The Assayer (Italian: Il saggiatore) is a book by Galileo Galilei, published in Rome in October 1623. It is generally considered to be one of the pioneering works of the scientific method, first broaching the idea that the book of nature is to be read with mathematical tools rather than those of scholastic philosophy, as generally held at the time. Despite the retroactive acclaim given to Galileo’s theory of knowledge, the empirical claim conjectured in the book — that comets and their observed properties are the product of optical phenomena — is incorrect.

In 1619, Galileo became embroiled in a controversy with Father Orazio Grassi, professor of mathematics at the Jesuit Collegio Romano. It began as a dispute over the nature of comets, but by the time Galileo had published The Assayer, his last salvo in the dispute, it had become a much wider controversy over the very nature of science itself.

The debate between Galileo and Grassi started in early 1619, when Father Grassi anonymously published the pamphlet, An Astronomical Disputation on the Three Comets of the Year 1618 (Disputatio astronomica de tribus cometis anni MDCXVIII), which discussed the nature of a comet that had appeared late in November of the previous year. Grassi concluded that the comet was a fiery, celestial body that had moved along a segment of a great circle at a constant distance from the earth, and since it moved in the sky more slowly than the Moon, it must be farther away than the Moon.

Grassi adopted Tycho Brahe's Tychonic system, in which the other planets of the Solar System orbit around the Sun, which, in turn, orbits around the Earth. In his Disputatio Grassi referenced many of Galileo's observations, such as the surface of the Moon and the phases of Venus, without mentioning him. Grassi argued from the apparent absence of observable parallax that comets move beyond the Moon. Galileo never explicitly stated that comets are an illusion, but merely wondered if they are real or an optical illusion.

Grassi's arguments and conclusions were criticised in a subsequent pamphlet, Discourse on Comets, published under the name of one of Galileo's disciples, a Florentine lawyer named Mario Guiducci, although it had been largely written by Galileo himself. Galileo and Guiducci offered no definitive theory of their own on the nature of comets, although they did present some tentative conjectures that are now known to be mistaken. (The correct approach to the study of comets had been proposed at the time by Tycho Brahe.) In its opening passage, Galileo and Guiducci's Discourse gratuitously insulted the Jesuit Christoph Scheiner, and various uncomplimentary remarks about the professors of the Collegio Romano were scattered throughout the work.

The Jesuits were offended, and Grassi soon replied with a polemical tract of his own, The Astronomical and Philosophical Balance (Libra astronomica ac philosophica), under the pseudonym Lothario Sarsio Sigensano, purporting to be one of his own pupils.

The Assayer was Galileo's devastating reply to the Astronomical Balance. It has been widely recognized as a masterpiece of polemical literature, in which "Sarsi's" arguments are subjected to withering scorn. It was greeted with wide acclaim, and particularly pleased the new pope, Urban VIII, to whom it had been dedicated. In Rome, in the previous decade, Barberini, the future Urban VIII, had come down on the side of Galileo and the Lincean Academy.

Galileo's dispute with Grassi permanently alienated many Jesuits, and Galileo and his friends were convinced that they were responsible for bringing about his later condemnation, although supporting evidence for this is not conclusive.