### Chapter XLIV. Dimensions and Complex Numbers

- § 351. Retrospect
- § 352. Geometry is the science of series of two or more dimensions
- § 353. Non-Euclidean geometry
- § 354. Definition of dimensions
- § 355. Remarks on the definition
- § 356. The definition of dimensions is purely logical
- § 357. Complex numbers and universal algebra
- § 358. Algebraical generalization of number
- § 359. Definition of complex numbers
- § 360. Remarks on the definition

### Chapter XLV. Projective Geometry

- § 361. Recent threefold scrutiny of geometrical principles
- § 362. Projective, descriptive, and metrical geometry
- § 363. Projective points and straight lines
- § 364. Definition of the plane
- § 365. Harmonic ranges
- § 366. Involutions
- § 367. Projective generation of order
- § 368. Möbius nets
- § 369. Projective order presupposed in assigning irrational coordinates
- § 370. Anharmonic ratio
- § 371. Assignment of coordinates to any point in space
- § 372. Comparison of projective and Euclidean geometry
- § 373. The principle of duality

### Chapter XLVI. Descriptive Geometry

- § 374. Distinction between projective and descriptive geometry
- § 375. Method of Pasch and Peano
- § 376. Method employing serial relations
- § 377. Mutual independence of axioms
- § 378. Logical definition of the class of descriptive spaces
- § 379. Parts of straight lines
- § 380. Definition of the plane
- § 381. Solid geometry
- § 382. Descriptive geometry applies to Euclidean and hyperbolic, but not elliptic space
- § 383. Ideal elements
- § 384. Ideal points
- § 385. Ideal lines
- § 386. Ideal planes
- § 387. The removal of a suitable selection of points renders a projective space descriptive

### Chapter XLVII. Metrical Geometry

- § 388. Metrical geometry presupposes projective or descriptive geometry
- § 389. Errors in Euclid
- § 390. Superposition is not a valid method
- § 391. Errors in Euclid (continued)
- § 392. Axioms of distance
- § 393. Stretches
- § 394. Order as resulting from distance alone
- § 395. Geometries which derive the straight line from distance
- § 396. In most spaces, magnitude of divisibility can be used instead of distance
- § 397. Meaning of magnitude of divisibility
- § 398. Difficulty of making distance independent of stretch
- § 399. Theoretical meaning of measurement
- § 400. Definition of angle
- § 401. Axioms concerning angles
- § 402. An angle is a stretch of rays, not a class of points
- § 403. Areas and volumes
- § 404. Right and left

### Chapter XLVIII. Relation of Metrical to Projective and Descriptive Geometry

- § 405. Non-quantitative geometry has no metrical presuppositions
- § 406. Historical development of non-quantitative geometry
- § 407. Non-quantitative theory of distance
- § 408. In descriptive geometry
- § 409. And in projective geometry
- § 410. Geometrical theory of imaginary point-pairs
- § 411. New projective theory of distance

### Chapter XLIX. Definitions of Various Spaces

### Chapter L. The Continuity of Space

- § 416. The continuity of a projective space
- § 417. The continuity of metrical space
- § 418. An axiom of continuity enables us to dispense with the postulate of the circle
- § 419. Is space prior to points?
- § 420. Empirical premisses and induction
- § 421. There is no reason to desire our premisses to be self-evident
- § 422. Space is an aggregate of points, not a unity

### Chapter LI. Logical Arguments Against Points

- § 423. Absolute and relative position
- § 424. Lotze's arguments against absolute position
- § 425. Lotze's theory of relations
- § 426. The subject-predicate theory of propositions
- § 427. Lotze's three kinds of Being
- § 428. Argument from the identity of indiscernibles
- § 429. Points are not active
- § 430. Argument from the necessary truths of geometry
- § 431. Points do not imply one another

### Chapter LII. Kant's Theory of Space

The Principles of Mathematics was written by Bertrand Russell, and published in in 1903. It is now available in the Public Domain.