After taking his doctorate, Valerio continued to live in Rome teaching both private pupils and having public teaching appointments. At first he taught rhetoric and Greek at the Collegio Greco. On a visit to Pisa in 1590 he met Galileo but it was not until around twenty years later that they entered into a correspondence. After his return to Rome he began teaching rhetoric and philosophy at the University of Sapienza. Sapienza was the name of the building which the University of Rome occupied at this time and it gave its name to the University. From about 1600 Valerio, still at Sapienza, began to teach mathematics which he continued to do for the rest of his life. In addition to these teaching positions, Valerio was also corrector of Greek at the Vatican library for many years.
Two of his private pupils must be mentioned for they played an important part in his life. His pupil Ippolito Aldobrandini held numerous church offices, then was made cardinal in 1585 by Pope Sixtus V and elected pope as Clement VIII on 30 January 1592. He certainly played a role in Valerio's appointment to Sapienza in 1591 and Valerio thanked him with a dedication in his work of 1604. In the same work he thanked Clement VIII's nephew Cardinal Petro Aldobrandini for his support. Clearly he was closely connected with the top people in the Roman Catholic Church. The second of his pupils who we must mention was Margherita Sarrocchi. She was a poet and by all accounts a flamboyant extrovert. This was in distinct contrast to Valerio who was described as a shy, very withdrawn and isolated person. The two became lovers but Margherita Sarrocchi totally dominated the relationship, being possessive and jealous of anything in Valerio's life that did not involve her.
Valerio's De centro gravitatis solidorum, written in 1603, applied methods of Archimedes to find volumes and centres of gravity of solid bodies, in particular solids of rotation and their segments. He used interesting early ideas concerning the quotient of limits showing that, if lim x = a, lim y = b and if x / y = c = constant, then a / b = lim x / lim y = lim (x / y ) = c. This is now known as 'Cavalieri's principle' although Valerio's work long predates that of Bonaventura Cavalieri. J L Berggren, reviewing  writes:-
When Luca Valerio published his 'De centro gravitatis' in 1603, he claimed he was opening up what he called a 'royal road' to the investigation of centres of gravity of solid figures. Although the topic had been treated earlier by his immediate predecessors, Commandino and Maurolico, Valerio felt, and the authors of this study agree, that his work marked an innovation in its investigation of all solids known at the time. One part of this innovation was his conception of a general class of objects (in this case, solids satisfying certain symmetry conditions), as opposed to more specific objects, such as conic sections. And the other was the development of a considerable body of theorems applicable to that class, including the 'invention' (italics in original) of the method of exhaustion, an invention usually ascribed to Eudoxus. (The authors mean this claim in the sense that Valerio was the first to systematize this ancient device, in the first three theorems of 'De centro'.) In applying these general theorems to the solution of a wide class of problems, Valerio thus advanced beyond his ancient Renaissance predecessors.The article  by Divizia aims to show how, in De centro gravitatis, Valerio anticipated the concept of limit and the methods of the integral calculus to calculate areas and volumes. Book I of De centro gravitatis follows Euclid in beginning with statements of the definitions and axioms. Divizia claims that the germ of the concept of limit can be found in Proposition 1 of Book II. Galileo was highly impressed by this work and, much later in 1638, described Valerio as "the greatest geometer, the new Archimedes of our age". Their mutual admiration following the publication of Valerio's book led to a renewal of their friendship begun in Pisa fourteen years earlier.
Among Valerio's other works was Quadratura parabolae (1606). In this work he used his ideas of the quotient of limits to find the centre of gravity of a segment of a parabola, deducing it from the known centre of gravity of a hemisphere.
From 1609 until 1616 Valerio corresponded with Galileo, each showing great respect for the mathematical ability of the other. On 7 June 1612 Valerio was elected to the Accademia dei Lincei. It was Galileo who proposed Valerio to Federico Cesi who founded the Academy. Freedberg writes :-
Valerio was the Lincean to whom the overall editorship of academic publications was assigned. He was given charge of the publication of one of the most critical documents for the history of the Linceans, the 'Lynceographum', the massive outline of the rules, regulations, aims, and ideals of the society. ... Valerio also played a crucial role in the publication of Galileo's 'Letters on the sunspots' of 1613. ... He [was] one of Galileo's most constant supporters and advisors. Galileo much admired him for his mathematical work.Cardinal Robert Bellarmine, the chief theologian of the Roman Catholic Church, issued a decree on 5 March 1616 which declared Copernicanism false and erroneous. Valerio appeared to take fright at this, ending his correspondence with Galileo, and resigning from the Accademia dei Lincei which he had been deeply involved with over the previous four years :-
Nothing was less expected than Valerio's abandonment of the views of his closest friend. The fact that the Jesuits of the Collegio Romano, who five years earlier had so lauded Galileo, were now silent was one thing: but from Valerio the Linceans expected greater consistency, to say the least. He had now aligned himself with the most blinkered and fanatical of Galileo's opponents. His astonished colleagues refused his resignation (for that was against their oath of loyalty); and in order to deal with this matter, as well as with the general crisis now facing them, they convoked what would turn out to be the most difficult meeting they ever had.This meeting of the Accademia dei Lincei took place on 24 March 1616 but, given the rules of the Academy, there was no doubt what the decision had to be:-
They censured him for having betrayed his oath of loyalty and for having, by his conduct, offended both Galileo himself and the fundamental Lincean principle of mutual solidarity, or Lyncealitas as they called it. Reiterating their complete solidarity with Galileo, they deprived Valerio of his voting rights and forbade him from ever again participating in the sessions of their society. Bitterly they noted that by expressing the wish to resign, Valerio had implied that the Academy itself had committed a crime or a grave error in supporting the view of the movement of the earth; and that in accusing his old friend Galileo of this same "error", he had overlooked the fact that Galileo had only held the movement of the earth to be a hypothesis.Now let us try to understand why Valerio behaved in the way that he did. It makes no sense to think that he might have changed his mind about the heliocentric theory for scientific reasons. He resigned immediately following the decree from the Vatican, and nothing in that decree would lead to a change of mind for scientific reasons. The most obvious reasons seem to relate to his two famous pupils that we mentioned above. His friendship with pope Clement VIII, and his resulting closeness to the Vatican, must be considered as being at least part of the reason. Clement VIII had expanded the Index of Forbidden Books (Index Librorum Prohibitorum) and intensified the activity and scope of the Inquisition. However, Clement VIII had died in 1605 and, after Pope Leo XI died after less than a month in office, Paul V had become pope. Although Paul V placed Copernicus's treatise on the Index of Forbidden Books, he was surprisingly undogmatic in doctrinal matters. It is possible that Valerio's closeness to the Vatican meant that he was aware that the Church was going to move more firmly against Galileo, yet his behaviour in resigning was out of character. The most likely explanation for Valerio's resignation relates to the poet Margherita Sarrocchi. She was undoubtedly jealous of Valerio's close association with the Accademia dei Lincei and it would appear that Valerio, being totally under her influence, had been pushed to act out of character to please Sarrocchi. Certainly if Valerio thought that his actions would prevent him from losing respect, he was totally wrong, for his spent the remaining two years of his life in obscurity and disgrace.
Baldini sums up Valerio's contribution as follows :-
Luca Valerio may without a doubt be considered one of the most important figures produced by the renaissance in mathematics and mechanics that took place in Italy in the 16th century. In fact, his work represents, at least for the Archimedean tradition, the apex of that intellectual movement, in the sense that with it a series of themes that had appeared throughout the mathematics of the Cinquecento were brought to maturity, a level to which subsequent research would necessarily have to raise the framework of Renaissance mathematics, which was oriented especially toward rediscovery and translation of, and commentary on, classical texts.As to his influence on the mathematicians who came after him, Stromholm writes :-
Among the mathematicians who studied him and spoke highly of him were Cavalieri, Torricelli and J C de la Faille. He also had a direct influence on Guldin, Gregorius Saint Vincent, and Tacquet.
Article by: J J O'Connor and E F Robertson