In order to get a more rigorous education, in addition to attending courses at the Polytechnical Institute, Brashman enrolled in courses at the University of Vienna. There he was taught by Joseph Johann Littrow (1781-1840), an Austrian who had worked in Russia at Kazan University before being appointed professor of astronomy at Vienna in 1819. At first Littrow felt that Brashman had such a poor educational background that he would not be able to succeed in studying at university level. However, Brashman soon showed that he did indeed have the ability not only to overcome his weak grounding but to produce work of high quality. Littrow and Brashman became good friends and the friendship continued until Littrow's death in 1840. Brashman graduated from the University of Vienna in 1821, but continued to undertake research at the university. Later in 1821, on the recommendation of Littrow, Brashman was given a position in the house of Prince Yablonovsky in Lemberg (now Lviv) as the tutor of his children. Two years later, in 1823, with several letters of recommendation and a small amount of money, Brashman went to St Petersburg in Russia.
In St Petersburg, Brashman was supported by Princess Evdokia Ivanovna Golitsyna (nee Izmailova) (1780-1850). She had been the wife of Sergey Mikhailovich Golitsyn (1774-1859) but by this time they had separated. She was known by the nickname Princesse Nocturne and owned the literary salon at 30 Millionnaya Street which was visited by Pushkin and other leading people. She was very enthusiastic about advanced mathematics and metaphysics and had a particular interest in mechanics writing an essay Analyse des forces Ⓣ. The Princess was friends with many leading mathematicians and this provided a good way for Brashman to become known. In January 1824 Brashman was appointed to teach mathematics and physics at Saint Peter and Saint Paul's School in St Petersburg. This school had a long history having been founded in 1709 as part of a Lutheran church and school in Millionnaya Street. He taught there for a year before accepting a post in the Faculty of Physics and Mathematics of the University of Kazan in March 1825. There he taught mathematics, spherical astronomy and mechanics. At Kazan, Brashman became a colleague of Nikolai Ivanovich Lobachevsky and in fact he taught mechanics using Lobachevsky's lecture notes. In 1827 Lobachevsky became rector of the University of Kazan and the university flourished with a vigorous programme of new building, with a library, an astronomical observatory, new medical facilities and physics, chemistry and anatomy laboratories being constructed. Brashman also took on a number of administrative roles but he was very definite that his greatest satisfaction was gained through teaching and research. During his nine years at Kazan he gained a high reputation both as a scientist and as a professor. The year 1830 proved a difficult one for everyone at the university when a cholera epidemic struck but Brashman played his part in minimising the damage.
Brashman became professor of applied mathematics in the University of Moscow in August 1834. His initial appointment was as an extraordinary professor but, in January 1835, he was promoted to a full professorship. This was a position that he held until he retired in 1864. At Moscow he promoted the subject which he loved most, namely mechanics. He did this by fine teaching, writing excellent textbooks and research articles. For example, he published the textbook Course in Analytical Geometry (Russian) in 1838. A T Grigorian, writing in , says:-
In his lectures on mechanics and in his articles Brashman not only tried to show the achievements of this science, but also worked out its most difficult sections. He also prepared textbooks for Russian institutions of higher education. His texts on mathematics and mechanics reflect the state of science at that time, and his proofs of important theorems show originality, clarity and comprehensiveness. Brashman wrote one of the best analytic geometry texts of his time, for which the Russian Academy of Sciences awarded him the entire Demidov Prize for 1836.The following year his textbook The theory of equilibrium of solid and liquid bodies (Russian) on mechanics, covering statics and hydrostatics using a highly original presentation, again won him the whole of the Demidov Prize. Brashman toured through Germany, France and England in 1842, where he met with leading European mathematicians. He was in England for the Twelfth Meeting of the British Association for the Advancement of Science held in Manchester in June 1842. He gave a talk entitled Considerations on the Principles of Analytical Mechanics. He began his talk as follows:-
The principle of virtual velocities, on which is based the theory of equilibrium and of motion, has not, in my opinion, been explained in a manner which is clear and unobjectionable ; and I am also inclined to believe that the problem of equilibrium has not been treated analytically in a point of view sufficiently general, and that there are still many observations to be made on the correctness of the application of the principle of virtual velocities to certain problems. Similar observations may be made also with regard to the theory of motion. Mikhail Vasilevich Ostrogradski brought forward, some years ago, some new and general ideas on the laws of equilibrium and of motion in two memoirs, one of which bears the title, "On the Momenta of Forces;" and the other, "On the instantaneous Displacements of the points of a System." Profiting by his enlightened views, I published, in 1837, a treatise in the Russian language on the equilibrium of solid and fluid bodies, from which I will now give a very short extract relating to the method I have there followed, and I shall add some observations which escaped me at the time of the publication of that work, respecting the number and the character of conditions of equilibrium.In 1844, back in Moscow, he set up a new course on practical mechanics which links theoretical and technical mechanics. Brashman wrote research articles on the Principle of Least Action which are important in the development of mechanics. In 1859 he published the article Principle of Minimum Action (Russian) and, in the same year, he published the textbook Theoretical Mechanics (Russian) which considered both the equilibrium and the motion of a point and of a system of points. In 1861 he published the article On the application of the Principle of Minimum Action to the determination of water volume in a spillway and, in the same year he published Note concernant la pression des wagons sur les rails droits et des courants d'eau sur la rive droite du movement en vertu de la rotation de la terre Ⓣ in Comptes rendus of the Paris Academy of Sciences. In this paper he tried to prove that the rotation of the Earth puts pressure on the same rail of a straight track of a railway irrespective of the direction of travel.
Another aspect of Brashman's work for which he is remembered is for his founding of the Moscow Mathematical Society which grew out of meetings held in Brashman's own home. The first meeting of the society was 15 September 1864 when Brashman was elected as the first president and August Yulevich Davidov was elected vice-president. Brashman held this position until his death in 1866 when Davidov became president. Brashman's aims for the Society were, at first, quite limited since it was intended only for those with a Master's Degree (or higher degree) in a mathematical discipline or for those with at least one important publication. In many ways the intention was to provide the members with mutual support in their research. At the first meeting of the Society only one aim was stated, namely that "the goal of the Society is mutual cooperation in the study of the mathematical sciences". At this stage, the Society was small with only 14 members and only one, namely Pafnuty Lvovich Chebyshev, holding a position outside Moscow. However Brashman quickly became more ambitious for the new Society and, at a meeting in January 1866, the aim had extended to become a Russian wide Society; "The goal of the organisation of the Society is to promote the development of mathematical sciences in Russia." Brashman also set up the Journal of that Society, Matematicheskii Sbornik, the first part of which appeared in the year of his death. This first part contains the paper Find the pressure of a river at its bank resulting from the Earth rotation about its axis (Russian) by Brashman.
Brashman had a number of outstanding students, including Pafnuty Lvovich Chebyshev and Osip Ivanovich Somov. We note that Brashman's students had a huge respect for him both as a mathematician and as a person. For example, Chebyshev felt that he had been inspired by Brashman and asked him for a photograph that he might keep with him; indeed he still had Brashman's photograph at the time of his death.
Finally, let us note that Brashman was a strong believer in the power of mathematics. For example he delivered the speech On the influence of the mathematical sciences on the development of mental facilities (Russian) on 17 June 1841 at a commemorative ceremony in Moscow University. This was intended as a refutation of William Hamilton's essay On the study of mathematics as an exercise for the mind published in the Edinburgh Review in 1836, in which Hamilton claims not only that mathematics is useless in developing mental facilities but he even claims that it is pernicious. Brashman gives a strong refutation of William Hamilton's ideas. The speech had considerable influence, for example Viktor Yakovlevich Bunyakovsky's 1846 book on probability Foundations of the mathematical theory of probability (Russian) was motivated by it as was Chebyshev's 1844 thesis An essay on elementary analysis of the theory of probabilities (Russian).
Brashman was honoured for his contributions with election to the St Petersburg Academy of Sciences in 1855. He is described in  as follows:-
Free from all prejudices, he led his life quietly not looking for practical help and not expecting or demanding gratitude from those who he helped. He was not upset when his honest and well-intentioned actions were wrongly interpreted and he walked the path of an honest man who could not be deviated from that path by any circumstances.
Article by: J J O'Connor and E F Robertson