A treatise on the analytical geometry of the point, line, by John Casey

By John Casey

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Additional info for A treatise on the analytical geometry of the point, line, circle, and conic sections, containing an account of its most recent extensions, with numerous examples.

Example text

Let the two distinct planes be E and F. 4, there are two points, A and B, such that fA; Bg  E \ F. 1, there is one and only one line, ! ! AB, containing A and B. 3, ! E \ F D AB. 5. Given a plane E and a point A belonging to E, there exists a line L such that L  E and A … L. Proof. 5 there exist three noncollinear points P, Q, and R belonging ! ! to E. 3, these lines are all contained in E; and since P, Q, and R are noncollinear, the lines are distinct. The proof now splits into two cases. ) In this case, the line determined by the other two points can be taken to be L.

It is well known that the isomorphic image of a group is a group, the isomorphic image of a field is a field, and the isomorphic image of a vector space is a vector space. Thus, if one can establish (as we do in later chapters) an isomorphism between a field F and another set F 0 which is equipped with two operations C and “ ” , the set F 0 is automatically a field, and likewise for a vector space. This relieves us of the tedium of proving all the various field (or vector space) properties on the second set.

Conversely, if U and V are the same space, they have the same dimensions. In this work we will be mainly concerned with vector spaces of dimension 1, 2, or 3. Since the vector space axioms are a subset of the field axioms, F is a vector space over itself, having dimension 1. If A ¤ O is a point of a vector space V, fxA j x 2 Fg (that is, a “line” through the origin) is a vector subspace of V having dimension 1. Thus, the word space in vector space may at times mean “line”; it may also mean “plane,” although not all lines (or planes) in a vector space are vector spaces.

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