¹ Galileo, , Opere (Ed. Naz.), x, 114.Google Scholar

² Opere, x, 115.Google Scholar

³ It is evident that velocità here means speed over an interval, and not instantaneous velocity, into which time does not enter.

⁴ In the autograph, Galileo first wrote la linea, which he cancelled and then wrote il lempo. The difference is one of thought rather than one of a plausible mistaken copying. The argument as we have it is thus probably not the result of successive revisions on paper, but a first record of an argument thought out. Possibly Galileo started to write “the line AD has to the line AC a subduplicate ratio to the speeds and a duplicate ratio to the times of motion over the distances AD and AC”, but decided instead to omit speeds and relate the times and distances directly.

⁵ Opere, viii, 373–374.Google Scholar

⁶ Études sur Leonard de Vinci (Paris, 1955), iii, 562–566.Google Scholar

⁷ Études Galiléennes (Paris, 1939), ii, 21–25.Google Scholar

⁸ “Another Galilean Error”, Isis, 1 (1959), 261–262.Google Scholar

⁹ “Galileo, Falling Bodies and Inclined Planes”, British Journal for the History of Science, iii, (1966–1967)Google Scholar, especially pp. 230–244.

¹⁰ This is the wording of Heath's Euclid. Commandino used the word “converse” instead of “inverse”, in which he was followed by the first English translation of 1570. Tartaglia used “contrary”. All these, of course, merely name the inversion of a ratio, or of both ratios in a proportionality. In 1638, Galileo used either the phrase “taken contrarily” or “taken permuted” after the second ratio named in a proportionality, when he wished to assert inverse proportion ality in the modern sense.

¹¹ Opere, viii, 361Google Scholar. Intended asan addition to the Discorsi, Galileo's treatment of the theory of proportion was first published by Viviani in 1674.

¹² Thus from the false assumption that all women are Greek, one might argue with a further error that since Socrates was a woman. Socrates was a Greek; or one might argue without further error that since Greeks are mortal, women are mortal.

¹³ A physical reason is invoked to support the crucial “contrary proportion” argument, discussed at length below.

¹⁴ Opere, vii, 75–76Google Scholar; Dialogue (tr. Drake), p. 51.Google Scholar

¹⁵ Contrary to the opinion of Koyré, the experiments described in 1638 were quite adequate to verify the law; cf. Settle, T., “An experiment in the history of science”, in Science, 133 (1961), pp. 19–23.CrossRefGoogle ScholarPubMed

¹⁶ Opere, i, 302Google Scholar “… hoc solum animadvertentes quod, sicut supra dictum de motu recto, ita etiam in his motibus super planis accidit non servari has proportiones quas posuimus …” Cf. Galileo on Motion and on Mechanics, ed. Drabkin, I. E. and Drake, S. (Madison, 1960), p. 69.Google Scholar

¹⁷ Opere, xiii, 348Google Scholar. S. Moscovici has translated this passage as saying that Galileo was unable to adduce a demonstration, perhaps having read però as potere. Baliani, however, implies if anything that Galileo had a demonstration but did not show it to him, giving only the argument mentioned.

¹⁸ Constantijn Huygens sent the youthful production of his son to Mersenne, and it was published in 1649 by Tenneur; it is a derivation of the odd-number rule without geometrical considerations, assuming only the concept of uniform acceleration.

¹⁹ Clagett, M., Science of Mechanics in the Middle Ages (Madison, 1959), p. 271Google Scholar; the quotation is from Heytesbury.

²⁰ Two New Sciences (tr. Crew and De Salvio), all editions p. 173.Google Scholar

²¹ Opere, viii, 208.Google Scholar

²² As previously mentioned, Galileo treated velocities as numbers. Adding the successive speeds as 1, 3, 5,… would give square numbers, as would adding the successive spaces traversed from rest. There is no reason to think that Galileo carried out such an operation, which would give some kind of undefined “cumulative speeds from rest,” but if he did, the proportionality to square numbers would not have disturbed him with regard to the restricted diagram shown above. By his hypothesis, the terminal velocities were as the total distances fallen from rest, so it was not necessary for him to determine them in any other way. Thus the anomalies that strike us could escape him, if he proceeded in the manner described here. I do not wish to conceal Galileo's error, but to show how it may have remained concealed from him in 1604.

²³ The figure references are altered here to conform to the previous diagram.

²⁴ Book V, Definition 3. Heath gives, “A ratio is a sort of relation in respect of size between two magnitudes of the same kind,” with a very extensive note on “ratio” and “proportion”: see Euclid's Elements (Dover, eds., N.Y., 1956), ii, pp. 116–119.Google Scholar

²⁵ This statement is to be amplified in a later paper. Until Galileo's discovery of the error and its correction, discussed below, it was universally assumed that space and time were proportionally related to one another in all matters of local motion. Until actual acceleration of falling bodies came under study by mathematicians such as Galileo, Beeckman, Descartes and Baliani, the error of that assumption was not recognized. Philosophers related speed to distance fallen, and mathematicians related uniform difform change to time elapsed, without anyone questioning whether the two were really compatible. Even Domingo de Soto, who related free fall to the Merton Rule, did not assert that this contradicted the numerous authors who by that time (1545) had related acceleration in free fall to the distances fallen.