Charles Howard Hinton

APPENDIX A.

This set of 100 names is useful for studying Plane Space, and forms a square 10 × 10.

Aion	Bios	Hupar	Neas	Kairos	Enos	Thlipsis	Cheimas	Theion	Epei
Itea	Hagios	Phaino	Geras	Tholos	Ergon	Pachus	Kion	Eris	Cleos
Loma	Etes	Trochos	Klazo	Lutron	Hedus	Ischus	Paigma	Hedna	Demas
Numphe	Bathus	Pauo	Euthu	Holos	Para	Thuos	Kare	Pyle	Spareis
Ania	Eon	Seranx	Mesoi	Dramo	Thallos	Akte	Ozo	Onos	Magos
Notos	Menis	Lampas	Ornis	Thama	Eni	Pholis	Mala	Strizo	Rudon
Labo	Helor	Rupa	Rabdos	Doru	Epos	Theos	Idris	Ede	Hepo
Sophos	Ichor	Kaneon	Ephthra	Oxis	Luke	Blue	Helos	Peri	Thelus
Eunis	Limos	Keedo	Igde	Mate	Lukos	Pteris	Holmos	Oulo	Dokos
Aeido	Ias	Assa	Muzo	Hippeus	Eos	Ate	Akme	Ore	Gua

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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.

S i x t h	F l o o r.	Fons	Plectrum	Vulnus	Arena	Mensa	Terminus
		Testa	Plausus	Uva	Collis	Coma	Nebula
		Copia	Cornu	Solum	Munus	Rixum	Vitrum
		Ars	Fervor	Thyma	Colubra	Seges	Cor
		Lupus	Classis	Modus	Flamma	Mens	Incola
		Thalamus	Hasta	Calamus	Crinis	Auriga	Vallum
F i f t h	F l o o r.	Linteum	Pinnis	Puppis	Nuptia	Aegis	Cithara
		Triumphus	Curris	Lux	Portus	Latus	Funis
		Regnum	Fascis	Bellum	Capellus	Arbor	Custos
		Sagitta	Puer	Stella	Saxum	Humor	Pontus
		Nomen	Imago	Lapsus	Quercus	Mundus	Proelium
		Palaestra	Nuncius	Bos	Pharetra	Pumex	Tibia
F o u r t h	F l o o r.	Lignum	Focus	Ornus	Lucrum	Alea	Vox
		Caterva	Facies	Onus	Silva	Gelu	Flumen
		Tellus	Sol	Os	Arma	Brachium	Jaculum
		Merum	Signum	Umbra	Tempus	Corona	Socius
		Moena	Opus	Honor	Campus	Rivus	Imber
		Victor	Equus	Miles	Cursus	Lyra	Tunica
T h i r d	F l o o r.	Haedus	Taberna	Turris	Nox	Domus	Vinum
		Pruinus	Chorus	Luna	Flos	Lucus	Agna
		Fulmen	Hiems	Ver	Carina	Arator	Pratum
		Oculus	Ignis	Aether	Cohors	Penna	Labor
		Aes	Pectus	Pelagus	Notus	Fretum	Gradus
		Princeps	Dux	Ventus	Navis	Finis	Robur
S e c o n d	F l o o r.	Vultus	Hostis	Figura	Ales	Coelum	Aura
		Humerus	Augur	Ludus	Clamor	Galea	Pes
		Civis	Ferrum	Pugna	Res	Carmen	Nubes
		Litus	Unda	Rex	Templum	Ripa	Amnis
		Pannus	Ulmus	Sedes	Columba	Aequor	Dama
		Dexter	Urbs	Gens	Monstrum	Pecus	Mons
F i r s t	F l o o r.	Nemus	Sidus	Vertex	Nix	Grando	Arx
		Venator	Cerva	Aper	Plagua	Hedera	Frons
		Membrum	Aqua	Caput	Castrum	Lituus	Tuba
		Fluctus	Rus	Ratis	Amphora	Pars	Dies
		Turba	Ager	Trabs	Myrtus	Fibra	Nauta
		Decus	Pulvis	Meta	Rota	Palma	Terra

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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

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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.

Colour.	Region of Tessaract.	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 of proportion
Orange-vermilion	Talus	Portio	?
Stone	Ops	Thyrsus	· (decimal point)
Quaker-green	Felis	Axis	? (factorial)
Leaden	Semita	Merces	? (parallel)
Dull-green	Mappa	Vulcan	? (not parallel)
Indigo	Lixa	Postis	p/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	8 (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	a
Crimson	Bolus	Sypho
Ochre	Limbus	Mica	?
Purple	Solia	Arcus	d
Leaf-green	Luca	Securis	e
Turquoise	Ancilla	Vinculum	?
Dark-grey	Orca	Colus	?
Fawn	NugÆ	Saltus	?
Smoke	Limus	Sceptrum	?
Light-buff	Mala	Pallor	?
Dull-purple	Sors	Vestis	?
Rich-red	Lucta	Cortex
Green-blue	Pator	Flagellum	?
Burnt-sienna	Silex	Luctus	?
Sea-blue	Lorica	Lacus	?
Peacock-blue	Passer	Aries	p
Deep-brown	Meatus	Hydra	?
Dark-pink	Onager	Anguis	s
Dark	Lensa	Laurus	t
Dark-stone	Pluvium	Cudo	?
Silver	Spira	Cervix	f
Gold	Corvus	Urna	?
Deep-yellow	Via	Spicula	?
Dark-green	Calor	Segmen	?

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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 ABCD be a pyramid on a triangular base ABC, and let abc be a section such that AB, BC, AC, are respectively not parallel to ab, bc, ac. It must be understood that a is a point on AD, b is a point on BD, and c a point on CD. Let, AB and ab, produced, meet in m. BC and bc, produced, meet in n; and AC and ac, produced, meet in o. These three points, m, n, o, are in the line of intersection of the two planes ABC and abc.

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 AB produced in m. If the lines bc and ac be projected in the same way on to the base, to the points b'c' and a'c'; then BC and b'c' produced, will meet in n, and AC and a'c' produced, will meet in o. The two triangles ABC 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 ABC 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.

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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.

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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.

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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: Z0 . X0 . Y¹/₂ 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 Z0 . X0 . Y¹/₄ and Z0 . X0 . Y1 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 Z0 . X¹/₂ . Y¹/₂.

This set of figures will be expressed

Z0 . X¹/₂ . Y¹/₂ Z0 . X1 . Y1 Z0 . X1¹/₂ . Y1¹/₂

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:—

ZO . XI . YI

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 ZO . XI . YII 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 Z1 . X1 . Y1 is also an equilateral triangle, and the figure Z1¹/₂ . X1¹/₂ . Y1¹/₂ 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 ZI . XII . YII or ZI . XII . YIII 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 (ZO . XI . YI) 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 ZI . XI . YI) 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.

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. ab is the line in Moena, and cd 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 ac and bd are made up of the X and Y directions, and the other two, ab 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 Z0 . X1¹/₂ . Y1¹/₂ 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 ab and cd (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 ab and ac (Fig. 25) meet, making an angle at a, and bc crosses them, and also crosses a'b' and a'c', these last two being parallel to ab and ac, then ab: ac? 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 ab, cd, ef, (Fig. 24)—ab and ef are sides, and cd 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 ZI . XII . YII or ZI . XI . YII

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.