theory of measurement see d'Ovidio, k< Le funzione metriche
fondamentali negli spazii di quantesivogliano dimension! e di curvatura costante,' 1
Atti detta Accademia de Lincei, ser. 3, vol. i, 1876, pp. 133-103; abstract in the
Mathematische Annalen, vol. 12, 1877, pp. 403-418.
On analysis situs there is an important series of memoirs by Poincarg Journal de
I'Ecole Poly technique, vol. 100, 1894; Rendiconti del Circolo Matematico di Palermo,
vol. 13, 1899; Proceedings of the London Mathematical Society, vol. 32, 1000; Bulletin
de la Societi Mathematique de France, vol. 30, 1002 ; Journal de mathematiques pure*
et appliquees, ser. 5, vol. 8, 1902; Rendiconti di Palermo, vol. 18, 1904; Comptes
Rendus, vol. 133, 1901.
The enumerative geometry has been developed chiefly by Schubert. H has
ESSENTIAL PART OF GEOMETRY 13
All these interpretations and methods that have been
applied to the study of the higher geometries, and all these
uses to which they have been put, are interesting and valu-
able to a greater or less degree ; but the greatest advantage
to be derived from the study of geometry of more than three
dimensions is a real understanding of the great science of
geometry. Our plane and solid geometries are but the
beginnings of this science. The four-dimensional geometry
is far more extensive than the three-dimensional, and all
the higher geometries are more extensive than the lower.
The number and variety of figures increases more and more
rapidly as we mount to higher and higher spaces, each space
extending in a direction not existing in the lower spaces,
each space only one of an infinite number of such spaces
in the next higher.
A study of the four-dimensional geometry, with its
hyperplanes like our three-dimensional space, enables us
to prove theorems in geometry of three dimensions, just
as a consideration of the latter enables us to prove theorems
in plane geometry. Such theorems may come from much
simpler theorems relating to the four-dimensional figures
of which the given figures are sections or projections.*
articles in the M ' athematische Annalen, vols. 26, 38, and 45, 1886, 1891 and 1894; in
the Acta Mathematica, vol. 8, 1886; and elsewhere.
In kinematics we may mention: Clifford, "On the Free Motion under No Forces
of a Rigid System in an A T -fold Homaloid," Proceedings of the London Mathematical
Society, vol. 7, 1876, Mathematical Papers, No 26, pp. 236-240; Beltrami, "Formules
fondamentales de cine"matique dans les espaces de courbure constante," Bulletin des
science mathematiques, vol. n, 1876, pp. 233-240, Opere, vol. Ill, 1911, pp. 23-29;
see also articles by Craig and Hatzidakis in the American Journal of Mathematics
vols. 20 and 22, 1898, and IQOO
Quaternions have been applied to geometry of four dimensions by Hathaway,
Bulletin of the American Mathematical Society, vol. 4, 1897, pp. 54-57; Trans-
actions of the American Mathematical Society, vol. 3, 1902, pp. 46-59; and by
Stringham, Transactions, vol. 2, 1901, pp. 183-214; Bulletin, vol. n, 1905, pp.
437-439-
Other methods are illustrated in memoirs already referred to.
* See Cayley's article in Crelle's Journal, vol. 31, and the articles of Veronese and
Segrc in the M athematische Annalen, to which we have already referred.
14 INTRODUCTION
Indeed, many theorems and processes are seen only partially
or not at all in the lower geometries, their true nature and
extent appearing hi the higher spaces. Thus in space of
four dimensions is found the first illustration of figures
which have two independent angles, and of different kinds
of parallelism and different kinds of perpendicularity.
Another example is the general theorem of which a partic-
ular case is given in Art 31, namely, that a section of a
simplex of n dimensions is one of the two parts into which
a simplex of n i dimensions (that is, its interior) may be
divided by a section.* There are also many properties
in which spaces of an even number of dimensions differ
from spaces of an odd number of dimensions, and these
differences would hardly be recognized if we had only the
ordinary geometries. Thus in spaces of an even number
of dimensions rotation takes place around a point, a plane,
or some other axis-space of an even number of dimensions,
while in spaces of an odd number of dimensions the axis
of a rotation is always of an odd number of dimensions
(see chap. IV).
The study of these geometries gives us a truer view of
the nature of geometrical reasoning, and enables us to
break away from intuition. This is especially true if we
adopt the synthetic method. The analytic geometry may
seem to be free from difficulty, and many feel a higher
degree of certainty in the results of their algebraic processes.
But we are apt to attach the terms of geometry to our
algebraic forms without any attempt at a realization of
their significance. There is, indeed, an abstract geometry
in which the terms are regarded as meaningless symbols;
but the interest and usefulness of geometry depend on the
clearness of our perception of the figures to which it may be
applied, and so we prefer to study some concrete geometry,
* See Schoute, Mehrdimensionale Geometric, vol. II, i, Nr. 6.
SYNTHETIC METHOD 15
some interpretation of the abstract geometry which we
could have obtained by giving a particular interpretation
to its terms. And then the abstract geometry and other
interpretations can all be obtained from the concrete
geometry.* There is really the same absolute certainty
to synthetic geometry if it is developed logically from the
axioms, and in the synthetic study of four-dimensional
geometry we are forced to give up intuition and rely entirely
on our logic, f
Although it is doubtful whether we can ever picture to
ourselves the figures of hyperspace in the sense that we can
picture to ourselves the figures of ordinary space, yet we
can reason about them, and, knowing that the validity of
our geometry depends only on the logical accuracy of our
reasoning, we can proceed to build it up without waiting
for a realization of it ; and then we may in time acquire
such facility in handling the geometrical proofs of the
theorems and in stating precisely the forms and properties
of the figures that it is almost as if we could see them. For
* Some portions of our study are treated by themselves as new interpretations
of geometries already studied. As soon as the fundamental propositions which
correspond to the axioms of some such geometry have been established, so as to
justify this mode of procedure, we have only to translate its theorems in accordance
with these propositions to have in our possession a complete development of the
particular subject considered. Examples are, the three-dimensional Point Geom-
etry (Art. 64), the two-dimensional Edge Geometry (Art. 78), and the theory of
systems of isocline planes at a point (Art. 112).
t We do not seek to know which of several geometries is the true geometry, and
in laying the foundations we do not seek for the true system of axioms, or even the
true system of elements and relations. All geometries are equally true, and some-
times a particular geometry may be built up equally well in several different ways.
A complete treatise on geometry should consider not only the different geometries,
but different methods of building up each geometry. An example of such a treat-
ment is the first volume of Fragen der Elcmentar geometric, edited by Enriques
(Leipzig, IQI i, German translation by H. Thieme of Question* riguardanti la geometria
clementare, Bologna, IQOO). See also the chapters on this subject by Enriques and
others in the French and German Encyclopedias (Encyklopadie der math. Wiss., vol.
IIIi, Leipzig, 1907; Encyclopedic des sci. math., vol. IIIi, Leipzig, IQII). A list of
different systems of fundamental elements and relations is given in a footnote at
the beginning of Coolidge's Non-Euclidean Geometry, Oxford, 1009.
1 6 INTRODUCTION
in studying the geometry synthetically our attention is
fixed upon the figures themselves, and this takes us directly
to the heart of the difficulty and keeps it before us until
we have mastered it. Thus in its results this geometry
greatly increases our power of intuition and our imagina-
tion*
The following pages have been written with the object
of meeting as far as possible the difficulties of the subject.
No knowledge of higher mathematics is necessary; yet
we do not believe that the simplest way is to avoid a math-
ematical treatment. The confidence gained from a study
of the proofs, if they can be made clear and precise, will
do more for the student than a mere description of four-
dimensional space. We will indicate how this purpose has
influenced us in our choice of subject-matter and the form
of presentation. .
We have adopted the synthetic method and made no
use of analytic proofs, feeling, as we have already explained,
that this study of the figures themselves will serve best to
help us understand them.
We have confined ourselves to the fourth dimension,
although it would have been easy to cover a much wider
field, f We hope that in this way the four-dimensional
space will be made to appear as a concrete matter to be
studied by itself, and not as one of an indefinite series of
spaces, each understood only in a vague general way.
We have wished to give to these pages a familiar appear-
ance, and so have endeavored to follow the popular text-
books and build up a structure that will rest on the founda-
tions laid in the schools. Our geometry might have been
adapted to the axioms of some modern investigation, or
* See C. J. Keyser, "Mathematical Emancipations/' Monist, vol. 16, 1006,
PP' 65-83, particularly pp. 81-82.
t See, for example, the Mehrdimensionalc Geometric of Schoute.
FOUNDATIONS 17
have attempted to establish a system of axioms, but either
course would have raised questions quite different from
those of four-dimensional geometry. The methods em-
ployed in this book are methods which the student has used
freely in the past, even though he may be ignorant of their
true significance and justification : there is nothing new
in their application here, and their employment without
question leaves him free to fix his attention upon the
difficulties inherent in the subject.
There is, however, one part of the foundations which has
been presented with considerable care, namely, that which
relates to the definitions and the intersections of certain
elementary figures. It is here that the four-dimensional
geometry begins to contradict our experience, declaring,
for example, that two complete planes may have only
a point in common, and that a line can pass through a point
of the interior of a solid without passing through any other
of its points. It is true that these facts and many others
not easy to realize are easily proved, and require only a few
of the theorems given in this connection. On the other
hand, the theorems for which most of these details are
needed are so "evident'' that they are usually ignored al-
together. Now a statement of these theorems, with a
realization of what is assumed and of what is to be proved,
and a logical working out of the proofs themselves, will
give the student more confidence in all the results of his
study. Similar considerations have led us in the fourth
chapter to take up symmetry, order, and motion in space
of two dimensions and in space of three dimensions.
Great assistance comes from the analogies that exist in
geometry, and so we have gone back in some cases and
given proofs which are not well known, and to which more
difficult proofs that follow are analogous;* and we have
* See, for example, the theorems of Arts. 61 and 62.
C
1 8 INTRODUCTION
tried to facilitate the comparison of chapters and sections
analogous to one another by adopting in them the same
arrangement of paragraphs and the same phraseology.
Not much use can be made of diagrams, and so far as
they are given they must be regarded as indicating the re-
lations of different parts of a figure rather than as showing
in any way its appearance. A figure can be accurately
determined by its projections, and the descriptive geometry
of four dimensions will be helpful to those who are familar
with the methods of descriptive geometry.* Much can
also be learned by studying the sections of a figure. A
section of a four-dimensional figure is that part which lies
in a three-dimensional space or hyperplane, and is, there-
fore, like the figures of our space. We can suppose that
we are able to place ourselves in any hyperplane, and so to
examine any hyperplane section: in connection with the
diagrams we shall sometimes call attention to those parts
which lie in any one section, speaking of them as "what
we can see in a hyperplane." One way of studying a
figure is to let it pass across our space, giving us a con-
tinuously varying section, as if time were the fourth di-
mension. Another way is to let it turn, or our section of it,
so that the direction of our view changes. It is along
these lines, if at all, that we are to acquire a perception of
hyperspace and its figures.
Some explanation should be made in regard to the arrange-
ment, the particular form chosen for the foundations so
far as they are considered, and the fundamental conceptions
as we have presented them.
We have given only the Euclidean Geometry, except that
the geometry of the hypersphere, and of the hyperplane
at infinity, and the geometry in a few less important cases,
* See Schoute, Mehrdimensionale Geometric, vol. I, 5.
COLLINEAR RELATION 19
are themselves non-Euclidean. It has been found, however,
that several chapters can be completed before we make
any hypothesis in regard to parallels, and that, too, without
much variation from the usual treatment. Perpendiculars
and all kinds of angles, symmetry and order, and those
hypersurfaces (the hyperpyramid, the hypercone, and the
hypersphere) which do not involve parallels in fact,
all of "restricted" geometry can be taken up before the
introduction of parallels.* In the chapter on the hyper-
sphere, its geometry, being elliptic, is stated as such, and
a group of theorems is given from the non-Euclidean geom-
etry; and in the last chapter the non-Euclidean prop-
erties of the hypersphere are used quite freely. Although
these portions of the book may be omitted, the student
will find it an advantage to make himself familiar with the
Hyperbolic and Elliptic geometries, f
We have started with points only as elements, regarding
all figures as classes of points, and so defining a figure
simply by stating what points constitute the class. To do
this we assume first a relation by which with any two points
certain points are said to be collinear. Then for line we
take two points and the class of points collinear with them,
add to the group all points collinear with any two that we
now have, and thus continue, at each step adding to our
class of points all that are collinear with any two already
in the group, so that the line includes every point which
it is possible to get in this way. Thus any two points
determine a class consisting of the points which are collinear
with them, and any two points determine a class of the
kind which we call a line.J By the axioms of Art. 3 the
* See the author's Non-Euclidean Geometry, Ginn and Co , Boston, 1001, chap. I ;
in particular, p. 6.
t The Hyperbolic and Elliptic geometries are the only non-Euclidean geometries
that we have referred to at all.
t That two points determine a line does not mean, as in some of our text-books,
"\
20 INTRODUCTION
two classes are identical : the line consists only of the points
collinear with the two given points, and there are no addi-
tional points to be obtained by taking any two of these
points. In fact, any two points of a line determine the
same class of points as collinear with them, and the same
line. But until we have adopted these axioms we must
suppose that the line might be a much more extended class :
that, if we have the points collinear with two given points,
the class of points collinear with any two of these might
be quite different; and that, while a line must contain
every point of the line determined by any two of its points,
the latter might not contain every point of the former.
Thus we make a distinction at the beginning between the
notion of collinear points and the notion of points of a line,
and this distinction makes line analogous to plane and
hyperplane, and to spaces of more than three dimensions.
But after we have adopted our first two axioms we are
able to employ the word collinear in its commonly accepted
sense, and thus to avoid the introduction of a new term for
one of these two relations.
A careful distinction has been made between the points
of a closed figure and the points of its interior. Thus
a triangle is made to consist of three vertices and the points
of its sides, a tetrahedron of its vertices and the points of
its edges and faces, and so on. This is only carrying to
the limit the tendency to regard a circle as a curve rather
than as the portion of the plane enclosed by the curve, and
a sphere as a surface. The figure of one-dimensional
geometry corresponding to the triangle and tetrahedron,
the one-dimensional simplex, is the segment. Therefore, we
have defined segment as consisting of two points, and let
that the line contains the two points, or that no other line contains them. A figure
may be determined in various ways. Thus a line in the ordinary plane geometry
may be determined by two points as the locus of points equidistant from them.
FIGURE AND ITS INTERIOR 21
the points between them constitute the interior of the seg-
ment.* On the other hand, a side of a polygon or an edge
of a polyhedron is the interior of a segment, consisting of
the points between two vertices and not including the ver-
tices themselves ; a face consists of the interior of a triangle
or polygon; a half -line is defined so as not to include its
extremity, a half-plane so as not to include its edge ; and
so on. We speak, indeed, of the sides and vertices of a face,
of the length of a segment, and of the area of a triangle,
just as, in general, we have used the terms of ordinary
geometry without definition, and employed freely all the
words and phrases of its everyday language. But the
distinction between the points of a closed figure and the
points of its interior is of great importance, and has been
carefully observed.
* Hilbert defines segment (Strecke) as a "system of two points," but he speaks
of the points between A and B as "points of the segment AB," although he also
speaks of them as points "situated within the segment" (Grundlagen der Geometric,
Leipzig, 1899, p 6, 4th ed , 10,13, p 5)
In the Encyclopedic des sciences mathematiques, vol. IIIi, p. 23, Enriques defines
segment upon a line as "having its extreme points at two given points A and B of
the line and containing the intermediate points." More definitely, in the Elementi
di Keometria of Enriques and Amaldi (Bologna, 1911), half-line is defined so as to
include its extremity, and then the segment AB is the part common to the half-
lines A B and BA (p 3)
E. 11. Moore defines the segment AB as consisting of points "distinct from A and
B" etc. ; that is, A and B are not included among the points of his segment ("On
the Project ive Axiom-* of Geometry," Transactions of the American Mathematical
Society, vol. 3, 1002, p ijy, Axiom 2], See also Veblen, "A System of Axioms
for Geometry," Transactions, vol 5, 1004, p. 354, Definition i, and "The Founda-
tions of Geometry," Monographs on Modern Mathematics, edited by J. W. A.
Young, New York, 1911, p. 5.
Most writers who use the word segment in this connection regard a segment as
an entity, a piece of a line, without considering whether the end-points are included
or not. Many writers speak of the segment as the "measure of the distance"
between the two points (see Schotten, Inkalt und Methode des Planimetrischen
Unterrichts, Leipzig, vol. II, 1893, chap, i, 2).
Veblen, in the " Foundations of Geometry" just referred to, defines triangle and
tetrahedron in the same way that we have defined them (pp. 29 and 45).
22 INTRODUCTION
* A remarkable memoir on geometry of n dimensions is Theorie der
vieljachen Kontinuitdt, by L. Schlafli, edited by J. H. Graph, Bern,
1911. This was written in the years 1850-1852, but the author did
not succeed in getting* it published, apparently on account of its
length, and it remained among his papers for fifty years, until after
his death (see Vorbemerkung).
Among other things he works out the theory of perpendicularity
and all kinds of angles, giving, in particular, a generalization of the
theorems which we have given in Arts. 66 and 67 (15). He proves the
polyhedroid formula and the corresponding formula for any number
of dimensions, and he constructs the six regular convex polyhedroids
and the three regular figures which exist in each of the higher spaces,
proving that these are the only regular figures of this kind ( 17).
He makes an extensive study of the hypervolume of a spherical sim-
plex, showing the difference between the cases of an even number and
of an odd number of dimensions, and giving the formula for a pen-
tahedroid to which we have referred at the end of Art. 165 ( 22).
In the third part of the memoir he takes up quadratic hypersurfaces,
the classification of these hypersurfaces, confocal hypersurfaces, etc.
The methods are analytical , but the language and conceptions are
purely geometrical.
* This note was written after the rest of the Introduction was in type.
CHAPTER I
THE FOUNDATIONS OF FOUR-DIMENSIONAL
GEOMETRY
I. POINTS AND LINES
1. Points. Figures regarded as classes of points.
The elements of geometry are points. We do not define
the term point. It is impossible to build up a system of
geometry without undefined terms, and if we can give
different meanings to this word we shall be able hereafter
to give to our geometry different interpretations (see, for
example, Arts. 64 and 78 and Introduction, p. 15).
The objects which we study are to be regarded as con-
sisting of points, that is, as classes of points selected accord-
ing to various laws from the class which includes all points.
Any selected class is said to constitute a figure, although
the word figure is also used to denote a drawing or picture.
The points of a figure may also be said to lie in the figure
or belong to it. One figure is said to lie in another when
all of its points are points of the second. It will often be
convenient to speak of a figure as consisting of certain other
figures, but this expression should always be understood
as meaning that it consists of the points of these other
figures.
Two figures intersect when they have a point or points in
common, and their intersection consists of such common
point or points.
2. The collinear relation. Geometries of x, a, 3, ... R,
. . . dimensions. Points have an undefined relation de-
23
24 POINTS AND LINKS [i. I.
noted by the term collinear. Given any two points, there
is a class consisting of the points collinear with them.
Geometries of i,-2, 3, . . . n, . . . dimensions are geom-
etries of figures lying in spaces 0/ i, 2, 3, . . . j, . . .
dimensions; that is, in a line, plane, hyperplane, and in
spaces of more than three dimensions.
A line * consists of the points that we get if we take two
distinct points, all points collinear with them, and all points
collinear with any two obtained by this process.
A plane consists of the points that we get if we take
three points not points of one line, all points collinear with
any two of them, and all points collinear with any two ob-
tained by this process.
A hyperplane consists of the points that we get if we take
four points not points of one plane, all points collinear with
any two of them, and all points collinear with any two
obtained by this process.
A space of four dimensions consists of the points that we
get if we take five points not points of one hyperplane,
all points collinear with any two of them, and all points
collinear with any two obtained by this process.
Continuing in this way, we can define a space of n + 1
dimensions after we have defined a space of n dimensions.
All spaces have this property: that the points collinear
with any two points of a space belong to the space.
Two distinct points are said to be independent. In
general, a point is independent of the points of a given
class if it is not included among the points that we can
get by taking these points, points collinear with any two
of them, and points collinear with any two obtained by
this process. The points of a given set are absolutely in-
* In the ordinary interpretation of these terms, line alone is used for straight line,
and the complete line is meant, not that part of a line which we shall speak of as the
interior of a segment (Art. 5).
2,3l AXIOMS OF COLLTNEARTTY 25
dependent if it is impossible to obtain them all in this way
from a smaller number of points.* The different kinds of
space are distinguished by the number of absolutely inde-
pendent points that they can contain.
3. Segments. Two axioms concerning the collinear
relation. A segment f consists of two distinct points.
Any two points are the points of a segment. The segment
consisting of the points A and B will be called the segment
AB.
A point is said to be collinear with a segment when it is
collinear with the two points of the segment.
Concerning the collinear relation we shall now make
two axioms :
AXIOM i . The class of points collinear with the segment
AB includes the two points of this segment.
AXIOM 2.J If a point O, not the point B, is collinear
with the segment AB, then any point P collinear with the seg-
ment AB will be collinear with the segment OB.
In other words, if is collinear with the segment AB
and is not the point B (so that there is a segment OB),
then all points collinear with the segment AB will be collin-
ear with the segment OB. In particular, A itself will be
collinear with the segment OB, and therefore, conversely,
all points collinear with the segment OB will be collinear
* If the points of a set are absolutely independent, each point will be independent
of the rest ; and in this geometry we shall find that the points of a set are absolutely
independent if each point is independent of the rest (Art. 4, Th. 2 ; Art. 10, Th. 2 ;
etc.). We may, therefore, speak of such points simply as independent points.
t Halsted calls this a sect (Elements of Geometry, New York, 1885, p. Q). He
also uses the word straight for line, taking these terms from the German Strecke
and Gerade (see Rational Geometry, New York, 1907, pp. i and 6).
tin the Elliptic Non-Euclidean Geometry this is assumed with certain re-
strictions. Thus on the sphere, whose geometry is the elliptic two-dimensional
geometry, collinear meaning "on a great circle with," the statement given here as
Axiom 2 is not true when A and B are opposite points
26 POINTS AND LINES [i. i.
with the segment AB. The class of points collinear with
one segment is the same as the class of points collinear
with the other segment. We can use A or O interchange-
ably in connection with B as one of the two points with
respect to which the class is selected.
In this second axiom we do not intend to make any dis-
tinction between the points A and B. Except when O is
one of these points, P will be collinear with both of the seg-
ments OA and OB ; and when O is the point -B, it will be
collinear with the segment OA .
4. Lines. Only one line contains two given points.
Given two distinct points A and 5, the line AB is the line
obtained when we start with these points and carry out
the process described in the definition of line (Art. 2).
If A' and B' are two distinct points of the line AB, then
all points of the line A'B' will be points of the line AB;
for the process of obtaining the second line is but a continu-
ation of the process of obtaining the first. We shall now
prove that all points of the line AB are points of the line
A'B' ; in other words, that the two are the same line, and
that two points can both be points of only one line.
THEOREM i. Any point O of the line AB is collinear
with the segment AB.
PROOF. We shall prove this theorem by induction.
We know that it is true of the points A and B (Art. 3, Ax. i).
Let be any other point of the line.
By hypothesis, O, being a point of the line AB, is collin-
ear with two points M and N of the line. We assume
that the theorem is true of M and N. M must be distinct
from one, at least, of the points A and J5, since they are
distinct. Let us suppose that M is not B. Then we can
substitute M for A , and say that all points collinear with
the segment AB are collinear with the segment MB.
3-sl ORDER ON A LINE 27
Again, since N is one of these points and is not the point
M, we can substitute N for B, and say that all of these
points are collinear with the segment MN. Now is
collinear with the segment MN. Therefore, reversing
this process of substitution, we have O collinear with the
segment MB, and finally with the segment AB.
THEOREM 2. If A' and B' are two distinct points of the
line AB, then the line A'B f is the same as the line AB.
PROOF. We may suppose that A' is not B. Then,
since A' is collinear with the segment AB,A will be collin-
ear with the segment A'B, and the line A'B will be the
same as the line AB. In the same way we prove that the
line A'B' is the same line.
It follows from these two theorems that the class of
points collinear with the segment AB constitute the line
A B, and that collinear with means lying on a line with. It
follows also that the two points A and B do not play any
particular part among the points of the line. We can
speak of a set of collinear points, or of points collinear with
one another, without specifying any two particular points
as special points of the class. We can also speak of two
or more points as collinear with one given point.
5. Order of points on a line. Half-lines. Another
relation in geometry, a relation of the points of a line, is
that of order* This may be explained somewhat in detail
as follows :
If A and B are two distinct points, then A comes before
B and B lies beyond A in one direction along the line AB,
while B comes before A and A lies beyond B in the opposite
direction. If A comes before B and B before C in a given
direction along a line, then A comes before C in the same
* Veblen uses the word order to mean order on a line, " System of Axioms," p. 344 ;
or "Foundations of Geometry," p. 5 (full references are given above, p. 21).
2 8 POINTS AND LINES [l T.
direction, and B is said to lie between A and C. Given any
three points of a line, one of them lies between the other
two.
This relation of order belongs to other classes of points be-
sides those of a line. For example, it belongs to the class of
points constituting what is commonly called a broken line.
The points of a line have also relations of density and
continuity, but it will not be necessary to explain these
terms here, nor to give the axioms and theorems by which
these relations and the relations of order are established.*
The interior of a segment consists of the points which lie
between the two points of the segment.
A half-line or ray is that portion of a line which lies in
one direction along the line from a given point of it. The
given point is called the extremity of the half-line, but this
point is not itself a point of the half-line. The half-line
which has the extremity A and contains B, the half -line
which we can describe as drawn from A through 5, is called
the half -line AB ; and that portion of this half-line which
lies beyond B is itself a half-line, called the continuation
of A B y or AB produced. The two half-lines into which
a given point of a line divides the rest of the line are called
opposite half-lines.
6. Cyclical order. There is another kind of order called
cyclical order. When the points of a class are in cyclical
order, two of these points A and B (unless they are con-
secutive points) divide the rest of them into two sub-classes,
those of one sub-class lying from A towards B in one direc-
tion and those of the other sub-class lying from A towards
B in the opposite direction. A and B are said to lie
* See Introduction, p. 16. For a treatment of these subjects we will refer to
Veblen's "System of Axioms " or " Foundations of Geometry," and to R. L. Moore,
"Sets of Metrical Hypotheses," Transactions of the American Mathematical Society,
vol. g, 1008, pp. 487-512
S-7l LINE AND TRIANGLE 29
between the points of the two sub-classes and to separate
them. If C is a point of one sub-class and D a point of
the other sub-class, that is, if we have these points in the
cyclical order ACS DA, then we can say that the segments
AB and CD separate each other. If we think of a class
of points in cyclical order as cut at some point, they will
then have the relations of order described in the pre-
ceding article. We shall have many illustrations of cycli-
cal order (see Arts. 7 and 14) .*t
II. TRIANGLES
7. Triangles. Axiom of Pasch. Intersection of a line
and a triangle. A triangle consists of three non-collinear
points and the interiors of the three segments whose points
are these points taken two at a time.
The three points are the vertices, and the interiors of the
segments are the sides.
Any three non-collinear points are the vertices of a tri-
angle. In particular, two vertices of a given triangle and
a point in a side which does not lie between them are them-
selves the vertices of a triangle ; so also are a point in each
of two sides and the vertex where these two sides meet.
The points of a triangle are in cyclical order in the tri-
angle (Art. 6).
For a complete treatment of the intersections of lines
and triangles the following axiom is required : f
* The points of a circle are in cyclical order. Indeed, the points of a line are in
cyclical order in Projective Geometry and in the Elliptic Non-Euclidean Geometry.
We do not wish to exclude the Elliptic Geometry by assuming that the points of the
entire line are not in cyclical order. In many cases of cyclical order there will be
an "opposite" to every element, and on a line of this kind we can determine the
two directions at any point by regarding the line as cut at the opposite point (see,
for example, the definition of "between" and "side produced" in Art. 122) ; or we
can suppose that we are considering only a "restricted" portion of the line; that is
one of the sub-classe* determined by two points suitably chosen in any given case.
t See Veblen, "System of Axioms," p. 351.
30 TRIANGLES [i. n.
AXIOM. A line intersecting one side of a triangle and
another side produced intersects the third side.
This will be referred to as the Axiom of Pasch. As stated,
it is a little broader than is necessary.
THEOREM. No line can intersect one side of a triangle
and two sides produced , nor can a line intersect all three sides
of a triangle.
PROOF. The first statement follows directly from the
Axiom of Pasch. To prove the second statement, let us
suppose that a line could intersect the three sides of the tri-
angle ABC, BC at A', AC at ', and AB at C", 5'coming
between A' and C' on the line. In the triangle A f BC' we
should then have the line AC intersecting one side, A'C'>
at 5', and the other two sides produced. But this is con-
trary to the first statement of the theorem. Therefore a
line cannot intersect all three sides of a triangle.
COROLLARY. No line can contain more than two points
of a triangle unless it contains one of the sides of the triangle.
8. Interior of a triangle. The interior of a triangle con-
sists of the interiors of all segments whose points are points
of the triangle, except of those segments which are collinear
with two vertices of the triangle, that is, whose interiors
also lie in the triangle. The interior of a triangle does not
include the triangle itself ; hence, whatever is in the triangle
cannot be in the interior of the triangle.
THEOREM i. If two segments lying in a triangle separate
each other in the cyclical order of the points of the triangle
(Art. 6), then their interiors intersect, unless the interior of
one of these segments lies in the triangle.
PROOF. When each of the two given segments has a
vertex for one of its points, each segment with the third
vertex determines a triangle to which we can apply the
7,8]
INTERIOR OF A TRIANGLE
Axiom of Pasch and thus prove that the interior of this
segment is intersected by the line containing the other
segment. From this it fol-
lows that the interiors of the
two segments have a point in
common.
When one of the two given
segments has a vertex for one
of its points and the other
does not, we form a triangle
by taking one point of the
latter and the two vertices of
the given triangle which are
collinear with its other point.
This triangle contains also
the first segment, or a seg-
ment whose interior is a part of the interior of the first
segment.* We can prove the theorem, then, by proving it
for this segment and the second given segment.
When neither of the two given segments contains a ver-
tex we proceed in the same way, reducing this case first to
the preceding.
THEOREM 2. A half -line drawn from any point P of a
triangle through a point O of the interior intersects the triangle
in a point of PO produced.
In proving this theorem we follow the methods of the
preceding proof, taking first the case where O is between
one vertex and a point of the opposite side and P is at
another vertex.
COROLLARY. // one of two opposite half -lines drawn from
a point O of the interior of a triangle intersects the triangle,
the other does also.
* According to theorems of order referred to in Art. 5 ; see Veblen, "System of
Axioms," p. 357, Lemma 6, or "Foundations of Geometry," p. n, Cor. 4.
TRIANGLES
[i. n.
9. The relation, collinear with a triangle. A point is
said to be collinear with a triangle when it is collinear with
any two points of the triangle.
This is true, in particular, of all points of the triangle,
of all points of the sides produced, and of all points of the
interior.
THEOREM i. If a point O is collinear with a triangle
A BC, and if P is any point of this triangle not a vertex and
not the point O, then the line PO will intersect the triangle at
least in a second point Q.
PROOF. The line PO may contain one side of the triangle.
Otherwise, if O is a point of the triangle, it will itself be the
second point Q, and if O is a point of a side produced, the
theorem is the same as the Axiom of Pasch. If O is a point
of the interior, the theorem is the same as Th. 2 of the
preceding article.
There remains, therefore, only the case where O is not a
point of any one of the lines AB, BC, or AC, nor a point of
the interior of the triangle. Let H and K then be the two
points with which O is collinear, H a point of the side A B,
or the vertex JB, and K a point of the side AC, or the vertex
C, H and K, not both vertices, however.
POINTS AND TRIANGLE
33
We may suppose also that O is a point of H K produced.
Then K will lie in the interior of the triangle OAB or on
the side OB, and the half-line AC, which passes through
K, will intersect the interior of the segment BO in a point
G. Now O is a point of BG produced. We have the tri-
angles A BG and CBG, and in one or both of these triangles
the line PO, intersecting BG produced and one other side,
will intersect the third side. Therefore, in all cases this
line will intersect the triangle A BC in a second point Q.
THEOREM 2. If a point O, not a point of the line
is collinear with the triangle ABC, then any point P collinear
with the triangle ABC will
be collinear with the triangle
OBC.
PROOF. First let O be a
point of the line AB. If
O is at A, the triangle OBC
is the same as the triangle
ABC. If P is a point of the
line AB, it is collinear with O
and B. Let us suppose then
that P is not a point of this
line, and that O is a point of
the side AB, a point of BA
produced, or a point of AB
produced.
If O is a point of the
side AB, then a line
through P and a point of
OB will intersect BC or
will pass through C,
or, intersecting AC, will
Pasch).
intersect OC (Axiom of
34 TRIANGLES [i. H.
If O is a point of BA produced, then a line through P
and a point of AB will intersect BC or will pass through C,
or, intersecting AC, will intersect OC.
If O is a point of AB produced, then a line through P
and a point of AB will intersect AC or will pass through C,
or, intersecting BC, will intersect OC. P will be collinear
with the triangle OAC, and therefore, by the first case, with
the triangle OBC.
In the same way we prove the theorem when O is a point
of the line AC.
If O is not a point of any one of the lines AB, BC, or
AC, then a line through O and a point D of the side
BC will pass through A or
will intersect one of the sides
AB or AC. Let the point of
intersection be E, and let us
suppose that it is the point
A or a point of AC. Then
P, being collinear with the
triangle ABC, will be collinear with the triangle A DC y
then with the triangle EDC, with the triangle ODC, and
finally with the triangle OBC.
In other words, if is collinear with the triangle ABC
and is not a point of the line BC (so that there is a triangle
OBC}, then all points collinear with the triangle ABC will
be collinear with the triangle OBC. In particular, A it-
self will be collinear with the triangle OBC, and therefore,
conversely, all points collinear with the triangle OBC will
be collinear with the triangle ABC. The class of points
collinear with one triangle is the same as the class of points
collinear with the other triangle. We can use A or O in-
terchangeably in connection with the segment BC as one of
the three points with respect to which the class is selected.
9, io] DETERMINATION OF A PLANE 35
In this theorem we do not intend to make any distinc-
tion between the point A and the points B and C. Except
when O is collinear with two of these three points, P will
be collinear with all three of the triangles OBC, OAC, and
OAB.
III. PLANES
10. Planes. Only one plane contains three given non-
collinear points. As already stated, a plane consists of the
points that we get if we take three points not points of
one line, all points collinear with any two of them, and all
points collinear with any two obtained by this process.
Given three non-collinear points, A, B, and C, the plane
ABC is the plane obtained when we take these points and
carry out the process described in the definition.
Now we can take for two of the three given points any
two points of their line. That is, if B' and C r are points
of the line EC, the plane AE f C f is the same as the plane
ABC (Art. 4, Th. 2). We can 'think of a line as one of the
things with which we start in the process of obtaining a plane.
Given the points of a line and a point not a point of the line,
we take all points collinear with any two of these and all
points collinear with any two obtained by this process.
If A', B', and C r are three non-collinear points of the
plane ABC, then all points of the plane A'B'C' will be
points of the plane ABC ; for the process of obtaining the
second plane is but a continuation of the process of obtain-
ing the first. We shall now prove that all points of the
plane ABC are points of the plane A'B'C'-, in other
words, that the two are the same plane, and that three
non-collinear points can all be points of only one plane.
THEOREM i. Any point of the plane ABC is collinear
with the triangle ABC*
* Collinear with a triangle is therefore the same as co planar with the triangle,
in the usual sense of the word coplanar.
36 PLANES [i. m.
PROOF. We prove this theorem just as we proved
Th. i of Art. 4. We, know that it is true of all points of
the triangle and of all points of the sides produced. We
let be any other point, and prove by induction that the
theorem is true for O.
This point is, by hypothesis, collinear with two points
M and N of the plane ABC. We assume that the theorem
is true of M and N. We can suppose also that M is not a
point of the line BC and that N is not a point of the line
MB. It follows (Art. 9, Th. 2) that we can substitute M
for A, and say that all points collinear with the triangle
A BC are collinear with the triangle MBC. Again, since N
is one of these points and is not a point of the line M B, we
can substitute N for C, and say that all of these points are
collinear with the triangle MNB. Now O is collinear with
the segment MN and so with the triangle MNB. There-
fore, reversing this process of substitution, we have O
collinear with the triangle MBC, and finally with the
triangle ABC.
THEOREM 2. // A', ', and C f are three non-collinear
points of the plane ABC, then the plane A'B'C* is the same
as the plane ABC.
PROOF. We will prove this theorem just as we proved
Th. 2 of Art. 4. We can suppose that A ' is not a point
of the line BC, and that B' is not a point of the line
A'C. Then, since A' is collinear with the triangle ABC,
A will be collinear with the triangle A'BC (Art. 9, Th. 2),
and the plane A'BC will be the same as the plane ABC.
In the same way we prove that the plane A'B'C, and
finally the plane A'B'C', is the same plane.
11. Intersection of a line and a triangle in a plane.
THEOREM. In the plane ABC any line intersecting a
side of the triangle ABC will intersect this triangle at least
THE TWO S1DKS OF A LINE 37
in a second point, and any half -line drawn from a point O
of the interior of the triangle will intersect the triangle.
PROOF. The first statement follows directly from Art. 9,
Th. i, since the line contains points collinear with the
triangle. To prove the second statement, let the half-line
AO intersect the side BC in a point /), a point of AO pro-
duced (Art. 8, Th. 2). Since the given plane can also be
regarded either as the plane ABD or as the plane ACD, it
follows that any line of this plane through O other than the
line A D will intersect both of these triangles again ; that
is, that any line through O, whether the line AD or some
other line of the plane, will intersect the given triangle;
and, therefore, that any half-line drawn in the plane from
0, as well as its opposite half-line, will intersect the tri-
angle (see Art. 8, Th. 2, Cor.).
12. The two parts of a plane lying on opposite sides of a
line.
THEOREM i. Any line of a plane divides the rest of the
plane * into two parts, so that the interior of a segment lying
one point in each part intersects the line, and the interior
of a segment lying both points in the same part does not inter-
sect the line.
PROOF, f Let a be the given line, and A a point of the
plane which is not a point of a. We divide the points of
the plane which are not points of a into two classes, putting
A into the first class, putting a point B into the first class
if the interior of the segment A B does not contain a point
of a, and putting a point B' into the second class if the
interior of the segment AB 1 does contain a point of a.
* Or at least any restricted portion of the plane through which it passes. The
same statement applies to the first theorem of Art. 23, and to Art. 28.
t This proof is given in Halsted's Rational Geometry, p. 8. The point A used
for purposes of proof does not play any particular part in the actual separation of
the points into two classes, any more than do any two points play a particular part
in the class of points which we call a line.
PLANES
[i. in.
Any line of the plane through A will contain only points
of the first class, or will contain a point of a by which
the rest of its points are separated into two classes, so
C
that the interior of a segment with one point in each
class itself contains a point of a, and the interior of a
segment with both points in the same class does not con-
tain a point of a.*
Now in a triangle ABB', B being a point of the first
class, and B f a point of the second class, the line a, intersect-
ing the side AB' and not AB, must intersect BB' (Art. n).
In a triangle ABC, B and C being points of the first
class, the line a does not intersect either side coming to A ,
and therefore it cannot intersect BC (same reference).
Finally, in a triangle AB'C' , B' and C' being points of
the second class, the line a intersects both of the sides
which meet at A, and therefore it cannot intersect B r C f
(Art. 7).
The two parts into which any line of a plane divides the
rest of the plane are said to lie on opposite sides of the line,
and the line is said to lie between them.
* This is one of the theorems of order referred to in Art. 5.
I2,i3l HALF-PLANES AND TRIANGLE 39
THEOREM 2. // two lines intersect, Hie opposite half -lines
of each, drawn from their point of intersection, lie in their
plane on opposite sides of the other.
13. Half-planes. The three half-planes of a triangle.
That portion of a plane which lies on one side of a line of it
is called a half-plane, and the line is the edge of the half-
plane. The half-plane which has the line AB for its edge
and contains the point C will be called the half -plane A B-C.
The two half -planes into which any line of a plane divides
the rest of the plane are called opposite half-planes.
THEOREM i. Every point of the plane ABC belongs to
one, at least, of the three half -planes BC-A, AC-B, or AB-C.
PROOF. All points of the plane which do not belong to
the half-plane BC-A belong to the line BC or to the half-
plane opposite to BC-A .
All points of the line BC belong to the half -line CB or to
the half-line BC, or to both, and so to one, at least, of the
half-planes AC-B or AB-C.
Let P be a point of the half -plane opposite to BC-A.
The interior of the segment AP has a point Q in common
with the line BC, a point of one or both of the half-planes
AC-B and AB-C. Now the interior of the segment PQ
cannot have points in common with both of the lines AC
and A B ; for PQ produced contains their common point A.
Therefore P belongs, with Q, to one or both of these two
half-planes.
That is, every point of the plane which is not a point of
the half-plane BC-A belongs to one, at least, of the two
half-planes AC-B and AB-C.
THEOREM 2. Any point O of the interior of the triangle
ABC belongs to all three of the half -planes BC-A, AC-B,
and AB-C.
40 CONVEX POLYGONS [i. w.
THEOREM 3. Conversely, if a point P belongs to all three
of the half-planes BC-A, AC-B, and AB-C, then it is a
point of the interior of the triangle ABC.
PROOF. Let O be any point other than P of the interior
of the triangle. Then O like P belongs to each of the half-
planes BC-A, AC-B, and AB-C, and the interior of the
segment PO cannot intersect the triangle (see statement
of Th. i of Art. 12). But the opposite half -lines drawn
from O, the half-line OP and the half-line PO produced,
do intersect the triangle (Art. n), intersecting it in the
two points of a segment whose interior contains O, and
therefore P 7 and lies entirely in the interior of the triangle.
Therefore P lies in the interior of the triangle.
IV. CONVEX POLYGONS
14. Polygons. The half-planes of a convex polygon.
A polygon consists of a finite number of points, three or
more, taken in a definite cyclical order, and the interiors
of the segments whose points are consecutive points of this
order.
The points are the vertices of the polygon, and the inte-
riors of the segments are its sides. If A, B, C, . . . are
the vertices in order, the sides are the interiors of the seg-
ments AB, BCj . . ., and the polygon may be described
as the polygon ABC . . . The entire class of points be-
longing to the polygon are in cyclical order (Art. 6).
The triangle is a particular case of the polygon.
A diagonal of a polygon is the interior of a segment whose
points are two non-consecutive vertices of the polygon.
A polygon must have at least four vertices to have a diag-
onal.
A plane polygon is a polygon which lies entirely in a plane.
If two points of such a polygon are on opposite sides of a
line of the plane, each of the two portions into which these
i3-i5l LINK AND POLYGON 41
points divide the rest of the polygon will intersect the line,
for in each portion there will be a vertex on the line or a
side that has points on both sides of the line.
A polygon is a simple polygon when no point occurs twice
as a point of the polygon. By polygon we shall always mean
a simple plane polygon.
A convex polygon is a simple plane polygon no point of
which is a point of a side produced.
THEOREM i . // each pair of consecutive vertices of a poly-
gon lie in the edge of a half-plane which contains all of the
other vertices, the polygon is a convex polygon.
THEOREM 2. Conversely, in a convex polygon each pair
of consecutive vertices lies in the edge of a half-plane which
contains all of the other vertices.
PROOF. Let A and B be two consecutive vertices, and
let D and E be any other vertices. If AB-D and AB-E
were opposite half-planes, each of the two portions into
which D and E divide the rest of the polygon would con-
tain at least one point of the line AB. But the side AB
lies entirely in one of these two portions, and no point
of the other portion can be a point of the line AB if the
polygon is a convex polygon.
COROLLARY. // A and B are two consecutive vertices of
a convex polygon, and if P is any point of the polygon not a
point of the line AB, then all points of the polygon except
those of the line AB will be points of the half -plane AB-P.
15. Intersection of a line and a convex polygon.
THEOREM i. No line can contain more than two points of
a convex polygon unless it contains one of the sides.
PROOF. Let us suppose that a line a, not containing a
side of the polygon, could contain the three points H, K,
and M of the polygon, M coming between H and K on
42 CONVEX POLYGONS [i. iv.
the line. That side of the polygon of which M is a point,
or one of the two sides which meet at M if M is a vertex,
is a part of a line i, the common edge of two opposite half-
planes which contain the points H and K respectively.
But this is contrary to the corollary of the preceding
theorem. Therefore, a line cannot contain three points
of the convex polygon unless it contains one of the sides.
THEOREM 2. If H and K are two points of a convex
polygon, with at least one vertex in each of the two portions
into which they divide the rest of the polygon, then H and K
and the interior of the segment H K will form with each of
these two portions a convex polygon.
PROOF. Let A, 5, ... be the vertices in one of these
portions, taken in order from H. These points with H
and K will then be the vertices of a polygon HAB . . .
KH. This is a simple polygon, for no point of the interior
of the segment HK is a point of the original polygon
(Th. i). It is also a convex polygon: no point of either
continuation of H K can be a point of the original polygon
(same reference) ; moreover, no point of H K can be a
point of some other side produced, for the line containing
such a side would be the common edge of two opposite
half-planes, one containing the point // and the other
the point K, which is contrary to the corollary in the last
article. In the same way we prove that the other polygon
is a convex polygon.
i Si LINE AND POLYGON 43
THEOREM 3. If a line intersects a convex polygon in two
points, and does not contain one of the sides of the polygon,
the two portions into which these points divide the rest of the
polygon lie on opposite sides of the line.
PROOF. If either of the two points is a point of a side
and not a vertex, the extremities of this side will lie on op-
posite sides of the given line ; and as the polygon can inter-
sect the line in only the two given points, the two portions
into which these points divide the rest of the polygon must
lie one entirely on one side and the other entirely on the
opposite side. It only remains, therefore, to prove the
theorem in the case of a diagonal.
Let A E be any diagonal, D, E, and F being three consecu-
tive vertices. No three of the points A, D, E, and F can
be collinear (Th. i).
Now Z), like all the vertices except E and F, is a point
of the half-plane EF-A (Art. 14, Th. 2). Again, D is a
point of the half-plane AF-E\ for D and , being con-
secutive vertices, belong to one of the two portions into
which A and F divide the rest of the polygon, and lie on
the same side of the line AF. Then if D and F were on the
same side of the line A , D would be a point of the half-
plane A E-F, and would therefore be in the interior of the
triangle AEF (Art. 13, Th. 3). But this would put A
44 CONVEX POLYGONS [i. iv.
and F on opposite sides of the line ED (see Art. 8, Th.
2), which is contrary to the second theorem of Art. 14. It
follows that D and P lie on opposite sides of the line AE\
and the same must be true of the entire portions into
which A and E divide the rest of the polygon.
THEOREM 4. Any line in the plane of a convex polygon
intersecting a side will intersect the polygon at least in a
second point.
If the polygon is not a triangle, a diagonal will form
with it two convex polygons, each having a smaller number
of vertices than the given polygon. Thus the theorem,
being true of a triangle (Art. n), can be proved by induc-
tion to be true of any convex polygon.
16. Interior of a convex polygon. The interior of a con-
vex polygon consists of the interiors of all segments whose
points are points of the polygon, except of those whose
interiors also lie in the polygon.
THEOREM i. // two segments lying in a convex polygon
separate each other in the cyclical order of the points of the
polygon (Art. 6), then their interiors will intersect unless the
interior of one of these segments lies in the polygon.
PROOF. Let EF and HK be two segments separating
each other in the cyclical order of the points of the polygon.
The points H and K lie on opposite sides of the line EF,
and the points E and F lie on opposite sides of the line HK,
unless one of these lines contains a side of the polygon
(Art. 15, Th. 3). The line of each segment therefore passes
between the points of the other, and the interiors of the two
segments intersect.
THEOREM 2. Any half -line drawn in the plane of a con-
vex polygon from a point O of the interior will intersect the
polygon.
iS-irl PLANE AND TETRAHEDRON 45
PROOF. The point O is in the interior of a segment H K
whose points are points of the polygon. It lies, then, in
the side H K of two convex polygons, all of which, except
this side and the points H and K, lie on opposite sides of
the line H K (Art. 15, Ths. 2 and 3). The theorem can
therefore be proved in the same way that the theorem of
Art. ii is proved.
THEOREM 3. Any half -plane whose edge contains two
consecutive vertices of a convex polygon, and which itself
contains the other vertices (Art. 14, Th. 2), contains also all
the points of the interior.
THEOREM 4. Conversely, if a point P lies in each of the
half-planes whose edges contain two consecutive vertices of
a convex polygon and which themselves contain the other ver-
tices, then P is in ttie interior of the polygon.
The proof is the same as that of Th. 3 of Art. 13.
V. TETRAHEDRONS
17. Tetrahedrons. Intersection of a plane and a tetra-
hedron. A tetraJiedron consists of four non-coplanar
points and the sides and interiors of the four triangles whose
vertices are these points taken three at a time.
The four points are the vertices of the tetrahedron, the
sides of the triangles are the edges, and the interiors of the
triangles are the faces. Any four non-coplanar points are
the vertices of a tetrahedron.
We shall sometimes speak of the vertices and sides of
a face, but it should always be remembered that a face of a
tetrahedron is the interior of a triangle and does not include
the triangle itself.
THEOREM i. The plane of three non-collinear points of
a tetrahedron, if not itself the plane of one of the faces, inter-
sects the tetrahedron in a triangle or a convex quadrilateral.
46 TETRAHEDRONS [i. v.
PROOF. Two points of the given intersection in the
plane of any one face determine a line which contains a
point in each of the planes of at least two other faces.
If then we have another point of the given intersection in
the plane of one of these faces, we have a second line;
and we can sometimes continue in this way and trace the
intersection completely around the tetrahedron.
This will always be the case when one of the given points
is a vertex. Any plane through a vertex and two points
of the tetrahedron not collinear with the vertex, if not
itself the plane of one of the faces, will intersect the tetra-
hedron in a triangle.
The following construction, with slight modifications,
will provide for all other cases:
Let A BCD be the given tetrahedron. Let H, F, and
K be three points of the given intersection : H a point of
the face ABC, F a point of the face BCD, and K a point
of the edge AD or of one of the faces ABD or ACD. A H
produced will meet the triangle BCD in a point H 1 ', and A K
produced in a point K', H' a point of the side BC y and the
interior of the segment H' K' lying in the interior of this
triangle.
I7l PLANE AND TETRAHEDRON 47
If F is a point of the line H f K', the given plane will pass
through A. Let us suppose that F lies on the opposite
side of this line from C. We have a point G common to
the interiors of the segments CF and H' K' (Art. 12, Th. i,
and Art. 8, Definition and Ths.) ; then in connection with
the triangle AH 1 ' K' we have a point M common to the
interiors of the segments AG and HK (Art. 8, Th. i) ;
and finally, with M in the interior of the triangle ACF,
we have in FM produced a point L of the side AC. But F
and the segment H K lie in the plane of the given intersec-
tion. Therefore, the line FM lies in this plane, and L
itself is a point of the given intersection.
Now assuming that the given plane does not pass through
a vertex, we have the following cases :
First, if the three given points lie in three different faces,
we can take any one of these points for the point H and
obtain a point of the intersection in one side of this face,
and then a line in which the given plane intersects the plane
of this face. These lines will intersect the edges of the tet-
rahedron in three points or in four points, and the inter-
section will be a triangle or a quadrilateral. The given
plane cannot intersect the three sides of a triangle (Art.
7, Th.), and therefore it cannot intersect more than four
of the edges of the tetrahedron.
Again, if two of the given points lie in two different faces
and the third lies in the edge which is not a side of either
of these faces, there will be lines in the planes of these two
faces intersecting in their common side, and intersecting
two other edges of the tetrahedron in two points which,
with the third given point, determine the rest of the inter-
section. The intersection will in this case be a quadri-
lateral.
In the third place, if one of the given points lies in a face
and the other two in edges which are not sides of this face,
48 TETRAHEDRONS [i. v.
let us suppose that H lies in the face ABC, F in the edge
BD, and K in the edge AD. This change in the position
of F will not affect th'e determination of the point L ; there
will be a line EL intersecting the edge BC, and the intersec-
tion will be a quadrilateral. The half -line LH cannot meet
the edge AB, for the plane already intersects AD and BD.
Finally, if the three given points lie in three edges not all
meeting at one vertex nor lying in one plane, we may sup-
pose that H is a point of BC, F a point of BD, and K a
point of AD. In the construction above, H and H f coin-
cide, but this will not affect the determination of the point
L ; we still have a point of the intersection in each of four
edges, and the intersection will be a quadrilateral.
All cases in which the intersection cannot be determined
directly from the given points can be reduced to one of these
four.
When the intersection is a quadrilateral, it is a convex
quadrilateral. No point of any face of the tetrahedron
lies in the plane of any other face, and no point of any edge
lies in the plane of any face except of the two faces which
have this edge for a common side. Therefore no point of
any side of the quadrilateral can be a point of a line con-
taining another side, and no vertex can be collinear with
two other vertices.
THEOREM 2. No line can contain more than two points of
a tetrahedron unless it lies in the plane of one of the faces.
PROOF. Let a given line a intersect the tetrahedron in
two points H and F, and let A be a vertex which is not a
point of a. The plane A HF will contain the line a and will
intersect the tetrahedron in a triangle, if it does not contain
one of the faces. The intersection of a and the tetrahedron
will then be the same as its intersection with the triangle,
and will consist only of the two points H and F (Art. 7,
Cor.).
17-iQl POINTS AND TETRAHEDRON 49
18. Interior of a tetrahedron. The interior of a tetra-
hedron consists of the interiors of all segments whose points
are points of the tetrahedron, except of those segments
whose interiors also lie in the tetrahedron.
THEOREM. A half -line drawn from any point P of a tetra-
hedron through a point O of the interior intersects the tetra-
hedron in a point of PO produced.
PROOF. The point O is given as in the interior of a seg-
ment whose points are points of the tetrahedron. If P is
one of these points, the other will be the point in which the
half-line PO intersects the tetrahedron, a point of PO
produced. If P is not one of these points, these points
and P will lie in a plane intersecting the tetrahedron in
a triangle or convex quadrilateral, and the theorem follows
as a theorem concerning this intersection (see Art. 16,
Th. 2).
19. The relation, collinear with a tetrahedron. A point
is said to be collinear with a tetrahedron when it is collinear
with any two points of the tetrahedron.
This is true, in particular, of all points of the tetrahedron,
of all points in the planes of its faces, and of all points of
its interior.
THEOREM i . If a point O is collinear with a tetrahedron
A BCD, and if P is any point of a face, not the point O, then
the line PO will intersect the tetrahedron at least in a second
point Q.
PROOF. Let H and K be the two points with which O
is collinear. Through these two points and P we can pass
a plane intersecting the tetrahedron in a triangle or convex
quadrilateral, or containing a face of the tetrahedron.
When the intersection is a triangle or quadrilateral, P will
be a point of a side, and thus in any case the line PO will
50 HYPERPLANES [i. vi.
contain at least a second point Q of the tetrahedron (Art.
9, Th. i, and Art. 15, Th. 4).
THEOREM 2. If a point O, not a point of the plane BCD,
is collinear with the tetrahedron A BCD, then any point P
collinear with the tetrahedron A BCD will be collinear with
the tetrahedron OBCD.
PROOF. Take a point H in the face BCD, not collinear
with O and P. The line HO will intersect the tetrahedron
in a second point F, and the line HP in a second point K.
The points H, F, and K are not collinear, and do not lie
in the plane of any one face of the tetrahedron A BCD.
The plane HFK intersects this tetrahedron in a triangle
or convex quadrilateral, one side of which lies in the face
BCD and contains the point H. That is, the plane inter-
sects the triangle BCD in a segment MN, and therefore
the tetrahedron OBCD in a triangle OMN, with H a point
of the side MN. The line HP must then intersect this
triangle in a second point, and P must be collinear with it.
Therefore P is collinear with the tetrahedron OBCD.
Remarks similar to those at the end of Art. 9 may be
made at this point. The class of points collinear with the
tetrahedron OBCD is the same as the class of points collin-
ear with the tetrahedron A BCD. We can use A and O
interchangeably in this connection.
VI. HYPERPLANES
20. Hyperplanes. Only one hyperplane contains four
given non-coplanar points. Our space a hyperplane. A
hyperplane consists of the points that we get if we take
four points not points of one plane, all points collinear
with any two of them, and all points collinear with any two
obtained by this process.
Given four non-coplanar points, A, B, C, and Z>, the
ig, 20] DETERMINATION OF A HYPERPLANE 51
hyper plane A BCD is the hyperplane obtained when we take
these points and carry out the process described in the
definition.
THEOREM i. // two points of a line lie in a given hyper-
plane, the line lies entirely in the hyperplane ; and if three
non-collinear points of a plane lie in a given hyperplane,
the plane lies entirely in the hyperplane.
For the line or plane can be obtained from these points
by the process used in obtaining the hyperplane.
It follows that a plane having two points in a given hy-
perplane, but not lying entirely in it, will intersect the
hyperplane in the line which contains these two points.
THEOREM 2. From the points of the figures given in each
of the following cases we can obtain just the points of a hyper-
plane if we take all points collinear with any two of them
and all points collinear with any two obtained by this process :
(1) A plane and a point not in it, or a plane and a line
that intersects it but does not lie in it;
(2) Two lines not in one plane;
(3) Three lines through one point but not in one plane;
(4) Two planes intersecting in a line.
We can, indeed, speak of a line or a plane as one of the
things with which we start in the process of obtaining a
hyperplane (compare this with Art. 10).
It follows from (i) that a line and a plane which do not
lie in a hyperplane do not intersect at all, and from (4) that
two planes which do not lie in a hyperplane cannot have
more than one point in common.
THEOREM 3. // A ', B', C", and D f are four non-coplanar
points of the hyperplane A BCD, then the hyperplane
A'B'C'D' is the same as the hyperplane A BCD.
52 HYPERPLANES [i. vi.
The proof follows the lines of proof of the two theorems
given in Art. 10.
In regard to Th. 2 we can now say that the hyperplane
obtained in each case is the only hyperplane that con-
tains the given figures.
Three non-collinear points can be points of two different
hyperplanes. The intersection of the hyperplanes will
then be the plane of the three points (see Art. 27, Th. 2).
Apparently we get all the points of ordinary space by
taking four non-coplanar points, all points collinear with
any two of them, and all points collinear with any two ob-
tained by this process. The space of our experience will
therefore be regarded as a hyperplane.
21. Intersection of a line or plane and a tetrahedron in
a hyperplane.
THEOREM i. In the hyperplane A BCD, any line inter-
secting a face of the tetrahedron A BCD will intersect this
tetrahedron at least in a second point, and any half -line drawn
from a point O of the interior of the tetrahedron will intersect
the tetrahedron.
PROOF. The first statement follows directly from Art. 19,
Th. i, since the line contains points collinear with the
tetrahedron. To prove the second statement, let the half-
line AO intersect the face BCD in a point P y a point of AO
produced (Art. 18, Th.). If we have given any other
half-line drawn from O in the hyperplane, any point 0' of
this half-line will be collinear with the tetrahedron (see
reference in connection with Art. 20, Th. 3), and the line
PO' will intersect the tetrahedron at least in a second point
Q not collinear with A and P (Art. 19, Th. i). The plane
APQ will intersect the tetrahedron in a triangle; and the
given half-line, being drawn in this plane from a point in
20-231 INTERSECTION OF PLANES 53
the interior of the triangle, will intersect the triangle, and
therefore the tetrahedron.
THEOREM 2. In the hyper plane ABCD, any plane in-
tersecting a face of the tetrahedron ABCD (if not itself the
plane of this face), or any plane containing a point O of the
interior of the tetrahedron, will intersect the tetrahedron in
a triangle or convex quadrilateral.
For in either case by drawing lines in the plane we can
obtain three non-collinear points of the intersection.
22. Intersection of two planes in a hyperplane.
THEOREM. // two planes lying in a hyperplane have
a point in common, they have in common a line through
O.
PROOF. Let a and /3 be the two given planes. Let C
and D be two points of a not collinear with O, and take P
a point between C and D, B a point of PO produced, and
A any point of the hyperplane not a point of a. The point
O lies in the face BCD of the tetrahedron ABCD, and we
can consider the given hyperplane as the hyperplane of
this tetrahedron. The plane /3, containing O, will intersect
the tetrahedron in a triangle or convex quadrilateral (Art.
21, Th. 2), and a, the plane of the face BCD, in the line
which contains one side of the triangle or quadrilateral,
a line through O.
23. Opposite sides of a plane. Half-hyperplanes. The
four half-hyperplanes of a tetrahedron.
THEOREM i. Any plane of a hyperplane divides the rest
of the hyperplane into two parts, so that tlie interior of a
segment lying one point in each part intersects the plane, and
the interior of a segment lying both points in the same part
does not intersect the plane.
The proof of Art. 12, Th. i, applies almost without
54 HYPERPLANES [i. VI.
change, the reference to Art. n holding true by virtue of
Art. 22.
The two parts into which any plane of a hyperplane
divides the rest of the hyperplane are said to lie on opposite
sides of the plane, and the plane is said to lie between them.
That portion of a hyperplane which lies on one side of
a plane of it is called a half -hyper plane, and the plane is
the face of the half-hyperplane. The half-hyperplane
which has the plane ABC for its face and contains the
point D will be called the half-hyperplane ABC-D. The
two half-hyperplanes into which any plane of a hyperplane
divides the rest of the hyperplane are called opposite half-
hyperplanes.
THEOREM 2. // two planes intersect in a line, the opposite
half-planes of each, which have this line for a common edge,
lie in their hyperplane on opposite sides of the other.
THEOREM 3. Every point of the hyperplane A BCD
belongs to one, at least, of the four half-hyperplanes BCD- A,
ACD-B, ABD-C, or ABC-D.
The proof is like that of Art. 13, Th. i.
THEOREM 4. Any point of the interior of the tetra-
hedron A BCD belongs to all four of the half-hyperplanes
BCD- A, ACD-B, ABD-C, and ABC-D \ and, con-
versely, if a point belongs to all four of these half-hyperplanes,
it is a point of the interior of the tetrahedron A BCD.
See Art. 13, Ths. 2 and 3.
We shall sometimes speak of a tetrahedron as a surface
and of its interior as a solid. A tetrahedron divides the
rest of its hyperplane into two portions, interior and ex-
terior to the tetrahedron.
23, 2 4 ] PLANE AND PYRAMID 55
VII. CONVEX PYRAMIDS AND PENTAHEDROIDS
24. Pyramids. Intersection of a plane and a convex
pyramid. A pyramid consists of a simple plane polygon
which has an interior, and a point not in the plane of the
polygon, together with the interior of the polygon and the
interiors of the segments formed by taking the given point
with the points of the polygon.
The point is called the vertex of the pyramid, and the
interior of the polygon is the base. The interiors of the
segments formed by taking the vertex and the vertices of
the polygon are the lateral edges, and the interiors of the
triangles determined by the vertex and the sides of the
base are the lateral faces. Often the term vertices is applied
to the vertex and the vertices of the polygon taken together,
the term edges to the lateral edges and the sides of the
polygon, and the term faces to the lateral faces and the
base. Sometimes, also, we shall speak of the vertices and
sides of the base or of a lateral face (see remark at the begin-
ning of Art. 17).
The tetrahedron is a particular case of the pyramid.
When the polygon is convex, the pyramid is a convex
pyramid. We shall consider only convex pyramids.
THEOREM. The plane of any three non-collinear points
of a convex pyramid, if not itself the plane of one of the faces,
intersects the pyramid in a convex polygon*
PROOF. If the plane passes through the vertex, the in-
tersection will be a triangle, as in the case of a tetrahedron
(Art. 17).
Let O be the vertex, and let A, B, C, . . . be the vertices
of the base.
Suppose the plane, not passing through O, intersects
the base in the interior of the segment H K, H and K
* It may be a triangle, which is a particular case of the convex polygon.
56 PYRAMIDS AND PENTAHEDROIDS [i. vn.
being points of the polygon AB . . . , dividing the rest
of this polygon into two portions with each of which they
and the interior of the segment HK form a convex polygon
(Art. 15, Th. 2). The triangle OHK will divide the rest
of the pyramid into two portions, with each of which it
and its interior will form a convex pyramid. Now the
base and the vertex of one of these pyramids will lie on
opposite sides of the given plane, while all of the other
pyramid except the segment HK and the points of its
interior lie on the same side of this plane. The given plane
will not intersect the second pyramid except along the
line H K, and the given intersection is the same as the
intersection of the plane with a pyramid whose vertex
and base lie on opposite sides of the plane.
Let us suppose, then, that we have a plane and a pyramid
O-ABC . . . whose vertex and base lie on opposite sides
of the plane. The plane may contain one vertex of the
base, or a side and two vertices, but the base itself does
not intersect the plane, and all the rest of the intersection
lies in the lateral faces and lateral edges. In particular,
any lateral edge OA will intersect the plane in a point A ' or
will meet the plane at A. Then the face OAB will in-
tersect the plane in the interior of a segment A'B f ', or AB',
unless (in one particular case) A B itself lies in the plane ;
and the entire intersection will be a polygon A'B'C' . . .
There is a one-to-one correspondence of the points of this
polygon and the points of the polygon ABC . . . , cor-
responding points being collinear with O. There is, indeed,
a correspondence of the entire planes of these two polygons
so far as there are points in both collinear with O. There
is also a correspondence in the order of the points of the
two polygons, and in the order of the points of any two
corresponding lines (Art. 8, Ths. i and 2). If, then,
a point P' were to be found twice among the points of the
24, 25l PLANE AND PENTAHEDROID 57
polygon A'B'C' . . ., the corresponding point P would be
found twice among the points of ABC . . . ; or if a point
of the former polygon were also a point of some side pro-
duced, the same would be true of the corresponding point
in the plane of the base. It follows that the intersection
must be a simple convex polygon, the polygon ABC . . .
being such a polygon (Art. 14, Defs.).
25. Pentahedroids. Intersection with a plane. In-
terior. The collinear relation. A pentahedroid consists
of five points not points of one hyperplane, and the edges,
faces, and interiors of the five tetrahedrons whose vertices
are these points taken four at a time.
The five points are the vertices, the edges and faces of
the tetrahedrons are edges and faces of the pentahedroid,
and the interiors of the tetrahedrons are its cells. Any five
points, not points of one hyperplane, are the vertices of
a pentahedroid.
We shall sometimes speak of the vertices, edges, and faces
of a cell, but it should always be remembered that a cell
of the pentahedroid is the interior of a tetrahedron and does
not include the tetrahedron itself.
THEOREM. The plane of three non-collinear points of
a pentaltedroid, if it does not itself lie in the hyperplane of one
of the cells, intersects the pentahedroid in a convex polygon.
PROOF. As in the case of a tetrahedron (Art. 17, Th. i,
first part of the proof), we can sometimes determine the
intersection directly from the three given points, two points
in the hyperplane of one cell determining a line which
contains a point in each of the hyperplanes of at least two
other cells. In particular, this will be true when one of
the given points is a vertex.
Let ABCDE be the pentahedroid, and let A be a vertex
not in the plane of the given points. If H is any point
58 PYRAMIDS AND PENTAHEDROIDS [i. vn.
of the pentahedroid other than A, the half- line A H will
meet the hyperplane of the opposite cell in a point //'.
This point we shall call the projection of // from A. //'is
the same as // when // itself is a point of the cell BCDE.
In all other cases //' is a point of the tetrahedron BCDE
(see Art. 8, Th. 2, and Art. 18).
Let the three given points be //, F, and A', and let //',
F', and K r be their projections from A. These last three
points are not collinear; for, if they were, the plane de-
termined by their common line and A would be a plane
containing the three given points and the vertex A . They
are not all in the plane of any one face of the tetrahedron
BCDE ; for, if they were, //, F, and K would be in the
hyperplane determined by this face and A, and the given
plane would lie entirely in this hyperplane. Therefore
the plane H'F' K' intersects the tetrahedron BCDE in
a triangle or convex quadrilateral (Art. 17, Th. i). This
triangle or quadrilateral is the base of a pyramid with
vertex at A which lies entirely in the hyperplane A H'F' K' ,
and which is, indeed, the intersection of the hyperplane
and pentahedroid (see Art. 31, Th. i).
The points //, F, and K are points of this pyramid, and
the plane HFK lies entirely in the hyperplane of this
pyramid. The intersection of the plane and the pentahe-
droid is the same as the intersection of the plane and this
pyramid: it is a convex polygon (Art. 24, Th.).
As there are only five cells, the intersection can only be
a triangle, a quadrilateral, or a pentagon.
We define interior of a pcnlahedroid and collinear with
a pentahedroid exactly as we define the corresponding ex-
pressions for the tetrahedron, and for the pentahedroid
we have theorems analogous to those of Arts. 18 and 19.
25, 26] RESTRICTION TO FOUR DIMENSIONS 59
In a pentahedroid each tetrahedron is adjacent to each of the other
four. If we move along a line through the interior of one of the tet-
rahedrons until we come to a face, a continuation of our movement
along this line will take us entirely away from the pentahedroid.
But we can change the direction of our path so as to pass into the
interior of an adjacent tetrahedron, thus following a broken line that
belongs entirely to the pentahedroid. This is like what we do when
we move along a line in one face of a polyhedron until we come
to an edge, and then change the direction of our path so as to pass
across an adjacent face.
It may not be very difficult to think of two adjacent tetrahe-
drons, even though they lie in different hyperplanes, but it is
quite impossible for us to form any picture of the pentahedroid
as a whole. All that we should try to do now is to remember in a
mechanical way the numbers and relations of the different parts.
VIII. SPACE OF FOUR DIMENSIONS
26. Space of four dimensions. The hyperspace of this
book. A space of four dimensions consists of the points
that we get if we take five points not points of one hyper-
plane, all points collinear with any two of them, and all
points collinear with any two obtained by this process.
By a series of theorems analogous to others already given
for the plane and hyperplane, we can prove that any five
points of a given space of four dimensions, not points of
one hyperplane, determine the same space of four dimen-
sions.
We shall limit ourselves to a discussion of space of four
dimensions. We shall assume that there is such a space,
and that all points lie in one such space.* The preceding
theorem can, therefore, be stated as follows :
THEOREM i. We get all points if we take any five points
not points of one hyperplane, all points collinear with any
* This is a mere matter of convenience, enabling us to avoid frequent repetition
of the phrase "in one space of four dimensions," and to state many theorems in a
simpler form.
60 SPACE OF FOUR DIMENSIONS [i. vm.
two of them, and all points collinear with any two obtained
by this process.
The word hypersfiace is used for any space of more than
three dimensions, but as the only hyperspace that we shall
consider is the space of four dimensions we shall use these
terms interchangeably.
THEOREM 2. Any line intersecting a cell of a pentahe-
droid will intersect the pentahedroid at least in a second point,
and any half -line drawn from a point O of the interior of
a pentahedroid will intersect the pentahedroid.
THEOREM 3. Any plane intersecting a cell of a pentahe-
droid, if it does not itself lie in the hyperplane of this cell,
or any plane containing a point O of the interior of the penta-
hedroid, will intersect the pentahedroid in a convex polygon.
These two theorems correspond to the two theorems of
Art. 21. For theorems on the intersections of pentahe-
droids and hyperplanes see Art. 31.
27. Intersection of a plane and a hyperplane and of two
hyperplanes. The linear elements of planes.
THEOREM i. If a plane and a hyperplane have a point
O in common, they have in common a line through O.
The proof is like that of the theorem of Art. 22. We
construct a pentahedroid with a cell lying in the given
hyperplane and containing O. The given plane intersects
the pentahedroid in a convex polygon and the given hyper-
plane in the line which contains one side of this polygon.
THEOREM 2. // two hyperplanes have a point O in com-
mon, they have in common a plane through 0.
PROOF. Let a and /8 be the two hyperplanes. Any
plane of a through O will intersect in a line through 0,
by the preceding theorem ; and a second plane of a through
26,27) PLANES IN HYPERSPACE 6 1
O, drawn so as not to contain the line in which the first
plane intersects /3, will intersect /? in a second line. We
have two lines through O common to the two hyperplanes.
The hyperplanes therefore intersect in a plane (see Art. 20).
Three hyperplanes having a point in common have in
common at least one line, a line lying in one hyperplane
and in the plane of intersection of the other two. Three
hyperplanes may also have a plane in common.
THEOREM 3. Two planes which do not lie in one hyper-
plane contain a set of lines, one and only one through each
point of either plane which is not a point of the other plane,
and any two of these lines coplanar.
PROOF. Let a and /3 be two planes which do not lie in
one hyperplane. Let A be any point of a which is not a
point of |8, and let B be any point of /? which is not a point
of a. The hyperplane determined by and A intersects
a in a line a, and the hyperplane determined by a and B
intersects /3 in a line b (Th. i). The lines a and ft, each
lying in both hyperplanes, lie in the plane of intersection
of the hyperplanes.
On the other hand, no two lines lying in one of the given
planes and coplanar with lines in the other can intersect
in a point which is not common to the two given planes ;
for, if they did, both of them and the entire plane in which
they lie would lie in the hyperplane determined by their
point of intersection and the other given plane.
The planes a and j8 are covered with these lines, and might
be said to consist of them. We shall call them the linear
elements of the two planes. When the two planes have a
point in common, the linear elements all pass through this
point. If any plane intersects the two planes in lines,
these lines are linear elements.
62 SPACE OF FOUR DIMENSIONS [i. vm.
The student may prove the following theorem :
If two planes have only one point in common, then through any
point not a point of either plane passes one and only one plane inter-
secting both the two planes in lines.
28. Opposite sides of a hyperplane. Half-hyperspaces.
A hyperplane divides the rest of hyperspace just as a plane
in a hyperplane divides the rest of the hyperplane (Art.
23). We can speak of the opposite sides of a hyperplane,
and of a half-hyperspace. We may have, for example, the
half -hyper space ABCD-E lying on one side of the hyper-
plane A BCD, and the opposite half-hyperspace lying on the
opposite side of this hyperplane. The hyperplane is the
cell of the half-hyperspace. If two hyperplanes intersect,
the opposite half-hyperplanes of each which have the plane
of intersection for a common face lie on opposite sides of
the other.
Given a pentahedroid, each of the five tetrahedrons
determines the cell of a half-hyperspace which contains
the opposite vertex and all points of the interior; and,
conversely, if a point lies in all five of these half-hyperspaces
it will lie in the interior of the pentahedroid. Every point
of hyperspace is a point of at least one of these half-hyper-
spaces.
We shall sometimes speak of a pentahedroid as a hyper-
surface (see Art. 33), and of its interior as a hyper solid.
A pentahedroid divides the rest of hyperspace into two
portions, interior and exterior to the pentahedroid.
As a line divides a plane into two parts, but not ordinary space (a
hyperplane), so a plane divides a hyperplane, but not hyperspace.
In hyperspace we can pass completely around a plane, keeping near
some one point of it, without touching the plane at all. We can do
this, for example, by keeping in another plane which has only one
point in common with the given plane. The student should bear
this in mind when he comes to consider absolutely perpendicular
planes (Art. 42) and rotation around a plane (Art. 81).
27-30] HYPER PYRAMID AS POLYHEDROID 63
IX. HYPERPYRAMIDS AND HYPERCONES
29. Introduction. Knowledge of polyhedrons presup-
posed. In this section will be introduced the hyper-
pyramids and hypercones of four-dimensional geometry,
with a few theorems analogous to those already given for
convex polygons and convex pyramids. We shall assume
that the term polyhedron has been defined, at least so far
as the hyper plane polyhedron is concerned, together with
vertices, edges, and faces, and the expressions convex poly-
hedron and interior of a polyhedron. We shall also speak
of curve and surface, and, in particular, of the circle, the
sphere, and the circular conical surface; and we shall use
such theorems as we need concerning these surfaces, and
concerning polyhedrons (see Introduction to chap. II).
We may find some difficulty in understanding the forms here con-
sidered, but that need not deter us from going on, as the next three
chapters do not depend on this part of our text. Indeed, the study
of hyperpyramids and hypercones could be omitted entirely until
we come again to their treatment in chap. V.
30. Hyperpyramids. Interior of a hyperpyramid. Fig-
ures in hyperspace which correspond to the polyhedrons
of geometry of three dimensions are called polyhedroids.
We shall not attempt to define this term, except to say that
a polyhedroid consists of vertices, edges, faces, and cells,
the cells being the interiors of certain hyperplane polyhe-
drons joined to one another by their faces so as to enclose
a portion of hyperspace, the interior of the polyhedroid.
We shall apply the term polyhedroid only to certain simple
figures which we shall define individually. The pentahe-
droid is the simplest polyhedroid.
A hyperpyramid consists of a hyperplane polyhedron
enclosing a portion of its hyperplane, and a point not a
64 HYPERPYRAMIDS AND HYPERCONES [i. ix.
point of this hyperplane, together with the interior of the
polyhedron and the interiors of the segments formed by
taking the given point with the points of the polyhedron.
The point is the vertex, and the interior of the polyhedron
is the base. The meaning of other terms used in connection
with the hyperpyramid may be readily inferred from the
definitions of Arts. 24 and 25. The pentahedroid is the
simplest hyperpyramid.
The interior of a hyperpyramid can be defined as con-
sisting of the interiors of the segments formed by taking
the vertex with the points of the base, but in the case of
a convex hyperpyramid the interior of any segment whose
points are points of the hyperpyramid will lie entirely in
the interior of the hyperpyramid unless it lies entirely in
the hyperpyramid itself. No line can intersect a convex
hyperpyramid in more than two points unless it lies in the
hyperplane of one of the cells, and any half-line drawn
from a point O of the interior will intersect the hyper-
pyramid in one and only one point.
In the last statement is involved one of the difficulties that we have
in attempting to understand geometry of four demensions. Since a
half-line can be drawn from O through any point of the hyperpyramid,
each point of the hyperpyramid plays a part in separating the interior
from that portion of hyperspace which is exterior to the hyperpyramid.
When we think of a solid as forming a part of the boundary of a figure,
we are apt to think of one face of the solid as coming next to the in-
terior and the other face as on the outside, and we think of the thick-
ness of the solid as representing the thickness of the boundary. It is
in a very different way that the cells of a polyhedroid separate the in-
terior from the outside. Each point, no matter how far within the
solid, is exposed both to the interior and to the outside, and its re-
moval would allow free passage to a half-line from 0.
31. Hyperplane sections of a hyperpyramid. A hyper-
pyramid or any polyhedroid can be cut by a hyperplane
30, 3i] SECTIONS 65
in a section* which divides the rest of the polyhedroid into
two parts lying on opposite sides of the hyperplane (Art.
28).
The sections of a polyhedroid are polyhedrons whose
faces are the sections of the cells of the polyhedroid made
by the planes in which the hyperplane intersects the
hyperplanes of the cells. Thus the following theorems
in regard to the sections of a hyperpyramid orpentahedroid
are proved by considering the plane sections of their cells :
THEOREM i. A section of a convex hyperpyramid made
by a hyperplane containing the vertex, is a convex pyramid
whose base is the corresponding plane section of the base of
the hyperpyramid. In the case of a pentahedroid this applies
to any vertex. When one vertex of a pentahedroid lies in
a hyperplane and two vertices of the opposite cell lie on each
side of the hyperplane, the section will be a quadrilateral
pyramid. In all other cases the section of a pentaliedroid
by a hyperplane containing one vertex and not containing
a cell will be a tetrahedron.
THEOREM 2. A hyperplane passing between one vertex
of a pentahedroid and the opposite tetrahedron will intersect
the pentahedroid in a tetrahedron.
THEOREM 3. // two vertices of a pentahedroid lie on one
side of a hyperplane and three on the opposite side, the section
will be a polyhedron in which there are two triangles separated
by three quadrilaterals.
PROO**. Let ABCDE be the pentahedroid, A and B on
one side of a hyperplane a, and C, D, and E on the opposite
side. The hyperplanes of the tetrahedrons ACDE and
BCDE are cut by a in planes which pass between the
* The word section is somewhat synonymous with intersection, but in general
it will not be used except for an intersection which divides the rest of a figure into
two parts and completely separates these parts. Thus a triangle or convex quadri-
lateral would be a section of a tetrahedron.
66 HYPERPYRAMIDS AND HYPERCONLS [i. ix.
triangle CDE and the points A and B (Art. 28), and which
therefore cut these tetrahedrons in triangles. The hyper-
planes of the other three tetrahedrons are cut by a. in planes
which pass between their common edge AB and the op-
posite edges CD, DE, and CE, and which therefore cut
these tetrahedrons in quadrilaterals. If we suppose the
triangles to be C'D'E' and C"D"E", the quadrilaterals
will be C'D'C'D", D'E'D"E", and C'E'C"E", and the
section of the pentahedroid will be C'D'E'-C"D"E".
If a. intersects the line AB in a point P, the three lines
C'C", D'D", and E'E" will pass through P, and the section
will be a polyhedron which may be called a truncated
tetrahedron. In any case the section will be a figure of
this type (see Introduction, p. 14, and the exercise at
the end of Art. 128).
A section of a figure in hyperspace is all that we can see in any
hyperplane (see Introduction, p. 18). We can, however, see any
section, and we can learn much about a figure by studying its various
sections.
32. Double pyramids. A hyperpyramid whose base is
the interior of a pyramid may be regarded in two ways
as a hyperpyramid of this kind, the vertex of the base in
one case being the vertex of the hyperpyramid in the other
case.
Thus there are two pyramids having themselves a com-
mon base, and we can say that the hyperpyramid is deter-
mined by a polygon and two points neither of which is
in the hyperplane containing the polygon and the other
point. Looked at in this way the hyperpyramid is called
a double pyramid.
A double pyramid consists, then, of the following classes
of points :
(i) the points of a convex polygon, or of any plane
31,32] WITH PYRAMID BASE 67
polygon which has an interior, and the points of its in-
terior ;
(2) two points not in *a hyperplane with the polygon,
the interior of the segment formed of these two points,
ind the interiors of the segments formed by taking each
3f these points with the points of the polygon ;
(3) the interiors of the triangles formed by taking each
point of the polygon with the two given points ;
(4) the interiors of two pyramids each formed by taking
the polygon with one of the two given points.
The interior of the segment of the two given points is
called the vertex -edge of the double pyramid, and the in-
terior of the polygon is the base. The interiors of the
triangles (3) are called elements, and, in particular, those
elements whose planes contain the vertices of the polygon
are lateral face elements or lateral faces of the double pyramid.
The two pyramids (4) are called the end-pyramids.
The vertex-edge and the sides of the base are opposite
edges of a set of tetrahedrons. These tetrahedrons are
in a definite cyclical order corresponding to the sides of
the polygon, and are joined, each to the next, by the faces
which are adjacent to the vertex-edge. They arc joined
to the end-pyramids by the faces which are adjacent to
the sides of the base. The interiors of these tetrahedrons
are the lateral cells, and these and the interiors of the end-
pyramids are the cells of the double pyramid.
The pentahedroid is the simplest double pyramid.
Certain cases of intersection of double pyramids are
given by the following theorems :
THEOREM i. A plane containing a point of the vertex-
edge and intersecting the base in the interior of a segment y
or a plane containing the vertex-edge and a point of the base y
will intersect the double pyramid in a triangle.
68 HYPERPYRAMIDS AND HYPERCONES [x. DC.
In the latter case two sides of the triangle are in the
interiors of the end-pyramids.
THEOREM 2. A hyper plane containing the base and a
point of the vertex-edge will intersect the double pyramid in
a pyramid.
THEOREM 3. A hyper plane containing the vertex-edge
and intersecting the base will intersect the double pyramid in
a tetrahedron.
In this case two faces of the tetrahedron are sections of
the interiors of the end-pyramids.
We have already referred to the importance of studying the sec-
tions of a figure of hyperspace (Art. 31). Another way of studying a
polyhedroid is to examine its cells. These cells can be cut apart
sufficiently to spread them out in a hyperplane where we can see
them. They are the interiors of polyhedrons; and we can think of
a polyhedroid as formed by taking a suitable set of polyhedrons, plac-
ing them upon one another, and folding them away from our hyper-
plane until they come together completely.
In a pentahedroid, for example, there are five tetrahedrons whose
twenty faces fit together in pairs, each tetrahedron having a face in
common with each of the other four. We can take any one of these
tetrahedrons and place the other four upon it, all in one hyperplane,
and then we can turn the four outside tetrahedrons away from this
hyperplane without separating them from the fifth or distorting them
in any way (Art. 81), until we have brought together every pair of
corresponding faces. The five tetrahedrons together with their
interiors now enclose a portion of hyperspace. This is analogous to
the process of forming a tetrahedron by placing three of its faces in a
plane around the fourth, turning them about the sides which lie upon
the sides of the fourth, and bringing them together completely.*
Another way of constructing a pentahedroid is by means of its
ten edges. Any tetrahedron will have six of these edges and will be
connected with the opposite vertex by the remaining four. We can
take these four and cause them to stand out from the vertices of the
tetrahedron, all in one hyperplane. In hyperspace these four edges
* Other examples of this method of studying polyhedroids are given in notes in
Arts. 135, 136, and 141.
32,331 HYPERSURFACE WITH LINE ELEMENTS 69
can be turned so that their ends come together at a point, and with
the faces of the given tetrahedron determine the other four tetrahe-
drons and the complete figure.
33. Hyperconical hypersurfaces. Hypercones. Hyper-
surface is the term applied to a figure in hyperspace which
corresponds to the surfaces of geometry of three dimen-
sions. We shall not attempt to define hypersurface, and
we shall use the word only in connection with certain
simple figures which we shall define individually. The
hyperplane is the simplest hypersurface.
A hyperconical hypersurface* consists of the lines deter-
mined by the points of a hyperplane surface and a point
not in the hyperplane of this surface.
The point is the vertex, the surface is the directing-
surface, and the lines are the elements. The hyperconical
hypersurface has two nappes.
The only hyperconical hypersurfaces which we have to
consider at present are those in which the directing-surface
is a plane, a sphere, a circular conical surface, or a part or
combination of parts of such surfaces. When the directing-
surface is a plane, the hypersurface is a hyperplane or a
portion of a hyperplane.
A hypercone consists of a hyperplane surface, or portions
of hyperplane surfaces, forming a closed hyperplane figure,
and a point not a point of the hyperplane of this figure,
together with the interior of the latter and the interiors
of the segments formed by taking the given point with
the points of the hyperplane figure.
The point is the vertex, the interiors of the segments are
elements, and the interior of the hyperplane figure is the
base.
* See foot-note, p. 220. We shall sometimes use the word conical for hyperconi-
cal when there is no ambiguity.
70 HYPERPYRAMIDS AND HYPERCONES [i. ix.
A hypercone, or the hypersolid which we call the interior of a hy-
percone (see below), can be somewhat inaccurately described as cut
from one nappe of a hyperconical hypersurface by the hyperplane of
the direct ing-surface.
The hyperpyramid may be regarded as a particular case
of the hypercone. The only other cases which we have
to consider at present are those in which the base is the
interior of a sphere or of a circular cone.
A plane containing the vertex of a hypercone and in-
tersecting the base in the interior of a segment, will inter-
sect the hypercone in a triangle; and a hyperplane con-
taining the vertex and intersecting the base, will intersect
the hypercone in a cone.
The interior of a hypercone consists of the interiors of the
segments formed by taking the vertex with the points of
the base, but in the case of a convex hypercone the interior
of any segment whose points are points of the hypercone
will lie entirely in the interior of the hypercone unless it
lies entirely in the hypercone itself. No line can intersect
a convex hypercone in more than two points if it passes
through a point of the interior, and any half-line drawn
from a point O of the interior will intersect the hypercone
in one and only one point.
34. Double cones. A hypercone whose base is the
interior of a cone may be regarded in two ways as a hyper-
cone of this kind, the vertex of the base in one case being
the vertex of the hypercone in the other case.
Thus there are two cones having themselves a common
base, and we can say that the hypercone is determined by
a closed plane curve and two points neither of which is in
the hyperplane containing the curve and the other point.
Looked at in this way the hypercone is called a double
cone.
33-351 HYPERSURFACE WITH PLANE ELEMENTS 71
A double cone consists, then, of the following classes of
points :
(1) the points of a closed plane curve and the points of its
interior ;
(2) two points not in a hyperplane with the curve, the
interior of the segment formed of these two points, and the
interiors of the segments formed by taking each of these
points with the points of the curve ;
(3) the interiors of the triangles formed by taking each
point of the curve with the two given points ;
(4) the interiors of two cones each formed by taking the
curve with one of the two given points.
The interior of the segment of the two given points is
the vertex-edge of the double cone, and the interior of the
curve is the base. The interiors of the triangles (3) are the
elements j and the two cones (4) are the end-cones.
35. Piano-conical hypersurfaces. A piano-conical hyper-
surface consists of the planes determined by the points of a
plane curve and a line not in a hyperplane with this curve.
The line is the vertex-edge, the curve is the directing-
curve, and the planes are the elements. Each element meets
the plane of the directing-curve in only one point, the point
where it meets the directing-curve itself. There are two
nappes to the hypersurface.
The only case which we have to consider at present is
the case where the directing-curve is a circle.
THEOREM. A hyperplane which contains the directing-
curve of a piano-conical hypersurface and a point of the vertex-
y intersects the hypersurface in a conical surface.
The line containing the vertex-edge of a double cone, and
the curve whose interior is the base, are the vertex-edge
and directing-curve of a piano-conical hypersurface.
72 HYPERPYRAMIDS AND HYPERCONES [MX.
A double cone, or the hypersolid which we call the interior of a
double cone, can be somewhat inaccurately described as cut from one
nappe of a piano-conical hype rsurf ace by two hyperplanes each of
which contains the directing-curve and a point of the vertex-edge.
There are theorems on certain cases of "intersection of
double cones corresponding to the theorems of Art. 32.
CHAPTER II
PERPENDICULARITY AND SIMPLE ANGLES
36. Introductory. From this point the theorems of
ordinary geometry presupposed. We shall now take up
perpendicularity and various kinds of angles in very much
the same way that these subjects are taken up in the
text-books. We shall find the relation of the perpendicular
line and hyperplane analogous to the relation of the per-
pendicular line and plane of three-dimensional geometry,
but a new kind of perpendicularity in the case of two planes
absolutely perpendicular in hyperspace, every line of one
plane through their point of intersection being perpendic-
ular to every line of the other through this point. The
relation of perpendicular planes in a hyperplane now takes
a very simple form, as that of a plane intersecting in a line
each of two absolutely perpendicular planes. This chapter
will conclude with a treatment of perpendicular planes
and hyperplanes, and of hyperplane angles, which are
analogous to dihedral angles.
Chap. Ill will take up what may be called two-dimen-
sional angles, and angles which correspond to polyhedral
angles; and chap. IV will consider questions of order,
symmetry, and motion. After finishing these chapters
we shall return to the polyhedroids and other hypersur-
faces already introduced.
As indicated at the beginning of the preceding section
(Art. 29), the three chapters which follow do not depend
on that section. Indeed, only a few simple facts are needed
from the entire first chapter besides what are already
73
74 LINES PERPENDICULAR TO A HYPERPLANE [n. x.
familiar to the student. These facts are easily stated and
understood, and a study of their application to the theorems
which follow will help us to realize their truth.
Except for the properties of points on a line referred to
in Art. 5, the development of most of chap. I has been
complete in itself, no theorem being based on theorems
not given there. The subjects now to be taken up belong
more particularly to metrical geometry; and we shall
assume the axioms of metrical geometry,* and employ
its terms without special definition. In fact we shall
assume all of the theorems of the ordinary geometry, except,
for the present, those which depend on the axiom of
parallels.!
I. LINES PERPENDICULAR TO A HYPERPLANE
37. Existence of perpendicular lines and hyperplanes.
THEOREM i. The lines perpendicular to a line at a given
point do not all lie in one plane.
PROOF. Every point in hyperspace lies in a plane with
the given line, and in every plane which contains the line
there is a perpendicular to the line at the given point. Now
if these perpendiculars were all in one plane, that plane
and the given line would determine a hyperplane con-
taining all of the planes which contain the line (Art. 20,
Th. i), and so all points of hyperspace. But the points
of hyperspace do not all lie in one hyperplane.
THEOREM 2. A line perpendicular at a point to each of
three non-coplanar lines, is perpendicular to every line through
this point in the hyperplane which the three lines determine.
* For a set of axioms of metrical geometry see R. L Moore, "Sets of Metrical
Hypotheses," referred to on p. 28, or Veblen, " Foundations of Geometry," pp. 27, ff .
f The theorems of plane geometry are true in any plane and of any plane figures,
and the theorems of geometry of three dimensions are true in any hyperplane and
of any hyperplane figures, for all the axioms are assumed to be true in every portion
of hyperspace.
36-38] LINES PERPENDICULAR TO A LINE 75
PROOF. Let m be a line perpendicular at a point O to
each of three non-coplanar lines, a, &, and c, and let d be
any other line through O in the hyperplane determined by
these three lines. The plane of cd will intersect the plane
of ab in a line h (Art. 22, Th.). The line w, being per-
pendicular to a and 6, is perpendicular to h lying in the
plane of aft; and then, being perpendicular to c and A,
it is perpendicular to d lying in the plane of ch*
THEOREM 3. All lines perpendicular to a given line at
a given point lie in a single hyperplane.
PROOF. Let m be the given line and O the given point.
Three non-coplanar lines perpendicular to m at O determine
a hyperplane a such that m is perpendicular to every line
of a through O. Now let a be any line perpendicular to
m at O. The plane of am intersects a in a line b (Art. 27,
Th. i), also perpendicular to m at O. In this plane, then,
we have a line m and the two lines a and b perpendicular to
m at O. Therefore a must coincide with b and lie in the
hyperplane a.
A line intersecting a hyperplane at a point is per-
pendicular to the hyperplane when it is perpendicular to
all lines of the hyperplane which pass through O ; the hyper-
plane is also said to be perpendicular to the line. The point
O is called the foot of the perpendicular.
38. One hyperplane through any point perpendicular to
a given line. Planes in a perpendicular hyperplane.
THEOREM i. At any point of a line there is one and only
one hyperplane perpendicular to the line.
This follows immediately from the theorems of the pre-
ceding article.
* A line m through a point O, perpendicular to each of two lines intersecting at
O, is perpendicular to every line through O in the plane which the two lines determine.
This is always true, for the plane and the line m lie in one hyperplane.
76 LINES PERPENDICULAR TO A HYPERPLANE [n. I.
THEOREM 2. Through any point outside of a line passes
one and only one hyperplane perpendicular to the line.
THEOREM 3. A line perpendicular to a hyperplane is
perpendicular to every plane of the hyperplane passing
through the foot of the line; and every plane perpendicular
to a line at a point lies in the hyperplane perpendicular to
the line at this point.
THEOREM 4. // a line and plane intersect, a line per-
pendicular to both at their point of intersection is perpendicular
to the hyperplane determined by them; or if two planes in-
tersect in a line, a line perpendicular to both at any point of
their intersection is perpendicular to the hyperplane deter-
mined by them.
39. Lines perpendicular to a given hyperplane.
THEOREM i. At a given point in a hyperplane there is
one and only one line perpendicular to the hyperplane.
PROOF. Consider three non-coplanar lines lying in
the hyperplane and passing through the given point. The
three hyperplanes perpendicular to these lines at the point
have in common at least a line (Art. 27, Th. 2 and remark),
and any such line must be perpendicular to the given
hyperplane because perpendicular to the three lines.
If there were two lines perpendicular to the hyperplane
at the given point, they would both be perpendicular to the
line in which their plane intersects the hyperplane. We
should have in a plane two lines perpendicular to a third
at the same point, which is impossible.
THEOREM 2. Two lines perpendicular to a hyperplane
lie in a plane.
PROOF. Any two lines lie in a hyperplane (Art. 20, Th.
2 (2)), and a hyperplane containing the two given lines
intersects the given hyperplane in a plane to which the
38, 39)
ONLY ONE THROUGH A POINT
77
lines are both perpendicular (Art. 38, Th. 3). Therefore,
as proved in geometry of three dimensions, the two lines
lie in a plane.*
THEOREM 3. Through any point outside of a hyper plane
passes one and only one line perpendicular to the hyperplane.
,P
o
-\
PROOF. Let P be the point, and a some line perpendic-
ular to the hyperplane. If a does not pass through P,
a and P determine a plane which intersects the hyperplane
in a line c, and in this plane there is a line b through P per-
pendicular to c, intersecting c at a point O. Let b f be the
line perpendicular to the hyperplane at O. a and 6' lie
in a plane (Th. 2), which is the plane containing a and the
point 0. But the plane determined by a and P passes
through O. Therefore b' lies in the plane determined by
a and P, and in this plane is perpendicular to c at 0. In
other words, b' coincides with 6, and b must be perpendic-
ular to the hyperplane.
If there were two lines through P perpendicular to the
hyperplane, we should have two lines through P per-
pendicular to the line which passes through their feet, and
this is impossible.
* Euclid proves this in connection with the theorem that two lines perpendic-
ular to a plane are parallel (Elements, Bk. XI, Prop. 6). The proof does not, how-
ever, depend on the axiom of parallels.
78 LINES PERPENDICULAR TO A HYPERPLANE [XL i.
40. Projection of a point upon a hyperplane. Distance
from a hyperplane. The projection* of a point upon a
hyperplane is the foot of the perpendicular from the point
to the hyperplane. The perpendicular may be called the
projecting line.
THEOREM i.f The distance from any point outside of a
hyperplane to its projection upon the hyperplane is less than
the distance from the point to any other point of the hyperplane.
The distance between a hyperplane and a point outside of
the hyperplane is the distance between the point and its
projection upon the hyperplane.
COROLLARY.J // the distance between two points is less
than the distance of one of them from a hyperplane, they lie
on the same side of the hyperplane in hyperspace.
THEOREM 2. Given any point P outside of a hyperplane,
and O its projection upon the hyperplane, then any two points
of the hyperplane equally distant from P will be equally dis-
tant from O, and any two points equally distant from O will
be equally distant from P; and if two points of the hyperplane
are unequally distant from either P or O, that point which is
nearer to one of them will be nearer to the other.
PROOF. The perpendicular PO, and the lines through P
and any two points of the hyperplane, lie in a second hyper-
plane which intersects the given hyperplane in a plane a.
The perpendicular is perpendicular to a, and the three
lines intersect a in the points where they intersect the given
hyperplane. The theorem is therefore a theorem in the
three-dimensional geometry of the second hyperplane.
* That is, orthogonal projection. We sometimes have cases of projection by
lines through a point (see, for example, the proof in Art. 25), but when we use the
word projection alone we shall mean by it orthogonal projection.
t The two theorems of this article are true at least when the distances referred
to are "restricted" (see Introduction, p. 19, and reference given there in the foot-
note).
I Used in the proof of the first theorem of Art. 96*
40,
PROJECTION OF A LINE
79
From any point of our three-dimensional space we can move off
into hyperspace without passing through or approaching any other
point of our space. A point at the centre of a sphere, for example,
moving off on a line perpendicular to our space, would not approach
any point of the sphere. It would, indeed, be moving farther and
farther from all points of the sphere, the distances from all of these
points being the same, and all increasing at the same rate. An object
completely enclosed within a surface in our three-dimensional space
would be entirely free if it could pass off in any direction out of this space.
41. Projection of a line upon a hyperplane. Angle of a
half-line and hyperplane. The projection of any figure
upon a hyperplane consists of the projections of its points.
THEOREM i . When a line and hyperplane are not per pen-
dicular, the projection of the line upon the hyperplane is a
line or a part of a line.
PROOF. Let m be the line, and a the perpendicular
projecting some point of m upon the hyperplane. Any
other perpendicular b projecting a point of m upon the hy-
perplane, lies in a plane with a (Art. 39, Th. 2), and this
plane, containing two points of m, is the plane determined
by m and a. Therefore, all the perpendiculars projecting
points of m upon the hyperplane lie in the plane determined
by m and a, and the projection of m upon the hyperplane
is the same as its projection upon the line in which this
plane intersects the hyperplane.*
* In plane geometry, when two lines are not perpendicular, the projection of
one upon the other is the latter line itself or a single connected part of it. That is, if
two given points are the projections on a
line a of points of another line m, any point
between them will be the projection of a
point of m. For the perpendiculars at the
three points of a do not intersect one
another, and therefore it follows that the
perpendicular which lies between the other
two must intersect m (Art. 12, Th. i) ;
that is, any point of a between the projec-
tions of two points of m is itself the projec-
tion of a point of m.
80 ABSOLUTELY PERPENDICULAR PLANES [u. n.
COROLLARY. When a half -line drawn from a point O of a
hyper plane does not lie in the hyperplane and is not perpen-
dicular to it, its projection upon the hyperplane is a half-
line drawn in the hyperplane from O, or the interior of a seg-
ment which has O for one of its points.
THEOREM 2. When a half -line drawn from a point O of
a hyperplane does not lie in the hyperplane and is not per-
pendicular to it, the angle which it makes with the half-line
drawn from O containing its projection is less than the angle
which it makes with any other half-line drawn in the hyper-
plane from O.
When a half-line drawn from a point O of a hyperplane
does not lie in the hyperplane and is not perpendicular to it,
the angle which it makes with its projection is called the
angle of the half-line and hyperplane. A half-line drawn
from a point O of a hyperplane perpendicular to the hyper-
plane is said to make a right angle with the hyperplane.
II. ABSOLUTELY PERPENDICULAR PLANES
42. Existence of absolutely perpendicular planes.
THEOREM i. A plane has more than one line perpendicu-
lar to it at a given point.
For the plane is the intersection of different hyperplanes,
and the lines in these hyperplanes perpendicular to the
plane at the given point must be different lines (see Art. 20).
THEOREM 2. Two lines perpendicular to a plane at a
given point determine a second plane, and the two planes are
so related that every line of one through the point is perpendicu-
lar to every line of the other through the point*
We cannot prove without the axiom of parallels that every point of a is the pro-
jection of a point of m, for the perpendiculars at some points of a may not intersect
m at all. The projection of m may be the entire line a, but it may be only a half-
line of a, or the interior of a definite segment of this line,
* See foot-note, p 75.
41-43) LINES PERPENDICULAR TO A PLANE 8l
THEOREM 3. All the lines perpendicular to a plane at
a given point lie in a single plane.
PROOF. Let a be the given plane, and the given point.
Two lines perpendicular to a at O determine a plane in
which every line through is a line perpendicular to a.
Now let a be any line perpendicular to a at O. The hyper-
plane determined by a and a intersects /3 in a line b (Art. 27,
Th. i), also perpendicular to a at O. But in a hyperplane
containing a. only one line can be perpendicular to a at O.
Therefore a coincides with b and lies in (3.
Two planes having a point in common are absolutely
perpendicular when every line of one through that point
is perpendicular to every line of the other through that
point.
These planes have only the point in common, and do not intersect
in a line. We can never see both planes in a single hyperplane like
the space in which we live. The most that we could see would be
one plane and a single line of the other.
43. One plane through any point absolutely perpendicu-
lar to a given plane.
THEOREM i. At any point of a plane there is one and
only one plane absolutely perpendicular to the given plane.
This follows immediately from the theorems of the pre-
ceding article.
A plane and a point outside of the plane lie in one and
only one hyperplane. Through any such point, therefore,
by three-dimensional geometry, passes one and only one
line perpendicular to the plane ; and the projection of the
point upon the plane, as in geometry of three dimensions, is
the foot of this perpendicular. A point that lies in a given
plane is its own projection upon the plane.
82 ABSOLUTELY PERPENDICULAR PLANES [n. u.
THEOREM 2. Through any point outside of a plane passes
one and only one plane absolutely perpendicular to the given
plane.
PROOF. Let P be the point, and O its projection upon the
plane. The absolutely perpendicular plane at O will then
contain the perpendicular line OP, and therefore P. More-
over, any absolutely perpendicular plane containing P
must contain a perpendicular line through P, and there is
only one such line.
44. Planes absolutely perpendicular to planes which
intersect in a line.
THEOREM i . // two planes intersect in a line and so lie
in a hyper plane, their absolutely perpendicular planes at
any point of their intersection intersect in a line and lie in
a hyperplane.
PROOF. Let a. and /3 be two planes intersecting in a line
a, and let a and /?' be the planes absolutely perpendicular
to a. and j3 respectively at a point O of a. a f and 0' are,
then, both perpendicular to the line a at 0, and lie in a
hyperplane perpendicular to a at (Art. 38, Th. 3). There-
fore a and ]8' intersect in a line (Art. 22, Th.).
THEOREM 2. // three planes have a line in common ,
their absolutely perpendicular planes at any point of this
line lie in a hyperplane; and if three planes lie in a hyper-
plane and have a point in common, their absolutely perpendicu-
lar planes at this point have a line in common.
This is proved by Art. 38, Th. 3, the line being perpen-
dicular to the hyperplane.
COROLLARY. // three planes have a line in common and
lie in a hyperplane, their absolutely perpendicular planes at
any point of this line have a line in common and lie in a hyper-
plane.
43~4S] TWO PLANES IN A HYPERPLANE 83
45. Planes absolutely perpendicular to a plane at differ-
ent points.
THEOREM. Two planes absolutely perpendicular to a
third lie in a hyperplane.
PROOF. Let a and ]8 be the planes absolutely perpendicu-
lar to a given plane 7 at two points and 0'. Let c be the
line OO', and let a and b be the lines perpendicular at O
and O f to the hyperplane determined by a and c. a and b
lie in a plane (Art. 39, Th. 2), and this plane is 7, for a,
being perpendicular to a, must lie in 7, and only one plane
can contain a and the point O'. b is then perpendicular to
|8 ; so that lies with a in the hyperplane to which b is per-
pendicular (Art. 38, Th. 3), the hyperplane determined by
a and c.
COROLLARY. All the planes absolutely perpendicular to
a plane at the points of any line of it, lie in a hyperplane.
It should be noted that the figure given here is merely to serve as
a diagram, and does not represent the actual appearance of two planes
absolutely perpendicular to a third. The most that we could ever
see in any one hyperplane would be the plane 7 and a line in each of
the other two planes, or the two planes a and and the single line c
of 7.
84 ABSOLUTELY PERPENDICULAR PLANES [n. n.
46. Projection of a line upon a plane. As in other cases
of projection, the projection of any figure upon a plane con-
sists of the projections of its points (Art. 41).
THEOREM i. The perpendiculars projecting the points
of a line upon a plane do not lie in a single plane, unless the
line itself lies in a hyperplane with the plane upon which it
is projected.
PROOF. If any two of the perpendiculars were in a plane,
that plane, having two points in common with the given
plane, would intersect the latter in a line and lie with it
in a hyperplane (Art. 20, Th. 2 (4)). Therefore, the given
line would lie in a hyperplane with the plane upon which it
is projected.
When two planes are absolutely perpendicular at a point
0, all the points of one project upon the other in the single
point O. We may regard projection upon a plane as made
by planes absolutely perpendicular to it, just as in geometry
of three dimensions we may regard projection upon a line
as made by planes perpendicular to the line. We shall
sometimes speak of projecting planes and think of a point
as projected upon a plane in this way.*
THEOREM 2. The projection of a line upon a plane is a
line or a part of a line, unless the given line lies in a plane
absolutely perpendicular to the given plane.
PROOF. Let m be the given line and y the given plane.
Let a be the plane absolutely perpendicular to 7 which
projects some point of m upon 7. Any other plane pro-
jecting a point of m upon 7 lies in a hyperplane with a
(Art. 45, Th.). But this hyperplane containing two
* In a hyperplane, on the other hand, where the perpendiculars projecting points
of a line upon a plane always lie in a single plane, the latter is sometimes called the
projecting plane, projecting the line as a whole. These two uses of the expression
"projecting plane" should be carefully distinguished.
46,47] PERPENDICULARITY AS INTERSECTION 85
points of m, is the hyperplane determined by m and a;
and the projection of m upon 7 is the same as its projection
upon the line in which this hyperplane intersects 7.*
The projecting lines form a curved surface which contains the
given line and its projection and lies in the hyperplane of these two
lines.
The student may prove the following theorem :
Given any two lines, we can pass through either a plane upon
which it will be the projection of the other.
III. SIMPLY PERPENDICULAR PLANES
47. Planes intersecting in lines two absolutely per-
pendicular planes. Two planes are perpendicular, or simply
perpendicular, when they lie in one hyperplane and in this
hyperplane form right dihedral angles. Each contains a
line in the other and lines perpendicular to the other, f
THEOREM i. A plane perpendicular to one of two abso-
lutely perpendicular planes, and passing through the point
where they meet, is perpendicular to the other.
PROOF. Let a and a' be two absolutely perpendicular
planes meeting in a point O, and let /3 be a plane passing
through and perpendicular to a. Two planes which are
perpendicular lie in a hyperplane, and, by a theorem of
geometry of three dimensions, a line in one perpendicular
to their intersection is perpendicular to the other. There
* See foot-note, p. 79.
t In a sense two planes are perpendicular whenever one of them contains a line
perpendicular to the other. Such planes have also been called half -perpendicular,
in distinction from absolutely perpendicular planes, which are then said to be com-
pletely perpendicular (see Schoute, M ehrdimensionale Geometric, vol. I, p. 49).
Two planes in a hyperplane forming a right dihedral angle might be described as
Perpendicular in a hyperplane. We shall find that such planes are of great impor-
tance in our geometry, and we shall, at least for the present, apply to them the word
perpendicular. Perpendicular, then, has with us the same meaning as in the ordi-
nary geometry ; and the only new concept with which we have to become famil-
iar at this point is that of absolute perpendicularity.
86 SIMPLY PERPENDICULAR PLANES [n. ra.
is, then, a line in /? perpendicular to a at O, that is, a line
common to ft and the absolutely perpendicular plane a',
so that ft intersects & in a line and lies with a' in a hyper-
plane. Now the line in which ft intersects a, like all the
lines of a through 0, is perpendicular to a! ; so in the hy-
perplane of ft and a! we have a line lying in ft perpendicular
to a'. Therefore ft is perpendicular to a'.
THEOREM 2. A plane intersecting in a line each of two
absolutely perpendicular planes, is perpendicular to both.
PROOF. Let a and a be two absolutely perpendicular
planes meeting in a point O, and let ft be a plane intersecting
both of these planes in lines. These two lines and the plane
ft pass through 0; otherwise the hyperplane determined
by ft and O would contain both a and a:', which is impos-
sible. Now the line in which ft intersects a! is that line
which is perpendicular to a at O in the hyperplane deter-
mined by a and ft. In this hyperplane, then, we have a
line lying in ft and perpendicular to a. Therefore is per-
pendicular to a.
In the same way, or by Th. i, we prove that ft is
perpendicular to a'.
THEOREM 3. // two planes are perpendicular, their ab-
solutely perpendicular planes at any point of their intersec-
tion are perpendicular.
PROOF. Let a and ft be two perpendicular planes, and
let a 1 and ft' be their absolutely perpendicular planes at
a point O of their intersection, a, being perpendicular
to /?, is perpendicular to 0' ; and then j3', being perpendicu-
lar to a, is perpendicular to a' .
The student may prove :
If a plane is perpendicular to one of two absolutely perpendic-
ular planes, and contains a point of the other, it is perpendicular to
both.
47-491 PLANES INTERSECTING IN A LINE 87
48. The different possible positions of two pairs of ab-
solutely perpendicular planes at a point. Let two pairs *
of absolutely perpendicular planes have their intersection
point in common. Then
(1) they may have only the point in common ;
(2) each plane of one pair may intersect in a line one
plane of the other pair ; or
(3) each plane of one pair may be perpendicular to both
planes of the other pair.
In the last case the four lines of intersection are mutually
perpendicular, and, taken two at a time, determine also a
third pair of absolutely perpendicular planes. The planes
of each of the three pairs are then perpendicular to all the
planes of the other two pairs.
We have, in fact, four mutually perpendicular lines, any
two of them determining a plane, and any three a hyper-
plane. Each line is perpendicular to the hyperplane de-
termined by the other three. In each hyperplane are three
of the six planes, three mutually perpendicular planes
forming a trirectangular trihedral angle.
We shall call this figure a rectangular system (see Art. 71).
49. Common perpendicular planes of two planes inter-
secting in a line.
THEOREM. Two planes which intersect in a line have at
any point O of that line one and only one pair of common
perpendicular planes.
PROOF. Let a and |8 be the two given planes, and let
a! and /?' be their absolutely perpendicular planes at O.
A plane which is perpendicular to a and /S at is a plane
which intersects these four planes in lines. The planes
* In general, we shall use the word "pair" in speaking of two planes only when
we have in mind two planes absolutely perpendicular to each other. Thus we shall
speak in this way of a pair of planes perpendicular to a given plane, and of a pair of
common perpendicular planes when we have two given planes.
88 SIMPLY PKRPKNDICULAR PLANKS IH. m.
a. and ft intersect in a line and lie in a hyperplane, and the
planes a' and & intersect in a line and lie in a hyperplane
(Art. 44, Th. i). The line of intersection of a! and ft' is
perpendicular to the hyperplane of a and /3, and the line of
intersection of a and ft is perpendicular to the hyperplane
of a' and/3' (Art. 38, Th. 4).
In a hyperplane two intersecting planes have a common
perpendicular plane at any point of their intersection,
perpendicular to the intersection. Thus we have one plane
perpendicular to a and ft at O lying in their hyperplane.
This plane cannot contain the intersection of a' and #',
for the intersection of a and ft' is perpendicular to the hy-
perplane of a and ft. The plane must therefore intersect
a' and ft' in separate lines, and lie also in their hyperplane.
In other words, the common perpendicular plane with which
we are familiar in the case of two intersecting planes,
perpendicular to their intersection at a point 0, is the plane
of intersection of their hyperplane with the hyperplane of
their absolutely perpendicular planes at O.
A second common perpendicular plane is the plane ab-
solutely perpendicular to the plane already found (Art. 47,
Th. i). This plane contains the lines of intersection of a
and ft and of a! and ft' and may be regarded as determined
by them.
It remains to prove that these two planes are the only
planes perpendicular to a and ft at 0.
Any plane intersecting a and ft in lines must pass through
their line of intersection or lie entirely in their hyperplane,
and any plane intersecting a and ft' in lines must pass
through their line of intersection or lie entirely in their
hyperplane (Art. 20, Th. i). But, as we have seen, a plane
lying entirely in the hyperplane of two of these planes can
not pass through the line of intersection of the other two.
Any common perpendicular plane, therefore, must lie in
4Q] PLANES INTERSECTING IN A LINE 89
both hyperplanes or pass through both lines of intersec-
tion : that is, it must be one of the two planes already found.
These are therefore the only planes which can be perpendic-
ular to a and /3 at O.
When the two given planes are perpendicular, the four
planes have all together four lines of intersection, and there
are four different hyperplanes which contain two of the
four planes. The planes can be associated in two ways so
that two planes shall intersect in a line and the other two
planes be their absolutely perpendicular planes. But the
plane determined by the two lines of intersection in one
case is the plane of intersection of the two hyperplanes in
the other case. Therefore we get only two planes perpen-
dicular to the two given planes.
That is, if a and /3 are perpendicular, so that a is also
perpendicular to /8' and /3 to a', then the plane determined
by the intersection of a and /3 and the intersection of a'
and /3' is the plane of intersection of the hyperplane deter-
mined by a and ft' and the hyperplane^determined by a'
and j8 ; and the plane determined by the intersection of a
and /3' and the intersection of a! and & is the plane of
intersection of the hyperplane determined by a and ft and
the hyperplane determined by a' and /3'. Thus we have
only two planes intersecting in lines the four planes, a, j8,
OL ', /3', and so perpendicular to the two given planes a
and/3.
The four planes and the two common perpendicular
planes in this case are the six planes of a rectangular sys-
tem (Art. 48).
COROLLARY. // each of two planes having a common point
O intersects in a line the plane absolutely perpendicular to
the other at O, then these planes have one and only one pair
of common perpendicular planes.
pO PERPENDICULAR PLANES AND HYPERPLANES [n. iv.
The student may prove the following theorem :
Given two planes with a point O in common, through any line con-
taining O in one of these planes can be passed a plane perpendicular
to this plane and intersecting the other in a line. In other words,
any line of one plane through is the projection upon this plane of a
line of the other.
If the two planes intersect in a line and the given line is this line
of intersection, an infinite number of planes can be drawn satisfying
the conditions of the theorem. Or, if one of the two planes lies in a
hyperplane with the plane absolutely perpendicular to the other at
O, and the given line is the intersection of the other given plane with
this hyperplane, there will be an infinite number of these planes.
Except in these two cases there is only one plane satisfying the
conditions of the theorem.
IV. PERPENDICULAR PLANES AND HYPERPLANES
50. Perpendicular planes and hyperplanes. The planes
perpendicular or absolutely perpendicular to planes lying
in the hyperplanes. A plane intersecting a hyperplane is
perpendicular to the hyperplane at a point of their intersec-
tion, if the plane absolutely perpendicular to the given plane
at this point lies in the hyperplane. The hyperplane is
also said to be perpendicular to the plane.
THEOREM i. If a plane is perpendicular to a hyperplane
at one point of their intersection, it is perpendicular all along
this line.
For the line of intersection and the plane absolutely per-
pendicular to the given plane at the given point determine
the hyperplane (Art. 20, Th. 2 (i) ), which, therefore, con-
tains the planes absolutely perpendicular to the given
plane at all points of the line of intersection (Art. 45, Th.
and Cor.).
THEOREM 2. If a plane a is perpendicular to a hyper-
plane along a line c, any plane in the hyperplane perpendic-
ular to c is absolutely perpendicular to a, and any plane ab-
49,5o] PLANES IN THE HYPERPLANES 91
solutely perpendicular to a, through a point of the hyperplane
lies entirely in the hyperplane.
PROOF. The planes absolutely perpendicular to ot at
the points of c lie in the hyperplane, and in the hyperplane
they are planes perpendicular to the line c. Now in the
hyperplane only one plane can be perpendicular to a given
line at a given point. Any plane in the hyperplane, there-
fore, perpendicular to the line c, must be one of the planes
absolutely perpendicular to a. at the points of c.
On the other hand, if a plane a! is given as passing through
a point of the hyperplane and absolutely perpendicular to
a, we can draw a plane /3 in the hyperplane through the same
point perpendicular to c. )3 is then absolutely perpendicu-
lar to a, and a must coincide with /3 and lie entirely in the
hyperplane, since we cannot have two planes through a
point absolutely perpendicular to a given plane (Art. 43,
Ths.).
THEOREM 3. If a plane a is perpendicular to a hyperplane
along a line c, any plane in the hyperplane passing through c
is perpendicular to a, and any plane perpendicular to a pass-
ing through c, or through any line which lies in the hyper-
plane and is not itself perpendicular to c, lies entirely in the
hyperplane.
PROOF. In the first case, the given plane, passing
through c, intersects a in this line ; and, lying in the given
hyperplane, it intersects in a line the plane absolutely per-
pendicular to a at any point of c. It is therefore perpendic-
ular to a.
On the other hand, if the plane is given as perpendicular
to a and passing through c, or through any line b which lies
in the hyperplane and is not itself perpendicular to c, it
must intersect in a line the plane absolutely perpendicular
to a through any point of the given line, a plane which lies
92 PERPENDICULAR PLANES AND 1IYPERPLANES [11. iv.
entirely in the hyperplane (Th. 2). It must, therefore,
contain two lines of the hyperplane and lie entirely in it
(Art. 20, Th. i).
THEOREM 4. // two hyper planes are perpendicular to
a plane at a point O, they intersect in the absolutely perpen-
dicular plane at O.
For the absolutely perpendicular plane lies in both hy-
perplanes, by hypothesis.
51. Lines lying in the plane and perpendicular to the
hyperplane, or in the hyperplane and perpendicular to the
plane.
THEOREM i. If a plane is perpendicular to a hyperplane,
any line in the plane perpendicular to their intersection is
perpendicular to the hyperplane, and any line perpendicular
to the hyperplane through a point of the plane lies entirely in
the plane.
PROOF. In the first case, the line, lying in the given
plane, is perpendicular not only to the intersection, but also
to the absolutely perpendicular plane at the point where
it meets the intersection. The line is therefore perpen-
dicular to the hyperplane (Art. 38, Th. 4).
On the other hand, if the line is given as passing through
a point of the plane and perpendicular to the hyperplane,
we can draw a line in the plane through the same point
perpendicular to the intersection, and the two lines must
coincide, since they are both perpendicular to the hyper-
plane (Art. 39, Ths. i and 3).
COROLLARY. // two planes are perpendicular to a hyper-
plane at a point 0, they intersect in a line which is perpen-
dicular to the hyperplane at O.
For, by the theorem, the line perpendicular to the hy-
perplane at O lies in both planes.
50-52] ONLY ONE THROUGH A LINE 93
THEOREM 2. If a plane is perpendicular to a hyper plane,
any line in the hyper plane perpendicular to their intersection
is perpendicular to the plane, and any line perpendicular
to the plane through a point of the hyper plane lies entirely
in the hyperplane.
PROOF. In the first case, the line, lying in the hyper-
plane and perpendicular to the intersection, lies also in
the plane which in the hyperplane is perpendicular to the
intersection at the same point. But this plane is absolutely
perpendicular to the given plane (Art. 50, Th. 2).
Therefore the given line is perpendicular to the given
plane.
On the other hand, if the line is given as passing through
a point of the hyperplane and perpendicular to the given
plane, it lies in the absolutely perpendicular plane which
passes through the same point, and therefore in the hyper-
plane (same reference).
52. Through any line one plane perpendicular to a given
hyperplane, or one hyperplane perpendicular to a given
plane.
THEOREM i. If a line is perpendicular to a hyperplane,
any plane passing through the line is perpendicular to the
hyperplane.
For the plane absolutely perpendicular to the given plane
at the point where the line meets the hyperplane is per-
pendicular to the line, and therefore lies in the hyperplane
(Art. 38, Th. 3).
THEOREM 2. Through a line not perpendicular to a hy-
perplane passes one and only one plane perpendicular to the
hyperplane.
PROOF. The plane perpendicular to the hyperplane
along the projection of the given line upon the hyperplane,
contains the given line (Art. 51, Th. i) ; and we cannot
94 PERPENDICULAR PLAJNEb AMD HYFfcKFLAIMfc& in. IV.
have two perpendicular planes containing the given line,
for that would make the line itself perpendicular to the hy-
perplane, which is contrary to hypothesis (same reference,
and Cor.).
THEOREM 3. Through a line not lying in a plane abso-
lutely perpendicular to a given plane, passes one and only
one hyperplane perpendicular to the given plane.
PROOF. The hyperplane perpendicular to the given plane
along the projection of the given line upon the plane, con-
tains the given line (Art. 51, Th. 2) ; and we cannot have
two perpendicular hyperplanes containing the given line,
for that would put the line itself into a plane absolutely
perpendicular to the given plane, which is contrary to hy-
pothesis (Art. 50, Th. 4).
53. Planes with linear elements all perpendicular to a
hyperplane.
THEOREM. Given two planes not in one hyperplane, if
any two of their linear elements (Art. 27) have a common
perpendicular line, they all have a common perpendicular
hyperplane, to which the two given planes are also perpen-
dicular.
PROOF. Let a. and ft be the two given planes, and let
a be one of the linear elements in a. and b one of the linear
elements in /?. If a and b have a common perpendicular
line, the hyperplane perpendicular to either of these lines
at the point where it meets the perpendicular line, is per-
pendicular to the plane of the two (Art. 52, Th. i), and
therefore perpendicular to the other (Art. 51, Th. i).
This hyperplane is also perpendicular to the planes a and
]8, as well as to every plane containing a or b (Art. 52,
Th. i). Now any other linear element in j8 is the inter-
section of |8 with a plane through a, and any other linear
element in a is the intersection of a with a plane through b.
52-54] HYPERPLANE ANGLES 95
The hyperplane is therefore perpendicular to all of these
elements (Art. 51, Th. i, Cor.).
If two elements in one of the two given planes are given
as having a common perpendicular line, then the perpen-
dicular hyperplane is perpendicular to the plane in which
they lie and to every plane containing either one of them.
But any element of the other plane is the intersection of
two planes through the two given elements, and is there-
fore, as before, perpendicular to the hyperplane. Thus
the hyperplane is perpendicular to elements in both planes,
and so perpendicular to all the elements and to the two
given planes.*
V. HYPERPLANE ANGLES
54. Definition. Interior. Plane angles. A hyper-
plane angle consists of two half-hyperplanes having a com-
mon face but not themselves parts of the same hyper-
plane, together with the common face.
The common face is the face of the hyperplane angle, and
the two half-hyperplanes are the cells.
The interior of a hyperplane angle consists of the interiors
of all segments whose points are points of the hyperplane
angle, except those segments whose interiors also lie in
the hyperplane angle ; that is, the interior of a hyperplane
angle consists of the interiors of those segments which have
a point in each cell. The hyperplane angle divides the
rest of hyperspace into two regions, interior and exterior
* The linear elements of two planes not in a hyperplane will all pass through a
point common to the two planes, or will all be perpendicular to a hyperplane per-
pendicular to the two planes, except in the Hyperbolic Non-Euclidean Geometry,
where there are lines in a plane which do not intersect and do not have a common
perpendicular, namely, the parallel lines of this geometry It follows from the
theorem that in this geometry the linear elements of two planes are all parallel if
two of them arc parallel. They will then be axes of a set of boundary-hyper surf aces,
and their geometry will correspond to the Euclidean three-dimensional geometry
of boundary-hypersurfaces (see the author's Non-Euclidean Geometry, p. 52).
gO HYPERPLANE ANGLES In. v.
to the hyperplane angle. Each cell of a hyperplane angle
lies on one side of the hyperplane of the other cell ; and the
portion of hyperspace which lies on that side of the hyper-
plane of each cell oh which lies the other cell, lies between
the two cells and constitutes the interior of the hyper-
plane angle.
Two half-lines drawn from a point O in the face of a
hyperplane angle, one in each cell, and each perpendicu-
lar to the face, are the sides of an angle which is called the
plane angle at of the hyperplane angle.
THEOREM i. The plane absolutely perpendicular at a
point O to the face of a hyperplane angle, intersects the hyper-
plane angle in the plane angle at O.
THEOREM 2. A hyperplane perpendicular to the face of
a hyperplane angle intersects the hyperplane angle in a dihe-
dral angle which at any point of its edge has the same plane
angle as the hyperplane angle.
PROOF. The intersection consists of two half^planes
with a common edge lying in the face of the hyperplane
angle. The plane absolutely perpendicular to the face
at any point of this edge, lies in the perpendicular hyper-
plane, and in this hyperplane is perpendicular to the edge
of the dihedral angle formed by the two half-planes (see Art.
50, Th. 2). This plane intersects the dihedral angle in the
same angle in which it intersects the hyperplane angle
an angle which is, therefore, the plane angle at O of both.
THEOREM 3. Two hyperplane angles are congruent if a
plane angle of one is equal to a plane angle of the other.
PROOF. We have given two hyperplane angles with
the plane angle at a point of one equal to the plane angle
at a point O' of the other. If we make these angles coincide,
the faces of the two hyperplane angles will coincide, each
face being absolutely perpendicular to the plane of the
54] THEIR PLANE ANGLES 97
common plane angle at the vertex of this angle. Now
one side of the common plane angle lies in one cell of each
of the hyperplane angles, and the other side lies in the other
cell of each. The two cells of one hyperplane angle, there-
fore, coincide with the two cells of the other, and the two
hyperplane angles coincide throughout and are congruent
(Art. 20, Th. 2 (i) and Th. 3).
If we make a cell of one of these hyperplane angles coin-
cide with a cell of the other, with the points O and O f
coinciding and with the other cells on the same side of the
hyperplane of the common cell, the hyperplane angles
will coincide throughout. For the planes of the given
plane angles will be absolutely perpendicular to the plane
of the common face at the same point and will coincide,
and the plane angles themselves will coincide, having one
side in common and the other sides in the same half-plane
with respect to the line of the common side.
When two hyperplane angles coincide, every point in the
face of one coincides with a point in the face of the other,
and the plane angles at any such common point coincide
and are equal.
THEOREM 4. The plane angle of a hyperplane angle is
the same at all points of the face.
PROOF. The absolutely perpendicular planes at any two
points of the face are in a perpendicular hyperplane (Art. 45,
Th.), in which the corresponding angles are two plane
angles of the same dihedral angle. Therefore the two
plane angles are equal.*
* The theorem that the plane angle of a dihedral angle is the same at all points
of the edge is usually made to depend on the axiom of parallels. It can be proved,
however, without the use of this axiom :
Let A and B be two points of the edge, and let be the point midway between
them. We can turn the dihedral angle so that the half-line OA will fall upon the
original position of the half -line OB and the half -line which bisects the plane angle
at O upon its own original position. This will make the plane angle at O coincide
98 HYPERPLANE ANGLES [n. v.
Hereafter we shall speak of the plane angle of a hyper-
plane angle without thinking of it as located at any particu-
lar point of the face.
COROLLARY. Two hyper plane angles which are congruent
in any position will always coincide as soon as they have a
common cell and the other cells lie on the same side of the
hyper plane of this common cell.
55. The hyperplane angle as a magnitude. Supple-
mentary hyper plane angles are those which can be placed
so as to have one cell in common while their other cells
are opposite half-hyperplanes. Each of them is then the
supplement of the other. A right hyperplane angle is one
which is congruent to its supplement. The hyperplanes
of the cells of a right hyperplane angle are said to be per-
pendicular.
If two hyperplane angles have a common face and the
cells of one lie in the interior of the other, or if they have
one cell in common while one cell of one lies in the interior
of the other, then the interior of one hyperplane angle is
a part of the interior of the other. We shall find it con-
venient to speak of one hyperplane angle as a part of the
other and as less in magnitude.
Let two hyperplane angles be placed so as to have one
cell in common while their other cells and their interiors
lie on opposite sides of the hyperplane of this common cell.
Then, if either hyperplane angle is less than the supple-
ment of the other, these two cells and the common face
will form a hyperplane angle which we can call the sum of
the given hyperplane angles.
as a whole with its original position and the dihedral angle as a whole will occupy
its original position. Then the plane angle at A will take the former position of
the plane angle at B; and the two plane angles must be equal.
This theorem should have been proved in the author's A' on- Euclidean Geometry at
some point before Art. 1 1, p. 26, as has been pointed out by Professor J L. Coolidge.
54-56] THEIR MEASUREMENT 99
THEOREM i. Two supplementary hyper plane angles have
supplementary plane angles; a right hyperplane angle has a
right plane angle; the smaller of two unequal hyperplane
angles has the smaller plane angle; and the plane angle of
the sum of two hyperplane angles is the sum of their plane
angles. The converse theorems are also true.
THEOREM 2. // we divide the plane angle of a hyperplane
angle into any number of equal parts, the lines of division will
determine additional cells by means of which the given hyper-
plane angle is divided into the same number of equal parts;
in particular, given a hyperplane angle a, any other hyper-
plane angle may be divided into a sufficient number of equal
parts so that one of these parts shall be less than a.
Thus we can build up a complete theory of the measure-
ment of hyperplane angles, taking any particular one as
the unit of measure.
The student will notice the analogy of the hyperplane angle to the
dihedral angle of the ordinary geometry. The theory of measure-
ment is identical in the two cases, and so does not involve any new
conceptions.
We may consider the class of half-hyperplanes with a common
face as having a geometry, a one-dimensional geometry, in which
these half-hyperplanes are the elements (compare " Point Geometry,"
Art. 64, and "Edge Geometry," Art. 78, and see Introduction, p.
15). Disregarding the common face, we may say in this geometry
that the hyperplane angle consists of two half-hyperplanes. The
hyperplane angle corresponds to the segment, and the magnitude of
the hyperplane angle to the length of the segment.
56. Hyperplane angles measured by their plane angles.
THEOREM. Two hyperplane angles are in the same ratio
as their plane angles, and the hyperplane angle may be meas-
ured by its plane angle.
PROOF. If we divide the plane angle of a hyperplane
angle into some number of equal parts, the half -lines of
100 HYPERPLANE ANGLES [n. v.
division, taken with the face of the hyperplane angle, will
determine half-hyperplanes which will divide the hyper-
plane angle (its interior) into the same number of equal
parts. We can therefore prove in the usual way that
hyperplane angles are proportional to their plane angles,
first when the plane angles are commensurable and then
(by the method of limits or some equivalent method) when
the plane angles are incommensurable.
Now a right hyperplane angle has a right plane angle
(Art. 55, Th. i). Therefore, the measure of the hyper-
plane angle in terms of a right hyperplane angle is always
the same as the measure of its plane angle in terms of a
right angle.
57. The bisecting half -hyperplane.
THEOREM i. The half -hyper plane bisecting a hyperplane
angle is the locus of points in the interior of the hyperplane
angle equidistant from the hyper planes of the cells.
PROOF. Through any point lying in the interior of a
hyperplane angle we can pass a plane absolutely perpendic-
ular to the face of the hyperplane angle (Art. 43, Th. 2).
This plane will intersect the cells of the hyperplane angle
in the sides of a plane angle, and the bisecting half -hyper-
plane in the half-line which bisects the plane angle. Now
this plane is perpendicular to the hyperplanes of the cells
of the hyperplane angle (Art. 50, Def.), and the distances
of the given point from these hyperplanes are the dis-
tances of the point from the sides of the plane angle (Art. 51,
Th. i). If, then, the point is in the half -hyperplane bi-
secting the hyperplane angle, it is in the half-line bisecting
the plane angle, and these distances are equal ; or, if these
distances are equal, the point is in the half-line bisecting
the plane angle, and therefore in the half -hyperplane
bisecting the hyperplane angle. That is, the bisecting
5&-581 PERPENDICULAR HYPKRPLANES IOI
half-hyperplane is the locus of points in the interior of the
hyperplane angle equidistant from the hyperplanes of its
cells.
THEOREM 2* The distance of a point in one cell of a
hyperplane angle from the bisecting half -hyper plane, is
greater than one-half the distance of the point from the hyper-
plane of the other cell.
This is proved, like the preceding theorem, by passing
a plane through the given point absolutely perpendicular
to the face of the hyperplane angle.
58. Perpendicular hyperplanes. Lines lying in one and
perpendicular to the other.
THEOREM i. // two hyperplanes are perpendicular, any
line in one perpendicular to their intersection is perpendicular
to the other, and any line through a point of one perpendicu-
lar to the other lies entirely in the first.
PROOF. In the first case, we have a line along which lies
one side of a plane angle of each of the four right hyper-
plane angles whose cells lie in the two hyperplanes, and,
since these plane angles are right angles, the line is per-
pendicular to that line of the second hyperplane along
which lie their other sides. That is, the given line is per-
pendicular to a line in the second hyperplane as well as to
the plane in which the two hyperplanes intersect, and
therefore it is perpendicular to the second hyperplane
(Art. 38, Th. 4).
The proof of the second part of the theorem is like the
corresponding proofs given in Art. 51.
THEOREM 2. If a line is perpendicular to a hyperplane,
any hyperplane which contains the line is perpendicular to
the hyperplane.
* Used in the proof of the first theorem of Art. 96.
102 HYPERPLANE ANGLES [n. v.
THEOREM 3. // three intersecting hyper planes with only
a line common to all three are perpendicular to a given hyper-
plane, the line of intersection is perpendicular to the given
hyper plane.
59. Planes lying in one and perpendicular to the other
of two perpendicular hyperplanes.
THEOREM i. // two hyperplanes are perpendicular, any
plane in one, perpendicular to their intersection, is perpen-
dicular to the other, and any plane through a line of one,
perpendicidar to the other, lies entirely in the first unless the
line itself is perpendicular to the second.
PROOF. In the first case, a line can be drawn in the
given plane perpendicular to the plane of intersection, and
therefore perpendicular to the second hyperplane. The
given plane must then be perpendicular to the second hy-
perplane (Art. 52, Th. i).
On the other hand, if the plane is given as perpendicular
to the second hyperplane, and contains a line b which lies
in the first hyperplane and is not perpendicular to the second,
we can pass a plane through b in the first hyperplane per-
pendicular to the intersection, and therefore perpendicular
to the second hyperplane, by the first part of this theorem.
But through a line not perpendicular to a hyperplane there
passes only one plane perpendicular to the hyperplane (Art.
52, Th. 2). Therefore the given plane must coincide with
the plane just drawn, and lies entirely in the first hyperplane.
THEOREM 2. If a plane is perpendicular to a hyperplane,
any hyperplane which contains the plane is perpendicular to
the hyperplane.
For in the plane are lines perpendicular to the given
hyperplane, and any hyperplane which contains the plane
will contain these lines, and must itself be perpendicular
to the hyperplane (Art. 58, Th. 2).
58-60] PROJECTION OF A PLANE 103
THEOREM 3. // two intersecting hyperplanes are per-
pendicular to a given hyper plane, their intersection is also
perpendicular to the given hyperplane.
60. Projection of a plane upon a hyperplane. Angle of
a half -plane and a hyperplane.
THEOREM i . The projection of a plane upon a hyperplane
is a plane or a part of a plane; it does not lie entirely in one
line unless the given plane is perpendicular to the hyperplane.
PROOF. The given plane and any point of its projection
determine a hyperplane which is perpendicular to the given
hyperplane (Art. 58, Th. 2). The lines which project
the points of the plane upon the given hyperplane are the
same as the lines which project the points of the plane
upon the plane of intersection of the two hyperplanes (Art.
58, Th. i, and Art. 38, Th. 3). The projection upon the
given hyperplane is therefore the same as the projection
upon the plane of intersection. Now the given plane is
not perpendicular to the plane of intersection if it is not
perpendicular to the given hyperplane (Art. 59, Th. i).
Therefore the projection is not a line, but is the plane of
intersection itself, or a part of this plane.*
COROLLARY. When a half-plane with its edge in a given
hyperplane does not lie in the hyperplane and is not perpen-
dicular to it, its projection upon the hyperplane is a half-
plane, or a portion of a half-plane, having the same edge.
THEOREM 2. When a half -plane with its edge in a given
hyperplane does not lie in the hyperplane and is not perpen-
dicular to it, the dihedral angle which it makes with the half-
plane having the same edge and containing its projection is
less than the dihedral angle which it makes with any other half-
plane of the hyperplane having the same edge.
* It is a single convex part, any point lying between the projections of two points
being also the projection of a point of the given plane. See foot-note, p. 79.
104 HYPERPLANK ANGLES [n. v.
PROOF. Let a be the given half-plane, /3 the half-plane
with the same edge containing the projection of a upon the
hyperplane, and 7 q.ny other half-plane lying in the hyper-
plane and having its edge in common with a and /8. A
hyperplane perpendicular to this common edge will inter-
sect the three half-planes in three half-lines, a, 6, and c,
and these half-lines, taken two at a time, are the sides of
plane angles of the dihedral angles whose faces are any
two of the three half-planes.
Now the last-named hyperplane and the hyperplane
of a and contain the half-lines a and 6, and are both per-
pendicular to the given hyperplane (the former by Art. 58,
Th. 2). Hence they intersect in a plane which contains a
and b and is perpendicular to the given hyperplane (Art.
59, Th. 3), and therefore contains the projection of a upon
the given hyperplane (Art. 51, Th. i). In other words,
the half-line b is the half-line which contains the projec-
tion of a upon the given hyperplane ; and a makes an angle
with b which is less than the angle which it makes with c
(Art. 41 , Th. 2). Therefore the dihedral angle which a makes
with )8 is less than the dihedral angle which it makes with 7.
When a half-plane with its edge in a given hyperplane
does not lie in the hyperplane and is not perpendicular to
it, the plane angle of the dihedral angle which it makes
with the half-plane having the same edge and containing
its projection is called the angle of the half -plane and hyper-
plane. When a half-plane with its edge in a given hyper-
plane is perpendicular to the hyperplane, it is said to make
a right angle with the hyperplane.
The student may prove the following theorem :
When a plane intersects a hyperplane but does not lie in it and is
not perpendicular to it, that plane of the hyperplane which passes
through the intersection and is perpendicular to the projection, is
perpendicular to the given plane.
CHAPTER III
ANGLES OF TWO PLANES AND ANGLES OF
HIGHER ORDER
I. THE COMMON PERPENDICULAR OF TWO LINES
61. Existence of minimum distance between two lines.
The theorems having to do with the common perpendicular
of two lines not in the same plane are usually made to de-
pend on the axiom of parallels. We shall give a form of
statement and proof for these theorems that will show
their independence of this axiom, so far as they are in-
dependent of it. We shall give the proofs in full, because
they will be a guide to us in proving a similar set of theo-
rems for planes in hyperspace.*
THEOREM. Given two lines a and b not in the same plane,
there is a point of a whose distance from b is less than or
equal to the distance from b of any other point of a.
PROOF. Let M be some point of a. The perpendicular
from M to the line b lies with b itself in a plane which inter-
sects a at the point M , so that the half-lines of a from M ,
forming an angle with this plane, move away from it indef-
initely. That is, we can take a definite portion of a, a
segment and its interior, and say that any point of a out-
side of this portion willbe distant from the plane, and there-
* In this method of treatment one or two modifications are necessary for the
Elliptic Geometry. Thus in the proof in this article the statement that the line a
moves off indefinitely is not true in the "third hypothesis" (see the author's Non-
Euclidean Geometry, p. 22). But in this case we can take the entire line for AB,
putting A and B together at some point and regarding them as the two ends of
the line Another slight modification will be pointed out In connection with Th. 2
of the next article.
105
106 COMMON PERPENDICULAR OF TWO LINES [in. i.
fore distant from ft, by more than the distance of M from
b*
Let AB denote such a portion of a, and let P be any
point of AB. The distance of P from the line b has for
all positions of P a lower limit / = o. By this we mean
that for every positive number e there are positions of P
where the distance, though never less than /, will be less
than l + e.
This is a theorem of irrational numbers, proved as follows :
If we separate all numbers into two classes, putting into
the first class all negative numbers and every positive
number which is never greater than the distance of P from
b y and into the second class every positive number which
is greater than the distance of some point P from 6, the
numbers of the first class will all be less than those of the
second class, and we determine in this way a number which
occupies the point of separation of the two classes. This
s the number /. / is, then, never greater than the dis-
tance of P from b ; but / + e is a number of the second
:lass, and there is some position of P for which its distance
from b is less than l + e.
On any portion of AB we prove in the same way that
the distance of a point P from b has a lower limit, and the
ower limit for any portion of AB taken is greater than or
*qual to the lower limit for any portion within which it is
xmtained, and so greater than or equal to /.
Now we separate the points of AB into two classes,
>utting A into the first class, and putting any other point
Z into the first class if for the points of AC the lower limit
>f the distance from b is greater than I, but into the second
:lass if this lower limit is equal to /. The points of the first
:lass all come before those of the second class; and by
,his separation we determine a point C', the point of sepa-
* See reference in previous foot-note.
DISTANCE BETWEEN LINES
107
ration, which may be A itself, or B y or some point between
A and B.
The distance of C" from b is exactly /.
For suppose this distance to be / + 5, where 6 is some
positive number. Take C\ on a towards A and Cz towards
B, each at a distance of ^ 5 from C'. If C' coincides with
A , or is at a distance from A less than | 5, we take C\ at -4 ;
or if C' coincides with 5, or is at a distance from B less
than ^5, we take C% at -B. In ACi, when Ci does not
coincide with A, the lower limit of the distance from b
is a number greater than /. In AC 2 this lower limit is /.
Therefore in CiC 2 the lower limit is /, and hence there is
in CiC 2 a point D whose distance from b is less than
/ + 2 5, 6 being a " positive number/' a value of e. Now
Therefore the distance of C' from the foot of the per-
pendicular drawn to b through D is less than I + d (one
side of a triangle less than the sum of the other two).
That is, the distance of C' from one point of 6, and there-
* In an equation or inequality "C'Z>" is used, as in our text-books, for the length
of the segment C'D (see Introduction, p. 21), that is, for the distance between the
points C' and D.
108 COMMON PERPENDICULAR OF TWO LINES [m. I.
fore certainly from b, is less than / + 5, whereas we have
assumed this distance to be equal to I + S.
This proves that the distance of C f from b is exactly /,
and is less than or equal to the distance from b of any other
point of a.
The distance / is not zero, since a and b do not intersect.
62. Existence of a common perpendicular. An infinite
number if more than one.
THEOREM i. Given two lines a and b not in the same
plane, the line along which we measure the minimum distance
from b of a point of a is perpendicular to both.
PROOF. Let M be a point of a whose distance from b
is less than or equal to the distance from b of any other
point of a, and let N be the projection of M upon b, so
that the line MN is perpendicular to b at N. If MN is
not perpendicular to a, then the projection of N upon a
will be a point whose distance from one point of b, and
therefore certainly from 6, is less than the distance of M
from b. But this is contrary to hypothesis; hence the
line MN must be a common perpendicular to the lines a
and b.
THEOREM 2. // two lines not in the same plane have
more than one common perpendicular, they have an infinite
number of common perpendiculars; along all of these per-
pendiculars the distance between them is the same, and any
two of the perpendiculars cut of the same distance on them*
* In the Elliptic Geometry we should say, " If two lines not in the same plane
have more than one common perpendicular within a restricted region," and in
the proof we must assume that the points M, N, M', and N' are all within
such a region.
This theorem will prove that it is only in the Elliptic Geometry (of the three prin-
cipal geometries, see Introduction, p. 19, foot-note) that two lines not in the
same plane can have more than one common perpendicular. For it is only in
this geometry that two lines not in the same plane can be everywhere equidistant
(see first paragraph of proof in Art. 61 and the reference in the foot-note on p. 105).
61,62]
WHEN THEY HAVE MORE THAN ONE
IOQ
N'
PROOF. Let a and b
be two lines not in the
same plane, and let
MN and M'N' be two
common perpendicu-
lars, perpendicular to
a at M and M ', and
perpendicular to b at
N and N'.
The distance M'N' is then equal to the distance MN.
For suppose we had
M'N f < MN.
Then in the right triangles which have NN' in common, we
should have
M'N < MN',
and in the right triangles which have MM' in common,
MN 1 < M'N.
Our hypothesis leads to a contradiction. We must have
M'N' = MN,
M'N = MN',
and in the right triangles which have MN in common
MM' = NN'.
Now on a and b lay off equal distances from M and N to
points P and Q respectively, both towards M 1 and N' or
both away from M' and N', and draw PQ, MQ, etc.
The right triangles MNP and MNQ are equal, having
the two legs of one equal respectively to the two legs of
the other. Therefore
MQ = PN.
In the same way we prove
M'Q = PN 9 .
110 COMMON PERPENDICULAR OF TWO LINES [m. i.
N Q^ N'
On b we take Q' in the same direction from N' that
Q is from N 9 so that
N'Q' = NQ,
and therefore so that
QQ' = NN'.
Then we have equal right triangles MNQ and M'N'Q',
M'Q' = MQ = PN,
the triangles PNN' and M'Q'Q mutually equilateral, and
Finally, now, the triangles PNQ and M'Q' N' have two
sides and the included angle of one equal respectively to
two sides and the included angle of the other, and
JLM'N'Q' = Z.PQN.
But the angle M'N'Q' is a right angle. Therefore PQN
is a right angle, and PQ is perpendicular to 6.
In the same way we prove that PQ is perpendicular
to a.
PQ is also equal to MN and to M 'N' : the lines a and b
are equidistant along all of these perpendiculars.
The last theorem is true and the proof holds good for
two lines in the same plane.
62,63] OF A LINE AND PLANE III
63. The common perpendicular line of a line and plane,
and the common perpendicular plane of two planes which
have a common perpendicular hyperplane.
THEOREM i . Given a line a and a plane /3 not in the same
hyperplane, there is a point of a whose distance from j8 is
less than or equal to the distance from j8 of any other point of a,
and the line along which we measure this minimum distance
is perpendicular to both.
PROOF. Let b be the line along which lies the projection
of a upon j8. Any line perpendicular to a and b will be
perpendicular to /3, and any line perpendicular to a and j3
will be perpendicular to b, and will be the projecting line
of a point of a. Now a and b lie in a hyperplane but not
in a plane (Art. 46, Th. i), and the distance of any point of
a from its projection upon j8 is less than its distance from
any other point of /3. Therefore, any point of a whose
distance from b is less than or equal to the distance from
6 of any other point of a, will be a point whose distance
from/3 is less than or equal to the distance from /3 of any
other point of a ; and the line along which we measure this
minimum distance will be perpendicular to a and |8, being
perpendicular to a and b.
THEOREM 2. If a line and plane not in one hyperplane
have more than one common perpendicular line they have
an infinite number of these perpendiculars, one through every
point of the line ; along all of these perpendiculars the dis-
tance between them is the same, and any two of the perpendic-
ulars intersect the line and plane at the same distance from
each other.
THEOREM 3. If two planes not in one hyperplane have
a common perpendicular hyperplane (Art. 53), they have
a common perpendicular line and a common perpendicular
plane.
112 POINT GEOMETRY [ni. n.
PROOF. The common perpendicular hyperplane inter-
sects the two planes in lines which do not lie in one plane
(Art. 27, Th. 3), and which have one common perpendicular
line. This line is perpendicular to the two given planes
(Art. 51, Th. 2), and the plane containing this line and the
linear elements which it intersects is perpendicular to the
two given planes.*
THEOREM 4. // two planes not in one hyperplane have
a common perpendicular hyperplane and more than one
common perpendicular plane, they have an infinite number
of common perpendicular planes, one through each linear
element; and any two of these planes are equidistant along the
intersections of the given planes with the given perpendicular
hyperplane or with all the perpendicular hyperplanes if
there can be more than one.\
II. POINT GEOMETRY
64. A geometry whose elements are the half-lines
drawn from a given point. We shall make a particular
study of the angles formed at a point O by the lines, planes,
and hyperplanes which pass through O ; and our presenta-
tion of the subject will be simpler if we omit all mention of
* In the Hyperbolic Non-Euclidean Geometry we have planes which do not
intersect and do not have a common perpendicular hyperplane, namely, planes
with parallel elements. These planes, however, do have a common perpendicular
plane. We can prove this by considering the geometry of an orthogonal boundary-
hypersurface. See foot-note, p. 95.
t It is only in the Elliptic Geometry that a line and plane not in one hyperplane
can have more than one common perpendicular line, and it is only in this geometry
that two planes not in one hyperplane having a common perpendicular hyperplane
can have more than one common perpendicular plane (see foot-note, p. 108). Two
planes in the Elliptic Geometry always have a point in common, a point common to
all their linear elements (Art. 27, Th. 3, remark). The above theorem should be
slightly "restricted," applied to some restricted region containing a part of the per-
pendicular hyperplane ; because two planes having a point in common always have
two common perpendicular planes at such a point But this case will be fully treated
in Arts. 66-69.
63,64] THE HALF-LINES AT A POINT 113
this point, leaving it to be understood that the lines, planes,
and hyperplanes which pass through O are the only ones
considered. In this way we can avoid frequent repetition
of the phrase " pass through O." Thus we can speak of
two planes as intersecting only when they intersect in a
line, always understanding that they intersect at O; and
since all lines, planes, and hyperplanes have O in common,
we can always say that two planes in a hyperplane inter-
sect, and that any plane and hyperplane or any two hyper-
planes intersect : it will not be necessary in each particular
case to assume such intersection or to mention the fact
that it exists.*
We shall call the geometry of the various kinds of angles
which we may have at a point Point Geometry.^
As O completely separates the two opposite half-lines
drawn from it along any line, it is better to consider the
half-line as one of the elements of Point Geometry, rather
than the entire line. To every half-line there is, then,
one opposite ; and a plane or hyperplane containing one
of two opposite half-lines always contains the other. Two
planes in a hyperplane, or any plane and hyperplane,
intersect in two opposite half-lines. Each hyperplane
has two perpendicular half-lines, one opposite the other.
Point Geometry in space of four dimensions is a three-
dimensional geometry. That is, we get all the elements
of the Point Geometry if we take four elements not in one
hyperplane, all elements coplanar with any two of them,
and all elements coplanar with any two obtained by this
process. We can interpret the two-dimensional and three-
dimensional geometries given in chap. I as point geom-
etries, if we give a proper meaning to their undefined
* Compare the remark on the limitation of hyperspace to four dimensions, Art.
26, and foot-note, p. 50.
t Klein, Nicht-cuklidische Geometric, Gottingen, 1 893, vol. I, p. 8, uses the ex-
pression "Metrik im Punktc."
I
114 THE ANGLES OF TWO PLANES [m. m.
terms and confine ourselves when necessary to a restricted
(angular) region.
III. THE ANGLES OF TWO PLANES*
65. Theorems in regard to perpendicular planes stated
in the language of Point Geometry. In this section we use
the language of Point Geometry, all planes and hyperplanes
being assumed to pass through a given point O, and all
half-lines to be drawn from O.
For perpendicular planes we have proved certain theorems
which can here be stated as follows (Arts. 43, 44, 47, and
49):
THEOREM i. Each plane has one and only one absolutely
perpendicular plane.
THEOREM 2. // two planes inter sect y their absolutely
perpendicular planes intersect.
THEOREM 3. A plane perpendicular to one of two ab-
solutely perpendicular planes is perpendicular to the other.
This theorem is included in the following.
THEOREM 4. Given two pairs of absolutely perpendicular
planes, if either plane of one pair is perpendicular to one plane
of the other pair, or if either plane of one pair intersects both
planes of the other pair, then each plane of either pair inter-
sects both planes of the other pair and is perpendicular to
both planes of the other pair.
THEOREM 5. // two planes have a common perpendicular
plane, the plane absolutely perpendicular to the latter is also
a common perpendicular plane to the two given planes.
* This subject is treated by C. J. Keyser, "Concerning the Angles and the
Angular Determination of Planes in 4-Space," Bulletin of the American Mathe-
matical Society, vol. 8, 1902, pp. 324-329, and by I. Stringham, "On the Geometry
of Planes in a Parabolic Space of Four Dimensions," Transactions of the American
Mathematical Society, vol. 2, IQOI, pp. 183-214. Stringham gets his results en-
tirely by quaternians.
5 4 -66] THE MINIMUM ANGLE 115
THEOREM 6. Two planes which intersect have one and
inly one pair of common perpendicular planes.
In the case of the last theorem, the two given planes
intersect in a pair of opposite half-lines and lie in a hyper-
Diane, and their absolutely perpendicular planes intersect
: n a pair of opposite half-lines and lie in a hyperplane.
3ne common perpendicular plane is the plane containing
these four half -lines, and the other is the plane of inter-
section of the two hyperplanes (see proof of theorem of
\rt. 49). The two given planes contain the faces of four
iihedral angles, and their absolutely perpendicular planes
:ontain the faces of four dihedral angles. The plane angles
)f all of these dihedral angles lie in the plane of intersection
)f the two hyperplanes, since this plane lies in both hyper-
planes and is perpendicular to the edges of the dihedral
ingles. The two hyperplanes contain the cells of four
lyperplane angles whose common face is their plane of
ntersection ; and each of the plane angles of these hyper-
)lane angles has for one side a half-line of the intersection
)f the two given planes, and for the other side a half-line
>f the intersection of their absolutely perpendicular
)lanes.
When the given planes are perpendicular, so that each
>f the common perpendicular planes can be determined
)y half-lines or as the intersection of two hyperplanes,
,hen in each we have angles which are at the same time
he plane angles of a set of dihedral angles and the plane
ingles of a set of hyperplane angles.
66. Existence of a minimum angle between two planes.
The theorems and proofs of this article and the next two
ollow the analogy of the theorems and proofs of Arts. 61
md 62. The two theories are, indeed, the same, as we
hall see later (Art. 123).
Il6 THE ANGLES OF TWO PLANES [m. in.
THEOREM. Given two planes a. and /3, there is a half -line
of a whose angle with /3 is less tlian or equal to the angle made
with j8 by any other ftalf-line of a*
PROOF. We will write angle ab to denote a certain angle
of 360 f in a (around the point O), putting two half -lines
a and b together in some position in a and regarding them
as the two sides of the angle.
Let m be any half-line of a, either a or b or a half-line in
some position between a and b. The angle which m makes
with /3 has for all positions of m a lower limit <f> ^ o.
That is, for every positive angle e there are positions of m
where this angle, though never less than </>, will be less
than <t> + This is a theorem of irrational numbers
proved as on p. 106.
Within any angl^e of a, that is, any portion of the angle
ab, we prove in the same way that the angle which m makes
with j8 has a lower limit : and the lower limit for any portion
of the angle ab is greater than or equal to the lower limit
for any portion within which it is contained, and so greater
than or equal to <f>.
Now we separate the half -lines of the angle ab into two
classes, putting a into the first class, and putting any other
half-line c into the first class if for all positions of m in the
angle ac the lower limit of the angle which it makes with
is greater than $, but into the second class if this lower
limit is equal to </>. The half-lines of the first class come
before those of the second class; and by this separation
we determine a half -line c' occupying the place of separa-
tion of the two classes, c 1 being a itself, or b, or some half-
line between a and b.
The angle which c' makes with /3 is exactly 0.
For suppose this angle to be <t> + 8, where 5 is some
* a and ^ pass through O, and a half-line is always a half-line drawn from 0.
t Using the term angle as it ib used in trigonometry.
66] THE MINIMUM ANGLE 117
positive angle. In a take half-lines c\ towards a and c 2
towards 6, each making an angle of \ 5 with c'. If c f
coincides with a, or makes with a an angle less than 6,
we take c\ along a ; or if c' coincides with 6, or makes with
b an angle less than 5, we take c 2 along ft. In the angle
aci, when c\ does not coincide with a, the lower limit of
the angle which the half-line m makes with |8 is greater
than 0. In the angle ac 2 this lower limit is 0. Therefore
in the angle c\c<t the lower limit is 0, and hence there is
a half-line m 9 in the angle c\c^ which makes an angle with
/? less than + | 5, \ S being a " positive angle," a value
of . Now
^c'fl*' 5 I 5,
and the half-line c', forming a trihedral angle with w' and
the projection of m' upon /?, makes with the projection an
angle less than the sum of the other two face angles of the
trihedral angle, that is, less than <f> + d. Thus c' makes
with one half-line of #, and therefore certainly with )8, an
angle less than + 5, whereas we have assumed this
angle to be equal to + 5.
This proves that the angle which c' makes with ]8 is
exactly 0, and is less than or equal to the angle made with
j8 by any other half-line of a.
Il8 THE ANGLES OF TWO PLANES [m. in.
When a and ft intersect, <t> = o ; in all other cases is
some definite positive angle, an acute angle unless a and
ft are absolutely perpendicular.
67. Existence of common perpendicular planes.
THEOREM. When two planes a and /3 do not intersect,
tfie plane of the minimum angle which a half -line of a makes
with ft is perpendicular to a and ft.
PROOF.* Let m be a half-line of a which makes with ft
an angle 0, less than or equal to the angle made with ft by
any other half-line of a, and let n be the projection of m
upon ft, so that the plane mn is perpendicular to ft along n.
If this plane is not perpendicular to a, the projection of n
upon a will be a half-line of a. which makes with one half-
line of ft, and therefore certainly with ft, an angle less than
0, But this is contrary to hypothesis ; and the plane mn
must therefore be a common perpendicular plane to a
and ft.
The plane mn, the common perpendicular plane of the
preceding theorem, intersects a. and ft in two pairs of op-
posite half-lines, and in this plane we have two pairs of
vertical angles, one pair of acute angles each equal to 0,
and one pair of obtuse angles (unless a and ft are absolutely
perpendicular).
The plane absolutely perpendicular to the plane mn is
also perpendicular to a and to ft (Art. 65, Th. 5). Let 0'
be one of the acute (or right) angles lying in the intersec-
tion of this plane with a and ft. We may let m' and n 1
be the half-lines forming the angle 0', m' in a and n' in ft.
n and n 1 (when 0' is not a right angle) are the projections
of m and m' upon ft, and that portion of a which lies within
the right angle mm' will be projected upon that portion of
* Given by C. J. Keyser, see reference on p. 114.
66-68] COMMON PERPENDICULAR PLANES IIQ
|8 which lies within the right angle nn r (see foot-note, p.
79)-*
68. Planes with an infinite number of common per-
pendicular planes. In the following proof we shall use
a figure drawn to represent points at a given distance from
O. Half-lines (drawn from 0) are represented in this
figure by points, and planes by lines. Any three points
of the figure not appearing on one line represent half-
lines which are the edges of a trihedral angle; or we can
think of the points as vertices of spherical triangles, the
different triangles of the figure lying on equal spheres but
not on the same sphere.
This figure is not all visible in any one hyperplane. Only four
points taken at random at the given distance from O could be seen at
any one time ; or along with a complete view of one spherical triangle
would appear only those points of the figure which lie on the sphere
of this triangle. The entire figure does, however, lie in a hypersphere
(chap. V), and it is not difficult to imagine a figure lying in a portion
of a hypersphere as a slightly curved space.
THEOREM. // two planes a and /3 cut out equal angles on
a pair of common perpendicular planes, they have an infinite
number of common perpendicular planes, the plane project-
ing any half -line of either upon the other being perpendicular
to both. On all of these common perpendicular planes they
cut out equal angles; and if a and /3 are not absolutely per-
pendicular, any two of these planes cut out on & angles equal
to the angles which they cut out on a.
Conversely, if a. and 0, not being absolutely perpendicular,
have more then two common perpendicular planes, the acute
angles which they cut out on any pair of these common per-
pendicular planes are equal.
* Combining this theorem with the theorem of Art. 49, with Th. 3 of Art. 63,
and with the first foot-note on p. 112, we can say that any two planes have a
common perpendicular plane.
I2O
THE ANGLES OF TWO PLANES
[ill. m.
PROOF. Two planes a and /3 have a pair of common
perpendicular planes on which they cut out the acute
angles <t> and </>'. Let the half-lines m and n be the sides
of one of the angles 0, and let m' and n' be the sides of one
of the angles </>', m and m' in a, and n and n' in /?. In the
right angles mm' and ww' lay off equal acute angles mp
and nq. The right trihedral angles O-mnp and O-mnq
have the face angles including the right dihedral angle of
one equal respectively to the face angles including the right
dihedral angle of the other. Therefore
Zmq = /.pn.
In the same way we prove
m'q = Z.pn'.
Now supposing <' to be equal to 0, we take the half -line
q r in the same direction (around O) from n 1 that q is from
n, so that
/.n'q'
and therefore so that
Z.qq f
Then we have right trihedral angles O-mnq and 0-m'n'q',
in which the face angles including the right dihedral angle
of one are equal respectively to the face angles including
the right dihedral angle of the other. Therefore
Z m'q' = Z mq = Z pn.
68] COMMON PERPENDICULAR PLANES 121
Now the trihedral angles O~pnn f and O-m'q'q have the
three face angles of one equal respectively to the three
face angles of the other, and so in these trihedral angles the
dihedral angles along the half-lines g r and n are equal.
Finally, the trihedral angles 0-pnq and 0-m f q'n r have
these same dihedral angles, and the adjacent face angles
of one equal respectively to the adjacent face angles of
the other, Z pn = Z.m'q', and Z qn = Z n'q'. There-
fore the dihedral angles along q and n r in these trihedral
angles are equal, and as the latter is a right dihedral angle
the former is also a right dihedral angle, and the plane
pq is perpendicular to /8.
In the same way we prove that the plane pq is perpendic-
ular to a.
The angle pq is equal to the angle m'n' and to the angle
mn, that is, it is equal to <.
Conversely, the angles mn and m'n f being the acute
angles <t> and <//, let p and q be half-lines within the angles
mm' and nn* ', q the projection of p upon 6 and p the pro-
jection of q upon a, so that the plane pq is perpendicular
both to a and to /3. Then the angle pq is equal to the
angle mn, that is to <, and the angles 4> and <j>' are equal.
Suppose, for example, that we had
Z pq < Z fif.
Then in the right trihedral angles which have the face
angle nq in common we should have
Z pn < Z w#,
but in the right trihedral angles which have the face angle
mp in common,
Z mq < Z /w.
Our hypothesis leads to a contradiction. We must have
Z pq = Z urn.
122 THE ANGLES OF TWO PLANES [m. m.
In the same way we prove
Z pq = Z m'n',
and therefore
Z m'n' = Z #w,
or <' = <.
Now in the right trihedral angles which have the face
angle mn in common, since also
Z pn = Z w<?,
we have
Z w/> = Z #.
69. The angles between two planes. Isocline planes.
The acute (or right) angles </> and </>' which two planes cut
out on their common perpendicular planes are called the
angles between the two planes. When one of these angles
is zero the two planes intersect and lie in a hypcrplane.
The other angle is then the measure of the acute (or right)
dihedral angles whose faces lie in the two planes. When
one angle is zero and one a right angle the planes are simply
perpendicular. When one angle is a right angle the planes
are sometimes said to be perpendicular even if the other
angle is not zero, but we have used the word perpendicular
alone as applied to planes only when the other angle is
zero (see foot-note, p. 85). When both angles are right
angles the planes are absolutely perpendicular.
THEOREM i. The angles which a plane makes with one
of two absolutely perpendicular planes are Hie complements
of the angles which it makes with the other; and any two
planes make the same angles as their absolutely perpendicular
planes.
PROOF. Let a and be two planes and a' and /3' their
absolutely perpendicular planes. In a plane perpendicular
to these four the complete intersection consists of two pairs
68, 6g]
ISOCLINE PLANES
123
of perpendicular lines; that is, of eight half -lines which
can be taken in the order a0a f /3'a j8 a'/?' around the point
O, and which are alternately perpendicular.
When the angles of two planes are equal the planes have
an infinite number of common perpendicular planes, but
they cut out the same angle on them all. The two planes
are then said to be isocline* Absolutely perpendicular
planes are always isocline, and a plane isocline to one of
two absolutely perpendicular planes is isocline to the other.
Any two lines taken one in each of two absolutely per-
pendicular planes determine a common perpendicular
plane (Art. 65, Th. 4), but in the case of two isocline planes
which are not absolutely perpendicular only one of the
common perpendicular planes passes through any line of
either. Any two of these common perpendicular planes
cut out equal angles on the two given planes and are them-
selves isocline.
THEOREM 2.f // two half -lines in a plane a make equal
angles with a plane |8, the half-line bisecting the angle between
* See p. 125, and foot-note.
t Used by Veronese in finding the perpendicular planes, Grundztige, 150.
124
THE ANGLES OF TWO PLANES
[in. m.
them and the half-line bisecting the angle between their pro-
jections upon /3 will lie in one of the planes perpendicular to
a and /3.
PROOF. Let p and p 1 be the half -lines in a, and q and q f
their projections upon /?, so that the angles pq and p'q*
are equal. Then if m is the half-line bisecting the angle
*
pp' and n the half-line bisecting the angle qq f , the plane
mn will be one of the common perpendicular planes of a
and j8.
To prove this we consider in succession the following
pairs of trihedral angles :
O-pqq' and O-p'q'q,
0-p'pq
O-pqn
0-pmn
O-mpq
O-mnq
0-pp'q',
0-p'q'n,
0-p'mn,
O-mp'q',
O-mnq 1 .
In each case two face angles and the included dihedral
angle of one trihedral angle are equal respectively to two
face angles and the included dihedral angle of the other,
or the three face angles of one trihedral angle are equal
respectively to the three face angles of the other.
COROLLARY. // more than two pairs of opposite half-
lines in one of two planes make any given angle with the other
plane, the two planes are isocline.
69)
THE "ISOCLINAL ANGLE"
125
The student may work m
out the details in proof of
the following theorems :
Given that the angles
' mn and m'n' are the angles
and 0' made by two
planes a and 0, let p be p
a half -line within the right n q n '
angle mm', and q its pro-
jection upon ft. Then if the angle pq is greater than it will be less
than 0'. That is, we have in succession the inequalities
and then, further,
/.pn
/.mp
/.mn,
/.mq,
/.pm' < /.qn',
/.pn' < Zw'<7,
/.pq < /.m'n'.
and the magnitude of
This requires that 0' shall be greater than ;
the angle pq is always between and 0'.
Now instead of taking m p
q as the projection of p
upon 0, let us take p and
q as in Art. 68, so that
/.mp = /.nq. Then we
shall have equal dihe-
dral angles at p and q, p
namely,
dihedral angle mpq dihedral angle nqp ;
and the plane of the half-lines bisecting the angles mn and m'n' will
intersect the plane pq at right angles and will contain the half-line
bisecting the angle pq.
The angle pq, because of the equal dihedral angles at p and g, has
been called an isoclinal angle of the planes a and 0, and the angles
and 0' are its minimum and maximum values.*
* See p. 1 09 of article by Stringham referred to at the beginning of this section.
Stringham says that "Two planes may be said to be mutually isoclinal when their
isoclinal angle is constant/' and he speaks of two such planes as "isoclines" (p. 210).
We have used the term isocline as an adjective. Stringham' s isoclinal angle is not
of special importance in this development of the subject.
126 POLYHEDROIDAL ANGLES [in. iv.
IV. POLYHEDROIDAL ANGLES
70. Polyhedroidal angles. Interior of a polyhedroidal
angle. Vertical polyhedroidal angles. A polyhedroidal
angle consists of the half-lines drawn through the points
of a polyhedron (Art. 29) from a given point not in the
hyperplane of the polyhedron, together with this given
point. The half-lines are called elements, the polyhedron
is the directing- polyhedron, and the pont is the vertex. The
elements which pass through the vertices of the polyhedron
are called edges, the elements which pass through the points
of any edge of the polyhedron lie in the interior of a face
angle, and the elements which pass through the points of
any face of the polyhedron lie within a polyhedral angle
of the polyhedroidal angle. The interiors of the polyhedral
angles are the cells. Adjacent polyhedral angles are joined
by their face angles, these lying in the planes of intersection
of their hyperplanes. A polyhedroidal angle is convex when
each of these hyperplanes contains no element of the
polyhedroidal angle except those which belong to the
polyhedral angle of this hyperplane and to its interior.
The polyhedroidal angle is convex when the directing-
polyhedron is convex. We shall consider only convex
polyhedroidal angles.
The interior of a polyhedroidal angle consists of the half-
lines drawn from the vertex through the points of the in-
terior of the directing-polyhedron. The interior of a
convex polyhedroidal angle lies within any one of its
hyperplane angles; and a point lying within all of the
hyperplane angles lies in the interior of the polyhedroidal
angle. The polyhedroidal angle separates the rest of
hyperspace into two portions, interior and exterior to it.
The polyhedroidal angle whose elements are half-lines
opposite to the elements of a given polyhedroidal angle is
70, ?i] TETRAHEDROIDAL ANGLES 127
vertical to the latter. In two vertical polyhedroidal angles
the face angles, dihedral angles, and polyhedral angles of one
are all vertical to the corresponding parts of the other, and
the face angles and dihedral angles of one are equal to the
corresponding face angles and dihedral angles of the other.
A polyhedroidal angle taken together with its vertical
polyhedroidal angle may be regarded as a particular case
of a hyperconical hypersurface (Art. 33).
A hyperpyramid, or the hypersolid which we call the interior of
a hyperpyramid, may be described as cut from the interior of a poly-
hedroidal angle by a hyperplane that cuts all of its elements and does
not pass through its vertex.
71. Tetrahedroidal angles. The rectangular system.
The simplest polyhedroidal angle is the tetrahedroidal angle,
having a tetrahedron for directing-polyhedron. Any four
hyperplanes with a point but not a line common to them
all are the hyperplanes of a tetrahedroidal angle, and any
four half-lines drawn from a point and not in one hyper-
plane are the edges of a tetrahedroidal angle. The four
hyperplanes, or the lines of the four half-lines, determine
a set of sixteen tetrahedroidal angles filling completely
the hyperspace about the point, and associated in eight
pairs of vertical tetrahedroidal angles. In a tetrahedroidal
angle there are six planes, three pairs* of opposites, the two
planes of a pair meeting only at the vertex. Otherwise
the six planes all intersect, three in each edge. We may
also speak of each half-line as opposite to the trihedral
angle formed by the other three half-lines.
A point is in the interior of a tetrahedron if it is within
any three of its dihedral angles whose edges lie in one plane,
or if it is within two dihedral angles whose edges contain
a pair of opposite edges of the tetrahedron. In the same
* We do not mean here that they are absolutely perpendicular.
128 POLYHEDROIDAL ANGLES [ra. iv.
way, a point is in the interior of a tetrahedroidal angle if
it is within any three of its hyperplane angles whose faces
lie in one hyperplane, or if it is within two hyperplane
angles whose faces are the planes of two opposite face angles.
At each vertex of a pentahedroid is a tetrahedroidal angle.
A particular case is the rectangular system: four half-
lines mutually perpendicular, six face angles which are
right angles lying in three pairs of absolutely perpendicular
planes, the trihedral angles rectangular trihedral angles,
the dihedral angles right dihedral angles, and the four
hyperplanes mutually perpendicular (see Art. 48).
72. Tetrahedroidal angles with corresponding face
angles equal. We can make two tetrahedrons correspond,
the four vertices of one to the four vertices of the other,
in any order, corresponding edges, faces, etc., being those
determined by corresponding vertices. Then we have
the following theorem in regard to tetrahedrons with cor-
responding edges equal :
THEOREM i. // each of the six edges of a tetrahedron is
equal to the corresponding edge of a second tetrahedron, when
the four vertices of one are made to correspond in some order
to the four vertices of the other, then the faces and dihedral
angles of one will be equal to the corresponding faces and
dihedral angles of the other.
For any two corresponding faces are triangles which are
mutually equilateral and therefore equal, the angles of one
equal to the corresponding angles of the other; then any
two corresponding trihedral angles have the face angles
of one equal to the corresponding face angles of the other,
and therefore corresponding dihedral angles are equal.*
* The axiom of parallels is usually employed in the proof of this theorem, but
the proof can be modified so as to make it independent of this axiom (compare proof
of Th. 2). Indeed, any theorem in regard to angles at a point must be independent
of the axiom of parallels (see the author's Non-Euclidean Geometry, p. 26).
7i,72) TETRAHEDROIDAL ANGLES 129
In the same way we can make two tetrahedroidal angles
correspond, the four edges of one to the four edges of the
other, in any order, corresponding parts being those de-
termined by corresponding edges. For these we have the
following theorem analogous to the theorem just given :
THEOREM 2. // each of the face angles of a tetrahedroidal
angle is equal to the corresponding face angle of a second
tetrahedroidal angle, when the four edges of one are made to
correspond in some order to the four edges of the other, then
corresponding dihedral angles and corresponding hyperplane
angles will all be equal.
PROOF.* Lay off convenient distances on the edges of
the two tetrahedroidal angles, the distances in one equal
to the corresponding distances in the other, forming two
pentahedroids. Let them be called OABCD and
O'A'B'C'D'. The five tetrahedrons of one correspond to
the five tetrahedrons of the other, and in all of these tetra-
hedrons corresponding edges, corresponding face angles,
and corresponding dihedral angles are equal. Thus, in
particular, the corresponding dihedral angles of the two
tetrahedroidal angles are all equal.
In the face OAB we take a convenient point H, say a
point on the line perpendicular to the edge AB at its middle
point, and through H pass a plane absolutely perpendicular
to the plane OAB. This absolutely perpendicular plane
will cut the half-hyperplanes OAB-C and OAB-D in half-
lines which are the sides of the plane angle at H of the hyper-
plane 'angle C-OAB-D (Art. 54, Th. i). In the half-
hyperplane OAB-C the half -line drawn from H perpen-
dicular to the plane OAB will meet the tetrahedron
OABC at a point P which is not a point of the plane
OAB (Art. 21, Th. i). If P is not a point of the half-plane
* It will help the student to note the analogy of this proof to the proof of the
corresponding theorem about trihedral angles as given in many of our text-books.
130 POLYHEDROIDAL ANGLES [m. iv.
AB-C it will Ue in the face OAC or in the face OBC or in
the edge OC, and the half-plane AB-P will intersect the
edge OC (Art. 8, Th. 2). We can then modify our construc-
tion, taking the point C at this intersection, so that the half-
plane AB-P will be the same as the half -plane AB-C, and
the perpendicular from H will in any case meet the half-
plane AB-C at the point P. Now let A' be the middle
point of AB. Then the angle at A' in the triangle HKP is
the plane angle of the dihedral angle O-AB-C.
In the same way we can modify the position of Z>, if nec-
essary, so that the half-line drawn from H in the half-hy-
perplane OAB-D perpendicular to the plane OAB will meet
the half-plane AB-D at a point Q\ and in the triangle
HKQ the angle at K will be the plane angle of the dihedral
angle O-AB-D.
It is the angle PHQ, formed by these two perpendicu-
lars, which is the plane angle at H of the hyperplane angle
C-OAB-D.
We repeat the construction in the second pentahedroid,
using the same letters with accents, making the distance
of H f from A'B' equal to the distance of H from AB, and
changing the positions of C f and D f ', if necessary, so that
their distances from O' are equal, respectively, to the final
distances of C and D from O. We have, then, to prove
that the angle P' H'Q' is equal to the angle PHQ.
In the tetrahedrons OABC and O'A'B'C' the dihedral
angle AB is equal to the dihedral angle A'B'. Therefore
their plane angles are equal, that is,
Z HKP = Z H'K'P';
and, as HK = H f A', the right triangles H KP and H'K'P'
are equal. Thus we have
KP = K'P'
and HP = H'P'.
72, 73l TETRAHEDROIDAL ANGLES 131
Similarly, triangles H KQ and H' K'Q' are equal,
KQ = K'Q 1 ,
and HQ = H'Q' .
Now in the tetrahedron A BCD the angle PKQ is the
plane angle of the dihedral angle AB. Thus
Z PKQ = Z P'#'(X,
the triangles PKQ and P' K'Q' are equal, and
Then, finally, the triangles PHQ and P'H'Q' are mu-
tually equilateral, and
Z P//(? = Z P'H'Q'.
In this way we prove that any hyperplane angle of one
tetrahedroidal angle is equal to the corresponding hyper-
plane angle of the other.
We shall see in the next section that the two theorems
of this article arc identical, and that the proof of Th. i,
interpreted as a proof in Point Geometry, holds for Th. 2
(Art. 76).
73. The bisectors of the hyperplane angles of a tetra-
hedroidal angle.
THEOREM. The half-hyperplanes bisecting the six hyper-
plane angles of a tetrahedroidal angle contain a half -line
common to them a//, tlie locus of points within the tetrahedroidal
angle equidistant from its four hyper planes.
PROOF. Let O be the vertex and a, 6, c, and d the four
edges of a tetrahedroidal angle. The half-hyperplane
bisecting the hyperplane angle c-ab-d is the locus of points
within this hyperplane angle equidistant from the hyper-
planes of its cells, namely, the hyperplanes abc and abd
132 POLYHEDROIDAL ANGLES [m. iv.
(Art. 57, Th. i). The half-hyperplane bisecting the hyper-
plane angle b-ac-d is the locus of points within this hyper-
plane angle equidistant from the hyperplanes abc and acd.
These bisecting half-hyperplanes intersect in a half-plane
a whose edge is the line which contains a. a lies entirely
within both of the hyperplane angles, and is the locus of
points within both hyperplane angles equidistant from the
three hyperplanes abc, abd, and acd.
Now points within the hyperplane angle c-ab-d are on
the same side of the hyperplane abd as the half-line c\
and points within the hyperplane angle b-ac-d are on the
same side of the hyperplane acd as the half-line 6. The
half-plane a, lying within both of these hyperplane angles,
is therefore within the hyperplane angle b-ad-c (Art. 54),
and so entirely in the half-hyperplane bisecting this hyper-
plane angle.
There are four such half-planes, a, 0, 7, and 5, having
for edges the lines containing a, 6, c, and </, respectively,
each lying within the three hyperplane angles whose faces
contain its edge, and each common to the half-hyperplanes
bisecting the three hyperplane angles.
The half-hyperplane bisecting the hyperplane angle
c-ab-d contains both a and 0. Therefore these half -planes,
whose edges pass through O, intersect in a half-line g
drawn from (Art. 22, Th.), and the points of g are equi-
distant from all four of the hyperplanes. The points of g
lie within five of the hyperplane angles of the tetrahedroidal
angle. In particular, they lie within each of the two oppo-
site hyperplane angles b-ac-d and a-bd-c. Therefore g
lies within all six of the hyperplane angles (Art. 71), is
common to the four planes a, /?, 7, and 5, and contains all
the points within the tetrahedroidal angle which are equi-
distant from the four hyperplanes.
73, 74] ANGLES FORMED WITH HALF-PLANES 133
V. PLANO-POLYHEDRAL ANGLES
74. Piano-polyhedral angles, analogous to polyhedral
angles and to polygons. Piano-trihedral angles. We
come now to another class of angles which are analogous
to polyhedral angles, more so, indeed, than the polyhedroidal
angles of the preceding section.
Polyhedral angles are themselves analogous to polygons.
Thus we may define a polyhedral angle as consisting of a
finite number of half-lines, drawn from a point O, and taken
in a definite cyclical order, together with the point O and
the interiors of the angles whose edges are consecutive
half-lines of this order (see Art. 14).
In the same way we may have a finite number of half-
planes having a common edge, and define a piano-polyhe-
dral angle as consisting of these half-planes taken in a defi-
nite cyclical order, together with their common edge and
the interiors of the dihedral angles whose faces are consecu-
tive half-planes of this order. The half-planes are the faces,
their common edge is the vertex-edge, and the interiors of
the dihedral angles are the cells. If a, /?, 7, . . . are the
faces in order, the cells are the interiors of the dihedral
angles a/3, 187, . . . , and the piano-polyhedral angle may
be described as the piano-polyhedral angle af3y ....
In the analogy of polyhedral angles and piano-polyhedral angles,
as in the analogy of dihedral angles and hyperplane angles, it is a
face in the hyperspace figure which corresponds to an edge in the
figure of the ordinary geometry, and a cell which corresponds to a
face. In the ordinary geometry the polyhedral angle lies, of course,
entirely in a hyperplane ; and so here the piano-polyhedral angle is
assumed to lie in one space of four dimensions.
Half -planes which lie within the cells and have the vertex-
edge for edge, and those which are the faces of the piano-
134 PLANO-POLYHEDRAL ANGLES [in. v.
polyhedral angle, are all called elements and are in cyclical
order (Art. 6).
A piano-polyhedral angle is simple when no half-plane
occurs twice as an element ; we shall always assume that it
is simple. A simple piano-polyhedral angle is convex when
the hyperplane of each cell contains no element except those
of the cell itself and the two faces of the cell. Each face
of a convex piano-polyhedral angle is a half-plane lying in
the common face of two half-hyperplanes which contain two
adjacent cells. These two half-hyperplanes are the cells
of a hyperplane angle, one of the hyperplane angles of the
piano-polyhedral angle.
In a polyhedroid each edge lies in the vertex-edge of a
piano-polyhedral angle whose cells contain the adjacent
cells of the polyhedroid, and the edges of a polyhedroidal
angle lie in the vertex-edges of piano-polyhedral angles
which belong to the polyhedroidal angle.
The piano-polyhedral angle whose elements are half-
planes opposite to the elements of a given piano-polyhedral
angle is vertical to the latter. In two vertical piano-polyhe-
dral angles the dihedral angles and hyperplane angles of one
are all vertical to the corresponding parts of the other.
A piano-polyhedral angle with three faces is called a piano-
trihedral angle. Any three half-planes having a common
edge but not lying in one hyperplane are the faces of a
piano-trihedral angle. Any three hyperplanes which inter-
sect but have only a line common to all three are the hyper-
planes of a piano-trihedral angle. The planes of three such
half-planes, or three such hyperplanes, determine eight
piano-trihedral angles, completely filling the hyperspace
about their line of intersection, and associated in four pairs
of vertical piano-trihedral angles.
The piano-polyhedral angles of a pentahedroid are all
piano-trihedral angles.
74, 7Sl DIRECTING POLYHEDRAL ANGLE 135
75. Polyhedral sections of a piano-polyhedral angle.
Right sections. A hyperplane intersecting the edge of a
piano-polyhedral angle, but not containing the edge, inter-
sects the faces in half-lines which are the edges of a poly-
hedral angle ; and the piano-polyhedral angle may be con-
sidered as determined by a polyhedral angle and a line
through its vertex not in its hyperplane. When either the
piano-polyhedral angle or the polyhedral angle is convex,
the other is convex.
A piano-polyhedral angle might be regarded as a polyhe-
droidal angle with a directing polyhedral angle instead of a
directing-polyhedron : that is, the half-lines drawn through
the points of a polyhedral angle from a given point not in
its hyperplane form a certain portion of a piano-polyhedral
angle.
THEOREM. A hyperplane perpendicular at a point to
the vertex-edge of a piano- polyhedral angle intersects the latter
in a polyhedral angle whose face angles are the plane angles
at O of the dihedral angles of the piano- polyhedral angle, and
whose dihedral angles have at O the same plane angles as the
hyperplane angles of the piano- polyhedral angle.
PROOF. The hyperplane, being perpendicular to the
vertex-edge at O, intersects the hyperplanes of the dihedral
angles in planes perpendicular to their common edge at
this point. Therefore the face angles of the polyhedral
angle are the plane angles of the dihedral angles of the piano-
polyhedral angle. The given hyperplane is also perpendicu-
lar to the planes of the faces of the piano-polyhedral angle
(Art. 52, Th. i). These planes are the faces of the hyper-
plane angles of the piano-polyhedral angle; therefore the
dihedral angles of the polyhedral angle have at O the same
plane angles as the hyperplane angles of the piano-polyhe-
dral angle (Art. 54, Th. 2).
136 PLANO-POLYHKDRAL ANGLES |m. v.
The polyhedral angle in which the piano-polyhedral angle
is cut by a hyperplane perpendicular to the edge is a right
section of the piano-polyhedral angle.
76. Theorems proved by means of a right section. The
piano-polyhedral angle in Point Geometry. Certain theo-
rems in regard to piano-polyhedral angles follow, as analo-
gous to theorems true of polyhedral angles :
THEOREM i. The sum of two dihedral angles of a piano-
trihedral angle is greater than the third.
THEOREM 2.* The sum of the dihedral angles of a convex
piano- polyhedral angle is less than the sum of four right dihe-
dral angles.
THEOREM 3.f // two piano-trihedral angles have the
three dihedral angles of one equal respectively to the three
dihedral angles of the other, their homologous hyperplane
angles are equal; and if two piano-trihedral angles have two
dihedral angles and tlte included hyperplane angle of one equal
respectively to two dihedral angles and the included hyper-
plane angle of the other, the remaining parts of one are equal
to the corresponding parts of the other.
The piano-polyhedral angle plays the part of polyhedral
angle in the three-dimensional Point Geometry at any point
of the edge, and the polyhedroidal angle plays the part of
polyhedron in the Point Geometry at its vertex. Thus
Th. i of Art. 72 and its proof themselves become a state-
ment and proof of the second theorem of the same article.
The student may prove the following theorems independently of
the axiom of parallels :
The area of a triangle is greater than the area of its projection on
any plane containing one of its sides, not the plane of the triangle
itself (see the author's Non-Euclidean Geometry, p. 7, Cor.).
* The proof of this theorem can be made independent of the axiom of parallels,
f See foot-note, p. 128.
75-77] D1RECTING-POLYGON 137
The area of any face of a tetrahedron is less than the sum of the
areas of the other three faces.
Any trihedral angle in a tetrahedroidal angle is less than the sum
of the other three.
77. The direc ting-polygon, and half-plane elements.
THEOREM. A convex polyliedral angle can be cut by a
plane in a convex polygon.
PROOF.* In a convex polyhedral angle each pair of con-
secutive edges lies in the face of a half-space which contains
all the other edges (compare Art. 14, Th. 2). Let the ver-
tex be and the edges the half -lines a, 6, c, d, . . ., and on
these edges take points A, B, C, D, . . . . All of these
points except A and B lie on one side of the plane ABO\
and the half-planes AB-C, AB-D, . . . , together with the
half-plane AB-O, are in a definite order around the line
ABj all but the half-plane AB-O on the same side of the
plane ABO. Therefore one of these half-planes, say A B-C,
comes next after AB-O around the line AB, and lies be-
tween AB-O and each of the others, so that it cuts the in-
teriors of the segments OZ), ....
That is, the plane ABC cuts all of the edges of the poly-
hedral angle in points, and the polyhedral angle itself in a
polygon, a convex polygon since the polyhedral angle is
convex.
It follows that a convex piano-polyhedral angle may be
considered as determined by a convex polygon and a line
not in a hyperplane with the plane of the polygon. The
elements of the piano-polyhedral angle (Art. 74) are then
half-planes having the given line for common edge and
containing each a point of the polygon. The polygon is
* This is a theorem of geometry of three dimensions, for a convex polyhedral
angle is denned as lying in a hyperplane, and in the proof we speak of the hyperplane
of the polyhedral angle as ''space," and of any half -hyperplane lying within this
hyperplane as a "half-space."
138 PLANO-POLYHEDRAL ANGLES [m. v.
called the directing-polygon, and each side of the polygon
lies in the interior of a cell of the piano-polyhedral angle.
Two vertical piano-polyhedral angles taken together
may be regarded as a particular case of a piano-conical
hypersurface (see Art. 35).
A double pyramid, or the hypersolid which we call the interior of
a double pyramid, may be described as cut from the interior of a
piano-polyhedral angle by two hyperplanes which contain the direct-
ing-polygon and each a point of the vertex-edge.
78. Edge Geometry. The elements half-planes with a
common edge. We can build up a geometry by taking for
elements the half-planes which have a given line for edge.
We will call this geometry an Edge Geometry. The half-
planes which lie in the same plane on opposite sides of this
common edge will be called opposite elements, and to every
element there is one which is opposite. Any two non-
opposite elements determine a hyperplane which contains
the given line. The geometry will be the geometry of these
half-planes and hyperplanes. We may think of a hyper-
plane as consisting of half-planes ; and in the development
of this geometry we shall omit all mention of the given line,
and think of two figures as intersecting only when they have
one or more half-plane elements in common (compare
Point Geometry, Art. 64).
By confining ourselves when necessary to a properly
restricted region (angular region about the vertex-edge), to
a region, for example, consisting of all the elements which
make an angle of less than 90 with any given element,
and by giving a suitable interpretation to the terms used,
we have the geometry developed in Arts. 1-16. Thus for
point we say half-plane, for line hyperplane, and for segment
dihedral angle. The half-planes in any hyperplane have
the relations of order referred to in Art. 5, all of the half-
77,78] EDGE GEOMETRY 139
planes in a hyperplane being, however, in cyclical order
(Art. 6). We get all of the half -planes of this geometry if
we take three not in a hyperplane, all that lie in a hyper-
plane with any two of them, and all that lie in a hyper-
plane with any two obtained by this process. This geom-
etry is, therefore, two-dimensional.
We have the following theorem equivalent to Th. 2 of
Art. 4, and so essentialy equivalent to the two axioms of
Art. 3 : If a' and /?' are two distinct non-opposite half-
planes of the hyperplane a/3, then the hyperplane <*'' is
the same hyperplane (Art. 20).
We can also prove a theorem corresponding to the Axiom
of Pasch. Suppose that a, j8, and 7 are three half-plane
elements not in one hyperplane, and that a hyperplane X,
not containing any one of these three, contains a half-plane
in the interior of the dihedral angle 187, and does not contain
a half-plane in the interior of the dihedral angle a/3. The
half-planes /3 and 7 are then on opposite sides of the hyper-
plane X, and the half -planes a and are on the same side.
Therefore the half-planes a and 7 are on opposite sides of
this hyperplane, and the hyperplane must contain a half-
plane lying in the interior of the dihedral angle ay (Art. 28).
A convex piano-polyhedral angle necessarily lies within
a restricted region, and therefore the geometry of convex
polygons holds true in this way of convex piano-polyhedral
angles. Such an angle divides all the elements which do
not belong to it into two classes, those in the interior and
those outside. The interior consists of the interiors of all
the dihedral angles whose faces are elements of the piano-
polyhedral angle except those whose interiors also lie in it ;
and any half-hyperplane whose face contains a half-plane
lying in the interior of a convex piano-polyhedral angle,
will itself contain one and only one element of the
latter.
140 PLANO-POLYHKDRAL ANGLES [HI. v.
Other theorems of triangles and convex polygons can be
interpreted in' the same way.
79. The interior of a convex piano-polyhedral angle.
Dropping the language of our Edge Geometry, we may say
that a convex piano-polyhedral angle divides the rest of
hyperspace into two parts, interior and exterior. The in-
terior contains the interiors of all segments whose points
are points of the piano-polyhedral angle except those whose
interiors also lie in it. The interior belongs to the interior
of each of its hyperplane angles ; and any point which is
in the interior of each of the hyperplane angles of a convex
piano-polyhedral angle is in the interior of the latter. In
the case of a piano-trihedral angle, if a point is in the in-
terior of two of the hyperplane angles it is in the interior
of the piano-trihedral angle.
THEOREM. Tfie three half -hyper planes which bisect the
hyperplane angles of a piano-trihedral angle intersect in a
half-plane, the locus of points in the interior of the piano-
trihedral angle equidistant from the hyper planes of its cells.
CHAPTER IV
SYMMETRY, ORDER, AND MOTION
I. ROTATION AND TRANSLATION
80. Rotation in a plane and rotation in a hyperplane. In
a plane a figure can rotate around a point, or we may think
of the entire plane as rotating on itself around one of its
points, one direction of rotation being to the right and the
other to the left.
In a hyperplane, or in the space of our experience, a plane
perpendicular to a given line at a point O, rotating on it-
self around O, always remains perpendicular to the line,
We say that the plane rotates around the line as axis, 01
axis-line. On the other hand, a plane can rotate around
one of its own lines as axis-line through a certain dihedral
angle ; and if a plane perpendicular to the axis rotates with
a plane which contains the axis, the former rotates through
an angle which is the plane angle of the dihedral angle
through which the latter rotates. Thus we can compare
the rotations of two planes perpendicular to an axis, since
the plane angles of a dihedral angle are the same at any twc
points of its edge (see foot-note, p. 97).
THEOREM. When all the planes perpendicular to a line
in a hyperplane rotate around the line in the same direction
and through the same angle or at the same rate, figures in the
hyperplane remain invariable, any two points being always
at the same distance from each other.
PROOF. Let A and B be two points, O and O' their pro-
jections upon the axis, and A' and B' their positions after
141
142
ROTATION AND TRANSLATION
[iv. i.
a rotation through an angle 0. We are to prove that A'B'
is equal to AB.
The half-plane OO'-B rotates in the same direction and
through the same dihedral angle <j> as the half-plane OO'-A.
That is,
dihedral angle A-OO'-A' = dihedral angle B-OO'-B',
and therefore
dihedral angle A-OO'-B = dihedral angle A'-OO'-B f .
Let a and be the planes in which A and B rotate, planes
perpendicular to the axis OO', and let C and C f be the pro-
jections of A and A ' upon /3. Then the angles CO'B and
C'O'B' are plane angles of the last mentioned dihedral
angles, and are equal. Thus we prove that the triangles
CO'B and C'O'B' are equal, and then that the right tri-
angles ACS and A 'C'B' are equal. Therefore
AB = A'B'.
That is, any figure in a hyperplane can rotate around a
line, and we can think of the entire hyperplane as rotating
on itself around one of its lines.
81. Rotation in hyperspace.* The axis-plane. Double
rotation. In space of four dimensions one of two absolutely
* This subject is treated by F. N. Cole, "On Rotations in Space of Four Dimen-
sions," American Journal 0} Mathematics, vol. 12, 1890, pp. 191-210.
8o, 8i] THE AXIS-PLANE 143
perpendicular planes rotating on itself around the point
where the two planes meet always remains absolutely per-
pendicular to the other. We can say, then, that the rotat-
ing plane rotates around the other plane, and we can call
the other plane an axis-plane. Two planes absolutely per-
pendicular to a given plane at points O and O' lie in a hy-
perplane in which they are perpendicular to the line OO r
(Art. 45, Th.). Thus we can compare the rotations of two
planes absolutely perpendicular to the axis-plane by con-
sidering them as rotations in a hyperplane around an axis-
line.
THEOREM i. When all the planes absolutely perpendicu-
lar to a given plane rotate around the given plane as axis-
plane in the same direction and through the same angle or at
the same rate, all figures remain invariable, any two points
being always at the same distance from each other.
PROOF. Any two points, with the absolutely perpendicu-
lar planes through them, lie in a hyperplane in which the
rotation takes place around a line perpendicular to these
planes. Therefore, by the theorem of the preceding article,
the distance between the two points remains unchanged.
Thus any figure in hyperspace can rotate around a plane.
THEOREM 2. Rotations around two absolutely perpen-
dicular planes are commutative; after two such rotations all
points of hyperspace take the same positions, whichever rota-
tion comes first.
PROOF. Let a and |8 be the two planes. A point in one
of these planes remains in it, and is subject only to the rota-
tion around the other, whichever rotation comes first. Let
P be any point outside of a and /3. Let us suppose that the
rotation around /3 turns P to P', and that the rotation
around ot turns P to P" and P' to (); then the rotation
around /3 will turn P" to Q, so that the final position of P
144
ROTATION AND TRANSLATION
[iv. i.
will be at Q, whether the rotation around a comes first, or
the rotation around j8. P and P" are in a plane j3i abso-
lutely perpendicular to a at a point B, and P' and Q are
o
in a plane 0'i absolutely perpendicular to a at the point
B f to which B moves in the rotation around /3.
In the rotation around the hyperplane containing P
and a. rotates on itself around its intersection with ft, and
the half-lines BP and B'P'> lying in this hyperplane, and
perpendicular to a, are coplanar. Also the hyperplane
containing P" and a rotates on itself and the half -line
BP" is coplanar with the half-line at B f into which it is
turned.
In the rotation around a the former hyperplane is turned
into the latter. But there is another hyperplane, the hy-
perplane containing fa and /3'i, which rotates on itself
around the line BB f , and in this hyperplane the half-
plane BB'-P is turned to the position of the half-plane
BB f -P", and the half-line B'P', which lies in the former,
to the position of B'Q, which must, therefore, lie in the latter.
Thus the half-line B'Q, and the half-line at B' into which
BP" is turned by the rotation around ]8, lying in the half-
plane -BjB'-P", and perpendicular to the edge 55', must
coincide.
8i,82]
TRANSLATION ALONG A LINE
145
We can break up the rotation around a. and the rotation
around into any number of equal parts and take these
smaller rotations alternately, and by a
limiting process we can derive a resultant
motion of which the two rotations are
the components.
We shall call a combination of rota-
tions around two absolutely perpendicular
planes a double rotation.
82. Translation along a line. Another
simple form of motion is translation along
a line. In this motion all the points of
the line move the same distance or at the
same rate, and in the same direction
along the line. In the translation of a plane or of a
plane figure along a line the plane as a whole remains
constantly coinciding with itself, and each point of the
plane or of the plane figure which is not a point of the
line remains constantly on the same side of the line and
at the same distance from it, moving in such a way that
its projection upon the line is translated along the line
as a point of the line itself. The line is called the line
of translation. The position of a point is fixed at each
instant, and any two points remain at a constant distance
from each other.
The translation of any figure along a line is a motion such
that each plane containing the line of translation, or the
intersection of the figure by any such plane, is itself trans-
lated along the line through the given distance or at the
given rate. Any two points, together with their projec-
tions upon the line of translation, if not all in one plane, are
the vertices of a tetrahedron .which remains congruent to
itself in the translation. Therefore, any two points remain
146 SYMMETRY [iv. n.
at a constant distance from each other, and any figure
remains invariable in the translation.*
In a hyperplane a translation along a line combined with
a rotation around the same line gives a screw motion. After
a screw motion the points of the hyperplane all take the
same positions, whether the translation and rotation are
simultaneous or taken in succession.
A screw motion in a hyperplane can be regarded as a
screw motion in hyperspace. In general, a screw motion
in hyperspace consists of a translation along a line com-
bined with a rotation around a plane containing the line.
Hyperplane figures in the hyperplane which is perpen-
dicular to the plane along the line (Art. 50) move in this
hyperplane in a screw motion along the same line.
II. SYMMETRY t
83. Symmetrical positions. Symmetry in a plane. Two
points are symmetrically situated with respect to the point
which lies midway between them. The point midway is
called their centre of symmetry. Two points are symmetri-
cally situated with respect to a line, plane, or hyperplane
which is perpendicular to the line of the two points at their
centre of symmetry. Such a line, plane, or hyperplane is
called the line, plane, or hyperplane of symmetry.
* For the points not on the line of translation the character of the motion depends
on the theory of parallels :
In the Elliptic Geometry translation along a line is a rotation around the pole,
polar line, or polar plane of this line, so that translation is not different from rotation.
In the Euclidean Geometry the translation takes place along a system of parallel
lines
In the Hyperbolic Geometry points which are not on the line of translation move
along a system of equidistant-curves. In the Hyperbolic Geometry we can also
have translation along a system of boundary-curves, the curves cutting orthogonally
a system of parallel lines, planes, or hyperplanes. In this case there is no line of
translation nor centre or axis of rotation (see the author's Non -Euclidean Geometry
chap. II, II).
t See Veronese on order and symmetry, Grundziige, 146, 147, and references.
82-84] IN A PLANE OR HYPERPLANE 147
Two figures are symmetrically situated with respect to a
point, line, plane, or hyperplane, when the points of the two
figures can be made to correspond in such a way that all
pairs of corresponding points are so situated.
THEOREM i. Two figures in a plane symmetrically situ-
ated with respect to a point can be made to coincide point for
point by a rotation of one of them through 180 around the
centre of symmetry.
THEOREM 2. Two figures in a plane symmetrically situ-
ated with respect to a line will not lose this relation of symme-
try, if they are rotated in the plane around any point of the
line through the same angle in opposite directions.
We can in this way bring any point of one figure into
coincidence with the corresponding point of the other, in the
line of symmetry, and then by a second rotation around
this new point we can bring a second pair of corresponding
points into coincidence.
THEOREM 3. When a plane figure is rotated in a hyper-
plane which contains the plane through 180 around some line
of the plane, it comes again into the same plane, to a position
which is symmetrical to its first position with respect to the
line; and so two figures in a plane symmetrically situated
with respect to a line of the plane can be made to coincide by
a rotation of one of them, in a hyperplane which contains the
plane, through 180 around the line of symmetry.
84. Symmetry in a hyperplane.
THEOREM i. In a hyperplane (or in ordinary space) any
two figures symmetrically situated with respect to a line can
be made to coincide point for point by a rotation of one of
them through 180 around the line of symmetry.
THEOREM 2. Two figures in a hyperplane symmetrically
situated with respect to a point O can be put into positions of
148
SYMMETRY
[iv. u.
symmetry with respect to any plane containing O by a rotation
of one of them through 180 around the line perpendicular to
this plane at O.
PROOF. Let a be the plane containing O, and c the line
perpendicular to a at this point. Let A and A 1 be two
corresponding points. If we rotate the figure to which A
belongs through 180 around c, we shall bring A again into
a plane with its first position and with A ' and c, to a posi-
tion where we will call it A i ; and then this point and A '
will be symmetrically situated with respect to the plane a.
THEOREM 3. Two figures in a hyper plane symmetrically
situated with respect to a plane will not lose this relation of
symmetry, if they are rotated in the hyperplane around any
line of the plane through the same angle in opposite directions.
THEOREM 4. When a hyperplane figure is rotated in hy-
per space through 1 80 around some plane of its hyperplane
(Art. 81), it comes again into tlie same hyperplane, to a posi-
tion which is symmetrical to its first position with respect to
the plane; and so two figures in a hyperplane symmetrically
situated with respect to a plane of the hyperplane can be made
to coincide by a rotation of one of them through 180 around
the plane of symmetry.
84, 8 S ]
WITH RESPECT TO A PLANE OR POINT
149
Thus figures which we call symmetrical could be made to coincide
if the space in which we live were a hyperplane in space of four dimen-
sions. This fact is mentioned and illustrated in nearly all popular
descriptions of the fourth dimension. See, for example, Fourth
Dimension Simply Explained (referred to on p. o), pp. 28, 48, 158,
214, etc.
85. Symmetry in hyperspace with respect to a plane and
with respect to a point. In space of four dimensions we have
symmetry with respect to a point, a line, a plane, and a
hyperplane (Art. 83).
THEOREM i. Any two figures symmetrically situated with
respect to a plane can be made to coincide point for point
by a rotation of one of them through 180 around this plane as
axis-plane.
THEOREM 2. Two figures symmetrically situated with
respect to a point can be made to coincide by a rotation of one
of them through 180 around each of two absolutely perpen-
dicular planes through the point.
PROOF. Let O be the centre of symmetry, and let a and
be two absolutely perpendicular planes through O. Let
A and A ' be two corresponding points. If we rotate the
figure to which A belongs through 180 around the plane a
SYMMETRY
[iv. n.
we shall bring A to a position where we will call it A\.
The three points A , A ', and A i are in a plane perpendicular
to a and /3, all three at the same distance from a and at the
same distance from /3. In this plane A 1 and A\ are sym-
metrically situated with respect to /?, and a second rotation,
around /8, will make the two figures coincide entirely.
We can take for a and /3 any two planes which are abso-
lutely perpendicular at O. Either figure can be rotated
around one of these two planes, and then the same figure
or the other figure can be rotated around the other plane.
86. Symmetry in hyperspace with respect to a line.
THEOREM. Two figures symmetrically situated with re-
spect to a line c can be put into positions of symmetry with
respect to any hyper plane containing c by a rotation of one of
them through 180 around the plane perpendicular to this
hyperplane along c.
AN
A,
PROOF. Let be the hyperplane containing c, and a
the plane perpendicular to 6 along this line. We are to
prove that a rotation of one figure through 180 around a
brings any point A of it into a position where this point
and the corresponding point A f of the other figure are
symmetrically situated with respect to 6.
The two points A and A' are symmetrically situated
with respect to the point O where the line A A' meets c.
8s, 86] WITH RESPECT TO A LINE OR HYPERPLANE 151
If we rotate the figure to which A belongs through 180
around a we shall bring A to a position where we will call
it Aij and we shall have A 1 and A\ symmetrically situated
with respect to the plane /? which is absolutely perpendicu-
lar to a at O (see proof of Th. 2 of Art. 85).
If A lies in /3, A\ will coincide with A', and these two
points will be coinciding points in 0.
If A does not lie in jS, the line A ' A\ is bisected by /? at a
point O' and is perpendicular to |8. The hyperplane 6,
perpendicular to a along c, contains the plane ]8 which is
absolutely perpendicular to a at O (Art. 50), and meets
the line A'Ai&t its middle point O' where this line meets 0.
We shall prove that the line A'Ai is perpendicular also to
another plane of 6 at O', namely, the plane determined by
c and the point O'. Thus we shall prove that the line A 1 A\
is perpendicular not only to /? but also to the hyperplane
9 (Art. 38, Th. 4).
A i lies in the line drawn through A perpendicular to a,
and therefore in a hyperplane with A, A r , and a. More-
over, the line c, lying in the plane a, is perpendicular to 00'
lying in the absolutely perpendicular plane /?, and to AA r
by hypothesis. Assuming that A does not lie in /3, we have
the line c perpendicular to the plane AA f A\, and then the
plane determined by c and O f perpendicular to the same
plane AA'A\\ for two planes in a hyperplane are perpen-
dicular when one contains a line perpendicular to the other.
Now A f A\ is perpendicular to OO', the intersection of these
two planes, since OO 7 lies also in ft. Therefore A* A\ is
perpendicular to the plane determined by c and O' ; for a
line in one of two perpendicular planes perpendicular to
their intersection is perpendicular to the other.
This proves that A 1 and A\ are symmetrically situated
with respect to 0; that is, that a rotation through 180
around a puts any point A of one of the two figures into
152 SYMMETRY [iv.it.
a position where this point and the corresponding point of
the other figure are symmetrically situated with respect
to 8, the hyperplane perpendicular to a along c.
In a single hyperplane we can see A, A', AI, the plane o, the line
of symmetry c, and the line OO' in ; but we cannot see the rest of
ft and of the hyperplane we can see only the plane determined by
c and O'.
87. Symmetry in hyperspace with respect to a hy-
perplane.
THEOREM i. Two figures symmetrically situated with
respect to a hyperplane will not lose this relation of symmetry,
if they are rotated around any plane of the hyperplane through
the same angle in opposite directions.
PROOF. Let A and A ' be two corresponding points, and
let a be some plane in the hyperplane of symmetry. The
line A A' is perpendicular to the hyperplane of symmetry,
and the plane through A absolutely perpendicular to a
is also perpendicular to this hyperplane (Art 50, Def.).
Therefore A A' lies in the plane (Art. 51, Th. i). in-
tersects the hyperplane of symmetry in a line c which
passes through O where meets a, and in the plane the
points A and A ' are symmetrically situated with respect to
the line c.
Now when the two figures rotate around a, the points
A and A ' rotate in around the point O, and if the rota-
tion takes place through the same angle in opposite direc-
tions these points remain symmetrically situated with
respect to the line c (Art. 83, Th. 2). In the new position
of the figures the line A A' \ lying in 0, is perpendicular to c,
and is bisected by c. This line is therefore perpendicular
to the hyperplane of symmetry (Art. 51, Th. i), and the
points A and A' are still symmetrically situated with re-
spect to this hyperplane.
86-89] ORDKR ON A LINE 153
THEOREM 2. // two figures are symmetrically situated
with respect to a hyper plane we can bring any Jour non-co-
planar points of one into coincidence with the corresponding
points of the other, in the hyperplane of symmetry, without
disturbing this relation of symmetry.
PROOF. Taking any plane in the hyperplane of symmetry
we can bring a point A into the hyperplane of symmetry,
and so into coincidence with its corresponding point A ' ;
then taking a plane through A in the hyperplane of sym-
metry we can bring a second point B into this hyperplane ;
a plane through A and B enables us to bring a third point
C into this hyperplane; and, finally, the plane ABC en-
ables us to bring the fourth point into this hyperplane.
THEOREM 3. // two figures are symmetrically situated
with respect to a point, line, plane, or hyperplane, any seg-
ment of one is of the same length as the corresponding segment
of the other, and any two corresponding angles, dihedral angles,
or hyperplane angles are equal.
III. ORDER
88. The two directions along a line. Two points which
are distinguished in some way from each other enable us
to distinguish the two directions along the line determined
by them (Art. 5). When we speak of the line AB we shall
often have in mind the direction on this line that B is from
A . We can speak of the line BA as a different line. Some-
times when we wish to call attention to a direction along
the line, or when there may be some question as to our
meaning, we shall say order AB instead of line AB.
89. Right and left in a plane. The two fundamental
principles of order. In a plane * the two half-planes which
are on opposite sides of a line (Art. 12) can be distinguished
*
* Or at least in a restricted portion of a plane, and so with all of this section.
1 54 ORDER [iv. m.
as right and left with respect to a particular direction along
the line. This notion we shall associate with the notion of
a right and left direction of rotation in the plane, by con-
sidering the rotation of half-lines. Two opposite half-lines
drawn in a plane from a given point would have to turn in
opposite directions around this point in order to pass to
the same side of the line on which they lie ; and two half-
lines drawn from a point along different lines would have to
turn in opposite directions in order to turn towards each
other through the interior of the angle of which they are
the sides.* We have then two principles on which we can
base the theory of order in a plane :
I. A and B being any two points of a plane, a point which
is on one side of the line AB is on the opposite side of the
line BA.
II. O, A , and B being any three non-collinear points of a
plane, B is on one side of the line OA and A is on the opposite
side of the line OB.
We shall speak of II as holding true of the half-lines OA
and OB, and we can express it by saying that two half-
lines drawn from a point along different lines lie in their
plane on opposite sides of each other.
Which shall be the right side of a line and which the left
is. an arbitrary matter that cannot be determined; but,
having assumed the two sides of one line, we shall be able
by means of I and II to determine the two sides of every
line in the plane. We prove this in what follows.
90. The right and left sides of lines through a point.
THEOREM i. Given a line BC and a half -line OA drawn
along some other line from a point O of the interior of the
* The half-lines drawn from a point in a plane are in cyclical order (Art. 6).
8g, QoJ IN A PLANE 155
segment BC, if II holds true of the half -lines OA and OB and
of the half-lines OA and OC, then I will hold true of the line
BC.
For, B and C are on opposite sides of the line OA (Art. 12,
Th. 2), and by II it follows that A must be on opposite
sides of the lines OB and OC, or, what is the same thing,
of the lines CB and BC.
COROLLARY. // / holds true of BC and II holds true of
OA and OB, then II will hold true of OA and OC.
THEOREM 2. Given three half-lines, OA, OB, and OC,
drawn in a plane from a point O along three different lines;
if I holds true of the line OA , and II holds true of the half-
lines OA and OB and of the half-lines OA and OC, then II
will hold true of the half -lines OB and OC.
PROOF. Let OA ' be the half -line opposite to OA . A and
A ' are on opposite sides of any other line through O (Art.
12, Th. 2), and if I holds true of the line OA, we can substi-
tute OA ' for OA in any application of II. Hence, if A and
C lie on the same side of the line OB, we can take A' in place
of A ; that is, we can assume that A and C lie on opposite
sides of the line OB, so that there is a point B\ of this line
lying in the interior of the segment AC, a point of the half-
line OB or of the opposite half-line. Then
I5^> ORUKR |iv. m.
Bi and C lie on the same side of the line OA ,
and B l " A " k< " OC;
or, since OA and OC lie on opposite sides of each other,
Bi lies on opposite sides of the lines OA and OC.
Now BI and B lie on the same side of every line through
O except the line B\B itself, or they lie on opposite sides of
every line through O except this line, and in any statement
like the last we can substitute B for B\. That is,
B lies on opposite sides of the lines OA and OC.
But by the conditions of the theorem
OA and OB lie on opposite sides of each other,
and by hypothesis
C and A lie on opposite sides of OB.
Therefore we must have, finally,
OB and OC lie on opposite sides of each other.
Given a half-line OA, we have only to assume that I
holds true of the line in which this half-line lies and that
II holds true of all the half-lines drawn from in other
lines of the plane taken with OA. By the first theorem I
will hold true of all the lines through O, and by the second
theorem II will hold true of any two non-opposite half-
lines drawn from O.
91. Right and left with respect to the sides of a triangle.
The notation, order ABC. Given three non-collinear
points of a plane, A, B, and C, we shall write order ABC
to denote that side of a line, right or left, on which A is
of the line BC.
Then if I holds true of the line J5C, order ACB is the oppo-
go, 91] WITH RESPECT TO A TRIANGLE 157
site of order A BC, and if II holds true of the half -lines BC
and BA, order CBA is the opposite of order ABC. In
other words, the opposite side of a line is indicated according
to I by interchanging the last two letters, and according
to II by writing the set of three letters backwards.
THEOREM. Given three non-collinear points, A, B, and
C, if I holds true of the lines BC, CA, and AB, and if II
holds true of the half-lines BC and BA and of the half-lines
CB and CA, then II will hold true of the half -lines AB and
AC.
PROOF. We have given that order CBA is the opposite
of order ABC, and that order BCA is the opposite of order
ACB\ also that ABC and ACB, BCA and BAG, and
CAB and CBA are pairs of opposite orders. We are to
prove that order CAB is the opposite of order BAC.
This will appear at once if we arrange these expressions
in a table. We find, in fact, that one side of a line, right
or left, is denoted by any one of the three expressions
order ABC, order BCA, order CAB,
and the other side by any one of the three expressions
order ACB, order BAC, order CBA.
When I holds true of all three of these lines and II of all
three pairs of half-lines, we can say that a cyclical per-
158 ORDER [iv. in.
mutation of the three letters does not change the meaning
of the expressions order ABC and order ACS.
Given the right a&d left sides of a line BC, we can assume
that I holds true of this line. Then by II we can determine
the right and left sides of all lines intersecting BC. I will
hold true of all these lines, and II will hold true of any
half-lines drawn from a common point along two of
these lines.
Now if a line A D does not intersect BC, we can draw a
transversal, say the line AB, and determine the right and
left sides of this new line by applying II to the half-line A B
and the two opposite half-lines drawn from A along this
line. If we draw any other transversal DC, II, being true
of AD and AC and of CA and CD (Art. 90, Th. 2), will
hold true of DA and DC, as also of the opposite half-line,
AD produced, and DC. Thus the right and left sides of
the line AD are determined by means of the line BC and
any transversal.
We shall say, therefore, that I and II constitute a defi-
nition of right and left for all lines of the plane, given the
right and left sides of one particular line.
Order A BC may be used to denote a direction of rotation
in the plane around the point B. If we traverse the tri-
angle ABC, passing along AB from A to B, and so on, we
shall turn at each vertex (through an exterior angle) in the
same direction, the direction indicated by order ABC.
92. Order in a plane unchanged by any motion in the
plane. Figures symmetrical in a plane.
THEOREM i. Given two triangles ABC and A 1 B'C' in
the same plane, order A'B'C' is the same as order ABC if
each of the three segments A A', BB', and CC' is less than
J h, where h is the shortest altitude of either of these triangles.
i, 92]
UNCHANGED BY ANY MOTION
159
PROOF. A and A ' are on the same side of BC, since A A '
is less than the distance of A from the line BC. Therefore
order A' EC is the same as order ABC.
Let m and n be the perpendicular lines which bisect the
four angles formed by the lines AC and BC. Then A is
at a distance greater than \ h from each of these bisecting
lines, the perpendicular
from A to a bisecting
line being one-half of an
oblique line from A to
the line BC. It follows
that A and A 1 lie to-
gether in one of the four
right angles formed by
the two bisecting lines. Likewise B and B' lie together
in one of these four right angles, in one of the two, indeed,
which are adjacent to the right angle containing A and
A 1 . Now the line A'C consists of the point C and two
opposite half-lines lying in two vertical right angles,
neither of which is the angle containing B and B'. There-
fore, no point of the interior of the segment BB' can be a
point of the line A'C, and order A'B'C is the same
as order A'BC.
Starting now with order A'B'C' , we prove that this is
the same as order A'B'C, and therefore that order A'B'C'
is the same as order ABC.
160 ORDER [iv. m.
When the two triangles are equal we can consider them
as representing two positions of a triangle moving in the
plane. We shall say, then, that the order of a triangle
cannot be changed by any motion of the triangle in its
plane,* regarding this statement, however, as in part a
definition of the phrase motion in a plane. In particular,
order is not changed by a rotation or translation of the
triangle in its plane.
THEOREM 2. Two figures in a plane symmetrically
situated with respect to a line cannot be made to coincide point
for point by any motion in the plane unless they are groups
of collinear points.
PROOF. We can make any two points of one coincide
respectively with the corresponding points of the other in
the line of symmetry, without disturbing the relation of
symmetry (Art. 83, Th. 2), but in this position any cor-
responding points not collinear with these two will be on
opposite sides of the line. Therefore any two corresponding
triangles are in opposite orders,f and the order of one can-
not be changed to the order of the other by any motion in
the plane.
If two figures in a plane symmetrically situated with
respect to a line have each a line of symmetry with respect
to which they are self -symmetrical (for example, two
isosceles triangles), one can be made to coincide with
the other as a whole by a motion in the plane, but not
point for point. Any point of one not a point of its line
of symmetry will coincide with that point of the other which
corresponds to its symmetrical point.
* Or at least in a restricted portion of the plane.
t When we say that two sets of points are in the same order or in opposite
orders we assume that each point of one set is associated with one and only one point
of the other set, and then, taking the points of one set in any order we please, we
compare this order with the corresponding order of the points of the other set.
92,93] IN A HYPERPLANE l6l
93. Order in a hyperplane. The two fundamental
principles. The notation, order ABCD. In a hyperplane
the two half-hyperplanes which are on opposite sides of
a plane (Art. 23) can be distinguished with respect to the
order of a triangle in the plane ; and by associating the two
sides of a plane with the two directions of rotation in the
hyperplane of a half -plane around its edge, we can derive,
as before, two principles which together will serve to define
the two sides of any plane when we have given the two sides
of a particular plane :
I. A, B, and C being any three non-collinear points of a
hyperplane, a point which is on one side of the plane of order
ABC is on the opposite side of the plane of order ACB.
II. A , B, C, and D being any four non-coplanar points
of a hyperplane, B is on one side of the plane of order ACD
and A is on the opposite side of the plane of order BCD.
We shall speak of II as holding true of the half-planes
CD- A and CD-B, and we can express it by saying that two
half-planes with a common edge,* lying in different planes,
lie in their hyperplanes on opposite sides of each other.
Theorems exactly analogous to those stated in Art. 90
of the half-lines drawn in a plane from a point O hold true
of the half -planes with a common edge lying in a hyperplane.
As to Art. 91, suppose that three planes intersect by
twos in three different lines, a, b, and c y and let A be a point
of a which is not a point of b or c, B a point of b which is
not a point of a or c y and C a point of c which is not a point
of a or b and not collinear with A and B. It will make no
difference whether the three lines, and so the three planes,
have a point common to them all or not. We can treat
the half -planes aB, bC> . . . bA, . . . just as in Art. 91
we have treated the half -lines AB, J8C, . . . BA, . . . .
* The edge taken in the same " order" in both.
II
162 ORDER [iv. m.
Given the two sides of a particular plane in a hyperplane,
and assuming that I holds true of this plane, we can by II
determine the two .sides of any plane in the hyperplane
intersecting this plane, and then of any other plane in the
hyperplane by using a plane which intersects this plane
and the given plane. That is, we determine the two sides
of every plane in the hyperplane by following exactly the
methods of Art. 91.
Given four non-coplanar points, A, B, C, and D, we shall
write order A BCD to denote that side of a plane on which
A is of the plane of order BCD.
In accordance with the definitions and theorems given
for order in a plane, we can say that a cyclical permutation
of the last three letters does not change the meaning of the
expression order A BCD, and that a non-cyclical permuta-
tion of these letters changes the expression to one denoting
the opposite order, if I, as stated for hyperplanes, holds true.
On the other hand, if II holds true, for example, of the
half-planes CD- A and CD-B, then order BACD is the
opposite of order A BCD. That is, the order is changed
to the opposite, according to II, by an interchange of the
first two letters.
By combining these operations we find that there are
twelve different arrangements * of the four letters for which
the expressions of the form order A BCD denote one
side of a plane, and twelve for which these expressions
denote the opposite side.
We cannot speak of the right and left sides of a line except as we
associate the line with some plane in which it lies ; but direction along
a line is a property of the line itself and is independent of any plane
or space that contains it.
Likewise, we cannot speak of one and the other side of a plane
except as we associate the plane with some hyperplane in which it
* Corresponding to the group of twelve even permutations of four things.
93, 94] UNCHANGED BY ANY MOTION 163
lies ; but " order in the plane," or direction of rotation, is independent
of any hyperplane.
94. Order in a hyperplane unchanged by any motion
in the hyperplane. Figures symmetrical in a hyperplane.
THEOREM i. Given two tetrahedrons, A BCD and
A'B'C'D', in the same hyperplane, order A'B'C'D' is the
same as order A BCD if each of tftefour segments A A', BB',
CC', and DD' is less than % h, where h is the shortest altitude
of either of these tetrahedrons.
PROOF. A and A' are on the same side of the plane
BCD, since A A' is less than the distance of A from this
plane. Therefore order A' BCD is the same as order
A BCD.
Let a and ,5 be the perpendicular planes which bisect the
four dihedral angles formed by the planes A CD and BCD.
Then we prove, exactly as in Art. 92, that A and A' lie
together in one of the four right dihedral angles formed by
these two bisecting planes, and that B and B f lie together
in an adjacent right dihedral angle. It follows that no
point of the interior of the segment BB' can be a point of
the plane A 'CD, and that order A'B'CD is the same as
order A' BCD.
Starting now with order A'B'C'D', we prove first that
this is the same as order A'B'C'D, and then the same as
order A'B'CD.
Thus we have, finally, order A'B'C'D' the same as order
A BCD.
We shall say that the order of four non-coplanar points
of a hyperplane figure cannot be changed by any motion
of the figure in its hyperplane, thus defining in part the
phrase motion in a hyperplane.
THEOREM 2. Two figures in a hyperplane symmetrically
situated with respect to a plane cannot be made to coincide
164 ORDER [iv. m.
by any motion in the hyperplanc unless they are plane
figures.
Figures which we tall symmetrical in the ordinary geometry,
figures "with the parts of one equal to the corresponding parts of
the other but arranged in the opposite order/ 1 can be placed in posi-
tions of symmetry with respect to a plane. They will be so placed
as soon as we have placed three non-collinear points of one upon the
corresponding points of the other. But it is better to define symmet-
rical figures in a hyperplane as figures which are not plane figures,
but can be placed in positions of symmetry with respect to a plane
by a motion of one or both in their hyperplane.
95. Order in hyperspace. The two fundamental principles.
The notation, order ABCDE. In hyperspace we shall
distinguish the two sides of a hyperplane (Art. 28) with
respect to the order of four non-coplanar points of it. We
proceed as before and write down two principles which
together will serve to define the two sides of any hyper-
plane when we have given the two sides of a particular
hyperplane :
I. A, B, Cj and D being any four non-coplanar points, a
point which is on one side of the hyperplane of order A BCD
is on the opposite side of the hyperplane of order ABDC.
II. A, B 9 C, Z), and E being any five points not in one
hyper plane , B is on one side of the hyperplane of order ACDE,
and A is on the opposite side of the hyperplane of order BCDE.
We can express II by saying that two half-hyperplanes
with a common face,* lying in different hyperplanes, lie
on opposite sides of each other.
Theorems analogous to those of Art.. 90 follow at once
for half-hyperplanes having a common face.
As to Art. 91, suppose that three hyperplanes intersect
by twos in three different ..planes, a, 0, and 7, and let A, be
* The face -taken in the same order in both.
94-96] IN HYPERSPACE 165
a point of a which is not a point of /8 or 7, B a point of ft
which is not a point of a or 7, and C a point of 7 which is
not a point of a or /? and not collinear with A and JB. We
can then carry through the methods of Art. 91 just as we
do in the case of a hyperplane (Art. 93).
Given five points, A , B, C, /?, and , not points of one
hyperplane, we shall write order ABODE to denote that
side of a hyperplane on which A is of the hyperplane of
order BCDE.
Order ABDEC denotes the same side of a hyperplane;
order ABC ED and order BACDE denote the opposite
side of a hyperplane. Thus we can obtain sixty different
permutations * of the five letters for which the expressions
of the above form denote one side of a hyperplane, and sixty
for which these expressions denote the opposite side.
96. Order in hyperspace unchanged by any motion.
Symmetrical figures.
THEOREM i. Given two pentahedroids, A BCDE and
A'B'C'D'E', order A'B'C'D'E' is the same as order
A BCDE if each of the five segments A A', BE', CC', DD',
and EE f is less than J h, where h is the shortest altitude of
either of these pentahedroids; that is, the shortest distance
between a vertex and the hyperplane of the opposite cell.
PROOF. A and A' are on the same side of the hyper-
plane BCDE, since A A 1 is less than the distance of A from
this hyperplane (Art. 40, Th. i, Cor.). Therefore order
A 'BCDE is the same as order A BCDE.
Now by considering the perpendicular hyperplanes
which bisect the four hyperplane angles formed by the
hyperplanes ACDE and BCDE, we prove, exactly as in
Art. 92, that order A'B'CDE is the same as order A 'BCDE
(using in this case Th. 2 of Art. 57).
* Corresponding to the group of sixty even permutations of five things.
1 66 ORDER [iv. in.
Moreover, we find in this proof that B is at a distance
greater than | h from the hyperplane A 'CDE. In the same
way we prove that C. is at a distance greater than f h from
the hyperplane A'BDE. Then, since the segments BB'
and CC f are each less than \ h, we can bisect the four
hyperplane angles formed by these two hyperplanes and
prove that no point of the segment CC" is a point of the
hyperplane A'B'DE. Therefore order A'B'C'DE is the
same as order A'B'CDE.
Starting now with order A'B'C'D'E', we prove first
that this is the same as order A'B'C'D'E, and then that
it is the same as order A'B'C'DE.
Thus we have, finally, order A'B'C'D'E' the same as
order ABCDE.
We shall say that in any figure the order of five points
not in a hyperplane cannot be changed by any motion of
the figure in hyperspace, thus defining in part the phrase
motion in hyperspace.
THEOREM 2. Two figures symmetrically situated with
respect to a hyperplane cannot be made to coincide point for
point by any motion in hyperspace unless they are hyperplane
figures.
In studying geometry of four dimensions we shall use
the word symmetrical only with reference to figures which
are not themselves hyperplane figures and which can be
placed in positions of symmetry with respect to a hyper-
plane by some motion in hyperspace. All the segments,
angles, dihedral angles, and hyperplane angles of one figure
are equal to the corresponding parts of the other (Art.
87, Th. 3), but the points of the two figures are arranged
in opposite orders.
We have in two vertical piano-polyhedral angles an
example of symmetrical figures, the two piano-polyhedral
g6, 97l MOTION IN A PLANE 167
angles being symmetrically situated with respect to a line
(Art. 86, Th.).
Two vertical polyhedroidal angles, on the other hand,
are equal and can be made to coincide, being symmetrically
situated with respect to a point (Art. 85, Th. 2).
IV. MOTION IN GENERAL
97. Motion of a plane on itself. In the ordinary geom-
etry we speak of the motion of a plane figure in its plane,
and of figures in space. With a plane figure we can associ-
ate the entire plane. Even in the case of motion in a plane
we can think of the entire plane as moved on itself. In the
motion of a plane on itself the order of any three non-
collinear points cannot be changed (Art. 92, Th. i, remark).
THEOREM i. Given any two positions of a plane, a
motion which takes three non-collinear points of it from their
first to their second positions will take every point of it from
its first to its second position.
PROOF. In the first place, each point of the line con-
taining two of the given points remains collinear with them
and at the same distances from them. Moreover, since
the plane as a whole after such a motion coincides with its
second given position, any point not a point of this line
remains at the same distance from it and with the same
projection upon it : it must come to its second position or
to a position symmetrical to its second position with respect
to this line. But the latter is impossible, as its distance
from the third given point would then be changed.
COROLLARY. // after a motion of a plane on itself one
point of the plane occupies the position that it occupied before,
then every point will occupy the position that it occupied before,
or every point will occupy a position that could have been
l68 .MOTION IN (iKXKRAL |iv. iv.
reached by a rotation of the plane on itself through a certain
angle around the given point.
THEOREM 2. In. a motion of a plane on itself, if two
positions of any point differ, let A be the first and B the
second position of such a point, and let C be the second posi-
tion of the point whose first position was B. If the line of
symmetry of B and C coincides with the line of symmetry of
A and B, that is, if C coincides with A, then the second
position could have been reached by a rotation of the plane on
itself through 180 around the centre of symmetry of the two
points. If the lines of symmetry do not coincide but do
intersect in a point P, the second position could have been
reached by a rotation of the plane around P. If the lines of
symmetry do not coincide but do have a common perpendic-
ular line, the second position could have been reached by
a translation along this perpendicular (see the author's
Non-Euclidean Geometry, chap. II, I, 8, proof, p. 42).*
98. Motion of a hyperplane on itself. We can now
speak of the motion of a figure in hyperspace. With a
hyperplane figure we can associate the entire hyperplane,
whether the motion of the figure be anywhere in hyper-
space or within the hyperplane itself. Indeed, in the ordi-
nary geometry we can associate with a figure any additional
points, and so speak of all space as moved about on itself,
even though we know of no hyperspace within which it
lies. In a motion of a hyperplane on itself the order of any
four non-coplanar points cannot be changed (Art. 94, Th. i,
remark).
THEOREM i. Given two positions of a hyper plane, a
motion which takes four non-coplanar points of it from their
* In the Hyperbolic Geometry the lines of symmetry may be parallel. The
second position is then one which could have been reached by a translation along a
system of boundary-curves (see the author's Non-Euclidean Geometry, chap. II, II).
97,98] IN A HYPERPLANE 169
first to their second positions will take every point of it from
its first to its second position.
PROOF. In the first place, each point of the plane con-
taining three of the given points will come to its second
position (Art. 97, Th. i). Moreover, since the hyperplane
as a whole coincides with its second given position, any
point not a point of this plane remains at the same distance
from it and with the same projection upon it: it must
come to its second position or to a position symmetrical
to its second position with respect to this plane. But the
latter is impossible, as its distance from the fourth given
point would then be changed.
COROLLARY i. Given bwo positions, one obtained from
the other by a motion of a hyperplane on itself, then any
motion of the hyperplane on itself which takes three non-
collinear points of it from their first to their second positions
will take every point from its first to its second position.
In particular, a motion of a plane on itself (Art. 97, Th. 2)
will determine a motion on itself of any hyperplane contain-
ing the plane.
COROLLARY 2. // after a motion of a hyperplane on itself
each of two points of the hyperplane occupies the position that
it occupied before, then every point will occupy the position
that it occupied before, or every point will occupy a position
that could have been reached by a rotation of the hyperplane
on itself through a certain angle around the line containing
the two given points.
THEOREM 2. // after a motion of a hyperplane on itself
one point of it occupies the position that it occupied before,
then every point will occupy the position that it occupied before,
or every point will occupy a position that could have been
reached by a rotation of the hyperplane on itself through a
certain angle around a line containing the given point.
170
MOTION IN GENERAL
[iv. iv.
The proof follows that of Th. 2 of Art. 97. The points
A y By and C lie on a sphere, on a circle whose axis is the
axis of the rotation. .
99. Every motion in a hyperplane equivalent to a motion
of a plane on itself or to a screw motion.
THEOREM. // after a motion of a hyperplane on itself
there is no point which occupies ttie position that it occupied
before, then every point will occupy a position that could
have been reached by the motion on itself of some plane of the
hyperplane or by a screw motion.
PROOF. Let A be the first position and B the second
position of some point, let C be the second position of the
point whose first position was B y and let D be the second
position of the point whose first position was C. We will
assume that A, B, C, and D are not collinear, and if they
are coplanar that the triangles ABC and BCD are in
opposite orders in their plane. Then
and
AB = BC = CD,
A ABC = A BCD.
Let b and c be the half-lines bisecting the angles ABC
and BCD. The lines containing these half-lines have a
common perpendicular m, even if coplanar, ABC and BCD
being in opposite orders (see the author's N on- Euclidean
Geometry, chap. I, I, 10). The figure DCBA taken with
c and b is congruent to the figure A BCD taken with b and c y
and if we place the former upon the latter, each of these
98,99] SCREW MOTION 171
half-lines will take the position which the other occupied
before. If the lines containing these half-lines have only
one common perpendicular m, this perpendicular will
also fall upon itself, and its distances from B and C are
equal. If these lines have more than one common per-
pendicular, one of these perpendiculars will be a line
through B intersecting the half-line c, and another will be
a line through C intersecting the half-line b. Then the
perpendicular mid- way between them, being equidistant
from B and C, can be taken for the line m (see Art. 62, Th. 2).
When A, B, C, and D are coplanar, a screw motion along
m with a rotation of 180 and a translation that puts A
upon B will put the entire hyperplane into its second
position.
Supposing that A, B y C, and D are not coplanar, let
A', B f j C' , and D' be their projections upon m. Then,
in the first place, the tetrahedrons CBB'C and BCC'B'
are congruent. Furthermore, A and C of the isosceles
triangle ABC are symmetrically situated with respect to
the line containing 6, and m is symmetrical to itself with
respect to this same line. Therefore A' and C' are sym-
metrically situated with respect to this line, and the tetra-
hedrons ABB' A' and CBB'C are congruent (Art. 84,
Th. i). It follows that the tetrahedrons ABB' A 9 and
BCC'B' are congruent. In the same way we prove that
these are congruent to the tetrahedron CDD'C'. Then
a screw motion along m putting A upon B will put ABB' A 9
upon BCC'B' and BCC'B' upon CDD'C' \ that is, it will
put B and C into their second positions and move the entire
hyperplane from its first to its second position.
The student may prove the following theorems :
Two successive rotations of a hyperplane on itself around two
different axis-lines are together equivalent to a single rotation around
an axis-line, if the two given lines intersect. When the two axis-
172 MOTION IN GENERAL [iv. iv.
lines are perpendicular to a plane the two rotations together will be
equivalent to some kind of motion which the plane can have on itself
(Art. 07, Th. 2).
Conversely, if two successive rotations of a hyperplane on itself
are together equivalent to a single rotation, the axes of the three
rotations will be the three lines of intersection of three planes inter-
secting two by two : they will meet in a point if two of them meet
in a point ; or they will be perpendicular to a plane if two of them
are perpendicular to a plane.*
100. Motion in hyperspace. In the motion of a figure
in hyperspace we can associate with it any additional
points, and so speak of all hyperspace as moved about on
itself. In this article we shall speak of a figure rather
than of all hyperspace, but we shall assume that all points
that we need to consider are included in the figure. In any
motion in hyperspace the order of five points not points of
one hyperplane cannot be changed (Art. 96, Th. i, remark).
THEOREM i. Given two positions of a figure, one of which
can be obtained from the other by a motion in hyperspace,
then any motion in hyperspace which takes Jour non-coplanar
points from their first to their second positions will take every
point of the figure from its first to its second position.
For each point of the hyperplane of the four given points
comes to its second position in this motion (Art. 98, Th. i)
and any point which is not a point of this hyperplane
remains at the same distance from it, on the same side of it,
and* with the same projection upon it, so that it must also
come to its second position.
COROLLARY i. In particular, a motion of a hyperplane
on itself will determine a motion in hyperspace of any figure
containing the hyperplane or containing four non-coplanar
points of it.
"The only other possibility is in the Hyperbolic Geometry, where the three
axes may be parallel.
99, ioo] IN HYPERSPACE 173
COROLLARY 2. // after a motion of a figure in hyper space
each of three non-collinear points occupies the position that
it occupied before, then every point of the figure will occupy
the position that it occupied before, or every point will occupy
a position that could have been reached by a rotation of the figure
through a certain angle around the plane of the three points.
THEOREM 2. // after a motion of a figure in hyper space
each of two points occupies the position that it occupied before,
then every point of the figure will occupy the position that it
occupied before, or every point will occupy a position that
could have been reached by a rotation of the figure around
a certain plane containing the two given points.
PROOF. Each point of the line containing the two given
points occupies the position that it occupied before, and
any hyperplane perpendicular to this line taken as a whole
occupies the position that it occupied before (Art. 38, Th. i).
Now such a hyperplane could be put into this second posi-
tion by a motion on itself, since each point of the given line
except its intersection with the hyperplane always remains
on the same side of the hyperplane with respect to any
given order of points in the hyperplane (Art. 96, Th. i).
But in the hyperplane one point occupies the position that
it occupied before, namely, its intersection with the given
line. Therefore, the second position of the hyperplane
could have been reached by a rotation around an axis-line
through this point (Art. 98, Th. 2), and the second position
of the figure could have been reached by a rotation around
the plane determined by this axis-line and the line contain-
ing the two given points.
This theorem might also be proved by interpreting Th. 2
of Art. 97 as a theorem of Edge Geometry (Art. 78).*
* Edge Geometry is Elliptic ; and in the Elliptic Geometry a translation along a
line is a rotation around the pole of the line. In other words, the lines of sym-
metry of AB and BC always meet in a point.
174 MOTION IN GENERAL [iv. iv.
101. Motion in which one plane remains fixed or moves
only on itself.
THEOREM i . // after a motion of a figure in hyper space
one point occupies the position that it occupied before, then
every point will occupy the position that it occupied before,
or every point will occupy a position that could have been
reached by a single or double rotation (Art. 81).
This can be proved by interpreting the theorem of Art. 99
as a theorem of the three-dimensional Point Geometry at
the given point. A, B, C, and D are to be interpreted as
half -lines, b and c as the bisecting half -planes of the dihedral
angles A BC and BCD, and m as a common perpendicular
plane of the plane containing b and the plane containing c.
The screw motion on m is then to be interpreted as a
double rotation around m and the plane absolutely perpen-
dicular to m at this point.
THEOREM 2. In a motion of a figure in hyper space, if
two positions of any points differ, let four of these points be
taken in such a way that A, B, C, and D are their first posi-
tions, and B, C, D, and E their second positions (see Art. 99).
There is then through C a plane y, a plane of symmetry for
A and E and for B and D. Let 8 be the corresponding plane
through D, lying in the second position of the plane whose
first position is that of y. Then if y and 8 intersect, or have
a common perpendicular hyperplane, the second position
of the figure is one which could have been reached from the
first by a motion in which one plane remains fixed or moves
only on itself*
* In the Hyperbolic Geometry we can have planes y and & which do not intersect
and do not have a common perpendicular hyperplane. These planes do, however,
have a common perpendicular plane *? (see first foot-note, p. 112). The theorem is
therefore true without restriction, whatever our theory of parallels, and can be
stated very simply as follows:
In a motion of a figure in hyperspace, if two positions of any points differ, the
second position of the figure is one which could have been reached from the first
by a motion in which one plane remains fixed or moves only on itself.
IQI] ONE PLANE MOVES ON ITSELF 175
PROOF. The proof need be given here only for the case
where the four points are non-coplanar and their positions
are in different hyperplanes or in opposite orders in the
same hyperplane. Then
AB = BC = CD = DE,
Z ABC = Z BCD = Z CDE,
and tetrahedrons A BCD and BCDE are congruent. More-
over, the tetrahedron DCBA is congruent to the tetra-
hedron A BCD, and EDCB to BCDE (Art. 93). The
hyperplane angle E-DCB-A is the same as the hyperplane
angle A-BCD-E, and the figure EDCB A is congruent to
the figure A BCDE. When these figures do not lie in one
hyperplane they have the same order (Art. 95), and when
they do lie in one hyperplane, A BCD and BCDE being in
opposite orders, any order of points of one figure in this
hyperplane is changed to the order of the corresponding
points of the other by a rotation of the hyperplane through
180 around one of its planes (Art. 84, Th. 4, and Art. 94).
The centre of symmetry of A and E and the centre of
symmetry of B and D do not change their positions when
we reverse the figure, placing EDCB A upon A BCDE.
Moreover, these two centres of symmetry cannot be
collinear with C. If they were, the entire figure A BCDE
would lie in one hyperplane, and, taken in two ways as
above, would have symmetrical positions with respect to
a line, so that corresponding points would be in the same
order in the hyperplane (Art. 84, Th. i). There is then
a plane 7 determined by these two points and C, a plane
of symmetry of A and E and of B and Z>, and the figure is
reversed by a rotation in hyperspace through 180 around 7.
At A, B, D, and E are also planes corresponding to 7,
say a, j8, 8, and e; 5, for example, being in the second
position of the plane whose first position is that of 7. The
176 MOTION IN GENERAL [iv. iv.
figure consisting of A , B, C, and 7 is congruent to the figure
consisting of J5, C, D, and 8. It is also congruent to the
figure consisting of /, Z>, C, and 7. Therefore the last
two figures are congruent to each other.
If all these points and planes lie in a hyperplane, then
the middle points of AB, BC, CD, and DE lie in a plane 77
perpendicular to all the planes a, /3, 7, 8, and e. For the
line determined by the middle points of AB and DE and
the line determined by the middle points of BC and CD
are both perpendicular to 7, and therefore lie in a plane
perpendicular to 7 (see foot-note, p. 77) ; and, since this
plane is determined by any three of the middle points, it
is perpendicular to 8 and to /3, and then also to a and to e.
The points A, B, C, D, and E are at the same distance
from rj, A y C, and E lying on one side of 17 and B and D
on the other side ; and the projections of AB, BC, CD, and
DE upon 77 are equal. Let A', B' ', C", >', and ' be the
projections upon t\ of A, B, C, D, and E. If we rotate
the hyperplane through 180 around the plane rj and at the
same time move this plane on itself so as to move A ' to B f
and B' to C', then we shall have moved A to B and B to C.
Four non-coplanar points of the hyperplane will be in
their second positions: that is, four non-coplanar points
of the given figure will be in their second positions, and
the given figure itself will be entirely in its second position
(Art. 100, Th. i).
The planes 7 and 8, not lying in one hyperplane, may
intersect in a point 0, a point common to 8 and ft, and there-
fore also to all five of the planes, a, j3, 7, 8, and . O cor-
responds to itself in the symmetry of the planes ft and 8
with respect to 7, and in the two congruent figures BCDS
and EDCy. O is therefore at the same distance from B,
C, and D, and so also from A and E. The projections of
O upon the hyperplanes of the tetrahedrons A BCD and
xoi] ONE PLANE MOVES ON ITSELF 177
EDCB will be corresponding points in the congruence of
these two tetrahedrons. O will lie within the hyperplane
angle A-BCD-E or within the vertically opposite hyper-
plane angle. That is, O will lie on opposite sides of the
hyperplanes of orders A BCD and EBCD, and so on the
same side of the hyperplanes of orders A BCD and BCDE
(Art. 93). O will then coincide with itself in the two
positions of the figure, and the second position is one which
could have been reached by a single or double rotation
around O (Th. i).
If 7 and 6, not lying in one hyperplane, do not intersect
but do have a common perpendicular hyperplane, they
have a common perpendicular plane rj (Art. 63, Th. 3) ;
and if there is only one common perpendicular plane, this
plane will coincide with itself when we place EDCy upon
BCDd. This plane cuts y and 5 in linear elements, and
is at the same distance from C and D. If y and 8 have
more than one common perpendicular plane,* one of these
planes passes through C and another through Z>, and there
is a plane mid-way between these two (Art. 63, Th. 4)
which can be taken for ?/, coinciding with itself in these
two positions of the figure, and therefore at the same
distance from C and D. The plane 17, being perpendicular
to y and passing through a line of 5, will intersect ft in a cor-
responding line and be perpendicular to ft also, since ft and
5 are symmetrically situated with respect to y. In other
words, the plane t\ is a common perpendicular plane to all
five of the planes a, /3, 7, 5, and , and is at the same distance
from all five of the points A,B,C,D, and . Let A', B', C",
Z?', and E' be the projections of these points upon 17. Then
A'B' = B'C' = CD 1 = />'';
* Two planes not in a hyperplane can have more than one common perpendicular
plane only in the Elliptic Geometry, and in the Elliptic Geometry two planes always
intersect at least in a point (see second foot-note, p. 112). Therefore it is not
really necessary to consider this case.
N
178 MOTION IN GENERAL [iv. iv.
B'C f and C'ZX, for example, are symmetrically situated
with respect to the plane 7, or, we may say, to the inter-
section of 7 and ?j. The hyperplane angles B-y-C and
C-y-D are equal, and so therefore are all four of the hyper-
plane angles A-rj-B, etc. These four hyperplane angles
are in the same order around the plane 77, and, if we rotate
the figure around rj through the hyperplane angle A-ij-B
the half-hyperplanes fj-A, rj-B, etc., will be turned to the
positions of the half-hyperplanes rj-B, ij-C, etc. If at the
same time we move TJ on itself so that A' shall move to B'
and B' to C", A will move to B and B to C : four non-
coplanar points of the figure will move to their second
positions, and the entire figure will move to its second
position (Art. 100, Th. i).
102. Composition of rotations.
THEOREM i. Two successive rotations around two dif-
ferent axis-planes are together equivalent to a single rotation
around an axis-plane, if the two axis-planes intersect in a
line (Art. 100, Th. 2).
THEOREM 2. // two successive rotations are together
equivalent to a single rotation around an axis-plane, the axis-
planes of the two rotations are in a hyperplane, and when
they have a point in common they intersect in a line.
PROOF. The points of the axis-plane of the third rota-
tion are left by the second rotation in their original posi-
tions. Let A be the position of one of these points. If
the first rotation moves this point to another position JB,
the second rotation will move it back from B to A . Then
the hyperplane of symmetry of A and B will contain the
axis-planes of both of these rotations.*
* The three axis-planes will all intersect in one line, or will all be perpendicular
to one plane, or (in the Hyperbolic Geometry) they will be parallel.
The first theorem of Art. 101 and the two theorems of this article, so far as ro-
tations around a fixed point are concerned, were proved by N. F. Cole, see reference
on p. 142.
IOI-I03] CONGRUENT ARRANGEMENTS 179
V. RECTANGULAR SYSTEMS
103. Ways in which a rectangular system is congruent
to itself. Four mutually perpendicular lines through a
point 0* forma rectangular system (Art. 48), a system of
rectangular axes in hyperspace. Distinguishing positive
and negative directions along lines through O, we let a, 6, c,
and d be the positive half-lines of such a system. Any
three of these half-lines form a rectangular trihedral angle
in a hyperplane perpendicular to the fourth ; and, without
disturbing the fourth, we can permute the three cyclically
by a rotation in the hyperplane around the half-line which
makes equal angles with them. By combining relations
of this type we find twelve different ways in which the
four half-lines can be thus permuted. The system is con-
gruent to itself in all of the twelve arrangements of the
same order, f
On the other hand, we can obtain any arrangement of
three of these half-lines by a rotation in hyperspace. If,
for example, we take an axis-plane through the first and
bisecting the angle formed by the second and third, a
rotation through 180 around this plane will permute the
second and third half-lines without disturbing the first,
but the direction of the fourth half-line will be reversed
by this process, and the system of four half-lines will not
completely occupy its original position: in fact, it will
* We can regard this section as a section in Point Geometry ; and in the next
section we shall use exclusively the language of Point Geometry.
t Arrangements of the four half-lines in the three-dimensional Point Geometry
at O. This geometry is "restricted," but is sufficiently extended to iiiclude four
mutually perpendicular half-lines.
If we take with four points A , B, C, and D on the four given half -lines, the
various arrangements of the half -lines will correspond to the various arrangements
ot the five points in which O comes first
Thus if we say order abed = order OABCD,.
then we have " abdc - " OABDC,
etc.
l8o ISOCLINE PLANES [iv. vi.
occupy a symmetrical position with respect to the hyper-
plane of the first three half-lines.
The rectangular . system determines three pairs of ab-
solutely perpendicular planes. We can write them as
the planes be and ad, ca and bd, and ab and cd. If we let
be denote that order of rotation which, turning b through
an angle of 90, would make it coincide with c, and so
for the other planes, then we can express the different
possible arrangements of four mutually perpendicular
half-lines as arrangements of order in these pairs of ab-
solutely perpendicular planes. In the case of any one pair
there are four congruent arrangements, and four other
congruent arrangements symmetrical to the first four.
Thus, in the pair of planes determined, one by b and c and
the other by a and d, we have the four arrangements
congruent to one another,
be and ad, ad and be, cb and da, and da and eb ;
and symmetrical to these the four other congruent arrange-
ments,
be and da, da and be, cb and ad, and ad and cb.
The three pairs be and ad, ca and bd, and ab and cd, are
congruent to one another, each pair taken in any one of
four different ways.
VI. ISOCLINE PLANES*
104. Rectangular systems used in studying the angles
of two planes. In this section, as in the third section of
chap. Ill (Arts. 65-69), we shall use the language of Point
Geometry, all lines, planes, and hyperplanes being assumed
to pass through a given point O.
* A study of the hypersphere and its relation to the Point Geometry at its centre
(Art. 123) will help the student to understand this section.
Many of the theorems of this section are proved by Stringham in the paper re-
ferred to on p. 114.
103, 104]
THK ANGLES OF TWO PLANES
181
Let a, b y c, and d be four mutually perpendicular half-
lines of a rectangular system. Suppose in the planes ba
and cd we lay off from b and c the angles and 0'. The
half-lines which terminate these angles determine a plane
a which makes with the plane of be the angles <t> and 0',
ba and cd being the pair of common perpendicular planes
perpendicular to a and to be.
Given any two planes a. and /3, with their common per-
pendicular planes 7 and 7', we can take for be the plane /3
and for ba and cd the planes y and 7'. ad will be the plane
#' absolutely perpendicular to /3, and the angles and 0'
will be laid off as above in the planes ba and cd.
When we say that the plane a makes with be the angles
and 0', we imply a sense of rotation in a corresponding
to the order be. If p and q are the terminal half-lines of
these angles, then a is the plane pq, with a sense of rotation
which turns p through 90 to the position of q. and 0'
can then be any angles whatever, positive or negative.
With a particular plane a (of order pq} each of the angles
and 0' can be changed by any multiples of 2 TT, or both
at the same time by odd multiples of TT. and 0' + fl-
are angles made with be by the plane qp.
1 82
ISOCLINE PLANES
[iv. vi.
The angles which a makes with the plane ad are the
complements of and 0'.
When 0' = <f>, a. is isocline to the planes be and ad; and
by giving different values to < we have an infinite number
of planes isocline to be and ad and isocline to one another.
These planes are all perpendicular to ba and to ed, and
constitute what is called a series of isocline planes.
105. The common perpendicular planes of an isocline
series. Conjugate series. When the plane a is isocline
to be, these two planes have an infinite number of common
perpendicular planes on which they cut out the same angle
0, and any two of the common perpendicular planes cut
out the same angle on a as on be (Art. 69).
If, for example, in the planes be and a we lay off an angle ^
from b and p, the terminal half-lines of these angles will
themselves form the angle and will determine a plane 7
perpendicular to the plane be and to a.
This plane y, being perpendicular to be, is perpendicular
to its absolutely perpendicular plane ad, and therefore
is one of the common perpendicular planes of a and ad.
The terminal half-lines of the angles \l/, forming in y an
angle 0, determine in y a sense of rotation corresponding
I04-io6] SERIES OF ISOCLINE PLANES 183
to ba ; and the half -line of 7, making an angle of + 90
with that terminal half-line which lies in be, will be a half-
line of ad, making with a the angle \l/ in the same way that
this angle was formed in the planes be and a.
The plane 7 may, then, be regarded as determined by
the angles equal to \f/ laid off on the planes be and ad. But
this construction is independent of the angle and the
position of a. The plane 7 is therefore perpendicular to
all the planes of the isocline series obtained by giving
different values to <f> and laying off these angles in ba and
cd from b and c.
By giving different values to \l/ we have an infinite
number of planes 7 perpendicular to all the planes a of the
isocline series. Starting with ba, we construct these planes
in the same way that the a-series was constructed, and
so they themselves form an isocline series with the planes
of the cK-series for their common perpendicular planes,
each plane a perpendicular to all of them.
Thus we have, associated with a rectangular system,
two series of isocline planes, each plane of either series
perpendicular to all the planes of the other series. We
shall call them conjugate series of isocline planes.
The planes a are not the only planes * which are isocline
to be. We can rotate the rectangular system around be
as an axis-plane, the half -lines a and d rotating in ad through
any angle to new positions, and in this new rectangular
system we can construct a new series of planes isocline to
be and ad, with a new series of common perpendicular
planes, perpendicular to all of these but not perpendicular
to any plane of the first series except to be and ad themselves.
106. The two senses in which planes can be isocline.
Conjugate series isocline in opposite senses. There are
* Not the only planes through O.
184 ISOCLINE PLANES [iv. vi.
two senses in which a plane can be isocline to a given plane
corresponding to the two possible arrangements of a rec-
tangular system. With a given rectangular system, using
the construction of Art. 104, we can say that the plane a
is isocline to be in one sense when we make <' = 0, and in
the opposite sense when we make 0' = 0.
Starting with a plane and a pair of absolutely per-
pendicular planes 7 and 7' perpendicular to /3, let b and c
be half -lines common to and to 7 and 7' respectively.
If we lay off two angles in the same direction from b in the
plane 7 and in the same direction from c in y', or if we lay
off two angles in opposite directions from b in 7 and in
opposite directions from c in 7', we shall have two planes
isocline to in the same sense. But if we take the same
direction in one of the two perpendicular planes and op-
posite directions in the other, we shall get two planes
isocline to /3 in opposite senses.
When two planes are isocline to a given plane in opposite
senses we can speak of one as positively isocline and the
other as negatively isocline.
If a is the plane pq of Art. 104 and is isocline to be, we
can determine the sense in which it is isocline by considering
the order of the four half-lines ft, c, p, and q. Now in this
determination we can take in each plane, instead of the two
given half-lines, any two non-opposite half-lines, determin-
ing their order by a positive rotation of less than 180.
That is, if p f and q f are two half-lines in the plane pq such
that a positive rotation of less than 180 turns p r to the
position of g', we shall have order bcp'q' = order bcpq;
for p and q can be turned to the positions of p' and q f
without becoming opposite, and so without changing this
order (Art. 94, Th. i). In the same way we can take for
b and c any two non-opposite half-lines in the plane be
such that a positive rotation of less than 180 will turn the
first to the position of the second.
io6J
THE TWO SENSES
Conversely, we can determine the order of four non-
coplanar half-lines drawn from O with reference to the
order of two isocline planes or of any two planes which
have only the point O in common.
THEOREM i. If a is isocline to ft /3 will be isocline to
a in the same sense.
THEOREM 2. Two conjugate series of isocline planes are
isocline in opposite senses.
For these correspond to two arrangements of the forms
bead and bacd, which are of opposite orders.
In fact, if we rotate
the rectangular system
around the plane which
passes through b and
bisects the angle ac, we
shall interchange a and
c and the planes ba and
be. d will have its direction reversed so that the plane ad will
coincide with the original position of (fc,not with the original
position of cd. One of the angles ^ (of Art. 105) is now laid
off in the direction of one of the angles < in the original
position of the figure, while the present position of the
1 86
ISOCLINE PLANES
[iv. vi.
other angle ^ and the original position of the other angle
<t> are opposite.
Two absolutely perpendicular planes are isocline in both
senses, but in only one sense when we distinguish in each
a particular direction of rotation. Thus in the rectangular
system ad and da are isocline to be in opposite senses.
107. Planes through any line isocline to a given plane.
Planes to which given intersecting planes are isocline.
THEOREM i. Through any half -line not in a given plane
nor perpendicular to it* two planes can be passed isocline,
one positively and the other negatively, to the given plane.
PROOF. Let p be a half-line not in a given plane /J nor
perpendicular to /8. If we pass a plane through p per-
pendicular to ]8, we can determine a rectangular system
with four mutually perpendicular half-lines, 0, 6, c, and d,
so taken that is the plane be and p a half-line in the in-
terior of the angle ba. Then we can take <t> equal to the
angle bp in this plane, and lay off <t> and <t> from c in the
* It is always to be remembered that in this section, as in the third section of
chap. Ill, all half-lines are supposed to be drawn from the point O and all planes
to pass through O.
io6, 107]
TWO ISOCLINE TO A THIRD
I8 7
plane cd. The terminal lines of these angles determine
with p two planes isocline to /3 in the two senses.
THEOREM 2. Two intersecting planes determine a pair
of planes (absolutely perpendicular to each other) to which
they are isocline in one way in the two senses respectively,
and another pair to which they are isocline in the other way
in the two senses respectively.
C ""-~^^
PROOF. Let p be one of the opposite half-lines in which
the given planes intersect, and let 7 and 7' be their common
perpendicular planes, 7 passing through the half -line p,
and 7' the plane of the plane angles of the dihedral angles
which they form (Art. 49). Let q and q' be the half -lines
which form one of these plane angles, and let c be the half-
line bisecting the angle qq l '.
In 7 and 7' we establish directions of positive rotation.
Then in 7' the half -lines q and q' form with c angles which
may be called <f> and <t>. If now in 7 we take a half-line
b so that the angle bp shall be equal to 0, we shall have the
plane be to which the two given planes are isocline in the
two senses, as also to its absolutely perpendicular plane.
If, on the other hand, we take b so that the angle bp shall
1 88 ISOCLINE PLANES [iv. vr.
be equal to </>, we shall have another plane be to which
the two given planes are isocline in the two senses, as also
to its absolutely perpendicular plane.
108. The common perpendicular planes when two
planes are isocline to a third.
THEOREM i. // two planes are isocline in the same sense
to a plane a, the common perpendicular planes which they
have with a through any half -line of a form a constant dihedral
angle, the same for all positions of the line in a.
V X
PROOF. Let )8 and 7 be two planes isocline to a in the
same sense, and let n be any half-line of a. The common
perpendicular planes of a and ]8 form a series of the opposite
sense, and the common perpendicular planes of a and 7
form a series of the opposite sense. Through the line con-
taining n passes one plane from each series, two planes
forming two pairs of vertical dihedral angles along this
line and intersecting the absolutely perpendicular plane a'
in two pairs of opposite half-lines which are the sides of
the plane angles of these dihedral angles. Let mm' be
one of these angles, m and m f so taken that /3 shall inter-
sect the interior of the right angle mn and 7 the interior
of the right angle m'n. Now to a second position of n,
107, io8] THE PERPENDICULAR PLANES 189
forming a certain angle with its first position, will correspond
second positions of m and m f forming the same angle with
their first positions, the angles being laid off in the same
direction (around O) in the plane a'. Therefore the angle
mm' will be the same in its second position as in its first,
and the corresponding dihedral angles along n formed in
the two positions of the perpendicular planes will be equal.
COROLLARY. // two planes isocline in the same sense to
a plane a have with a a single pair of common perpendicular
planes, perpendicular to all three, then all the common per-
pendicular planes of either and a are perpendicular to all
three, and the two planes with a. belong to a single series of
isocline planes.
THEOREM 2. // two planes are isocline in opposite senses
to a plane a, the two planes with a. have one and only one pair
of common perpendicular planes, perpendicular to all three.
V, \
/\ \
PROOF. Let and 7 be two planes isocline to a. in oppo-
site senses, and let n be any half-line of a. The common
perpendicular planes of a and /3 and the common perpendic-
ular planes of ot and 7 form two series of opposite senses.
One plane of each series contains the half-line n, and these
1 90 ISOCLINE PLANES [iv. vi.
two contain half-lines, m and m r respectively, in the abso-
lutely perpendicular plane a', so taken that j8 shall inter-
sect the interior of the right angle mn and 7 the interior
of the right angle m'n. Then the half-line which bisects
the angle mm' will lie in a plane common to the two series ;
for this half-line forms equal angles in opposite directions
with m and m', and the planes of the two series which con-
tain this half-line contain the half-lines of a which form
the same angle in the same direction from n, and which
therefore coincide (Art. 106). We have then a plane, and
so a pair of planes, perpendicular to and y and to a ; that
is, a pair of planes common to the two series of perpendic-
ular planes.
These are the only planes that can be perpendicular to
j8 and 7 and to a. Any such plane will be perpendicular
to all the planes of the a/3-series and to all the planes of the
7-series. Through any half-line of it not a half-line of a
or a! pass two distinct planes, one belonging to each of
these series; that is, there are planes of one series that
intersect planes of the other series, and two intersecting
planes can have only one pair of common perpendicular
planes (Art. 49).
COROLLARY. Two planes which are isocline to a plane a
in opposite senses and make the same angle with a. always
intersect.
For they intersect each of the two perpendicular planes
which they have in common with a, and the common angle
which they form with a is laid off in the same direction from
a in one of these perpendicular planes, though in opposite
directions in the other.
109. Two planes isocline to a third in the same sense
isocline to each other.
io8, 109]
TWO ISOCLINE TO A THIRD
IQI
THEOREM. Two planes isocline to a third in the same
sense are isocline to each other in this sense also.
q'
P'
PROOF. Let /3 and 7 be two planes isocline in the same
sense to a plane a. )8 and 7 have at least one pair of com-
mon perpendicular planes mn and m'n' , where m, n, m',
and n' are half -lines taken in the intersections of /3 and 7
with these perpendicular planes in such a way that the
angles mm' and nn r are two positive right angles in these
two planes, n and n' lie in planes np and n'p' perpendic-
ular to 7 and a, p and p r being taken in a in such a way
that the angles pn and p'n' shall be two equal angles of
the isocline planes 7 and a. Then p and p f lie in planes
pq and p'q' perpendicular to a and /?, yand q' being taken
in 13 in such a way that the angles pq and '#' shall be
two equal angles of the isocline planes a and /3.
Since mm' and ' are right angles, pp' and 99 'are also
right angles, qq' a positive right angle like mm' (see Art.
104). Then we have
Z mq = Z wV
Again, the dihedral angle along p formed by the half-
planes containing n and g, is equal to the dihedral angle
along p' formed by the half-planes containing n f and q',
since j8 and 7 are isocline to a. in the same sense (Art. 108,
Th. i). Therefore, the trihedral angles npq and n'p'q'
IQ2 ISOCLINE PLANES [iv. vi.
have two face angles and the included dihedral angle of
one equal respectively to two face angles and the included
dihedral angle of the other, so that the third face angles
are equal ; namely, '
Z nq = Z n'q f .
Then in the right trihedral angles mnq and m'n'q' there
arc two face angles, one adjacent and one opposite to the
right dihedral angle, which have the same values in one
trihedral angle as in the other. The third face angles in
these two trihedral angles are therefore equal ; namely,
Z mn = Z m'n'.
This proves that ft and 7 are isocline.
Now ft and 7 must be isocline to each other in the same
sense as to a. Otherwise a and 7, being isocline to ft in
opposite senses, would have with ft one pair of common
perpendicular planes perpendicular to all three (Art. 108,
Th. 2). But as ft and 7 are isocline to a in the same sense,
there can be no planes perpendicular to all three unless
the three belong to the same isocline series.
110. Poles and polar series.
THEOREM i. Given two conjugate series of isocline planes
(Art. 105), there is a pair of absolutely perpendicular planes
to which the planes of the two series are isocline in opposite
senses, all at an angle of 45.
PROOF. Let ft be any plane of the first series and 7 a
plane of the conjugate series. These planes intersect at
right angles and are therefore both isocline, in opposite
senses, to a pair of planes a and a', each forming an angle
of 45 with a and the same angle with a' (Art. 107, Th. 2).
All the planes, then, of the first series are isocline to a and
a' in the same sense as )8, and all the planes of the conjugate
series in the same sense as 7 (Art. 109, Th.). Now every
plane of the conjugate series has one pair of opposite half-
109,110] POLKS AND POLAR SKRFKS 193
lines in common with /3, and every plane of the first series
has one pair of opposite half-lines in common with 7, and
these half-lines all make an angle of 45 with a and the same
angle with a'. Therefore, all the planes of both series
are isocline to a and a! at an angle of 45.
THEOREM 2. All the planes isocline to a given plane at
an angle of 45 lie in two conjugate series.
PROOF. Two planes which are isocline to a plane a in
opposite senses, and make with a an angle of 45, intersect
and are perpendicular to each other (Art. 108, Th. 2, Cor.
and Art. 107). Thus all the planes /3, isocline in one sense,
are perpendicular to any one of the planes 7, isocline in the
opposite sense ; and through each half-line of 7 passes one
and only one of these planes (Art. 107, Th. i). But any
two that are not absolutely perpendicular determine a
series of planes all perpendicular to 7, one through each
pair of opposite half -lines of 7, and all making an angle of
45 with a. Therefore these are the planes ]8, and the
planes /3 all belong to one series. In the same way we prove
that the planes 7, isocline in the opposite sense, all belong
to one series.
The two isocline planes forming an angle of 45 with all
the planes of a given series may be called the poles of the
series, and the series may be called one of their polar series.
Two absolutely perpendicular planes always have two polar
series, conjugate to each other.
COROLLARY. When two series of the same sense have a
pair of planes in common, these planes are tfie poles of the
series determined by tJie poles of the two given series.
THEOREM 3. In a complete system of all planes * isocline
to a given plane in a given sense, any two series have a pair
of planes in common.
*AtO.
194
ISOCLINE PLANES
[iv. vi.
For the poles of the two given series determine a pair of
conjugate series whose poles, being 45 from each of them,
are planes common to the two given series.
111. Planes intersecting two isocline planes.
THEOREM i. If a plane intersects two isocline planes,
each of the four dihedral angles formed about one line of in-
tersection is equal to the corresponding dihedral angle of the
four formed about the other line of intersection (see Art. 104).
PROOF, Let a and /3 be two isocline planes, and let a
and b be half-lines lying in the intersections of these planes
with a third plane. Through each of the half-lines a and
b passes a plane perpendicular to a and (8. Let n and m
be half-lines in the intersections of these planes with ft and
a respectively, so taken that the angles na and bm shall
be two of the equal angles formed by the isocline planes.
Then the angles am and nb will also be equal, and the two
trihedral angles abn and bam will have the three face angles
of one equal respectively to the three face angles of the
other. It follows that the dihedral angle along a of the
trihedral angle bam is equal to the dihedral angle along
b of the trihedral angle abn. But the latter dihedral angle
is vertical to the dihedral angle which corresponds to the
former. These corresponding dihedral angles are there-
fore equal, and each of the four dihedral angles along a is
equal to the corresponding dihedral angle along b.
no, in] LIKE PARALLEL LINES 195
THEOREM 2. Given two isocline planes a. and |3, not
absolutely perpendicular, with non-opposite half-lines a and
a' in a and b and V in $ so taken that
then the planes ab and a'b' will be isocline in the opposite
sense.
PROOF. If these planes are isocline, they are isocline in
the sense opposite to that of a and /3, for the orders aba'b'
and aa'W are opposite (Art. 106).
Now the half -lines a' and &' make equal angles with the
plane ab, for their projections upon this plane form with
the four given half -lines two equal trihedral angles. More-
over, there is a plane X to which a and j8 are isocline in the
given sense and ab is isocline in the opposite sense (Art.
107, Th. 2). These planes and the four given half -lines all
form the same angle with X. Then the plane /?, and the
plane through a' isocline in the opposite sense to ab and to
X, form this same angle with X and must intersect (Art.
108, Th. 2, Cor.) in two opposite half-lines which form with
the plane ab the angle that a' forms with ab. In other
words, the plane a'b' is this isocline plane through a',
isocline to ab in the sense opposite to that of a and ]3.
196 ISOCLINE PLANES [iv, vi.
The four half -lines a, b, a', and b' are the edges of a
polyhedral angle in hyperspace, not in a hyperplane, hav-
ing properties somewhat analogous to those of the parallel-
ogram of the Euclidean Plane Geometry.
112. Isocline rotation. In a simple rotation around a,
or in a double rotation around a and its absolutely per-
pendicular plane a f (Art. 81), the complete system of all
planes isocline to a in a given sense is transformed into
itself, the planes generally into one another.
When the two rotations around a and a' are equal, all
the planes isocline in the sense corresponding to the rota-
tion rotate on themselves, the series conjugate to any series
of these planes moving as a series on itself. Every half-
line (drawn from O) rotates in a plane isocline to a, and any
one of these planes and its absolutely perpendicular plane
can be regarded as the axis-planes of the rotation, no partic-
ular pair of planes playing in this way a special part.
We shall call this rotation isocline rotation, and the com-
mon angle of the two rotations the angle of the isocline
rotation.
THEOREM i. In an isocline rotation every plane that
does not rotate on itself remains isocline to itself in the sense
opposite to the rotation.
PROOF. Let 7 be any plane rotated to a position 7'.
Two half-lines a and b of 7 rotate to positions a f and 6',
rotating in two isocline planes, and we have
Z aa'= Z W.
Therefore 7 and 7' ^re isocline in the sense opposite to the
rotation (Art. in, Th. 2).
THEOREM 2. A simple rotation of angle 9 around a
plane a moves any plane of a polar series to a position where
it makes an angle of % 6 with its original position.
m,ii2] THEIR TWO-DIMENSIONAL GEOMETRY 197
PROOF. In an isocline rotation of angle 0, any plane
conjugate to a series of the system with respect to which
the rotation takes place is rotated on the planes of this
series through an angle 6. Now the isocline rotation can
be decomposed into two equal simple rotations around the
two poles, say a and a', of the series. Either of these
simple rotations, therefore, moves the given plane to a
position where it makes an angle of 6 with its original
position. But the effect of a simple rotation around a is
the same on both of its polar series. After such a rotation
any plane of either series in its final position makes an angle
of 5 with its original position.
The planes isocline to a given plane, in a given sense,
and at a given angle, constitute a conical hypersurface of
double revolution. The hypersurface contains also all the
planes isocline to the given plane in the opposite sense and
at the same angle, but the planes of either set contain all
the half-lines which make this angle with the given plane,
and therefore completely fill the hypersurface (see Art.
1 1 8). In particular, two conjugate series lie together in
such a hypersurface, which therefore may be said to con-
sist of the planes of either of these series (see Art. 124).
The student will find it useful to think of the planes
isocline in a given sense to a given plane as the elements
of a two-dimensional geometry which is exactly like the
geometry of the sphere, absolutely perpendicular planes
corresponding to opposite points on the sphere, and series
of planes to great circles. For distance between two ele-
ments we should take double the angle between the two
planes, and to measure the angle between two series we
can measure the dihedral angle between conjugate planes
intersecting in one of the common planes of the series (Art.
108, Th. i). These conjugate planes are not themselves
198 ISOCLINE PLANES liv. vi,
a part of the two-dimensional geometry, but the measure
of the angle so determined will be the same as the distance
intercepted on the, polar series of its vertex-plane. An iso-
cline rotation with respect to the planes of this system is
to be regarded as no motion at all in this geometry, but a
simple rotation or a double rotation which is not an isocline
rotation corresponds to a rotation of the sphere, in the
latter case through an angle equal to the difference of the
two component rotations.*
* Stringham calls a series of planes an ordinal system, and the set of planes isocline
to a given plane at a given angle forms with him a cardinal system. He uses these
terms, however, with reference to a particular pair of absolutely perpendicular planes
They correspond to meridian and parallel circle taken on the sphere with reference
to a particular axis. See p. 212 of the paper referred to on p. 1 14.
CHAPTER V
HYPERPYRAMIDS, HYPERCONES, AND THE HYPER-
SPHERE
I. PENTAHEDROIDS AND HYPERPYRAMIDS*
113. Pentahedroids : the point equidistant from the
five vertices, the point equidistant from the five cells, and
the centre of gravity.
THEOREM i. In a pentahedroid, if two of the tetrahedrons
can be inscribed in spheres, the lines drawn through the centres
of these spheres perpendicular to their hyperplanes lie in a
plane; when they meet in a point this point is equidistant
from the five vertices of the pentahedroid, the five tetrahedrons
can all be inscribed in spheres, and the five lines drawn through
the centres of these spheres perpendicular to their hyperplanes
all pass through the same point .f
PROOF. The common face of the two tetrahedrons is
the interior of a triangle inscribed in a circle common to
the two spheres. The absolutely perpendicular plane at
the centre of this circle contains the centres of the spheres
and is perpendicular to the hyperplanes of the .two tetra-
hedrons. It therefore contains the lines perpendicular to
the hyperplanes at these points (Art. 51, Th. i).
Now the line drawn through the centre of a sphere per-
pendicular to its hyperplane is the locus of points equidis-
* This section and the next are continuations of the last section of chap. I, and
the latter should be read again at this point.
t There are other possibilities in the Hyperbolic Geometry, which the student
may investigate if he is familiar with this geometry.
199
200 PENTAHEDROIDS AND HYPERPYRAMIDS [v. z.
tant from the four vertices of any tetrahedron inscribed
in the sphere (Art. 40, Th. 2). If, then, the two perpendic-
ulars meet, the point where they meet must be at the
same distance from the five vertices of the pentahedroid ;
and a line through this point perpendicular to the hyper-
plane of any one of the five tetrahedrons contains a point
of this hyperplane equidistant from the vertices of the
tetrahedron.
THEOREM 2. The half -hyper planes bisecting the ten
hyperplane angles of a pentahedroid all pass through a point
within the pentahedroid, a point equidistant from the hyper-
planes of its five cells.
PROOF. At a vertex A we have a tetrahedroidal angle
with six hyperplane angles, and the bisecting half-hyper-
planes of these hyperplane angles have in common a half-
line a, the locus of points within the tetrahedroidal angle
equidistant from its four hyperplanes (Art. 73, Th.).
Drawn from another vertex B we have another half-line
by the locus of points within the tetrahedroidal angle at
this vertex equidistant from the four hyperplanes of this
tetrahedroidal angle. Now the piano-polyhedral angle
AB contains three hyperplane angles which belong to the
tetrahedroidal angle at A and also to the tetrahedroidal
angle at 5, and the bisecting half-hyperplanes of these
three hyperplane angles intersect in a half -plane a which
must contain both a and b (Art. 79, Th.). The plane of a
intersects the pentahedroid in a triangle, and, as the half-
lines a and b pass within the angles at A and B of this
triangle, they must intersect in a point O within the penta-
hedroid (see Art. 8, Th. i). We have, then, a point O
equidistant from the five hyperplanes of the pentahedroid,
lying within the pentahedroid, and lying in each of the ten
half-hyperplanes bisecting the ten hyperplane angles of the
pentahedroid.
113] THE CENTRE OF GRAVITY 2OI
THEOREM 3. The half -lines drawn from the vertices of a
pentahedroid through the centres of gravity of the opposite
cells meet in a point.
PROOF. Given the pentahedroid ABCDE, we will
write half -plane AB to denote the half -plane whose edge
is the line AB and which itself contains the centre of
gravity of the triangle CDE, and we will write triangle AB
to denote the triangle whose vertices are A and B and this
same centre of gravity. In this way we can speak of the
half-plane and triangle determined by any two vertices
of the pentahedroid and the centre of gravity of the opposite
face. All points in the interior of the triangle AB are in
the interior of the pentahedroid.
The four half -planes AB, AC, AD, and AE contain
respectively the half-lines drawn from the vertices J5, C, D,
and E through the centres of gravity of the opposite faces
in the tetrahedron BCDE. They contain, therefore, the
centre of gravity of this tetrahedron, as well as the vertex
A. In the same way the four half-planes AB, BC, BD,
and BE contain the vertex B and the centre of gravity
of the tetrahedron ACDE. Thus we have in the half-
plane AB two half-lines drawn from the vertices A and B
through points in the opposite sides of the triangle AB.
These two half-lines, therefore, intersect in a point P in the
interior of the triangle AB y and so in the interior of the
pentahedroid (Art. 8, Th. i). But A and B are any two
vertices. Hence the half-lines drawn from the vertices
of the pentahedroid through the centres of gravity of the
opposite cells must all intersect one another in the interior
of the pentahedroid.
These intersections all coincide. If, for example, the
half-lines from A, B, and C intersected in three different
points, these half -lines, and so the vertices A, B, and C,
would lie in the plane of these three points. But this is
202 PENTAHEDROIDS AND HYPERPYRAMIDS [v. i.
impossible, the intersections being in the interior of the
pentahedroid.*
The point where these half -lines intersect is called the
centre of gravity of the pentahedroid.
114. Pentahedroids with corresponding edges equal.
Regular pentahedroids. We can make two pentahedroids
correspond, the five vertices of one to the five vertices of
the other, in any order, just as we have made two tetra-
hedrons correspond in Art. 72.
THEOREM. // each of the edges of a pentahedroid is equal
to the corresponding edge of a second pentahedroid, when the
five vertices of one are made to correspond in some order to the
five vertices of the other, the pentahedroids will be congruent
or symmetrical.
PROOF. All the faces and all the face angles, dihedral
angles, and hyperplane angles of one are equal to the corre-
sponding parts of the other, as proved in the two theorems
of Art. 72 ; and any two corresponding tetrahedrons can
be made to coincide, even though they happen to be in a
hyperplane in opposite orders (Art. 84, Th. 4). Then by
putting two such tetrahedrons together we can prove that
the pentahedroids will coincide entirely if their orders are
the same, and that they will be symmetrically situated
* The corresponding theorem for tetrahedrons is proved in the same way.
The theorem which determines the centre of gravity of a triangle is usually
made to depend on the axiom of parallels. The theorem is true, however, in the
Hyperbolic and Elliptic Geometries, and therefore is independent of the axiom of
parallels. It can be proved very simply in these two geometries and in the Euclid-
ean Geometry by means of trigonometrical formulae (see Chauvenet's Trigonom-
etry, gth ed , Philadelphia, 1881, Part II, Art. 188, p. 253). Or, having proved the
theorem for Euclidean Geometry by means of parallels, we can prove it for any
spherical triangle by projection from the centre of the sphere upon the plane of the
plane jtriangle which has the same vertices, and then in the Non-Euclidean Geom-
etries for any plane triangle which can be inscribed in a circle, and so in a sphere,
by reversing this projection.
The term centre of gravity is used here without any reference to the physical
properties of the point. Another name is centroid.
II3-H5] RIGHT AND REGULAR 203
with respect to the common hyperplane of these tetrahe-
drons if their orders are opposite.
When the pentahedroids are symmetrical, corresponding
tetrahedroidal angles and corresponding piano-polyhedral
angles are symmetrical.
If we take a regular tetrahedron and draw a line through
its centre perpendicular to its hyperplane, every point of
this line will be equidistant from the four vertices of the
tetrahedron, and we can take a point at a distance from
the four vertices equal to one of the edges of the tetrahedron.
We have then a pentahedroid in which the ten edges are
all equal. All the parts of any one kind, face angles, di-
hedral angles, faces, etc., are equal; for the pentahedroid
is congruent to itself in sixty different ways (Art. 95), and
can be made to coincide with itself, any part coinciding
with any other part of the same kind. Such a penta-
hedroid is called a regular pentahedroid (see Art. 166).
115. The terms right and regular as used of hyperpyra-
mids and double pyramids. When the base of a hyper-
pyramid is the interior of a regular polyhedron, the interior
of the segment consisting of the vertex and the centre of
the base is called the axis of the hyper pyramid ; and when
the line containing the axis is perpendicular to the hyper-
plane of the base the hyperpyramid is regular.
THEOREM i. In a regular hyperpyramid the lateral
pyramids are equal regular pyramids. The axis of any
one of these lateral pyramids is the hypothenuse of a right
triangle whose legs are the axis of the hyperpyramid and a
radius of the sphere inscribed in the base.
The slant height of a regular hyperpyramid is the altitude
of any one of the lateral pyramids.
204 HYPERCONES AND DOUBLE CONES [v. n.
When the base of a double pyramid (see Art. 32) is the
interior of a regular polygon, the interior of the triangle
determined by the vertex-edge and the centre of the base
is called the axis-element of the double pyramid; and when
the plane of this triangle is absolutely perpendicular to
the plane of the base we have a right double pyramid. A
right double pyramid is isosceles when the extremities of
the vertex-edge are at the same distance from the plane of
the base. Such a double pyramid is also called regular.
THEOREM 2. In a right double pyramid (the base being
regular) the lateral faces are congruent, the lateral cells are
congruent, and the two end-pyramids are regular. In a
regular double pyramid the end-pyramids are congruent.
II. HYPERCONES AND DOUBLE CONES
116. Spherical hypercones and right hypercones. A
spherical hyper cone is one whose base is the interior of a sphere.
The axis of a spherical hypercone is the interior of a segment
consisting of the vertex and the centre of the base. A right
spherical hypercone, or simply a right hypercone, is one whose
axis lies in a line perpendicular to the hyperplane of the
base.
A section of a spherical hypercone by a hyperplane con-
taining the vertex and any point of the base is a circular
cone.
THEOREM i. When a right triangle takes all possible posi-
tions with one leg fixed, the vertices and the points of the other
two sides of the triangle make up a right spherical hypercone.
The fixed side is the axis, the hypothenuse is an element, and
the other leg is a radius of the base.
THEOREM 2. // in the hyperplane of a cone of revolution
we pass a plane through its axis and rotate around this plane
IIS-"?] RIGHT DOUBLE CONE 205
that portion of the cone which lies on one side of it, we shall
have all of a right spherical hyper cone except that portion which
makes up the section of the cone by the plane.
The slant height of a right spherical hypercone is the dis-
tance from the vertex to any point of the sphere whose
interior is the base; it is the length of the hypothenuse
of the right triangle of Th. i.
117. Circular double cones and right double cones. A
circular double cone is one whose base is the interior of a
circle. The axis-element of a circular double cone is the
interior of the triangle determined by the vertex-edge and
the centre of the base. A right circular double cone, or
simply a right double cone, is one whose axis-element lies
in a plane absolutely perpendicular to the plane of the
base, and the double cone is also isosceles when the extremi-
ties of the vertex-edge are at the same distance from the
plane of the base.
THEOREM i. In a right double cone the elements are
congruent, and the two end-cones are cones of revolution. In
an isosceles right double cone the end-cones are congruent.
THEOREM 2. A right double cone may be generated by
the rotation of a tetrahedron which has an edge and face in a
perpendicular line and plane, the rotation taking place around
the latter.
Let A BCD be the tetrahedron with the line CD per-
pendicular to the plane ABC. In the rotation around this
plane the face ABD will generate the set of elements of
the double cone, the face ABC will be the axis-element,
the faces ACD and BCD will generate the interiors of the
end-cones, and the edge CD will generate all of the base
except the centre.
206 HYPERCONES AND DOUBLE CONES [v. n.
118. Hypersurfaces consisting of planes through a
point with only this point common to any two of them.
The conical hypersurface of double revolution. When
the directing-surface of a hyperconical hypersurface con-
sists of lines (as, for example, in the case of a conical sur-
face), the hypersurface consists of planes or portions of
planes, every point collinear with the vertex and a point
of any one of these lines being a point of the hypersurface.
If the entire planes are not included in this way, we can
consider the hypersurface which does consist of the entire
planes, defining it as consisting of these planes and calling
the planes elements.
We can also form a hypersurface of planes through a
fixed point O and the points of a plane curve, the planes
being determined in some way so that they shall have only
the point O common to any two of them and shall intersect
the plane of the directing-curve only in the points of this
curve. For example, the planes can be isocline to a given
plane through O.
These various hypersurfaces should all be regarded as
hyperconical hypersurfaces (see foot-note, p. 220).
We shall consider only the hypersurface generated by the
rotation of one of two isocline planes around the other,
the conical hypersurface of double revolution of Art. 112.
O being the vertex (where the two planes meet), the plane
through any other point of the hypersurface, absolutely
perpendicular to the axis-plane, will intersect the hyper-
surface in a circle which can be taken for the directing-circle.
As we have seen, there are two sets of planes in the hyper-
surface, and two generating planes, and the hypersurface
rotates on itself in any double rotation around the given
axis-plane and its absolutely perpendicular plane at O,
or in any simple rotation around either one of these planes.
Thus the hypersurface has a pair of axis-planes.
n8, 119] THE HYPERSPHERE, SECTIONS 207
The hypersurface can also be described as consisting of
all the half-lines drawn from which make a given angle
with a given plane through O. All other half-lines drawn
from O are divided into two classes, those which make a
smaller angle with the given plane, and those which make
a smaller angle with its absolutely perpendicular plane, than
do the half-lines of the hypersurface itself. The hyper-
surface, therefore, divides all the remaining points of hyper-
space into two classes, orue containing all the points except
O of one of the axis-planes, and the other containing all
the points except O of the other axis-plane.
We can pass around either axis-plane without passing through a
point of the hypersurface, just as in a hyperplane we can pass around
the axis-line of a conical surface of revolution without passing through
a point of the surface. In the case of the hypersurface, however, each
of the two regions into which it divides the rest of hyperspace is com-
pletely connected, so that in either one of these regions we can pass
from any point to any other point without passing through a point
of the hypersurface ; while the axis of a conical surface of revolution
in a hyperplane lies partly in one and partly in the other of two verti-
cal regions that are completely separated.
III. THE HYPERSPHERE
119. Spheres and circles in a hypersphere. Tangent
hyperplanes. A hypersphere consists of the points at a
given distance from a given point. The terms centre,
radius, chord, and diameter are used as with circles and
spheres.
THEOREM i. Any hyperplane section of a hypersphere
is a sphere having for centre the projection of the centre of the
hypersphere upon the hyperplane (Art. 40, Th. 2).
When the hyperplane passes through the centre of the
hypersphere the section is a great sphere. Other spheres
of the hypersphere are small spheres.
208 THE HYPERSPHERE [v. m.
THEOREM 2. Four non-coplanar points of a hyper sphere
determine a sphere of the hypersphere, and three points not
coplanar with the centre of the hypersphere determine a great
sphere.
THEOREM 3. Any plane having more than one point
in a hypersphere intersects the hypersphere in a circle having
for centre the projection of the centre of the hypersphere upon
the plane.
This is proved by considering the hyperplane which
contains the plane and the centre of the hypersphere.
A circle of a hypersphere is a great circle when its plane
passes through the centre of the hypersphere.
THEOREM 4. Three points of a hypersphere determine a
circle, and two points not collinear with the centre of the
hypersphere determine a great circle.
THEOREM 5. Two great circles on the same great sphere
intersect, and two great circles which intersect lie on one great
sphere (Art. 22, Th., and Art. 20, Th. 2 (4) ).
A great circle and a great sphere always intersect, inter-
secting in the extremities of a diameter, and two great spheres
intersect in a great circle (Art. 27, Ths. i and 2).
Distance in a hypersphere between two points not the
extremities of a diameter is always measured on the arc
less than 180 of the great circle containing them. The
distance between the extremities of any diameter is 180.
THEOREM 6. All the circles of a hypersphere which pass
through a given point are perpendicular, that is, their tangents
are perpendicular, to the radius of the hypersphere at this
point. These tangent lines, therefore, all lie in the hyper-
plane which is perpendicular to the radius at this point.
A hyperplane perpendicular to a radius of a hypersphere
at its extremity is tangent to the hypersphere.
i ig, 120] THEOREMS OF VOLUMES 209
120. Spherical dihedral angles and spherical tetrahe-
drons. A great circle of a sphere, dividing the rest of the
sphere into two hemispheres, may be called the edge of
either of these hemispheres. Two hemispheres of great
spheres in a hypersphere having a common edge form a
figure somewhat like a double convex lens, and enclose
a portion of the hypersphere, a definite volume. Along
the edge we have a spherical * dihedral angle, which we can
think of as consisting of a restricted portion of the edge and
restricted portions of the hemispheres. The tangent half-
planes which have a common edge tangent to the edge of
the spherical dihedral angle form an ordinary dihedral
angle whose measure can be taken as the measure of the
former.
THEOREM i. A spherical dihedral angle has the same
measure at all points of its edge.
THEOREM 2. The volume enclosed by the hemispheres of a
spherical dihedral angle is to the volume of the hypersphere
as the dihedral angle is to four right dihedral angles.
The edge of a spherical dihedral angle has on each face
a pole, and the arcs of great circles drawn through these
poles from any point of the edge determine a spherical
angle by which the spherical dihedral angle can be measured,
just as the dihedral angle formed by two half-planes is
measured by its plane angle. The spherical angle is itself
measured by the distance between the two poles of the edge,
so that this distance can be considered a measure of the
spherical dihedral angle, and also, if we take correspond-
ing units, as a measure of the volume enclosed by its
hemispheres.
It is hardly necessary to define spherical trihedral angle
* We might have said hyperspherical, but we shall use the shorter word where
there is no ambiguity.
210 THE HYPERSPHERE [v. ill.
and spherical tetrahedron. We shall suppose that the
sides of a triangle and the edges of a tetrahedron are less
than 1 80. The four great spheres which contain the faces
of a spherical tetrahedron determine a set of sixteen tetra-
hedrons, eight pairs, the two tetrahedrons of a pair being
symmetrically situated with respect to the centre of the
hypersphere, and therefore congruent (Art. 85, Th. 2). That
half-hypersphere which lies on one side of any one of the
four great spheres (on one side of the hyperplane of the
great sphere) contains the interiors of eight tetrahedrons,
one from each pair.
A spherical tetrahedron has six edges, each lying in the
edge of a spherical dihedral angle whose interior contains
the interior of the tetrahedron. The interior of one of these
spherical dihedral angles contains also the interiors of three
of the fifteen tetrahedrons associated with the given tetra-
hedron as explained above, and its volume is equal to the
sum of the volumes of the four tetrahedrons whose interiors
are within it.
Writing A ' for the opposite point to A , the other extrem-
ity of the diameter to A, and so for other points, we let T
denote the volume of the tetrahedron A BCD, Ti the volume
of A 'BCD, Ti 2 the volume of A 'B'CD, and so on. A BCD'
is congruent to A'B'CD, and we have r 34 = r ]2 , etc.
The interior of the dihedral angle C-AB-D contains the
interiors of the four tetrahedrons whose volumes are T, TI,
T%, and TU. If 612 is the measure of the dihedral angle
AB in terms of a right dihedral angle, and if we take for
unit of volume one-sixteenth of the volume of the hyper-
sphere, we shall have the relation
T + T! + T 2 + T
12
There are six of these equations, and in addition one equa-
tion expressing the fact that the sum of the eight different
120,121] POLES AND POLAR CIRCLES 21 1
volumes is equal to the volume of a half-hypersphere,
namely,
r + 2 Ti + 2 Ti 2 = 8.
These seven equations reduce to the following equivalent
system :
T + T\ =2(612 + 0i3 + 0i4 2), etc., four equations,
T Ti2 = 2(013 + 0H + 023 + 024 ~ 4), etc., three equations.
Given the volume of one tetrahedron, we can find the
volumes of the others; but we have no simple formula
for the volume of a single tetrahedron as we have for the
area of a spherical triangle.*
121. Poles and polar circles. Duality in the hyper-
sphere. The diameter of a hypersphere perpendicular to
the hyperplane of any sphere of the hypersphere is called
the axis of the sphere, and the extremities of the axis are the
poles of the sphere.
THEOREM i. Each pole of a sphere of a hypersphere is
equidistant from all the points of the sphere.
The plane through the centre of a hypersphere absolutely
perpendicular to the plane of any circle of the hypersphere
is called the axis-plane of the circle, and the great circle iii
which this plane intersects the hypersphere is the polar
circle of the given circle.
THEOREM 2. Each point of the polar circle of a circle of a
hypersphere is equidistant from all the points of the given
circle.
THEOREM 3. A great circle of a hypersphere is itself
polar to the great circle which is its polar, and the distance
between any two points, one in each of two polar great circles,
is a quadrant.
* See Coolidge, Non-Euclidean Geometry, p. 181.
212 THE HYPERSPHERE [v. m.
THEOREM 4. A great sphere contains all the points at a
quadrant's distance from either of its poles, and each oj two
polar great circles contains all points at a quadrant's distance
from the other.
THEOREM 5. Great circles which pass through the poles
of a sphere are perpendicular to the sphere, and any great
circle perpendicular to a sphere passes through the poles of
the sphere.
If a point moves a given distance along an arc of a great
circle, its polar great sphere will rotate around the polar
great circle and generate a spherical dihedral angle whose
measure is this same distance. If a great circle rotates on
a great sphere through a given spherical angle around one
of its points (and the opposite point), its polar great circle,
lying on the polar great sphere of the given point and pass-
ing through the pole of the given great sphere, will rotate
around the latter through the same angle. In fact, the
rotation takes place in the hypersphere around the great
circle (see Art. 124) determined by the two fixed points, and
in hyperspace around the plane of this great circle.
We have a principle of duality in the hypersphere, points
and great spheres corresponding to each other, and great
circles to great circles, one great circle considered as made
up of points and the other as common to a set of great
spheres. Corresponding figures are called reciprocal figures.
This correspondence can be realized by taking the relations
of pole and polar as its basis.
The student may investigate the properties of polar spherical
tetrahedrons.
122. Geometry of the hypersphere as an independent
three-dimensional geometry. Starting with its points and
great circles, and with certain fundamental theorems, we
i2i, 122] AN INDEPENDENT GEOMETRY 213
can build up the geometry of the hypersphere without
further reference to the hyperspace in which it lies. These
fundamental theorems play the part of axioms and we shall
statue them here as axioms.*
The points of this geometry are paired, the two points
of a pair being called opposites. We express this as an
axiom : namely,
AXIOM i . To each point there is one and only one oppo-
site point.
With any two non-opposite points A and B is associated a
class of points as points of the great circle AB, a great circle
determined by these two points. For great circles we have
the following axioms :
AXIOM 2. The great circle determined by any two non-
opposite points of a given great circle is the given great circle
itself (Art. 10, Th. 2).
AXIOM 3. All great circles which contain a given point
contain also its opposite point.
The points of a great circle are in cyclical order. In
particular, any two pairs of opposite points separate each
other (Art. 6).
A spherical triangle consists of three points not points of
one great circle and all points between any two of them on
that portion of a great circle which does not contain their
opposites.
On a great circle containing one side of a spherical tri-
angle, that portion which lies between one vertex and the
opposite point to the other will be called the side produced,
produced in one direction or the other as the case may be.
AXIOM 4. A great circle intersecting one side of a spheri-
cal triangle and another side produced intersects the third side.
* Compare the chapter on "Pure Spherics" in Halsted's Rational Geometry
(chap. XVI of the second edition).
214 THE HYPERSPHERE [v. m.
The two axioms of Art. 3 are included in Ax. 2, modified
only by the term " non-opposite." Ax. 4 is the same as
the Axiom of Pasch (Art. 7), with the restriction placed
above on the phrase " side produced." It follows that in
any portion of the hypersphere thus restricted the theorems
of the first four sections of chap. I all hold true, if in place
of the word " collinear " we use the phrase " on a great
circle with/' and for " line " say " great circle."
Certain other forms of expression, also, are changed as a
matter of convenience. Thus we shall speak of a great
circle as a transversal to a triangle when it contains at least
two points of the triangle, and then we can speak of a point
as on a transversal to a triangle where in chap. I we have
said " collinear with a triangle." Again, in place of the
word " plane " we must now say great sphere. This we
define as consisting of the points that we get if we take
three points not points of one great circle, all points on a
great circle with any two of them, and all points on a great
circle with any two non-opposite points obtained by this
process.
We prove then that any point of the great sphere ABC
is on a transversal to the triangle ABC, and that any three
points of a great sphere, not points of one great circle, de-
termine the same great sphere.
Now in order to confine our geometry to a single hyper-
sphere we make this axiom :
AXIOM 5. We gel all the points of a hypersphere if we
take four points not points of one great sphere, all points of
the great circles determined by any two of them, and all points
of the great circles determined by any two non-opposite points
obtained by this process.
Thus we see that the hypersphere is a space of three di-
mensions, and its geometry is a geometry of three dimen-
sions.
122] ITS GEOMETRY DOUBLE ELLIPTIC 215
Finally, we are able to consider the entire hypersphere,
and determine its relation to the theory of parallels, by
introducing the following axiom :
AXIOM 6. Any two great circles of the same great sphere
intersect.
By passing a great sphere through a given great circle
and a point of a given great sphere we prove from these
axioms that the circle and sphere always intersect (Art. 22,
Th.), and then that any two great spheres intersect in a
great circle.
These " axioms/' and such axioms as are necessary to
establish the properties of points on a great circle (see Arts.
5 and 6) , determine the nature of the hyperspherical geome-
try. The result we express in the following theorem :
THEOREM. The geometry of the hypersphere is the same
as the Double Elliptic Non-Eitclidean Geometry of Three
Dimensions* the great circles and great spheres of the hyper-
sphere being taken for lines and planes.
* See the author's Non-Euclidean Geometry, chap. III. This book has been
very properly criticised for giving only the Single Elliptic Geometry ; but in any
"restricted region" the two geometries are the same.
In the Double Elliptic Geometry two lines in the same plane intersect in two
points, and a line meets any plane in which it does not lie in two points, the distance
between the two points in each of these cases being one-half of the entire length
of the line. The length of the line is most conveniently taken as 2n.
If we start at an intersection of two lines and follow one of them until we come
again to the other, we shall come, not to the same intersection point, but to an
"opposite" point. We have traversed only one-half of the line, and we arrive at
the starting point only when we have gone the same distance further. The geome-
try of the sphere is the same as the Double Elliptic Geometry of Two Dimensions.
In the Double Elliptic Geometry a line divides a plane in which it lies, and a plane
divides space of three dimensions, into two entirely separate parts, which is not the
case with the Single Elliptic Geometry. Symmetrical figures in the Single Elliptic
Geometry can be made to coincide by moving one of them along the entire length of
a line, but in the Double Elliptic Geometry a movement from intersection point
to intersection point of two lines only puts a figure into the opposite region of space,
and two symmetrical figures can never be made to coincide.
Two polar lines are still everywhere at a quadrant's distance from each other,
and the locus of points at a given distance from a given line is also the locus of points
2l6 THE HYPERSPHERE [v. m.
123. Point Geometry the same as the geometry of the
hypersphere.
THEOREM. Th$ Point Geometry at the centre of a hyper-
sphere is the same as the geometry of the hypersphere.
PROOF. In the hypersphere points, great circles, and
great spheres are its intersections with half-lines drawn
from the centre and with planes and hyperplanes through
the centre, and the distances and angles in the hypersphere
are the same as the corresponding angles at the centre.
Therefore the two geometries are the same.
In particular, to a great sphere of the hypersphere and its
poles correspond at the centre a hyperplane and its per-
pendicular half -lines ; to two polar great circles correspond
two absolutely perpendicular planes; and two simply
perpendicular planes correspond to two great circles inter-
secting at right angles.
The theorems of Point Geometry can, then, be stated
as theorems of the geometry of the hypersphere. We
shall mention only some of the more important results
(Arts. 67, 68, 106, 107, and 109) :
Any two great circles have a pair of common perpendic-
ular great circles, two polar circles which intersect them at
right angles.
When two great circles cut out equal arcs on a polar pair
of common perpendicular great circles, they have an infinite
number of common perpendicular great circles, on all of
which they cut out the same arc. Conversely, if two great
circles have more than two common perpendicular great
circles, the arcs not greater than a quadrant which they
cut out on any one of them and on its polar circle are equal.
at the complementary distance from its polar line, namely, a surface of double revo-
lution with the two polar lines for axes. But a line intersecting the axes meets the
surface in four points instead of two ; and a plane through one axis, and so perpendic-
ular to the other, cuts the surface in two circles instead of one.
123] AND POINT GEOMETRY 217
There are two distances between two great circles, the dis-
tances not greater than a quadrant measured along a polar
pair of common perpendicular great circles.
When the distances are equal the given circles are parallel
in the sense used by Clifford.* Parallel great circles, there-
fore, correspond to isocline planes of the Point Geometry.
There are two senses in which great circles can be parallel,
and two great circles perpendicular to both of two parallel
great circles (which are not polar) are themselves parallel
in the opposite sense. Through any point not a point of a
given great circle nor a point of its polar great circle pass
two great circles parallel in the two senses to the given
circle and to its polar. Two great circles parallel to a given
great circle in the same sense are parallel to each other in
this sense also ; and the set of all the great circles parallel
to a given great circle in a given sense completely fills the
hypersphere, one and only one such circle passing through
each point.
We can prove in another way that the geometry of the
hypersphere and the Double Elliptic Geometry of Three
Dimensions are the same : f
The geometry of the hypersphere is the same as the Point
Geometry at its centre. But Point Geometry is the same
whatever our theory of parallels, and therefore the geometry
of /the hypersphere is the same whatever our theory of par-
allels,
. Now in- the Double Elliptic Geometry of Four Dimensions
all the lines perpendicular to a hyperplane meet in a pair
of opposite points, the poles of the hyperplane. This is true
because any two of these lines lie in a plane (Art. 39, Th. 2)
in which they are perpendicular to a line, and therefore
they meet in two points each at a given distance on both
* See the author's Non-Euclidean Geometry, p. 68.
t See the author's Non-Euclidean Geometry, pp. 63 and 26.
2l8 THE HYPERSPHERE [v. ra.
of them. Thus the hyperplane is a particular case of a
hypersphere having either pole as centre. Any point is
one of the poles of a hyperplane, and the Point Geometry
is the same as the hyperplane geometry : it is the Double
Elliptic Non-Euclidean Geometry of Three Dimensions.
124. Rotation of the hypersphere. The surface of
double revolution. Rotation of the hypersphere on itself
is the same as the rotation of the Point Geometry at its
centre. In any simple rotation a certain great circle, the
axis of rotation, remains fixed in all of its points; while
its polar great circle, the circle of rotation, rotates or slides
on itself (Art. 81).*
Rotation around a great circle in the hypersphere is not a dis-
torted rotation, such as we might have in the case of a flexible object
rotated around a curved axis, but an actual hyperspace rotation
around the plane of the great circle.
A double rotation is a combination of two simple rotations
around two polar great circles. A double rotation can also
be regarded as a double rotation along the two polar great
circles, or as a screw motion along either one of them.
When the two rotations of a double rotation are equal it
is a parallel motion, corresponding to an isocline rotation
at the centre of the hypersphere. In a parallel motion all
great circles parallel to the circles of rotation in the sense
of the rotation rotate on themselves, and the motion can
be regarded as a parallel motion along any polar pair taken
* A hypersphere can be moved freely on itself. Therefore, without attempting
to define curvature in hyperspace, we can say that a hypersphere is a space of con-
stant curvature. If our space were a hypersphere in Euclidean space of four dimen-
sions we should realize the Elliptic Geometry. The Elliptic Geometry is therefore
sometimes supposed to assume that our space is a space of constant curvature like
a hypersphere, not a space of no curvature like a hyperplane. Elliptic Geometry
of Three Dimensions, however, does not depend on any assumption of a Euclidean
space of four dimensions. We might suppose our space to be an elliptic space lying
in elliptic space of four dimensions and with no curvature whatever.
123, 124] SURFACE OF DOUBLE REVOLUTION 219
from this set of circles. We can also think of the motion
as a parallel motion with respect to the set of circles, with-
out thinking of any particular pair as the circles of rotation
or as the axes of rotation (Art. 112).
THEOREM i. Any position of a hypersphere can be ob-
tained from any other position with the same centre by a simple
or a double rotation (Art. 101, Th. i).
THEOREM 2. Two simple rotations are equivalent to a
single simple rotation when and only when their axes inter-
sect (Art. 102, Ths. i and 2).
A surface of double revolution consists of the points of a
hypersphere at a given distance from a given great circle,
and so at the complementary distance from the polar great
circle.* These two great circles are the axes and the two
distances are the two radii of the surface. The surface is
covered with two sets of parallel great circles, those of one
set parallel in one sense and those of the other set parallel
in the other sense. Through each point of the surface
passes one and only one circle of each set, and the surface
can be regarded as consisting of the circles of either one of
these sets. Any great sphere containing one of the axes
intersects the surface in a meridian circle ; and any circle of
either of the two sets of parallel great circles forms with the
meridian circle at any point of it an angle equal to the cor-
responding radius of the surface (see proof of Th. 2 of Art.
107, or the author's Non- Euclidean Geometry , p. 67). The
meridian circles with respect to one axis are the parallel
circles of the surface regarded as a surface of revolution
around the other axis ; and two great circles, one from each
* See the author's Non-Euclidean Geometry, p. 68. The surface of double revolu-
tion is somewhat like the anchor ring of the ordinary geometry. This surface is
of importance in the theory of functions of two complex variables. See Poincare*,
" Sur les residus des integrates doubles," A eta Mathematica, vol. 9, 1886-1887, p. 359.
220 THE HYPERSPHERE [v. m.
of the two sets of parallel great circles, form at either of
their two points of intersection two pairs of vertical angles,
the angles of one pair the double of one radius of the sur-
face, and the angles of the other pair the double of the other
radius.
The surface of double revolution is the intersection of the
hypersphere and a conical hypersurface of double revolu-
tion having its vertex at the centre of the hypersphere
(Art. 118). When the two radii of the surface are equal,
each being equal to -> the circles of each set of parallel
4
great circles on the surface are perpendicular to the circles
of the other set. They correspond to two conjugate series
of isocline planes at the centre (Art. 105).*
* We have defined hyperconical hypersurface as consisting of lines through a
point with a hyperplane directing-surface (Art. 33), and this is sufficient for any
hypercone with a hyperplane base ; but unless we make provision for certain special
cases, like that of the conical hypersurface of double revolution, we should get a
more general definition by making the hypersurface a "surface" of the three-dimen-
sional Point Geometry, or by defining it as consisting of the half-lines drawn from the
centre of a hypersphere through the points of any hyperspherical surface, together
with the centre itself.
CHAPTER VI
EUCLIDEAN GEOMETRY. FIGURES WITH PARALLEL
ELEMENTS
125. The axiom of parallels first introduced at this point.
The development of the preceding chapters has been made
independent of the axiom of parallels. They may be called
chapters in Pangeometry* We shall now make a study of
parallels and of figures with parallel elements, confining
ourselves, however, to the geometry of Euclid. In other
words, we shall assume an axiom of parallels which it will
be convenient to put in the following form :
AXIOM. Through any point not a point of a given line
passes one and only one line that lies in a plane with the given
line and does not intersect it.
I. PARALLELS
126. Parallel lines and parallel planes. Lines and
planes are parallel to one another as in the ordinary geome-
try : two lines when they lie in one plane and do not inter-
sect, a line and a plane or two planes when they lie in one
hyperplane and do not intersect.
THEOREM i. Two lines perpendicular to the same hy-
per plane are parallel (see Art. 39, Th. 2).
THEOREM 2. A hyperplane perpendicular to one of two
parallel lines is perpendicular to the other.
THEOREM 3. // two planes through a point are parallel
to a given line they intersect in a parallel line.
* A title used by Lobachevsky in 1855.
222 PARALLELS [vi. i.
THEOREM 4. // a hyperplane intersects one of two par-
allel planes and does not contain it, the hyperplane intersects
the other plane also y and the two lines of intersection are
parallel.
For the hyperplane intersects the hyperplane of the
parallel planes in a plane which intersects the parallel
planes in parallel lines.
THEOREM 5. If a plane meets one of two parallel planes
in a single pointy it will meet the other in a single point.
THEOREM 6. Two planes absolutely perpendicular to a
third are parallel (see Art. 45, Th.).
THEOREM 7. A plane absolutely perpendicular to one of
two parallel planes is absolutely perpendicular to the other.
THEOREM 8. Two planes parallel to a third are parallel
to each other.
For a plane absolutely perpendicular to the third is ab-
solutely perpendicular to the first two, and they are par-
allel by Th. 6.
THEOREM 9. // three parallel planes all intersect a given
line, they all lie in one hyperplane.
THEOREM 10. Two planes absolutely perpendicular to
two parallel planes are parallel, and two planes parallel re-
spectively to two absolutely perpendicular planes are absolutely
perpendicular.
THEOREM n. // two planes intersect in a line, planes
through any point parallel to them intersect in a parallel line
and form dihedral angles equal to the dihedral angles formed by
the two given planes.
PROOF. The parallel planes are parallel to the line of
intersection of the two given planes, and therefore intersect
in a parallel line, by Th. 3. Now a hyperplane perpendicu-
lar to these parallel lines (see Th. 2) cuts the planes in lines
126] LINES AND PLANES 223
which contain the sides of the plane angles of the various
dihedral angles formed about the two parallel lines. Cor-
responding plane angles, and therefore corresponding dihe-
dral angles,* are equal.
COROLLARY. // two planes are perpendicular, planes
through any point parallel to them are also perpendicular.
THEOREM 12. // two planes have a point in common,
parallel planes through any other point make the same angles
(Art. 69).
PROOF. Let a and ft be the two given planes having
a point O in common, and let OL and ft' be planes through a
second point Q' parallel respectively to a and ft. The
planes through O' parallel to the common perpendicular
planes of a. and ft are themselves common perpendicular
planes of a r and ft' (Th. n, Cor.). On each of these common
perpendicular planes the same angles are cut out as on the
corresponding planes at O, since the intersection of any two
planes intersecting in a line at O' is parallel to the intersec-
tion of the parallel planes at O, and two intersecting lines
at O' lie in a hyperplane with the parallel lines at 0, forming
angles equal to the angles formed by the latter.
COROLLARY. A plane isocline to one of two parallel planes
is isocline to the other and makes the same angle with both.
THEOREM 13. Two lines not in the same plane have only
one common perpendicular line (see Art. 62).
Since the two lines lie in a hyperplane this is always a
theorem of geometry of three dimensions, and is proved
as in the text-books.
THEOREM 14. // a line and plane do not lie in one hy-
perplane, they have only one common perpendicular line.
See proof of Th. i of Art. 63.
* Corresponding dihedral angles have corresponding faces. Two corresponding
faces are parallel half-planes lying in their hyperplane on the same side of the plane
determined by the two parallel lines.
224 PARALLELS [vi. i.
127. Half-parallel planes. Two planes which do not
lie in one hyperplane and do not intersect are said to be
half-parallel or semi-parallel.
THEOREM i. The linear elements of two half-parallel
planes are all parallel to one another (see Art. 27).
THEOREM 2. The linear elements which lie in one of two
half-parallel planes are parallel to the other plane, and these are
the only lines which lie in one plane and are parallel to the other.
THEOREM 3. Through any point passes one and only
one hyperplane perpendicular to each of two half-parallel
planes (see Art. 53).
THEOREM 4. Two half-parallel planes have one and only
one common perpendicular plane.
PROOF. There is one such plane, by Th. 3 of Art. 63.
Suppose, then, we have given a plane perpendicular to each
of two half-parallel planes. It will intersect these planes
in linear elements, the edges of various right dihedral angles,
each with one face in the perpendicular plane and one in
one of the half-parallel planes. A perpendicular hyper-
plane intersects the planes in lines which contain the sides
of the plane angles of these dihedral angles, that is, it in-
tersects the perpendicular plane in the common perpen-
dicular line of the lines in which it intersects the two half-
parallel planes. There is only one such common perpen-
dicular line, and the given plane is the plane determined as
in Art. 63 by this common perpendicular line and the linear
elements which it intersects.
THEOREM 5. The only common perpendicular lines of
two half-parallel planes are those which lie in the common
perpendicular plane.
The perpendicular distance or simply the distance, be-
tween two half-parallel planes is the distance between
1271 HALF-PARALLEL PLANES 22$
the points where they are cut by a common perpen-
dicular line. It is the same for all of these lines, since
the common perpendicular plane cuts the given plane in
parallel lines (Th. i).
THEOREM 6. The perpendicular distance between two
half-parallel planes is less than the distance measured along
any line which intersects both and is not perpendicular to both.
PROOF. The perpendicular distance between two ele-
ments lying one in each of the two given planes is the dis-
tance measured along some line lying in a perpendicular
hyperplane, the distance measured along some line between
the intersections of the given planes and this hyperplane.
It is less than the distance between the two elements along
any line which does not lie in a perpendicular hyperplane.
But the intersections of the given plane and the perpen-
dicular hyperplane have for common perpendicular only
the line in this hyperplane which is perpendicular to the two
given planes. Therefore the perpendicular distance be-
tween the two given planes is the perpendicular distance
between these two intersections, and is less than the dis-
tance between the two planes measured on any line that is
not perpendicular to both.
THEOREM 7. Two planes through a point parallel respec-
tively to two half -parallel planes intersect in a line which u
parallel to their linear elements.
For the line through the point parallel to the linear ele-
ments is parallel to the two given planes, and therefore
lies in both of the two planes which are parallel to them
through the point.
THEOREM 8. // a plane distinct from each of two parallel
planes intersects one in a line and does not intersect the other
in a line, it will be half-parallel to the second.
Q
226 PARALLELS [vi. i.
PROOF. If the given plane were in a hyperplane with the
second parallel plane, this hyperplane, containing the line
in which the given plane intersects the first parallel plane,
must be the hyperplane of the parallel planes; or if the
given plane intersected the second parallel plane in a point,
it would lie entirely in the hyperplane of the parallel planes.
Thus, in either case, we should have a plane lying in the hy-
perplane of the two parallel planes, intersecting one in a
line, and therefore the other in a line. As the given plane
does not intersect the second parallel plane in a line, it
cannot lie in a hyperplane with it nor intersect it at all.
They must, therefore, be half -parallel.
THEOREM 9. // a plane perpendicular to one of two ab-
solutely perpendicular planes does not contain their point
of intersection, it is half-parallel to the other.
PROOF. Let a and a' be two absolutely perpendicular
planes intersecting in a point O, and let /3 be a plane perpen-
dicular to a but not containing O. Then cannot lie in
a hyperplane with a', for such a hyperplane would inter-
sect a only in a line through 0. Nor can /3 intersect a!
even in a point, for then it would contain the line through
such a point perpendicular to a, and so contain the point 0.
j8 is therefore half-parallel to a'.
128. Lines and planes parallel to a hyperplane. Parallel
hyperplanes. A line and a hyperplane, a plane and a hy-
perplane, or two hyperplanes, are parallel when they do
not intersect.
THEOREM i. If a line, not a line of a given hyperplane,
is parallel to a line of the hyperplane, it is parallel to the
hyperplane; and if a plane, not a plane of a given hyperplane,
is parallel to a plane of the hyperplane, it is parallel to the
hyperplane.
127, i2] HYPERPLANES 227
THEOREM 2. // a line is parallel to a hyper plane, it is
parallel to the intersection of the hyperplane with any plane
through it or with any hyperplane through it; and if a plane
is parallel to a hyperplane, it is parallel to the intersection of
the hyperplane with any hyperplane through it.
THEOREM 3. If a line is parallel to a hyperplane, a line
through any point of the hyperplane parallel to the given
line lies wholly in the hyperplane; and if a plane is parallel
to a hyperplane, a plane or line through any point of the hy-
per plane parallel to the given plane lies wholly in the hyper-
plane.
THEOREM 4. Two hyperplanes perpendicular to the same
line are parallel.
THEOREM 5. // one of two parallel hyperplanes is per-
pendicular to a line, the other is also perpendicular to the line.
THEOREM 6. Through a point, not a point of a given hy-
perplane, can be passed one and only one parallel hyperplane.
In general, we can pass through a point a hyperplane
parallel to a given hyperplane, to a given line and plane,
or to three given lines ; through a line, a hyperplane parallel
to a given plane or to two given lines ; through a plane, a
hyperplane parallel to a given line. In some cases, how-
ever, the construction will give us a hyperplane containing
some or all of the given figures, and in some cases more
than one hyperplane can be obtained.
THEOREM 7. All the lines and planes in one of two par-
allel hyperplanes are parallel to the other, and all the lines
and planes through a point, parallel to a hyperplane, lie in
a parallel hyperplane.
THEOREM 8. // a plane intersects two parallel hyper-
planes, or if a hyperplane intersects two parallel planes, the
228 PARALLELS [vi. I.
lines of intersection are parallel; and if a hyper plane inter-
sects two parallel hyperplanes, the planes of intersection are
parallel.
THEOREM g. // three non-coplanar lines through a point
are respectively parallel to three other non-coplanar lines
through a point, the two sets of lines determine the same hyper-
plane or parallel hyperplanes; or if an intersecting line and
plane are respectively parallel to another intersecting line and
plane, they determine the same hyperplane or parallel hy-
perplanes.
THEOREM 10. Two trihedral angles having their sides
parallel each to each and extending in the same direction *
from their vertices are congruent.
For the corresponding face angles are equal, f
THEOREM n. If a line is parallel to a hyperplane, all
points of the line are at the same distance from the hyperplane;
or if a plane is parallel to a hyperplane, all points of the
plane are at the same distance from the hyperplane.
THEOREM 12. Two parallel hyperplanes are everywhere
equidistant.
The student may prove the following theorem :
Let ABCDE be a pentahedroid cut by a hyperplane a so that the
edge AB lies on one side of a and the face CDE on the other side.
Then if a is parallel to the line AB and to the plane CDE, the section
will be a prism ; if a is parallel to the line but not to the plane, the
section will be a truncated prism ; if a is parallel to the plane but not
to the line, the section will be a frustum of a pyramid ; or if a is not
parallel to the line nor to the plane, the section will be a truncated
pyramid (see Art. 31, Th. 3).
* Two parallel half-lines extend in the same direction when in their plane they
tie on the same side of the line determined by their extremities.
t The proof given in our text-books that the two dihedral angles are equal does
not require that the' trihedral angles shall lie in one hyperplane. , .
128, i2 9 ] ISOCLINE PROJECTION 2 29
129. Projection from a plane upon an isocline plane
produces similar figures.
THEOREM i. Any plane polygon and its projection upon
an isocline plane are similar.
PROOF. Let be the point of intersection of the two
isocline planes. Let A and B be any two points of the orig-
inal polygon, and A' and B' their projections. 0-4-4'
and OBB' are two right triangles with equal acute angles
at O (Art. 69). They are similar, and the sides OA and
OB are proportional to the sides OA 1 and OB'. The
angle AOB is also equal to the angle A'OB f (same refer-
ence). Therefore the triangles OAB and OA'B' are them-
selves similar triangles. Now if the triangles formed by
joining the vertices of a plane polygon to a point O in its
plane are respectively similar to the triangles formed in the
same way from another polygon, the two polygons are
similar. Therefore, the given polygon and its projection
are similar.*
COROLLARY. The projection of a circle upon a plane iso-
cline to its plane is a circle.
THEOREM 2. Conversely, if a plane polygon is similar
to its projection upon another plane , the two planes are iso-
cline or parallel.
PROOF. The planes (Art. 46) projecting a figure upon
one of two parallel planes project the same figure upon the
other (Art. 126, Th. 7), and the two projections are equal.
We can then suppose that the plane of the projection passes
through a vertex of the given polygon, and complete our
proof by reversing the steps of the preceding proof and using
the corollary to the second theorem of Art. 69.
* It can be proved in any case of projection of a plane polygon upon another
plane that the area of the projection is equal to the area of the original polygon multi-
plied by cos 4> cos ', where < and <' are the angles between the planes of the two
figures.
230 THE "HYPERPLANE AT INFINITY" [vi. n.
II. THE "HYPERPLANE AT INFINITY"
130. The sense in which expressions in regard to in-
finity are used. Just as we sometimes speak of points,
lines, and a plane at infinity, so now we can speak of planes
at infinity in different hyperplanes, and of a hyperplane
at infinity. This we shall regard only as a matter of lan-
guage. We introduce these expressions without introduc-
ing any new elements into our geometry or a number " in-
finity " into our number-system. We express certain facts
of parallelism as if they were matters of intersection, from
which, indeed, they are derived by limiting processes.
Thus, we say that two lines intersect at infinity only as
another way of saying that they are parallel. These forms
of expression appear to simplify the conception of parallel-
ism, and they enable us to generalize certain theorems of
intersection. We must be careful to remember, however,
that from this point of view we are not really introducing
any points, lines, and planes, nor a region to be called in-
finity, and that we are not really making any change in our
conception of parallelism.*
Points at infinity are sometimes called ideal points,
lines at infinity ideal lines, and so on.
We shall give a brief account of the geometry at infinity,
and point out the relation to infinity of certain figures
studied in the following pages; but the rest of this book
will be entirely independent of the present section, and all
reference to infinity can be omitted without disturbing the
continuity of our chain of theorems. f
* It would be perfectly legitimate to assume points, lines, etc , at infinity, and a
number infinity, just as we assume other points, lines, and numbers. Such a course
may seem no more unreasonable than the assumptions which distinguish the Elliptic
and Hyperbolic geometries from that of Euclid, or the assumption of a fourth di-
mension. But the point of view here presented is better for this geometry. See
note by Professor Bdcher, Bulletin of the American Mathematical Society, vol. 5,
1898-1899, p, 182.
t The elements at infinity are made the basis of many of the proofs in the Geome-
try of Veronese.
130,131] A SINGLE HYPERPLANE 231
131. The elements at infinity ; all comprised in a single
hyperplane. A line has a single point at infinity, its in-
tersection with any parallel line. A plane has a line at
infinity, its intersection with any parallel plane. The line
at infinity of any plane is made up of the points at infinity
of its lines, and is determined by any two of them.
In space of three dimensions, and so in any hyperplane,
we have a plane at infinity, the intersection of the hyper-
plane with any parallel hyperplane. The plane at infinity
of any hyperplane consists of the points at infinity of its
lines, and contains the lines at infinity of its planes. Or,
we can say that a plane at infinity consists of the points
that we get if we take any three non-collinear points at
infinity, all points collinear with any two of them, and all
points collinear with any two obtained by this process.
Lines which intersect a given hyperplane, but do not lie
in it, have points at infinity which do not lie in the plane
at infinity of this hyperplane. That is, in space of four
dimensions the points at infinity are not all points of a
single plane at infinity. Thus we have a hyperplane at
infinity.
A hyperplane at infinity consists of the points that we get
if we take four non-coplanar points at infinity, all points
collinear with any two of them, and all points collinear
with any two obtained by this process.
THEOREM. All points at infinity in space of four dimen-
sions lie in a single hyperplane.
PROOF. Through a given point O can be drawn a line
parallel to any given line. Therefore, all points at infinity
are the points at infinity of the lines through O. Through
any point of hyperspace passes a line that goes through O
and has a point at infinity.
Now we can get all the lines through O by taking four
232 THE "HYPERPLANE AT INFINITY" [vi. n.
of these lines which are not in one hyperplane, all coplanar
with any two of them, and all coplanar with any two ob-
tained by this process (see last paragraph of Art. 64). But
any four non-coplanar points at infinity are the points at
infinity of four lines through O not in a hyperplane, and if
a line through O is coplanar with two others, its point at
infinity is collinear with their points at infinity. There-
fore we can get all the points at infinity by taking four non-
coplanar points at infinity, all points collinear with any two
of them, and all points collinear with any two obtained by
this process. Hence all the points at infinity lie in a single
hyperplane.
132. Distance and angle at infinity. We can define
distance between two points at infinity as proportional to
the angle which they subtend at any point O which is not
a point at infinity. There are two supplementary angles
at 0, and so two distances between two points at infinity.
We shall usually mean the smaller of these two distances
when they are not equal. Taking the entire length of a
line at infinity as equal to TT, we shall give to all distances
at infinity the same measures as to the corresponding angles
atO.
Two lines in a plane at infinity are the lines at infinity
of two non-parallel planes in a hyperplane, and always in-
tersect, intersecting in a single point. They divide the
rest of their plane into two separate regions, each a lune-
shaped region with the extremities coinciding and the two
angles actually vertical angles. A half -line at infinity is
the same as an entire line, but it will be more convenient
here to regard it as a " restricted " portion of a line, lying
in one direction from one of its points, the extremity of
the half-line. An angle will then be formed by two such
half-lines with a common extremity lying along different
131-133] ANGLES AT INFINITY 233
lines. The lines through the points of such an angle and
a point O not at infinity will determine a definite pair of
vertical dihedral angles, and the measure of the angle at
infinity will be the same as the measure of either of these
dihedral angles, the sum of the angles around a point in
a plane being always equal to 2 TT.
Two planes at infinity intersect in a line, and divide the
rest of the hyperplane at infinity into two regions each some-
what like a double convex lens.* Along the edge of such a
region we have what may be called a dihedral angle at in-
finity. We can think of a half-plane at infinity as a " re-
stricted " portion of a plane lying on one side of a line, the
edge of the half-plane, and think of a dihedral angle at in-
finity as formed by two such half-planes with a common
edge but not lying in the same plane (compare Art. 120).
To the dihedral angle will then correspond a hyperplane angle
whose measure can be taken as the measure of the former.
Since two lines in a plane at infinity intersect in a single
point, the geometry at infinity is the Single Elliptic Geome-
try (see foot-note, p. 215). This geometry is, indeed, the
same as the Point Geometry of the half-lines from a point
O, except that two opposite half-lines correspond to a single
point at infinity. Thus all the theorems of Point Geometry
can be stated as theorems of the geometry at infinity, just
as they can be stated as theorems of the geometry of the
hypersphere (Art.
133. Some generalizations now made possible. In-
tersection and perpendicularity. We will now show how
* The volume of one of these regions is to the entire volume of the hyperplane at
infinity as the corresponding dihedral angle is to two right dihedral angles (not to
four right dihedral angles, because the geometry at inanity is the Single Elliptic
Geometry).
t We could have assumed that a line has two points at infinity, a separate point
for each of two opposite half -lines. The geometry at infinity would then be the
Double Elliptic Geometry. See Veronese, Grundzugc, Ft. I, Bk. I.
234 THE "HYPERPLANE AT INFINITY" [vi. n.
the forms of expression introduced in this section enable
us to generalize certain theorems of geometry :
Two points determine a line, three non-collinear points
determine a plane, and four non-coplanar points determine
a hyperplane, even when some or all of the points are at
infinity.
Two lines in a plane always intersect, and any two lines
which intersect lie in a plane. A line and a plane in a hy-
perplane intersect, and any line and plane which intersect
lie in a hyperplane. Thus a line and plane which do not
lie in a hyperplane do not intersect even at infinity.
Two planes which lie in a hyperplane intersect in a line.
Two planes which do not lie in a hyperplane intersect in
a point ; when the point is at infinity the planes are half-
parallel.
A line and a hyperplane always intersect, intersecting
in a point ; a plane and a hyperplane always intersect, in-
tersecting in a line ; and two hyperplanes always intersect,
intersecting in a plane.
A dihedral angle at infinity is the intersection of a hyper-
plane angle by the hyperplane at infinity. A polyhedral
angle at infinity is the intersection of a piano-polyhedral
angle by the hyperplane at infinity. The parts of the
polyhedral angle have the same relation to the piano-
polyhedral angle as do those of any right section (Art. 75).
Two perpendicular lines, not lines at infinity, are lines
whose points at infinity are at a quadrant's distance from
each other. A line and plane are perpendicular when their
point and line at infinity are pole and polar in the plane at
infinity of their hyperplane. Two planes are absolutely
perpendicular when their lines at infinity are polar lines.
When a line at infinity intersects both of two polar lines
it is perpendicular to both. Any two lines at infinity have
133, i34l PRISMOIDAL HYPERSURFACES 235
two common perpendicular lines, and the distances between
two lines are measured on these common perpendicular
lines.
III. HYPERPRISMS
134. Prismoidal hypersurfaces. Their interiors. Sec-
tions. Axes. A prismoidal hypersurface consists of a
system of parallel lines passing through the points of a given
polyhedron but not lying in the hyperplane of the polyhe-
dron. The polyhedron is called the directing-polyhedron,
the parallel lines are the elements, and the elements which
pass through the vertices are lateral edges. We shall
assume that the directing-polyhedron is a simple convex
polyhedron.
The elements which pass through the points of a face of
the directing-polyhedron constitute the interior of a pris-
matic surface and a cell of the hypersurface. The elements
which pass through the points of an edge of the directing-
polyhedron constitute what may be called a strip, which
is that portion of a plane that lies between two parallel
lines. This strip is a face of the hypersurface, the common
face of two adjacent prismatic surfaces.
The interior of a prismoidal hypersurface consists of the
lines which pass through the points of the interior of the
directing-polyhedron and are parallel to the elements.
The hypersurface being convex, the interior of any segment
whose points are points of the hypersurface will lie entirely
in the interior of the hypersurface unless it lies entirely in
the hypersurface itself, and a half-line drawn from a point
of the interior and not parallel to the elements will inter-
sect the hypersurface in one and only one point.
THEOREM i. A hyperplane passing through a point of
the interior of a prismoidal hypersurface and parallel to the
elements intersects the hypersurface in a prismatic surface.
236 HYPERPRISMS [vi. m.
For the hyperplane intersects the directing-polyhedron
in a convex polygon, and intersects the hypersurface in
the elements which pass through the points of this polygon.
THEOREM 2. A hyperplane which is not parallel to the
elements of a prismoidal hypersurface intersects the hyper-
surface in a polyhedron, and any such polyhedron can be
taken as directing-polyhedron.
Each element of the hypersurface, and, indeed, each line
of hyperspace which is parallel to the elements, meets
the hyperplane in a point. We have a correspondence
between the points of the hyperplane and the lines which
are parallel to the elements of the hypersurface, as also
between the points of the intersection and the points of
the given directing-polyhedron. Each cell of the hyper-
surface intersects the given hyperplane in a face, each face
of the hypersurface in an edge, and each lateral edge in a
vertex, of the given intersection.
A right section is a directing-polyhedron whose hyperplane
is perpendicular to the elements.
THEOREM 3. Directing- polyhedrons lying in parallel hy-
perplanes are congruent, and any two homologous points of
two such polyhedrons lie in a line parallel to the elements.
THEOREM 4. // a prismoidal hypersurface has a paral-
lelopiped for directing- poly hedr on, it will have three pairs of
equal opposite lateral cells lying in parallel hyper planes, and
all of its directing-polyhedrons will be parallelopipeds.
PROOF. Any two opposite faces of the given parallelepiped
are equal parallelograms lying in parallel planes. They
are, then, directing-polygons of equal prismatic surfaces
lying in parallel hyperplanes (Art. 128, Th. 9). Therefore,
any directing-polyhedron will have three pairs of parallel
opposite faces (Art. 128, Th. 8), and will be a parallelepiped.
134, i3Sl THE PARTS OF A HYPERPRISM 237
THEOREM 5. // any directing-polydedron of a prismoidal
hypersurface has a centre of symmetry, the line through this
point parallel to the elements is an axis of symmetry, meeting
the hyper plane of every directing-polyhedron in a point which
is a centre of symmetry of this polyhedron. Each point of
the line is, in fact, a centre of symmetry for the entire hyper-
surface, and the line as a whole is a line of symmetry.
For this line lies mid-way between the two lines in which
any plane containing it intersects the hypersurface, and any
line intersecting it determines with it such a plane.
135. Hyperprisms. Interior of a hyperprism. A
hyperprism consists of that portion of a prismoidal hyper-
surface which lies between two parallel directing-polyhe-
drons, together with the directing-polyhedrons themselves
and their interiors.
The interiors of the directing-polyhedrons are the bases.
In each hyperplane of the hypersurface we have a prism
whose interior is one of the lateral cells of the hyperprism.
The lateral faces and edges of these prisms are the lateral
faces and lateral edges of the hyperprism. The lateral edges
are all equal; the bases are congruent (Art. 134, Th. 3).
The interior of a hyperprism consists of that portion of
the interior of the prismoidal hypersurface which lies be-
tween the bases. The hypersurface being convex, the
interior of any segment whose points are points of the hy-
perprism will lie entirely in the interior of the hyperprism
unless it lies entirely in the hyperprism itself, and a half-
line drawn from a point of the interior will intersect the
hyperprism in one and only one point.
A hyperprism is a right hyperprism when the lateral
edges are perpendicular to the hyperplanes of the bases.
When also the bases are the interiors of regular polyhedrons
the hyperprism is regular.
HYPERPRISMS
[vi. ra.
We can cut apart a hyperprism, cutting it along its faces, suffi-
ciently to spread it out into a single hyperplane. We can do this, for
example, so that the lateral prisms shall remain attached to one base,
while the other base rests upon one of them. The figure below repre-
sents a tetrahedroidal hyperprism cut apart and spread out in this
way. The four prisms rest upon the four faces of a tetrahedron, and
a second tetrahedron equal to the first (symmetrical in this hyper-
plane) rests on the other end of one of these prisms. Now in hyper-
space we can turn these prisms around the faces of the tetrahedron
upon which they rest away from the hyperplane of this tetrahedron,
and the other tetrahedron around the face by which it is attached to
one of the prisms: we can do this without separating any of the
figures or distorting them in any way, until we bring them all together,
each prism with a lateral face resting upon a lateral face of each of
the others, and each of the four faces of the second tetrahedron rest-
ing upon one of the prisms. The figure will then enclose completely
a portion of hyperspace (see note in Art. 32).
The student may investigate the conditions necessary in order
that three given prisms and two tetrahedrons may be the lateral
prisms and tetrahedrons of a tetrahedroidal hyperprism.
i3S i3fl HYPERPARALLELOPIPEDS 239
136. Special forms of hyperprisms. Hyperparallelo-
pipeds. The hypercube. A hyperprism whose bases are
the interiors of prisms can be regarded in two ways as a
hyperprism of this kind ; for the lateral prisms correspond-
ing to the ends of the bases are parallel (Art. 128, Th. 9)
and congruent, and the remaining lateral prisms are parallel-
epipeds, which can be regarded as having their bases on this
second pair of prisms and their lateral edges those edges
which belong also to the first pair of prisms. This figure
is a particular case of a double prism, and will be studied
in the next section (see Art. 144).
A hyper parallelepiped is a hyperprism whose bases are
the interiors of parallelepipeds. In a hyperparallelopiped
there are four pairs of opposite equal parallel parallelepi-
peds whose interiors are the cells, and the interiors of any
pair can be taken as bases. There are four sets of eight
parallel edges, each set joining the vertices of two opposite
cells, becoming the lateral edges when these cells are taken
as bases. The section of a hyperparallelopiped made by a
hyperplane intersecting all eight of the edges of a set will be
a parallelepiped (Art. 134, Th. 4).
THEOREM i. The diagonals of a hyperparallelopiped bi-
sect one another, all passing through a point which is a centre
of symmetry for the hyperparallelopiped.
A right hyperparallelopiped whose base is a rectangular
parallelepiped is a rectangular hyperparallelopiped. The
edges which meet at any vertex lie in the lines of a rectangu-
lar system (Art. 48). The lengths of these four edges are
the dimensions of the hyperparallelopiped.
THEOREM 2. The square of the length of a diagonal of a
rectangular hyperparallelopiped 4s equal to the sum oj the
squares of the four dimensions.
240
HYPERPRISMS
[vi. in.
A hypercube is a rectangular hyperparallelopiped whose
base is the interior of a cube and whose altitude is equal to
the edge of the cube ; that is, its four dimensions are all
equal. The hypercube is a regular polyhedron (see Art.
1 66) : it has eight equal cubical cells, twenty-four equal
faces each a common face of two cubes, thirty- two equal
edges, and sixteen vertices. There are four axes, lying in
lines which also form a rectangular system.
THEOREM 3. The diagonal of a hypercube is twice as
long as the edge.
Six squares which can be folded so as
to form a cube.
Eight cubes which can be folded so as
to form a hypercube.
If we place six equal cubes upon the six faces of a cube, and one
more outside of one of these, just as we put together four prisms and
two tetrahedrons in the note in Art. 135, we can turn these cubes
around the faces upon which they rest and bring them together so
as to form a hypercube. This is analogous to the process of forming
a cube by folding six squares together.
136-138] PLANO-PRISMATIC HYPERSURFACES 241
There are two ways of projecting a hypercube that will both assist
us in forming some conception of it. One is by an oblique projec-
tion, and the other is by projection from a point at a little distance
from the hypercube in the line of one of its axes. We can think of
the first as representing the appearance of the hypercube when we
stand a little to one side, and the second as we look down into it. In
each case we can pick out the projections of the eight cubes whose
interiors form the cells of the hypercube.
The hypercube has become one of the most familiar of the figures
of hyperspace. The reader will find it mentioned in nearly all popu-
lar descriptions of the fourth dimension. See, for example, Fourth
Dimension Simply Explained, pp. 46, 72, 88, 92, and 113.
137. Relation of the prismoidal hypersurface to infinity.
Taking the point of view explained in Art. 130, we can say
that a prismoidal hypersurface is a polyhedroidal angle
whose vertex is at infinity.
The eight hyperplanes of a hyperparallelopiped inter-
sect the hyperplane at infinity in four planes which contain
the faces of a tetrahedron. Each vertex of this tetrahedron
is the vertex of one of the prismoidal hypersurfaces con-
nected with the hyperparallelopiped, and the opposite face
of the tetrahedron is the plane of intersection of the hyper-
planes of the corresponding pair of bases * (see Art. 145).
IV. DOUBLE PRISMS
138. Piano-prismatic hypersurfaces. Sections. We
shall use the word layer to denote that portion of a hyper-
plane which lies between two parallel planes, and call the
parallel planes the faces of the layer.
A piano-prismatic hypersurface consists of a finite number
of parallel planes taken in a definite cyclical order, and the
layers which lie between consecutive planes of this order.
* There are, indeed, four tetrahedrons at infinity, each of which has itS vertices
at these four points and its faces in these four planes, but we do not need to consider
them here.
R
242 DOUBLE PRISMS [vi. IV.
The parallel planes are faces and the layers are cells of the
hypersurface. If a, /3, 7, ... are the faces in order, the
cells can be described as the layers a/3, 187, . . . , and
the hypersurface as the piano-prismatic hypersurface a(3y
.... The faces and all parallel planes within the layers
are the elements of the hypersurface, and are in cyclical
order (Art. 6).
The hypersurface is a simple piano-prismatic hypersurface
when no plane occurs twice as an element. It is convex
when, also, the hyperplane of each cell contains no element
except those of this cell and the two which are its faces.
We shall consider only hypersurfaces which are simple
and convex.
THEOREM i. A hyperplane containing an element of a
piano-prismatic hypersurface or any parallel plane will in-
tersect the hypersurface in elements if at all (Art. 128, Th. i).
THEOREM 2. A hyperplane intersecting but not contain-
ing an element of a piano-prismatic hypersurface will in-
tersect the hypersurface in a prismatic surface.
For the hyperplane intersects all the elements in parallel
lines (Art. 126, Th. 4), and so the cells in strips (see Art.
134), which are the faces of a prismatic surface.
Thus the hypersurface can be regarded as a prismoidal
hypersurface, with a directing prismatic surface instead
of a directing-polyhedron.
THEOREM 3. Two parallel prismatic sections of a piano-
prismatic hypersurface are congruent.
.PROOF. The strips in which the hyperplanes of the
two parallel sections intersect any one of the cells of the
hypersurface lie in parallel planes (Art. 128, Th. 8) and be-
tween parallel planes (the faces of the cell), and are there-
fore of the same width. They lie also on the same side of
i3, i39l DIRECTING- POLYGONS 243
either pair of corresponding edges, lying as they do in the
same layer. Therefore corresponding dihedral angles of
the two prismatic sections are equal (Art. 126, Th. n;
see also foot-note, p. 223), and the sections themselves are
congruent.
139. Directing-polygons of a piano-prismatic hypersur-
face. Interior of the hypersurface.
THEOREM i. A plane which contains a line parallel to
the elements of a piano-prismatic hypersurface, but is not
itself parallel to them, will intersect in a line every element
which it intersects at all.
THEOREM 2. A plane which does not contain a line
parallel to the elements of a piano-prismatic hypersurface
will intersect every element in a point, and will intersect the
hypersurface, the latter being convex, in a convex polygon.
PROOF. The plane will intersect each element in a
point, by Th. 5 of Art. 126. In particular, it will inter-
sect the faces in points which are the vertices of a polygon,
and the layers in the sides of this polygon. The polygon
is a simple convex polygon, since no two elements can inter-
sect the same plane in the same point.
The polygon in which a plane containing no line parallel
to the elements intersects the hypersurface can be called
a directing-polygon, and the hypersurface can be described
as consisting of a system of parallel planes passing through
the points of a given polygon and intersecting the plane of
the polygon only in these points. We can think of the
polygon as going around the hypersurface.
THEOREM 3. Through any line which is not parallel to
the elements of a piano- prismatic hypersurface can be passed
planes intersecting the hypersurface in directing-polygons.
PROOF. Through a point A of the line pass a plane
244 DOUBLE PRISMS [vi. iv.
parallel to the elements. The line intersects the plane
only in this point, and the line and plane determine a
hyperplane. Arty plane through the line and a point which
is not a point of the hyperplane will intersect the hyper-
plane only in the given line, and the plane which we have
drawn parallel to the elements only in the point A : it will
be the plane of a directing-polygon.
THEOREM 4. Two parallel directing- polygons of a piano-
prismatic hypersurface are congruent, and any two fwmologous
points lie in one of the elements.
In fact, their interiors are the bases of a prism whose
lateral surface is cut out from the prismatic surface in
which their hyperplane intersects the hypersurface.
THEOREM 5. If a piano- prismatic hypersurface has a
parallelogram for directing-polygon, it will ham two pairs
of equal opposite cells (layers of the same width) lying in
parallel hyper planes, and all of its directing- polygons will be
parallelograms (see Art. 128, Ths. 8 and 9).
We can study the properties of these hypersurfaces as a
part of the two-dimensional geometry' whose elements are
a set of parallel planes. The theorems in chap. I which
relate to triangles and convex polygons can all be inter-
preted as theorems concerning these hypersurfaces (com-
pare this with Arts. 78 and 112). But we can also prove
these theorems by means of a directing-polygon, making
the elements and all parallel planes correspond to the points
where they intersect the plane of the polygon.
Thus we have the interior of the hypersurface as consisting
of those planes which correspond to the interior of the
directing-polygon. The interior of any segment whose
points are points of the hypersurface will lie entirely in
the interior of the hypersurface unless it lies in the hyper-
surface itself, and a half-line drawn from a point of the
139, 140] DIRECTING-POLYGONS 245
interior and not parallel to the elements will intersect the
hypersurface in one and only one point. Also, that portion
of a plane between two parallel lines of the hypersurface
(a strip, Art. 134), or a layer between any two elements,
lies entirely in the interior unless it lies entirely in the
hypersurface.
THEOREM 6. When the layer between two elements of a
piano-prismatic hypersurface lies entirely in the interior of
the hypersurface, it separates the rest of the interior into two
portions lying on opposite sides of the hyperplane, and with
its faces and each of the two parts into which they separate
the rest of the hypersurface it forms a convex piano-prismatic
hypersurface (see Art. 15, Ths. 2 and 3).
COROLLARY. By taking the diagonal layers which have
in common one of the lateral faces of the hypersurface we can
form a set of triangular piano-prismatic hyper surf aces, their
interiors, together with the diagonal layers, making up the
interior of the given hypersurface.
140. Right direc ting-polygons. Axis-planes. A direct-
ing-polygon whose plane is absolutely perpendicular to the
planes of the elements of a piano-prismatic hypersurface
is called a right directing-polygon.*
THEOREM i. The projection of any directing-polygon
upon the plane of a right directing-polygon is the right direct-
ing-polygon itself (see Art. 46).
THEOREM 2. A plane isocline to the plane of a right direc t-
ing-polygon, but not parallel to the elements, intersects the
hypersurface in a polygon similar to the right directing-poly-
gon; and, conversely, any directing-polygon similar to a
* We will not call it a right section, since it does not completely separate any
two parts of the hypersurface, nor even of a restricted portion of the hypersurface
(see foot-note, p, 65).
246 DOUBLE PRISMS [vi. iv.
right directing-polygon lies in a plane which is isocline to the
plane of the latter (see Art. 129, Ths. i and 2). The right
directing-polygon is the minimum of all these similar polygons.
THEOREM 3. // any directing-polygon of a piano- pris-
matic hypersurface has a centre of symmetry, the plane
through this point parallel to the elements is an axis-plane of
symmetry, meeting the plane of every directing-polygon in a
point which is a centre of symmetry of this polygon. Each
point of the plane is, in fact, a centre of symmetry for the
entire hypersurface, every line of the plane is a line of sym-
metry, and the plane as a whole is a plane of symmetry.
For the plane lies mid- way between the two planes in
which any hyperplane containing it intersects the hyper-
surface, and any line intersecting it but not lying in it
determines with it such a hyperplane.
141. Intersection of two piano-prismatic hypersurfaces.
The two sets of prisms. When the elements of a piano-
prismatic hypersurface intersect the elements of a second
piano-prismatic hypersurface only in points, the intersec-
tion of the two hypersurfaces consists of the lateral sur-
faces of a set of prisms joined together in succession by
their bases, together with the polygons whose interiors
are these bases. In another way, also, the same intersec-
tion consists of the lateral surfaces of a set of prisms joined
together by their bases, together with the polygons whose
interiors are these bases.
In fact, the faces of the first hypersurface are parallel
planes intersecting the second hypersurface in a set of equal
parallel directing-polygons of the latter, and the cells of
the first are layers, each layer intersecting the second in
the lateral surface of a prism whose bases are interiors of
two of these directing-polygons (see Art. 139, Th. 4).
In the same way the faces of the second hypersurface
i 4 o, i4i] TWO HVPERSURFACES 247
are parallel planes intersecting the first hypersurface in a
set of parallel directing-polygons, and each cell of the second
intersects the first in the lateral surface of a prism whose
bases are the interiors of two of these directing-polygons.
Thus the entire intersection consists in two ways of the
lateral surfaces of a set of prisms joined in succession by
their bases, together with the polygons whose interiors are
these bases.
The lateral faces of any prism of the first set are the
interiors of parallelograms, and are the intersections of one
particular cell of the first hypersurface with the different
cells of the second. A set of corresponding faces of these
prisms, one from each prism, are, then, the intersections
of the different cells of the first hypersurface with one
particular cell of the second. Thus the faces of any partic-
ular prism of either set form a set of corresponding faces
of the different prisms of the other set, and every lateral
face of a prism of one set is a lateral face of some prism of
the other set. The lateral edges of the prisms of one set
are the sides of the bases of the prisms of the other set, and
the prisms of one set can be said to be joined crosswise by
their lateral faces to the prisms of the other set. The in-
teriors and the bases of the prisms of the first set lie in the
first hypersurface and in the interior of the second, and the
interiors and bases of the prisms of the second set lie in the
second hypersurface and in the interior of the first. We
can think of the first set of prisms as going around the first
hypersurface, while any base or interior of a cross section
of any of these prisms is a piece cut out of an element of
the first hypersurface. In the same way the second set
of prisms goes around the second hypersurface.
A double prism consists of the intersection of two piano-
prismatic hypersurfaces whose elements intersect only in
24$
DOUBLE PRISMS
[vi. iv.
points, together with all that portion of each hypersurface
which lies in the interior of the other ; that is, it consists
of all of the prisms of both sets described above and their
interiors.
When the elements of one hypersurface are absolutely
perpendicular to the elements of the other the double prism
is a right double prism. When also the prisms of the two
sets are regular the double prism is regular.
C
In a right double prism the prisms of each set can be put together
in one hyperplane so as to form a single right prism with lateral edge
equal to the perimeter of the base of the prisms of the other set. In
forming the double prism these two single prisms
are folded towards each other in such a way that
the upper base of each occupies the same position
as its lower base and the lateral surface of one
coincides as a whole with the lateral surface of
the other. Thus one single prism alone, folded
in this way until the ends come together, gives
us the other prism and the entire figure.
When we have an oblique double prism and
spread out the prisms of one set in a single
hyperplane, we have, not a prism, but a set of
prisms with equal bases resting upon one another
somewhat like a broken column. The upper
base of the highest prism, however, and the
i 4 i, 142] THE INTERIOR 249
lower base of the lowest prism will lie in parallel planes and will be
so placed that each is the projection of the other.
142. Interior of a double prism. The directing-polygons.
The interior of a double prism consists of the points which
are common to the interiors of its two hypersurfaces. A
plane lying in the interior of one of the hypersurfaces parallel
to its elements intersects the other in a directing-polygon
whose interior belongs to the interiors of both, and so to
the interior of the double prism. The vertices of this
polygon are a set of corresponding points of the bases, and
the sides lie in the interiors, of the prisms of the set which
goes around the former hypersurface.
THEOREM i. Any plane through a point of the interior
of a convex double prism intersects the double prism in a
convex polygon.
PROOF. The plane will intersect each hypersurface in a
convex polygon, in two parallel lines, or not at all (Art.
139, Ths. i and 2). When the plane intersects each hyper-
surface in a pair of parallel lines the intersection with the
double prism will be a parallelogram. In all other cases
there will be at least one convex polygon, and by applying
one or more times the second theorem of Art. 15 we can
prove in all cases that the intersection of the double prism
will be a convex polygon.
COROLLARY. The interior of any segment whose points
are points of a double prism will lie entirely in the interior
of the double prism unless it lies entirely in the double prism,
and a half -line drawn from a point of the interior will inter-
sect the double prism in one and only one point.
In a double prism the directing-polygons of each hyper-
surface whose planes are elements of the other are called
the directing-polygons of the double prism (just as we say
2$0 DOUBLE PRISMS [vi. iv.
the vertex of a pyramid or hyperpyramid). Any two
polygons intersecting in a single point and lying in planes
which have only this point in common can be taken each
as a directing-polygon with the plane of the other as ele-
ment of a piano-prismatic hypersurface, and so the two
together as the directing-polygons of a double prism.
We can say that the surface of intersection of the two hypersur-
faces, the common lateral surfaces of the two sets of prisms as de-
scribed in the preceding article, is generated by moving one of these
polygons kept parallel to itself around the other. Each point of
the polygon moves along the prisms of one set and around one of
the prisms of the other set. The interior of the polygon generates the
interiors of the prisms along which it moves. The interiors of the
other prisms will be generated by the interior of the other polygon
moving in the same way around the first. The surface of intersec-
tion is covered with the polygons of each set, the two sets forming on
it a net.
143. Cutting a double prism so as to form two double
prisms. Doubly triangular prisms.
THEOREM. When a double prism is cut by a hyperplane
passing through points of the interior and containing elements
of one hypersurface, the intersection is a prism, and the rest
of it is separated into two portions, which, each combined
with the prism and its interior, form two double prisms whose
interiors, with that of the prism, make up the whole interior.
PROOF. The layer between the two elements of the
hypersurface lies entirely in the interior of this hypersurface,
and forms with it two hypersurfaces of the same kind (Art.
139, Th. 6). This layer and its faces intersect the other
hypersurface in the lateral surface of a prism and two direct-
ing-polygons whose interiors are the bases of the prism.
The polygons belong to both hypersurfaces, their interiors,
the bases, belong to the first hypersurface and to the interior
of the second, and the lateral surface belongs to the second
142, i43l SUBDIVISION 251
hypersurface and to the interior of the first. The entire
prism, therefore, belongs to the double prism, and is its
intersection with the hyperplane of the layer, separating
the rest of it into two portions which lie on opposite sides
of this hyperplane. The prism and its interior form with
these two portions two double prisms, whose hypersurfaces
are the two hypersurfaces formed from the first given hyper-
surface each taken with the second given hypersurface.
The interior of the prism lies entirely in the interior of the
given double prism, and separates the rest of this interior
into two portions which are the interiors of the two new
double prisms.
The prisms of one set, the set which goes around the first hyper-
surface, are separated, some of them going to one of the two new
double prisms and the rest to the other. One or two of them may
be divided, cut into two shorter prisms, one of these shorter prisms
going to each of the new double prisms. The prisms of the other set
are all cut lengthwise, one part of each prism going to one of the new
double prisms and the other part to the other.
COROLLARY. By cutting a double prism diagonally we can
form double prisms in which the prisms of one set are triangular,
so that those of the other set are three in number; and then,
cutting these in another way diagonally, we can form double
prisms in which the prisms of both sets are triangular, the
interiors of all of these double prisms together with the interiors
of the prisms of intersection making up the whole interior.
A double prism in which the prisms of both sets are
triangular is a doubly triangular double prism, or simply a
doubly triangular prism. Such a double prism is formed
when any three hyperplanes intersecting by twos in three
parallel planes are cut by three other hyperplanes which
intersect by twos in three parallel planes, any plane of one
set intersecting any plane of the other set only in a single
point.
252 DOUBLE PRISMS [vi. IV.
In Art. 163 it will be shown that a doubly triangular
prism can be cut by hyperplanes so as to form six penta-
hedroids.
144. Hyperprisms with prism bases as double prisms.
Hyperparallelopipeds. Centre of symmetry. A hyper-
prism whose bases are the interiors of prisms is a double
prism, the two prisms of the bases and the two lateral
prisms corresponding to their ends forming one of the two
sets of prisms of the double prism, while the prisms of the
other set are parallelepipeds (see Art. 136).
Conversely, a double prism in which the prisms of one
set are parallelepipeds (and therefore the prisms of the
other set are four in number, two pairs of opposites) can
be regarded in two ways as a hyperprism, the bases in each
case being the interiors of a pair of opposite prisms of the
second set.
THEOREM i. When the prisms of both sets in a double
prism are parallelopipeds, or, what is the same thing, when
both sets of directing- polygons are parallelograms, the figure
is a hyperparallelo piped. Indeed, the hyperparallelopiped
can be regarded in three ways as a double prism, the parallel-
opipeds of two pairs of opposite cells forming one of the sets
of prisms and the other four parallelepipeds the other set.
THEOREM 2. When the two hyper surf aces of a double
prism have axis-planes of symmetry, the point of intersection
of these planes is a centre of symmetry of the double prism;
and any hyper plane through this point intersects the double
prism in a polyhedron which divides the rest of the double
prism into two congruent parts (Art. 85, Th. 2).
146. Relation of double prisms to infinity. Taking
the point of view of Art. 130, we can say that a piano-
143-146] HYPERCYLINDRICAL HYPERSURFACES 253
prismatic hypersurf ace is a piano-polyhedral angle with ver-
tex-edge at infinity. Two such hypersurfaces give us a
double prism when the vertex-edge of one does not inter-
sect the vertex-edge of the other, both being lines at infinity.
When the prisms of one set in a double prism are parallel-
epipeds, so that the double prism can also be regarded in
two ways as a hyperprism, there will be two points at
infinity which are the vertices of two prismoidal hyper-
surfaces, and the line containing these two points will
be the vertex-edge of a piano-prismatic hypersurface whose
directing-polygons are parallelograms.
As stated in Art. 137, the eight hyperplanes of a hyper-
parallelopiped intersect the hyperplane at infinity in the
planes of the faces of a tetrahedron whose vertices are
the vertices of the four prismoidal hypersurfaces in which
the hyperparallelopiped lies. Each of the six edges of the
tetrahedron lies in the vertex-edge of a piano-prismatic
hypersurface, the three pairs of opposite edges corresponding
to the three ways in which the hyperparallelopiped can be
taken as a double prism.
The tetrahedron at infinity corresponding to a rectangu-
lar hyperparallelopiped is a rectangular tetrahedron. Each
vertex is the pole of the opposite face, opposite edges are
polar lines, and at each vertex there is a trirectangular
trihedral angle.
V. HYPERCYLINDERS
146. Hypercylindrical hypersurfaces. Their interiors.
Sections. Axes. A hypercylindrical hypersurface consists
of a system of parallel lines passing through the points of a
hyperplane surface, but not lying in the hyperplane of the
surface. The surface is called the directing-surface, and the
parallel lines are the elements.
We shall consider only those cases in which the directing-
254 HYPERCYLINDERS [vi. v.
surface is a surface of elementary geometry, a plane or a
sphere, or a conical or cylindrical surface with directing-
circle, or a part or combination of parts of such surfaces.
A prismoidal hypersurf ace can also be regarded as a partic-
ular case of a hypercylindrical hypersurface.
Many of the properties of the hypersurface correspond
to the properties of the directing-surface. In particular,
the hypersurface has an interior when the directing-surface
has an interior, the interior of the hypersurface consisting
of the lines which pass through the points of the interior
of the directing-surface and are parallel to the elements.
Sections of the hypersurface are like those of the pris-
moidal hypersurface: a hyperplane passing through a
point of the interior and parallel to the elements intersects
the hypersurface in a cylindrical surface, and a hyperplane
which is not parallel to the elements intersects the hyper-
surface in a surface, or at least in a system of points , which
can serve as directing-surface. A right section is a directing-
surface whose hyperplane is perpendicular to the elements.
THEOREM. Sections of a hypercylindrical hypersurface
made by parallel hyperplanes not parallel to the elements are
congruent.
PROOF. The distance between any two points of one
section is equal to the corresponding distance in the other
section. Then any tetrahedron whose vertices are points
of one section will be congruent to the corresponding tetra-
hedron in the other section (see Art. 72, Th. i), and if we
make these tetrahedrons coincide, every point of one section
(forming a tetrahedron with three of these points) will
coincide with the corresponding point of the other section
(see proof of Th. i of Art. 98).
147. Hypercylinders. Special forms. Relation of the
hypercylindrical hypersurface to infinity. A hypercylinder
146, i47l SPECIAL FORMS 255
consists of that portion of a closed hypercylindrical hyper-
surface which lies between two parallel directing-surfaces,
together with the directing-surfaces themselves and their
interiors.
The interiors of the directing-surfaces are the bases, and
that portion of the hypercylindrical hypersurface which
lies between the directing-surfaces is the lateral hyper-
surface of the hypercylinder. The interior of the hyper-
cylinder consists of that portion of the interior of the hyper-
cylindrical hypersurface which lies between the bases.
A hypercylinder is a right hypercylinder when the ele-
ments are perpendicular to the hyperplanes of the bases.
A spherical hypercylinder is one whose bases are the
interiors of spheres. The axis of a spherical hypercylinder
is the interior of the segment whose points are the centres
of the bases.
THEOREM i. When a spherical hypercylinder is cut by a
hyperplane which passes through a point of the interior and
is parallel to the elements, the intersection is a circular cylinder.
THEOREM 2. When a rectangle takes all positions possible
with one side fixed, the vertices and the points of the other
three sides make up a right spherical hypercylinder. The
fixed side is the axis, the opposite side is an element, and the
other two sides are radii of the bases.
THEOREM 3. // we pass a plane through the axis of a
cylinder of revolution and rotate around this plane that por-
tion of the cylinder which lies on one side of it, we shall form
all of a right spherical hypercylinder except that portion which
is the intersection of the cylinder and plane.
A hypercylinder whose bases are the interiors of cylinders
can be regarded in two ways as a hypercylinder of this
kind ; for there are two lateral cylinders corresponding
256 PRISM CYLINDERS AND DOUBLE CYLINDERS [vi. vi.
to the ends of the bases, and these can be taken as the bases
and the given bases as parts of the lateral hypersurface.
In fact, the two lateral cylinders are congruent and lie
in parallel hyperplanes, with the elements of one parallel
to the elements of the other and the planes of the bases of
one parallel to the corresponding planes of the other.
Moreover, those elements of the hypercylinder whose
lines intersect any element of one of its bases lie in the
interior of a parallelogram which bears the same relation
to both pairs of cylinders. This figure is a particular case
of a prism cylinder, and will be studied in the next section
(see Art. 150).
Taking the point of view of Art. 130, we can say that a
hypercylindrical hypersurface is a hyperconical hypersur-
face with vertex at infinity (see Art. 153).
VI. PRISM CYLINDERS AND DOUBLE CYLINDERS
148. Piano-cylindrical hyper surfaces. Sections. Right
directing-curves. A piano-cylindrical hypersurface con-
sists of a system of parallel planes passing through the
points of a plane curve ajid intersecting the plane of the
curve only in these points.* The curve is the directing-
curve, and the planes are the elements. A piano-prismatic
hypersurface can be regarded as a particular case of a
piano-cylindrical hypersurface. Except for this, we shall
consider only the case in which there is a directing-circle.
As the piano-cylindrical hypersurface is analogous to
the piano-prismatic hypersurface, many of the theorems
correspond. We shall only state some of them briefly
(see Arts. 138-140) :
* If we had undertaken to give a definition and some of the properties of curves,
we might have followed the analogy of the piano-prismatic hypersurface and defined
this hypersurface independently of any directing-curve, as a part of the two-dimen-
sional geometry of a system of parallel planes (see Art. 138).
147, US] PLANO-CYLINDRICAL HYPERSURFACES 257
We have the interior of the hypersurface consisting of
those planes parallel to the elements which pass through
the points of the interior of the directing-curve, with the
usual theorems in regard to the interior. The section
made by a hyperplane containing an element will be one
or two elements, but the section made by a hyperplane
which does not contain an element will be a cylindrical
surface, so that the hypersurface can also be regarded as a
hypercylindrical hypersurface with a directing cylindrical
surface. Likewise, as in the case of the piano-prismatic
hypersurface, a plane which do.es not contain a line parallel
to the elements will intersect every element in a point, and
the hypersurface in a curve which can be taken as directing-
curve; and directing-curves which lie in parallel planes
are congruent.
A directing-curve whose plane is absolutely perpendic-
ular to the planes of the elements is called a right directing-
curve ; and any plane isocline to the plane of a right direct-
ing-curve but not parallel to the elements, or, what is the
same thing, any plane isocline to the elements, intersects
the hypersurface in a curve which is similar * to the right
directing-curves.
When the hypersurface has a directing-circle, the plane
through its centre parallel to the elements is an axis-plane
of the hypersurface, and every point of it is a centre of
symmetry. When the right directing-curve is a circle,
the hypersurface can be generated by the rotation of one
of the elements around the axis-plane, that is, by the rota-
tion of one of two parallel planes around the other. It is
then a piano-cylindrical hypersurface of revolution. For
such a hypersurface we can say that any plane isocline to
the elements, or to the axis-plane, intersects the hyper-
* Two curves are similar when they can be placed in positions where they cut
proportionally all half -lines which intersect them from a given point.
258 PRISM CYLINDERS AND DOUBLE CYLINDERS [vi. vi.
surface in a circle ; and through any line which is not per-
pendicular nor parallel to the elements pass two such
planes (Art. 107, Th. i).
THEOREM. Any directing-curve of a piano-cylindrical
hypersurface of revolution is a directing-curve of a circular
cylindrical surface*
PROOF. Through some line of the plane of the direct-
ing-curve pass a plane isocline to the planes of the elements,
and so intersecting the hypersurface in a circle. This
plane and the given plane lie in a hyperplane, and the direct-
ing-curves in which they intersect the hypersurface are
directing-curves of the cylindrical surface in which the
hyperplane intersects the hypersurface. Since one of
these is a circle, the surface is a circular cylindrical surface ;
and the given directing-curve is a directing-curve of a
circular cylindrical surface.
149. Intersection of a piano-prismatic hypersurface and
a piano-cylindrical hypersurface. When the elements of a
piano-prismatic hypersurface intersect the elements of a
piano-cylindrical hypersurface only in points, the inter-
section of the two hypersurfaces consists of the lateral sur-
face of a set of cylinders lying in the cells of the prismatic
hypersurface and joined together by their bases, together
with the curves whose interiors are these bases.
In fact, the faces of the prismatic hypersurface are parallel
planes intersecting the cylindrical hypersurface in a set
of equal parallel directing-curves of the latter, and the
cells of the prismatic hypersurface are layers, each layer
intersecting the cylindrical hypersurface in the lateral
surface of a cylinder whose bases are the interiors of two
of these directing-curves. The interiors of the bases of
the cylinders lie in the prismatic hypersurface and in the
* And is therefore a circle or ellipse.
i 4 8, 149] PRISM CYLINDERS 259
interior of the cylindrical hypersurface. A set of correspond-
ing elements of the cylinders, one from each cylinder, are
the sides of a polygon which is a directing-polygon of the
prismatic hypersurface ; and the interiors of these polygons
lie in the cylindrical hypersurface and in the interior of the
prismatic hypersurface. We can think of the set of cylin-
ders as going around the prismatic hypersurface, while any
base or interior of a cross section of any of these cylinders is
a piece cut out of an element of the prismatic hypersurface.
A prism cylinder consists of the intersection of a piano-
prismatic hypersurface and a piano-cylindrical hypersurface
whose elements intersect only in points, together with all
that portion of each hypersurface which lies in the interior
of the other.
The interior of a prism cylinder consists of the points
which are common to the interiors of its two hypersurfaces.
A plane lying in the interior of the cylindrical hypersurface
parallel to its elements intersects the prismatic hypersur-
face in a directing-polygon whose interior belongs to the
interiors of both hypersurfaces, and so to the interior of
the prism cylinder. The vertices of this, polygon are a
set of corresponding points of the bases, and the sides lie
in the interiors, of the set of cylinders described above.
When the elements of one hypersurface are absolutely
perpendicular to the elements of the other the prism cylin-
der is a right prism cylinder. When also the cylinders are
cylinders of revolution, and when any set of corresponding
elements, one from each cylinder, form a regular polygon,
the prism cylinder is regular.
In a right prism cylinder the cylinders of the surface of inter-
section of the two hypersurfaces can be put together in one hyper-
plane so as to form a single right cylinder. On the other hand, if
this surface is cut along a set of corresponding elements of the cylinder,
and so along a directing-polygon of the prismatic hypersurface, it can
260 PRISM CYLINDERS AND DOUBLE CYLINDERS [vi. vi.
be spread out so as to form a single right prism. The cylinder and
prism will each have their elements equal to the perimeters of the
bases of the other.. In forming the prism cylinder this single right
cylinder is folded on the planes of certain cross sections until the ends
are brought together, and the prism is bent around the cylinder.
The two are linked together, the lateral surface of one coinciding as a
whole with the lateral surface of the other (see note to Art. 141).
150. Directing-polygons and directing- curves. Triangu-
lar prism cylinders. Prism cylinders of revolution. In a
prism cylinder the directing-polygons of the prismatic
hypersurface whose planes are elements of the cylindri-
cal hypersurface, and the directing-curves of the cylindrical
hypersurface whose planes are elements of the prismatic
hypersurface, are called the directing-polygons and directing-
curves of the prism cylinder.
We can say that the surface of intersection of the two hypersur-
faces is generated by moving the polygon around the curve, or by
moving the curve around the polygon. In the first case each point
of the polygon moves around one of the cylinders ; in the second case
each point of the curve moves along them all. The interior of the
curve generates the interiors of the cylinders; the interior of the
polygon generates that portion of the prism cylinder which belongs
to the cylindrical hypersurface and to the interior of the prismatic
hypersurface. The surface of intersection is covered with the poly-
gons and with the curves, the two sets forming on it a net.
THEOREM i. When a prism cylinder is cut by a hyper-
plane passing through points of the interior and containing
elements of the prismatic hypersurface, the intersection is a
cylinder, and the rest of it is separated into two portions,
which, each combined with the cylinder and its interior,
form two prism cylinders whose interiors, with that of the
cylinder, make up the whole interior.
The proof is the same as that of Art. 143, with only
such changes in the terms used as are necessary because
the second hypersurface is cylindrical and not prismatic.
149-151] SUBDIVISION 261
The cylinders of the prism cylinder are separated, some of them
going to one of the two new prism cylinders and some to the other.
The other portion of the prism cylinder, that portion which goes
around the cylindrical hypersurface, is cut lengthwise, one part going
to each of the new prism cylinders.
COROLLARY. By cutting a prism cylinder diagonally
we can form prism cylinders in which the directing-polygons
are triangles and the cylinders, three in number, triangular
prism cylinders.
A hypercylinder whose bases are the interiors of cylinders
is a prism cylinder; and a prism cylinder in which the
directing-polygons are parallelograms (and therefore the
cylinders are four in number) can be regarded in two ways
as a hypercylinder.
When the two hypersurfaces of a prism cylinder have
axes-planes of symmetry, the point of intersection of these
planes is a centre of symmetry (as in Art. 144, Th. 2).
THEOREM 2. // we rotate a right prism around the plane
of one base, the rest of the prism will generate a right prism
cylinder having circles for its directing-curves. The lateral
edges generate the bases of the cylinders of ttie prism cylinder,
each lateral face generates the interior of one of these cylinders,
and the moving base generates that portion of the prism cylin-
der which lies in the cylindrical hypersurface and in the
interior of the prismatic hypersurface. The fixed base and
the interior of the prism belong to the interior of the prism
cylinder.
151. Intersection of two piano-cylindrical hypersurfaces.
When the elements of a piano-cylindrical hypersurface
intersect the elements of a second piano-cylindrical hyper-
surface only in points, each element of one hypersurface
intersects the other hypersurface in a directing-curve, and
262 PRISM CYLINDERS AND DOUBLE CYLINDERS [vi. vi.
the surface of intersection * consists of the curves of either
one of these sets. The interiors of the curves of each set
lie in one of the hypersurfaces and in the interior of the
other.
A double cylinder consists of the intersection of two piano-
cylindrical hypersurfaces whose elements intersect only in
points, together with that portion of each which lies in
the interior of the other. The directing-curves of each
hypersurface whose planes are elements of the other are
called the directing-curves of the double cylinder.
The interior of a double cylinder consists of the points
which are common to the interiors of its two hypersurfaces.
A plane lying in the interior of one hypersurface parallel
to its elements intersects the other hypersurface in a direct-
ing-curve whose interior belongs to the interior of the double
cylinder.
When the elements of one hypersurface of a double cylin-
der are absolutely perpendicular to the elements of the
other the double cylinder is a right double cylinder.
The surface of intersection of the two hypersurfaces is generated
by moving a directing-curve of one system around a directing-curve
of the other. If this surface is cut along a directing-curve, it can be
spread out in a hyperplane, and in the case of a right double cylin-
der it will then form the lateral surface of a right cylinder. This can
be done in two ways, and in the double cylinder we have two cylin-
ders bent around each other, the lateral surface of one coinciding as
a whole with the lateral surface of the other, and this common surface,
together with the interiors and bases of the two cylinders, making
up the double cylinder. When the two bases of a single cylinder
come together, any two corresponding points of these bases and the
points which were on a line between them become the points of a
directing-curve whose interior belongs to the interior of the double
cylinder.
* Without defining surface in general, we assume that the intersection of two
hypersurfaces is a surtace.
iSi, IS*] DOUBLE REVOLUTION 263
152. Cylinders of double revolution.
THEOREM i. If we rotate a cylinder of revolution around
the plane of one base, the rest of the cylinder will generate a
right double cylinder with directing-circles ; and the double
cylinder can be generated in two ways by the rotation of a
cylinder of revolution around one of its bases.
The right double cylinder with directing-circies is there-
fore called a double cylinder of double revolution, or simply a
cylinder of double revolution.
THEOREM 2. In a cylinder of double revolution the in-
tersection of the two hypersurfaces lies in a hypersphere, and
in this hypersphere it is a surface of double revolution (Art.
124).
PROOF. Any point of the intersection, its projection upon
one of the axis-planes, and the centre of the double cylinder,
are the vertices of a right triangle whose legs are radii of
directing-circles of the two systems. The hypothenuse,
therefore, and the angle which it makes with the axis-
plane are the same for all points of the intersection, so that
the intersection lies entirely in a hypersphere, and in this
hypersphere is a surface of double revolution whose axis-
circles are the two great circles in which the axis-planes of
the double cylinder intersect the hypersphere.
A plane passing through the hypothenuse of the triangle
just considered, and isocline to the axis-planes of the
double cylinder, intersects the hypersurfaces in the same
circle. The system of all these planes forms the conical
hypersurface of double revolution whose intersection with
the hypersphere is this same surface of double revolution.
THEOREM 3. Conversely, any surface of double revolu-
tion in a hypersphere is the surface of intersection of the two
hypersurfaces of a cylinder of double revolution.
264 PRISM CYLINDERS AND DOUBLE CYLINDERS [vi. vi.
For the points of this surface, being all at the same dis-
tance from each of the axis-circles of the surface, are in
hyperspace all at the same distance from each of the planes
of these circles, and lie, therefore, in two piano-cylindrical
hypersurfaces of revolution which have these planes for
axis-planes.
The cylinder of double revolution can be regarded as
inscribed in the hypersphere.
153. Relation of prism cylinders and double cylinders to
infinity. Taking the point of view explained in Art. 130,
we can say that a piano-cylindrical hypersurface is a piano-
conical hypersurface with vertex-edge at infinity. We
have a prism cylinder or double cylinder when the vertex-
edge of one hypersurface does not intersect the vertex-edge
of the other. We have a right prism cylinder or double
cylinder when one vertex-edge is polar to the other.
When the bases of a hypercylinder are cylinders there
are two points at infinity, the vertices of two hypercylindri-
cal hypersurfaces which we get by taking the figure in two
ways as a hypercylinder. The line determined by these
two points is the vertex-edge of the piano-cylindrical hyper-
surface which belongs to the figure regarded as a prism
cylinder. The other vertex-edge in this case is the line
of intersection of the planes of the bases of the four cylinders.
CHAPTER VII
MEASUREMENT OF VOLUME AND HYPERVOLUME
IN HYPERSPACE
I. VOLUME
164. Lateral volumes of hyperprisms and hyperpyramids.
Volume of the double prism. The cells of the polyhedroids
that we have studied are polyhedrons of three-dimensional
geometry, and it is only necessary to state the theorems
which concern their volumes.
THEOREM i. The lateral volume of a hyper prism is equal
to the area of a right section multiplied by the lateral edge.
The lateral volume of a right hyperprism is equal to the
area of the surface of the base * multiplied by the altitude.
THEOREM 2. The lateral volume of a regular hyper pyramid
is equal to the area of the surface of the base multiplied by one-
third of the slant height, the common altitude of the lateral
bvramids.
THEOREM 3. The lateral volume of a frustum of a regular
hyperpyramid is equal to the sum of the surface areas of the
bases plus a mean proportional between them, multiplied by
one-third of the slant height.
THEOREM 4. In a double prism the total volume of one
set of prisms is equal to the common area of their bases multi-
plied by the perimeter of a right directing-polygon of the
hypersurface around which the set of prisms extends.
* Area of the polyhedron whose interior is the base. Many forms of expression
commonly used in mensuration will be employed freelv in -this chapter.
265
266 VOLUME [vn. i.
PROOF. Any prism of the given set has its bases in the
two faces of a cell of the hypersurface around which this
set of prisms extends, and its altitude is the distance be-
tween these two faces. Now a right directing-polygon
of the hypersurface is a polygon whose plane is absolutely
perpendicular to the elements, and the side which lies in
this cell is perpendicular to the faces and measures the dis-
tance between them. Therefore the volume of this prism
is equal to the area of its base multiplied by this side of
the right directing-polygon of the hypersurface, and the
total volume of the given set of prisms is equal to the
common area of their bases multiplied by the perimeter
of the right directing-polygon.
COROLLARY. The total volume of a right double prism
is equal to the area of one directing-polygon multiplied by the
perimeter of the other, plus the area of the second multiplied
by the perimeter of the first.
165. Lateral volumes of cylindrical and conical hyper-
surfaces. In the case of curved hypersurfaces we have to
employ the theory of limits or some other equivalent
theory, and, in fact, to extend our definition of volume.
Without going into details, we state the following theorems :
THEOREM i. The lateral volume of a right spherical
hypercylinder is equal to the area of the base multiplied by
the altitude. It is given by the formula
R being the radius and H the altitude.
THEOREM 2. The lateral volume of a right spherical hyper-
cone is equal to the area of the base multiplied by one-third of
the slant height. Its formula is
H' being the slant height.
154-156] THEOREMS OF VOLUME 267
THEOREM 3. The lateral volume of a frustum of a right
spherical hyper cone is given by the formula
r being the radius of the upper base.
THEOREM 4. In a right prism cylinder with directing-
circles the total volume is equal to the area of the directing-
polygon multiplied by the circumference of the directing-
circle, plus the area of the directing-circle multiplied by the
perimeter of the directing-polygon.
The total volume of a cylinder of double revolution is equal
to the area of one directing-circle multiplied by the circum-
ference of the other, plus the area of the second multiplied by
the circumference of the first. It is given by the formula
2 T 2 RR'(R + R'),
R and R' being the radii of the two circles.
156. Volume of the hypersphere.
THEOREM. The volume of a hypersphere is given by the
formula
2 TT 2 R*,
R being the radius.
PROOF. Let AB be a quadrant of a great circle. Two
hyperplanes perpendicular to the radius OB cut the hyper-
sphere in two spheres, whose interiors are the bases of
a frustum of a hypercone inscribed in the hypersphere.
Let H f be the slant height of this frustum, and r\ and r 2
the radii of the bases. H ' is the length of a chord of the
great circle. The lateral volume of the frustum is given
by the formula
Let H be the altitude of the frustum, and K the distance
from to the middle point of the chord which represents
H'. From similar triangles we have
268
VOLUME
[vii. i.
H
or
O
= 2_KH_
ri + r, H , .
Also, from right triangles,
H' 2 = H 2 + (r, - r 2 ) 2 ,
R* = ^ + (J ff O 2 -
and
Therefore
r,r,
= (r, + r 2 ) 2
For an inscribed hypercone with vertex at B and r\ the
radius of the base, we have only to make r^ = o in all of
these expressions.
If we take two arcs symmetrically situated on the arc
AB with respect to P, the middle point of this arc, we shall
have two right triangles with hypothenuse equal to H'
symmetrically situated with respect to the radius OP,
and therefore equal. But the legs denoted by H are non-
homologous sides in the two triangles, and so the sum of
269
their squares is H ' 2 . Therefore, if we write down the for-
mula given above for each of two divisions of AB sym-
metrically situated with respect to P, the sum of the two
expressions will be
and the sum of the volumes of the frustums corresponding
to the two arcs will be
Now if we divide the arc AB into 2 n equal parts, we
shall have a hypercone and frustum, and n i pairs of
frustums, all inscribed in a half-hypersphere ; and when
n is increased indefinitely the sum of their lateral volumes
will have for limit the volume of the half-hypersphere.*
This sum is
But
and
lim nH' = arc AP = ^,
lim
n = 00
Un,^,o.
vt*
= 00
n'
* Here again we omit details involving essentially a definition of the volume of the
hypersphere.
270 HYPERVOLUME [vn. n.
Thus the volume of the half-hypersphere is
4
and the volume of the hypersphere is
2 7T 2 3 .
COROLLARY. The volume of a cylinder of double revolution
circumscribed about a hypersphere is twice the volume of the
hypersphere.
II. HYPERVOLUME
167. The terms hypersolid and hypervolume. We shall
use the term hypersolid for that portion of hyperspace
which constitutes the interior of a polyhedroid or of a simple
closed hypersurface such as a hypercone, hypersphere, or
double cylinder. A hypersolid is supposed to have hyper-
volume, which can be computed from the measurements
of certain segments and angles, and which can be expressed
in terms of the hypervolume of a given hypercube taken
as a unit. The theory of hypervolume is exactly the same
as the theory of volume in the ordinary geometry. We
shall omit all discussion of this matter, and, as in the pre-
ceding section, we shall use freely the forms of expression
commonly employed in mensuration. The distinction
between hypersurface and hypersolid is important, but
we shall often use these terms interchangeably, speaking,
for example, of the hypervolume of a given hypersurface,
and, on the other hand, of the vertices, edges, faces, or
cells of a hypersolid.
By the ratio of two hypersolids we mean the ratio of their
hypervolumes. Thus the ratio of any hypersolid to the
unit hypercube is the same as the hypervolume of the
hypersolid. Two hypersolids which have the same hyper-
volume are equivalent; and if a hypersolid is divided into
two or more parts, the hypervolume of the whole is equal
156-159] HYPERPARALLELOP1PED 271
to the sum of the hypervolumes of the parts. Two hyper-
solids which are congruent are equivalent.
168. Congruent and equivalent hyperprisms.
THEOREM i. Two right hyperprisms are congruent when
they have congruent bases and equal altitudes.
PROOF. A given base of one can be made to coincide
with either base of the other, and in one of these two
positions the hyperprisms will lie on the same side of the
hyperplane of the coinciding bases and will coincide through-
out.
THEOREM 2. An oblique hyper prism is equivalent to a
right hyperprism having for its base a right section and for
its altitude a lateral edge of the oblique hyperprism.
COROLLARY. Any two hyperprisms cut from the same
prismoidal hypersurface with equal lateral edges are equivalent.
159. Hypervolume of a hyperparallelopiped.
THEOREM i. The hypervolume of a rectangular hyper-
parallelopiped is equal to the product of its four dimensions.
PROOF. In the first place, two rectangular hyper-
parallelopipeds having congruent bases are to each other
as their altitudes. This we can prove when the altitudes
are commensurable and then when the altitudes are in-
commensurable.
Then we prove that when they have two dimensions in
common they are to each other as the products of the other
two dimensions; when they have one dimension in com-
mon they are to each other as the products of the other
three dimensions; and, finally, in any case, they are to
each other as the products of their four dimensions.
From the last statement, by taking for the second hyper-
parallelopiped the unit hypercube, we have the theorem
as stated.
272 HYPERVOLUME [vii. n.
THEOREM 2. The hypervolume of any hyperparallelopiped
is equal to the volume of any base multiplied by the correspond-
ing altitude.
PROOF. We shall prove the theorem by proving that
we can construct an equivalent rectangular hyperparallel-
opiped with base equivalent to the base and altitude equal
to the altitude of the given hyperparallelopiped.
We shall speak of a pair of opposite cells and the eight
edges which join the vertices of one to the vertices of the
other as a corresponding pair of bases and set of edges (Art.
136)-
We produce a set of edges and cut off an equivalent
hyperparallelopiped by two hyperplanes perpendicular
to these edges (Art. 158, Th. 2). The set of edges which
are produced are taken on the same lines in the two hyper-
parallelopipeds, and are of equal lengths. The other
three sets of edges of the given figure are replaced by edges
perpendicular to the set produced. If the edges of any
set were already perpendicular to the edges produced, they
are replaced by a set of edges parallel and equal to them,
and if the edges of any set were perpendicular to the cor-
responding bases, and so to all of the other edges, the same
will be true of the edges by which they are replaced. The
pair of bases corresponding to the edges produced is re-
placed by a pair of bases perpendicular to these lines. The
bases of the other three pairs are parallelepipeds replaced
by equivalent right parallelepipeds lying in the same
hyperplanes.
Starting with the second hyperparallelopiped, we produce
a second set of edges, forming a third hyperparallelopiped
in the same way; and, finally, from the third hyper-
parallelopiped, producing a third set of edges, we form a
fourth hyperparallelopiped, the four being all equivalent.
The edges of the first set in all four hyperparallelopipeds
159, 160] HYPERPRISM WITH PRISM BASES 273
are equal and parallel, and in the second, third, and fourth
they are perpendicular to the corresponding bases, and
therefore to all the other edges. The edges of the second
set in the second, third, and fourth hyperparallelopipeds
are equal and parallel, and in the third and fourth they
are perpendicular to the corresponding bases, and there-
fore to all the other edges. The edges of the third set in
the third and fourth hyperparallelopipeds are equal and
taken along the same lines, and in the fourth they are
perpendicular to the corresponding bases.
Therefore, in the fourth hyperparallelopiped the edges
of each set are perpendicular to the edges of all the other
sets, and the hyperparallelopiped is rectangular.
Now the hyperplanes of the fourth pair of bases remain
the same throughout this process, and with these bases
the four hyperparallelopipeds all have the same altitude
(Art. 128, Th. 12). Moreover, these bases do not differ
in volume, and in the last two they are congruent; they
are, in fact, parallelepipeds formed by this same process
of producing successively different sets of edges.
We have constructed an equivalent rectangular hyper-
parallelopiped with base equivalent to the base and alti-
tude equal to the altitude of the given hyperparallelopiped,
and this is sufficient to prove the theorem.
160. Hypervolume of any hyperprism.
THEOREM i. The hypervolume of a hyperprism whose
base is the interior of a prism is equal to the volume of its
base multiplied by its altitude.
PROOF. As any prism can be divided into triangular
prisms,* it is only necessary to prove the theorem when
the base is a triangular prism.
* We are using the term prism here for the solid (see Art. 157) ; we mean that a
set of triangular prisms can be formed whose volumes together make up the volume
of the given prism.
274 HYPERVOLUME [vn. n.
On a triangular prism we can build a parallelepiped by
joining an equal triangle to its base so as to form a par-
allelogram, and drawing a fourth lateral edge. We join
to the given prism a second triangular prism, and the two
are symmetrically situated with respect to the centre of
the parallelepiped, and therefore equal (Art. 84, Ths. 2
and 4). On the hyperprism we can, then, build a hyper-
parallelopiped having this parallelepiped as base. We
join to the given hyperprism a second hyperprism, and the
two hyperprisms are indeed congruent, since the centre
of symmetry of the hyperparallelopiped lies in the diagonal
hyperplane along which the two prisms are joined (Art.
136, Th. i, and Art. 85, Th. 2). The hypervolume of the
given hyperprism is therefore equal to one-half of the
hypervolume of the hyperparallelopiped, and so to the
volume of its own base multiplied by its altitude.
THEOREM 2. Two right hyperprisms are equivalent if
they ham equal altitudes and if their bases are the interiors
of tetrahedrons which can be so placed that as triangular
pyramids they shall have equivalent bases and equal altitudes.
PROOF. The lateral cells are the interiors of triangular
prisms. We will rest the hyperprism upon one of these
cells, which may be called a prismatic base. The opposite
lateral edge will then be a vertex-edge, and the altitude will
be the distance of the vertex-edge from the prismatic base.
The original base is now at one end, and is itself the in-
terior of a triangular pyramid with one of the ends of the
prismatic base for its base, and for its altitude the altitude
of the hyperprism in its present position.
Looking at the hyperprism in this way, we have a series
of theorems analogous to the theorems by which we de-
termine the volume of a pyramid in geometry of three
dimensions :
i6o] TETRAHEDRAL HYPERPRISMS 275
(1) A hyperplane section parallel to the prismatic base
is itself a prism of the same length as the prismatic base,
and its end is a section of the end-pyramid parallel and
similar to the base of the pyramid. The volume of this
parallel hyperplane section is, then, proportional to the
square of its distance from the vertex-edge.
(2) If two right hyperprisms with tetrahedral ends, when
placed as explained above, have equivalent prismatic bases
and equal altitudes, hyperplane sections parallel to the
prismatic bases and at the same distance from the vertex-
edges are equivalent.
(3) Given the two hyperprisms just considered, we divide
the common altitude into equal parts and construct a series
of inscribed and circumscribed prisms to the pyramids at
the ends, and so a series of hyperprisms having these
prisms for ends, inscribed and circumscribed to the original
hyperprisms. The hypervolume of either of the given
hyperprisms is the limit of the sum of the hypervolumes of
the hyperprisms inscribed or of the hyperprisms circum-
scribed to it, when the number of subdivisions of the alti-
tude is increased indefinitely. Any inscribed or cir-
cumscribed hyperprism of one figure is equivalent to the
corresponding hyperprism of the other figure, and there-
fore the two given hyperprisms must be equivalent.
Now when two right hyperprisms have equal altitudes,
and for bases the interiors of triangular pyramids with
equivalent bases and equal altitudes, the hyperprisms
satisfy the conditions of (3) and are equivalent.
THEOREM 3. The hypervolume of any hyperprism is
equal to the volume of its base multiplied by its altitude.
PROOF. As any polyhedron can be divided into tetra-
hedrons, it is only necessary to prove the theorem when
the base is a tetrahedron.
276 HYPERVOLUME [vii. n.
On a tetrahedron we can build a triangular prism by
joining to it two other equivalent tetrahedrons, and so
on the given hyperprism we can build a hyperprism with
a triangular prism for base, composed of three hyperprisms
with tetrahedral bases. If we produce the lateral edges,
we can form a right hyperprism with a triangular prism
for base, composed of three right hyperprisms with tetra-
hedral bases, the four right hyperprisms equivalent re-
spectively to the hyperprisms from which they were pro-
duced (Art. 158, Th. 2). The three tetrahedral bases of
the right hyperprisms are equivalent, and any two of them
can be so placed that as triangular pyramids they shall
have equivalent bases and equal altitudes. Hence the
three right hyperprisms with tetrahedral bases are equiv-
alent, and the first three hyperprisms with tetrahedral
bases are equivalent. The hypervolume of the given
hyperprism is, therefore, equal to one-third of the hyper-
volume of the hyperprism with triangular prism for base
of which it forms a part, and so it is equal to the volume
of its own base multiplied by its altitude.
COROLLARY. The hypervolume of a hyperprism is equal
to the volume of a right section multiplied by the lateral edge.
161. Hypervolume of a hyperpyramid. For the hyper-
volume of a hyperpyramid we have, as in the proof of Th. 2
of the preceding article, a series of theorems analogous to
the theorems by which we determine the volume of a
pyramid in geometry of three dimensions :
THEOREM i. A hyper plane section of a hyperpyramid
parallel to the base is similar to the base, and its volume is
proportional to the cube of its distance from the vertex.
THEOREM 2. // two hyperpyramids have equivalent bases
and equal altitudes, hyperplane sections parallel to the bases
and at the same distance from the vertices are equivalents
i6o, 161] HYPERPYRAMID 277
THEOREM 3. Two pentahedroids are equivalent if they
can be so placed that as hyper pyramids they shall have equiv-
alent bases and equal altitudes.
PROOF. Dividing the altitude into some number of
equal parts, we can construct a series of inscribed and cir-
cumscribed hyperprisms and prove that the hypervolume
of either pentahedroid is the limit of the sum of the hyper-
volumes of the set of hyperprisms inscribed or circumscribed
to it when the number of subdivisions of the altitude is in-
creased indefinitely. Thus we prove our theorem in the
same manner as we prove the corresponding theorem in
geometry of three dimensions.
THEOREM 4. The hypervolume of any hyperpyramid is
equal to the volume of its base multiplied by one-fourth of its
altitude.
PROOF. As any polyhedron can be divided into tetra-
hedrons, it is only necessary to prove the theorem when
the base is the interior of a tetrahedron, that is, to prove
it for pentahedroids.
On a pentahedroid taken as a hyperpyramid we can
build a hyperprism having the same base and one lateral
edge the same. In doing this we join to the given penta-
hedroid a hyperpyramid with a triangular prism for base,
a hyperpyramid which can be divided into three penta-
hedroids, the triangular prism being divided into three
tetrahedrons. One of these pentahedroids can then be
proved equivalent to the given pentahedroid, so that the
four pentahedroids are all equivalent.
Let A-A'B'C'D' be the given pentahedroid. Drawing
lines through J3', C", and D' parallel to A' A, and a hyper-
plane through A parallel to the hyperplane of the tetrahe-
dron A'B'C'D', we have a hyperprism ABCD-A'B'C'D'
composed of the given pentahedroid and the hyperpyramid
278 HYPERVOLUME [vn. n.
A-BCDB'C'D'. This hyperpyramid we divide into three
equivalent pentahedroids by dividing the prism BCD-
B'C'D' into three equivalent tetrahedrons, the common
altitude of the three pentahedroids being the distance of
the vertex A from the hyperplane of the triangular prism.
Now of these three pentahedroids one can be regarded as
having A BCD as its base, and as its vertex one of the points
B', C", or D r of the lower base of the hyperprism. Re-
garded in this way it is seen to be equivalent to the original
pentahedroid, since the bases of the two are the bases of
the hyperprism, and their common altitude is the altitude
of the hyperprism.
The given pentahedroid is, therefore, one of four equiv-
alent^ pentahedroids which go to make up the hyperprism ;
and its hypervolume is one-fourth of the hypervolume of
the hyperprism, and so equal to the volume of its own base
multiplied by one-fourth of its altitude.
The figure on page 238 represents the cells of the hyperprism spread
out into a single hyperplane. Dividing the three prisms on the left
by the planes AB'C, AB'D', and AC'D', we have all the cells except
the common cell AB'C'D' of the given pentahedroid and of the
hyperpyramid A-BCDB'C'D'.
162. Hypervolume of a frustum of a hyperpyramid.
THEOREM. The hypervolume of a frustum of a hyper-
pyramid is given by the formula
i H(B + # + #V + ft),
where B and b are the volumes of the bases and H is the
altitude.
PROOF. Let B be the volume of the lower base, and let
r be the ratio of an edge of the upper base to the correspond-
ing edge of the lower base, so that we have b = Br 3 . The
formula of the theorem can, then, be written
| HB(i + r + r 2 + r 3 ).
161,162] FRUSTUM OF A HYPERPYRAMID 279
By dividing the bases into tetrahedrons we can divide
the frustum into frustums of pentahedroids,* in all of
which H and r have the same values. Therefore, it is
only necessary to prove the theorem when the bases are
the interiors of tetrahedrons, that is, for a frustum of a
pentahedroid.
Now the frustum of a pentahedroid is of the same general
form as a hyperprism with tetrahedral bases, and can be
divided into four pentahedroids in the same way that the
hyperprism of the preceding proof is divided into four
pentahedroids. f Then it can be proved that the hyper-
volumes of these four pentahedroids form a geometrical
progression in which the ratio is r and the first term \ HB,
the hypervolume of the pentahedroid whose base is the
lower base of the frustum.
Let the given frustum be ABCD-A'B'C'D* ', ABCD
being the upper base and A'B'C'D' the lower base. We
divide this into four pentahedroids by the three hyperplanes
AB'C'D', ABC'D', and ABCD';
namely, into
AA'B'C'D', ABB'C'D', ABCC'D', and ABCDD'.
The first hyperplane, cutting off the pentahedroid
AA'B'C'D', leaves the hyperpyramid A-BCDB'C'D r .
That is, the last three of the four pentahedroids together
form a hyperpyramid with vertex at A and base the frustum
BCD-B'C'D' of a tetrahedron. When the hyperpyramid
is divided into three pentahedroids this frustum is divided
into three tetrahedrons whose volumes are proportional
to i, r, and r 2 . Therefore, the hypervolumes of the three
pentahedroids are proportional to i, r, and r 2 .
* See Art. 157, and foot-note, p. 273.
t A truncated pentahedroid is divided in this way by Sylvester in the memoir,
"On the Centre of Gravity of a Truncated Triangular Pyramid" (p. 172), referred
to on p. 5.
280 HYPERVOLUME [vn. n.
Similarly, the first three pentahedroids form a hyper-
pyramid with vertex at D' and base the frustum ABC-
A'B'C' of a tetrahedron, and their hypervolumes are
proportional to i, r, and r 2 .
In other words, the hypervolumes of the four pentahe-
droids are proportional to i, r, r 2 , and r 3 .
But the hypervolume of the first pentahedroid is \ HB.
Therefore the hypervolume of the given frustum is
i HB(i + r + r* + r).
The formula can also be derived algebraically by sub-
tracting from the hypervolume of a hyperpyramid the
hypervolume of a smaller hyperpyramid cut off so as to
leave a frustum.
163. Hypervolume of a double prism.
THEOREM i. The hypervolume of a doubly triangular
prism is equal to six times the hypervolume of the pentahe-
droid whose vertices are the points obtained by taking the
vertices of a base in one of the two sets of prisms together with
the vertices of a base in the other set (Art. 141).
PROOF. We shall prove this theorem by dividing the
double prism into two hyperpyramids with triangular
prisms for bases, and then into six equivalent pentahedroids.
Certain of these pentahedroids will be found to have among
their faces a base from each of the two sets of prisms of
the double prism.
Let the nine vertices of the double prism be
ABC
A' B' C
A" B" C",
where the three lines as written represent three equal
parallel triangles, and the three columns represent three
A
B
B'
B"
C
C
C"
and
A
A'
A"
B'
B"
162, 163] DOUBLE PRISM 281
equal parallel triangles. Any two triangles of either set
lie in a hyperplane, and their interiors are the bases of a
prism.
The point A and the plane of the parallelogram B r C'B"C"
determine a hyperplane which divides the double prism
into two hypersolids. We can indicate this by writing
their vertices
C
The hyperplane cannot contain any of the points J5, C, -4',
or A" \ for it contains at least one vertex of each of the
six triangles, and if it contained any one of these triangles,
it would contain the two triangles parallel to it and so all
the nine vertices of the double prism. Now the hyper-
plane intersects the hyperprism ABC-A'B'C' in the plane
AB'C r which separates B and C from A 1 ', and it intersects
the hyperplane of the prism ABC-A"B"C lf in the plane
AB"C" which separates B and C from A". Therefore
B and C are separated in hyperspace from A r and A 11 by
this hyperplane (Art. 28).
The two hypersolids are hyperpyramids, each having
its vertex at A and a triangular prism as base. Each of
these hyperpyramids can be divided into three equivalent
pentahedroids, the base being divided into three equivalent
tetrahedrons. This can be done in such a way that one
of the pentahedroids of the first set shall be the penta-
hedroid ABCC'C", and one of the pentahedroids of the
second set the pentahedroid AA'A"B"C". We write
these
ABC and A
C A'
C' 1 A" B" C".
282 HYPERVOLUME [vn. n.
We will take the first pentahedroid as a hyperpyramid
with vertex at C" and its base as a pyramid with vertex at
C", and the second pentahedroid as a hyperpyramid with
vertex at A' and its base as a pyramid with vertex at A.
The two pyramids will then have equal bases, ABC and
A"B"C", and equal altitudes, the distance between the
planes of these bases. In fact, they are two of the three
equivalent tetrahedrons into which the prism ABC-
A"B"C" can be divided. The altitudes of the two
hyperpyramids will be equal to the distance of the plane
A'B'C' from the hyperplane of this prism, to which it is
parallel. Therefore, these two hyperpyramids are equiv-
alent pentahedroids ; that is, the six pentahedroids into
which the double prism has been divided are all equivalent,
and the hypervolume of the double prism is equal to six
times the hypervolume of any one of these pentahedroids, for
example, of the pentahedroid ABCC'C" , which has among
its faces the triangles ABC and CC'C".
The figure on page 248 represents the cells of the double prism
spread out in a single hyperplane. Dividing the prisms on the right
and left by the planes AB'C' and AB"C", and the upper two prisms
in front by the planes AB'B" and ACC" \ we have all the cells except
the common cell AB'CB"C" of the two hyperpyramids into which
we first divide the double prism.
THEOREM 2. The hypervolume of a double prism is equal
to the area of a base of any of the prisms of either set, multi-
plied by the area of a right directing-polygon of the hyper-
surface around which this set of prisms extends.
PROOF. We can divide the bases of the prisms of the
given set into triangles, the given prisms into triangular
prisms, and the double prism into double prisms having
each a set of triangular prisms for the given set of prisms.
The hypersurface around which these prisms extend is
the same for the given double prism and for all of the double
iC 3 ] DOUBLE PRISM 283
prisms into which it is divided (Art. 143, Th. and Cor.).
Therefore it is only necessary to prove the theorem for
double prisms in which the given set of prisms is a set of
triangular prisms.
Again, taking a double prism with a given set of triangular
prisms for one of its sets of prisms, we can divide the right
directing-polygons of the hypersurface around which these
prisms extend into triangles. The diagonals which divide
one of these directing-polygons into triangles, together
with the faces of the hypersurface at their extremities,
determine layers which form, with the parts into which
they divide the hypersurface, triangular hypersurfaces,
and so divide the double prism into doubly triangular
prisms. If the theorem is true of doubly triangular prisms,
it is true of any double prism in which the given set of
prisms is a set of triangular prisms. Therefore, it is only
necessary to prove the theorem for doubly triangular prisms.
Proceeding as in the proof of Th. i, we have a penta-
hedroid ABCC'C", one of six equivalent pentahedroids
into which the double prism can be divided. The volume
of the tetrahedron ABCC" is equal to the area of the
triangle ABC multiplied by one-third of the distance of
C" from the plane of this triangle ; and the hypervolume
of the pentahedroid is equal to the volume of the tetrahe-
dron A BCC" multiplied by one-fourth of the distance of C'
from the hyperplane of this tetrahedron. That is, the
hypervolume of the pentahedroid is equal to the area of
the triangle ABC multiplied by one- twelfth of the product
of these two distances.
Now the plane absolutely perpendicular to the plane ABC
intersects the hypersurface which has this plane for one
of its faces in a right directing-triangle. One side of this
triangle measures the distance between the planes ABC
and A"B"C", and the corresponding altitude of the
284 HYPERVOLUME [vn. n.
triangle measures the distance of the plane A'B'C' from
the hyperplane of the other two planes. The area of the
triangle, that is, of the right directing-triangle of the hyper-
surface which has the plane ABC for one of its faces, is
then equal to one-half of the product of these two distances ;
and the hypervolume of the pentahedroid ABCC'C" is equal
to one-sixth of the product of the areas of the two triangles.
But the hypervolume of the pentahedroid is also one-
sixth of the hypervolume of the double prism. Therefore,
the latter is exactly equal to the product of the areas of
the two triangles, that is, to the area of a base in one set
of prisms multiplied by the area of a right directing-
triangle of the hypersurface around which this set of
prisms extends.
COROLLARY. The hyperwlume of a right double prism is
equal to the product of the areas of its two directing-polygons.
164. Hypervolumes of cylindrical and conical hyper-
surfaces. For the hypercylinder, hypercone, prism
cylinder, and double cylinder we have the following
theorems, derived from the corresponding theorems of the
preceding articles :
THEOREM i. The hyperwlume of a spherical hyper-
cylinder is equal to the volume of the base multiplied by the
altitude. It is given by the formula
R being the radius of the base and H the altitude, as in Art.
155-
THEOREM 2. The hyperwlume of a spherical hypercone
is equal to the volume of the base multiplied by one-fourth of
the altitude. It is given by the formula
163-165] CURVED HYPERSURFACES 28$
THEOREM 3. The hypervolume of a frustum of a spherical
hypercone is given by the formula
*H(R* + R 2 r
or
THEOREM 4. The hypervolume of a right prism cylinder
with directing-circle is equal to the product of the areas of the
directing-polygon and directing-circle.
The hyperwlume of a cylinder of double revolution is equal
to the product of the areas of its directing-circles. It is given
by the formula
166. Hypervolume of the hypersphere.
THEOREM. The hypervolume of a hypersphere is equal
to its volume multiplied by one-fourth of its radius.
PROOF. We use the construction and notation of Art.
156. We inscribe a frustum of a hypercone entirely on
one side of the centre 0, and on each base we place a hyper-
cone with vertex at 0. The figure formed by adding to
the frustum the hypercone with larger base and then
taking away the hypercone with smaller base is a hyper-
solid which we can use to determine the hypervolume of
the hypersphere. If we divide the arc AB into some num-
ber of equal parts and form these figures for all the chords
#', we shall have a set of hypersolids fitting together
within the half-hypersphere, and the limit of the sum of
their hypervolumes we assume to be the hypervolume of
the half-hypersphere.
Let x denote the distance from O of the nearer base of
the inscribed frustum, and let this be the base whose
radius is r\. The hypervolumes of the two hypercones
are then
286
HYPERVOLUME
[vn. n.
and f TT(X + H)r 2 3 ,
and the hypervolume of the frustum is
Adding to this last the hypervolume of the first hypercone
and substracting the hypervolume of the second, we have
i T ! H(n + r 2 )(r! 2 + r 2 2 ) + xrf -(x
Now
Hence
and
r 2
(*+ H}H - r! 2 -
That is,
2 H
But
and (r\
r 2 2 )
- H* (Art. 156).
i6s] HYPERSPHERE 287
Hence, finally, our hypervolume becomes
r 2 2 )
- # 2 (n 2 - nr, + r 2 2 ) I
Now the lateral volume of the frustum is
so that the hypervolume of our hypersolid is equal to the
lateral volume of the frustum multiplied by \ K.
Consider now all the subdivisions of the arc AB. When
the number of subdivisions is increased indefinitely the
sum of the hypervolumes will, as we have assumed, ap-
proach as limit the hypervolume of the half-hypersphere,
the sum of the lateral volumes will approach what we have
called the volume of the half-hypersphere, and K will
approach R. Therefore, passing to the limit and express-
ing the result for the entire hypersphere, we have the
hypervolume of the hypersphere equal to its volume
multiplied by one-fourth of the radius.
COROLLARY i. The hypervolume of the hypersphere is
given by the formula
COROLLARY 2. The hypervolume of a hypersphere is equal
to one-half the hypervolume of the circumscribed double
cylinder, and twice the hypervolume of the inscribed double
cylinder with equal radii. It is equal to the hypervolume of
any inscribed double cylinder plus the hypervolumes of two
hyperspheres whose radii are the radii of the double cylinder.
The student may investigate the hypervolume of a pentahedroid
in elliptic hyperspace, following the analogy of Art. 120. He will
288 HYPERVOLUME [vn. n.
find that there are eleven equations connecting the sixteen different
hypervolumes of a set of associated pentahedroids with the measures
of the five hyperplane angles. If he introduces also what we may
call the hypervolumes of the five tetrahedroidal angles, he will have
five more equations,, so that he can express the hypervolume of the
pentahedroid in terms of the hyperplane angles and the tetrahedroidal
angles.* But a tetrahedroidal angle is like a spherical tetrahedron :
we cannot measure it directly, and we have no simple formula for its
hypervolume.
* M. Dehn, " Die Eulersche Formel in Zusammenhang mit dem Inhalt in der Nicht-
Euklidische Geometric," Mathematische Annalen, vol, 61, 1905, pp. 561-586, in par-
ticular, pp. 583-584.
CHAPTER VIII
THE REGULAR POLYHEDROIDS *
I. THE FOUR SIMPLER REGULAR POLYHEDROIDS
166. Definition of regular polyhedroid. The regular
pentahedroid. A regular polyhedroid f consists of equal
regular polyhedrons together with their interiors, the poly-
hedrons being joined by their faces so as to enclose a por-
tion of hyperspace, and the hyperplane angles formed at
the faces by the half-hyperplanes of adjacent polyhedrons
being all equal to one another.
We have already had two regular polyhedroids, the regu-
lar pentahedroid (Art. 114) and the hypercube (Art. 136).
The interior of the segment which measures the altitude
of a regular pentahedroid is one leg of a right triangle whose
hypothenuse is the edge of one of its cells and whose other
leg is the radius of the sphere circumscribed about the cell.
The radius of the hypersphere circumscribed about the
pentahedroid is equal to four-fifths of the altitude, and the
radius of the inscribed hypersphere is equal to one-fifth of
the altitude. These theorems are proved in the same way
that the corresponding theorems are proved for the regular
tetrahedron and the triangle.
Radii perpendicular to the cells of a regular pentahedroid
meet the circumscribed hypersphere in five points which
are the vertices of a second regular pentahedroid symmetri-
* This subject is treated by I. Stringham, "Regular Figures in -dimensional
Space/' American Journal of Mathematics, vol. 3, 1880, pp. 1-14.
t That is, a regular convex polyhedroid. We shall consider only convex polyhe-
droids in this chapter.
U 289
2QO THE SIMPLER REGULAR POLYHEDROIDS [vm. i.
cally situated to the first with respect to the centre, and
therefore equal to it (Art. 85, Th. 2).
The pentahedroid has
5 vertices, 10 edges, 10 faces, and 5 cells.
167. The hypercube. The hypercube can be generated
by the motion of a cube in a direction perpendicular to its
hyperplane through a distance equal to its edge. The
centfe of the cube generates the interior of a segment
whose middle point is the centre of the hypercube, equally
distant from all of the sixteen vertices. The distance of
the centre from any cell (the radius of the inscribed hyper-
sphere) is equal to one-half of the edge, the radius of the
circumscribed hypersphere is equal to the edge, and the
diagonal of the hypercube (the diameter of the circum-
scribed hypersphere) is twice the edge (Art. 136, Th. 3).
The hypercube has
1 6 vertices, 32 edges, 24 faces, and 8 cells.
As the regular polyhedroids are usually named from the
number of cells, the hypercube is also called a regular
octahedroid.
We can fill all hyperspace with a set of hypercubes and
their interiors, sixteen hypercubes coming together at any
vertex. Moreover, the centres of these hypercubes are
themselves the vertices of a second set of hypercubes of
the same kind, and the two sets are in this way recipro-
cally related. Two hypercubes of one set whose centres
are the extremities of an edge of the other set have in
common a cube, and the centre of the cube is the projec-
tion upon its hyperplane of the centres of the two hyper-
cubes. The edge and the cube lie, therefore, in a perpen-
dicular line and hyperplane, intersecting at a point which
is the middle point of the edge and the centre of the cube.
i66-i68] HYPERCUBE AND I6-HEDROID 291
Four hypercubes of one set whose centres are -the vertices
of a face of the other set have in common a square. The
hypercubes are in cyclical order around the plane of this
square, and the centre of the square is the projection upon
its plane of the centres of the four hypercubes. More-
over, the vertices of this square are in turn the centres of
four hypercubes of the second set, arranged in cyclical
order around the plane of the given face of this set. The
planes of the two faces are therefore absolutely perpendic-
ular, intersecting at a point which is the centre of both
(see Arts. 46 and 179).
A set of polyhedroids filling hyperspace without over-
lapping is called a net : every point of hyperspace is either
a point of two or more of the polyhedroids or a point of
the interior of only one. The two sets of hypercubes just
described are reciprocal nets.
168. The hexadekahedroid or 16-hedroid. If we lay
off a given distance in both directions on each of four
mutually perpendicular lines intersecting at a point 0,
the eight points so obtained are the vertices of a regular
polyhedroid which has four diagonal^ along the four given
lines. In fact, the rectangular system contains sixteen
rectangular tetrahedroidal angles, and the four vertices
which lie on the edges of one of these angles are the ver-
tices of a regular tetrahedron congruent to the tetrahedron
whose vertices lie on the edges of any other one of these
tetrahedroidal angles (see Art. 72, Th. i). Thus the sixteen
tetrahedrons together with their interiors form a polyhe-
droid and enclose a portion of hyperspace about the point
0. Now any face of this polyhedroid can be taken as the
base of each of the two tetrahedrons which have this face
in common and lie in the cells of a hyperplane angle of the
polyhedroid. Thus the opposite vertices of these two tet-
2Q2 THE SIMPLER REGULAR POLYHEDROIDS [vm. i.
rahedrons lie in the plane angle at the centre of the given
face, and as they lie also in one of the four lines of the rec-
tangular system', they and the centre of the given face are
the vertices of an isosceles triangle whose sides and angles
are the same whatever face be taken. The hyperplane
angles of the polyhedroid are, therefore, equal,* and the
polyhedroid is regular. As there are sixteen cells the poly-
hedroid is called a hexadekahedroid, or as we shall usually
write it, a i6-hedroid.
Each tetrahedron has four faces, and each face is common
to two tetrahedrons. The number of faces of the 16-
hedroid is therefore
jc6_X_4 _
2 ~ 32 '
Each vertex, being one extremity of a diagonal, is a
common extremity of six of the edges. The other extremi-
ties of these edges are the extremities of the other three
diagonals. The number of edges is therefore
8X6 - 2 A
24.
2
Thus the regular i6-hedroid has
8 vertices, 24 edges, 32 faces, and 16 cells.
The diagonals of the i6-hedroid are also called its axes.
169. Reciprocal relation of the hypercube and the 16-
hedroid. When a hypercube is inscribed in a hypersphere,
the radii perpendicular to the hyperplanes of its eight cells
are the radii to the eight vertices of a regular i6-hedroid
inscribed in the same hypersphere, the two polyhedroids
having their axes along the same lines. Each vertex of
the i6-hedroid and the centre of the hypersphere are sym-
* They are, in fact, angles of 120.
168-170] HYPERCUBE AND I6-HEDROID 293
metrically situated with respect to the hyperplane of a
cube, and the vertex is at a distance r from each of the
eight vertices of the cube, r being the radius of the hyper-
sphere. We shall speak of the vertex as corresponding to
the cube.
This relation is reciprocal. At a vertex of the hyper-
cube there are four cubes lying in hyperplanes perpendicu-
lar respectively to the four axes ; and the four vertices of
the i6-hedroid corresponding to these four cubes are the
vertices of one of its tetrahedrons. These four vertices
are at a distance r from the vertex of the hypercube, and
the radius to the latter point is perpendicular to the hy-
perplane of the tetrahedron, passing through its centre.
Therefore, the radii perpendicular to the hyperplanes of
the sixteen cells of a i6-hedroid are the radii to the sixteen
vertices of a hypercube. The hypercube and the 16-
hedroid are said to be reciprocal polyhedroids (see Art. 177).
170. The diagonals of the hypercube and the 16-hedroid
forming three rectangular systems. The hypercube has
eight diagonals joining the eight vertices of any one cube
to the eight vertices of the opposite cube. Since any
vertex of a i6-hedroid is at a distance r from each of the
eight vertices of the corresponding cube, the radius to the
former makes an angle of 60 with each of the eight radii
to the latter.
Now the eight radii to the vertices of a cube can be asso-
ciated in two sets of four each ; for on a cube the common
vertex of three adjacent squares and the three opposite
vertices of these squares are the vertices of a tetrahedron
whose edges are diagonals of the six faces of the cube,
one in each face, while the other four vertices of the cube
are the vertices of a second tetrahedron of the same kind.
The diagonal of a square subtends a right angle at the
2Q4 THE SIMPLER REGULAR POLYHEDROIDS [vm. i.
centre of the hypersphere, and the four radii of a set are
mutually perpendicular. Two radii taken, one from each
set, are radii to the extremities of an edge or to the extremi-
ties of a diagonal of the cube. Now an edge subtends
an angle of 60 at the centre, and a diagonal, its length
being r Vj, subtends an angle of 120. Therefore, any
radius of one set makes an angle of 60 with three of the
radii of the other set, and with the fourth an angle of 120.
In fact, the radius opposite to this fourth radius goes to
the other extremity of the fourth edge of the hypercube
at the given point. Thus we can say that the radii to the
extremities of the four edges which go out from a vertex of
a hypercube are a set of four mutually perpendicular radii.
Putting these results together, we find that the four
diagonals of the i6-hedroid and the eight diagonals of the
hypercube lie in three sets of four mutually perpendicular
lines, each line of one set making with each line of the other
two sets the two supplementary angles of 60 and 120.
The lines of any one of these three sets can be taken as
the axis-lines of a hypercube, and the eight lines of the other
two sets will pass through its vertices, for there are only
eight lines that can make angles of 60 and 120 with the
four lines of a rectangular system. A half-line making
an angle of 60 with a half-line of a rectangular system will
make an angle of 30 with the hyperplane to which the latter
is perpendicular, and the point at a distance r on it will
be at a distance - from the hyperplane. But there are just
2
sixteen points at the distance - from each of the four hy-
2
perplanes of a rectangular system, the sixteen points of
intersection of four pairs of parallel hyperplanes, and so
the sixteen vertices of a hypercube which has its axes along
the four lines of the rectangular system.
I7Q> 171] 24-HEDROID 295
Associated with the twelve lines described above, there
are, therefore, three hypercubes and three regular 16-
hedroids.
171. The 24-hedroid associated with a hypercube and
a 16-hedroid. Given a hypercube and a regular i6-hedroid
inscribed in the same hypersphere and with their axes
lying in the same rectangular system, let P and P 1 be two
vertices of the latter corresponding to the two cubes which
have in common the square A BCD. Let O' be the centre
of the square. Since P is at the distance r from each of
the four vertices of the square, it determines with the
square one-half of a regular octahedron. The same is
true of the point P 1 ', and indeed it is also true of the centre
of the hypersphere. The altitude O'P of the pyramid
P-ABCD is equal to -=, and O'P 1 is of the same length.
V2
But the segment PP 1 subtends at O an angle of 90, and is
of twice this length. Hence the line PP f passes through O',
and the points P and P', together with the square, lie in
one hyperplane, and are the vertices of a regular octahe-
dron with centre at 0'.
From a cube can be formed six equal pyramids having a
common vertex at the centre of the cube and the six faces
of the cube as bases. The interior of the square ABCD
is the common base of two such pyramids, one from each
of the two cubes which have this square in common. These
two pyramids do not lie in one hyperplane, nor does either
of them belong to a regular octahedron, but the two pyramids
of the octahedron PP f are the projections of these pyramids
upon the hyperplane of the octahedron, projected from the
centre of the hypersphere. At a vertex P there are six
quadrangular pyramids, belonging to six regular octahe-
drons, and the projections of the six pyramids which have
296 THE SIMPLER REGULAR POLYHEDROIDS [vra. i.
a common vertex at the centre of the corresponding cube.
The interiors of these six pyramids at P are the lateral
cells of a hyperpyramid whose base is the interior of the
cube, that is, a cell of the hypercube.
From the cubes of the hypercube are formed in all
twenty-four pairs of pyramids, the two pyramids of a pair
having a common base. The interiors of these pyramids
and their bases can be projected as above from the centre
O into the interiors of the twenty-four regular octahedrons,
the cells of a polyhedroid which can be built up by placing
a hyperpyramid as described above upon each cell of the
hypercube. This polyhedroid is called an ikosatetrahe-
droid or 24-hedroid. Its vertices are the sixteen vertices
of the hypercube and the eight vertices of the i6-hedroid,
twenty-four in all. This polyhedroid can be built up from
any one of the three hypercubes associated with a set of
twelve lines such as is described in the preceding article,
and it has the same number and arrangement of parts at
a vertex of the hypercube that it has at a vertex of the 16-
hedroid.
To get the plane angle of a hyperplane angle of the 24-
hedroid we draw half-lines through the centres of two ad-
jacent octahedrons from the centre of their common face.
One common vertex of the two octahedrons can be taken
as the point P, and their centres are the centres of two adja-
cent squares of the cube to which P corresponds. These
two points and the centre of the common face of the octa-
hedrons are the vertices of an isosceles triangle whose
sides are equal respectively to the sides of the isosceles
triangle formed in the same way from any other pair of
adjacent octahedrons. Therefore the hyperplane angles
of the polyhedroid are all equal,* and the polyhedroid is
regular.
* These are angles of 120, like those of the i6-hedroid.
i/i, 172] RECIPROCAL 24-HEDROIDS 297
The octahedron has eight faces. The number of faces
of the 24-hedroid is therefore
24 X 8
7- =96.
Eight edges meet at a vertex, and the number of edges is
likewise 96.
Thus the regular 24-hedroid has
24 vertices, 96 edges, 96 faces, and 24 cells.
172. Reciprocal 24-hedroids. There are twenty-four
points O', the centres of the twenty-four octahedrons,
the centres of the twenty-four squares of the hypercube,
and the middle points of the twenty-four edges of the 16-
hedroid. These twenty-four points lie on twelve lines
through 0.
Since the points 0' are the middle points of the edges
of a i6-hedroid, the half-lines OO' bisect the twenty-four
right angles formed by the four axes of this i6-hedroid.
Now these right angles lie in three pairs of absolutely
perpendicular planes, and the half -lines which bisect the
eight right angles in any one of these three pairs of planes
are themselves the half-lines of a set of four mutually
perpendicular lines. Our twelve lines, therefore, consist
of three sets of four mutually perpendicular lines.
If two right angles lie in perpendicular planes and have
a common side along the intersection of these planes,
their bisectors form an angle of 60. Now any two of the
six planes of a rectangular system, if not absolutely per-
pendicular, are perpendicular, intersecting in one of the four
lines. Therefore each of the bisectors of the four right
angles in one of these planes makes with each of the bisec-
tors of the four right angles in the other an angle of 60 or
an angle of 120.
208 THE SIMPLER REGULAR POLYHEDROIDS [vm. i.
This proves that a line in any one of our three sets of
four mutually perpendicular lines makes with each of the
eight lines of the other two sets the supplementary angles
of 60 and 120, and that the radii of the hypersphere drawn
through the points O f meet the hypersphere in twenty-
four points which are the vertices of a second regular 24-
hedroid. The two 24-hedroids are related in the same way
as a hypercube and a i6-hedroid whose axes lie along the
same four lines; namely, the radii perpendicular to the
hyperplanes of the cells of one are the radii to the vertices
of the other (Art. 169). The two 24-hedroids are said to
be reciprocal (see Art. 177).
173. The reciprocal nets of 24-hedroids and 16-hedroids.
The interior of a cube of a hypercube is the base of a hyper-
pyramid with vertex at the centre O, congruent to the
hyperpyramid built upon the same base with vertex at
the corresponding point P *. From the hypercube can
be formed eight such hyperpyramids, and therefore eight
polyhedroidal angles like that of the 24-hedroid at P can
be placed together with a common vertex so as to fill the
hyperspace about this vertex. In other words, eight 24-
hedroids can be placed together at a point with their in-
teriors filling the hyperspace about this point, and we have
in hyperspace a net of 24-hedroids like the net of hyper-
cubes described in Art. 167. The centres and vertices
of the 24-hedroids are, indeed, the centres and vertices of
a net of hypercubes. The eight polyhedroidal angles at
O have their axes lying in the axis-lines of the hypercube.
Therefore, the eight 24-hedroids put together at any point
have axis lines through this point forming a rectangular
system, and their centres are the vertices of a i6-hedroid
whose centre is this point.
* The two hyperpyramids are symmetrically situated with respect to the centre
of the cube, and are congruent. Thus the hypervolume of the 24-hedroid is twice
that of the hypercube.
i?2, i73l RECIPROCAL NETS 299
Let us suppose that the given 24-hedroid with centre at
O is one of eight 24-hedroids put together at a vertex P,
so that P is the centre of the i6-hedroid and one of its
vertices is at O. The edges of the i6-hedroid pass through
the centres of certain cells of the 24-hedroids. In partic-
ular, the edges which come to O pass through points
which we have called 0', through those six which lie on
the cube corresponding to P. Now the points O' which
lie on a single cube lie on radii to the vertices of a single
cell of the reciprocal 24-hedroid constructed in the preced-
ing article. This cell is therefore the base of a hyper-
pyramid with vertex at O whose polyhedroidal angle at O
is the polyhedroidal angle at this point of the i6-hedroid
just mentioned. Therefore, twenty-four such polyhe-
droidal angles, and so twenty-four i6-hedroids, can be
put together at a point with their interiors filling the hy-
perspace about this point; and we have in hyperspace a
net of i6-hedroids reciprocal to the net of 24-hedroids.
Two adjacent 24-hedroids have in common an octahe-
dron PP', with its centre 0' the projection upon its hy-
perplane of the centres of the two 24-hedroids. These
centres are the extremities of an edge in the net of i6-he-
droids. Thus the edges of the i6-hedroids correspond to
the cells of the 24-hedroids, an edge and a cell lying in a
perpendicular line and hyperplane, and intersecting at a
point which is the middle point of the edge and the centre
of the cell. Three 24-hedroids whose centres are the ver-
tices of a face of the net of i6-hedroids are in cyclical order,
each having a cell in common with the next, and the hyper-
planes of the three cells, being perpendicular to the sides
of the given face at their middle points, pass through the
centre and intersect in the plane absolutely perpendicular
to its plane at this point. In other words, the three 24-
hedroids have in common a triangle and the face which is
300 THE POLYHEDROID FORMULA [vra. n.
the interior of this triangle. The vertices of this triangle
are in turn the centres of three of the i6-hedroids having
the given face in 'common, and arranged in cyclical
order around the plane of this face. Each of these three
i6-hedroids has in common with the next a cell whose
centre is the projection upon its hyperplane of the
centres of the two i6-hedroids, so that each edge of the net
of 24-hedroids and the corresponding cell of the net of 16-
hedroids lie in a perpendicular line and hyperplane, and
intersect at a point which is the middle point of the edge
and the centre of the cell (see Art. 179).
II. THE POLYHEDROID FORMULA
174. Extension of the polyhedron formula. In a simple
polyhedron the number of vertices and faces taken together
is 2 more than the number of edges.* This relation can
be expressed very conveniently in the form
i - No + #1 - Ni + i = o,
where N Q is the number of vertices, Ni the number of
edges, and Nz the number of faces.
For the five regular polyhedrons this formula becomes
for the tetrahedron i 4+ 6 4 + 1=0,
" hexahedron or cube i 8 + 12 6 + 1=0,
" octahedron i 6 + 12 8 + 1=0,
" dodekahedron 1 20 + 30 12 + 1 = o,
" ikosahedron 1 12+30 20 + 1 =o.
*This theorem was discovered by Euler about 1750 and usually goes by his
name. It was known to Descartes more than a century earlier at least it follows
directly from formulae in a manuscript, "De Solidorum Elementis, " left by Des-
cartes. This memoir was not published, however, until 1860, a copy having been
found only a few years earlier among the papers of Leibnitz (CEuvres infdites de
Descartes, par M. le Comte Foucher de Careil, Paris, 1860, vol. II, p. 214. See com-
munication by Prouhet in the Comptes Rendus, vol. 50, 1860, p. 779, and several
by E. de Jonquieres in vol. no, i8go).
173, i74l PROOF FOR POLYHEDROIDS 301
In the first three cases the first members of these equations
are the expansions of
(i - i) 4 , i -(2-1)', and(i -2)^ + 1.
The polyhedrons take their names from the numbers N%.
The above formula can be generalized so as to apply
to certain more complicated figures and to polyhedroids
in space of four dimensions.
For a simple polygon we can write a similar formula,
namely,
i - No + Ni - i = o.
In proving the formula for a polyhedron we think of the poly-
hedron as built up by putting together a set of polygons,
taking them in succession in such order that each is joined
to those already taken by a side or by two or more sides
forming a single broken line. As long as we have not com-
pletely enclosed any portion of space and formed a poly-
hedron we have the relation
i - # + Ni - Ni = o.
In the same way we can build up a polyhedroid by putting
together a set of polyhedrons. We take them in succes-
sion in such order that each is joined to those already
taken by a set of polygons like the incomplete polyhedron.
At each stage of the process we add to the number of ver-
tices, edges, and faces already obtained the number of
vertices, edges, and faces of the new polyhedron, and to
the number of cells we add i ; and we subtract the number
of vertices, edges, and faces of the connecting figure, which
otherwise would be counted twice.
Assuming that the figure had an equation of the form
I - #0 + # 1 - #2 + # 3 = O,
302 THE POLYHEDROID FORMULA [vm. n.
where N* is the number of cells, we add all but the first
term of the equation of the new polyhedron, and then sub-
tract all but the first term of the equation of the connecting
figure. Thus we prove by induction that as long as the
polyhedroid is incomplete its equation is of the form as-
sumed.
When we come to the last polyhedron there are no new
vertices, edges, or faces. Only the number N 9 is increased
by i, and the equation will be true if we write it
i - tfo + N l - N* + #3 - i = o.
This equation is true for a simple polyhedroid, and we shall
call it the polyhedroid formula. The relation can be stated
as a theorem in the following words :
THEOREM. In a simple polyhedroid the number of cells
plus the number of edges is equal to the number of faces plus
the number of vertices*
For the four regular polyhedroids already considered
the polyhedroid formula becomes
for the pentahedroid i 5 + 1010+ 5 1=0,
" ' hypercube i 16 + 32 24 + 8 i = o,
" i6-hedroid i 8 + 24 32 + 16 i = o,
" 24-hedroid 1 24 + 96 96 + 24 1 = o.
In the first three cases the first numbers of these equa-
tions are the expansions of
(i -i), i -( 2 -i)*, and(i-2) 4 -i.
In all cases the name of the polyhedroid comes from the
number N*.
* An interesting discussion of this law and its extension to geometry of higher
dimensions is given in Schoute's Mehrdimensionale Geometric, vol. II, 2. The corre-
sponding law for simple polyhedroids of any number of dimensions is proved by
Stringham in the article referred to on p. 289. The theorem will not be used in prov-
ing the existence of any regular polyhedroid, nor even in computing the number of
any of its elements. Its proof may be omitted by the student, but we have in
the formula itself a convenient mode of expressing these numbers.
174, i75l NETS ON THE SPHERE 303
III. RECIPROCAL POLYHEDROIDS AND RECIPROCAL
NETS OF POLYHEDROIDS
175. Reciprocal polyhedrons and nets of polyhedral
angles. In geometry of three dimensions a regular poly-
hedron can always be inscribed in a sphere. This is proved
in the same way that it is proved for tetrahedrons, and the
proof holds also in the Elliptic Non-Euclidean Geometry,
and so in the geometry of the hypersphere. The vertices
of the polyhedron are the vertices of a net of equal regular
spherical polygons, a net of spherical polygons being a set
covering the sphere so that every point of the latter is
either a point of two or more of the polygons or a point
of the interior of only one. Conversely, the vertices of a
net of equal regular spherical polygons are always the ver-
tices of a regular polyhedron inscribed in the sphere.
Since the sphere is the same in the Elliptic Geometry, the
regular polyhedrons of this geometry are of the same
types as those of Euclidean Geometry.*
When we have a net of equal regular spherical polygons,
the centres and vertices of these polygons are the vertices
and centres of a second net of equal regular polygons
reciprocal to the first. And so when a regular polyhedron
is inscribed in a sphere, radii perpendicular to the planes
of the faces are radii to the vertices of a second regular
polyhedron. Two polyhedrons which can be so placed
that half-lines from the centre perpendicular to the planes
of the faces of one are half -lines from the centre through the
vertices of the other, are said to be reciprocal polyhedrons.
* It can also be proved in the Hyperbolic Geometry that a regular polyhedron
can be inscribed in a sphere, and that we have just the same types of regular poly-
hedrons. But in the Hyperbolic Geometry we have also boundary-surfaces and
equidistant-surfaces on which we can form other nets of polygons ; and the vertices
of such a net are the vertices of an infinite broken surface formed of equal regular
polygons and their interiors, like a regular polyhedron.
304 RECIPROCAL POLYHEDROIDS AND NETS [vm. m.
They have the same number of edges, and the number of
vertices of one is the same as the number of faces of the
other. The polyhedron formula of one is the polyhedron
formula of the other written backwards.
Half-lines from the centre of a regular polyhedron
through the vertices are the edges of equal regular polyhe-
dral angles forming a net of polyhedral angles in the two-
dimensional Point Geometry at this point, every half-
line element of the Point Geometry being either a half-line
of two or more of the polyhedral angles, or a half-line of
the interior of only one. Corresponding to two reciprocal
polyhedrons and to two reciprocal nets on the sphere we
have two reciprocal nets of polyhedral angles, the edges of
the polyhedral angles of one net being the axes of the
polyhedral angles of the other net. Conversely, a net of
equal regular polyhedral angles in a hyperplane intersects
any sphere of the hyperplane with centre at the vertex of
the net in a net of equal regular spherical polygons, and
the edges of the polyhedral angles pass through the ver-
tices of a regular inscribed polyhedron which corresponds
to the net.
176. Reciprocal nets of polyhedrons. In a net of equal
regular spherical polygons the angles at a vertex P are
equal, vertices adjacent to P are vertices of a regular
spherical polygon, and the corresponding inscribed poly-
hedron has a regular polyhedral angle at P. If, then, a
set of equal regular polyhedrons with their interiors fills
a three-dimensional space about a point so that their
polyhedral angles at this point form a net, the centres of
the polyhedrons, lying at a given distance on the axes
of these polyhedral angles, are themselves the vertices
of a regular polyhedron, reciprocal to the polyhedron which
corresponds to the net. With any given net of regular
I7S.I76] NETS OF POLYHEDRONS 305
polyhedrons is associated a reciprocal net, the vertices
of the polyhedrons of one net being the centres of the
polyhedrons of the other. Any edge of a polyhedron of
one net, joinitfg the centres of two polyhedrons of the
other net, and the common face of these two polyhedrons,
lie in a perpendicular line and plane and intersect at a point
which is the middle point of the edge and the centre of the
face. The number of vertices in a polyhedron of one
net is equal to the number of polyhedrons of the other net
at a vertex. The polyhedrons of two reciprocal nets are
not, in general, reciprocal polyhedrons.
Now the polyhedral angles of the polyhedrons of a net
must be such as occur in a net of polyhedral angles at the
centre of a regular polyhedron. Thus we have
at the centre of a tetrahedron 4 trihedral angles,
" a cube 6 tetrahedral "
" an octahedron 8 trihedral "
" a dodekahedron 12 pentahedral "
" an ikosahedron 20 trihedral "
The polyhedral angles of the five regular polyhedrons are
as follows :
in the tetrahedron trihedral angles,
" cube
" octahedron tetrahedral "
" dodekahedron trihedral "
" ikosahedron pentahedral "
Therefore the only sets of regular polyhedrons that can be
used to form nets are
4 tetrahedrons, cubes, or dodekahedrons at a point,
O it U
20
6 octahedrons "
12 ikosahedrons "
x
306 RECIPROCAL POLYHEDROIDS AND NETS [vm. m.
In two reciprocal nets the number of vertices in a poly-
hedron of one net is equal to the number of polyhedrons of
the other net at a- point. These nets of polyhedrons are
associated, therefore, as follows :
4 cubes reciprocal to 8 tetrahedrons,
4 dodekahedrons " 20 "
8 " " 20 cubes;
also
4 tetrahedrons reciprocal to a net of the same kind,
8 cubes
20 dodekahedrons "
6 octahedrons
12 ikosahedrons " " "
These theorems are true of the Non-Euclidean Geometries
as well as of Euclidean Geometry, but in the Hyperbolic
Geometry the angles of a regular polyhedron are smaller,
and in the Elliptic Geometry they are larger, than they are
in Euclidean Geometry. Moreover, we can make the
angles in the first case as small as we please, and in the
second case as large as we please up to 180, by taking the
figures sufficiently large. Thus any of these combinations
is possible in one of the three geometries, at least for a
restricted portion of space, and two reciprocal nets must
occur in the same kind of geometry. Any combination
which more than fills the part of Euclidean space about
a point belongs to Hyberbolic Geometry, and any combina-
tion which does not fill the part of Euclidean space about
a point belongs to Elliptic Geometry.
Now in Euclidean Geometry we have a net of cubes,
eight at a point. Then eight dodekahedrons, twenty cubes,
or twenty dodekahedrons would more than fill Euclidean
space, and nets of these types must belong to the Hyper-
bolic Geometry ; while nets with four cubes, four tetrahe-
NETS OF POLYHEDRONS
307
drons, or eight tetrahedrons will belong to the Elliptic
Geometry. We can also prove that the polyhedral angles
of a regular octahedron in Euclidean Geometry are smaller
than those of a net at the centre of a cube, so that the net
of octahedrons, six at a point, belongs to Elliptic Geometry.
For example, the faces of the octahedron are equilateral
triangles, while the centre and two adjacent vertices of a
cube are the vertices of an isosceles triangle in which the
legs, each being the half of a diagonal, are shorter than the
base.
There remains to be considered the net of ikosahedrons,
twelve at a point, and the reciprocal nets of four dodeka-
hedrons and twenty tetrahedrons. In the net of twelve
pentahedral angles and in the net of four trihedral angles
there are three of these angles around an edge, and the
dihedral angles must all be angles of 120. Therefore in
the net of ikosahedrons and in the net of dodekahedrons
the dihedral angles must be angles of 120. Thus we have
to determine in which geometry
the dihedral angles of the ikosa-
hedron, and in which geometry
the dihedral angles of the do-
dekahedron, are angles of 120.
Let ABC be a spherical tri-
angle of the net corresponding
to the ikosahedron, its centre
(on the sphere), and D the A D B
middle point of the arc AB. Then 2 OD is the supplement
of the dihedral angle of the ikosahedron, and 2 AD is the
supplement of the dihedral angle of the dodekahedron,
the dodekahedron being reciprocal to the ikosahedron.
Now two sides of the triangles ABC on opposite sides of
the sphere lie on a great circle which also crosses four of
these triangles. Hence, we have four arcs equal to OD,
308 RECIPROCAL POLYHEDROIDS AND NETS [vra. m.
four equal to AD, and four equal to A O on such a circle.
Therefore,
OQ + AD + AO = 90.
But in the triangle AOD the angles are 36, 60, and 90,
OD< AD< AO,
and OD < 30.
Again, in the triangle ACD the angles are 72, 36, and 90,
and
CD< AC.
That is, OD + AO < 2 AD,
and therefore AD > 30.
It follows that in the Euclidean Geometry the dihedral
angles of the ikosahedron are greater than 120, and the net
of ikosahedrons belongs to Hyperbolic Geometry ; but the
dihedral angles of the dodekahedron are less than 120, so
that the net of dodekahedrons and the reciprocal net of
tetrahedrons belong to Elliptic Geometry.*
Summing up we find that
Euclidean space can be filled with cubes, eight at a point ;
Hyberbolic space can be filled with dodekahedrons, eight
at a point or twenty at a point, with cubes, twenty at a
point, or with ikosahedrons, twelve at a point ; and
Elliptic space, or at least any restricted portion of elliptic
space, can be filled with tetrahedrons, four at a point,
eight at a point, or twenty at a point, with cubes, four at a
point, with dodekahedrons, four at a point, or with octa-
hedrons, six at a point.
* In the net of twenty trihedral angles the dihedral angles are angles of 72, and
it can be proved that the dihedral angles of the tetrahedron in Euclidean Geometry
are less than 72, so that the net of twenty tetrahedrons'at a point belongs, indeed,
to the Elliptic Geometry.
i?6, i77l NETS IN THE HYPERSPHERE 309
177. Reciprocal polyhedroids. A regular polyhedroid
can always be inscribed in a hypersphere. For the perpen-
diculars to the hyperplanes of the cells at the centres of the
cells all pass through a point which is at the same distance
on each of them, and, therefore, at the same distance from
all the vertices of the polyhedroid. This is proved in the
same way that it is proved for pentahedroids (see Art. 113,
Th. i). Since the hyperplane angles are all equal, no two
adjacent cells lie in the same hyperplane, and the perpen-
diculars to the hyperplanes of two adjacent cells cannot be
parallel and must therefore intersect.*
When a regular polyhedroid is inscribed in a hypersphere,
its vertices are the vertices of a net of equal regular hyper-
spherical polyhedrons. For the vertices of any polyhedron
of the polyhedroid lie on a sphere which lies entirely in the
hypersphere, and so they are the vertices of a regular hyper-
spherical polyhedron of the same type. The polyhedrons
are equal, for the spheres are equal, and with their interiors
they completely fill the hypersphere. They can, indeed,
be regarded as the projections of the polyhedrons of the
given polyhedroid, projected by radii from the centre of
the hypersphere. Conversely, the vertices of a net of equal
regular hyperspherical polyhedrons are always the vertices
of a regular polyhedroid inscribed in the hypersphere.
The vertices of any one of the polyhedrons lie on a sphere
and are the vertices of a regular hyperplane polyhedron in
the hyperplane of the sphere. These hyperplane polyhe-
drons are all equal, with their interiors they enclose a por-
tion of hyperspace, and the hyperplane angles formed by
the hyperplanes of any two which are adjacent are all equal.
When a regular polyhedroid is inscribed in a hypersphere,
* The theorem is true in Elliptic Geometry of four dimensions, since any two
lines in a plane of Elliptic Geometry intersect. It can also be proved in Hyperbolic
Geometry.
310 RECIPROCAL POLYHEDROIDS AND NETS [vm. m.
so that its vertices are the vertices of a net of equal regular
hyperspherical polyhedrons, radii perpendicular to the hy-
perplanes of its colls will be radii to the centres of these
hyperspherical polyhedrons ; that is, they will be radii to
the vertices.of the reciprocal net, to points which are, there-
fore, the vertices of a second regular polyhedroid. The
relation of the two polyhedroids is reciprocal ; and two regu-
lar polyhedroids which can be so placed that half -lines from
the centre perpendicular to the hyperplanes of the cells of
one are half-lines from the centre through the vertices of
the other, are called reciprocal polyhedroids. The number
of vertices of one is equal to the number of cells of the other,
and the number of edges of one is equal to the number of
faces of the other. The polyhedroid formula of one is
the polyhedroid formula of the other written backwards.
Moreover, the number of cells at a vertex of one equals the
number of vertices to a cell of the other, the number of
edges at a vertex of one equals the number of faces to a cell
of the other, and so on. Whenever we have constructed
a regular polyhedroid, or proved its existence, we have
proved the existence of a reciprocal polyhedroid. The two
may, however, be polyhedroids of the same type.
The hypercube and the regular i6-hedroid are reciprocal
polyhedroids. The regular pentahedroid and the regular
24-hedroid are self-reciprocal. These cases correspond
to the two reciprocal nets of four cubes at a point and eight
tetrahedrons at a point, and to the two self-reciprocal nets
of four tetrahedrons at a point and six octahedrons at a
point. Now we have found that in Elliptic space of three
dimensions, or at least in a restricted portion of such space,
and so in the hypersphere or in a restricted portion of the
hypersphere, there exists another pair of reciprocal nets,
nets with four dodekahedrons at a point and twenty tetra-
hedrons at a point ; and so in Euclidean space of four di-
1771 THE N- AND SN-HEDROTD 311
mensions there can be at most only two types of regular
polyhedroids besides the four which we have already found.
In the next section we shall construct a regular polyhedroid
with twenty tetrahedrons at a point and a regular polyhe-
droid with four dodekahedrons at a point, one of these con-
structions being necessary to complete the proof of the
existence of the two.
Assuming that these polyhedroids exist, we know that
they are reciprocal and that the polyhedroid equation of
one is the polyhedroid equation of the other written back-
wards. Let us suppose that the second polyhedroid has N
cells, that is, that it contains N dodekahedrons, and call
it an N-hedroid. Each dodekahedron has twenty vertices,
and at each vertex there are four dodekahedrons. There-
fore the number of vertices is
4
The first polyhedroid is, then, a $ N-hedroid, containing
5 N tetrahedrons.
The dodekahedron has twelve faces, and in the AT-hedroid
each face is common to two dodekahedrons. Therefore
the number of faces in the A^-hedroid is
In the same way the number of faces in the
is found to be
2
Thus the polyhedroid formulae of the two are
i- N+ 6N - loN + $N - i = o,
and i-sN+ioN- 6AT + N - i - o.
312 RECIPROCAL POLYHEDROIDS AND NETS [vin. in.
178. Regular polyhedroidal angles. A regular poly-
hedroidal angle is one subtended at the centre of a hyper-
sphere by a regular hyperspherical polyhedron whose
circumscribed sphere is not a great sphere. But the hy-
perspherical polyhedron is the projection from the centre of
the hypersphere of a hyperplane polyhedron which has the
same vertices, and through the vertices of a given hyper-
plane polyhedron we may pass a hypersphere having its
centre at any point in the line perpendicular at the centre
of the polyhedron to its hyperplane. Therefore the polyhe-
droidal angle subtended by a regular polyhedron at any
point in the line drawn through the centre of the polyhe-
dron perpendicular to its hyperplane, except at the centre
itself, is a regular polyhedroidal angle. The polyhedroidal
angle at the vertex of a regular hyperpyramid is a regular
polyhedroidal angle.
The half-line drawn from the vertex of a regular poly-
hedroidal angle through the centre of the subtending poly-
hedron, or, in the case of a regular hyperpyramid, the half-
line which contains the axis of the hyperpyramid, is the
axis of the polyhedroidal angle.
Let P be the vertex of a regular polyhedroidal angle and
0' the centre of the subtending polyhedron. If we project
the polyhedroidal angle by orthogonal projection upon the
hyperplane of the polyhedron, the vertex will be projected
at the centre 0', and the polyhedroidal angle will be pro-
jected into the net of equal regular polyhedral angles at
this point. Take a point on PO r produced, and with
as centre construct a hypersphere passing through P.
Radii from O will project the polyhedroidal angle into a
net of equal regular hyperspherical polyhedral angles at
P congruent to the net at O'. The polyhedral angles at P
in the hypersphere can be regarded as in the tangent
hyperplane at P. Any one of the half-lines from P in this
178] POLYHEDROIDAL ANGLES 313
hyperplane, and the corresponding half -line from O', lie
in the intersections of a projecting plane * through O with
two parallel hyperplanes, and are parallel (Art. 128, Ths.
4 and 8). Conversely, suppose we have a net of equal
regular hyperspherical polyhedral angles at P. If we pro-
ject these by radii upon a hyperplane perpendicular to OP
at some point O f between O and P, we shall have a con-
gruent net at O', and any sphere in this hyperplane with
centre at O' will intersect the edges of the polyhedral angles
in the vertices of a regular polyhedron which subtends at
P a regular polyhedroidal angle.
A regular polyhedroid is projected by radii upon the
circumscribed hypersphere in a net of equal regular hyper-
spherical polyhedrons, and the edges of the polyhedroid
which come to a vertex P are projected into arcs which
lie in the edges of the hyperspherical polyhedral angles
of the net at P. Now these edges are of the same length,
their extremities are the vertices of a -regular polyhedron,
and the polyhedroidal angle at P of the given polyhedroid
is a regular polyhedroidal angle subtended by this poly-
hedron. In other words, the polyhedroidal angles at the
vertices of a regular polyhedroid are regular polyhedroidal
angles.
The half-line from a vertex of a regular polyhedroid
through the centre is the axis of the polyhedroidal angle
at the vertex.
Half -lines drawn from the centre of a regular polyhedroid
through the vertices are the edges of a set forming a net
of polyhedroidal angles in the Point Geometry at this point,
every half-line of the Point Geometry being a half-line of
two or more of the polyhedroidal angles or a half-line of
the interior of only one. Corresponding to two reciprocal
nets of polyhedrons in the hypersphere, we have two re-
* See foot-note, p. 84.
314 RECIPROCAL POLYHEDROIDS AND NETS [vnnn.
ciprocal nets of polyhedroidal angles, the edges of the poly-
hedroidal angles of one net being the axes of the polyhedroi-
dal angles of the. other net. Conversely, a net of equal
regular polyhedroidal angles intersects any hypersphere
with centre at the vertex of the net in a net of equal regu-
lar polyhedrons, and the edges of the polyhedroidal angles
pass through the vertices of a regular inscribed polyhe-
droid which corresponds to the net.
179. Reciprocal nets of polyhedroids. If a set of equal
regular polyhedroids having a common vertex at P, with
their interiors, fill the part of hyperspace about this point
so that their polyhedroidal angles at P form a net, the
centres of the polyhedroids, lying at a given distance from P
on the axes of these polyhedroidal angles, are themselves
the vertices of a regular polyhedroid with centre at P,
reciprocal to the polyhedroid which corresponds to the
net. With any given net of regular polyhedroids is asso-
ciated a reciprocal net, the vertices of the polyhedroids of
one net being the centres of the polyhedroids of the other.
Any edge of a polyhedroid of one net, joining the centres
of two polyhedroids of the other net, and the common cell
of these two polyhedroids, lie in a perpendicular line and
hyperplane, and intersect at a point which is the middle
point of the edge and the centre of the cell. Those polyhe-
droids of one net whose centres are the vertices of a face
of the other net, have in common a face in a plane abso-
lutely perpendicular to the plane of the given face at a
point which is the centre of both faces. The vertices of
the second face are in turn the centres of polyhedroids of
the other net which have the given face in common (see
Arts. 167 and 173). The number of vertices in a polyhe-
droid of one net is equal to the number of polyhedroids
of the other net at a vertex. The polyhedroids of two
I7, i79l NETS OF POLYHEDROIDS 315,
reciprocal nets are not, in general, reciprocal polyhedroids.
Whenever we have constructed a net of equal regular poly-
hedroids, or proved the existence of such a net, we have
proved the existence of a reciprocal net.
Now the polyhedroidal angles of the polyhedroids of a
net must be such as occur in a net of polyhedroidal angles
at the centre of a regular polyhedroid. Thus we have
at the centre of a pentahedroid 5 tetrahedroidal angles,
" a hypercube 8 6-hedroidal "
" a i6-hedroid 16 tetrahedroidal "
a 24- " 24 8-hedroidal
as#- " sN tetrahedroidal
an N- " N i2-hedroidal
The polyhedroidal angles of the six regular polyhedroids
are as follows :
in the pentahedroid tetrahedroidal angles,
" hypercube
" i6-hedroid 8-hedroidal
" 24- " 6-
" SN- " 20- "
N- " tetrahedroidal "
Therefore the only sets of regular polyhedroids that can be
used to form nets are
5 pentahedroids, hypercubes, or ^V-hedroids at a point,
16
8 24-hedroids "
24 16- "
The number of vertices in a polyhedroid of one of two re-
ciprocal nets is equal to the number of polyhedroids of the
other at a point. These nets of polyhedroids are associated,
therefore, as follows :
316 RECIPROCAL POLYHEDROIDS AND NETS [viu. in.
5 hypercubes reciprocal to 16 pentahedroids,
S #-hedroids " s#
16 N- " - " $N hypercubes,
8 24- " " 24 i6-hedroids;
also
5 pentahedroids reciprocal to a net of the same kind,
16 hypercubes
These theorems are true of the Non-Euclidean Geometries
as well as of Euclidean Geometry. As in the case of poly-
hedrons (Art. 176), those combinations which more than
fill the part of Euclidean hyperspace about a point belong
to Hyperbolic Geometry, and those which do not fill the
part of Euclidean hyperspace about a point belong to Ellip-
tic Geometry.
Now in Euclidean Geometry we have a net of hyper-
cubes, sixteen at a point (Art. 167). Then sixteen AT-he-
droids, or 5 N of either of these polyhedroids, would more
than fill the Euclidean hyperspace, and nets of these types
must belong to Hyperbolic Geometry; while nets with
five hypercubes, five pentahedroids, or sixteen pentahe-
droids, will belong to Elliptic Geometry.
We have also in Euclidean hyperspace the reciprocal
nets of eight 24-hedroids and twenty- four i6-hedroids
(Art. 173). There remain, therefore, to be considered only
the reciprocal nets of five N-hedroids and 5^ pentahe-
droids. We shall find when we have constructed the 5 N-
hedroid that its edge subtends an angle of 36 at the centre
(see Art. 182), and therefore the hyperplane angles of the
AMiedroid in Euclidean hyperspace are the supplements
of 36, or 144, the ^-hedroid being reciprocal to the
S^V-hedroid. But in a net of five ^-hedroids there are
four of these around any face, and the hyperplane angles
PLATE I.
FIG. i.
FIG. 2.
FIG. 4.
FIG. 3.
FIG. 5.
FIG. 6.
179, i8o] 600-HEDROID 317
must be angles of 90. Therefore these reciprocal nets
belong to the Hyperbolic Geometry. Or, we may say that
the face angles of the pentahedroid in Euclidean hyperspace
are angles of 60, so that 5JV pentahedroids would more
than fill Euclidean hyperspace about a point.
Summing up, we find that
Euclidean hyperspace can be filled with hypercubes,
sixteen at a point, with 24-hedroids, eight at a point, or
with i6-hedroids, twenty-four at a point;
Hyperbolic hyperspace can be filled with AMiedroids,
five at a point, sixteen at a point, or 5 N at a point, with
pentahedroids, 5# at a point, or with hypercubes, $N
at a point ; and
Elliptic hyperspace, or at least any restricted portion
of elliptic hyperspace, can be filled with pentahedroids,
five at a point, or sixteen at a point, or with hypercubes,
five at a point.*
IV. CONSTRUCTION OF THE REGULAR 6oo-HEDROID
AND THE REGULAR I20-HEDROID
180. First half of the 600-hedroid : proving its exist-
ence. We shall construct in a hyper sphere a net of equal
regular tetrahedrons, twenty at a point, with their interiors
completely filling the hypersphere. We have already
proved that we can in this way fill the part of the hyper-
sphere about a point, and so any " restricted " portion of
the hypersphere, with tetrahedrons. What we have to
show now is that a certain number of these tetrahedrons
will fill the entire hypersphere without overlapping.
* Thus in space of five dimensions there are only three possible types of regular
(convex) figures:
the simplex, corresponding to the tetrahedron and pentahedroid,
the orthogonal, corresponding to the cube and hypercube, and
the figure reciprocal to the latter, constructed on a set of mutually perpendicular
diagonals and corresponding to the octahedron and i6-hedroid.
318 THE 600-HEDROID AND I2O-HEDROID [vm. iv.
The hypersphere is a three-dimensional space (Art. 122),
and by fixing our attention at any one time upon a suffi-
ciently small portion of it we can carry on our processes
as if it were the space of our experience. We shall use the
language of the ordinary three-dimensional geometry and
say "line" and "plane" instead of "great circle" and
" great sphere. "
We start with a regular ikosahedron made up of twenty
tetrahedrons (Plate I, fig. i). Let A be the centre and B
any one of the vertices. The radii A B, as
well as the edges BB, are edges of the
tetrahedrons. About each edge there are
five tetrahedrons.
Two adjacent tetrahedrons form a double
triangular pyramid,* the common face of
the two tetrahedrons being the interior of what we may
call a cross section of the double pyramid.
Five tetrahedrons about an edge
form a double pentagonal pyramid *
with this edge as axis. The cross
section is a pentagon whose sides
and vertices are edges and vertices
of the tetrahedrons.
Let a denote any one of the component tetrahedrons of
the ikosahedron. We have
20 tetrahedrons a,
i vertex A and 12 vertices J5,
12 edges AB " 30 edges BB,
30 faces ABB" 20 faces BBS.
* The term double pyramid is not used here in the technical sense defined in
Art. 32. However, this may be regarded as a limiting case, obtained by rotating
the end-pyramids around the plane of their common base until they come into a
hyperplane with their vertices on opposite sides of this plane.
i8o] 600-HEDROID 319
To each face BBB attach a tetrahedron /3 (Fig. 2). The
a and /3 which have this face in common form a double
triangular pyramid. We have a new vertex C, three new
edges BC, and three new faces BBC. We have added
20 tetrahedrons j8,
20 vertices C,
60 edges J3C,
and 60 faces BBC.
Along an edge BB we have now two tetrahedrons a and
two tetrahedrons 0. Along such an edge, therefore, we
can put one more tetrahedron 7 (fig. 3). This gives us a
new edge CC and two new faces BCC. The vertices C
and the edges CC are the vertices and sides of a set of
pentagons like those on the original ikosahedron. The
pentagons of the ikosahedron, however, overlap, while
these form a regular dodekahedron. We have added
30 tetrahedrons 7,
30 edges CC,
and 60 faces BCC.
Along an edge BC we have now one tetrahedron /3 and
two tetrahedrons 7. At a vertex B we have five tetra-
hedrons a, five tetrahedrons /?, and five tetrahedrons 7.
We can fill the space about B by inserting a double pentag-
onal pyramid made up of five new tetrahedrons 5 (fig. 4).
We have a new vertex D. The axis forms an edge BD, a
continuation of AB y and there are five new edges CD.
There are five faces BCD between the tetrahedrons of
the double pyramid, and five faces CCD coming to the
point D. We have added
60 tetrahedrons 5,
12 vertices D,
12 edges BD and 60 edges CD,
60 faces BCD " 60 faces CCD.
320 THE 600-HEDROID AND I2O-HEDROID [vra. IV.
Along CC we have one tetrahedron 7 and two tetrahe-
drons 5. Along this edge there is room, therefore, for two
more tetrahedrons * forming a double triangular pyramid
(fig. 5). This gives us a new vertex E, two new edges
CJE, and two new edges DE. There is one new face CCE
between the two tetrahedrons, and there are four new faces
CDE coming to the point E. We have added now
60 tetrahedrons e,
30 vertices E,
60 edges CE and 60 edges DE,
30 faces CCE " 120 faces CDE.
Along CD we have two tetrahedrons 8 and two tetra-
hedrons . Along CD there is room, therefore, for one
more tetrahedron , with one new edge EE and two new
faces CEE and DEE (Fig. 6). The vertices E and the
edges EE are the vertices and sides of a third set of pen-
tagons like the two sets already mentioned. These pen-
tagons, however, touch only at their vertices, and are
separated by triangles. We have added
60 tetrahedrons f ,
60 edges EE,
60 faces CEE and 60 faces DEE.
Along each edge CE we have two tetrahedrons and two
tetrahedrons ?, and along each edge DE we have one
tetrahedron e and two tetrahedrons f .
Now the pentagons and triangles just mentioned lie
all in one plane.* If at C we insert a tetrahedron rj having
the interior of the triangle EEE for face, and at D the half
of a double pentagonal pyramid formed by taking a half
of each of five tetrahedrons 0, with the interior of the
* We say plane instead of great sphere, as explained at the beginning of this
article.
i8o, i8ij 600-HEDROID 321
pentagon for base, we shall have along the edge EE one
tetrahedron ?, one tetrahedron 17, and a half of a tetrahedron
0, forming dihedral angles whose sum is just equal to two
right dihedral angles. Therefore it is not necessary to
continue our process. We have a plane completely filled
with triangles and pentagons and their interiors, and the
half of space on one side of this plane completely filled with
the tetrahedrons which we have taken, and their interiors.
If we continued our process we should have the same
figure on the other side of the plane, and the two together
would completely fill the elliptic space.
There are
20 tetrahedrons rj and 60 tetrahedrons 0.
The total number of tetrahedrons in the entire figure will
be the number of each of the kinds a, /?, . . . 77 counted
twice, and the number of the tetrahedrons counted once.
That is, it will be
2(20 + 20 + 30 + 60 + 60 + 60 + 20) 4- 60 = 600.
This construction in the hypersphere of 600 equal regular
tetrahedrons, which, with their interiors, fill the hyper-
sphere, determines in space of four dimensions a regular
polyhedroid containing 600 equal regular tetrahedrons.
Its name is hexakosioihedroid, or 6oo-hedroid.
We can count the number of vertices, edges, and faces,
but from any one of these numbers the other three can be
computed directly. In fact, the " N" of Art. 177 is 120,
and the polyhedroid formula of this polyhedroid is
i 120 + 720 1200 + 600 1=0.
181. Completion of the 600-hedroid. Although it is
not necessary to do so, we shall complete the figure by the
process employed for the first half. The different parts,
322 THE 600-HEDROID AND I2O-HEDROID [vili. iv.
as we come to them, will correspond to the parts already
formed for the first half, and we shall denote them by the
same letters with accents.
After adding the tetrahedrons f we had a figure (Fig. 6)
on which were triangular cavities at the points C and pen-
tagonal cavities at the points D. In the former we insert
double triangular pyramids each made up of two tetra-
hedrons 17 and 77', and at each point D we put a double
pentagonal pyramid formed by taking five tetrahedrons
(Plate II, fig. 7). The double triangular pyramid gives
us a new vertex C", three new edges EC', a new face EEE
separating the two tetrahedrons, and three faces EEC'
coming to C'. The double pentagonal pyramid gives us a
new vertex ZX, an axis edge DD' in a line with A B and BD y
and five new edges ED' coming to D f . It gives us five
faces DED', common faces of successive tetrahedrons
around the axis, and five faces RED 1 coming to D'. Corre-
sponding to themselves in the two halves of the figure are
the tetrahedrons 0, the vertices E, the edges EE and Z)ZX,
and the faces EEE and DED'.
Along EE there is room for one tetrahedron ', with a
new edge D'C and two new faces ED'C (fig. 8).
At the point E we can still insert a double triangular
pyramid (fig. 9), corresponding to the one which first
produced the vertex E (fig. 5). This is formed of tetra-
hedrons e', giving us the new edge C'C', a dividing face
EC'C, and two new faces D f C f C f . The points C" are the
vertices of twelve pentagons forming a regular dodekahe-
dron, outside of which our figure lies.
At the point D f we insert a double pentagonal pyramid
(fig. 10), corresponding to that which first produced the
vertex D. This is composed of five tetrahedrons 8'. We
have a new vertex B', a new edge, the axis D'B', and five
new edges C'B'. We have five new faces D'C'B', common
i8i, 182] THE SEVENTY-TWO DEKAGONS 323
faces of successive tetrahedrons around the axis, and five
new faces C'C'B' coming to the point B r .
Along the edge C'C" we have two tetrahedrons ' and
two tetrahedrons 5'. We have room, here, then, for one
more tetrahedron 7', giving us one new edge B'B' and two
new faces C'B'B' (fig. n).
At the point C' we can still insert one tetrahedron /3',
with the face B'B'B' (fig. 12).
Our figure now encloses a regular ikosahedron like that
with which we started. It contains twenty tetrahedrons
a', with one new vertex A 1 , twelve new edges B'A', and
thirty new faces B f B' A' (fig. 13).
On the following page is a table of all the parts of the
regular 6oo-hedroid.
182. The seventy-two dekagons in a 600-hedroid.
The angle subtended at the centre by an edge. We have
noticed that the edges AB, BD, and DD' lie along the
same line. From A to A ' one of these lines contains five
edges, and the entire line must contain ten edges ; the ten
edges and their extremities make up the entire line. In the
complete figure there are seventy-two such lines, each line
running along ten of the 720 edges.
In the hypersphere there are seventy- two great circles;
and in hyperspace there are seventy-two planes through
the centre of the 6oo-hedroid, each intersecting the poly-
hedroid in a regular dekagon whose sides are all edges of
the polyhedroid. An edge of the 6oo-hedroid, therefore,
subtends an angle of 36 at the centre ; and a set of regu-
lar pentahedroids, 600 at a point, having as they do face
angles of 60, with their interiors would more than fill the
part of Euclidean hyperspace about the point, so that
such a set must belong to Hyperbolic Geometry (see
Art. 179).
3 2 4
THE 600-HEDROID AND I2O-HEDROID [vin. iv.
TETRAHEDRONS
VERTICES
EDGES
FACES
<* 2O
;1 I
AB 12
.455 30
'B 12
BB 30
EBB 20
ft 2O
C 20
5C 60
&BC 60
y 30
CC 30
5CC 60
5 60
Z> 12
BD 12
BCD 60
45
CD 60
CCZ> 60
2
QO
60
30
CE 60
CCE 30
Z> 60
CDJS 120
324
2
648
r 60
E 60
CEE 60
DEE 60
560
2
1 1 2O
V 20
270
EEE 20
2
540
60
Z>Z>' 12
DEZX 60
600
1 2O
720
1 200
183. Construction of the 120-hedroid. In the manner
in which we have constructed the regular 6oo-hedroid
we can construct a regular polyhedroid with four dodeka-
hedrons at a point.
Starting with a dodekahedron a, we attach a dodeka-
hedron /8 to each face (Plate III, fig. 14). About any
vertex A of a we have three new dodekahedrons, with a
common edge which extends outwards from A to a
vertex B. The face common to two adjacent /3's has for
its vertices two -4's, two 5's, and a vertex C, the highest
PLATE II.
D' c'
FIG. 7.
FIG. 8.
FIG. 9.
FIG. 10.
FIG. ii.
B ;
B'
FIG. 12.
FIG. 13.
182, 183] I20-HEDROID 325
point of this face. Each dodekahedron /3 has at the top
a face DD. . . .
At B we have now a triangular cavity in which we can
insert a dodekahedron 7 (Fig. 15). Each 7 is attached to
three others. The face common to two adjacent 7*5 has
for vertices one C, two Z?'s, and two new vertices E. The
two 7*s form a figure with a neck across which this face
cuts. Each y has three new vertices F, and at the very
top one new vertex G.
The outer face of the dodekahedron j8 now becomes the
base of a pentagonal cavity in which we can insert a dodeka-
hedron 8 (Fig. 1 6). Each 8 has at the top a face with five
new vertices H. Two S's resting above two adjacent faces
of the original dodekahedron a are separated by the neck
joining two adjacent 7*5. The upper edge of this neck,
EE, is now at the base of a cavity in which we can insert a
dodekahedron e edgewise.
If instead of a dodekahedron c we take a half, cut off by
the plane of two opposite edges, we shall have along the
edge H H one dodekahedron 5 and a half of a dodekahedron
e, forming dihedral angles whose sum is just equal to two
right Dihedral angles (Fig. 17). The section of the e will
lie in a plane with the upper face of the 5, and the half of
space on one side of this plane will be filled.* The plane
contains a set of regular pentagons, and a set of hexagons
with two sides equal to the sides of a pentagon and four
sides equal to the altitudes of a pentagon, the pentagons
and hexagons with their interiors filling the plane.
The highest point of a dodekahedron 7 does not appear
in this plane, being at a distance equal to one-half of
the edge beneath the point common to three adjacent
hexagons.
We have one dodekahedron a, twelve j8's, twenty 7 J s,
* See foot-note, p. 320.
326
THE 600-HEDROID AND I2O-HEDROID [vm. w.
and twelve 6's, making forty-five besides the half-dodeka-
hedrons e, of which there are thirty. Thus the total
number in the entire figure will be
2 X 45 + 3 = I2 -
The name of the figure is hekatonikosahedroid or i2o-hedroid y
and its polyhedroid formula is
i 600 + 1200 720 + 120 i
o.
Below is a table for the i2o-hedroid, corresponding to
the table on p. 324.
DODEKAHEDRONS
VERTICES
EDGES
FACES
I
A
20
AA
30
AA
. . . 12
ft 12
B
20
AB
20
AA
. .C 30
C
3<>
BC
60
B .
DD 60
D
60
CD
60
DD
. . . 12
DD
60
y 20
E
60
DE
60
C . .
EE 30
F
60
EE
30
DD
. .F 60
G
20
EF
1 20
EE
..G 60
270
FG
60
2
540
a 12
H
60
Ftt
60
E..
EE 60
45
560
324
2
2
2
QO
1 1 20
648
EE
60
EH
. . . 12
30
GG 1
20
G. .
G' 60
1 2O
600
1 200
720
FIG. 14.
PLATE III.
FIG. 15.
FIG. 17.
FIG. 16.
TECHNICAL TERMS
In this list are some of the terms of four-dimensional geometry not used in the
text and of it-dimensional geometry, also terms equivalent to some that are used,
and the principal abbreviations. In most cases a reference is added. For terms
used or explained in the text, see Index.
The following are the authors most frequently mentioned, many of the references
being given in full in the preceding pages: Cayley, Math. Papers (p. 5) ; Clifford,
Math. Papers (p. 5) ; Cole (p. 142) ; Dehn (p. 288) ; Enriques, Encyclopedic, vol.
IIIi (p- IS) ; Jouffret (p. 9) ; Loria (p. 9) ; Pascal, Repertorium der hbheren Mathe-
matik, Ger. trans, by Schepp, vol. II, Leipzig, 1902 ; Poincare", Proc. London Math.
So., vol. 32 (p. 12); Riemann (p. 6); Schlafli (p. 22); Schoute (p. 9); Stringham
(p. 289); Sylvester, 1851 and 1863 (p. 5); Veronese, Grundzuge, etc. (p. 9); Wil-
son and Lewis (p. 12).
The numbers refer always to pages.
Achtzell, Z 8 , regulare, hypercube;
netz; Schoute, II, 202, 242.
See Zell.
Allomorph, allomorphic, two poly-
hedrons having the same number
of vertices and edges, and the
same number of faces of each
kind that they have; similarly
of polyhedroids; Schoute, II,
22-23. See Isomorph.
Ankugel, Ankugelraum, Anradius, see
Kugel.
Apothema (of a hypercone), slant
height, Schoute, II, 302.
Arele, edge, Jouffret, 96.
Axe (of a piano-polyhedral angle),
vertex-edge, see Kant.
Basis, base, of a pyramid, hyperpyra-
mid, etc., Schoute, II, 35 ;
-raum, Schoute, II, 242 ; used also
of the base of a linear system of
spaces (e.g., the line common to
the linear system formed from the
equations of three hyperplanes in
RJ, Schoute, I, 141.
Bildraum, the space of the figure of
descriptive geometry, the space
in which all the different projec-
tions are placed together, Schoute,
I, 88, 124.
Bipiano (Ital.), #- 2 in R n , Pascal,
577-
C 5 , C 8 , C 16 , C 24 , C 120 , C 600 , the six
regular polyhedroids, Jouffret, 103.
Case, cell; hypercase, corresponding
term in space of five dimensions;
Poincare", 278. See Jouffret, 96,
103-
Cell, case, Grenzraum, Seitenraum,
Zell; (of a hyperplane angle)
Schenkelraum.
-cell, -hedroid, Maschke, Am. Jour.
Math., vol. 1 8, 181.
Configuration, Cay ley- Veronese (p. 5),
Carver, Trans. Am. Math. So., vol.
6, 534-
Confine, poly hedroid of n dimensions;
face of a , (n i)-boundary;
prime , simplex; rectangular
prime , with edges at one vertex
equal and perpendicular to one
another; Clifford, 603.
Cylinderraum, (Cy), of k dimen-
sions, hypercylinder ; spharisch,
with spherical bases; zweiter
Stufe, with cylinders for bases (Art,
327
3*8
TECHNICAL TERMS
147) ; 5-ter Stufe (k + $)-ter
Dimension, Cy[(Fo)i> RV(s)], what
a prism of this kind becomes when
the bases, (P0)fc are no longer
entirely linear (see Prisma)
Schoute, II, 293. Kreiscylinder
piano-cylindrical hypersurface with
right directing-circle (Art. 148)
Veronese, 557.
Decke, vertex-face, analogous to ver-
tex-edge, Schoute, II, 4.
Demi-, half-; espace, half -hyper-
plane; Jouffret, 60.
Difcdre d'espaces, hyperplane angle,
Jouffret, 60.
Ditheme, surface, see Theme.
Differentielle, g&>m6trie m6trique,
restricted geometry, see Restreint.
Dreikant, trihedral angle, see Kant.
Droite-sommet, vertex-edge, Jouffret,
92.
Eben, flat; Ebene, plane; used by
Pascal for an (n i) -dimensional
flat, E*-i, the same as R n -i, 577;
Dreieben, vierdimensionale Eben-
tripel, piano-trihedral angle, Schoute,
II, 8, 4.
Eigentlich, proper, not at infinity, see
UneigenUich, Schoute, I, 20.
Entendue, hyperspace (of four dimen-
sions), Jouffret, i.
Entgegengesetzte Punkte, opposite
points (of the Double Elliptic
Geometry), see Gegen-Punkte.
Espace, hyperplane, Jouffret, 2.
Face: a deux dimensions, face
angle (of a tetrahedroidal angle),
half-plane (of a plano-trihedr.al
angle); a trois dimensions,
dihedral angle; a quatre dimen-
sions, hyperplane angle; Jouffret,
62-63.
First, vertex-edge, Schoute, II, 4.
Flat noun and adjective), linear,
homaloid (or omaloid), eben, flach.
Fluchtpunkt (of a line), the point at
infinity (the "vanishing point" of
perspective), Schoute, I, 2 ; Fiucht-
raum (of R n ), Schoute, I, 124.
Fold: two-fold, w-fold, applied to
figures, angles, boundaries, spheres,
etc., to indicate the number of
their dimensions, Stringham; see
Manifold. A k-iold relation in
space of m dimensions gives an
(m k) -dimensional locus, Cay-
ley, VI, 458; seeOwo/.
Ftinfzell, Z 6 , pentahedroid, Schoute,
11,4; seeZell.
g^, line at infinity, Schoute, I, 21.
Gegen-Punkte or entgegengesetzte
Punkte, opposite points (as in the
Double Elliptic Geometry, see
footnote, p. 215), Veronese, 237.
Gegenuber, opposite (as in a tri-
hedral angle each edge is opposite
the face which contains the other
two), Veronese, 449; see also
Schoute, I, 268.
Gemischt, having both proper and
improper points; gemischtes Sim-
plex, S g (d) ; Schoute, I, 29.
Gerade, right, Schoute, II, 108, 293;
schief when not gerade.
Gleichwinklige Ebene, isocline planes,
Veronese, 539.
Grad von Parallelismus, Orthogonal-
itatsgrad, see Parallel, Orthogonal.
Grenzraum, cell, Schoute, I, 10;
Grenztetraeder, Schoute, II, 218.
Half-, semi-, demi-, halb-.
He'catonicosae'drolde, C 120 , i2O-he-
droid, Jouffret, 105, 169.
-hedroid, Stringham.
HexacosildroTde, C 600 , 6oo-hedroid,
Jouffret, 105, 169.
Hexad6cae*drolde, C 16 , i6-hedroid t
Jouffret, 105, 128.
Homaloid, flat, represented by an
equation or by equations of the
first degree, Sylvester, 1851 ; writ-
ten also omaloid; see Theme.
uf, polyhedron in which two faces
TECHNICAL TERMS
329
are polygons of the same num-
ber of sides with one side in com-
mon (Ferse) and the remaining
sides of one connected with the
remaining sides of the other by
two triangles and by quadri-
laterals; in R n a polyhedroid
formed in a similar way. Partic-
ular cases are the Prismenkeil
and the Pyramidenkeil. Schoute,
II, 26, 41, 43.
Hundertzwanzigzell, Zmo, i2O-he-
droid, Schoute, II, 213.
Hyper-: hyperlocus, Sylvester, 1851,
8, 12; plane, planar,
pyramid, pyramidal, geome-
try, theory, ontological,
Sylvester, 1863, 172-177; Ital.
iper-.
Hypercone, hyper cone de premiere
espece, Kegelranm.
Hypercone de premiere espfcce, hy-
per cone; de seconde espece, double
cone; Jouffret, 92.
Hypercube, tessaract, octaedroide,
Achtzell, Masspolytop, Oktaschem
(see -schem).
Hypercylinder, Cylinderraum.
Hyperebene, -R w _i (or _!, Pascal,
577).
Hyperparallelopiped, paralUlepipede
d quatre dimensions, Parallelotop,
Paralleloschem (see -schem).
Hyperplane, lineoid, quasi-plane, es-
pace, plan, Hyperebene, Raum.
Hyperplane angle, diedre d'espaces,
Raumwinkel.
Hyperprism, Prisma.
Hyperpyramid, Pyramide.
Hypersolid, Confine, Polytop.
Hyperspace, 4-space, I'entendue, Hy-
perraum (Pascal, 577).
Hypersphere, quasi-sphere, Kugel-
raum, n-Sphdre, Polysphare.
(Hyper) 1 *"" 8 surface, of p i dimen-
sions in space of p dimensions;
e.g., in space of five dimensions,
hyper-hyper-surface. H. R. Greer,
"Question 2503," Math. Questions
from the Educational Times, vol.
10 : 100.
Icosat6tra6droide, C 24 , 24-hedroid,
Jouffret, 105, 137.
Ideal, improper, uneigentlich.
Inkugel, see Kugel.
Ineunt points, the points of a locus,
Cayley, VI, 469. In the same way
he uses the expression, "tangent
omals of an envelope."
Inhalt, volume, hypervolume, etc.,
Schoute, II, 94.
Iper- (Ital.), hyper-, Loria, 302.
Isocline planes, plans d'angles igaux,
plans a une infinite d' angles,
gleichwinklige Ebene.
Isomorph, isomorphic, allomorphic
polyhedrons or polyhedroids are
isomorphic when faces or cells
which come together in one always
correspond to faces or cells which
come together in the other.
Schoute, II, 22-23. See Allo-
morph.
Isomorphic geometries are different
interpretations of the same ab-
stract geometry (see p. 15).
KI, Kugelraum, see this word.
Kant, edge, Schoute, I, 9; Dreikant,
trihedral angle, Schoute, I, 271 ;
Vierkant, tetrahedroidal angle,
Schoute, I, 267 ; w-Kant, Vielkant,
Schoute, I, 279, 286; Scheitelkant,
vertex-edge, Schoute, I, 268; Drei-
kant zweiter Art, piano-trihedral
angle, Axe, its vertex-edge, Veronese,
540, 544; regulare w-Kant ^-ter
Art, Schoute, II, 140.
Kantenwinkel (of a Vierkant), face
angle, Schoute, I, 268.
Kegel : Kreiskegel erster Art, piano-
conical hypersurface of revolution;
zweiter Art, conical hypersurface of
double revolution (Art. 112), Vero-
nese, 557-
Kegelraum, (Ke)i, of k dimensions,
hypercone; zweiter Stufe, double
330
TECHNICAL TERMS
cone, j-ter Stufe (k + $)-ter Dimen-
sion, Ke[(Po)i, S(s)], what a
pyramid of this kind becomes when
the base, (Po)t, is no longer en-
tirely linear (see Pyramide) ; the
Kegelraum (of any kind) is sphar-
isch, if the base is spherical, gerade
if also the centre of the base is
the projection of the entire vertex-
simplex upon the space of the base,
regulare if, further, the vertex-
simplex is regular and its centre
is the projection upon its space of
the centre of the base, schief if
not gerade; Schoute, II, 292-293.!
Keil, dihedral angle, Veronese, 444;!
hyperplane angle (Keil von vierj
Dimensionen), Veronese, 544 ;!
piano- polyhedral angle, Dehn, 571.}
Prismenkeil, Pyramidenkeil, see
these words.
Kiste, (Ki)ij of k dimensions, rec-
tangular par allelo piped or hyper-
parallelo piped, Schoute, II, 94.
Kontinuum, the aggregate of all
solutions of an equation or of a
system of equations, locus or
spread, Schlafli, 6.
Kreiscylinder, Kreiskegel, see Cylin-
der, Kegel.
Kreuzen, used of lines not parallel
and not intersecting, also of other
spaces ; often used with senkrecht,
Schoute, I, 43-44-
Kugel, hyperspherc (of any number of
dimensions), Veronese, 592. With
Veronese Kugel denotes the in-
terior, Kugeloberflache the hyper-
surface; see Kugelraum; -netz,
net of regular polygons on a
sphere, Schoute, II, 154; Ankugel,
sphere tangent to the edges of a
regular polyhedron, Inkugel, in-
scribed sphere, Umkugel, circum-
scribed sphere; Ankugelraum, An-
radius, etc.; Schoute, II, 151, 199,
245-
Kugelraum, K n -.\. or K n , hyper sphere,
Schoute, I, 127. With Schoute
Kugelraum is the hypersurface,
not the interior. In space of n
dimensions it is of n i dimen-
sions, and he writes K n ,\, but later
he writes K n , the subscript denot-
ing the number of dimensions of
the space. See foot-note, II, 263.
Leit-, directing-; strahl, kurve,
raum; Schoute, I, 98, 117, 199,
etc.
Linear, flat, spaces as denned in
Art. 2, represented by equations of
the first degree.
Lineoid, hyperplane, Cole, 192; col-
lineoidal, Keyser, Bull. Am. Math.
So., vol. 9 : 86.
Losung, point, used by Schlafli to
denote a set of values of the vari-
ables satisfying given equations,
and then for any set of values,
like Cauchy's "analytical point"
(p. 6), Schlafli, 6.
Lot, perpendicular line, Schoute, I,
44; Letvielcck, simplex with all
angles right angles (in Elliptic
Geometry), Schoute, I, 47.
Manifold, space of any number of
dimensions, variety, Mannigfaltig-
keit, Grassmann, Riemann, and
others.
Mantel, lateral boundary, of a hyper-
pyramid, etc., Schoute, II, 35.
Masspolytop, hypercube, Schoute,
n, 93-
Monotheme, line, Sylvester; sec
Theme.
Netz, Achtzellnetz, etc., Schoute,
II, 242.
Normal, perpendicular; (of planes)
absolutely perpendicular; stereo-
metrisch normale Ebenen, per-
pendicular in a hyperplane;
Schoute, I, 70-72 ; see Senkrecht.
Oberflache, Inhalt of the boundary,
Schoute, II, 95, 263.
TECHNICAL TERMS
331
Octa6drolde, C 8 , hypercube, Jouffrct,
83, 118.
Omal or omaloid (both noun and
adjective), line, plane, etc., linear,
the same as homaloid. If the
relation is linear or omal, the locus
is a fc-fold or (m k) -dimensional
(in space of m dimensions) omaloid.
Omaloid is used absolutely to de-
note the onefold or (m i)-dimen-
sional omaloid, Cayley, VI, 463.
Opera tionsraum, R n , the space in
which all the figures considered
are supposed to lie, as if there
were no space of higher dimensions
(see Art. 26), Schoute, I, 4.
Opposite, gegenuber, entgegengesetzte.
Order, Richtung, Sinne.
Orthogonal, teilweise, Orthogonali-
tatsgrad, Schoute, I, 49; see
Senkrecht.
Orthogonal figure, cube, hypercube,
etc., Stringham, 5.
I?ao> point at infinity, Schoute, I, 21.
Parallel: planes are parallel "von
der ersten Art" if they have the
same line at infinity, "von der
zweiten Art" if they have only a
point in common and this point
is at infinity, Veronese, 516;
planes are "paralleles suivant le
premier mode ou incompletement
paralleles" if their lines at infinity
have a single point in common,
"paralleles suivant le deuxieme
mode ou completement paralleles "
if their lines at infinity coincide,
Jouffret, 31 (we should notice
that these two writers use "first"
and "second" in opposite senses);
teilweise parallel or halb parallel
are terms used by Schoute; in a
space of more than four dimensions
lower spaces may be "ein Viertel,
zwei Viertel or drei Viertel parallel,"
etc., Schoute, I, 34.
Parallel6pipfcde a quatre dimensions,
Jouffret, 82.
Paralleloschem, hyperparallelopiped,
see -schem.
Parallelotop, (Pa)*, of k dimensions,
hyperparallelopiped, Schoute, II,
39-
Pentae"drolde, C 6 , penlahedroid, Jouf-
fret, 105, 132.
Perpendicular, absolutely, simply,
Cole, 195, 198; simplement ou in-
complStement perpendiculaires,
absolument ou completement ,
Jouffret, 34; see Normal, orthog-
onal, senkrecht.
Plagioschem, spherical simplex, see
-schem.
Plan (Fr.), plane; used by Cayley
in 1846 for hyperplane, demi-plan
for ordinary plane (I, 321), see
also plane; plans d'angles 6gaux
ou plans a une infinite" d'angles,
isocline planes, Jouffret, 77.
Plane, used by Cayley in five-di-
mensional geometry for a space
of four dimensions: "In five-
dimensional geometry we have:
space, surface, subsurface, super-
curve, curve, and point-system ac-
cording as we have between the six
coordinates, o, i, 2, 3, 4, or 5 equa-
tions : and so when the equations
are linear, we have: space, plane,
subplane, superline, line, and
point," IX, 79.
Piano-trihedral angle, triedre de sec-
onde espece, Dreieben (see Eben).
Planoid, hyperplane, Wilson and
Lewis, 446.
Point: r point (Fr.), the space of
r i dimensions determined by r
independent points (Art. 2); bi-
point, line, d'Ovidio, Math. An-
nalen, vol. 12, 403-404.
Polvierkant, the edges perpendicular
to the cells of a given Vierkant
(analogous to supplementary tri-
hedral angle), Schoute, I, 268.
(Po)*, of k dimensions, Poly top; when
the boundaries are not all linear
CP0)* is used, Schoute, II, 28, 292.
332
TECHNICAL TERMS
Polyschem, polyhedroid, see -schem
Polysphare, see Sphere of n dimen-
sions.
Polytop, (P0)t, a limited portion of
any space, the boundaries (Po)t
it is generally understood that
the boundaries are linear, then it
is a polyhedroid; Simplexpolytop,
one bounded by simplexes, such
in R* is the Tetraederpolytop
or Vierflachzell ; Schoute, II, i,
28.
Prisma, (Pr)i, of k dimensions,
prism, hyper prism, etc.;
zweiter Stufe, with prisms for
bases (Art. 136); s-ter Stufe
(k + *)-ter Dimension, Pr[(Po)i,
RV(s)], the bases parallel (Po)*,
the lateral elements parallel Rs,
Schoute, II, 37-39; pyramidales
Prisma, hyperprism with pyramids
for bases, Schoute, II, 41.
Prismenkeil, the Huf formed when a
hyperprism is cut into two parts
by a hyperplane (or by an Rn
which intersects a base, Schoute,
II, 43-
Prismoid, w-dimensional polyhedroid
bounded by two (Po) n -\ in parallel
/Zn-i, an< * a Mantel of simplexes,
S(n), Schoute, II, 44.
Proper, see EigenUich.
Punktwert of a space, the number
of independent points in it (Art.
2) ; for R# it is d -f i ; Schoute,
1,12.
Pyramide, (Py)i, of k dimensions,
pyramid, hyperpyramid, etc.;
zweiter Stufe, double pyramid;
5-ter Stufe (k + s)-ter Dimension,
Py((Po) t , S(s)], with a polyhe-
droid, (Po)t, for base and a vertex-
simplex, S(s); Schoute, II, 35-
36.
Pyramidenkeil, the Huf formed when
a hyperpyramid is cut into two
parts by a hyperplane (or by an
R n -\) which intersects the base,
Schoute, II, 41.
Quadri&dre a quatre dimensions,
tetra&iroide, tetrahedroidal angle;
it has six "faces a deux dimen-
sions," four "faces a trois dimen-
sions," and four "tri&ires de
seconde espce," Jouffret, 63.
Quasi-plane, /? n _i in /?n, Sylvester,
1863, 173; quasisphere, sphere of
any number of dimensions, Clif-
ford, 604, 605.
R n , linear space of n dimensions ; in
Schoute R n is usually the Opera-
tionsraum, R& any space in R n ,
Rt+i * R<t, but R w is used for
R w , "w" referring to the Punkt-
wert ; Schoute, 1, 4, 13.
RV(s), the lateral elements, R 9 , of a
prism or cylinder "j-ter Stufe,"
Schoute, II, 39.
Raum, space; used alone for #3,
Schoute, I, i ; Basis-, Bild-,
Cylinder-, Kegel-, Kugel-, Seiten-,
etc., see these words.
Raumwinkel, hyperplane angle,
Schenkelraume, its cells, Schoute,
I, 268.
Region limite'e, restricted region.
Restreint (Fr.), restricted, region
limite'e, gtomitrie mitrique dif-
fer entielle, Enrique, 48, 112.
Richtung, Sinne, order, Veronese,
381, 456, 498, 550.
Rotation, simple, double, Cole, 201,
209 ; Rotation von den (p i)-ten
Potenz, when each point describes
a K 9 (see Art. 116, Th. i, and Art.
147, Th. 2), Schoute, II, 297.
SQ, Si, ^2, etc., point, line, plane, etc.
(Stern), Veronese, 509.
S(d), simplex with d vertices, Schoute,
I, 10 ; S u (d), improper simplex,
S0(d), gemischtes Simplex, Schoute,
I, 28, 29.
St(P, q)> Simplotop.
Scheitel, vertex, Scheitelkant, vertex-
edge, Schoute, I, 268.
Schein, contour of a figure as seen
TECHNICAL TERMS
333
from an outside point, Schoute,
II, 17-
-schem, -hedroid, used by Schlafli
with qualifying prefixes to denote
various polyhedroids : Polyschem,
19 ; paralleloschem, hyper parallele-
piped, 12; in space of four dimen-
sions the six regular polyhedroids
are, Pentaschem, Oktaschem, Hek-
kaidekaschem, Eikositetraschem,
Hekatonkaieikosaschem, and Hexa-
kosioschem, 46-52; a spharisches
Polyschem is a polyhedroid be-
longing to the geometry of the n-
sphere; it is a Plagioschem when
it has n boundaries, the same as
simplex; the single parts of the
boundary of a spharisches Poly-
schem are Perrischeme, 58.
Schenkel (of an angle) side, Schen-
kelraum (of a hyperplane angle)
cell, Schoute, I, 268.
Schief, oblique, Schoute, II, 293.
Sechshundertzell, Zm> 6oo-hedroid;
Schoute, II, 213.
Sechszehnzell, Zi$, t6-hedroid; Sechs-
zehnzellnetz, Schoute, II, 202, 242.
Seiten (of a poiyhedroidal angle),
ebenen , face angles, von drei
Dimensionen, trihedral angles, Ve-
ronese, 544; (of a pentahedroid)
faces and cells, Veronese, 547.
Seitenraum, the # n _i of a simplex
S(n -f i), Schoute, I, 142.
Semi-parallel, half -parallel*
Senkrecht, perpendicular; kreu-
zen, schneiden ; zugeordnet,
polar, used of P^ ; halb , in R\
used of planes with one angle a
right angle (see Art. 69) ; in space
of more than four dimensions
lower spaces may be "ein Viertel,
zwei Viertel, or drei Viertel ,"
etc., Schoute, I, 40-49. Veronese
uses senkrecht or "senkrecht von
der ersten Art" for planes abso-
lutely perpendicular in 4, and
" senkrecht von der zweiren Art"
for planes half -perpendicular, 521.
Sense, order, Sime.
Simplex, triangle, tetrahedron, and in
general a polyhedroid whose ver-
tices are all independent (Art. 2).
S(d -f i) the simplex in Ra with
d + i vertices, Schoute, I, 9-10,
Simplicissimum, prime confine.
Simplex, spharisch, spherical tetra-
hedron, etc. Schoute, II, 291.
Simplexpolytop, bounded by sim-
plexes, Schoute, II, 28.
Simplicissimum, simplex, Sylvester,
"Question 8242,*' Math. Questions
from Educational Times, 47 : 53.
Simplotop, St(p, q), obtained by
forming two simplexes S(p + i)
and S(q -f- i) with one vertex
in common from a set of p -f q + i
independent points (and so lying
in a space Rp+q), and then letting
each move parallel to itself over
the other. A particular case is
the doubly triangular prism (see
Art. 143), Schoute, II, 44.
Sinne, sense, order, see Richtung.
Situs (Lat.), used by Gauss for direc-
tion of a plane, like Stellung,
" Disquisitioncs generates circa
superficies curvas," Werke, IV,
219.
Space, 3-space, 4-space, etc., for /?a,
/?4, etc. ; the word space is used by
Cayley for the highest space con-
sidered, like Schoute 's Opera-
tionsraum, see Plane.
Sphere of n dimensions, hypersphere,
n-sphere; called by Schlafli Poly-
sphare or n-sphare, Disphare, cir-
cle, Trisphare, sphere, 58.
Spharisch Cylinderraum, Kegelraum,
Simplex, see these terms.
Spitze, vertex (point, line, or higher
space), Veronese, 466, 557, 606.
Spread, surface, hypersurface, repre-
sented by an equation or by a
system of equations, used with a
number to denote the number
of its dimensions as 2-spread,
3-spread, etc.; see articles in
334
TECHNICAL TERMS
Am. Jour. Math, by Carver, vo
31, Sisam, 33, Eisenhart, 34
Eisland, 35.
Stellung, direction of a t plane, space
etc., Schoute, I, 26, 75.
Stern, consists of all points collinea
with the points of a space and a
point outside of the space, used b>
Veronese to define the differen
spaces. erster Art., # 3 > zweiter
Art., /<U, etc., Veronese, 424, 507
Schoute, I, 190-192.
Strahl, line, Haupt , Schoute, II
161 ; Halb , Schoute, I, 85.
Straight, flat; vector, line-vector
plane-vector, etc. ; straight 3-space
Euclidean, Lewis, "Four Dimen-
sional Vector Analysis," Proc. Am,
Acad. Arts and Sci., v. 46 : 166, 1 73,
Stumpf (of an angle), obtuse; (of a
prism), truncated, etc., Schoute, II
43, 127-
Subplane, subsurface, see Plane.
Superline, supercurve, see Plane.
Surface (Cayley), see Plane.
Teilweise parallel, orthogonal or
senkrecht, see these terms.
Tessaract, hypercube, Hinton, The
Fourth Dimension, London, 1904,
159. Tessaract belongs to a ter-
minology in which the name of a
figure designates the number of its
axes: pentact, a figure with five
axes, penta-tessaract, a regular
i6-hedroid, T. Proctor Hall, Am.
Jour. Math., vol. 15: 179.
Tetrahedroidal angle, quadriedre a
quatre dimensions, tetraedro'ide,
Vierkant.
Theme, spread; mono-, line, or
curve, di-, surface, keno-, system of
points, homaloid theme, flat; terms
used by Sylvester in 1851.
Totalitat, hyperspace, Schlafli, 6.
Trifcdre de seconde espfcce, piano-
trihedral angle, it has three "faces
a deux dimensions, 1 ' three "faces
a trois. dimensions," and three
"faces a quatre dimensions," Jouf-
fret, 62.
Ua-i, the infinitely distant (uneigent-
_ lich) part of R d , Schoute, I, 22.
V* - ^*-i (see R n ), Schoute, I, 45.
Umkugel, see Kugel.
Unabhangig, independent (points),
Veronese, 256.
Uneigentlich, improper, ideal, at
infinity; Punkt, Fluchtpunkt;
also unendlich; Schoute, I, 2, 21.
Variety, variett, Enrique, 66 ; Variet&t,
Schoute, I, 209; manifold.
Vertex-edge, droite-sommet, First,
Scheitelkant.
Vierflachzell, polyhedroid in /? 4
bounded by tetrahedrons, Tetrae-
derpolytop, Schoute, II, 28.
Vierkant, vierdimensionale , letra-
hedroidal angle, Schoute, I, 267.
Vierundzwanzigzell, Z 24 , 24-hedroid;
netz; Schoute, II, 203, 242.
Volumeinheit, unit of volume, hyper-
volume, etc., Schoute, II, 95.
Volum, Inhalt, Schoute, I, 156.
, line or curve, Schlafli, 6.
Winkel, angle, Kan ten-, Flachen-,
Raum-, see these words; kor-
perlich von vier Kan ten, tetra-
hedroidal angle, Veronese, 544.
Z 5 , Z 8 , Z 16 , Z 24 , 120, Zeoo, the regular
polyhedroids, Schoute, II, 203, 213.
Zell, cell, polyhedroid, Schoute, II,
196. Schoute uses the term Zell
alone for polyhedroid of four dimen-
sions, but with prefixes (Achtzell,
Fiinfzell, etc.) it seems to refer to
the three-dimensional boundaries.
See Jouffret, 96.
rweiflach, used of a polygon in space
regarded as having two faces,
Schoute, II, 148, 182.
Iweiraum, hyperplane angle, Schoute,
II, 8; used also of a polyhedron
regarded as having two sides in
Rt, Schoute, II, 186.
INDEX
Numbers refer to pages.
Absolutely independent points, 24.
Absolutely perpendicular planes, 81 ;
see Perpendicularity.
Abstract geometry and different
interpretations, 14 ; see Geometries,
examples of diferent kinds.
Alembert, d', on time as the fourth
dimension, 4.
Analogies, their assistance, 17; hy-
perplane angles and dihedral
angles, 99; tetrahedroidal angles
and trihedral angles, 129; piano-
polyhedral angles and polyhedral
angles, 133 ; isocline planes and
parallel lines, 194; piano-pris-
matic hypersurfaces and polygons,
242.
Analytic development of the higher
geometry, 6.
Analytical point, locus, terms used
by Cauchy, 6.
Angle, hyperplane, 95; see Hyper-
plane angle.
Angle of a half-line and a hyperplane,
80; of a half -plane and a hyper-
plane, 104 ; minimum between two
planes, 116.
Angles at infinity, 232-234.
Angles, the two between two planes,
122; the associated rectangular
system, the associated sense of rota-
tion, 181.
Applications of the higher geometries :
to a problem in probability (Clif-
ford) , 5 ; geometries with other ele-
ments, lines in space, spheres, etc.,
10 ; in connection with complex
variables, n; to mechanics, n;
in proofs of theorems in three-di-
mensional geometry, 13.
Aristotle on the three dimensions of
magnitudes, i.
Axes of a hypercube and a i6-hedroid,
240, 292.
Axiom of Pasch, 30 ; of parallels, 221 ;
see Parallels, axiom of.
Axioms of collinearity, restrictions in
Elliptic Geometry, 25.
Axis of a sphere in a hypersphere,
21 1 ; of a regular polyhedroidal
angle, 312.
Axis-element of a double pyramid,
204; of a double cone, 205.
Axis-plane of a rotation, 142, see
Rotation; of a circle in a hyper-
sphere, 211 ; of a piano-cylindrical
hypersurface of revolution, 257.
Axis-planes of a conical hypersurface
of double revolution, 206.
Base, prismatic, of a hyperprism
with tetrahedral ends, 274.
Beginnings of geometry of more than
three dimensions, synthetic, 4;
analytic, 6.
Beltrami, Hyperbolic Geometry on
certain surfaces, 7; kinematics,
13-
Bibliographies, Loria, Sommerville, 9.
B6cher, our use of the term "in-
finity," 230.
Boundaries of a hypersolid are three-
dimensional, 64.
Boundary-hypersurface of Hyperbolic
Geometry, 95, 112.
Cauchy, applied the language of
geometry to analysis, 6.
Cayley, early papers, 5; "Memoir
on Abstract Geometry," 8.
335
336
INDEX
Cell of a half-hyperspace, 62.
Cells of pentahedroid, 57; polyhe-
droid, 63; double pyramid, 67;
hyperplane angle, ' 95 ; poly-
hedroidal angle, 126; piano-poly-
hedral angle, 133; prismoidal hy-
persurface, 235 ; piano-prismatic
hypersurface, 242.
Centroid, 202 ; see Gravity, centre of.
Classes of points constitute figures,
19, 23.
Clifford, problem in probability, 5;
"On the Classification of Loci,"
8; kinematics, 13,
Closed sphere, passing out of, 79.
Collinear relation, 23; distinguished
at first from "on a line," 19, 27;
the two axioms, 19, 25.
Collinear with a segment, 25; tri-
angle, 32 ; tetrahedron, 49 ; pen-
tahedroid, 58.
Complex variables represented in
space of four dimensions, n, 219.
Cone, double, 70 ; see Double cone.
Configurations of points (Cayley,
Veronese), 5, 8.
Conical hypersurface of double revo-
lution inscribed in a hypersphere,
intersecting it in the same surface
as the inscribed cylinder of double
revolution, 263.
Conical, sometimes used for hyper-
conical, 69.
Conjugate series of isocline planes,
183 ; see Isocline planes.
Continuity of points on a line, 28.
Coolidge, list of systems of geometry,
IS-
Corresponding dihedral angles of
parallel planes, 223.
Craig, kinematics of four dimensions,
13-
Curvature, Riemann, 7; of the hy-
persphere, constant, 218.
Cyclical order, 28.
Cylinder, double, 262; see Double
cylinder.
Cylinder, prism, 259; see Prism
cylinder.
Cylinders, the set in a prism cylinder,
258.
Cylindrical, sometimes used for hy-
percylindrical or for piano-cylin-
drical, 258, 266, 284.
Darboux, hesitated to use geometry
of four dimensions, 9.
Dekagons, the seventy-two in a 600-
hedroid, 323.
Density, of points on a line, 28.
Descartes, use of "sursolid," 2 ; knew
the polyhedron formula, 300.
Descriptive geometry of four dimen-
sions, 1 8.
"Determine," meaning in geometry,
19.
Diagrams only indicate relations, 18.
Dihedral angle, its plane angle the
same at all points, proof indepen-
dent of the axiom of parallels, 97 ;
in a hypersphere, its volume, 209.
Dimensions, early references to the
number, i ; differences in spaces of
an even number and of an odd
number, 14; of a rectangular
hyperparallelopiped, 239 ; only
three regular figures in a space of
five dimensions, 317.
Diophantus, use of "square-square,"
etc., 2.
Directing-curve of a piano-conical
hypersurface, 71 ; 'piano-cylindrical
hypersurface, 256; similar direct-
ing-curves, 257.
Directing-polygon of a piano-poly-
hedral angle, 137 ; piano-prismatic
hypersurface, 243; similar direct-
ing-polygons, 245.
Directing polyhedral angle of a piano-
polyhedral angle, 135.
Directing-polyhedron of a poly-
hedroidal angle, 126; prismoidal
hypersurface, 235.
Directing-surface of a hyperconical
hypersurface, 69; hypercylindrical
hypersurface, 253.
Direction on a line, opposite direc-
tions, 27.
INDEX
337
Distance between a point and a hy-
perplane, 78; the minimum be-
tween two lines, 105 ; in a hyper-
sphere, 208 ; at infinity, 232.
Distances between two great circles
in a hypersphere, 217; two lines
at infinity, 235.
Dodekahedrons, the net, four at a
point, 324.
Double cone, 70; vertex-edge, base,
elements, end-cones, 7 1 ; cut from
a piano-conical hypersurface, 72;
circular, axis-element, right, isos-
celes, generated by the rotation of
a tetrahedron, 205.
Double cylinder, directing-curves,
interior, right, generated by the
directing-curves and their interiors,
spread out in a hyperplane, 262;
cylinder of double revolution,
inscribed in a hypersphere, 263;
relation to infinity, 264; volume,
267; hypervolume, 285; ratio to
circumscribed and to inscribed
hypersphere, 287.
Double Elliptic Geometry, 215; see
Elliptic Non-Euclidean Geometry.
Double prism, the two sets of prisms,
246 ; right, regular, its cells spread
out in a hyperplane, 248 ; interior,
directing-polygons, 249; generated
by the directing-polygons and their
interiors, cut into two double
prisms, 250; doubly triangular,
251 ; hyperprisms with prisms
for bases as double prisms, rela-
tion to infinity, 252; volume, 265;
doubly triangular double prisms
cut into six equivalent pentahe-
droids, 280; hypervolume, 282.
Double pyramid, 66; vertex-edge,
base, elements, end-pyramids, lat-
eral faces, lateral cells, intersection
with a plane, 67; with a hyper-
plane, 68; cut from a piano-poly-
hedral angle, 138; axis-element,
right, isosceles, regular, 204; in a
hyperplane, 318.
Double revolution, conical hyper-
surface of, 197; with plane ele-
ments, 206; its interior all con-
nected, 207; cylinder of, 263, see
Double cylinder; surface of, in a
hypersphere, its importance in the
theory of functions, 219; the in-
tersection of the hypersphere with
an inscribed cylinder of double
revolution and with an inscribed
conical hypersurface of double
revolution, 263.
Double rotation, 145 ; in the hyper-
sphere, 218.
Doubly triangular prism, 251.
Duality in the hypersphere, recipro-
cal figures, 212.
Edge Geometry, the elements half-
planes with a common edge, 138;
applied to the theory of motion
with two points fixed, 1 73.
Edge of a polyhedron, how defined,
2 1 ; of a half -plane, 39 ; hemi-
sphere, 209.
Edges of a tetrahedron, 45 ; pyramid,
55 ; pentahedroid, 57; polyhe-
droid, 63.
Elements, linear, of two planes, 61.
Elements of geometry, points, 19,
23; of Point Geometry, 113.
Elements of a double pyramid,
67 ; hyperconical hypersurface, hy-
percone, 69; piano-conical hyper-
surface, double cone, 71 ; poly-
hedroidal angle, 126; piano-poly-
hedral angle, 134 ; piano-prismatic
hypersurface, 242; piano-cylindri-
cal hypersurface, 256.
Elliptic Non-Euclidean Geometry :
due to Riemann, 7 ; restrictions to
the axioms of collinearity, 25 ; the
points of a line are in cyclical order,
29; modification of proof of min-
imum distance between two lines,
105; lines with more than one
common perpendicular line, 108;
lines and planes with more than
one common perpendicular line,
planes with more than one common
338
INDEX
perpendicular plane, 112; Edge
Geometry is elliptic, 173 ; the most
general motion in hyperspace, 174,
177 ; volume of a tetrahedron, 211;
the geometry of the hypersphere is
the Double Elliptic, 215, 217;
difference between the Single El-
liptic and the Double Elliptic, 215 ;
poles of a hyperplane in four
dimensions, 217 ; space of constant
curvature, 218; the geometry at
infinity is the Single Elliptic, 233 ;
hypervolume of a pentahedroid,
287; the possible nets of poly-
hedrons in Elliptic Geometry, 306 ;
a regular polyhedroid can be in-
scribed in a hypersphere, 309 ; the
possible nets of polyhedroids, 316;
see also Non-Euclidean Geometry,
Parallel axiom, Restrictions.
Enriques, the foundations of geom-
etry, 15; definition of segment, 21.
Euler's name usually associated with
the polyhedron formula, 300.
Face angles of a polyhedroidal angle,
126.
Face of a half-hyperplane, 54 ; hyper-
plane angle, 95.
Faces of a tetrahedron, 45 ; pyramid,
55; pentahedroid, 57; polyhe-
droid, 63 ; piano-polyhedral angle,
, 133 '
Figure is regarded as a class of points,
19, 23; belong to, lie in, 23.
Five dimensions, only three regular
figures, 317.
Foundations, different systems, 15;
definitions and intersections of
elementary figures particularly con-
sidered, 17.
Four dimensions, space of, 24; our
restriction to, 16, 59.
Fourth dimension as time, 4, n ; The
Fourth Dimension Simply Ex-
plained, 9.
Gauss considered the higher space a
possibility, 10.
Geometries of i, 2, 3, ...,...
dimensions, 24.
Geometries of different kinds, dif-
ferent interpretations of an abstract
geometry, 14, 15; the geometry of
half-hyperplanes with a common
face, 99; Point Geometry, 113;
Edge Geometry, 138; system of
isocline planes, 197 ; the .hyper-
sphere, 212; system of parallel
planes, 244.
Grassmann, Au$dehmmgslehre, 7.
Gravity, centre of, memoir by Syl-
vester, 5 ; of a pentahedroid, 201 ;
tetrahedron, triangle, 202.
Green, problem in attraction, 6.
Half-hyperplane, * ' half-hyperplane
ABC-D" face, opposite half-
hyperplanes, 54; half-hyperplanes
with a common face form a one-
dimensional geometry, 99.
Half -hyper space, " half -hyperspace
ABCD-E," cell, opposite half-
hyperspaces, 62.
Half-hypersphere, 210.
Half-line, "half-line AB," "AB pro-
duced," opposite half -lines, 28.
Half -parallel planes, 224; see Paral-
lelism.
Half-perpendicular planes, 85 ; see
Perpendicularity.
Half-plane, "half-plane AB-C" edge,
opposite half-planes, 39.
Halphen, geometry of n dimensions, 8.
Halsted, use of the terms "sect" and
"straight," 25; proof that a line
divides a plane, 37.
Hathaway, application of quaternians
to geometry of four dimensions, 13.
Hatzidakis, kinematics of four di-
mensions, 13.
Hekatonikosahedroid, 326; see 120-
hedroU.
Hexadekahedroid, 291 ; see 16-
hedrold.
Hexakosioihedroid, 321; see 600-
hedroid.
Hilbert, definition of segment, 21.
INDEX
339
Hyperbolic Non-Euclidean Geom-
etry : planes with parallel elements,
95, 112; their common per-
pendicular plane, 112; boundary-
hypersurfaces, 95, 112; translation
along a line, 146, along boundary-
curves, 146, 168; rotations in a
hyperpiane around parallel axes,
172;. the most general motion in
hyperspace, 174; rotations around
parallel axis-planes, 1 78 ; pentahe-
droids which have no point equi-
distant from the five vertices, 199 ;
the possible nets of polyhedrons,
306 ; a regular polyhedroid can be
inscribed in a hypersphere, 309;
the pdssible nets of polyhedroids,
316 ; see also Non-Euclidean Ge-
ometry.
Hypercone, 69; intersections, in-
terior, with a cone for base, dif-
ferent ways of regarding it, 70;
axis, generated by the rotation of
a half-cone, 204; lateral volume,
266; lateral volume of a frustum,
267 ; hypervolume, 284 ; hyper-
volume of a frustum, 285.
Hyperconical hypersurface, direct-
ing-surface, elements, 69 ; of double
revolution, 197 ; its plane elements,
206 ; its interior all connected, 207 ;
intersection with a hypersphere,
spherical directing-surface, 220.
Hypercube, the diagonal twice the
edge, its cells spread out in a hyper-
plane, two forms of projection,
240; as a regular polyhedroid,
reciprocal nets, 290; reciprocal
relation to the i6-hedroid, 292;
diagonals and axes form three
rectangular systems, 293; the
associated 24-hedroid, 295.
Hypercylinder, 254; lateral hyper-
surface, interior, spherical, axis,
generated by a rectangle, by rota-
tion of a half-cylinder, 255 ; with
cylinders for bases, different ways
of regarding it, 255, 261 ; lateral
volume, 266; hypervolume, 284.
Hypercylindrical hypersurface, di-
recting-surface, elements, 253; in-
terior, sections, 254; relation to
infinity, 256.
Hyperparallelopiped, its diagonals all
bisect one another; rectangular,
its dimensions, the square of the
length of its diagonal equals the
sum of the squares of its four
dimensions, 239; as a double
prism, 252; hypervolume when
rectangular, 271, when oblique,
272.
Hyperpiane, 24 ; " hyperpiane
A BCD" 50; figures which deter-
mine it, only one contains four-
given non-coplanar points, 51;
ordinary space a hyperpiane, 52;
divided by a plane, 53; intersec-
tion with a plane, 60 ; intersection
of two hyperplanes, 52, 60; oppo-
site sides of a hyperpiane, 62;
at infinity, 231.
Hyperpiane angle, face, cells, interior,
divides the rest of hyperspace,
95; intersection with a hyper-
plane perpendicular to its face, 96 ;
plane angle, 96, 98; two hyper-
plane angles are congruent when
they have two equal plane angles,
96 ; the plane angle is the same at
all points of the face, 97; as a
magnitude, supplementary hyper-
plane angles, right hyperpiane
angles, the sum of two, 98 ; anal- %
ogous to a dihedral angle, in the
geometry of half-hyperplanes,
measured by the plane angle, 99;
the bisecting half-hyperplane, 100.
Hyperprism, lateral cells, etc., 237;
its cells spread out in a hyperpiane,
238; with prisms for bases, dif-
ferent ways of regarding it, 239,
252; lateral volume, 265; con-
gruent and equivalent hyperprisms,
271 ; hypervolume when the bases
are prisms, 273 ; when the bases
are tetrahedrons, 274; hyperprism
with tetrahedral ends, prismatic
340
INDEX
base and vertex edge, 274; hyper-
volume of any hyperprism, 275.
Hyperpyramid, 63; base, interior,
sections, 64; with a pyramid for
base, different ways of regarding it,
66 ; cut from a polyhedroidal angle,
127; axis, regular, 203; lateral
volume, of a frustum, 265; hy-
pervolume, 276; frustum cut
into pentahedroids, hypervolume,
278.
Hypersolid, the interior of a pentahe-
droid as a hypersolid, 62; boun-
daries are three-dimensional, 64;
hypervolume, ratio of two, equiva-
lent hypersolids, 270.
Hyperspace, term used to denote the
space of four dimensions, 60; di-
vided by a hyperplane, not di-
vided by a plane, 62.
Hypersphere, great spheres and small
spheres, 207; great circles and
small circles, 208; distance in a
hypersphere, tangent hyperplanes,
208; spherical dihedral angle, its
volume, 209; tetrahedron, the
sixteen associated tetrahedrons,
their volumes, 210; axis and poles
of a sphere, axis-plane and polar cir-
cle of a circle, 211 ; their motion in
a rotation of the hypersphere, 212 ;
duality, reciprocal figures, 212;
the geometry of the hypersphere as
an independent three-dimensional
geometry, 212; it is the Double
Elliptic Non-Euclidean Geometry,
215; the Point Geometry at the
centre, 216 ; the distances between
two great circles, parallel great
circles, 217; proof from Point
Geometry that the geometry of the
hypersphere is the Double Elliptic,
217; rotation, double rotation,
screw motion, parallel motion, 218 ;
curvature constant, 218; inter-
section with a conical hypersurface
of double revolution, 220; inter-
section with an inscribed cylinder
of double revolution a surface of
double revolution, 263; volume,
267; hypervolume, 285.
Hypersurface, 69 ; of a pentahedroid,
62.
Hypervolume, 270; of a rectangular
hyperparallelopiped, 271 ; any
hyperparallelopiped, 272; hyper-
prism with prisms for bases, 273 ;
with tetrahedral bases, 274; any
hyperprism, 275 ; hyperpyramid,
276 ; frustum, 278 ; double prism,
280; cylindrical and conical hy-
persurfaces, 284 ; hypersphere,
285; ratio of the hypersphere to
inscribed and to circumscribed
double cylinders, 287; pentahe-
droid in elliptic hyperspace, 287.
Ideal points, lines, etc., at infinity,
230.
Ikosatetrahedroid, 296, see 24-he-
droid.
Independent points, 24.
Infinity, sense in which the term is
used, 230; points, lines, etc., at
infinity, all points at infinity
in a single hyperplane, 231 ; dis-
tance, angle, 232; dihedral angle,
233; the geometry at infinity is
the Single Elliptic, generalizations
made possible by the use of these
forms of expression, 233; dis-
tances between two lines, 235;
relation to infinity of the prismoidal
hypersurface, 241 ; hyperparallelo-
piped, 241, 253; piano-prismatic
hypersurface, double prism, 252;
hypercylindrical hypersurface, 256 ;
piano-cylindrical hypersurface,
prism cylinder, double cylinder,
264.
Interior of a figure as distinguished
from the figure itself, 20 ; see Seg-
ment, Triangle, Polygon, etc.
Intersect, intersection, 23.
Isoclinal angle (Stringham), 125.
Isocline planes, 123, 180; have an
infinite number of common per-
pendicular planes, 123, 182; series
INDEX
341
of isocline planes, 182; conjugate
series, 183 ; the two senses in which
planes can be isocline, 184; con-
jugate series isocline in opposite
senses, 185 ; through any line pass
two isocline to a given plane in
opposite senses, 186; two inter-
secting planes are isocline to two
pairs of planes, 187; when two
planes are isocline to a given plane
in the same sense the common per-
pendicular planes which they have
with the given plane form a con-
stant dihedral angle, 188; when
two planes are isocline to a given
plane in opposite senses, there is
only one pair of common perpen-
dicular planes, perpendicular to all
three, 189; two planes isocline to
a third in the same sense are
isocline to each other in this sense
also, 190; poles and polar series,
all the planes of two conjugate
series are isocline at an angle of
45 to a single pair of planes, 192 ;
the converse also true, 193; in a
system of planes isocline in a given
sense any two series have a pair
of planes in common, 193 ; if a
plane intersects two isocline planes
in lines the corresponding dihedral
angles are equal, analogy to parallel
lines, 194-196; a system of planes
isocline in a given sense forms a
two-dimensional geometry, 197 ;
"ordinal" and "cardinal" system
(Stringham), 198; a series cuts a
hypersphere (with centre at O) in
a surface of double revolution with
equal radii, 220; projection from
one upon the other of two isocline
planes produces similar figures, 229.
Isocline rotation, every plane remains
isocline to itself, 196.
Isosceles double pyramid, 204 ; double
cone, 205.
Jacobi, generalizations of geometrical
formulae, 6.
Jordan, geometry of n dimensions, 8.
Jouffret, Gtomitrie a quatre dimen-
sions, 9.
Kant, reference to the number of
dimensions of space, 3.
Keyser, the four-dimensional geom-
etry of spheres, 1 1 ; our intuition
of hyperspace, 16; the angles of
planes, 114; proof that two planes
have a common perpendicular
plane, 118.
Kinematics of four dimensions,
articles by Clifford, Beltrami,
Craig, Hatzidakis, 13.
Kwietnewski, complex variables rep-
resented in space of four dimen-
sions, ii.
Lagrange, time as the fourth dimen-
sion, 4.
Lateral edges, faces, cells, hypersur-
face, etc., see Pyramid, Hyper-
pyramid, Hypercone, etc.
Layer, 241.
Left, right and left in a plane, 154.
Lewis, G. N., Wilson and Lewis on
relativity, 12.
Line, 24; "line AB," only one line
contains two given points, 26 ; prop-
erties of its points, order, 27 ; den-
sity and continuity, 28; opposite
sides in a plane, 38 ; at infinity, 231.
Linear elements of two planes, 61.
Lobachevsky, Pangeometry, 221.
Loria, bibliography, 9.
McClintock, interpretations of Non-
Euclidean Geometry, 10.
Methods of studying the higher
geometries, 12.
Minkowski developed application to
relativity, 12.
M6bius, symmetrical figures, 4.
Moore, E. H., definition of segment,
21.
Moore, R. L., properties of points on
a line, 28; axioms of metrical
geometry, 74.
342
INDEX
More, Henry, spirits are four-di-
mensional, 3.
Motion in a plane does not change
order in the plane, 160; in a hy-
perplane, does not change order in
the hyperplane, 163; in hyper-
space, does not change order in
hyperspace, 166; in a plane with
one point fixed, 167; the most
general, 168; in a hyperplane, two
equivalent if equivalent for three
non-collinear points, motion with
one point fixed, 169; every motion
in a hyperplane equivalent to a
motion of a plane on itself or to a
screw motion, 170; in hyperspace,
two equivalent if equivalent for four
non-coplanar points, 172; motion
with two points fixed, 173; with
one point fixed, 1 74 ; every motion
equivalent to a motion in which one
plane remains fixed or moves only
on itself, 174.
n dimensions, space of, 24.
^-hedroid and 5 JV-hedroid, 311 ; the
same as i2o-hedroid and 600-
hedroid, 321.
Nets of hypercubes, 290 ; 24-hedroids
and i6-hedroids, 298; spherical
polygons, 303; polyhedral angles,
polyhedrons, 304 ; polyhedroidal
angles, 313; polyhedroids, 314;
net of twenty tetrahedrons at a
point, 317 ; four dodekahedrons at
a fx>int, 324.
Non-Euclidean Geometry used in the
theory of relativity, 12; not par-
ticularly considered in this text,
18; translations in, 146; nets of
spherical polygons, 303; see also
Elliptic Geometry, Hyperbolic Geom-
etry, Parallelism, Restrictions.
Non-Euclidean Geometry by the au-
thor, 19.
Ndther, birational transformations, 8.
Object and plan of this book, 16, 73.
Octahedroid, regular, 290; see Hy-
percubc.
Opposite directions on a line, 27;
half-lines, 28; in cyclical order,
29; sides of a line in a plane,
38; half-planes, 39; sides of a
plane in a hyperplane, half-hyper-
planes, 54; sides of a hyperplane,
half-hyperspaces, 62; elements of
Point Geometry, 113; elements of
Edge Geometry, 138; points in a
hypersphere, 210, 213.
Order of points on a line, 27 ; "order
AB" 153 ; Veblen's use of the term
" order," 27 ; cyclical, 28 ; order in a
plane, 1 53 ; two fundamental prin-
ciples, right and left sides of lines
through a point, 154; with respect
to a triangle, "order ABC," 156;
unchanged by any motion in the
plane, 158; independent of any
hyperplane, 162 ; order in a hyper-
plane, 161 ; with respect to a tetra-
hedron, "order ABCD" 162; un-
changed by any motion in the
hyperplane, 163; order in hyper-
space, 164; with respect to a pen-
tahedroid, "order ABCDE" un-
changed by any motion, 165 ; order
in Point Geometry, 179.
Ovidio, d', projective geometry, 12.
Ozanam, higher products imaginary,
3-
Paciuolo, use of "primo relato," etc.,
2.
Pangeometry, term used by Loba-
chevsky, 221.
Parallel axiom, proofs which do not
depend on it, 77, 97, 105, 128, 136,
202; restrictions due to its omis-
sion, 37, 78, 79, 103, 108, 112, 138,
*39 iS3 l6 J s 66 Elliptic Geom-
etry and Restrictions.
Parallelism taken up after many other
subjects, 19 ; parallel great circles
in a hypersphere, 217; parallel
motion in a hypersphere, 218;
axiom of parallels, 221 ; parallel
lines and planes, 221 ; half-parallel
planes, their common perpendicular
INDEX
343
lines and planes, 224; their mini-
mum distance, their linear ele-
ments, 225; lines and planes par-
allel to a hyperplane, parallel
hyperpianes, 226.
Pasch, Axiom, 30.
Pentahedroid, edges, faces, cells, 57 ;
intersection with a plane, 57, 60;
interior, coliinear with, 58; pass-
ing from cell to cell, 59 ; intersection
with a line, 60; the five half-hy-
perspaces and the interior, 62 ;
sections, 65 ; its cells or its edges
spread out in a hyperplane, 68;
the point equidistant from its
vertices, 199; the point equidis-
tant from its cells, 200 ; its centre
of gravity, 201 ; pentahedroids
with corresponding edges equal,
202 ; hypervolume in elliptic hyper-
space, 287; regular, 203, 289;
radii of circumscribed and in-
scribed hyperspheres, reciprocal
pentahedroids, 289.
Perpendicularity : lines perpendicular
to a line at a point, 74; perpen-
dicular line and hyperplane, 75;
planes perpendicular to a line at a
point; two lines perpendicular
to a hyperplane lie in a plane, 76 ;
lines perpendicular to a plane at a
point, 80; absolutely perpendicu-
lar planes, 81 ; if two planes inter-
sect in a line, their absolutely per-
pendicular planes at any point of
this line intersect in a line, 82 ;
two planes absolutely perpendicular
to a third lie in a hyperplane, 83 ;
perpendicular planes, simply per-
pendicular, half-perpendicular, or
perpendicular in a hyperplane;
a plane perpendicular to one of two
absolutely perpendicular planes
at their point of intersection is per-
pendicular to the other, 85; a
.plane intersecting two absolutely
perpendicular planes in lines is
perpendicular to both, 86; the
common perpendicular planes of
two planes intersecting in a line,
87; perpendicular planes and hy-
perplancs, perpendicular along a
line, 90; the planes perpendicular
or absolutely perpendicular to
planes lying in the hyperpianes, 91 ;
lines lying in either and perpendic-
ular to the other, 92 ; planes with
linear elements all perpendicular
to a hyperplane, 94 ; perpendicular
hyperpianes, 98; lines or planes
lying in one and perpendicular to
the other, 101 ; the common per-
pendicular line of two lines not in
one plane; lines with more than
one common perpendicular line,
1 08; the common perpendicular
line of a line and plane ; the com-
mon perpendicular plane of two
planes which have a common
perpendicular hyperplane, in;
the common perpendicular planes
of two planes which intersect
only in a point, 118; planes
with an infinite number of com-
mon perpendicular planes, 119,
182.
Plan and object of this book, 16, 73.
Plane, 24; "plane ABC" only one
contains three given non-collinear
points, 35 ; divided by a line, 37 ;
two planes with only odd point in
common, 51, 81 ; intersection of
two in a hyperplane, 53 ; opposite
sides of a plane in a hyperplane,
54; intersection with a hyper-
plane, 60; linear elements of two
planes, 61 ; absolutely perpendicu-
lar planes, 81; perpendicular, 85;
see Perpendicularity; if two not in a
hyperplane have a common per-
pendicular line, they have a common
perpendicular hyperplane, 94 ;
isocline planes, 123; see Isocline
planes; planes at infinity, 231.
Plane angle of a hyperplane angle, 96 ;
see Hyperplane angle.
Piano-conical hypersurface, vertex-
edge, directing-curve, elements,
344
INDEX
intersection with a hyperplane, 71 ;
see Double cone.
Piano-cylindrical hypersurface, di-
recting-curve, elements, 256 ;
interior, right directing-curves, sim-
ilar directing-curves, hypersurface
of revolution, axis-plane, 257;
intersection with piano-prismatic
hypersurface, the set of cylinders,
258; intersection of two piano-
cylindrical hypersurfaces, 261 ; the
surface of intersection, 262; rela-
tion to infinity, 264; see Prism
cylinder and Double cylinder.
Piano-polyhedral angle, faces, vertex-
edge, cells, 133; elements, simple,
convex, its hyperplane angles, 134 ;
vertical piano-polyhedral angles,
134; directing polyhedral angles,
polyhedral angles which are right
sections, 135 ; theorems proved by
means of them, 136; directing-
polygons, 137 ; interior, 139, 140.
Piano-prismatic hypersurface, 241 ;
faces, cells, elements, simple, con-
vex, sections, 242; directing-poly-
gons, 243; triangular, similar
directing-polygons, 245; intersec-
tion of two piano-prismatic hyper-
surfaces, the two sets of prisms,
246; intersection with a piano-
cylindrical hypersurface, 258; see
Double prism and Prism cylinder.
Piano- trihedral angle, 134.
Plucker, the four coordinates of a line
in space, 10.
Poincare' avoided use of geometry
of four dimensions, 10 ; on analysis
situs, 12; double integrals, 219.
Point, 23; independent and abso-
lutely independent points, 24; at
infinity, 231.
Point Geometry, 113; theorems in
regard to perpendicular planes
stated in the language of Point
Geometry, 114; applied to the
study of the angles of two planes,
114; piano-polyhedral angles and
polyhedroidal angles, 136; Point
Geometry of a rectangular system,
179; in the theory of isocline
planes, 180 ; the same as the geom-
etry of the hypersphere, 216.
Poles and polar series of isocline
planes, 193 ; poles of a sphere and
polar circles in a hypersphere, 211 ;
their motion in a rotation, 212 ; of
a hyperplane in Elliptic Geometry
of Four Dimensions, 217. *
Polygon, sides, diagonals, cyclical
order, 40; simple, convex, inter-
section with a line, 41 ; divided into
two polygons, 42 ; interior, 44; the
half -planes and interior, 45.
Polyhedral angle, 133; convex, can
be cut in a convex polygon, 137 ;
nets of polyhedral angles, 304.
Polyhedroid, edges, faces, cells, in-
terior, 63 ; regular, definition, 289 ;
can be inscribed in a hypersphere,
the associated net of hyperspherical
polyhedrons, 309 ; reciprocal poly-
hedroids, 310; its polyhedroidal
angles are regular, 313; nets of
polyhedroids, 314; list of possible
nets, 315; the nets in each of the
non-Euclidean geometries, 316.
Polyhedroid formula, 302 ; proved by
Schlafli, 22 ; by Stringham, 302.
Polyhedroidal angle, elements, di-
recting-polyhedron, face angles,
polyhedral angles, cells, interior,
126; vertical polyhedroidal angles,
127; regular, axis, 312; the poly-
hedroidal angles of a regular poly-
hedroid are regular, net, reciprocal
nets, 313.
Polyhedron, 63; regular, can be in-
scribed in a sphere ; the associated
net of spherical polygons ; recipro-
cal polyhedrons, 303; nets, 304;
list of possible nets, 305 ; the nets
in each of the non-Euclidean geom-
etries, 306.
Polyhedron formula, Descartes,
Euler, 300.
Popular interest in the fourth dimen-
sion, 9.
INDEX
345
Powers of a number in early algebra, 2.
Prism, the two sets of prisms in a
double prism, 246; see Double
prism.
Prism cylinder, the set of cylinders,
right, regular, spread out in a hy-
perplane, 259 ; the directing-poly-
gons and the directing-curves,
generated by them and their
interiors; cut into two prism cyl-
inders, 260; triangular, hyper-
cylinder with cylinders for bases as
a prism cylinder, generated by the
rotation of a prism, 261 ; volume,
267; hypervolume, 285.
Prismatic base of a hyperprism with
tetrahedral ends, 274.
Prismoidal hypersurface, directing-
polyhedron, edges, faces, cells, in-
terior, sections, 235 ; with parallel-
epiped for directing-poiyhedron,
236.
Projecting line, 78 ; plane, 84 ; factor
for area, 229.
Projection upon a hyperplane, 78;
of a line is a line or a part of a line,
79, 84 ; upon a plane, 81 ; a line
and its projection upon a plane not
coplanar, 84; of a plane upon a
hyperplane, 103; from a plane
upon an isocline plane produces
similar figures, 229.
Protective geometry, points of a line
in cyclical order, 29.
Ptolemy, the number of distances, i.
Pyramid, base, edges, faces, inter-
section with a plane, 55; double
pyramid, 66 ; see Double pyramid.
Quaternions applied to the study of
geometry of four dimensions by
Hathaway and Stringham, 13.
Ray or half-line, 28.
Reciprocal figures in a hypersphere,
212; pentahedroids, 289; hy-
percube and i6-hedroid, 293; 24-
hedroids, 297; polyhedrons, 303;
polyhedroids, 309 ; nets, see Nets.
Rectangular hyperparallelopiped,
239-
Rectangular system, 87, 89; as a
tetrahedroidal angle, 128; ways
in which it is congruent to itself,
179; the different arrangements,
notation, used in studying the
angles of two planes, 180; three
belonging to the hypercube and
i6-hedroid, 293.
Regular hyperpyramid, 203; pen-
tahedroid, 203, 289; double pyr-
amid, 204; hyperprism, 237; the
hypercube is regular, 240, 289;
regular double prism, 248; prism
cylinder, 259; polyhedroid, 289;
octahedroid (hypercube), 290; 16-
hedroid, 291; 24-hedroid, 295;
a regular polyhedroid can be in-
scribed in a hypersphere, the asso-
ciated net of spherical polyhedrons,
309; regular polyhedroidal angle,
312; in space of five dimensions
only three regular figures, 317;
6oo-hedroid ,317; 1 2o-hedroid, 3 24.
Relativity and the fourth dimension,
ii.
Restricted geometry, 19.
Restrictions to the second axiom of
collinearity in Elliptic Geometry,
25 ; necessary in Edge Geometry,
138; a convex piano-polyhedral
angle is restricted, 139; in Point
Geometry a rectangular system is
restricted, 179; restrictions due
to omission of the axiom of paral-
lels, see Parallel axiom; see also
Elliptic Geometry.
Revolution, surface of double revolu-
tion in a hypersphere of importance
in the theory of functions, 219;
see Double revolution.
Riemann on the foundations of
geometry, Elliptic Geometry due
to him, 6.
Right and left in a plane, 154; see
Order.
Rotation in a plane, in a hyperplane,
figures remain invariable, 141;
346
INDEX
in hyperspace, the axis-plane, 142 ;
figures remain invariable, rota-
tions around absolutely perpen-
dicular planes commutative, 143 ;
double rotation, 145 ; right and left
in a plane, 154 ; when two rotations
are equivalent to a single one in a
hyperplane, 171, in hyperspace,
178; isocline, 196; of the hyper-
sphere, the axis-circle and the circle
of rotation, double rotation, screw
motion, parallel motion, 218.
Rudolph, use of terms representing
powers of a number, 2.
Schlafli, multiple integrals, 6; mul-
tiple continuity, 22.
Schotten, definitions of segment, 21.
Schoute, Mehrdimensionale Geom-
etrie, g ; sections of a simplex, 14 ;
descriptive geometry, 18 ; different
kinds of perpendicularity, 85 ; the
polyhedroid formula, 302.
Schubert, enumerative geometry, 12.
Screw motion, the translation and
rotation are commutative, 146;
in the hypersphere, 218.
Sect, used by Halsted for segment, 25.
Sections, study of a figure by them,
1 8 ; divide a figure into completely
separated parts, 65, 245; of a
pentahedroid, hyperpyramid, etc.,
see these terms; of a piano-poly-
hedral angle, 135.
Segment as defined by different
writers, Hilbert, Enriques, E. H.
Moore, Veblen, Schotten, 21 ;
definition, "segment AB," col-
linear with, 25 ; interior, 28.
Segre, the use of geometry of four
dimensions, 10, 13.
Semi-parallel, the same as half-paral-
lel, 224.
Separate, of cyclical order, 29.
Series of isocline planes, 182; see
Isocline planes.
Sides of a polygon, how defined, 21 ;
triangle, 29; polygon, 40; of a
line in a plane a property of the
plane, of a plane !n a hyperplane a
property of the hyperplane, 162.
Similar figures produced by projec-
tion from one upon the other of
two isocline planes, 229; directing-
polygons of a piano-prismatic hy-
persurface, 245 ; directing-curves
of a piano-cylindrical hypersurface,
257.
Simplex, sections, 14.
Simplicius, reference to Aristotle and
Ptolemy, i.
Solid, the interior of a tetrahedron as
a solid, 54.
Sommerville, bibliography, 9.
Space of i, 2, 3, ... w, ... dimen-
sions, 24; of four dimensions, 24,
59; ordinary space a hyperplane,
52.
Sphere, its geometry is elliptic, 25;
closed, passing out of, 79; in a
hypersphere, 207.
Spherical sometimes used for hyper-
spherical, spherical dihedral angle,
tetrahedron, 209.
Stifel regards the higher powers as
"against nature," 3.
Straight, used by Halsted for line, 25.
Stringham, application of quaternions
to geometry of four dimensions, 13 ;
on the angles of two planes, 114;
use of the term "isoclinal angle,"
125; "ordinal" and "cardinal
systems," of isocline planes, 198;
gave a proof of the polyhedroid
formula, 302.
Strip, the portion of a plane between
two parallel lines, 235.
Surface, the tetrahedron as a surface,
54 ; of double revolution in a hyper-
sphere, 219; see Double revolution.
Sursolid in early algebra, sursolid
loci, 2.
Sylvester, early papers, 5 ; defence of
the use of geometry of four dimen-
sions, 10.
Symmetrical figures congruent,
Mobius, 4, 149; defined as those
that can be placed in positions of
INDEX
347
symmetry with respect to a plane,
164.
Symmetry, 146 ; in a plane, in a hy-
perplane, 147, in hyperspace, 149 ;
rotations which leave the sym-
metrical relation undisturbed in a
plane, 147, in a hyperplane, 148,
in hyperspace, 152; rotations
whicji bring into coincidence figures
symmetrical in a plane, 147, in a
hyperplane, 148; symmetrical fig-
ures of ordinary geometry are really
congruent, 148; symmetry in a
hyperplane with respect to a point
can be changed by rotation to
symmetry with respect to a plane,
147 ; figures symmetrical in hyper-
space with respect to a point or
plane are congruent, 149; sym-
metry in hyperspace with respect
to a line can be changed by rotation
to symmetry with respect to a hy-
perplane, 150; in every kind of
symmetry corresponding segments
and angles are equal, 153; figures
symmetrical in a plane cannot be
made to coincide by any motion in
the plane, 160 ; figures symmetrical
in a hyperplane cannot be made to
coincide by any motion in the hy-
perplane, 163 ; figures symmetrical
in hyperspace cannot be made to
coincide by any motion in hyper-
space, 1 66.
Synthetic development of the higher
geometry, 4; advantages of the
synthetic method over the analytic,
14.
Tetrahedroidal angle, 127; two with
corresponding face angles equal,
1 29 ; the bisecting half-hyperplanes
of its hyperplane angles have a
common half -line ,131.
Tetrahedron, edges, faces, intersec-
tion with a plane, 45 ; with a line,
48, 52 ; interior, collinear with, 49;
the four half-hyperplanes and the
interior, 54; correspondence of
two, 128; spherical, its volume,
210; net of tetrahedrons, twenty
at a point, 317.
Time as the fourth dimension, La-
grange, d'Alembert, 4; relativity,
n.
Translations along a line, figures
remain invariable, 145; different
kinds of translation in Non-Euclid-
ean Geometry, 146.
Triangle, sides, cyclical order, 29 ; in-
tersection with a line, 30, 37 ; inte-
rior, 30; collinear with, 32 ; the three
half-planes and the interior, 39.
Use of studying the higher geometries,
13 ; see also Applications.
Veblen, definition of segment, 21 ; use
of the term "order," 27 ; the prop-
erties of points on a line, 28 ; Axiom
of Pasch, 29; axioms of metrical
geometry, 74.
Vector analysis of Grassmann, 8.
Veronese, Fondamenti, 5, 8, 9; ap-
plication of the higher geometry to
theorems of ordinary geometry, 13 ;
use of the elements at infinity, 230.
Vertex-edge of a double pyramid,
67 ; double cone, piano-conical hy-
persurface, 71 ; piano-polyhedral
angle, 133 ; hyperprism with tetra-
hedral ends, 274.
Vertical polyhedroidal angles, 127;
can be made to coincide, 167;
piano-polyhedral angles, 134; can-
not be made to coincide, 166.
Vieta, use of terms representing
powers of a number, 2.
Volume of a spherical dihedral angle,
209; spherical tetrahedron, 210;
at infinity, 233; lateral volume
of a hyperprism, hyperpyramid,
frustum of a hyperpyramid, 265;
hypercylinder, hypercone, 266, frus-
tum of a hypercone, 267 ; volume
of a double prism, 265, 266, prism
cylinder, double cylinder, 267;
hypersphere, 267.
348
INDEX
Wallis on the geometrical names of
the higher powers, 3.
Wilson and Lewis, relativity, 12.
i6-hedroid or hexadekahedroid, 291 ;
axes, reciprocal relation to the hy-
percube, 292; diagonals of the
hypercube and the i6-hedroid form
three rectangular systems, 293 ; the
associated 24-hedroid, 295; re-
ciprocal nets of i6-hedroids and
24-hedroids, 298.
24-hedroid or ikosatetrahedroid, asso-
ciated with a hypercube and a
i6-hedroid, 295 ; reciprocal 24-
hedroids, 297; reciprocal nets
of i6-hedroids and 24-hedroids,
298.
i2o-hedroid or hekatonikosahedroid,
construction, 324; table of its
parts, 326.
6oo-hedroid, or hexakosioihedroid,
construction of the first half, 317;
completed, 321 ; its seventy- two
dekagons, 323; table of its parts,
324.
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