The English works of Thomas Hobbes of Malmesbury, Volume 01 (of 11)
Thomas Hobbes · en
Again, seeing by supposition A Z. Z B:: C D F E. C I A P E are
proportionals; A B. A Z:: C D F E + C I A P E. C D F E will also, by
compounding, be proportionals. And seeing A L is the half of A B, A L. A
Z:: C D F E + C I A P E. 2 C D F E will also be proportionals. But the
proportion of C D F E + C I A P E to 2 C D F E is compounded, as was but
now shown, of the proportions of moment to moment, &c., and therefore
the proportion of A L to A Z is compounded of the proportion of the
moment of the complete figure C D F E to the moment of the deficient
figure C I A P E, and of the proportion of the weight of the deficient
figure C I A P E to the weight of the complete figure C D F E; but the
proportion of A L to A Z is compounded of the proportions of A L to B Z
and of B Z to A Z. Now the proportion of B Z to A Z is the proportion of
the weights reciprocally taken, that is to say, of the weight C I A P E
to the weight C D F E. Therefore the remaining proportion of A L to B Z,
that is, of L B to B Z, is the proportion of the moment of the weight C
D F E to the moment of the weight C I A P E. But the proportion of A L
to B Z is compounded of the proportions of A L to A Z and of A Z to Z B;
of which proportions that of A Z to Z B is the proportion of the weight
C D F E to the weight C I A P E. Wherefore (by art. 5 of this chapter)
the remaining proportion of A L to A Z is the proportion of the
distances of the points Z and L from the centre of the scale, which is
A. And, therefore, (by art. 6) the weight C I A P E shall hang from O in
the strait line O Z. So that O Z is one diameter of equiponderation of
the weight C I A P E. But the strait line A B is the other diameter of
equiponderation of the same weight C I A P E. Wherefore (by the 7th
definition) the point Z is the centre of the same equiponderation; which
point, by construction, divides the axis so, that the part A Z, which is
the part next the vertex, is to the other part Z B, as the complete
figure C D F E is to the deficient figure C I A P E; which is that which
was to be demonstrated.
Coroll. I. The centre of equiponderation of any of those plane
three-sided figures, which are compared with their complete figures in
the table of art. 3, chap. XVII, is to be found in the same table, by
taking the denominator of the fraction for the part of the axis cut off
next the vertex, and the numerator for the other part next the base. For
example, if it be required to find the centre of equiponderation of the
second three-sided figure of four means, there is in the concourse of
the second column with the row of three-sided figures of four means this
fraction 5⁄7, which signifies that that figure is to its parallelogram
or complete figure as 5⁄7 to unity, that is, as 5⁄7 to 7⁄7, or as 5 to
7; and, therefore the centre of equiponderation of that figure divides
the axis, so that the part next the vertex is to the other part as 7 to
5.