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Geometry In 1741, Alexis Clairaut wrote a book called Èléments de Géométrie. The book outlines the basic concepts of geometry. Geometry in the 1700's was complex to the average learner. It was considered to be a dry subject. Clairaut saw this trend, and wrote the book in an attempt to make the subject more interesting for the average learner. He believed that instead of having students repeatedly work problems that they did not fully understand, it was imperative for them to make discoveries themselves in a form of active, experiential learning. [1] He begins the book by comparing geometric shapes to measurements of land, as it was a subject that most anyone could relate to. He covers topics from lines, shapes, and even some three dimensional objects. Throughout the book, he continuously relates different concepts such as physics, astrology, and other branches of mathematics to geometry. Some of the theories and learning methods outlined in the book are still used by teachers today, in geometry and other topics. [2]

Èléments de Géométrie

Subsections:

Part One: Of the Means That It Was Most Natural to Employ for the Measurement of Land Geometric theories:

 Lines:
   Straight Lines
   Perpendicular lines
   Parallel Lines
   Angles

 Shapes:
   Rectangles
   Squares
   Triangles
   Parallelograms
   Circles
   Other polygons

Part two: Of the Geometrical Method of Comparing Rectilinear Figures

Part three: Of the Measurement and Properties of Circular Figures

Part 4: Of the Measurements of Solids and Their Surfaces

The shape of the Earth:

In 1736, Clairaut sent his findings to the Royal Society of London. The writing was later published the following year in a volume of Philosophical Transactions. Initially, Clairaut disagrees with Newton's theory on the shape of the earth. In the article, he outlines several key problems that effectively disprove Newton's calculations, and provides some solutions to the complications. The issues addressed include calculating gravitational attraction, the rotation of an ellipsoid on its axis, and the difference in density of an ellipsoid on its axes. At the end Clairaut writes that: "It appears even Sir Isaac Newton was of the opinion, that it was necessary the Earth should be more dense toward the center, in order to be so much the flatter at the poles: and that it followed from this greater flatness, that gravity increased so much the more from the equator towards the Pole." This conclusion suggests not only that the Earth is of an oblate ellipsoid shape, but it is flattened more at the poles and is wider at the center.

Focus on Astronomical Motion:

One of most controversial issues of the 18th century was the three body problem, or how the Earth, moon, and Sun are attracted to one another. With the use of the recently founded Leibnizian calculus, Clairaut was able to solve the problem using four differential equations. However, these equations only offered approximate measurement, and no exact calculations. Another issue still remained with the three body problem; how the moon rotates on its "apsides--that is, the apogee, or point at which the Moon is farthest from Earth, and the perigee, at which it is nearest. Those two points rotate around Earth with a period of nine years, as illustrated in figure 2. Even the great Newton could account for only half of the motion of the apsides."

Clairaut also used applied mathematics to study Venus. He took very accurate measurements of the planets size and distance from the Earth. This was the first precise reckoning of the planet's size

Subsection: Arguments on lunar motion

Clairaut's theorem: History:

In the 18th century, there was much controversy over the shape of the Earth. In Sir Issac Newton's "Principia", he outlines his theory and calculations on the shape of the Earth. Newton theorized that the Earth had an ellipsoid shape, with flattening at the poles. Using geometric calculations, he gave a concrete argument as to the hypothetical Ellipsoid shape of the Earth.

The goal of "Principia" was not to provide exact answer for natural phenomena, but to theorize potential solutions to these unresolved factors in science. He pushed for scientists to look further into the unexplained variables. Two prominent researchers that he inspired were Alexis Clairaut and Pierre Louis Maupertuis. They both sought to prove the validity of Newton's theory on the shape of the Earth. In order to do so, they went on an expedition to Lapland, a region in Northern Europe shared by Sweden, Noway, and Finland, in an attempt to accurately measure the meridian arc. By doing so, they could hypothetically gauge the shape of the Earth. Clairaut found that Newton's calculations were incorrect, and wrote a letter to the Royal Society of London with his findings. The society published an article in "Philosophical Transactions" the following year in 1737 that revealed his findings. Clairaut showed how Newton's equations were incorrect, and did not prove an ellipsoid shape to the Earth. However, he corrected problems with the theory, that in effect would prove Newton's theory correct. The article did not provide an valid equation to back up his argument. This created much controversy in the scientific community.

It was not until Clairaut wrote "Théorie de la figure de la terre" in 1943 that a proper answer was provided. In it, he promulgated what is more formally known today as Clairaut's theorm.

Edit Meridian Arc