I have a strong emotional reaction when I read the works of Euler. I have seen many extremely beautiful and intriguing identities in the notebook of Ramanujan, so much so that I think he is indeed a Euler of th 20th century.
But what I notice is that mathematics have changed since the 19th century. Math seems to develop more theory and become much more rigorous due to the development of logic (think of Gottlob Frege). I realize that math during the day of Euler, Lagrange and d'Alembert is very calculational, computational. They trust algebraic symbols and are much more formal.
From Littlewood's Miscellany (page 95, 96):
But the great day of formulae seems to be over. No-one, if we are again to take the highest standpoint, seems to be able to discover a radically new type, though Ramanujan comes near in his work of partition series; it is futile to multiply examples in the sphere of Cauchy's theorem and elliptic function theory, and some general theory dominates, if in a less degree, every other field. A hundred years or so ago, his power would have had ample scope. Discoveries alter the general mathematical atmosphere and have very remote effects, and we are not prone to attach great weight to rediscoveries, however independent they seem. How much are we allow for this, how great a mathematician might Ramanujan have been 100 or 150 years ago; and what would have happened if he had come in touch with Euler at the right moment?
In the article "Exceptions and counterexamples: Understanding Abel's comment on Cauchy's Theorem", the author also wrote that:
From the mid-18th century to the mid-19th century, the style of mathematical research in analysis underwent changes from a formula-centered approach epitomized by L. Euler (1707–1783) to a concept-centered style presented in works of G. P. L. Dirichlet (1805–1859) and G. F. B. Riemann (1826–1866). The transition manifested itself in multiple aspects of the mathematical enterprise including notations, questions, results, methods, and techniques. It was felt by the active and creative mathematicians of the nineteenth century who spotted a difference between the computational machinery associated with the formula-centered approach and the decidedly mental analysis belonging to the concept-centered approach. We find this distinction seized upon for instance in Dirichlet's obituary of C. G. J. Jacobi (1804–1851) where Dirichlet noticed1
[…] the constantly increasing tendency of the new analysis to put thoughts in the place of calculations, […],2
in the methodological principle attributed by D. Hilbert (1862–1943) to Riemann near the end of the century, I have tried to avoid the large computational apparatus of Kummer such that also here Riemann's principle should be observed, according to which one should conquer proofs not by computations but solely through thoughts,3
or in what H. Minkowski (1864–1909) called the “second Dirichlet principle” heralding the modern times in mathematics, according to which problems should be conquered with a minimum of blind calculations and a maximum of enlightening thoughts.4
Is it true that formulae is no longer in its heyday as compared to the 18th, early 19th century? After all, we have Roger Apéry proving that $\zeta(3)$ is an irrational number and probably cannot be expressed in closed form (I think I am wrong here).