The purpose of this study (1) is to reconsider Galileo's treatment of motion on inclined planes in his earliest written works devoted to motion, known as De motu antiquiora (hereafter DMA), avoiding the usual clear-cut separation with the treatment given, in the same work, to the problem of motion in media. It is believed that some clarifications arise from this unconventional approach concerning the relations between the young Galileo's views on dynamics and his terminology on the one hand, and the scholastic tradition on the other hand.

In the 14th chapter of the version in 23 chapters of DMA " in quo agitur de proportionibus motuum eiusdem mobilis super diversa plana inclinata ", Galileo considers two questions (here and in the following (N,M;ch.P) stands for Opere, vol.I, from page N line M onwards, chapter P) (2):

(296,12;ch.14) Quaeritur enim cur idem mobile grave, naturaliter descendens per plana ad planum horizontis inclinata, in illis facilius et celerius movetur quae cum horizonte angulos recto propinquiores continebunt; et, insuper, petitur proportio talium motuum in diversis inclinationibus factorum.

He claims that he has succeeded in solving these problems, " ex notis et manifestis naturae principis ".

His answer to the second question, viz. to the quantitative problem, is first given in the form:

(298,23;ch.14) Eandem ergo proportionem habebit celeritas in ef ad celeritatem in gh, quam linea da ad lineam pa. Est autem sicut da ad ap ita qs ad sp, hoc est obliquus descensus ad rectum descensum.

He gives the following figure

Here the motion on an inclined plane is compared to the vertical free fall over the same difference in height. This is easily transformed into the comparison of the motions on two inclined planes with different slopes, answering precisely the problem as it is stated:

(301,23;ch.14) Constat ergo, eiusdem mobilis in diversis inclinationibus celeritates esse inter se permutatim sicut obliquorum descensuum, aequales rectos descensus compraehendentium, longitudines.

This answer, as well as its derivation, have both been considered faulty by modern critics.

The proposition (301,23) itself is considered to be grossly wrong because it is supposed to imply, in its very wording, the uniformity of the motion on an inclined plane (3). I did challenge this conclusion elsewhere (4); this led me to see in this "de motu theorem" nothing but an earlier formulation of the isochronism of the descent along the cords of a circle the final form of which is the "Theorem VI" on accelerated motion in the Discorsi. This question - the meaning of the proposition - will not be considered further here.

The lengthy demonstration which Galileo provides for his Proposition is also claimed by the modern critics to be unsatisfying, either for a lack of well defined concept of celeritas (5) or for the use of an unjustified proportionality (proportionalitas) (6).

Such analyses by distinguished scholars of course reflect the considerable complexity of the conceptual and the terminological aspects of the text, and one finds noticeable variations of interpretation between authors. It is clear that a definite understanding of this text implies definite interpretations of such crucial terms as celeritas, tarditas, facilitas and of different occurrences of vis in the text (e.g. ... tanto maiori vi fertur; ... descendat, quanto minori vi trahi sursum, ...)

It is worthwhile noticing that these alleged weaknesses in Galileo's text have not hitherto been considered to arise in his discussion of natural motion in media, which constitutes the main part of the treatise. If I can show that both theories are but two instances of a unique theory, then we shall be confronted with two possibilities, either that some of the flaws found in the theory of the inclined plane must also occur in the theory motions in media, or that the "flaws" are artifacts due to a misunderstanding of Galileo's text. In both cases some new insight should arise from a detailed comparison of these two theories as they are expounded in DMA. I now consider such a comparison.

I must be emphasized that I shall restrict the comparison in this study to Galileo's analysis of the motion of one and the same body in different situations, although the full problem includes of course the general case of different bodies in different situations. Besides the conditions set by the limited format of this contribution, there is an historical justification for a separate presentation of this restricted problem: the full problem is actually solved by Galileo in the DMA for the case of motions in media, but not for the case of motions on inclined planes. For this latter case Galileo contents himself with relying on the skill of his reader to make such generalizations: haec et similia ab his, qui quae supra dicta sunt intellexerint, facile inveneri possunt. However, the generalization consistent with the limited problem as solved by Galileo is far from evident and raises interesting problems which will be considered in a forthcoming paper.

Motions of a body in media and on inclined planes

Let us first consider the logical structure of Galileo's demonstrations of his propositions concerning motion in media (chapters 7; 8; 9) and on inclined planes (chapter 14). I claim that close analysis shows that both situations are analyzed within a unique demonstrative scheme. To support this claim I shall in the following consider in parallel the successive steps of the arguments as they appear in the DMA, and give for each of the two problems the relevant (or representative) passages.

1.- The problem is first set, in similar terms, as the comparison of the motions of one and the same body in different media, or on different planes.

a) in media

A traditional demonstrative scheme

Both problems are seen to be treated in a fully consistent way along the following demonstrative scheme:

A dynamical rule - or law - is first accepted in the generic form of the proportionality (proportionalitas) between causes and effects (item 2.2), and is then transformed into a quantitative form by introducting the adopted measures for cause and effect (in item 3 above for the cause, and according to the preclassical tradition for the celeritas (8) designated as the effect).

This general dynamical statement is nothing but a variant of the standard dynamical rule discussed by scholastic commentators, stating that the velocitas is proportional to the potentia and inversely proportional to the resistentia (9). In the restricted problem considered here of the motions of one and the same body the resitentia plays no role, but I shall just mention that a concept of resistentia is actually implied and used by Galileo in the DMA in a quantitative way, although not explicitly, when he proves that a large piece of a given material rises in a heavier media with the same velocity as a larger piece of the same material, or when he let his reader extend his analysis to the case of the fall along different inclined planes of bodies of different gravitates (10).

Some consequences of the hypothesis of internal consistancy of DMA chapter 14 on the interpretation of Galileo's scientific terminology

This analysis, together with the consistency hypothesis that Galileo's demonstrations and discussions in this chapter 14 of the DMA are logically deductive rather than somewhat metaphoric at critical places, allows for a clarification of some terminological problems. The consistency hypothesis is accepted here but is by no mean trivial; in my view, the stronger argument for it is precisely the unity of Galileo's dynamical discussions pointed to above.

Consider first two crucial occurrences of vis. The consistency hypothesis implies that in in item (4,b), i.e. (296,26;ch.14), tanto minori vi trahi sursum idem pondus the term vis has a dynamical meaning referring to the cause of the motion, rather than a kinematical meaning referring to the effect (11), while in the same sentence tanto maiori vi descendere must be understood in the kinematical meaning of "descends with a velocity so much larger".

More difficult is a correct interpretation of Galileo's use of facilitas which occurs in the DMA at least five times in expressions such as facilius et citius, as hereabove in item 1.

We may first notice that the statement of the problem in item 1 above, as "quaeritur ... cur ... facilius et celerius movetur ..." does actually introduce the two questions investigated in the text following, specifically the variations of the gravitas in plano and of the celeritas with the slope of the planes if and only if facilitas refers to the cause and celeritas to the effect.

The same conclusion arises with still more strength when applying the consistency hypothesis to the much debated section of DMA chapter 14 where Galileo puts together his arguments before concluding to the proportionality between the celeritates and, say, the slopes of the planes. The text goes as follow (where I emphasize the terms discussed, and the figure is as above):

(298,16;ch.14) Sed quanto maiori vi moveatur per ef quam per gh, ita innotescet: extensa, scilicet, linea ad extra circulum, quae secet lineam gh in puncto q. Et quia tanto facilius descendit mobile per lineam ef quam per gh, quanto gravius est in puncto d quam in puncto s; est autem tanto gravius in puncto d quam in s, quanto longior est linea da quam linea ap; ergo mobile eo facilius descendet per lineam ef quam per gh, quo linea da longior est ipsa pa. Eandem ergo proportionem habebit celeritas in ef ad celeritatem in gh, quam linea da ad lineam pa.

If we are to recognize in this short section a synthetic restatement of the arguments developed at length in the preceding discussion, then facilitas must be understood here as the denomination of the cause of the motion; it is then what scholastics used to call inclinatio, and what Galileo himself in the tract Le meccaniche written soon after called propensione or momento.

According to the above analysis Galileo's demonstration of the inclined plane theorem in DMA is devoid of the ambiguities which modern critics find in it.

An essential implication of my interpretation concerns Galileo's scientific terminology. More specifically, it implies the understanding of the term facilitas as it appears in the DMA is as a cause of motion, as an evolution of the term inclinatio of scholastic dynamics. And the latter, as is well known, came to have the same meaning as the propensione or momento which Galileo was soon to use in Le meccaniche and further works. This interpretation accords with my previous claim that chapter 14 of the DMA belongs to what I called the velocitas- momentum class of Galileo's texts (12).

The hitherto unnoticed unity of Galileo's treatment of the motion of a body on different inclined planes and in different media which the present paper demonstrated, means that these two physical problems are treated in DMA with one and the same theory of motion, which is nothing but a variant - however important it may be - of the so-called traditional "scholastic dynamics".