Geometry of Complex Numbers (Dover Books on Mathematics) by Hans Schwerdtfeger

By Hans Schwerdtfeger

Illuminating, greatly praised publication on analytic geometry of circles, the Moebius transformation, and 2-dimensional non-Euclidean geometries. "This publication could be in each library, and each specialist in classical functionality conception may be accustomed to this fabric. the writer has played a different carrier by means of making this fabric so with ease obtainable in one book." — Mathematical Review.

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Geometry of Complex Numbers (Dover Books on Mathematics)

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Account of the texts in [Waerden 1962]). Speaking purely geometrically, we can deal with this issue easily when generalising the problem of doubling the square (cf. 1). Speaking algebraically, we are dealing with the extension of extracting the square root to extracting the cubic root, which is an issue that had already been addressed by the Babylonians. 1) This formation of two geometrical means x, y between two given quantities (here: a, 2a) corresponds to a pure cubic equation, as springs to mind easily.

The diagonals of a rectangle are equal and halve each other. 6. The peripheral angle in a semi-circle is a right one. 36 2 Geometry in the Greek-Hellenistic era and late Antiquity The theorem stated last, known as Thales’ theorem, has been passed on by the female historian Pamphile (1st century AD), as reported by Diogenes Laertius (3rd century). In order to express his gratitude for recognising this fundamental fact, Thales is said to have sacrificed an ox to honour the gods! As we already know from the paragraphs on pre-Greek mathematics, the line segment and/or the straight line and the circle (next to the point) belong to the oldest geometrical elements.

Hence, it is clear that we must be dealing with a transcendental curve! Yet, we can easily describe it, since it is constructed by two simple movements. Imagine a square, the upper side of which moves parallel to the starting position with constant speed to the lower side. At the same time, the left side of the square turns clockwise around the lower corner point with constant angle speed in a manner such that both movements start and end simultaneously. In this case, the upper end of the turning line segment describes a quadrant within the square.

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