His father was Juan Torroja, a Professor of Geography and History.
He continued his studies at Complutense University, where he obtained the degrees of Bachelor of Science (1864), Masters of Science (1866), Architect (1869) and Doctor of Science (1873) in Mathematics.
Very early in his studies he became a disciple of Karl Georg Christian von Staudt, whose ideas of synthetic geometry he embraced and promoted among his fellow mathematicians for the rest of his life.
The strong presence of geometry in Spain's mathematical curriculum, even to this day, can be traced back to Torroja's influence.
In 1869, Torroja became a research fellow at the Astronomical Observatory in Madrid, where he contributed to the geodesic triangulation of Spain, initiated by the General Carlos Ibåñez.
While working at that Research Institute, he became Assistant Professor at the Faculty of Sciences.
In 1873 he became a full professor of Algebra, Geometry, Trigonometry and Analytic Geometry at the University of Valencia.
His books became highly regarded textbooks, expanding the scope of his influence and that of his inspiration, Karl von Staudt.
At that time Complutense University was the only University in the Spanish Empire entitled to grant Doctor degrees, so that most Spanish mathematicians can trace their academic genealogy to Dr.