Table of Contents

1. Chapter XLIV. Dimensions and Complex Numbers

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

2. Chapter XLV. Projective Geometry

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

3. Chapter XLVI. Descriptive Geometry

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

4. Chapter XLVII. Metrical Geometry

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

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

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

6. Chapter XLIX. Definitions of Various Spaces

   1. § 412. All kinds of spaces are definable in purely logical terms
   2. § 413. Definition of projective spaces of three dimensions
   3. § 414. Definition of Euclidean spaces of three dimensions
   4. § 415. Definition of Clifford's spaces of two dimensions

7. Chapter L. The Continuity of Space

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

8. Chapter LI. Logical Arguments Against Points

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

9. Chapter LII. Kant's Theory of Space

   1. § 432. The present work is diametrically opposed to Kant
   2. § 433. Summary of Kant's theory
   3. § 434. Mathematical reasoning requires no extra-logical element
   4. § 435. Kant's mathematical antinomies
   5. § 436. Summary of Part VI