I remember hearing once that the first "proof" that the angle to maximise projectile range gave the correct answer, 45 degrees, but was later found that the proof was wrong. I can't remember many other details, but I believe the name was Italian. Does anybody know a reference or a name that I could use to find out more about this?

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    $\begingroup$ It wasn't "wrong." It assumes zero air resistance, which is a perfect first-order approximation. $\endgroup$ Aug 10, 2020 at 10:11

1 Answer 1


This probably refers to Galileo's "derivation" of Tartaglia's observation that cannon balls achieve maximal range when fired at 45°. Tartaglia's theory of projectile motion was wrong, he assumed that fired balls follow a line segment going up, then an arc of a circle to change direction, and finally fall vertically down, but the observation was experimental. Galileo obtained the same angle from his theory of accelerated motion in Discorsi intorno a Due Nuove Scienze.

But, as he came to realize, this only applied to slow projectiles launched without gunpowder (black powder different from gunpowder of today), e.g from bows and catapults, or even with it from mortars, but not from cannons. Because for fast projectiles air resistance alters trajectories considerably. And Galileo had no theory to take it into account. For example, in modern guns maximal range is attained at angles nearer to 30° rather than 45°.

Ironically, fast projectiles have trajectories shaped more like Tartaglia's than like Galileo's parabolas. Nonetheless, Galileo fudged the difference to make his result look more general and significant, see Rose, Galileo's Theory of Ballistics:

In his letter of March I637, Galileo had enthusiastically proclaimed a table usable for both cannon and mortar shooting. Between writing that letter and submitting the final version of the Fourth Day to Elsevir three months later, Galileo had come to realize that because of air- resistance a general solution to the ballistic problem of ranging was beyond him; in the Discorsi he had to be content with a partial solution applying to mortars only.

How then has the notion arisen that the table in the Discorsi represents a general solution? Undoubtedly part of the blame can be attached to Galileo's obliqueness about the exact purpose of the table that he does print. This equivocation was perhaps born from a desire to seem to have discovered the long sought after general solution and yet still to retain his integrity. This equivocation was in fact noticed by Descartes... Other readers were less perceptive than Descartes, and rushed to hasty optimistic conclusions. Torricelli and Mersenne, both friends of Galileo, though aware of the existence of air-resistance, still thought that the resistance was so slight as to be disregarded in the preparation of general range-tables."

When Torricelli's tables were tested by Renieri in 1646 he expressed astonishment that "such a well-founded theory should respond so poorly in practice". In response, Torricelli invoked Galileo's authority. The error came to be commonly attributed to Galileo himself. Descartes was quite derisive in his commentary:

"It ought to be noticed that in proposing his suppositions he excepts from them artillery so that he may demonstrate them more easily. But that, all the same, towards the end it is principally to artillery that he applied his conclusions. In a word, il a tout basti en l'air".

The last phrase is a pun, "he did everything in the air", as in, he screwed everything up.

  • $\begingroup$ "slow projectiles launched without gunpowder, e.g from mortars" -- what propellant did mortars use if not gunpowder? $\endgroup$
    – nanoman
    Aug 9, 2020 at 1:38
  • $\begingroup$ @nanoman Black powder, often confused with the smokeless powder which is now called gunpowder. $\endgroup$
    – Conifold
    Aug 9, 2020 at 6:08
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    $\begingroup$ Okay, but smokeless powder did not exist until the 1800s, long after Galileo, so why do you say that Galileo "came to realize, this only applied to slow projectiles launched without gunpowder, e.g from mortars, not from cannons"? The cannons of his time used black powder too, right? So I fail to understand "without gunpowder" as a distinction Galileo could have made between mortars and cannons. $\endgroup$
    – nanoman
    Aug 9, 2020 at 19:20
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    $\begingroup$ @nanoman Sorry for confusion. Salviati first separates "projectiles shot by firepower (as I believe, by the force of gunpowder), from those other projectiles shot from crossbows, bows or catapults", but later adds that "velocita soprannaturale" derived from gunpowder is only achieved in cannons, while mortar bombs are closer to catapults as far as their velocita (Rose estimates it as 300 ft/s). I edited the text to clarify. $\endgroup$
    – Conifold
    Aug 10, 2020 at 4:30

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