Euclidean geometry, elementary geometry of two and three dimensions (plane and solid geometry), is based largely on the Elements of the Greek mathematician Euclid (fl. c.300 B.C.). In 1637, René Descartes showed how numbers can be used to describe points in a plane or in space and to express geometric relations in algebraic form, thus founding analytic geometry, of which algebraic geometry is a further development (see Cartesian coordinates). The problem of representing three-dimensional objects on a two-dimensional surface was solved by Gaspard Monge, who invented descriptive geometry for this purpose in the late 18th cent. differential geometry, in which the concepts of the calculus are applied to curves, surfaces, and other geometrical objects, was founded by Monge and C. F. Gauss in the late 18th and early 19th cent. The modern period in geometry begins with the formulations of projective geometry by J. V. Poncelet (1822) and of non-Euclidean geometry by N. I. Lobachevsky (1826) and János Bolyai (1832). Another type of non-Euclidean geometry was discovered by Bernhard Riemann (1854), who also showed how the various geometries could be generalized to any number of dimensions.

These tidbits, would have been evidence as projects predceding as "towers across valleys" amd "between mountain measures," to become what they are today. Allows us to se in ways that we are not used too, had we not learnt of this progression and design that lead from one to another.

**8.6 On Gauss's Mountains**

One of the most famous stories about Gauss depicts him measuring the angles of the great triangle formed by the mountain peaks of Hohenhagen, Inselberg, and Brocken for evidence that the geometry of space is non-Euclidean. It's certainly true that Gauss acquired geodetic survey data during his ten-year involvement in mapping the Kingdom of Hanover during the years from 1818 to 1832, and this data included some large "test triangles", notably the one connecting the those three mountain peaks, which could be used to check for accumulated errors in the smaller triangles. It's also true that Gauss understood how the intrinsic curvature of the Earth's surface would theoretically result in slight discrepancies when fitting the smaller triangles inside the larger triangles, although in practice this effect is negligible, because the Earth's curvature is so slight relative to even the largest triangles that can be visually measured on the surface. Still, Gauss computed the magnitude of this effect for the large test triangles because, as he wrote to Olbers, "the honor of science demands that one understand the nature of this inequality clearly". (The government officials who commissioned Gauss to perform the survey might have recalled Napoleon's remark that Laplace as head of the Department of the Interior had "brought the theory of the infinitely small to administration".) It is sometimes said that the "inequality" which Gauss had in mind was the possible curvature of space itself, but taken in context it seems he was referring to the curvature of the Earth's surface.

One had to recognize the process that historically proceeded in our overviews "to non-euclidean perspectives," "geometrically enhanced" through to our present day headings, expeirmentallly.

**Michelson interferometer**(27 Mar 2006 wikipedia)

Michelson interferometer is the classic setup for optical interferometry and was invented by Albert Abraham Michelson. Michelson, along with Edward Morley, used this interferometer for the famous Michelson-Morley experiment in which this interferometer was used to prove the non-existence of the luminiferous aether. See there for a detailed discussion of its principle.

But Michelson had already used it for other purposes of interferometry, and it still has many other applications, e.g. for the detection of gravitational waves, as a tunable narrow band filter, and as the core of Fourier transform spectroscopy. There are also some interesting applications as a "nulling" instrument that is used for detecting planets around nearby stars. But for most purposes, the geometry of the Mach-Zehnder interferometer is more useful.

A quick summation below leads one onto the idea of what experimental validation has done for us. Very simply, the graduation of interferometer design had been taken to astronomical proportions?

Today the Count expands on this for us by showing other information on expeirmental proposals. How fitting that this historical drama has been shown here, in a quick snapshot. As well the need for understanding the "principal inherent" in the project below.

VLBI is a geometric technique: it measures the time difference between the arrival at two Earth-based antennas of a radio wavefront emitted by a distant quasar. Using large numbers of time difference measurements from many quasars observed with a global network of antennas, VLBI determines the inertial reference frame defined by the quasars and simultaneously the precise positions of the antennas. Because the time difference measurements are precise to a few picoseconds, VLBI determines the relative positions of the antennas to a few millimeters and the quasar positions to fractions of a milliarcsecond. Since the antennas are fixed to the Earth, their locations track the instantaneous orientation of the Earth in the inertial reference frame. Relative changes in the antenna locations from a series of measurements indicate tectonic plate motion, regional deformation, and local uplift or subsidence.

**See**: