I was writing a paper on the basics of calculus, and of course the study of velocities plays a big part in that. In introducing the problem statement, I started with a classic word problem, "Alice gets in her car and drives 100 miles at a constant speed. It takes her two hours to do so. How fast did she travel?" This is the kind of word problem shows up very early on in our education. It's quite a simple problem.

I got curious how far back in history this idea of "speed" went. I figured I'd find a story about one of the famous Greek philosophers, but Wikipedia says otherwise:

Italian physicist Galileo Galilei is usually credited with being the first to measure speed by considering the distance covered and the time it takes. Galileo defined speed as the distance covered per unit of time. In equation form, that is

$$v=\frac d t$$ where $v$ is speed, $d$ is distance, and $t$ is time.

This surprised me a bit. I didn't expect our definition to have come about that recently. The statement is cited as coming from a page in a book I do not own (Hewitt (2006), p. 42), so I can't check the specifics.

It seems to me that any decent civil planner would have a sense of how long it takes to travel between cities by foot, by horse, by chariot, etc. I feel the Greeks would have had such concepts, and in hindsight, knowing the definition attributed to Galileo, it seems like this would be a very natural way to capture this data.

So, before Galileo's work, how was the concept of speed treated? Is the only thing "new" about Galileo's version the fact that we put it into an equation form? Or does the history of that concept really that new?

  • $\begingroup$ I agree, and I suspect the Hewitt reference is taken out of context. By the way, where is the identifying information (full author, title, year, etc.) for the Hewitt book? I get only three hits there for a search-in-page for "Hewitt", none of which gives even the most basic identifying information for the book. $\endgroup$ Commented Dec 8, 2022 at 15:50
  • $\begingroup$ @DaveLRenfro sadly what I typed is all the information I have. The Wikipedia page sadly only gives that in it's citation. $\endgroup$
    – Cort Ammon
    Commented Dec 9, 2022 at 2:41
  • $\begingroup$ FWIW, the book is probably Paul G. Hewitt's Conceptual Physics, now in its 13th edition. I could not get hold of a copy to check, though. $\endgroup$
    – Tom Heinzl
    Commented Dec 9, 2022 at 16:51
  • $\begingroup$ hsm.stackexchange.com/questions/14418/… $\endgroup$
    – releseabe
    Commented Dec 11, 2022 at 6:01
  • $\begingroup$ See Galileo's Discussion of Uniform Motion for reference to Galileo's formulations. As you can see, there is no "equation form" in the modern sense in Galileo's version. And this is the basic reason: modern algebraic form (equation) emerged only in the Renaissance and were common in mathematics from Euler. $\endgroup$ Commented Dec 12, 2022 at 16:08

2 Answers 2


Are you asking about the concept of speed in general or the equational relationship? I can highlight a few developments.

The concept of speed was debated by the Greeks. For instance, Zeno of Elea presented two famous paradoxes regarding speed, and Aristotle addresses them in his Physics. From Wiki:

Achilles and the Tortise

Achilles is in a footrace with the tortoise. Achilles allows the tortoise a head start of 100 meters, for example. Suppose that each racer starts running at some constant speed, one faster than the other. After some finite time, Achilles will have run 100 meters, bringing him to the tortoise's starting point. During this time, the tortoise has run a much shorter distance, say 2 meters. It will then take Achilles some further time to run that distance, by which time the tortoise will have advanced farther; and then more time still to reach this third point, while the tortoise moves ahead. Thus, whenever Achilles arrives somewhere the tortoise has been, he still has some distance to go before he can even reach the tortoise. As Aristotle noted, this argument is similar to the Dichotomy. It lacks, however, the apparent conclusion of motionlessness.

Arrow Paradox

Zeno states that for motion to occur, an object must change the position which it occupies. He gives an example of an arrow in flight. He states that at any one (duration-less) instant of time, the arrow is neither moving to where it is, nor to where it is not. It cannot move to where it is not, because no time elapses for it to move there; it cannot move to where it is, because it is already there. In other words, at every instant of time there is no motion occurring. If everything is motionless at every instant, and time is entirely composed of instants, then motion is impossible.

Whereas the [Achilles and the Tortise paradox] divide[s] space, this paradox starts by dividing time—and not into segments, but into points.

These are interesting paradoxes to consider, especially in the context of calculus.


According to science and math historian Marshal Clagett, Aristotle "is the first one whose treatise we have to give us rules for comparing the speeds of bodies in terms of the space traversed in given times...we can derive from [four inequalities between space, time, and speed] our general expression V=S/T once we have decided that velocity can be represented as a ratio."


This is probably more what you're looking for.

Also, it's worth noting that Aristotle proposed that the speed at which two identically shaped objects sink or fall is directly proportional to their weights and inversely proportional to the density of the medium through which they move. This was a major point of contention for Galileo as he showed this proposition to be false.

In the Third Day: Uniform Motion of Two New Sciences, Galileo restates the definition of uniform speed: "distances traversed by the moving particle during any equal intervals of time, are themselves equal."

He further adds the condition "the word 'any,' meaning by this, all equal intervals of time; for it may happen that the moving body will traverse equal distances during some equal intervals of time and yet the distances traversed during some small portion of these time-intervals may not be equal, even though the time-intervals be equal."

From there, he states that the definition of uniform speed requires Axioms I-IV (the inequalities mentioned earlier), then uses them to construct ratios of speed, distance, and time in Theorems I-III, and then shows the product (i.e., compound) of these ratios in Theorems IV-VI.


Although Galileo famously presents his version of the mean value theorem for uniform acceleration, the mean value theorem was actually demonstrated by the Oxford Calculators in the 14th century, and was later proved by Nicole Oresme. The theorem is also known as the "Merton Rule": https://en.wikipedia.org/wiki/Mean_speed_theorem

Lastly, I should note that, in all the developments above, the arguments were presented using geometry and not algebra. The equation V=S/T would be objectionable to everyone mentioned above because dividing one magnitude (distance) by a different magnitude (time) to produce a third magnitude (speed) was considered absurd -- it would be analogous to dividing angles by areas or arc lengths by volumes. Instead, in Euclidean geometry, like magnitudes were compared to like magnitudes through ratios. For example, Galileo presents the relationship V1/V2 = (S1/S2)x(T2/T1) where "/" signifies ratios and not algebraic division, and "x" signifies a compounding of ratios and not a multiplication or product.

  • 2
    $\begingroup$ That is extremely helpful, thank you! It definitely shows that the quote I was working from wasn't quite capturing the entire story. I very much appreciate the last paragraph in particular. As someone raised on the modern concept of functions, I'd known they did things "geometrically" before Calculus, but your walk through of how they would think of such things was tremendously helpful for understanding what it means to do things geometrically. $\endgroup$
    – Cort Ammon
    Commented Dec 9, 2022 at 5:12
  • $\begingroup$ Glad to hear it! There was a big push to algebratize geometry in the 19th century, and the result is what's taught in most schools now. It may come as a surprise that defining areas as A=L*W is relatively new -- you won't find this equation in Euclid's Elements because equating a product of two lines to an area is absurd to Euclid; however, the relationship works if expressed through ratios, as is shown in Elements Book VI, Prop. 23. This proposition gives a broader relationship than just the algebra expression because it holds true, not just for rectangles, but for any parallelogram. $\endgroup$
    – Andrew R.
    Commented Dec 9, 2022 at 17:02
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    $\begingroup$ Seeing that you're working through calculus, you should check out Newton's Principia -- his original calculus and his laws of gravity were all demonstrated geometrically. Geometry can be difficult and is often pedantic, but I found it rewarding and useful for understanding Galileo's and Newton's original arguments. Elements is free online, and this site comes with some really great commentary: aleph0.clarku.edu/~djoyce/java/elements/elements.html $\endgroup$
    – Andrew R.
    Commented Dec 9, 2022 at 17:14

This discussion comes from confusion of two notions: what we call now the average speed and the momentary speed. The concept of average speed (distance divided by time) is very old indeed. For example, it was used by sailors in Dead reckoning since the times immemorial, and in general to estimate time of travel. The concept of momentary speed (the limit of the above ratio) was only made clear in the 17 century. Zeno'a paradoxes show how the ancient Greeks struggled with this notion.


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