The Discourses and Mathematical Demonstrations Relating to Two New Sciences (Italian: Discorsi e dimostrazioni matematiche intorno a due nuove scienze pronounced [diˈskorsi e ddimostratˈtsjoːni mateˈmaːtike inˈtorno a dˈduːe ˈnwɔːve ʃˈʃɛntse]) published in 1638 was Galileo Galilei's final book and a scientific testament covering much of his work in physics over the preceding thirty years.

After his Dialogue Concerning the Two Chief World Systems, the Roman Inquisition had banned the publication of any of Galileo's works, including any he might write in the future.

[3] Discourses was written in a style similar to Dialogues, in which three men (Simplicio, Sagredo, and Salviati) discuss and debate the various questions Galileo is seeking to answer.

The likeness between the topics discussed, specific questions that are hypothesized, and the style and sources throughout give Galileo the backbone to his First Day.

Page numbers at the start of each paragraph are from the 1898 version,[7] presently adopted as standard, and are found in the Crew and Drake translations.

Sagredo (taken to be the younger Galileo) cannot understand why with machines one cannot argue from the small to the large: "I do not see that the properties of circles, triangles and...solid figures should change with their size".

[85] The difference between a fine dust and a liquid leads to a discussion of light and how the concentrated power of the sun can melt metals.

He also did not believe that motion in a void was possible, but since air is much less dense than water Salviati asserts that in a medium devoid of resistance (a vacuum) all bodies—a lock of wool or a bit of lead—would fall at the same speed.

[128] Measuring the speed of a fall is difficult because of the small time intervals involved and his first way round this used pendulums of the same length but with lead or cork weights.

He gives an erroneous solution to the brachistochrone problem, claiming to prove that the arc of the circle is the fastest descent.

He shows in detail how to construct the parabolas in various situations and gives tables for altitude and range depending on the projected angle.

Following a letter from Cardinal Bellarmine in 1615 Galileo distinguished his arguments and Copernicus' as natural suppositions as opposed to the "fictive" that are "introduced only for the sake of astronomical computations," such as Ptolemy's hypothesis on eccentrics and equants.

These notes mirrored those of his contemporaries at the Collegio as well as contained an "Aristotelian context with decided Thomistic (St. Thomas Aquinas) overtones.

"[9] These earlier papers are believed to have encouraged him to apply demonstrative proof in order to give validity to his discoveries on motion.

Discovery of folio 116v gives evidence of experiments that had previously not been reported and therefore demonstrated Galileo's actual calculations for the Law of Falling Bodies.

[11] The discussion begins with a demonstration of the reasons that a large structure proportioned in exactly the same way as a smaller one must necessarily be weaker known as the square–cube law.

Later in the discussion this principle is applied to the thickness required of the bones of a large animal, possibly the first quantitative result in biology, anticipating J.

Galileo expresses clearly for the first time the constant acceleration of a falling body which he was able to measure accurately by slowing it down using an inclined plane.

This clock was a large vessel of water placed in an elevated position; to the bottom of this vessel was soldered a pipe of small diameter giving a thin jet of water, which we collected in a small glass during the time of each descent, whether for the whole length of the channel or for a part of its length.

To compensate, he conducted experiments using a shallowly inclined ramp, smoothed so as to eliminate as much friction as possible, on which he rolled down balls of different weights.

In this manner, he was able to provide empirical evidence that matter accelerates vertically downward at a constant rate, regardless of mass, due to the effects of gravity.

[13] The unreported experiment found in folio 116V tested the constant rate of acceleration in falling bodies due to gravity.

These discrepancies forced Galileo to assert that the postulate held only under "ideal conditions," i.e., in the absence of friction and/or air resistance.

He poses the question of what happens to a ball dropped from the mast of a sailing ship or an arrow fired into the air on the deck.

The resolution to the problem may be generalized by considering Galileo's first definition of what it means for sets to have equal sizes, that is, the ability to put them in one-to-one correspondence.

Eventually, he concludes "the line traversed by the larger circle consists then of an infinite number of points which completely fill it; while that which is traced by the smaller circle consists of an infinite number of points which leave empty spaces and only partly fill the line," which would not be considered satisfactory now.

Renyi said that, having removed this 2000-year-old stumbling block, Galileo went on to introduce his mathematical laws of motion, anticipating Newton.

[20] Pierre Gassendi defended Galileo's opinions in his book, De Motu Impresso a Motore Translato.

In fact, Galileo's water clock (described above) provided sufficiently accurate measurements of time to confirm his conjectures.

Later research into Galileo's unpublished working papers from 1604 clearly showed the reality of the experiments and even indicated the particular results that led to the time-squared law.