Jacob Bernoulli's contributions to mathematics include calculus, differential equations, summation of infinite series, analytic geometry, probability theory and variational methods. Jacob Bernoulli's most outstanding contribution to mathematics is in the fields of probability theory and variational method. The results of his work in probability theory are included in his thesis "The Art of Speculation". In this book, he made some important contributions to probability theory, including the discovery of the so-called law of large numbers. This paper also records Jacob Bernoulli's work in permutation and combination. The Bernoulli family always likes to argue about academic issues. Jacob Bernoulli and his younger brother johann bernoulli had a heated argument about finding the steepest descent line, that is, the shortest path for a particle to pass through two fixed points under the sole action of gravity. The result of this serious debate has produced a variational method. In addition, Jacob Bernoulli also made important contributions to the study of catenary, and he also applied this achievement to the design of bridges. 1694, he gave the formula of curvature radius in rectangular coordinates and polar coordinates for the first time, which was also the beginning of using polar coordinates in the system. Jacob Bernoulli and his younger brother johann bernoulli played an important role in the development and spread of calculus just invented by Newton and Leibniz at that time, and both made important contributions to the creation of calculus. Jacob Bernoulli's special contribution to calculus is that he pointed out how to apply this technology to the broad field of applied mathematics. The word "integral" was first used by him in 1690. Jacob Bernoulli's most creative work in his life is Conjecture published in 17 13, which is a great event in the history of combinatorial mathematics and probability theory. The Bernoulli number given in this paper has many applications. Bernoulli theorem in probability theory is put forward, which is the earliest form of the law of large numbers. Because the Bernoulli brothers argued so fiercely on scientific issues, their families also participated in it, so that after Jacob Bernoulli's death, his manuscript of Guess was hidden by his widow and son for many years, and it was not published until 17 13, which almost damaged the value of this classic work. Due to the extreme importance of the law of large numbers, the Petersburg Academy of Sciences held a celebration meeting on February 19 13 to commemorate the 200th anniversary of the birth of the law of large numbers.