The Cubists are entitled to the serious attention of all who find enjoyment in the colored puzzle pictures of the Sunday newspapers. Of course there is no reason for choosing the cube as a symbol, except that it is probably less fitted than any other mathematical expression for any but the most formal decorative art. There is no reason why people should not call themselves Cubists, or Octagonists, or Parallelopipedonists, or Knights of the Isosceles Triangle, or Brothers of the Cosine, if they so desire; as expressing anything serious and permanent, one term is as fatuous as another.

I... began... with simple series... of quantities in arithmetic proportion, or... their squares, cubes, etc. and then... their square roots, cube roots, etc. and powers composed of these... square roots of cubes etc. or... whatever... composites, whether the power was rational or... irrational. ...Whence a general theorem emerged... Proposition 64. But also... the quadrature... of the simple parabola... of all higher parabolas, and their complements, which no-one before... achieved. I... had enlarged geometry; for... there may now be taught by a single proposition the quadrature or all higher of infinitely many kinds... by one general method. ...I felt it would be welcome ...to the mathematical world ...also I saw ...the same doctrine widened ...I have related everything, whether conoids or pyramids, either erect or inclined, to cylinders and prisms. ...I saw ...as a direct consequence an almost completed teaching of spirals; and indeed I have taught the comparison with a circle... But also that teaching... was capable of extension...

Cubism is not a formula, it is not a school. Cubism is a philosophy, a point of view in the universe. It is like standing at a certain point on a mountain and looking around. If you go higher, things will look different; if you go lower, again they will look different. It is a point of view.

Many think that Cubism is an art of transition, an experiment which is to bring ulterior results. Those who think that way have not understood it. Cubism is not either a seed or a foetus, but an art dealing primarily with forms, and when a form is realized it is there to live its own life. A mineral substance, having geometric formation, is not made so for transitory purposes, it is to remain what it is and will always have its own form.

Mathematics, trigonometry, chemistry, psychoanalysis, music, and what not have been related to cubism to give it an easier interpretation. All this has been pure literature, not to say nonsense, which brought bad results, blinding people with theories. Cubism has kept itself within the limits and limitations of painting, never pretending to go beyond it.

He... gave thirteen forms of the cubic which have positive roots, these having already been given by Omar Kayyam.

The Circle is a Geometrical Line, not because it may be express'd by an Æquation, but because its Description is a Postulate. It is not the Simplicity of the Æquation, but the Easiness of the Description, which is to determine the Choice of our Lines for the Construction of Problems. For the Æquation that expresses a Parabola, is more simple than That that expresses a Circle, and yet the Circle, by reason of its more simple Construction, is admitted before it. The Circle and the Conick Sections, if you regard the Dimension of the Æquations, are of the fame Order, and yet the Circle is not number'd with them in the Construction of Problems, but by reason of its simple Description, is depressed to a lower Order, viz. that of a right Line; so that it is not improper to express that by a Circle that may be expressed by a right Line. But it is a Fault to construct that by the Conick Sections which may be constructed by a Circle. Either therefore you must take your Law and Rule from the Dimensions of Æquations as observ'd in a Circle, and so take away the Distinction between Plane and Solid Problems; or else you must grant, that that Law is not so strictly to be observ'd in Lines of superior Kinds, but that some, by reason of their more simple Description, may be preferr'd to others of the same Order, and may be number'd with Lines of inferior Orders in the Construction of Problems.

Circles are the only curvilineal plane figures considered in the elements of geometry. If they could have allowed... these as similar polygons of an infinite number of sides (as some have done who pretend to abridge their demonstrations), after proving that any similar polygons inscribed in circles are in the duplicate ratio of the diameters, they would have immediately extended this to the circles themselves and would have considered the second proposition of the twelfth book of the Elements as an easy corollary from the first. But there is ground to think that they would not have admitted a demonstration of this kind. It was a fundamental principle with them, that the difference of any two unequal quantities, by which the greater exceeds the lesser, may be added to itself till it shall exceed any proposed finite quantity of the same kind: and that they founded their propositions concerning curvilineal figures upon this principle... is evident from the demonstrations, and from the express declaration of Archimedes, who acknowledges it to be the foundation...[of] his own discoveries, and cites it as assumed by the antients in demonstrating all their propositions of this kind. But this principle seems to be inconsistent with... admitting... an infinitely little quantity or difference, which, added to itself any number of times, is never supposed to become equal to any finite quantity whatsoever.

Cubism is an anatomical chart of a way of seeing external objects. But I want to confuse the meaning of the act of looking.

PUPIL: Are the roots of words square? PROFESSOR: Square or cube. That depends. PUPIL: I've got a toothache.

A nonsingular cubic surface F can be rationally represented upon a plane α if and only if F contains a rational point.

Cubism ('multi-locationalism') is one of the painterly forms of acoustic space.

[L]et us assign the figures that have come into being in our theory to fire and earth and water and air. To earth let us give the cubical form; for earth is least mobile of the four and most plastic of bodies: and that substance must possess this nature in the highest degree which has its bases most stable. Now of the triangles which we assumed as our starting-point that with equal sides is more stable than that with unequal; and of the surfaces composed of the two triangles the equilateral quadrangle necessarily is more stable than the equilateral triangle... Now among all these that which has the fewest bases must naturally in all respects be the most cutting and keen of all, and also the most nimble, seeing it is composed of the smallest number of similar parts... Let it be determined then... that the solid body which has taken the form of the pyramid [tetrahedron] is the element and seed of fire; and the second in order of generation let [octahedron] us say to be that of air, and the third [icosahedron] that of water. Now all these bodies we must conceive as being so small that each single body in the several kinds cannot for its smallness be seen by us at all; but when many are heaped together, their united mass is seen...

The Pythagoreans were fascinated by the regular solids, symmetrical three-dimensional objects all of whose sides are the same regular polygon. The cube is the simplest example, having six squares as sides. There are an infinite number of regular polygons, but only five regular solids. (The proof of this statement, a famous example of mathematical reasoning, is given in Appendix 2.) For some reason, knowledge of a solid called the dodecahedron having twelve pentagons as sides seemed to them dangerous. It was mystically associated with the Cosmos. The other four regular solids were identified, somehow, with the four “elements” then imagined to constitute the world; earth, fire, air and water. The fifth regular solid must then, they thought, correspond to some fifth element that could only be the substance of the heavenly bodies. (This notion of a fifth essence is the origin of our word quintessence.) Ordinary people were to be kept ignorant of the dodecahedron.