Alexis Claude Clairaut

(May 7, 1713 ? May 17, 1765)

 

Clairaut was a leading French mathematician (geometer) who expressed his mathematical abilities in early childhood. He was one of the most renown mathematicians and physicists of the 18th century.? At age 10, Alexis knew infinitesimal calculus [Clairaut?s Differential Equation: ^{y = xD}_x^{y + f(D}_x^{y) => y = cx + f(c)].} ?http://www.math.uiuc.edu/~inik/teaching/math341s03/lectmaterial/clairaut.pdf

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He was a mathematical genius who already at the age of twelve had been called to visit the Academy of Sciences in Paris. At twelve he submitted his first paper to the Academy of Sciences, and at eighteen he published a book containing important additions to geometry that won him membership in the Academy in 1731.

 

Clairaut was one of the scientists who accompanied Maupertuis to Lapland to collect data that was used to determine the shape of the earth. In 1743 he published his Theorie de la figure de la terre (Theory on the shape of the earth) that calculated, more precisely than Newton had done, the form that a rotating body mechanically assumes from the natural gravitation of its parts. http://www.visitvoltaire.com/e_alexis-claude_clairaut.htm

 

He was famous for replacing the fifth postulate by his own postulate in 1741 in the text "Elements De Geometrie" (Elements of Geometry). So, we are going to prove that Clairaut?s axiom is logically equivalent in neutral geometry to the parallel postulate:

 

 

 

 

 

 

Assume the existence of rectangles.

 

Conversely, assume Clairaut?s axiom.

 

All triangles have angle sum 180 degrees (By NG Th. 7)

 

By introducing diagonal, all convex quadrilaterals have angle sum 360 degrees (NG Th.4, Corollary 2).

 

Let S be a perpendicular bisector of Y to line PQ. (NG. Prop. 5)

 

S is the same side of m as Y and Q because line SY is parallel to m (NG. Th.1, Corollary 1)

 

Quadrilateral PXYS(which has 3 right angles)? is know as a rectangle (def. of rectangle).

 

PS = XY (The opposite sides of a rectangle are congruent ? Def. of rectangle)

 

Y can be chosen on the given ray of n so that XY>PQ? (Aristotle?s Axiom)

 

Then, PS>PQ and P*Q*S.

 

Y is on the same side of l as S, but on the opposite side of l from P.

 

Therefore, l meets n at some point between P and Y.