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When the assumption is that no such line exists, you enter a geometry that is called the elliptical geometry. In multivariate analysis the Mahalanobis distance weights the differences in variability of the variables. Distances are measured in units of standard deviation. It is easy to see that the lines of constant standard deviation are ellipses. Brocard pointed out the construction of a specific ellipse, inscribed in a triangle: Brocard's ellipse. The Steiner ellipse is the ellipse through the three vertices of a triangle and with the centroid of the triangle as its center.

3) Newton came to 72 species, including the conic sections . This classification was criticized by Euler, because of it's lack of generality. And in fact, Newton missed sex species of curves. There are also other classifications, ranging from 57 to 219 types. The one with the 219 classes was given by Plücker. 4) Dick Nickalls describes in the Mathematical Gazette a new approach to solve the cubic. It gives new geometric insights into how and why the standard Cardan algebra works. I distinguish the following cubic curves: • circular cubic ♦ Apollonian cubic ♦ conchoid of de Sluze • cissoid • cubic spiral • divergent parabola • folium of Descartes th • Newton's diverging parabola (3 species of Newton) • oblique strophoid • resonance curve • serpentine • strophoid th • trident of Newton (66 species of Newton) • trisectrix of MacLaurin notes 1) Curves by Sir Isaac Newton in Lexicon Technicum by John Harris, published in London in 1710.

This definition − which gives the essence of the circle − was already formulated by Euclid (300 BC) in book III of his 'Elements'. That's why you can draw the curve with a pair of compasses. The circle's form remains intact while turning, what makes her very useful as lid for closing jars. And also for a watch with turning hands. The diameter of a screw is a circle too. But man got the greatest advantage from the insight that for a wheel of a cart not the square, but the circle is the most suitable form.

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