CHAPTER XI.

Chapter 339,805 wordsPublic domain

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.

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.

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.

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.

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

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.

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.

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.

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.

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′.

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

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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 at www.gutenberg.org); some versions and narrow windows may not display all elements of the book as intended, depending on the hard- and software used and their settings. For this text file, several diagrams and tables have been split or otherwise re-arranged in order 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:

Footnotes, tables, diagrams and illustrations have been moved outside 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 side grid axes

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