# Galileo’s Dialogues Concerning Two New Sciences

One afternoon, during a particularly difficult day, I found my way to Galileo’s Dialogues Concerning Two New Sciences. It was a PDF version of the 1914 edition translated by Crew and de Salvio that I had randomly stumbled upon. (It has been made freely available by archive.org).

I’ve learned that studying is one of the only things that brings me comfort and pleasure, and I found great joy and satisfaction in reading this edition of Two New Sciences. As I proceeded toward the work without definite aim, there were so many moments that emerged worthy of memory. One of my many favourite parts of the dialogues was Galileo’s account of the speed of light using flickering lanterns. But the dialogues are filled with many incredible moments. Take, for instance, some of the geometrical demonstrations, such as the theorem of how “the volumes of right cylinders having equal curved surfaces are inversely proportional to their altitudes”. Or the theorem presented by Galileo on the area of a circle as “a mean proportional between any two regular and similar polygons of which one circumscribes it and the other is isoperimetric with it”. [As an aside,  isoperimetric inequalities and ratios are very interesting. So, too, is the isoperimetric problem, which I’ve just begun digging into].

I am beginning to feel that bringing everything back to geometry is an important recourse we too often take for granted. The dialogues are fascinating in that they weave together and connect so many concepts and theories, like any great book – from geometry, ballistics, and acoustics to astronomy, the dialogues flow in a way that seems so rare today. Galileo’s presence, or voice, also emerges through the pages, and I find the work offers a rare opportunity to spend time with one of the great masters. Perhaps it is the clarity of the edition, but it is easy to follow Galileo from thought to thought, as though sitting beside him pondering some of the pressing physical questions of the 17th century. I like it, too, because of momentary engagements with his compatriots, even the philosopher Simplicio, a fictitious straw man, created to perform the perfect mediate that keeps the discussion between Sagredo and Galileo (Salviati) unfolding. One of Simplicio’s great passages is as follows:

What a sea we are gradually slipping into without knowing it! With vacua and infinities and
indivisibles and instantaneous motions, shall we ever be able, even by means of a
thousand discussions, to reach dry land? – Simplicio

It is a marvellous moment in the context of the first day of the dialogues, in which Galileo ponders the role of infinite numbers and issues pertaining to the Aristotelian school mechanics, among other things. It makes me think of some of the theoretical issues currently facing us in contemporary physics, as though, in some way, we’re continuously having to search for and reach dry land, and then, once we find it, the tide comes in a little bit more and pushes us a little bit further.

As a whole, it is obviously one of the great works ever produced by a human being, and certainly a work that anyone interested in physics or is studying to become a professional physicist ought to read. Galileo is one of those masterful scientists and philosophers that we hear about as kids, along with Newton, Franklin, and others. But encouragement to actually read his and other’s work would seem rare, and that is unfortunate.

# Exponents, History, Civilization

I decided to make a video the other day on exponent properties. I like to think about mathematical concepts and to explore first principles. I spend a lot of my spare time working through proofs and also studying the history of mathematics and physics (as well as science in general). A natural extension of these interests is the idea of making videos, where I can talk about my studies and explore different concepts and problems.

In making the recent video on exponent properties, this also gave me the opportunity to briefly touch on the fascinating history of exponents, which also raises a wider discussion on human evolution and the history of civilization.

For instance, math historians have suggested that the concept of exponents could date as far back as the 23rd century BC. This depth of history when it comes to basic mathematical ideas is something that I find so astounding. There is something both beautiful and intriguing about the idea of ancient Babylonians thinking about various mathematical concepts, not least exponents.

Of course, the context in which ancient Babylonians were thinking about exponents was very different than how we think of exponents today. For starters, the numbering system was in Sumerian, which is now an extinct language. Secondly, and perhaps most interestingly, ancient Babylonians would use symbols to denote mathematical formulae (the image below was borrowed from this article).

One of the most famous tablets that we have discovered is a cuneiform tablet known as Plimpton 322. I believe current estimates suggest that this particular tablet was created around 1900–1600 BCE. Although it is one of thousands of clay tablets that date as far back as 4000 years ago, during the Babylonian period in Mesopotamia, what I enjoy about it most is that it represents some of the most advanced mathematics prior to our now widely cherished and celebrated ancient Greek mathematicians.

In short, the tablet lists what we now term Pythagorean Triples. Basically, what a Pythagorean Triple consists of is three positive integers a, b, and c, such that a^2 + b^2 = c^2. One might be immediately reminded of Pythagorean Theorem, and you would be right to think of this famous formula. The name Pythagorean Triple is derived from Pythagorean Theorem, which one will know states that every right triangle has side lengths satisfying the formula a^2 + b^2 = c^2.

What we see in the Plimpton 322 tablet is a very cool table of numbers, with four columns and fifteen rows.

In the image above, which was borrowed from this article, we see a very nice illustration of what Robson (2002) has most recently claimed to be “a list of regular reciprocal pairs.” Robson has suggested that, when one considers all of the historical, cultural and linguistic evidence, the author of the tablet was most likely “a teacher and Plimpton 322 a set of exercises.” This account of the tablet deepens my fascination, as one can imagine an ancient teacher going over lessons that consist of very early ideas of mathematics and the relationship of numbers.

All of that aside, the wider history of exponents is fascinating. If it is true that human beings were already thinking about and experimenting with exponents as far back as 23rd century BC, this fact would contribute to an already beautiful picture that is the history of mathematics. There is something objectively human about the development and evolution of our mathematical ideas – a history that is intimately entwined with human evolution and the development of human civilization.

The ancient Babylonians were known to be fantastic record keepers, compiling ledgerbooks that act as sort of Sumerian spreadsheets. If the development of agriculture actuated the birth of human civilization approximately 12,000 years ago, as many experts agree, the emergence of basic things that we now take for granted – such as writing, town planning, the division of labour, administration, law, commerce – were able to crystalize as future possibilities.  Free from the precariousness of sustenance living, people were allowed more free time, with greater access to resources. New technologies were eventually conceived, and human pursuit was existentially freed from basic survival to expand beyond that which was unavailable to hunter-gathers. With all of this came writing, and thus also things like Plimpton 322.

There are many detailed, well-research and richly scholarly books available today that explore this particular moment in history. In many ways, it is an important chapter within the broader history of human thought, science and technological pursuit.

To close: The earliest known form of writing, as we already touched on, was cuneiform – a Sumerian system thought to have emerged roughly five-thousand years ago. The cuneiform tablets, such as Plimpton 322, allowed for the ability of human beings to begin to harness the power of knowledge; because, in essence, the most basic idea of writing and written language allowed for the keeping of records. Human thought could be recorded, shared among generations, and thus concepts and ideas could develop and evolve. The Babylonians, at the start of human civilization, harnessed this power as much as possible: they were not only avid record keepers, but they developed lists in order to keep track of acreages of wheat and quantities of livestock. They also kept track of taxes and legal disputes. And from the basis of these very humble foundations, using soft clay tablets, early ideas in mathematics could be pursued.