The direction passed to Guido Castelnuovo who held it from 1921 to 1922:[8] with the beginning of and during the publishing of the whole second and third series of the journal, from 1922 to 1935 Federigo Enriques was the director.

[9] During fourth series of the journal, at the starting of which it underwent its first name change, the direction passed to Gaetano Scorza, who held it until his death in 1939.

[10] The founding of the Istituto Nazionale di Alta Matematica in 1939, under the decisive influence of Francesco Severi,[11] had important consequences on the "Rendiconti del Seminario Matematico".

On 23 November 1939, during its first meeting, the Scientific Council[12] of the then newborn institute,[13] considering the need of a means to publish the results of the research done by the members and the students working at the institute,[14] decided to merge the needed means and the already existing journal in one single entity, giving birth to the "Rendiconti di Matematica e delle sue Applicazioni".

[14] Entrusted with this double function the fifth series of the journal, from the first volume published in 1940[15] to the first issue of the fourth volume published in 1943,[16] included a section listing the programs of the INdAM courses of the current academic year[17] as well as section listing the research problems proposed by the lecturers currently working at the institute.