# First attempt to measure speeds using distance divided by time?

Did the ancient Greeks, for example, speak of number of paces per heartbeat? I could see them only saying, something moved with the speed of an arrow or a running horse or a running man but that they lacked a unifying concept of speed being distance covered in a constant time. I am not even sure the Greeks understood there to be an abstract time unit that did not vary -- the time keeper for legal court that was invented divided a day into 12 (or some number) of equal sections but the length of each unit varied according to the time of year because days were shorter.

EDIT: Even if the ancients understood time as being measured in something like seconds, when speed or velocity was first thought as distance divided by time remains my question. My guess is that this occurred after 1000 AD -- Merton's Theorem I think would imply a modern way of looking at velocity: https://www.britannica.com/science/Merton-theorem and I also wonder how scientists in China or India thought of velocity or even in pre-Columbian America.

• I am not even sure the Greeks understood there to be an abstract time unit that did not vary --- This web page should help in finding appropriate search terms whose use will likely change your mind. There's also this recently published book: A Brief History of Timekeeping: The Science of Marking Time, from Stonehenge to Atomic Clocks by Chad Orzel (2022). Commented May 1, 2022 at 17:11
• @DaveLRenfro: Thanks -- how ancients thought about time should be a very interesting read. Commented May 1, 2022 at 17:48
• Galileo was the first one, using some "home made" devices; see Stillman Drake's detailed studies. The reason is simple: no timekeeping device available in ancient times. Commented May 2, 2022 at 8:32
• It might come down to a question of awareness & also subdivisions of time. An ancient Greek or Roman might want to travel 10 leagues. Walking might take one day. Riding a chariot might take half a day, while riding a horse might take a third of a day. They would have been aware of the differences in time by each mode of travel, but would they then have that leap of thought that states, walking speed would be 10 leagues per day, chariot 20 leagues per day & horse 30 leagues per day? Just a comment to contemplate.
– Fred
Commented May 4, 2022 at 1:46
• So you are looking for Merton theorem but for constant speed? Commented Sep 6, 2022 at 11:43

Greek astronomers of the Hellenistic epoch had a good understanding of time and speed in astronomy. They understood that neither Sun nor Moon move with constant speed, had the notion of average speed (still called "mean motion" in astronomy) and deviation of the actual speed from this average (which they called "equation", see Equation of time). So they had the concept of uniform time.

In daily life they used unequal, seasonal hours, but astronomers understood that they are unequal and took into account the necessary corrections.

I believe that minutes and seconds of time were used only by astronomers. I do not know of any Greek texts with a quantitative discussion of time and speed in physics. Or any discussion of measuring speed.

Edit. To address the comment about Greek and Roman artillery. 5 ancient technical books on this subject survive. None of them addresses the subject which we would call "ballistics". There is even no attempt to discuss the motion of a projectile, so these books confirm what I wrote above. All known sources confirm that Greek physics was limited to statics (including hydrostatics). But there is no hint of (mathematical, that is quantitative) dynamics, or even kinematics.

• I believe both the Greeks and Romans had artillery that fired bolts and in order to aim these accurately, it would have been useful to have some notion of speed and how it related to force but perhaps aiming accurately was not something they did with such weapons. Commented May 2, 2022 at 9:14
• Did the Hellenistic astronomers really consider speed? According to Toomer's translation of the 'Almagest', Ptolemy considered motion, recognizing both uniform ('homalos') and non-uniform ('anomalos') motion, which would cover the distinction you point to. It could be quantified by the equality or inequality of the changes of place occurring in equal increments of time, without necessarily adopting an explicit concept of speed. I would point to the reference and explanation cited in my answer below, about Euclid's theory of proportion forbidding the division of distance by time. Commented May 4, 2022 at 0:25
• Yes, their concept of motion is equivalent to our concept of speed. The units of measurement of "motion" are degrees/day. Or degrees/year, etc. And it is important that they distinguished "true motion" (actual speed) from "mean motion" (average speed), unlike the Babylonian astronmers, who also had a concept of mean motion. Commented May 4, 2022 at 21:21
• Are there sources for the use of such units? Commented May 5, 2022 at 12:58
• The main source of Greek astronomy is Ptolemy. Commented May 5, 2022 at 23:21

I'm not sure about Greeks, but Ossendrijver argues that Babylonian astronomers ca. 350-50 BCE used the trapezoidal rule to integrate Jupiter's motion along the ecliptic. The velocity is measured in degrees per day. The method of calculation resembles the trapezoidal rule and the much later “Mertonian mean speed theorem,” all corresponding to the formula $$s = t\cdot(v_0 + v_1)/2$$. Apparently, as Ossendrijver argues, the method was applied to determine the time that Jupiter crossed the ecliptic.

Reference

Mathieu Ossendrijver, Ancient Babylonian astronomers calculated Jupiter’s position from the area under a time-velocity graph, Science, 29 Jan 2016, Vol 351, Issue 6272. pp. 482-484

• Just amazing! What such great minds of the past would think of our sending probes to Jupiter and its moons. Imagine if they could look through a decent amateur telescope as I once did at Jupiter -- they would be astounded. Commented May 3, 2022 at 6:22

The Euclidean theory of proportion appears to speak against the ancient use of speed as a quotient between distance and time. Ratios, or proportions, according to that theory, were seen as legitimate only if they were established between quantities of like kind -- we might say, similarly dimensioned.

Explanation is given for example by N Guicciardini in "Mathematics and the New Sciences", chapter 8, (esp. sec. 8.3.2) in 'Oxford Handbook of the History of Physics' ed J Buchwald et al., 2013. Guicciardini wrote (in the context of discussing the heritage of antiquity still operative down ro the time of Galileo):

Since antiquity, continuous magnitudes had been approached through the theory of proportions, codified in Book 5 of Euclid’s Elements. [...] Note that the theory of proportions does not allow the formation of a ratio between two heterogeneous magnitudes. This is particularly important for kinematics, since it is not possible, for instance, to define speed as a ratio between distance and time.

And

The relationship between time, distance and speed in equable motion might {now} be expressed as

    s = v · t.


This was not possible within the framework of the theory of proportions, since one had to express a magnitude, speed, as being equal to the ratio between two heterogeneous magnitudes, distance and time. One has instead to state a series of proportions allowing only two of the three magnitudes involved (distance, time and speed) to vary.

What happened was that a kind of circumlocution was in use : (from the same source):

‘when the speed is the same and we compare two equable motions the distances are as the times’:

  s1/s2 = t1/t2,


where s1 (s2) is the distance covered in time t1 (t2). Or, ‘when the distance covered is the same, the speeds are as the times inversely’:

  v1/v2 = t2/t1.


It looks as though this doctrine had a considerable constraining effect on the ways in which motion might be conceptualised. The inhibitions appear to have lasted until the 17th century. According to Guicciardini in the work cited above, "Descartes’ geometry replaced the theory of proportions". It seems to have been then that more flexible patterns of thought about motion, including velocity, gained currency and accessibility.

(Some evidence in support of this late date for clear thinking about velocity comes from old works of astronomy, especially tables of motions. There are clear signs in many of them that motion and position were not clearly distinguished from each other. There was for example a concept of 'mean motion' that -- from the evidence of the numbers associated with it -- covered both place at a given point in time, and change of place in a given increment of time.)

(edit:) About the Babylonian astronomy: there may be evidence in the cuneiform tablets that the Babylonian astronomers expressly considered speed. Modern knowledge of this, however, dates only from the late 19th and early 20th century. The Greeks obtained some numerical parameters from ultimately Babylonian sources. It's not clear how much they obtained besides some numerical relationships that we might express as ratios, but they expressed as pairs of numbers of events of different types happening in the same interval of time.

• Interesting about how apparently a ratio between time and distance would be an unusual thing for the ancients to consider. Commented May 4, 2022 at 8:15
• Further to my above: It is pretty sophisticated to divide distance by time when all of your previous divisions have indeed been length by another length. It is sort of related to Bertrand Russel's assertion about a pair or apples and a brace of pheasants having something in common, I think. If you talk to beginners about math, you see that much of what one eventually takes for granted is not intuitive, at least not for everyone. Commented Feb 19 at 4:42