CHAPTER XI.

A TESSARACTIC FIGURE AND ITS PROJECTIONS.

We will now consider a fourth-dimensional shape composed of tessaracts,

and the manner in which we can obtain a conception of it. The operation

is precisely analogous to that described in chapter VI., by which a

plane being could obtain a conception of solid shapes. It is only a

little more difficult in that we have to deal with one dimension or

direction more, and can only do so symbolically.

We will assume the shape to consist of a certain number of the 81

tessaracts, whose names we have given on p. 168. Let it consist of the

thirteen tessaracts: Urna, Moles, Plebs, Frenum, Pallor, Tessera, Cudo,

Vitta, Cura, Penates, Polus, Orcus, Lacerta.

Firstly, we will consider what appearances or projections these

tessaracts will present to us according as the tessaractic set touches

our space with its (_a_) Mala cubes, (_b_) Vesper cubes, (_c_) Pluvium

cubes, or (_d_) Lar cubes. Secondly, we will treat the converse

question, how the shape can be determined when the projections in each

of those views are given.

Let us build up in cubes the four different arrangements of the

tessaracts according as they enter our space on their Mala, Vesper,

Pluvium or Lar sides. They can only be printed by symbolizing two of the

directions. In the following tabulations the directions Y, X will at

once be understood. The direction Z (expressed by the wavy line)

indicates that the floors of nine, each printed nearer the top of the

page, lie above those printed nearer the bottom of it. The direction W

is indicated by the dotted line, which shows that the floors of nine

lying to the left or right are in the W direction (Ana) or the -W

direction (Kata) from those which lie to the right or left. For

instance, in the arrangement of the tessaracts, as Malas (Table A) the

tessaract Tessara, which is exactly in the middle of the eighty-one

tessaracts has

Domitor on its right side or in the X direction.

Ocrea on its left „ „ -X „

Glans away from us „ „ Y „

Cudo nearer to us „ „ -Y „

Sacerdos above it „ „ Z „

Cura below it „ „ -Z „

Lacerta in the Ana or W „

Pallor in the Kata or -W „

Similarly Cervix lies in the Ana or W direction from Urna, with Thyrsus

between them. And to take one more instance, a journey from Saltus to

Arcus would be made by travelling Y to Remus, thence -X to Sector,

thence Z to Mars, and finally W to Arcus. A line from Saltus to Arcus is

therefore a diagonal of the set of 81 tessaracts, because the full

length of its side has been traversed in each of the four directions to

reach one from the other, _i.e._ Saltus to Remus, Remus to Sector,

Sector to Mars, Mars to Arcus.

TABLE A.

Mala presentation of 81 Tessaracts.

Z W------------------------------------------------------- -W

|

| Y Y Y

| | | |

| | Block A | Block B | Block C

| | | |

| +---------------X +---------------X +---------------X

|

| Y Y Y

| | | |

| | Block D | Block E | Block F

| | | |

| +---------------X +---------------X +---------------X

|

| Y Y Y

| | | |

| | Block G | Block H | Block I

| | | |

| +---------------X +---------------X +---------------X

-Z

Block A:

Arcus Ovis Portio

Laurus Tigris Segmen

Axis Troja Aries

Block B:

Ara Vomer Pluma

Praeda Sacerdos Hydra

Cortex Mica Flagellum

Block C:

Mars Merces Tyro

Spicula Mora Oliva

Comes Tibicen Vestis

Block D:

Postis Clipeus Tabula

_Orcus_ _Lacerta_ Testudo

Verbum Luctus Anguis

Block E:

Pilum Glans Coins

Ocrea _Tessera_ Domitor

Cardo _Cudo_ Malleus

Block F:

Ala Cortis Aer

Uncus‡ _Pallor_‡ Tergum

Ostrum Bidens‡ Scena

Block G:

Telum Nepos Angusta

_Polus_ _Penates_ Vulcan

Cervix Securis Vinculum

Block H:

Agmen Lacus Arvus

Crates _Cura_ Limen

Thyrsus _Vitta_ Sceptrum

Block I:

Sector Hama Remus

_Frenum_‡ _Plebs_‡ Sypho

_Urna_‡ _Moles_‡ Saltus

TABLE B.

Vesper presentation of 81 Tessaracts.

Z W------------------------------------------------------ -W

|

| Y Y Y

| | | |

| Block A | Block B | Block C |

| | | |

| -X---------------+ -X---------------+ -X---------------+

|

| Y Y Y

| | | |

| Block D | Block E | Block F |

| | | |

| -X---------------+ -X---------------+ -X---------------+

|

| Y Y Y

| | | |

| Block G | Block H | Block I |

| | | |

| -X---------------+ -X---------------+ -X---------------+

-Z

Block A:

Portio Pluma Tyro

Segmen Hydra Oliva

Aries Flagellum Vestis

Block B:

Ovis Vomer Merces

Tigris Sacerdos Mora

Troja Mica Tibicen

Block C:

Arcus Ara Mars

Laurus Praeda Spicula

Axis Cortex Comes

Block D:

Tabula Colus Aer

Testudo Domitor Tergum

Anguis Malleus Scena

Block E:

Clipeus Glans Cortis

_Lacerta_* _Tessera_* _Pallor_*

Luctus* _Cudo_* Bidens*

Block F:

Postis Pilum Ala

_Orcus_* Ocrea* Uncus*

Verbum† Cardo† Ostrum†

Block G:

Angusta Arvus Remus

Vulcan Limen Sypho

Vinculum Sceptrum Saltus

Block H:

Nepos Lacus Hama

_Penates_* _Cura_* _Plebs_*

Securis* _Vitta_* _Moles_*

Block I:

Telum Agmen Sector

_Polus_* Crates* _Frenum_*

Cervix* Thyrsus* _Urna_*

TABLE C.

Pluvium presentation of 81 Tessaracts.

Z W------------------------------------------------------- -W

|

| +----------------X +----------------X +---------------X

| | | |

| | Block A | Block B | Block C

| | | |

| -Y -Y -Y

|

| +----------------X +----------------X +---------------X

| | | |

| | Block D | Block E | Block F

| | | |

| -Y -Y -Y

|

| +----------------X +----------------X +---------------X

| | | |

| | Block G | Block H | Block I

| | | |

| -Y -Y -Y

-Z

Block A:

Mars Merces Tyro

Ara Vomer Pluma

Arcus Ovis Portio

Block B:

Spicula Mora Oliva

Praeda Sacerdos Hydra

Laurus Tigris Segmen

Block C:

Comes Tibicen Vestis

Cortex Mica Flagellum

Axis Troja Aries

Block D:

Ala Cortis Aer

Pilum Glans Colus

Postis Clipeus Tabula

Block E:

Uncus* _Pallor_* Tergum

Ocrea* _Tessera_* Domitor

_Orcus_* _Lacerta_* Testudo

Block F:

Ostrum† Bidens† Scena

Cardo† _Cudo_* Malleus

Verbum† Luctus† Anguis

Block G:

Sector Hama Remus

Agmen Lacus Arvus

Telum Nepos Angusta

Block H:

_Frenum_* _Plebs_* Sypho

Crates* _Cura_* Limen

_Polus_* _Penates_* Vulcan

Block I:

_Urna_* _Moles_* Saltus

Thyrsus* _Vitta_* Sceptrum

Cervix† Securis† Vinculum

TABLE D.

Lar presentation of 81 Tessaracts.

Z W------------------------------------------------------- -W

|

| Y Y Y

| | | |

| | Block A | Block A | Block A

| | | |

| +---------------X +---------------X +---------------X

|

| Y Y Y

| | | |

| | Block A | Block A | Block A

| | | |

| +---------------X +---------------X +---------------X

|

| Y Y Y

| | | |

| | Block A | Block A | Block A

| | | |

| +---------------X +---------------X +---------------X

-Z

Block A:

Mars Merces Tyro

Spicula Mora Oliva

Comes Tibicen Vestis

Block B:

Ala Cortis Aer

Uncus _Pallor_* Tergum

Ostrum Bidens Scena

Block C:

Sector Hama Remus

_Frenum_* _Plebs_* Sypho

_Urna_* _Moles_* Saltus

Block D:

Ara Vomer Pluma

Proeda Sacerdos Hydra

Cortex Mica Flagellum

Block E:

Pilum Glans Colus

Ocrea _Tessera_* Domitor

Cardo _Cudo_* Malleus

Block F:

Agmen Laurus Arvus

Crates _Cura_* Limen

Thyrsus _Vitta_* Sceptrum

Block G:

Arcus Ovis Portio

Laurus Tigris Segmen

Axis Troja Aries

Block H:

Postis Clipeus Tabula

_Orcus_* _Lacerta_* Testudo

Verbum Luctus Anguis

Block I:

Telum Nepos Angusta

_Polus_* _Penates_* Vulcan

Cervix Securis Vinculum

The relation between the four different arrangements shown in the tables

A, B, C, and D, will be understood from what has been said in chapter

VIII. about a small set of sixteen tessaracts. A glance at the lines,

which indicate the directions in each, will show the changes effected by

turning the tessaracts from the Mala presentation.

In the Vesper presentation:

The tessaracts--

(_e.g._ Urna, Ostrum, Comes), which ran Z still run Z.

(_e.g._ Urna, Frenum, Sector), „ Y „ Y.

(_e.g._ Urna, Moles, Saltus), „ X now run W.

(_e.g._ Urna, Thyrsus, Cervix), „ W „ -X.

In the Pluvium presentation:

The tessaracts--

(_e.g._ Urna, Ostrum, Comes), which ran Z still run Z.

(_e.g._ Urna, Moles, Saltus), „ X „ X.

(_e.g._ Urna, Frenum, Sector), „ Y now run W.

(_e.g._ Urna, Thyrsus, Cervix), „ W „ -Y.

In the Lar presentation:

The tessaracts--

(_e.g._ Urna, Moles, Saltus), which ran X still run X.

(_e.g._ Urna, Frenum, Sector), „ Y „ Y.

(_e.g._ Urna, Ostrum, Comes), „ Z now run W.

(_e.g._ Urna, Thyrsus, Cervix), „ W „ -Z.

This relation was already treated in chapter IX., but it is well to have

it very clear for our present purpose. For it is the apparent change of

the relative positions of the tessaracts in each presentation, which

enables us to determine any body of them.

In considering the projections, we always suppose ourselves to be

situated Ana or W towards the tessaracts, and any movement to be Kata or

-W through our space. For instance, in the Mala presentation we have

first in our space the Malas of that block of tessaracts, which is the

last in the -W direction. Thus, the Mala projection of any given

tessaract of the set is that Mala in the extreme -W block, whose place

its (the given tessaract’s) Mala would occupy, if the tessaractic set

moved Kata until the given tessaract reached our space. Or, in other

words, if all the tessaracts were transparent except those which

constitute the body under consideration, and if a light shone through

Four-space from the Ana (W) side to the Kata (-W) side, there would be

darkness in each of those Malas, which would be occupied by the Mala of

any opaque tessaract, if the tessaractic set moved Kata.

Let us look at the set of 81 tessaracts we have built up in the Mala

arrangements, and trace the projections in the extreme -W block of the

thirteen of our shape. The latter are printed in italics in Table A, and

their projections are marked ‡.

Thus the cube Uncus Mala is the projection of the tessaract Orcus,

Pallor Mala of Pallor and Tessera and Tacerta, Bidens Mala of Cudo,

Frenum Mala of Frenum and Polus, Plebs Mala of Plebs and Cura and

Penates, Moles Mala of Moles and Vitta, Urna Mala of Urna.

Similarly, we can trace the Vesper projections (Table B). Orcus Vesper

is the projection of the tessaracts Orcus and Lacerta, Ocrea Vesper of

Tessera, Uncus Vesper of Pallor, Cardo Vesper of Cudo, Polus Vesper of

Polus and Penates, Crates Vesper of Cura, Frenum Vesper of Frenum and

Plebs, Urna Vesper of Urna and Moles, Thyrsus Vesper of Vitta. Next in

the Pluvium presentation (Table C) we find that Bidens Pluvium is the

projection of the tessaract Pallor, Cudo Pluvium of Cudo and Tessera,

Luctus Pluvium of Lacerta, Verbum Pluvium of Orcus, Urna Pluvium of Urna

and Frenum, Moles Pluvium of Moles and Plebs, Vitta Pluvium of Vitta and

Cura, Securis Pluvium of Penates, Cervix Pluvium of Polus. Lastly, in

the Lar presentation (Table D) we observe that Frenum Lar is the

projection of Frenum, Plebs Lar of Plebs and Pallor, Moles Lar of

Moles, Urna Lar of Urna, Cura Lar of Cura and Tessara, Vitta Lar of

Vitta and Cudo, Penates Lar of Penates and Lacerta, Polur Lar of Polus

and Orcus.

Secondly, we will treat the converse problem, how to determine the shape

when the projections in each presentation are given. Looking back at the

list just given above, let us write down in each presentation the

projections only.

Mala projections:

Uncus, Pallor, Bidens, Frenum, Plebs, Moles, Urna.

Vesper projections:

Orcus, Ocrea, Uncus, Cardo, Polus, Crates, Frenum, Urna, Thyrsus.

Pluvium projections:

Bidens, Cudo, Luctus, Verbum, Urna, Moles, Vitta, Securis, Cervix.

Lar projections:

Frenum, Plebs, Moles, Urna, Cura, Vitta, Polus, Penates.

Now let us determine the shape indicated by these projections. In now

using the same tables we must not notice the italics, as the shape is

supposed to be unknown. It is assumed that the reader is building the

problem in cubes. From the Mala projections we might infer the presence

of all or any of the tessaracts written in the brackets in the following

list of the Mala presentation.

(Uncus, Ocrea, Orcus); (Pallor, Tessera, Lacerta);

(Bidens, Cudo, Luctus); (Frenum, Crates, Polus);

(Plebs, Cura, Penates); (Moles, Vitta, Securis);

(Urna, Thyrsus, Cervix).

Let us suppose them all to be present in our shape, and observe what

their appearance would be in the Vesper presentation. We mark them all

with an asterisk in Table B. In addition to those already marked we must

mark (†) Verbum, Cardo, Ostrum, and then we see all the Vesper

projections, which would be formed by all the tessaracts possible from

the Mala projections. Let us compare these Vesper projections, viz.

Orcus, Ocrea, Uncus, Verbum, Cardo, Ostrum, Polus, Crates, Frenum,

Cervix, Thyrsus, Urna, with the given Vesper projections. We see at once

that Verbum, Ostrum, and Cervix are absent. Therefore, we may conclude

that all the tessaracts, which would be implied as possible by their

presence, are absent, and of the Mala possibilities may exclude the

tessaracts Bidens, Luctus, Securis, and Cervix itself. Thus, of the 21

tessaracts possible in the Mala view, there remain only 17 possible,

both in the Mala and Vesper views, viz. Uncus, Ocrea, Orcus, Pallor,

Tessera, Lacerta, Cudo, Frenum, Crates, Polus, Plebs, Cura, Penates,

Moles, Vitta, Urna, Thyrsus. This we call the Mala-Vesper solution.

Next let us take the Pluvium presentation. We again mark with an

asterisk in Table C the possibilities inferred from the Mala-Vesper

solution, and take the projections those possibilities would produce.

The additional projections are again marked (†). There are twelve

Pluvium projections altogether, viz. Bidens, Ostrum, Cudo, Cardo,

Luctus, Verbum, Urna, Moles, Vitta, Thyrsus, Securis, Cervix. Again we

compare these with the given Pluvium projections, and find three are

absent, viz. Ostrum, Cardo, Thyrsus. Hence the tessaracts implied by

Ostrum and Cardo and Thyrsus cannot be in our shape, viz. Uncus, Ocrea,

Crates, nor Thyrsus itself. Excluding these four from the seventeen

possibilities of the Mala-Vesper solution we have left the thirteen

tessaracts: Orcus, Pallor, Tessera, Lacerta, Cudo, Frenum, Polus,

Plebs, Cura, Penates, Moles, Vitta, Urna. This we call the

Mala-Vesper-Pluvium solution.

Lastly, we have to consider whether these thirteen tessaracts are

consistent with the given Lar projections. We mark them again on Table D

with an asterisk, and we find that the projections are exactly those

given, viz. Frenum, Plebs, Moles, Urna, Cura, Vitta, Polus, Penates.

Therefore, we have not to exclude any of the thirteen, and can infer

that they constitute the shape, which produces the four different given

views or projections.

In fine, any shape in space consists of the possibilities common to the

projections of its parts upon the boundaries of that space, whatever be

the number of its dimensions. Hence the simple rule for the

determination of the shape would be to write down all the possibilities

of the sets of projections, and then cancel all those possibilities

which are not common to all. But the process adopted above is much

preferable, as through it we may realize the gradual delimitation of the

shape view by view. For once more we must remind ourselves that our

great object is, not to arrive at results by symbolical operations, but

to realize those results piece by piece through realized processes.

APPENDICES.

APPENDIX A.

This set of 100 names is useful for studying Plane Space, and forms a

square 10 × 10.

Aiōn Bios Hupar Neas Kairos Enos Thlipsis Cheimas Theion Epei

Itea Hagios Phaino Geras Tholos Ergon Pachūs Kiōn Eris Cleos

Loma Etēs Trochos Klazo Lutron Hēdūs Ischūs Paigma Hedna Demas

Numphe Bathus Pauo Euthu Holos Para Thuos Karē Pylē Spareis

Ania Eōn Seranx Mesoi Dramo Thallos Aktē Ozo Onos Magos

Notos Mēnis Lampas Ornis Thama Eni Pholis Mala Strizo Rudon

Labo Helor Rupa Rabdos Doru Epos Theos Idris Ēdē Hepo

Sophos Ichor Kaneōn Ephthra Oxis Lukē Blue Helos Peri Thelus

Eunis Limos Keedo Igde Matē Lukos Pteris Holmos Oulo Dokos

Aeido Ias Assa Muzo Hippeus Eōs Atē Akme Ōrē Gua

APPENDIX B.

The following list of names is used to denote cubic spaces. It makes a

cubic block of six floors, the highest being the sixth.

_ F Fons Plectrum Vulnus Arena Mensa Terminus

S l Testa Plausus Uva Collis Coma Nebula

i o Copia Cornu Solum Munus Rixum Vitrum

x o Ars Fervor Thyma Colubra Seges Cor

t r. Lupus Classis Modus Flamma Mens Incola

h _ Thalamus Hasta Calamus Crinis Auriga Vallum

_ F Linteum Pinnis Puppis Nuptia Aegis Cithara

F l Triumphus Curris Lux Portus Latus Funis

i o Regnum Fascis Bellum Capellus Arbor Custos

f o Sagitta Puer Stella Saxum Humor Pontus

t r. Nomen Imago Lapsus Quercus Mundus Proelium

h _ Palaestra Nuncius Bos Pharetra Pumex Tibia

_

F F Lignum Focus Ornus Lucrum Alea Vox

o l Caterva Facies Onus Silva Gelu Flumen

u o Tellus Sol Os Arma Brachium Jaculum

r o Merum Signum Umbra Tempus Corona Socius

t r. Moena Opus Honor Campus Rivus Imber

h _ Victor Equus Miles Cursus Lyra Tunica

_ F Haedus Taberna Turris Nox Domus Vinum

T l Pruinus Chorus Luna Flos Lucus Agna

h o Fulmen Hiems Ver Carina Arator Pratum

i o Oculus Ignis Aether Cohors Penna Labor

r r. Aes Pectus Pelagus Notus Fretum Gradus

d _ Princeps Dux Ventus Navis Finis Robur

_

S F Vultus Hostis Figura Ales Coelum Aura

e l Humerus Augur Ludus Clamor Galea Pes

c o Civis Ferrum Pugna Res Carmen Nubes

o o Litus Unda Rex Templum Ripa Amnis

n r. Pannus Ulmus Sedes Columba Aequor Dama

d _ Dexter Urbs Gens Monstrum Pecus Mons

_ F Nemus Sidus Vertex Nix Grando Arx

F l Venator Cerva Aper Plagua Hedera Frons

i o Membrum Aqua Caput Castrum Lituus Tuba

r o Fluctus Rus Ratis Amphora Pars Dies

s r. Turba Ager Trabs Myrtus Fibra Nauta

t _ Decus Pulvis Meta Rota Palma Terra

APPENDIX C.

The following names are used for a set of 256 Tessaracts.

FOURTH BLOCK. THIRD BLOCK.

_Fourth Floor._ _Fourth Floor._

Dolium Caballus Python Circaea Charta Cures Quaestor Cliens

Cussis Pulsus Drachma Cordax Frux Pyra Lena Procella

Porrum Consul Diota Dyka Hera Esca Secta Rugæ

Columen Ravis Corbis Rapina Eurus Gloria Socer Sequela

_Third Floor._ _Third Floor._

Alexis Planta Corymbus Lectrum Arche Agger Cumulus Cassis

Aestus Labellum Calathus Nux Arcus Ovis Portio Mimus

Septum Sepes Turtur Ordo Laurus Tigris Segmen Obolus

Morsus Aestas Capella Rheda Axis Troja Aries Fuga

_Second Floor._ _Second Floor._

Corydon Jugum Tornus Labrum Ruina Culmen Fenestra Aedes

Lac Hibiscus Donum Caltha Postis Clipeus Tabula Lingua

Senex Palus Salix Cespes Orcus Lacerta Testudo Scala

Amictus Gurges Otium Pomum Verbum Luctus Anguis Dolus

_First Floor._ _First Floor._

Odor Aprum Pignus Messor Additus Salus Clades Rana

Color Casa Cera Papaver Telum Nepos Angusta Mucro

Spes Lapis Apis Afrus Polus Penates Vulcan Ira

Vitula Clavis Fagus Cornix Cervix Securis Vinculum Furor

SECOND BLOCK. FIRST BLOCK.

_Fourth Floor._ _Fourth Floor._

Actus Spadix Sicera Anser Horreum Fumus Hircus Erisma

Auspex Praetor Atta Sonus Anulus Pluor Acies Naxos

Fulgor Ardea Prex Aevum Etna Gemma Alpis Arbiter

Spina Birrus Acerra Ramus Alauda Furca Gena Alnus

_Third Floor._ _Third Floor._

Machina Lex Omen Artus Fax Venenum Syrma Ursa

Ara Vomer Pluma Odium Mars Merces Tyro Fama

Proeda Sacerdos Hydra Luxus Spicula Mora Oliva Conjux

Cortex Mica Flagellum Mas Comes Tibicen Vestis Plenum

_Second Floor._ _Second Floor._

Ardor Rupes Pallas Arista Rostrum Armiger Premium Tribus

Pilum Glans Colus Pellis Ala Cortis Aer Fragor

Ocrea Tessara Domitor Fera Uncus Pallor Tergum Reus

Cardo Cudo Malleus Thorax Ostrum Bidens Scena Torus

_First Floor._ _First Floor._

Regina Canis Marmor Tectum Pardus Rubor Nurus Hospes

Agmen Lacus Arvus Rumor Sector Hama Remus Fortuna

Crates Cura Limen Vita Frenum Plebs Sypho Myrrha

Thyrsus Vitta Sceptrum Pax Urna Moles Saltus Acus

APPENDIX D.

The following list gives the colours, and the various uses for them.

They have already been used in the foregoing pages to distinguish the

various regions of the Tessaract, and the different individual cubes or

Tessaracts in a block. The other use suggested in the last column of the

list has not been discussed; but it is believed that it may afford great

aid to the mind in amassing, handling, and retaining the quantities of

formulae requisite in scientific training and work.

_Region of _Tessaract

_Colour._ Tessaract._ in 81 Set._ _Symbol._

Black Syce Plebs 0

White Mel Mora 1

Vermilion Alvus Uncus 2

Orange Cuspis Moles 3

Light-yellow Murex Cortis 4

Bright-green Lappa Penates 5

Bright-blue Iter Oliva 6

Light-grey Lares Tigris 7

Indian-red Crux Orcus 8

Yellow-ochre Sal Testudo 9

Buff Cista Sector + (plus)

Wood Tessaract Tessara - (minus)

Brown-green Tholus Troja ± (plus or minus)

Sage-green Margo Lacerta × (multiplied by)

Reddish Callis Tibicen ÷ (divided by)

Chocolate Velum Sacerdos = (equal to)

French-grey Far Scena ≠ (not equal to)

Brown Arctos Ostrum > (greater than)

Dark-slate Daps Aer < (less than)

Dun Portica Clipeus ∶ } signs

Orange-vermilion Talus Portio ∷ } of proportion

Stone Ops Thyrsus · (decimal point)

Quaker-green Felis Axis ∟ (factorial)

Leaden Semita Merces ∥ (parallel)

Dull-green Mappa Vulcan ∦ (not parallel)

Indigo Lixa Postis π⁄2 (90°) (at right

angles)

Dull-blue Pagus Verbum log. base 10

Dark-purple Mensura Nepos sin. (sine)

Pale-pink Vena Tabula cos. (cosine)

Dark-blue Moena Bidens tan. (tangent)

Earthen Mugil Angusta ∞ (infinity)

Blue Dos Frenum a

Terracotta Crus Remus b

Oak Idus Domitor c

Yellow Pagina Cardo d

Green Bucina Ala e

Rose Olla Limen f

Emerald Orsa Ara g

Red Olus Mars h

Sea-green Libera Pluma i

Salmon Tela Glans j

Pale-yellow Livor Ovis k

Purple-brown Opex Polus l

Deep-crimson Camoena Pilum m

Blue-green Proes Tergum n

Light-brown Lua Crates o

Deep-blue Lama Tyro p

Brick-red Lar Cura q

Magenta Offex Arvus r

Green-grey Cadus Hama s

Light-red Croeta Praeda t

Azure Lotus Vitta u

Pale-green Vesper Ocrea v

Blue-tint Panax Telum w

Yellow-green Pactum Malleus x

Deep-green Mango Vomer y

Light-green Lis Agmen z

Light-blue Ilex Comes α

Crimson Bolus Sypho β

Ochre Limbus Mica γ

Purple Solia Arcus δ

Leaf-green Luca Securis ε

Turquoise Ancilla Vinculum ζ

Dark-grey Orca Colus η

Fawn Nugæ Saltus θ

Smoke Limus Sceptrum ι

Light-buff Mala Pallor κ

Dull-purple Sors Vestis λ

Rich-red Lucta Cortex μ

Green-blue Pator Flagellum ν

Burnt-sienna Silex Luctus ξ

Sea-blue Lorica Lacus ο

Peacock-blue Passer Aries π

Deep-brown Meatus Hydra ρ

Dark-pink Onager Anguis σ

Dark Lensa Laurus τ

Dark-stone Pluvium Cudo υ

Silver Spira Cervix φ

Gold Corvus Urna χ

Deep-yellow Via Spicula ψ

Dark-green Calor Segmen ω

APPENDIX E.

A THEOREM IN FOUR-SPACE.

If a pyramid on a triangular base be cut by a plane which passes through

the three sides of the pyramid in such manner that the sides of the

sectional triangle are not parallel to the corresponding sides of the

triangle of the base; then the sides of these two triangles, if produced

in pairs, will meet in three points which are in a straight line,

namely, the line of intersection of the sectional plane and the plane of

the base.

Let A B C D be a pyramid on a triangular base A B C, and let a b c be a

section such that A B, B C, A C, are respectively not parallel to a b,

b c, a c. It must be understood that a is a point on A D, b is a point

on B D, and c a point on C D. Let, A B and a b, produced, meet in m. B C

and b c, produced, meet in n; and A C and a c, produced, meet in o.

These three points, m, n, o, are in the line of intersection of the two

planes A B C and a b c.

Now, let the line a b be projected on to the plane of the base, by

drawing lines from a and b at right angles to the base, and meeting it

in a′ b′; the line a′ b′, produced, will meet A B produced in m. If the

lines b c and a c be projected in the same way on to the base, to the

points b′ c′ and a′ c′; then B C and b′ c′ produced, will meet in n, and

A C and a′ c′ produced, will meet in o. The two triangles A B C and

a′ b′ c′ are such, that the lines joining A to a′, B to b′, and C to c′,

will, if produced, meet in a point, namely, the point on the base A B C

which is the projection of D. Any two triangles which fulfil this

condition are the possible base and projection of the section of a

pyramid; therefore the sides of such triangles, if produced in pairs,

will meet (if they are not parallel) in three points which lie in one

straight line.

A four-dimensional pyramid may be defined as a figure bounded by a

polyhedron of any number of sides, and the same number of pyramids whose

bases are the sides of the polyhedron, and whose apices meet in a point

not in the space of the base.

If a four-dimensional pyramid on a tetrahedral base be cut by a space

which passes through the four sides of the pyramid in such a way that

the sides of the sectional figure be not parallel to the sides of the

base; then the sides of these two tetrahedra, if produced in pairs, will

meet in lines which all lie in one plane, namely, the plane of

intersection of the space of the base and the space of the section.

If now the sectional tetrahedron be projected on to the base (by drawing

lines from each point of the section to the base at right angles to it),

there will be two tetrahedra fulfilling the condition that the line

joining the angles of the one to the angles of the other will, if

produced, meet in a point, which point is the projection of the apex of

the four-dimensional pyramid.

Any two tetrahedra which fulfil this condition, are the possible base

and projection of a section of a four-dimensional pyramid. Therefore, in

any two such tetrahedra, where the sides of the one are not parallel to

the sides of the other, the sides, if produced in pairs (one side of the

one with one side of the other), will meet in four straight lines which

are all in one plane.

APPENDIX F.

EXERCISES ON SHAPES OF THREE DIMENSIONS.

The names used are those given in Appendix B.

Find the shapes from the following projections:

1. Syce projections: Ratis, Caput, Castrum, Plagua.

Alvus projections: Merum, Oculus, Fulmen, Pruinus.

Moena projections: Miles, Ventus, Navis.

2. Syce: Dies, Tuba, Lituus, Frons.

Alvus: Sagitta, Regnum, Tellus, Fulmen, Pruinus.

Moena: Tibia, Tunica, Robur, Finis.

3. Syce: Nemus, Sidus, Vertex, Nix, Cerva.

Alvus: Lignum, Haedus, Vultus, Nemus, Humerus.

Moena: Dexter, Princeps, Equus, Dux, Urbs, Pullis, Gens, Monstrum,

Miles.

4. Syce: Amphora, Castrum, Myrtus, Rota, Palma, Meta, Trabs, Ratis.

Alvus: Dexter, Princeps, Moena, Aes, Merum, Oculus, Littus, Civis,

Fulmen.

Moena: Gens, Ventus, Navis, Finis, Monstrum, Cursus.

5. Syce: Castrum, Plagua, Nix, Vertex, Aper, Caput, Cerva, Venator.

Alvus: Triumphus, Tellus, Caterva, Lignum, Haedus, Pruinus, Fulmen,

Civis, Humerus, Vultus.

Moena: Pharetra, Cursus, Miles, Equus, Dux, Navis, Monstrum, Gens,

Urbs, Dexter.

ANSWERS.

The shapes are:

1. Umbra, Aether, Ver, Carina, Flos.

2. Pontus, Custos, Jaculum, Pratum, Arator, Agna.

3. Focus, Omus, Haedus, Tabema, Vultus, Hostis, Figura, Ales, Sidus,

Augur.

4. Tempus, Campus, Finis, Navis, Ventus, Pelagus, Notus, Cohors,

Aether, Carina, Res, Templum, Rex, Gens, Monstrum.

5. Portus, Arma, Sylva, Lucrum, Ornus, Onus, Os, Facies, Chorus,

Carina, Flos, Nox, Ales, Clamor, Res, Pugna, Ludus, Figura, Augur,

Humerus.

FURTHER EXERCISES IN SHAPES OF THREE DIMENSIONS.

The Names used are those given in Appendix C; and this set of exercises

forms a preparation for their use in space of four dimensions. All are

in the 27 Block (Urna to Syrma).

1. Syce: Moles, Frenum, Plebs, Sypho.

Alvus: Urna, Frenum, Uncus, Spicula, Comes.

Moena: Moles, Bidens, Tibicen, Comes, Saltus.

2. Syce: Urna, Moles, Plebs, Hama, Remus.

Alvus: Urna, Frenum, Sector, Ala, Mars.

Moena: Urna, Moles, Saltus, Bidens, Tibicen.

3. Syce: Moles, Plebs, Hama, Remus.

Alvus: Uma, Ostrum, Comes, Spicula, Frenum, Sector.

Moena: Moles, Saltus, Bidens, Tibicen.

4. Syce: Frenum, Plebs, Sypho, Moles, Hama.

Alvus: Urna, Frenum, Uncus, Sector, Spicula.

Moena: Urna, Moles, Saltus, Scena, Vestis.

5. Syce: Urna, Moles, Plebs, Hama, Remus, Sector.

Alvus: Urna, Frenum, Sector, Uncus, Spicula, Comes, Mars.

Moena: Urna, Moles, Saltus, Bidens, Tibicen, Comes.

6. Syce: Uma, Moles, Saltus, Sypho, Remus, Hama, Sector.

Alvus: Comes, Ostrum, Uncus, Spicula, Mars, Ala, Sector.

Moena: Urna, Moles, Saltus, Scena, Vestis, Tibicen, Comes, Ostrum.

7. Syce: Sypho, Saltus, Moles, Urna, Frenum, Sector.

Alvus: Urna, Frenum, Uncus, Spicula, Mars.

Moena: Saltus, Moles, Urna, Ostrum, Comes.

8. Syce: Moles, Plebs, Hama, Sector.

Alvus: Ostrum, Frenum, Uncus, Spicula, Mars, Ala.

Moena: Moles, Bidens, Tibicen, Ostrum.

9. Syce: Moles, Saltus, Sypho, Plebs, Frenum, Sector.

Alvus: Ostrum, Comes, Spicula, Mars, Ala.

Moena: Ostrum, Comes, Tibicen, Bidens, Scena, Vestis.

10. Syce: Urna, Moles, Saltus, Sypho, Remus, Sector, Frenum.

Alvus: Urna, Ostrum, Comes, Spicula, Mars, Ala, Sector.

Moena: Urna, Ostrum, Comes, Tibicen, Vestis, Scena, Saltus.

11. Syce: Frenum, Plebs, Sypho, Hama.

Alvus: Frenum, Sector, Ala, Mars, Spicula.

Moena: Urna, Moles, Saltus, Bidens, Tibicen.

ANSWERS.

The shapes are:

1. Moles, Plebs, Sypho, Pallor, Mora, Tibicen, Spicula.

2. Urna, Moles, Plebs, Hama, Cortis, Merces, Remus.

3. Moles, Bidens, Tibicen, Mora, Plebs, Hama, Remus.

4. Frenum, Plebs, Sypho, Tergum, Oliva, Moles, Hama.

5. Urna, Moles, Plebs, Hama, Remus, Pallor, Mora, Tibicen, Mars,

Merces, Comes, Sector.

6. Ostrum, Comes, Tibicen, Vestis, Scena, Tergum, Oliva, Tyro, Aer,

Remus, Hama, Sector, Merces, Mars, Ala.

7. Sypho, Saltus, Moles, Urna, Frenum, Uncus, Spicula, Mars.

8. Plebs, Pallor, Mora, Bidens, Merces, Cortis, Ala.

9. Bidens, Tibicen, Vestis, Scena, Oliva, Mora, Spicula, Mars, Ala.

10. Urna, Ostrum, Comes, Spicula, Mars, Tibicen, Vestis, Oliva, Tyro,

Aer, Remus, Sector, Ala, Saltus, Scena.

11. Frenum, Plebs, Sypho, Hama, Cortis, Merces, Mora.

APPENDIX G.

EXERCISES ON SHAPES OF FOUR DIMENSIONS.

The Names used are those given in Appendix C. The first six exercises

are in the 81 Set, and the rest in the 256 Set.

1. Mala projection: Urna, Moles, Plebs, Pallor, Cortis, Merces.

Lar projection: Urna, Moles, Plebs, Cura, Penates, Nepos.

Pluvium projection: Urna, Moles, Vitta, Cudo, Luctus, Troja.

Vesper projection: Urna, Frenum, Crates, Ocrea, Orcus, Postis, Arcus.

2. Mala: Urna, Frenum, Uncus, Pallor, Cortis, Aer.

Lar: Urna, Frenum, Crates, Cura, Lacus, Arvus, Angusta.

Pluvium: Urna, Thyrsus, Cardo, Cudo, Malleus, Anguis.

Vesper: Urna, Frenum, Crates, Ocrea, Pilum, Postis.

3. Mala: Comes, Tibicen, Mora, Pallor.

Lar: Urna, Moles, Vitta, Cura, Penates.

Pluvium: Comes, Tibicen, Mica, Troja, Luctus.

Vesper: Comes, Cortex, Praeda, Laurus, Orcus.

4. Mala: Vestis, Oliva, Tyro.

Lar: Saltus, Sypho, Remus, Arvus, Angusta.

Pluvium: Vestis, Flagellum, Aries.

Vesper: Comes, Spicula, Mars, Ara, Arcus.

5. Mala: Mars, Merces, Tyro, Aer, Tergum, Pallor, Plebs.

Lar: Sector, Hama, Lacus, Nepos, Angusta, Vulcan, Penates.

Pluvium: Comes, Tibicen, Mica, Troja, Aries, Anguis, Luctus, Securis.

Vesper: Mars, Ara, Arcus, Postis, Orcus, Polus.

6. Mala: Pallor, Mora, Oliva, Tyro, Merces, Mars, Spicula, Comes,

Tibicen, Vestis.

Lar: Plebs, Cura, Penates, Vulcan, Angusta, Nepos, Telum, Polus,

Cervix, Securis, Vinculum.

Pluvium: Bidens, Cudo, Luctus, Troja, Axis, Aries.

Vesper: Uncus, Ocrea, Orcus, Laurus, Arcus, Axis.

7. Mala: Hospes, Tribus, Fragor, Aer, Tyro, Mora, Oliva.

Lar: Hospes, Tectum, Rumor, Arvus, Angusta, Cera, Apis, Lapis.

Pluvium: Acus, Torus, Malleus, Flagellum, Thorax, Aries, Aestas,

Capella.

Vesper: Pardus, Rostrum, Ardor, Pilum, Ara, Arcus, Aestus, Septum.

8. Mala: Pallor, Tergum, Aer, Tyro, Cortis, Syrma, Ursa, Fama, Naxos,

Erisma.

Lar: Plebs, Cura, Limen, Vulcan, Angusta, Nepos, Cera, Papaver,

Pignus, Messor.

Pluvium: Bidens, Cudo, Malleus, Anguis, Aries, Luctus, Capella, Rheda,

Rapina.

Vesper: Uncus, Ocrea, Orcus, Postis, Arcus, Aestus, Cussis, Dolium,

Alexis.

9. Mala: Fama, Conjux, Reus, Torus, Acus, Myrrha, Sypho, Plebs,

Pallor, Mora, Oliva, Alpis, Acies, Hircus.

Lar: Missale, Fortuna, Vita, Pax, Furor, Ira, Vulcan, Penates, Lapis,

Apis, Cera, Pignus.

Pluvium: Torus, Plenum, Pax, Thorax, Dolus, Furor, Vinculum, Securis,

Clavis, Gurges, Aestas, Capella, Corbis.

Vesper: Uncus, Spicula, Mars, Ocrea, Cardo, Thyrsus, Cervix, Verbum,

Orcus, Polus, Spes, Senex, Septum, Porrum, Cussis, Dolium.

ANSWERS.

The shapes are:

1. Urna, Moles, Plebs, Cura, Tessara, Lacerta, Clipeus, Ovis.

2. Urna, Frenum, Crates, Ocrea, Tessara, Glans, Colus, Tabula.

3. Comes, Tibicen, Mica, Sacerdos, Tigris, Lacerta.

4. Vestis, Oliva, Tyro, Pluma, Portio.

5. Mars, Merces, Vomer, Ovis, Portio, Tabula, Testudo, Lacerta,

Penates.

6. Pallor, Tessara, Lacerta, Tigris, Segmen, Portio, Ovis, Arcus,

Laurus, Axis, Troja, Aries.

7. Hospes, Tribus, Arista, Pellis, Colus, Pluma, Portio, Calathus,

Turtur, Sepes.

8. Pallor, Tessara, Domitor, Testudo, Tabula, Clipeus, Portio,

Calathus, Nux, Lectrum, Corymbus, Circaea, Cordax.

9. Fama, Conjux, Reus, Fera, Thorax, Pax, Furor, Dolus, Scala, Ira,

Vulcan, Penates, Lapis, Palus, Sepes, Turtur, Diota, Drachma, Python.

APPENDIX H.

SECTIONS OF CUBE AND TESSARACT.

There are three kinds of sections of a cube.

1. The sectional plane, which is in all cases supposed to be infinite,

can be taken parallel to two of the opposite faces of the cube; that is,

parallel to two of the lines meeting in Corvus, and cutting the third.

2. The sectional plane can be taken parallel to one of the lines meeting

in Corvus and cutting the other two, or one or both of them produced.

3. The sectional plane can be taken cutting all three lines, or any or

all of them produced.

Take the first case, and suppose the plane cuts Dos half-way between

Corvus and Cista. Since it does not cut Arctos or Cuspis, or either of

them produced, it will cut Via, Iter, and Bolus at the middle point of

each; and the figure, determined by the intersection of the Plane and

Mala, is a square. If the length of the edge of the cube be taken as the

unit, this figure may be expressed thus:

Z X Y

0 . 0 . ¹⁄₂

showing that the Z and X lines from Corvus are not cut at all, and that

the Y line is cut at half-a-unit from Corvus.

Sections taken

Z X Y

0 . 0 . ¹⁄₄

and

Z X Y

0 . 0 . 1

would also be squares.

Take the second case.

Let the plane cut Cuspis and Dos, each at half-a-unit from Corvus, and

not cut Arctos or Arctos produced; it will also cut through the middle

points of Via and Callis. The figure produced, is a rectangle which has

two sides of one unit, and the other two are each the diagonal of a

half-unit squared.

If the plane cuts Cuspis and Dos, each at one unit from Corvus, and is

parallel to Arctos, the figure will be a rectangle which has two sides

of one unit in length; and the other two the diagonal of one unit

squared.

If the plane passes through Mala, cutting Dos produced and Cuspis

produced, each at one-and-a-half unit from Corvus, and is parallel to

Arctos, the figure will be a parallelogram like the one obtained by the

section

Z X Y

0 . ¹⁄₂ . ¹⁄₂.

This set of figures will be expressed

Z X Y Z X Y Z X Y

0 . ¹⁄₂ . ¹⁄₂ 0 . 1 . 1 0 . 1¹⁄₂ . 1¹⁄₂

It will be seen that these sections are parallel to each other; and that

in each figure Cuspis and Dos are cut at equal distances from Corvus.

We may express the whole set thus:--

Z X Y

O . I . I

it being understood that where Roman figures are used, the numbers do

not refer to the length of unit cut off any given line from Corvus, but

to the proportion between the lengths. Thus

Z X Y

O . I . II

means that Arctos is not cut at all, and that Cuspis and Dos are cut,

Dos being cut twice as far from Corvus as is Cuspis.

These figures will also be rectangles.

Take the third case.

Suppose Arctos, Cuspis, and Dos are each cut half-way. This figure is an

equilateral triangle, whose sides are the diagonal of a half-unit

squared. The figure

Z X Y

1 . 1 . 1

is also an equilateral triangle, and the figure

Z X Y

1¹⁄₂ . 1¹⁄₂ . 1¹⁄₂

is an equilateral hexagon.

It is easy for us to see what these shapes are, and also, what the

figures of any other set would be, as

Z X Y

I . II . II

or

Z X Y

I . II . III

but we must learn them as a two-dimensional being would, so that we may

see how to learn the three-dimensional sections of a tessaract.

It is evident that the resulting figures are the same whether we fix the

cube, and then turn the sectional plane to the required position, or

whether we fix the sectional plane, and then turn the cube. Thus, in the

first case we might have fixed the plane, and then so placed the cube

that one plane side coincided with the sectional plane, and then have

drawn the cube half-way through, in a direction at right angles to the

plane, when we should have seen the square first mentioned. In the

second case

(Z X Y)

(O . I . I)

we might have put the cube with Arctos coinciding with the plane and

with Cuspis and Dos equally inclined to it, and then have drawn the cube

through the plane at right angles to it until the lines (Cuspis and Dos)

were cut at the required distances from Corvus. In the third case we

might have put the cube with only Corvus coinciding with the plane and

with Cuspis, Dos, and Arctos equally inclined to it (for any of the

shapes in the set

Z X Y)

I . I . I)

and then have drawn it through as before. The resulting figures are

exactly the same as those we got before; but this way is the best to

use, as it would probably be easier for a two-dimensional being to think

of a cube passing through his space than to imagine his whole space

turned round, with regard to the cube.

We have already seen (p. 117) how a two-dimensional being would observe

the sections of a cube when it is put with one plane side coinciding

with his space, and is then drawn partly through.

Now, suppose the cube put with the line Arctos coinciding with his

space, and the lines Cuspis and Dos equally inclined to it. At first he

would only see Arctos. If the cube were moved until Dos and Cuspis were

each cut half-way, Arctos still being parallel to the plane, Arctos

would disappear at once; and to find out what he would see he would have

to take the square sections of the cube, and find on each of them what

lines are given by the new set of sections. Thus he would take Moena

itself, which may be regarded as the first section of the square set.

One point of the figure would be the middle point of Cuspis, and since

the sectional plane is parallel to Arctos, the line of intersection of

Moena with the sectional plane will be parallel to Arctos. The required

line then cuts Cuspis half-way, and is parallel to Arctos, therefore it

cuts Callis half-way.

[Illustration: Fig. 21.]

Next he would take the square section half-way between Moena and Murex.

He knows that the line Alvus of this section is parallel to Arctos, and

that the point Dos at one of its ends is half-way between Corvus and

Cista, so that this line itself is the one he wants (because the

sectional plane cuts Dos half-way between Corvus and Cista, and is

parallel to Arctos). In Fig. 21 the two lines thus found are shown. a b

is the line in Moena, and c d the line in the section. He must now find

out how far apart they are. He knows that from the middle point of

Cuspis to Corvus is half-a-unit, and from the middle point of Dos to

Corvus is half-a-unit, and Cuspis and Dos are at right angles to each

other; therefore from the middle point of Cuspis to the middle point of

Dos is the diagonal of a square whose sides are half-a-unit in length.

This diagonal may be written d (¹⁄₂)². He would also see that from the

middle point of Callis to the middle point of Via is the same length;

therefore the figure is a parallelogram, having two of its sides, each

one unit in length, and the other two each d (¹⁄₂)².

He could also see that the angles are right, because the lines a c and

b d are made up of the X and Y directions, and the other two, a b and

d, are purely Z, and since they have no tendency in common, they are at

right angles to each other.

[Illustration: Fig. 22.]

If he wanted the figure made by

Z X Y

0 . 1¹⁄₂ . 1¹⁄₂

it would be a little more difficult. He would have to take Moena, a

section halfway between Moena and Murex, Murex and another square which

he would have to regard as an _imaginary_ section half-a-unit further Y

than Murex (Fig. 22). He might now draw a ground plan of the sections;

that is, he would draw Syce, and produce Cuspis and Dos half-a-unit

beyond Nugæ and Cista. He would see that Cadus and Bolus would be cut

half-way, so that in the half-way section he would have the point a

(Fig. 23), and in Murex the point c. In the imaginary section he would

have g; but this he might disregard, since the cube goes no further than

Murex. From the points c and a there would be lines going Z, so that

Iter and Semita would be cut half-way.

[Illustration: _Groundplan of Sections shown in Fig. 22._

Fig. 23.]

He could find out how far the two lines a b and c d (Fig. 22) are apart

by referring d and b to Lama, and a and c to Crus.

In taking the third order of sections, a similar method may be followed.

[Illustration: Fig. 24.]

Suppose the sectional plane to cut Cuspis, Dos, and Arctos, each at one

unit from Corvus. He would first take Moena, and as the sectional plane

passes through Ilex and Nugæ, the line on Moena would be the diagonal

passing through these two points. Then he would take Murex, and he would

see that as the plane cuts Dos at one unit from Corvus, all he would

have is the point Cista. So the whole figure is the Ilex to Nugæ

diagonal, and the point Cista.

Now Cista and Ilex are each one inch from Corvus, and measured along

lines at right angles to each other; therefore, they are d (1)² from

each other. By referring Nugæ and Cista to Corvus he would find that

they are also d (1)² apart; therefore the figure is an equilateral

triangle, whose sides are each d (1)².

Suppose the sectional plane to pass through Mala, cutting Cuspis, Dos,

and Arctos each at unit from Corvus. To find the figure, the plane-being

would have to take Moena, a section half-way between Moena and Murex,

Murex, and an imaginary section half-a-unit beyond Murex (Fig. 24). He

would produce Arctos and Cuspis to points half-a-unit from Ilex and

Nugæ, and by joining these points, he would see that the line passes

through the middle points of Callis and Far (a, b, Fig. 24). In the last

square, the imaginary section, there would be the point m; for this is

1¹⁄₂ unit from Corvus measured along Dos produced. There would also be

lines in the other two squares, the section and Murex, and to find these

he would have to make many observations. He found the points a and b

(Fig. 24) by drawing a line from r to s, r and s being each 1¹⁄₂ unit

from Corvus, and simply seeing that it cut Callis and Far at the middle

point of each. He might now imagine a cube Mala turned about Arctos, so

that Alvus came into his plane; he might then produce Arctos and Dos

until they were each unit long, and join their extremities, when he

would see that Via and Bucina are each cut half-way. Again, by turning

Syce into his plane, and producing Dos and Cuspis to points 1¹⁄₂ unit

from Corvus and joining the points, he would see that Bolus and Cadus

are cut half-way. He has now determined six points on Mala, through

which the plane passes, and by referring them in pairs to Ilex, Olus,

Cista, Crus, Nugæ, Sors, he would find that each was d (¹⁄₂)² from the

next; so he would know that the figure is an equilateral hexagon. The

angles he would not have got in this observation, and they might be a

serious difficulty to him. It should be observed that a similar

difficulty does not come to us in our observation of the sections of a

tessaract: for, if the angles of each side of a solid figure are

determined, the solid angles are also determined.

There is another, and in some respects a better, way by which he might

have found the sides of this figure. If he had noticed his plane-space

much, he would have found out that, if a line be drawn to cut two other

lines which meet, the ratio of the parts of the two lines cut off by the

first line, on the side of the angle, is the same for those lines, and

any other two that are parallel to them. Thus, if a b and a c (Fig. 25)

meet, making an angle at a, and b c crosses them, and also crosses a′ b′

and a′ c′, these last two being parallel to a b and a c, then a b ∶

a c ∷ a′ b′ ∶ a′ c′.

[Illustration: Fig. 25]

If the plane-being knew this, he would rightly assume that if three

lines meet, making a solid angle, and a plane passes through them, the

ratio of the parts between the plane and the angle is the same for those

three lines, and for any other three parallel to them.

In the case we are dealing with he knows that from Ilex to the point on

Arctos produced where the plane cuts, it is half-a-unit; and as the Z,

X, and Y lines are cut equally from Corvus, he would conclude that the X

and Y lines are cut the same distance from Ilex as the Z line, that is

half-a-unit. He knows that the X line is cut at 1¹⁄₂ units from Corvus;

that is, half-a-unit from Nugæ: so he would conclude that the Z and Y

lines are cut half-a-unit from Nugæ. He would also see that the Z and X

lines from Cista are cut at half-a-unit. He has now six points on the

cube, the middle points of Callis, Via, Bucina, Cadus, Bolus, and Far.

Now, looking at his square sections, he would see on Moena a line going

from middle of Far to middle of Callis, that is, a line d (¹⁄₂)² long.

On the section he would see a line from middle of Via to middle of Bolus

d (1)² long, and on Murex he would see a line from middle of Cadus to

middle of Bucina, d (¹⁄₂)² long. Of these three lines a b, c d, e f,

(Fig. 24)--a b and e f are sides, and c d is a section of the required

figure. He can find the distances between a and c by reference to

Ilex, between b and d by reference to Nugæ, between c and e by reference

to Olus, and between d and f by reference to Crus; and he will find that

these distances are each d (¹⁄₂)².

Thus, he would know that the figure is an equilateral hexagon with its

sides d (¹⁄₂)² long, of which two of the opposite points (c and d) are d

(1)² apart, and the only figure fulfilling all these conditions is an

equilateral and equiangular hexagon.

Enough has been said about sections of a cube, to show how a plane-being

would find the shapes in any set as in

Z X Y

I . II . II

or

Z X Y

I . I . II.

He would always have to bear in mind that the ratio of the lengths of

the Z, X, and Y lines is the same from Corvus to the sectional plane as

from any other point to the sectional plane. Thus, if he were taking a

section where the plane cuts Arctos and Cuspis at one unit from Corvus

and Dos at one-and-a-half, that is where the ratio of Z and of X to Y is

as two to three, he would see that Dos itself is not cut at all; but

from Cista to the point on Dos produced is half-a-unit; therefore from

Cista, the Z and X lines will be cut at ²⁄₃ of ¹⁄₂ unit from Cista.

It is impossible in writing to show how to make the various sections of

a tessaract; and even if it were not so, it would be unadvisable; for

the value of doing it is not in seeing the shapes themselves, so much as

in the concentration of the mind on the tessaract involved in the

process of finding them out.

Any one who wishes to make them should go carefully over the sections of

a cube, not looking at them as he himself can see them, or determining

them as he, with his three-dimensional conceptions, can; but he must

limit his imagination to two dimensions, and work through the problems

which a plane-being would have to work through, although to his higher

mind they may be self-evident. Thus a three-dimensional being can see at

a glance, that if a sectional plane passes through a cube at one unit

each way from Corvus, the resulting figure is an equilateral triangle.

If he wished to prove it, he would show that the three bounding lines

are the diagonals of equal squares. This is all a two-dimensional being

would have to do; but it is not so evident to him that two of the lines

are the diagonals of squares.

Moreover, when the figure is drawn, we can look at it from a point

outside the plane of the figure, and can thus see it all at once; but

he who has to look at it from a point in the plane can only see an edge

at a time, or he might see two edges in perspective together.

Then there are certain suppositions he has to make. For instance, he

knows that two points determine a line, and he assumes that three points

determine a plane, although he cannot conceive any other plane than the

one in which he exists. We assume that four points determine a solid

space. Or rather, we say that _if_ this supposition, together with

certain others of a like nature, are true, we can find all the sections

of a tessaract, and of other four-dimensional figures by an infinite

solid.

When any difficulty arises in taking the sections of a tessaract, the

surest way of overcoming it is to suppose a similar difficulty occurring

to a two-dimensional being in taking the sections of a cube, and, step

by step, to follow the solution he might obtain, and then to apply the

same or similar principles to the case in point.

A few figures are given, which, if cut out and folded along the lines,

will show some of the sections of a tessaract. But the reader is

earnestly begged not to be content with _looking_ at the shapes only.

That will teach him nothing about a tessaract, or four-dimensional

space, and will only tend to produce in his mind a feeling that “the

fourth dimension” is an unknown and unthinkable region, in which any

shapes may be right, as given sections of its figures, and of which any

statement may be true. While, in fact, if it is the case that the laws

of spaces of two and three dimensions may, with truth, be carried on

into space of four dimensions; then the little our solidity (like the

flatness of a plane-being) will allow us to learn of these shapes and

relations, is no more a matter of doubt to us than what we learn of two-

and three-dimensional shapes and relations.

There are given also sections of an octa-tessaract, and of a

tetra-tessaract, the equivalents in four-space of an octahedron and

tetrahedron.

A tetrahedron may be regarded as a cube with every alternate corner cut

off. Thus, if Mala have the corner towards Corvus cut off as far as the

points Ilex, Nugæ, Cista, and the corner towards Sors cut off as far as

Ilex, Nugæ, Lama, and the corner towards Crus cut off as far as Lama,

Nugæ, Cista, and the corner towards Olus cut off as far as Ilex, Lama,

Cista, what is left of the cube is a tetrahedron, whose angles are at

the points Ilex, Nugæ, Cista, Lama. In a similar manner, if every

alternate corner of a tessaract be cut off, the figure that is left is a

tetra-tessaract, which is a figure bounded by sixteen regular

tetrahedrons.

[Illustration: (i)

Fig. 26.

Fig. 27.

Fig. 27.

Fig. 26.]

[Illustration: (ii)

Fig. 28.

Fig. 29.

Fig. 30.]

[Illustration: (iii)

Fig. 31.

Fig. 32.]

[Illustration: (iv)

Fig. 33.

Fig. 34.

Fig. 35.]

[Illustration: (v)

Fig. 36.

Fig. 37.

Fig. 38.]

[Illustration: (vi)

Fig. 39.

Fig. 40.

Fig. 41.]

The octa-tessaract is got by cutting off every corner of the tessaract.

If every corner of a cube is cut off, the figure left is an octa-hedron,

whose angles are at the middle points of the sides. The angles of the

octa-tessaract are at the middle points of its plane sides. A careful

study of a tetra-hedron and an octa-hedron as they are cut out of a cube

will be the best preparation for the study of these four-dimensional

figures. It will be seen that there is much to learn of them, as--How

many planes and lines there are in each, How many solid sides there are

round a point in each.

A DESCRIPTION OF FIGURES 26 TO 41.

Z X Y W

Z X Y W {26 is a section taken 1 . 1 . 1 . 1

I . I . I . I {27 „ „ „ 1¹⁄₂ . 1¹⁄₂ . 1¹⁄₂ . 1¹⁄₂

{28 „ „ „ 2 . 2 . 2 . 2

Z X Y W

Z X Y W {29 is a section taken 1 . 1 . 1 . ¹⁄₂

II . II . II . I {30 „ „ „ 1¹⁄₂ . 1¹⁄₂ . 1¹⁄₂ . ³⁄₄

{31 „ „ „ 2 . 2 . 2 . 1

32 „ „ „ 2¹⁄₂ . 2¹⁄₂ . 2¹⁄₂ . 1¹⁄₄

The above are sections of a tessaract. Figures 33 to 35 are of a

tetra-tessaract. The tetra-tessaract is supposed to be imbedded in a

tessaract, and the sections are taken through it, cutting the Z, X and Y

lines equally, and corresponding to the figures given of the sections of

the tessaract.

Figures 36, 37, and 38 are similar sections of an octa-tessaract.

Figures 39, 40, and 41 are the following sections of a tessaract.

Z X Y W

Z X Y W {39 is a section taken 0 . ¹⁄₂ . ¹⁄₂ . ¹⁄₂

O . I . I . I {40 „ „ „ 0 . 1 . 1 . 1

{41 „ „ „ 0 . 1¹⁄₂ . 1¹⁄₂ . 1¹⁄₂

It is clear that there are four orders of sections of every

four-dimensional figure; namely, those beginning with a solid, those

beginning with a plane, those beginning with a line, and those beginning

with a point. There should be little difficulty in finding them, if the

sections of a cube with a tetra-hedron, or an octa-hedron enclosed in

it, are carefully examined.

APPENDIX K.

[Illustration: MODEL 1. MALA.]

COLOURS: MALA, LIGHT-BUFF.

_Points_: Corvus, Gold. Nugæ, Fawn. Crus, Terra-cotta. Cista, Buff.

Ilex, Light-blue. Sors, Dull-purple. Lama, Deep-blue. Olus, Red.

_Lines_: Cuspis, Orange. Bolus, Crimson. Cadus, Green-grey. Dos, Blue.

Arctos, Brown. Far, French-grey. Daps, Dark-slate. Bucina, Green.

Callis, Reddish. Iter, Bright-blue. Semita, Leaden. Via, Deep-yellow.

_Surfaces_: Moena, Dark-blue. Proes, Blue-green. Murex, Light-yellow.

Alvus, Vermilion. Mel, White. Syce, Black.

[Illustration: MODEL 2. MARGO.]

COLOURS: MARGO, SAGE-GREEN.

_Points_: Spira, Silver. Ancilla, Turquoise. Mugil, Earthen. Panax,

Blue-tint. Felis, Quaker-green. Passer, Peacock-blue. Talus,

Orange-vermilion. Solia, Purple.

_Lines_: Luca, Leaf-green. Mappa, Dull-green. Mensura, Dark-purple.

Opex, Purple-brown. Pagus, Dull-blue. Onager, Dark-pink. Vena,

Pale-pink. Lixa, Indigo. Tholus, Brown-green. Calor, Dark-green.

Livor, Pale-yellow. Lensa, Dark.

_Surfaces_: Silex, Burnt-sienna. Sal, Yellow-ochre. Portica, Dun.

Crux, Indian-red. Lares, Light-grey. Lappa, Bright-green.

[Illustration: MODEL 3. LAR.]

COLOURS: LAR, BRICK-RED.

_Points_: Spira, Silver. Ancilla, Turquoise. Mugil, Earthen. Panax,

Blue-tint. Corvus, Gold. Nugæ, Fawn. Crus, Terra-cotta. Cista, Buff.

_Lines_: Luca, Leaf-green. Mappa, Dull-green. Mensura, Dark-purple.

Opex, Purple-brown. Ops, Stone. Limus, Smoke. Offex, Magenta. Lis,

Light-green. Cuspis, Orange. Bolus, Crimson. Cadus, Green-grey. Dos,

Blue.

_Surfaces_: Lotus, Azure. Olla, Rose. Lorica, Sea-blue. Lua,

Bright-brown. Syce, Black. Lappa, Bright-green.

[Illustration: MODEL 4. VELUM.]

COLOURS: VELUM, CHOCOLATE.

_Points_: Felis, Quaker-green. Passer, Peacock-blue. Talus,

Orange-vermilion. Solia, Purple. Ilex, Light-blue. Sors, Dull-purple.

Lama, Deep-blue. Olus, Red.

_Lines_: Tholus, Brown-green. Calor, Dark-green. Livor, Pale-yellow.

Lensa, Dark. Lucta, Rich-red. Pator, Green-blue. Libera, Sea-green.

Orsa, Emerald. Callis, Reddish. Iter, Bright-blue. Semita, Leaden.

Via, Deep-yellow.

_Surfaces_: Limbus, Ochre. Meatus, Deep-brown. Mango, Deep-green.

Croeta, Light-red. Mel, White. Lares, Light-grey.

[Illustration: MODEL 5. VESPER.]

COLOURS: VESPER, PALE-GREEN.

_Points_: Spira, Silver. Corvus, Gold. Cista, Buff. Panax, Blue-tint.

Felis, Quaker-green. Ilex, Light-blue. Olus, Red. Solia, Purple.

_Lines_: Ops, Stone. Dos, Blue. Lis, Light-green. Opex, Purple-brown.

Pagus, Dull-blue. Arctos, Brown. Bucina, Green. Lixa, Indigo. Lucta,

Rich-red. Via, Deep-yellow. Orsa, Emerald. Lensa, Dark.

_Surfaces_: Pagina, Yellow. Alvus, Vermilion. Camoena, Deep-crimson.

Crux, Indian-red. Croeta, Light-red. Lua, Light-brown.

[Illustration: MODEL 6. IDUS.]

COLOURS: IDUS, OAK.

_Points_: Ancilla, Turquoise. Nugæ, Fawn. Crus, Terra-cotta. Mugil,

Earthen. Passer, Peacock-blue. Sors, Dull-purple. Lama, Deep-blue.

Talus, Orange-vermilion.

_Lines_: Limus, Smoke. Bolus, Crimson. Offex, Magenta. Mappa,

Dull-green. Onager, Dark-pink. Far, French-grey. Daps, Dark-slate.

Vena, Pale-pink. Pator, Green-blue. Iter, Bright-blue. Libera,

Sea-green. Calor, Dark-green.

_Surfaces_: Pactum, Yellow-green. Proes, Blue-green. Orca, Dark-grey.

Sal, Yellow-ochre. Meatus, Deep-brown. Olla, Rose.

[Illustration: MODEL 7. PLUVIUM.]

COLOURS: PLUVIUM, DARK-STONE.

_Points_: Spira, Silver. Ancilla, Turquoise. Nugæ, Fawn. Corvus, Gold.

Felis, Quaker-green. Passer, Peacock-blue. Sors, Dull-purple. Ilex,

Light-blue.

_Lines_: Luca, Leaf-green. Limus, Smoke. Cuspis, Orange. Ops, Stone.

Pagus, Dull-blue. Onager, Dark-pink. Far, French-grey. Arctos, Brown.

Tholos, Brown-green. Pator, Green-blue. Callis, Reddish. Lucta,

Rich-red.

_Surfaces_: Silex, Burnt-Sienna. Pactum, Yellow-green. Moena,

Dark-blue. Pagina, Yellow. Limbus, Ochre. Lotus, Azure.

[Illustration: MODEL 8. TELA.]

COLOURS: TELA, SALMON.

_Points_: Panax, Blue-tint. Mugil, Earthen. Crus, Terra-cotta. Cista,

Buff. Solia, Purple. Talus, Orange-vermilion. Lama, Deep-blue. Olus,

Red.

_Lines_: Mensura, Dark-purple. Offex, Magenta. Cadus, Green-grey. Lis,

Light-green. Lixa, Indigo. Vena, Pale-pink. Daps, Dark-slate. Bucina,

Green. Livor, Pale-yellow. Libera, Sea-green. Semita, Leaden. Orsa,

Emerald.

_Surfaces_: Portica, Dun. Orca, Dark-grey. Murex, Light-yellow.

Camoena, Deep-crimson. Mango, Deep-green. Lorica, Sea-blue.

[Illustration: MODEL 9. SECTION BETWEEN MALA AND MARGO.]

COLOURS: INTERIOR OR TESSARACT, WOOD.

_Points_ (_Lines_): Ops, Stone. Limus, Smoke. Offex, Magenta. Lis,

Light-green. Lucta, Rich-red. Pator, Green-blue. Libera, Sea-green.

Orsa, Emerald.

_Lines_ (_Surfaces_): Lotus, Azure. Olla, Rose. Lorica, Sea-blue. Lua

Bright-brown. Pagina, Yellow. Pactum, Yellow-green. Orca, Dark-grey.

Camoena, Deep-crimson. Limbus, Ochre. Meatus, Deep-brown. Mango,

Deep-green. Croeta, Light red.

_Surfaces_ (_Solids_): Pluvium, Dark-stone. Idus, Oak. Tela, Salmon.

Vesper, Pale-green. Velum, Chocolate. Lar, Brick-red.

[Illustration: MODEL 10. SECTION BETWEEN LAR AND VELUM.]

COLOURS: INTERIOR OR TESSARACT, WOOD.

_Points_ (_Lines_): Pagus, Dull-blue. Onager, Dark-pink. Vena,

Pale-pink. Lixa, Indigo. Arctos, Brown. Far, French-grey. Daps,

Dark-slate. Bucina, Green.

_Lines_ (_Surfaces_): Silex, Burnt-sienna. Sal, Yellow-ochre. Portica,

Dun. Crux, Indian-red. Pagina, Yellow. Pactum, Yellow-green. Orca,

Dark-grey. Camoena, Deep-crimson. Moena, Dark-blue. Proes, Blue-green.

Murex, Light-yellow. Alvus, Vermilion.

_Surfaces_ (_Solids_): Pluvium, Dark-stone. Idus, Oak. Tela, Salmon.

Vesper, Pale-green. Mala, Light-buff. Margo, Sage-green.

[Illustration: MODEL 11. SECTION BETWEEN VESPER AND IDUS.]

COLOURS: INTERIOR OR TESSARACT, WOOD.

_Points_ (_Lines_): Luca, Leaf-green. Cuspis, Orange. Cadus,

Green-grey. Mensura, Dark-purple. Tholus, Brown-green. Callis,

Reddish. Semita, Leaden. Livor, Pale-yellow.

_Lines_ (_Surfaces_): Lotus, Azure. Syce, Black. Lorica, Sea-blue.

Lappa, Bright-green. Silex, Burnt-sienna. Moena, Dark-blue. Murex,

Light-yellow. Portica, Dun. Limbus, Ochre. Mel, White. Mango,

Deep-green. Lares, Light-grey.

_Surfaces_ (_Solids_): Pluvium, Dark-stone. Mala, Light-buff. Tela,

Salmon. Margo, Sage-green. Velum, Chocolate. Lar, Brick-red.

[Illustration: MODEL 12. SECTION BETWEEN PLUVIUM AND TELA.]

COLOURS: INTERIOR OR TESSARACT, WOOD.

_Points_ (_Lines_): Opex, Purple-brown. Mappa, Dull-green. Bolus,

Crimson. Dos, Blue. Lensa, Dark. Calor, Dark-green. Iter, Bright-blue.

Via, Deep-yellow.

_Lines_ (_Surfaces_): Lappa, Bright-green. Olla, Rose. Syce, Black.

Lua, Bright-brown. Crux, Indian-red. Sal, Yellow-ochre. Proes,

Blue-green. Alvus, Vermilion. Lares, Light-grey. Meatus, Deep-brown.

Mel, White. Croeta, Light-red.

_Surfaces_ (_Solids_): Margo, Sage-green. Idus, Oak. Mala, Light-buff.

Vesper, Pale-green. Velum, Chocolate. Lar, Brick-red.

Transcriber’s Notes

Lay-out and formatting have been optimised for browser html (available

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to fit the available width. Wherever possible, these splits and

re-arrangements have been done so that the meaning of the lay-out has

been retained. The contents of Tables A through D have been replaced

with place holders; their meanings are listed below the diagrams.

Inconsistencies in spelling (Mœnas v. Moenas; Praeda v. Proeda),

hyphenation (Deep-blue v. Deep blue, etc.) have been retained.

Page 197, row starting Sophos: the last letter of Blue has been

assumed.

Changes made:

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text paragraphs. Indications for the location of illustrations (To

face p. ...) have been removed; the illustrations concerned have been

moved to where they are discussed.

Some minor obvious typographical errors have been corrected silently.

Page 42: ... the flat, being ... changed to ... the flat being ...

Page 127: Cube itself: considered to be the table header rather than a

table element

Page 175: is all Ana our space changed to is all Ana in our space

Page 187: Clipens changed to Clipeus; legend Y added to right-hand

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Page 219: Part II. Appendix K. changed to Appendix K. cf. other

Appendices.

End of Project Gutenberg's A New Era of Thought, by Charles Howard Hinton