The long term problems, such as the study of the Euler-Mascheroni constant Gamma, are a test for the continuity and discontinuity in mathematics: continuity in the subject, discontinuity in the methods. Considering the fact that the transcendence, and, therefore, the irrationality, of e and p was demonstrated in the nineteenth century, the transcendence and the irrationality of Gamma remains the most important open question in this kind of studies. In 1734 Euler discovered that the series of reciprocals of natural numbers differed at the limit of the logarithm for a constant value for which he calculated the first five decimal digits. Unfortunately, the convergence is very slow. Even with 4500 terms the approximation, using the definition, is only good for three decimal digits. Euler carried out most of his scientific activity in Russia and had all his memoirs on this constant published in Commentarii of St. Peterburg, where his Institutiones Calculi Integralis was also published in 1768-70. This work aroused great interest in Lorenzo Mascheroni, whose Adnotationes were then published together with the Institutiones in Euler’s Opera Omnia. Mascheroni took up the calculus in the Adnotationes of 1790 in which he started from the results of Euler’s differential calculus and integral calculus. He was able to rectify Euler’s value for C, reaching the calculus of thirty-two decimal digits, of which only the first nineteen have been revealed to be exact: 0.577215 664901 532860 618112 090082 39. In Munich, 1809, Soldner published his theory on the transcendent function of integral logarithm (Théorie d’une nouvelle fonction trascendante) calculating 22 digits for the Euler-Mascheroni constant with a different value from Mascheroni’s at the twentieth digit. In 1813 Gauss, in his famous memoir on the hypergeometric series: Disquisitiones generales circa seriem infinitam reported the calculus that his pupil, Nicolai, carried out of the Euler-Mascheroni constant, which was different from the twentieth digit calculated by Mascheroni and in agreement with Soldner’s calculation. In 1857 Lindemann, famous for his demonstration of the transcendence of e and p, obtained a value for the constant in a different way calculating with 34 and 24 digits. In 1869 Shanks obtained 59 decimal digits, and in 1871 calculated 110.