The Discourses and Mathematical Demonstrations Relating to Two New Sciences (Discorsi e dimostrazioni matematiche, intorno à due nuove scienze, 1638) was Galileo's

Galileo Galilei

Galileo Galilei , was an Italian physicist, mathematician, astronomer, and philosopher who played a major role in the Scientific Revolution. His achievements include improvements to the telescope and consequent astronomical observations and support for Copernicanism...

 final book and a sort of scientific testament covering much of his work in physics over the preceding thirty years.

After his Dialogue Concerning the Two Chief World Systems

Dialogue Concerning the Two Chief World Systems

The Dialogue Concerning the Two Chief World Systems was a 1632 Italian language book by Galileo Galilei comparing the Copernican system with the traditional Ptolemaic system. It was translated to Latin as Systema cosmicum in 1635 by Matthias Bernegger...

, the Roman Inquisition

Roman Inquisition

The Roman Inquisition was a system of tribunals developed by the Holy See during the second half of the 16th century, responsible for prosecuting individuals accused of a wide array of crimes related to heresy, including Protestantism, sorcery, immorality, blasphemy, Judaizing and witchcraft, as...

 had banned publication of any work by Galileo, including any he might write in the future. After the failure of attempts to publish the work in France

France

The French Republic , The French Republic , The French Republic , (commonly known as France , is a unitary semi-presidential republic in Western Europe with several overseas territories and islands located on other continents and in the Indian, Pacific, and Atlantic oceans. Metropolitan France...

, Germany

Germany

Germany , officially the Federal Republic of Germany , is a federal parliamentary republic in Europe. The country consists of 16 states while the capital and largest city is Berlin. Germany covers an area of 357,021 km2 and has a largely temperate seasonal climate...

, or Poland

Poland

Poland , officially the Republic of Poland , is a country in Central Europe bordered by Germany to the west; the Czech Republic and Slovakia to the south; Ukraine, Belarus and Lithuania to the east; and the Baltic Sea and Kaliningrad Oblast, a Russian exclave, to the north...

, it was picked up by Lodewijk Elzevir

Lodewijk Elzevir

Lodewijk Elzevir , originally Lodewijk or Louis Elsevier or Elzevier, was a significant Dutch printer...

 in Leiden, The Netherlands

Netherlands

The Netherlands is a constituent country of the Kingdom of the Netherlands, located mainly in North-West Europe and with several islands in the Caribbean. Mainland Netherlands borders the North Sea to the north and west, Belgium to the south, and Germany to the east, and shares maritime borders...

, where the writ of the Inquisition was of little account (see House of Elzevir

House of Elzevir

Elzevir is the name of a celebrated family of Dutch booksellers, publishers, and printers of the 17th and early 18th centuries. The duodecimo series of "Elzevirs" became very famous and very desirable among bibliophiles, who sought to obtain the tallest and freshest copies of these tiny...

).

The same three men as in the Dialogue carry on the discussion, but they have changed. Simplicio, in particular, is no longer the stubborn and rather dense Aristotelian; to some extent he represents the thinking of Galileo's early years, as Sagredo represents his middle period. Salviati remains the spokesman for Galileo.

The Science of materials

The sciences named in the title are the strength of materials

Materials science

Materials science is an interdisciplinary field applying the properties of matter to various areas of science and engineering. This scientific field investigates the relationship between the structure of materials at atomic or molecular scales and their macroscopic properties. It incorporates...

 and the motion of objects

Kinematics

Kinematics is the branch of classical mechanics that describes the motion of bodies and systems without consideration of the forces that cause the motion....

. Galileo worked on an additional section on the force of percussion, but was not able to complete it to his own satisfaction.

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

Square-cube law

The square-cube law is a principle, drawn from the mathematics of proportion, that is applied in engineering and biomechanics. It was first demonstrated in 1638 in Galileo's Two New Sciences...

. 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

Biology

Biology is a natural science concerned with the study of life and living organisms, including their structure, function, growth, origin, evolution, distribution, and taxonomy. Biology is a vast subject containing many subdivisions, topics, and disciplines...

, anticipating J.B.S. Haldane's seminal work On Being the Right Size, and other essays, edited by John Maynard Smith

John Maynard Smith

John Maynard Smith,His surname was Maynard Smith, not Smith, nor was it hyphenated. F.R.S. was a British theoretical evolutionary biologist and geneticist. Originally an aeronautical engineer during the Second World War, he took a second degree in genetics under the well-known biologist J.B.S....

.

The Law of falling bodies

Thomas Bradwardine

Thomas Bradwardine

Thomas Bradwardine was an English scholar, scientist, courtier and, very briefly, Archbishop of Canterbury. As a celebrated scholastic philosopher and doctor of theology, he is often called Doctor Profundus, .-Life:He was born either at Hartfield in Sussex or at Chichester, where his family were...

 was the first to formulate the equation for the displacement s of a falling object, which starts from rest, under the influence of gravity for a time t (the essential principle had been previously stated by the Oxford Calculators

Oxford Calculators

The Oxford Calculators were a group of 14th-century thinkers, almost all associated with Merton College, Oxford, who took a strikingly logico-mathematical approach to philosophical problems....

):

In Two New Sciences Galileo (Salviati speaks for him) used a wood

Wood

Wood is a hard, fibrous tissue found in many trees. It has been used for hundreds of thousands of years for both fuel and as a construction material. It is an organic material, a natural composite of cellulose fibers embedded in a matrix of lignin which resists compression...

 molding

Molding (decorative)

Molding or moulding is a strip of material with various profiles used to cover transitions between surfaces or for decoration. It is traditionally made from solid milled wood or plaster but may be made from plastic or reformed wood...

, "12 cubits long, half a cubit wide and three finger-breadths thick" as a ramp

Inclined plane

The inclined plane is one of the original six simple machines; as the name suggests, it is a flat surface whose endpoints are at different heights. By moving an object up an inclined plane rather than completely vertical, the amount of force required is reduced, at the expense of increasing the...

 with a straight, smooth, polished groove to study rolling balls ("a hard, smooth and very round bronze ball"). He lined the groove with "parchment

Parchment

Parchment is a thin material made from calfskin, sheepskin or goatskin, often split. Its most common use was as a material for writing on, for documents, notes, or the pages of a book, codex or manuscript. It is distinct from leather in that parchment is limed but not tanned; therefore, it is very...

, also smooth and polished as possible". He inclined the ramp at various angle

Angle

In geometry, an angle is the figure formed by two rays sharing a common endpoint, called the vertex of the angle.Angles are usually presumed to be in a Euclidean plane with the circle taken for standard with regard to direction. In fact, an angle is frequently viewed as a measure of an circular arc...

s, effectively slowing down the acceleration enough so that he could measure the elapsed time. He would let the ball roll a known distance down the ramp, and used a water clock

Water clock

A water clock or clepsydra is any timepiece in which time is measured by the regulated flow of liquid into or out from a vessel where the amount is then measured.Water clocks, along with sundials, are likely to be the oldest time-measuring instruments, with the only exceptions...

 to measure the time taken to move the known distance; 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; the water thus collected was weighed, after each descent, on a very accurate balance; the differences and ratios of these weights gave us the differences and ratios of the times, and this with such accuracy that although the operation was repeated many, many times, there was no appreciable discrepancy in the results."

Infinity

The book also contains a discussion of infinity

Infinity

Infinity is a concept in many fields, most predominantly mathematics and physics, that refers to a quantity without bound or end. People have developed various ideas throughout history about the nature of infinity...

. Galileo considers the example of numbers, and their squares. He starts by noting that it cannot be denied that there are as many as there are numbers because every number is a root of some square:

1 <--> 1, 2 <--> 4, 3 <--> 9, 4 <--> 16, and so on.

(In modern terms, it is possible to have a one-to-one correspondence between the elements of a set N and the elements of a proper subset S of N). But he notes the contradiction: Yet at the outset we said there are many more numbers than squares, since the larger portion of them are not squares. Not only so, but the proportionate number of squares diminishes as we pass to larger numbers.

He resolves the contradiction by denying the possibility of comparing infinite numbers: we can only infer that the totality of all numbers is infinite, that the number of squares is infinite, and that the number of their roots is infinite; neither is the number of squares less than the totality of all numbers, nor the latter greater than the former; and finally the attributes "equal," greater," and "less," are not applicable to infinite, but only to finite, quantities. Indeed, he denies that an infinite quantity can meaningfully be said to be greater than a finite quantity. This is a possible resolution, and it implicitly recognises that he has no definition of comparison for infnite numbers, but is less powerful than the modern resolution.

How does this arise in the dialogues? Galileo is discussing problems arising from rolling circles: if two concentric circles of different radius roll along lines, then if the larger does not slip, it appears clear that the smaller must slip. But in what way? Galileo attempts to clarify the matter by considering hexagons, and then extending to rolling polygons with 100,000 sides, or any arbitrarily large number, where he shows tha a finite number of finite slips occur on the inner shape. Eventually he concludes that 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.

Reactions by Commentators

"So great a contribution to physics was Two New Sciences that scholars have long maintained that the book anticipated Isaac Newton

Isaac Newton

Sir Isaac Newton PRS was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian, who has been "considered by many to be the greatest and most influential scientist who ever lived."...

's laws of motion

Laws of motion

In physics, a number of noted theories of the motion of objects have developed. Among the best-known are:* Classical mechanics** Newton's laws of motion**Euler's laws**Cauchy's equations of motion** Kepler's laws of planetary motion ** General relativity...

."

"Galileo ... is the father of modern physics

Physics

Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

—indeed of modern science

Science

Science is a systematic enterprise that builds and organizes knowledge in the form of testable explanations and predictions about the universe...

"—Albert Einstein

Albert Einstein

Albert Einstein was a German-born theoretical physicist who developed the theory of general relativity, effecting a revolution in physics. For this achievement, Einstein is often regarded as the father of modern physics and one of the most prolific intellects in human history...

.

The flow of time

• The water clock
  Water clock
  A water clock or clepsydra is any timepiece in which time is measured by the regulated flow of liquid into or out from a vessel where the amount is then measured.Water clocks, along with sundials, are likely to be the oldest time-measuring instruments, with the only exceptions...
  
   mechanism described above was engineered to provide laminar flow
  Laminar flow
  Laminar flow, sometimes known as streamline flow, occurs when a fluid flows in parallel layers, with no disruption between the layers. At low velocities the fluid tends to flow without lateral mixing, and adjacent layers slide past one another like playing cards. There are no cross currents...
  
   of the water during the experiments, thus providing a constant flow of water for the durations of the experiments, and embodying what Newton called duration. In particular, Galileo ensured that the vat of water was large enough to provide a uniform jet of water.
• Galileo's experimental setup to measure the literal flow of time (see above), in order to describe the motion of a ball, was palpable enough and persuasive enough to found the sciences of mechanics and kinematics. Time in physics
  Time in physics
  Time in physics is defined by its measurement: time is what a clock reads. It is a scalar quantity and, like length, mass, and charge, is usually described as a fundamental quantity. Time can be combined mathematically with other physical quantities to derive other concepts such as motion, kinetic...
  
  , in particular, could be founded on the notion of the linear flow of time.

• The law of falling bodies was discovered in 1599. But in the 20th century some authorities challenged the reality of Galileo's experiments, in particular the distinguished French historian of science
  History of science and technology
  The history of science and technology is a field of history which examines how humanity's understanding of the natural world and ability to manipulate it have changed over the centuries...
  
   Alexandre Koyré
  Alexandre Koyré
  Alexandre Koyré , sometimes anglicised as Alexander Koiré, was a French philosopher of Russian origin who wrote on the history and philosophy of science.-Life:...
  
  . The experiments reported in Two New Sciences to determine the law of acceleration of falling bodies, for instance, required accurate measurements of time, which appeared to be impossible with the technology of 1600. According to Koyré, the law was arrived at deductively, and the experiments were merely illustrative thought experiments.
  • Later research, however, has validated the experiments. The experiments on falling bodies (actually rolling balls) were replicated using the methods described by Galileo (Settle, 1961), and the precision of the results was consistent with Galileo's report. Later research into Galileo's unpublished working papers from as early as 1604 clearly showed the reality of the experiments and even indicated the particular results that led to the time-squared law (Drake, 1973).

External links

Italian text with figures

• English translation by Crew and de Salvio, with original figures
• Another on-line copy of Crew and de Salvio's translation
• Galileo Galilei: The Falling Bodies Experiment - Background and experiments.