From the standpoint of the History of Mathematics, beginning with pioneering work of Hurwitz on invariant integrals (or invariant measures) on Lie groups, we pick up epoch-making works successively and draw the main stream among so many contributions to the study of invariant integrals on groups, due to Hurwitz, Schur, Weyl, Haar, Neumann, Kakutani, Weil, and Kakutani-Kodaira, and explain their contents and give the relationships among them.

The evolution of the orientations of the orbital planes of the planets is calculated under the approximation of circular orbits. The inclination and the longitude of the ascending node of each orbital plane are then described by means of a linear combination of complex exponentials of time with periods of several thousand years. The evolution of these orbital elements for Mercury, Jupiter and Saturn is displayed as well as that of the ecliptic. Finally, the obliquity of the ecliptic is computed from $-2\,000\,000$ to $+2\,000\,000$ years since J2000. It ranges from $10^\circ$ to $35^\circ$ in this time interval.

Here we demonstrate how the very definition of the numerical range leads to its direct geometrical identification. The procedure which we follow can be even slightly refined by making use of the famous Jacobi's method for diagonalization in reverse direction. From mathematical point of view, the Jacobi's idea here is used to reduce the number of the independent parameters from three to two which simplifies significantly the problem. As a surplus we have found an explicit recipe how to associate a Cassinian oval with the numerical range of any real $2\times 2$ matrix. Last, but not least, we have derived their explicit parameterizations.