Dialogues
The life of Galileo Galilei (1564-1642) has a certain dramatic - one might almost say operatic - character, and this element of theatrical showmanship is evident even in the form of his scientific writings. His most famous works were dialogues, i.e., they are presented as plays, representing discussions between advocates of two opposing points of view. For example, in 1632 he published Dialogue on the Two Chief World Systems, in which fictional proponents of the ancient Ptolemaic/Aristotelian theory and the modern Copernican theory present their cases, each trying to win over a third character who represented an unbiased observer. The "modernist" is called Salviati, presumably named after Filipo Salviati, who until his death in 1614 had been a close friend of Galileo's in Florence. The clear-thinking representative of the educated public is called Sagredo, after Galileo's former student Giovanfrancesco Sagredo of Venice, who had died in 1620. The pedantic Aristotelian of the trio is called Simplicio, apparently named after Simplicius, a Greek philosopher of the 6th century AD known for his commentaries on Aristotle.
It's natural to think of Plato's classic dialogues as the inspiration for this mode of presentation. In addition, it has often been pointed out that the dialogue form enabled Galileo to present the case for the Copernican world view while maintaining "plausible deniability", i.e., he could claim that he was not advocating any particular point of view, but was merely presenting a fictional discussion among advocates of different points of view. Of course, it's perfectly clear from reading the dialogues that Salviati (the character who advocates Copernicus' theory) speaks for Galileo, but since the Copernican thesis was considered heretical in those days, it is suggested that Galileo chose the dialogue to insulate himself from direct attacks, especially since he had promised to Church officials in 1616 "not to hold, teach, or defend Copernicanism". In Galileo's mind, this promise did not preclude him from presenting an "unbiased" review of the issues, as long as he did not explicitly endorse one side or the other. (As is well known, the Church took a different view.)
The model provided by Plato's dialogues, and the usefulness of the dialogue form for surreptitious advocacy, may both have influenced Galileo's choice of format, but it's interesting to note that in his use of the dialogue form Galileo was actually carrying on a family tradition. Galileo's father, the musician and scholar Vincenzio Galilei (1525-1591), was one of the founding members of the Florentine Camerata, a group of men who met in the home of Count Giovanni Bardi to discuss topics mainly related to musical theory, but also touching on science and the arts. The modern art form known as opera was created by this group, and the very first operas were composed by its members as part of their campaign to restore what they believed to have been the classical Greek forms in music, with simple monodal melodies emphasizing the words. Vincenzio wrote a book on musical theory, entitled A Dialogue on Ancient and Modern Music.
It's also interesting to note that one of Vincenzio's guiding principles was that we should not follow authority in matters that can be directly checked by experience. For example, the accepted wisdom in those days was that the frequency of the tone produced by a vibrating string in tension would be cut in half if the string were doubled in length (with the same tension). This is true enough, but it was also commonly believed that the frequency produced by a string of a given length would be reduced by half if the tension was reduced by half. Vincenzio conducted a series of experiments to test these beliefs, and found that although the frequency is indeed proportional to the length (for a given tension), the frequency is actually proportional to the square root of the tension (for a given length). He demonstrated this by attaching weights to strings, showing that in order to double the frequency of the tone, it is necessary to quadruple the weight.
By these experiments he overturned a belief that had been attributed to the ancient Pythagoreans, whose obsession with mathematical patterns Vincenzio despised. Throughout his writings, Vincenzio criticized the tendency to mathematize music, and he argued against the kind of numerological and idealist reasoning associated with Plato and the Pythagoreans. In view of this, it's not surprising to learn that Vincenzio was opposed to his eldest son studying mathematics, prevailing upon Galileo to study medicine instead. Of course, Galileo ultimately gave up medicine in favor of science, and became a professor of mathematics at the Universitis of Pisa and Padua. It's fascinating that Galileo's work embodies and fulfills his father's emphasis on direct experimentation for the determination of scientific truth (it's been speculated that Galileo used a musical metronome to time his dynamical experiments), and yet he combined this with a profound appreciation for the importance and value of mathematics in our efforts to understand the natural world.
Following the publication of Galileo's dialogues on Copernicanism in 1633 (for which he was charged and convicted of heresy and disobedience to the Catholic Church), he composed his second great work, entitled Dialogues Concerning Two New Sciences, in which the three fictional interlocutors - Salviati, Sagredo, and Simplicio - are re-united for four days of discussion on the fundamentals of quantitative reasoning as applied to two sciences: (1) the resistance of solids to fracture, and (2) the motions of objects (what we would call dynamics and mechanics). The First Day of discussion digresses over a wide range of topics, including several propositions relating to musical instruments and the numerical relations between the lengths and tensions of strings and the tones produced. The three characters also confront the subtleties of continuity when they consider two concentric polygons "rolling" along parallel surfaces. (This remains a popular mathematical conundrum even today.) The Second Day is devoted mainly to the strengths of solids.
The Third and Forth Days are devoted to a discussion of various kinds of motion. On the Third Day Salviati argues that "constant acceleration" is wrongly believed by many people to consist of motion in which the speed is proportional to the distance traveled, whereas (he argues) it should really be defined as motion in which the speed is proportional to the time traveled. Obviously Galileo (speaking through Salviati) was correct in identifying the change in speed with time (rather than space) as being the more natural and useful definition of acceleration, but the argument he gives is rather seriously flawed, because he fails to distinguish between average speed and instantaneous speed. He says we can immediately rule out the idea that constant acceleration consists of velocity changing in proportion to distance traveled, because this would imply that a constantly accelerated object would travel four meters at an average speed of four meters per second, and it would travel eight meters at an average speed of eight meters per second, and so on. Hence, the object would take one second to reach every distance, i.e., it would have to move instantaneously. But clearly this is not an accurate representation of motion in which the speed is proportional to the distance traveled, at least not if we interpret this to mean that the instantaneous speed is always proportional to the distance traveled. In modern terms, beginning with zero velocity at x = 0, there is some constant g such that
The real difficulty with this relation is not that it implies discontinuous motion (it doesn't), but that it provides no motive for escape from the static solution x = 0. Since Galileo wasn't undertaking to identify the actual cause of accelerated motion, but simply to describe it quantitatively, he could simply have noted (if he had known how to solve an ordinary first-order differential equation) that the assumption of "speed proportional to distance" implies motion of the form x(t) = Aegt. Conversely, t = ln(x/A) / g.
Galileo's preferred definition of constant acceleration, with speed proportional to time, would be expressed as
Differentiating gives d2x/dt2 = k, and integrating gives the general motion x(t) = (1/2)gt2. Ultimately Galileo argues that the distance traveled by a uniformly accelerating object is proportional to the square of the times (which he quaintly expresses by saying the distance traveled in successive equal intervals of time are proportional to the odd numbers 1, 3, 5, ..., since the sum of the first n odd integers is n2), and so the definition based on time is the one he selects. However, to some extent this actually just constitutes a definition of time. In other words, we define the measure of time in such a way as to make the second definition of uniform acceleration true.