Arithmetic and geometrical progression

Use in the book

Arithmetic first opens an unintended door in §152. Nine unit Squares form a Square three inches on each side, and the grandfather uses the arrangement to explain the second power. The child asks what the third power means geometrically. Since calculating a third power is already part of his instruction, the denial that it can signify a figure exposes an awkward division in the curriculum: arithmetic is allowed an operation whose possible spatial meaning geometry forbids him to investigate.

The Sphere then reconstructs the same ladder in dialogue. His initial terminal-point series begins with one, the Point itself, and proceeds through two and four to eight. His boundary series begins with zero, since a Point has no lower-dimensional side, and proceeds through two and four to six. Abbott makes the Square name the progressions and give their next terms. Nothing in the difficulty is due to inability to calculate. The pupil knows that the next numbers are eight and six, but cannot yet turn those counts into a coherent perceptual object.

When the Square subsequently reasons towards four dimensions, he omits the opening terms and restates the operative sequences as two, four, eight, sixteen and two, four, six, eight. The projected object has sixteen corners and eight cubic boundary cells. The eight are not its square faces or its edges. Here ordinary side has already been stretched to mean a boundary of the immediately lower dimension, and keeping the vocabulary straight prevents an apparent numerical contradiction.

The serial pattern also changes the relation of speaker and listener. What had looked like a closed examination, with the visitor holding the answer, becomes a procedure capable of producing an answer he refuses to hear. The Square quotes his master's own formula against him. Mathematics gives the pupil a resource that does not depend on rank or permission, although calculation alone cannot make the object visible or establish that a corresponding inhabited world exists.

The later failed lessons retain this distinction. The Council excludes the enumeration of imaginary sides and angles from the kind of demonstration it will accept. That restriction is politically interested, but the reader has already seen the genuine gap between a count and an image. Abbott therefore makes arithmetic powerful without presenting it as an automatic substitute for every other kind of evidence.

Places to read

1. §152“raise numbers to the third power” — An authorized operation provokes an unauthorized question
2. §162345“1, 2, 4” — Terminal points form the opening geometrical series
3. §162345“0, 2, 4” — Lower-dimensional boundaries form the arithmetical series
4. §192345“sixteen terminal points” — The anticipated fourth-dimensional count
5. §2223“other than the enumeration of imaginary sides and angles” — The Council narrows acceptable proof

In this edition

The mathematical notes identify the object now generally called a tesseract, but Abbott does not use that later term in the romance. The lexicon retains his language and explains the counts without silently replacing it with modern terminology.

Arithmetical

Increasing by a fixed added amount when applied to the progression here. Zero, two and four boundaries lead to six square faces for the Cube. Printed form: Arithmetical (§ 162345). See Arithmetic and geometrical progression.

Geometrical Progression

A sequence with a constant multiplying ratio. The Sphere uses successive doubling to predict the Cube's eight terminal points from the earlier counts one, two and four. Printed form: Geometrical Progression (§ 162345). See Arithmetic and geometrical progression.

Terminal Points

Endpoints or corners used in the dimensional sequence. Their number doubles from Point through Line and Square to Cube, and is distinguished from the count of lower-dimensional boundary faces. Printed form: terminal Points (§ 162345). See Arithmetic and geometrical progression.

Third power

A number multiplied by itself twice, as three times three times three. The grandson asks for the geometrical meaning of an operation he has already learned arithmetically. Printed form: third power (§ 152). See Arithmetic and geometrical progression.

One Dimension behind the thing (contextual phrase)

§162345. “one Dimension behind the thing”

The Sphere uses behind metaphorically for a decrease in dimensional degree, not for a position to the rear. The boundary of a Line consists of Points, that of a Square of Lines, and that of a Cube of square faces. This is why the same word side cannot keep an identical geometrical object as its referent through the lesson. The phrase lets familiar language carry a general relation, but the guide repeatedly has to explain that his sides are what the Square would call solids. In §192345 the pupil extends this relation to eight bounding Cubes for a four-dimensional figure. The construction is coherent even though the proposed world is not witnessed.

See also. Arithmetic and geometrical progression · Fourth Dimension