Shape of the Earth, Motion of the Planets and the Method of Least Squares. Probal Chaudhuri

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1 Shape of the Earth, Motion of the Planets and the Method of Least Squares Probal Chaudhuri Indian Statistical Institute, Kolkata IMS Public Lecture March 20, 2014

2 Shape of the Earth French astronomer Jean Reicher in South America in 1672 Reicher s pendulum clock was losing 2 minutes 28 seconds every day in French Guyana.

3 Shape of the Earth Isaac Newton and his Principia in 1687 Newton claimed that the Earth s rotation around its axis made it an Oblate Sphere. Rotation caused flattening at the Poles and bulging at the Equator. Many French Scientists, however, believed that Newton was wrong. There was yet another conflict between the English and the French scientists.

4 Shape of the Earth The Earth viewed from a Satellite

5 Shape of the Earth (Contd.)

6 Relationship between the Arc Length and the Latitude Let Y be the length of a 1 0 meridian arc centered at latitude θ. Then, assuming the Earth to be an ellipsoid, Y m sin 2 θ + c = mx + c, where X = sin 2 θ. Ellipticity of the Earth can be measured by m/3c. Newton estimated the ellipticity of the Earth to be 1/230.

7 Shape of the Earth French arc measurements around 1720 over a range of 9 0 latitude. The project was led by Dominico Cassini and Jacques Cassini of the Royal Observatory in Paris. Their measurements and analysis contradicted Newton s model for the Earth. French Geodesic Mission during Expeditions to Quito in Ecuador and Lapland in Finland.

8 Meridian Arc Length Data

9 The Problem of Inconsistent Equations m 12 =(y 2 y 1 )/(x 2 x 1 ) (x 2,y 2 ) (x 5,y 5 ) m 34 =(y 4 y 3 )/(x 4 x 3 ) (x 1,y 1 ) (x 4,y 4 ) (x 3,y 3 ) m 45 =(y 5 y 4 )/(x 5 x 4 )

10 Roger Boscovich Roger Joseph Boscovich ( )

11 Boscovich s Least Absolute Deviations Line Fitting We have data points (X 1, Y 1 ), (X 2, Y 2 ),, (X 5, Y 5 ). Y = arc length, X = sin 2 (latitude) To find the line : Y = mx + c that fits the data best. Minimize m,c 5 Y i mx i c. i=1 This is the least absolute deviations problem formulated by Roger Boscovich in Boscovich worked on the problem during

12 Least Absolute Deviations Line Fitting (Contd.) Minimize m Y 1 mx 1 c + + Y n mx n c = Minimize m Y i mx i c. i=1 Y axis X axis

13 The Pairwise Solution m 12 =(y 2 y 1 )/(x 2 x 1 ) (x 2,y 2 ) (x 5,y 5 ) m 34 =(y 4 y 3 )/(x 4 x 3 ) (x 1,y 1 ) (x 4,y 4 ) (x 3,y 3 ) m 45 =(y 5 y 4 )/(x 5 x 4 )

14 The Pairwise Solution (Contd.)

15 The Solar System The Solar System

16 The Problem of the Saturn and the Jupiter : In 1676, Edmund Halley was analyzing the data on the positions of the Jupiter and the Saturn that were tabulated over many centuries. Halley s analysis implied that the Jupiter was accelerating while the Saturn was retarding. The Academy of Sciences in Paris announced a prize in 1748 for a scientific resolution of this crisis in the solar system. Leonhard Euler was the winner of the prize but only for making some interesting attempts to solve the problem.

17 Leonhard Euler Leonhard Euler ( )

18 Pierre-Simon Laplace Pierre-Simon Laplace ( )

19 The Problem of the Saturn and the Jupiter (Contd.) : The solar system was eventually rescued by Pierre Simon Laplace in Laplace showed that the inequalities in the motions of the Jupiter and the Saturn are NOT Secular but Periodic. Laplace was dealing with 24 linear equations in 4 unknowns. He created a system of 4 equations by forming linear combinations of original 24 equations.

20 Laplace s First Attempt to Least Absolute Deviations Problem: Laplace changed Boscovich s formulation of the problem and worked on it during He wanted to minimize max 1 i n Y i mx i c w.r.t. m and c. In 1799 Laplace decided to go back to Boscovich s original least absolute deviations problem and concentrated on that.

21 Laplace s Second Attempt to Boscovich s Problem: Question 1 : How to minimize n i=1 Y i m w.r.t. m? Answer 1 : The minimizing m will be the middle most Y -value after the Y i s are ordered. Question 2 : How to minimize n i=1 Y i mx i w.r.t. m? Answer 2 : Note that Y i mx i = (Y i /X i ) m X i = i=1 i=1 Z i m W i, where Z i = Y i /X i and W i = X i. Order the Z i s, where Z i has weight W i. The minimizing m is the middle most Z -value in the weight distribution. i=1

22 Laplace s Solution (Contd.) : Question 3 : How to minimize n i=1 Y i mx i c w.r.t. m and c? Partial Answer 3 : Assume that the minimizing line passes through ( X, Ȳ ). Then we need to choose m only by minimizing (Y i Ȳ ) m(x i X). i=1

23 The Metric System in Europe : In 1792, after the French Revolution, France and many other European counties decided switch to a new system of measurement - the Metric System. One Meter was defined to be 1/10, 000, 000 of a Meridian Quadrant, which is the distance from the Equator to the North Pole. In 1792, Adrien Marie Legendre, was associated with the French Commission charged with the measurement of a Meridian Quadrant. Legendre invented and published the Principle of Least Squares in 1805.

24 Adrien-Marie Legendre Adrien-Marie Legendre ( )

25 French Meridian Arc Measurements

26 French Meridian Arc Measurements (Contd.)

27 Legendre s Least Squares Line Fitting We have data points (X 1, Y 1 ), (X 2, Y 2 ),, (X n, Y n ). To find the line : y = mx + c that fits the data best. Minimize m,c (Y i mx i c) 2. i=1 (Y i mx i c) 2 = {(Y i Ȳ ) m(x i X) + (Ȳ m X c)} 2. i=1 i=1 = (Y i Ȳ )2 +m 2 (X i X) 2 2m (Y i Ȳ )(X i X)+n(Ȳ m X c) 2. i=1 i=1 i=1 (Ȳ m X c)} 2 is minimized if c = Ȳ m X. So, the optimal line (Y Ȳ ) = m(x X) passes through the point ( X, Ȳ ).

28 Least Squares Line Fitting (Contd.) (Y i Ȳ )2 + m 2 (X i X) 2 2m (Y i Ȳ )(X i X) i=1 i=1 i=1 The optimal m that minimizes the above expression is m = (Y i Ȳ )(X i X) i=1 = (X i X) 2 i=1 (Y i Y j )(X i X j ) i=1 j=1 i=1 j=1. (X i X j ) 2

29 Least Squares Line Fitting (Contd.) m = 1 i<j n ( ) Yi Y j (X i X j ) 2 X i X j. (X i X j ) 2 i=1 j=1 So, optimal m is the weighted average of the slopes of the lines joining pairs of points like (X i, Y i ) and (X j, Y j ).

30 French Meridian Arc Measurements

31 Carl Friedrich Gauss Carl Friedrich Gauss ( )

32 Gauss and the Ceres : In January-February 1801, Italian astronomer Joseph Piazzi observed and recorded data on the dwarf planet Ceres for 41 days before it vanished in the glare of the Sun. Astronomers wanted to predict the position and the time of re-appearance of the Ceres. Many scientists including Laplace thought that it is an impossible problem due to inadequate data. Carl Friedrich Gauss made a very accurate prediction of the time and the position of re-appearance. The Ceres was observed again in November-December 1801.

33 Gauss and the Method of Least Squares : Gauss finished writing his work related to the Method of Least Squares in 1807, and the book was published in Gauss claimed Least Squares as his method and said that he was using it since This is one of the most famous priority disputes in the history of Statistical Science.

34 A Major Breakthrough due to Gauss : Recall Question 3 : How to minimize n i=1 Y i mx i c w.r.t. m and c? Solution due to Gauss : One of the elemental lines is an optimal line.

35 The Linear Programming Formulation : Problem : To minimize n i=1 Y i mx i c w.r.t. m and c Equivalent Linear Programming Problem : To minimize n i=1 R i subject to the constraints R i Y i mx i c R i + mx i + c Y i R i Y i + mx i + c R i mx i c Y i for all i = 1, 2,..., n

SECOND, METRE AND SPEED OF LIGHT. HISTORICAL AND EXPERIMENTAL APPROACH

SECOND, METRE AND SPEED OF LIGHT. HISTORICAL AND EXPERIMENTAL APPROACH Francis Berthomieu EAAE Summer School Working Group (France) Abstract For millenniums, units of time were defined in astronomical