Depiction of the pump problem: the man (top right) is operating the piston of the suction pump by raising and lowering the opposite end of the beam to which the piston rod is attached; the ability of the pump, however, is limited to a depth of 32 feet. [3]
In science, pump problem refers to the 1630 puzzle as to why pumps, or rather suction pumps, or siphons, could not operate, in their ability to pump water out of mines, if the bottom of the well were more than 32 feet deep.

Gallileo
On 27 Jul 1630, Giovanni Baliani, a physicist and experimenter, wrote to Galileo asking him why a siphon, made to carry water over a hill, failed to work if the hill was more than 18 cubits (27 feet)?

Baliani asked Galileo, in short, why his pump would not work past a depth of 27 feet. Galileo is said to have concluded that: [1]

(a) Atmosphere has weight
(b) Air weight was not heavy enough to balance a column of water more than 18 cubits.

Galilei, stressed from his inquisitions, met his reaction end in 1642, but before doing so assigned this pump problem to his assistant Italian physicist Evangelista Torricelli.

Torricelli
In 1643, Evangelista Torricelli, in effort to test Galileo's answer to the pump solution, proposed inverting a liquid, water first, supposedly, then mercury, into a dish of the same liquid to see if a vacuum would form, or something to this effect. That year the experiment was performed by him or by his colleague Vincenzo Viviani. [5] The following is a rendition of this experiment:

On 11 Jun 1644, Torricelli wrote to his friend Michelangelo Ricci, who like Torricelli had been a student of Benedetto Castelli, wherein he stated the following: [5]
 A depiction of a “Torricelli tube”, the empty space at the top of the tube called a “Torricellian vacuum”, both named after Italian physicist Evangelista Torricelli who conducted this so-called vacuum testing experiment in 1643. [2]

“I have already called attention to certain philosophical experiments that are in progress ... relating to vacuum, designed not just to make a vacuum but to make an instrument which will exhibit changes in the atmosphere, which is sometimes heavier and denser and at other times lighter and thinner. Many have argued that a vacuum does not exist, others claim it exists only with difficulty in spite of the repugnance of nature; I know of no one who claims it easily exists without any resistance from nature.”

Sometime hereafter, Torricelli began to use mercury, and therein developed his famous "mercury column experiment" where he inverted a column of mercury into a second bowl of mercury, without letting air seep in, and found that a "space" was created at the top of the inverted tube and concluded that it was a vacuum later to be called a “Torricellian vacuum” (or Torricelli's emptiness).

Torricelli found that the level of the mercury varied from day to day and conclude that this was due to variations in atmospheric pressure. This was the first barometer or atmospheric pressure measuring device.

Guericke
In 1654, German polymath Otto Guericke, who had already been working on his own vacuum experiments, learned of the inverted mercury experiment from his acquaintance with Valerianus Magnus (ΡΊ), who claimed that he had invented the inverted mercury experiment, and gave Guericke his 1647 book Visual Demonstrations of a Vacuum. [7] Sometime therein, while reading this book, and others on the same subject, Guericke learned that Torricelli was the first one to have done the inverted mercury vacuum experiment.

Quotes
The following are related quotes:

“Let us suppose that the cylinder has an elevation of more than 32 feet: the liquid column will rise until it attains about this height. At this moment its weight is in equilibrium with the pressure of the atmosphere; if the piston continues to rise, the water will not follow it. This is precisely the obstacle which the Florentine workmen encountered, and which caused the physicists belonging to the Court of the Grand Duke to believe that nature ceased to abhor a vacuum beyond 32 feet. Such is the principle of the pump to which is given the name of suction-pump, because the piston appears to suck up the liquid as it rises.”
— Amedee Guillemin (1872), The Force of Nature (pg. 131)

References
1. Maury, Mathew F. (1864). Physical Geography (pg. 41). Science.
2. Evangelista Torricelli – RobertNowlan.com.
3. Agricola, pseudonym of Georg Baue. (1959). De re Metallica (image; Draining a mine by means of a suction pump). Basil.
4. Craik, George L. (1831). The Pursuit of Knowledge under Difficulties (pg. 81). C. Knight.
5. Torricelli vacuum – Vacuum-Guide.com.
6. Tokaty, Grigori A. (1971). A History and Philosophy of Fluid Mechanics (§: Evangelista Torricelli (1608-1647) and Otto von Guericke (1602-1686), pgs. 56-59). Dover.
7. (a) Magnus, Valerianus. (1647). Visual Demonstrations: of a Space Without Something Contained in it, of a Body Moved Gradually in a Vacuum, and of Light Which Adheres to No Body (Demonstratio ocularis, Loci sine locato, Corporis successive moti in Vacuo, Lumimis nulli corpori inhorentis) (GB). Publisher.
(b) Guericke, Otto. (1663). The New (So-Called) Magdeburg Experiments of Otto von Guericke (Experimenta Nova (Ut Vocantur) Magdeburgica de Vacuo Spatio). Amsterdam, Netherlands: Johann Jansson, 1672 (Latin); English translation by Margaret Ames (GB) (Amz) (pg. 181). Springer, 1993.
8. Guericke, Otto. (1663). The New (So-Called) Magdeburg Experiments of Otto von Guericke (Experimenta Nova (Ut Vocantur) Magdeburgica de Vacuo Spatio). Amsterdam, Netherlands: Johann.