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The Measurement of Everything

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The Measurement of Everything The measurement of everything Image:IGM from the shape of the Earth to that of a statue Fernando Sansò 1 Academic authorities, ladies and gentlemen, dear friends and colleagues, I am sure that you all expect me to start this talk by saying how much I’m honoured and happy to be here. And indeed, I’m profoundly honoured to be awarded a “laurea honoris causa” by the University of Aristoteles and happy to be here, celebrating more than me, more than my modest personal contribution to a science, the new greening of this science in itself, the old lady, Geodesy, born together with the theory of mechanics and astronomy developed by the great scientists of the XVII, XVIII century, now regenerated into a young beautiful lady with her feet on the blossoming ground of the virtual representation of the geometry of our everyday world and its head high in space, looking down to the earth and its global evolution and even further to the moon and the planets. This science with a modern name is called Geodesy and Geomatics, at least here in Greece, certainly in Italy and in many other countries, though not everywhere. But I will return later on this point. So let me start with the title of my talk “The measurement of every thing; from the shape of the earth to that of a statue”. The two key words here are “measuring and shape”. One could define Geodesy and Geomatics to be the science of measuring and modelling the geometry of the world at all scales, from planetary down to the scale of a small manufact, or even to the single cell of a leaving body. Measuring means translating our perception of reality into numbers, with a modern word we could say digitizing. Modelling, in a physical conceptual environment, means building a logical system composed by a certain number of quantities bound one to the other by a system of mathematical relations, such that knowing one part of the quantities through the measurements we can arrive at predicting the other quantities. 2 Without observations a model is just a mathematical game, a nice toy maybe, but disconnected from reality. Without a model you cannot even think of performing a measurement; this is a deep concept that was taught to me by P.Caldirola, my teacher of theoretical physics, one of the pupils of the school of Enrico Fermi. Think that already the world digitizing is an interesting anthropomorphic “model” meaning making a one to one correspondence of a group of objects with the fingers in my hands. But let us make a small example of different nature: we want to prove that the measure of the distance between two points A and B is just 3 meters, having available a ruler 1 meter long. And then you know that all what I have to do is to report my ruler three times along the line from A to B as in the figure. Yet this elementary experiment is based on quite fundamental hypotheses (model), namely: • that it is possible to draw a unique straight line through A and B, what a mathematician would call a geodesic, • that A and B are not moving while I am measuring, • that my ruler is so stiff that it does not change its length when I report it on the line AB . 3 By the way, before I continue with more pertinent examples, let me make a remark that we shall use later on: this “scientific activity” of the human bride is not so different, after all, to other activities of cultural nature, for instance the production of art. Also when we create an artistic “object” we start from the experience , the sensation, the feeling and use a model, a language specific of the artistic object we want to create, be it a poem, a painting or a sculpture, and we predict, verify whether we succeed in communicating the message, the emotion we want to convey through it. And notice; with no feeling there is no art, only dry technique; with no language there is no art; with a wrong prediction of the effect of communication there is no art. A poem could be sincere, written in a “proper” language, yet unable to generate feeling into the audience. We come back now to some other examples much closer to Geodesy and Geomatics. One is the measurement of the radius of earth according to Erathostenes. The idea, illustrated by the Figure, is that if on the same day and even at the same hour in the day, at noon, in S (Syene), the modern Aswuan, we have no shadow of the column , while in A (Alessandria) we have the shortest shadow in the year measurement A R a S εεε 4 so that S and A are on a sphere (model) on the same meridian (model) then we can use the arc a (measurement) and the angle ε (measurement) to derive R (model) as a R = . ε Another example much more related to an engineering problem, is the measurement of the height of a bell tower, without climbing it. The idea is shown in Figure S b h O a D where the problem is solved by the principle of similarity of two triangles and we can see that by measuring a, b, D we can derive h as a proportion b h = ⋅ D a Consider how many hypotheses (model) are implied by this simplistic reasoning: 5 • that the tower is vertical, for static reasons, • that the square (a, b) is disposed with b along the vertical, • that the vertical direction in 0 is parallel to that through S, • that the line of sight of 0A is a straight line. As simple as they are these examples can already illustrate one of the distinctive characters of Geodesy and Geomatics considered as a science; mathematical models in this discipline are never considered to be right or wrong rather they are considered to have a certain degree of likelihood, to contain a certain quantity of truth, to be able to explain reality in a certain bounded and identifiable environment. I would say that a fuzzy logic or, if you like a Bayesian view point on learning models from experiments is naturally cast into the genetic structure of Geodesy and Geomatics, because we are so much used to treat measurements at the limits of the technical capability, that we are aware that any improvement in our instrumentation, by lowering the background noise, reveals new unexpected and reach relations between quantities previously not considered into our models. Let me try to illustrate this concept through the history of one of the most classical fields of geodesy; that of determination of the figure of the earth. Of the spherical model of Erathostenes we already said. Incredibly enough it took 2000 years before scientists could be able to reason effectively on the shape of the earth in terms of models and measurements. The models was drawn in XVIII century from the intuition of Pierre Simon Laplace that the earth was formed by “the same gravitational forces that had spun the solar system out of a disk of luminous nebular dust”. According to the work of Alexis Claude Clairaut on self- gravitating bodies (1743) the model was that of an ellipsoid of revolution. 6 It will not enter into the interesting history of the expeditions organized by the French Academy of Sciences to measure the length of arcs of meridians, in Ecuador (Charles-Marie de La Condamine, Louis Godin, Pierre Bouguer), in Lapland (Pierre Louis Moreau de Maupertuis) and in France itself, among which I want to recall the one organized by Jean Baptiste Delambre and Pierre-François-André Méchain (1792-99). But let me only report about an interview of Jean Dominique Cassini (the fourth in the family that directed the Observatoire de Paris over one century) with King Louis XVI, to explain why it was necessary a new measurement of the arc of meridian in France: “my father and grandfather’s instruments could but measure to within 15 seconds of arc, the instrument of Monsieur Borda here can measure to within one second”. So it took more or less one century of discussions on models, measurements and their accuracy to decide that the shape of the earth was that of an ellipsoid of revolution, what is nowadays well know from satellite dynamics and even obvious if we look at the earth from space. 7 Yet the earth surface is not that of an ellipsoid. The thing is obvious on continents, where topographic irregularities are part of our common experience, but even oceans, with their smooth surface, averaged with respect to short periodic motions (tides and waves) are not following exactly the shape of an ellipsoid because gravity, generated by a somewhat irregular distribution of masses inside the planet, impresses on the ocean the shape of an equipotential surface; what is called the geoid. In order to survey the oceanic surface we have nowadays available measurements from satellites bearing a radar altimeter that allows the reconstruction of the mean sea surface with an accuracy to within few centimeters. And in this way we have understood that the actual mean sea surface can be far away from the geoid by meters, as predicted by the oceanographic models, when permanent oceanic currents, like the Gulf Stream or the Kuroshyo, are taken into account. 8 So we have to distinguish from the gravimetric shape of the earth, the geoid, and its geometric surface, namely the mean sea surface on the ocean and the topography on the earth.
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