I. When two bodies are equally poised, if weight be added to one of them and not to the other, their equiponderation ceases.
II. When two ponderants of equal magnitude, and of the same species or matter, press the beam on both sides at equal distances from the centre of the scale, their moments are equal. Also when two bodies endeavour at equal distances from the centre of the scale, if they be of equal magnitude and of the same species, their moments are equal.
Two planes
of equiponderation
are
not parallel.
2. No two planes of equiponderation are parallel.
Let A B C D (in [fig. 1]) be any ponderant whatsoever; and in it let E F be a plane of equiponderation; parallel to which, let any other plane be drawn, as G H. I say, G H is not a plane of equiponderation. For seeing the parts A E F D and E B C F of the ponderant A B C D are equally poised; and the weight E G H F is added to the part A E F D, and nothing is added to the part E B C F, but the weight E G H F is taken from it; therefore, by the first supposition, the parts A G H D and G B C H will not be equally poised; and consequently G H is not a plane of equiponderation. Wherefore, no two planes of equiponderation are parallel; which was to be proved.
The centre of equiponderation is in every plane of equiponderation.
3. The centre of equiponderation is in every plane of equiponderation.
For if another plane of equiponderation be taken, it will not, by the last [article], be parallel to the former plane; and therefore both those planes will cut one another. Now that section (by the 6th definition) is the diameter of equiponderation. Again, if another diameter of equiponderation be taken, it will cut that former diameter; and in that section (by the 7th definition) is the centre of equiponderation. Wherefore the centre of equiponderation is in that diameter which lies in the said plane of equiponderation.
The moments
of equal ponderants
are to one
another as their
distances from
the centre of
the scale.
4. The moment of any ponderant applied to one point of the beam, to the moment of the same or an equal ponderant applied to any other point of the beam, is as the distance of the former point from the centre of the scale, to the distance of the latter point from the same centre. Or thus, those moments are to one another, as the arches of circles which are made upon the centre of the scale through those points, in the same time. Or lastly thus, they are as the parallel bases of two triangles, which have a common angle at the centre of the scale.