The equality of the ratios MX/NZ and DK/DA shows that we can replace movements on inclined
planes by vertical displacements. Jordanus goes no further. But we can spontaneously give a
physical interpretation of the inverse proportionality between weights g and h, and the vertical
displacements NZ and MX: “that which suffices to raise h along XM also suffices to raise g along
ZN” or, as Tartaglia writes, “the force or the power of h on the plane DA is equal to the force or the
power of g on the plane DK”.
33
This interpretation is not explicit in Jordanus, but his reasoning as
follows seems to require it. In fact, he continues, since e cannot raise g (indeed, by construction,
planes DC and DK have the same slope and weights g and e are equal to each other), e cannot raise
h either. Thus e and h will remain in equilibrium.
34
An ancient mechanical system and a new model
After having set certain physical definitions, Jordanus presents a “purely formal” mechanical proof;
in one decisive point, it even avoids any involvement in a physical interpretation. It is nonetheless
incomprehensible if we do not identify the underlying mechanical system: if e and h were not tied
by a line passing over D (or better yet, through a pulley attached at D), there would be no sense in
supposing that e, when it descends to L, would draw h to M. We do not know if certain manuscripts
32
Ibid., p. 190: “Quia igitur proportio NZ ad NG sicut DY ad DG, et ideo sicut DB ad DK, et
quia similiter MX ad MH sicut DB ad DA, erit MX ad NZ sicut DK ad DA, et hoc est sicut g ad h”.
33
Tartaglia, Quesiti et inventioni diverse, fol. 97v: “E pero [h e g] si vengono ad egualiar in
virtu, over potentia, E per tanto quella virtu, over potentia, che sara atta à far ascendere luno de detti
dui corpi, cioe à tirarlo in suso, quella medesima sara atta, over sofficiente à fare ascendere anchora
l’altro, adunque sel corpo e (per laversario) è atto, E sofficiente à far ascendere il corpo h per fin in
M, el medesimo corpo e saria adunque sofficiente à far ascendere anchora il corpo g a lui equale, E
inequale declinatione, la qual cosa è impossibile per la precedente propositione (Thus, if [h and g]
come to be equal in force, or in power, for in as much as the force, or the power, which will be apt
or sufficient to make the other rise, thus if the body e (according to the opponent) is apt and
sufficient to make the body h rise to m, this same body e would be by this fact sufficient to make
the body g also rise to equal to itself, and equal in inclination, which, according to preceding
proposition, is impossible”. Duhem fills in this gap by applying here an “implicit postulate” by
which that which can raise a weight P by a height h can also raise nP by a height h/n (Duhem, Les
origines de la statique, vol. I, p. 142, p. 147). This postulate is also made explicit in prop. 6 of De
ratione ponderis, in ibid., p. 182.
34
Ibid., p. 190: “Sed quia e non sufficit attollere g in N, nec sufficiet attollere h in M. Sic
ergo manebunt”.