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An Elementary Course In Synthetic Projective Geometry by Derrick Norman Lehmer

By Derrick Norman Lehmer

Meant to offer, As easily As attainable, The necessities of artificial Projective Geometry - Chapters: One-To-One Correspondence - kinfolk among basic types In One-To-One Correspondence With one another - blend of 2 Projectively comparable basic varieties - Point-Rows Of the second one Order - Pencils Of Rays Of the second one Order - Poles And Polars - Metrical homes Of The Conic Sections - Involution - Metrical houses Of Involutions - at the background of man-made Projective Geometry - Index

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21] 22 An Elementary Course in Synthetic Projective Geometry If, now, two forms are perspectively related to a third, any four harmonic elements of one must correspond to four harmonic elements in the other. We take this as our definition of projective correspondence, and say: 36. Definition of projectivity. Two fundamental forms are protectively related to each other when a one-to-one correspondence exists between the elements of the two and when four harmonic elements of one correspond to four harmonic elements of the other.

It must be remembered, however, that the expression "infinitely distant point" must not be taken literally. When we say that two parallel lines meet "at infinity," we really mean that they do not meet at all, and the only reason for using the expression is to avoid tedious statement of exceptions and restrictions to our theorems. We ought therefore to consider the drawing of a line parallel to a given line as a different accomplishment from the drawing of the line joining two given points. It is a remarkable consequence of the last theorem that a parallel to a given line and the mid-point of a given segment are equivalent data.

FIG. 7 41. Parallels and mid-points 25 40. Projective theorems and metrical theorems. Linear construction. This theorem is the connecting link between the general protective theorems which we have been considering so far and the metrical theorems of ordinary geometry. Up to this point we have said nothing about measurements, either of line segments or of angles. Desargues's theorem and the theory of harmonic elements which depends on it have nothing to do with magnitudes at all. Not until the notion of an infinitely distant point is brought in is any mention made of distances or directions.

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