## The Axioms of Projective Geometry |

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### Common terms and phrases

A. N. Whitehead AA'U anharmonic ratio annexed figure axioms of order axis belongs to segm class g class of points coincide commutative law complete quadrangle concord concurrent considering the quadrangle coplanar deductions defined definition Desargues Descriptive Geometry distinct collinear points distinct points enlarged segment entities equation equivalent everywhere dense existence theorem finite number further axioms Harm ABCD harmonic conjugates harmonic system homogeneous coordinates intersect Lemma lies in segm Math member of G0 member of K modified lines number of points numeration-system OEU OA U2 OAU2 OBU2 OCU2 OEU2 Pappus Pascal's theorem Pieri point-pairs Projective Geometry projective relation projective transformation proof propositions prosp prospectivity proved quad quadrangular transformation quaternions respect S-concord S-orders ABC satisfied segm ABC self-corresponding point set of axioms set of points Staudt straight lines subclass of G0 system of magnitudes tf-concord three distinct collinear three non-collinear points Type

### Popular passages

Page iv - Thus the points mentioned in the axioms are not a special determinate class of entities ; but they are in fact any entities whatever, which happen to be inter-related in such a manner, that the axioms are true when they are considered as referring to those entities and their inter-relations. Accordingly—since the class of points is undetermined—the axioms are not propositions at all: they are propositional functions*.

Page vii - If A , B, C are three non-collinear points, the plane ABC is the class of points lying on lines joining C to the points of the line AB.

Page vi - This general sciences may be defined thus: given any class of entities K, the subclasses of K form a new class of classes, the science of classification is the study of sets of classes selected from this new class so as to possess certain assigned properties. For example, in the traditional Aristotelian branch of classification by species and genera, the selected set from the class of subclasses of K are...

Page 33 - A to A, B to B, C to C, and so on, without ever allowing one line to cross another or pass through another company's station.

Page v - Law of Contradiction,' a set of entities cannot satisfy inconsistent axioms. Thus the existence theorem for a set of axioms proves their consistency. Seemingly this is the only possible method of proof of consistency. But the only rigid proofs of existence theorems are those which are deductions from the premises of formal Logic. Thus there can be no formal proof of the consistency of the logical premises themselves.

Page vii - This set of subclasses is to be such that any two points lie on one and only one line, and that any line possesses at least three points.

Page 1 - Definition. If A, B, C are three non-collinear points, the plane ABC is the class of points lying on the lines joining A to the various points on BC.

Page 24 - Dedekind t. axiom, or as enunciating the Dedekind property, is as follows. XIX. If u is any segment of a line, there are two points A and B, such that, if P be any member of u distinct from A and B, segm (APB) is all of u with the possible exception of either or both of A and B which may also belong to u. Note that the axioms of order, viz. XVI, XVII, XVIII, and this axiom need only be enunciated for one line. Then by projection they can be proved for every line.

Page 29 - C, and A, B', C', and A', B, C', and A', B', C, are such that the three sides of the quadrangle EFGH through any set are concurrent in one of the angular points of the quadrilateral; while the four sets, A', B', C', aud A', B, C, and A, B' C, and A, B, C' are such that the three sides through any set form a triangle.

Page 10 - When two figures can be derived one from the other by a single projection, they are said to be 'in perspective