Electronic edition published by Cultural Heritage Langauge Technologies (with permission from Charles Scribners and Sons) and funded by the National Science Foundation International Digital Libraries Program. This text has been proofread to a low degree of accuracy. It was converted to electronic form using data entry.
JORDANUS DE NEMORE (fl. ca. 1220), mechanics,
mathematics.
equal force.
FIGURE 4
He first assumes that a weight g, equal
to e, is on another plane, DK, whose obliquity is equal
to that of DC and then assumes that if e moves to L
through vertical distance ER, it will also draw h up
to M. Should this occur, however, it would follow by
the principle of work that what is capable of moving
h to M can also move g to N, since it can be
shown
that MX/NZ = g/h. But g is equal to e and at the
same
inclination; hence, by Book I, Proposition 9, they are
of equal force because they will intercept equal
segments of vertical DB. Therefore e is incapable of
raising g to N and, consequently, unable to raise
h to M. By substituting a straight line for an arcal
path and utilizing Postulate 5 of the Elementa (see
above), Jordanus was, in modern terms, measuring
the obliquity of descent, or ascent, by the sine of the
angle of inclination. The force along the rectilinear
oblique path, or incline, is thus equivalent to
F = W sin a,
where W is the free weight and a is the angle of inclination
of the oblique path.
The principle of work, which was but a vague
concept prior to Jordanus, was not only used effectively
in the proof of the inclined plane and, as indicated
above, in the proof of the law of the lever in the
Elementa, a proof repeated in Book I, Proposition 6,
of the De ratione ponderis, but was also applied
successfully to the first proof of the bent lever in
Book I, Proposition 8, of the De ratione ponderis,
which reads: “If the arms of a balance are unequal,
and form an angle at the axis of support, then, if their
ends are equidistant from the vertical line passing
through the axis of support, equal weights suspended
from them will, as so placed, be of equal heaviness.”15
In this proof there is also an anticipation of the concept
of static moment, that the effective force of a weight
is dependent on the weight and its horizontal
distance from a vertical line passing through the
fulcrum.16
Over and above his specific contributions to the
advance of statics, Jordanus marks a significant
departure in the development of that science. He
joined the dynamical and philosophical approach
characteristic of the dominant Aristotelian physics
of his day with the abstract and rigorous mathematical
physics of Archimedes. Thus the postulates of the
Elementa and De ratione ponderis were derived from,
and consistent with, Aristotelian dynamical concepts
of motion but were arranged in a manner that permitted
the derivation of rigorous proofs within a
mathematical format modeled on Archimedean
statics and Euclidean geometry.
The extensive commentary literature on the statical
treatises ascribed to Jordanus began in the middle
of the thirteenth century and continued into the
sixteenth. Through printed editions of the sixteenth
century, the content of this medieval science of
weights, identified largely with the name of Jordanus,
became readily available to mechanicians of the
sixteenth and seventeenth centuries. Dissemination
was facilitated by works such as Peter Apian's Liber
Iordani Nemorarii ... de ponderibus propositiones
XIII et earundem demonstrationes (Nuremberg, 1533);
Nicolo Tartaglia's Questii ed invenzioni diverse (Venice,
1546, 1554, 1562, 1606; also translated into English,
German, and French), which contained a variety of
propositions from Book I of the De ratione ponderis;
and Jordani Opusculum de ponderositate (Venice,
1565), a version of the De ratione ponderis published
by Curtius Trojanus from a copy owned by Tartaglia,
who had died in 1557.
Concepts such as positional gravity, static moment,
and the principle of work, or virtual displacement,
were now available and actually influenced leading
mechanicians, including Galileo, although some preferred
to follow the pure Greek statical tradition of
Archimedes and Pappus of Alexandria. In commenting
on Guido Ubaldo del Monte's Le mechaniche (1577),
which he himself had translated into Italian, Filippo
Pigafetta remarked that Guido Ubaldo's more immediate
predecessors
... are to be understood as being the modern writers
on this subject cited in various places by the author
[Guido Ubaldo], among them Jordanus, who wrote on
weights and was highly regarded and to this day has
been much followed in his teachings. Now our author
[Guido Ubaldo] has tried in every way to travel the road
of the good ancient Greeks, ... in particular that of
Archimedes of Syracuse ... and Pappus of Alexandria
...17