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I would assume that the early estimates were wildly off and wonder how this measurement was done.

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Some prehistory involving Aristotle, Tartaglia, Galileo and Torricelli can be found in The Impact of Ballistics on Mathematics by McMurran and Rickey. It was more speculative than experimental, and focused more on the trajectories and range of projectiles rather than speed. Euler wrote a short note Meditatio in experimenta explosione tormentorum nuper instituta (1727-28) (better known for the employment of letter $e$ for the base of natural logarithms) reporting on some early experiments with cannon balls, possibly conducted by Daniel Bernoulli. According to Johnson's Encounters between Robins, and Euler and the Bernoullis:

"The circumstance of its employment there described was to report on seven experiments conducted on cannon balls of different weights projected by different weights of gun powder, for different angles of elevation; range and time of flight (time of ascent and descent were calculated separately) based on some of Euler's own calculations which rest on his assumptions about air resistance. The original title of the 10 page paper written in Latin, when translated, is Meditation upon Experiments made recently in the firing of Cannon. From the use of the word 'recently' it is deduced that the original article was written in 1727 or 1728. However, the manuscript of the paper was first printed at Petrograd [sic!] only in 1862 in his Postumous Mathematical and Physical Work (pp. 468-477), and it would seem therefore to have been published much too late to be of value for artillerists at large...

It is likely that these experiments were made by Daniel Bernoulli because all we see presented are various quantities (given) which are substituted into certain equations and then magnitudes such as time of flight, etc., determined. No experimental details are reported, e.g. details about the gun used, charges, shot, calibre and the weather, etc."

But historians of ballistics usually credit Robins with the first reliable measurements, conducted using a clever contraption he invented, known as ballistic pendulum. The momentum of a fired bullet was either measured directly, by firing it into the pendulum, or by attaching the gun to the pendulum and measuring the recoil. The firing velocity could be found by dividing the momentum by the mass. Robins published his results, on this and many other ballistic topics, in New Principles of Gunnery (1742), which became the bible of modern ballistics. Euler translated it into German in 1745, with a commentary four times the size of the original that mathematized the subject. McMurran and Rickey give a detailed discussion of both. Here is from Exterior Ballistics in the Plane of Fire by Ingalls (1886):

"Benjamin Robins was the first to execute a systematic and intelligent series of experi ments to determine the velocity of projectiles and the effect of the resistance of the air, not only in retarding but in de flecting them from the plane of fire. He was the inventor of the ballistic pendulum, an instrument for measuring the momenta of projectiles and thence their velocities. He also invented the Whirling Machine for determining the resistance of air to bodies of different forms moving with low velo- cities. His " New Principles of Gunnery," containing the results of his labors, was published in 1742, and immediately attracted the attention of the great Euler, who translated it into French [sic!].

The next series of experiments of any value were made toward the close of the last century by Dr. Hutton, of the Royal Military Academy, Woolwich. He improved the apparatus invented by Robins, and used heavier projectiles with higher velocities. His experiments showed that the resistance is approximately proportional to the square of the diameter of the projectile, and that it increases more rapidly than the square of the velocity up to about 1440 f. s., and nearly as the square of the velocity from 1440 f. s. to 1968 f. s.*"

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  • $\begingroup$ I would wonder if Aristotle would have understood a metric like feet per second -- the relationship between distance traveled per unit time (when say an arrow left the bow) and impact or range would not have been obvious. In fact, I wonder when FPS would have first been understood in this way. $\endgroup$ – releseabe Jan 26 at 12:28
  • $\begingroup$ @releseabe That sounds like a separate question, but Aristotle and other ancients only dealt with speed for uniform motion where these issues did not arise. $\endgroup$ – Conifold Jan 26 at 12:30
  • $\begingroup$ yes but related. your answer makes it clear that it was not until fairly modern times that speed was measured and of course the thing that the military was interested in would have been range which after Newton would have been understood to be related to speed. $\endgroup$ – releseabe Jan 26 at 12:32
  • $\begingroup$ @releseabe Are you asking how the speed/range of arrows was measured? $\endgroup$ – Conifold Jan 26 at 12:34
  • $\begingroup$ i am saying the speed of arrows would not have been of interest to anyone in ancient times, i don't think. range would have been but the relationship between speed and range would not have been understood. $\endgroup$ – releseabe Jan 26 at 12:41

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