1. Definitions and suppositions.—2. Two planes of equiponderation are not parallel.—3. The centre of equiponderation is in every plane of equiponderation.—4. The moments of equal ponderants are to one another as their distances from the centre of the scale.—5, 6. The moments of unequal ponderants have their proportion to one another compounded of the proportions of their weights and distances from the centre of the scale.—7. If two ponderants have their weights and distances from the centre of the scale in reciprocal proportion, they are equally poised; and contrarily.—8. If the parts of any ponderant press the beams of the scale every where equally, all the parts cut off, reckoned from the centre of the scale, will have their moments in the same proportion with that of the parts of a triangle cut off from the vertex by strait lines parallel to the base.—9. The diameter of equiponderation of figures, which are deficient according to commensurable proportions of their altitudes and bases, divides the axis, so that the part taken next the vertex is to the other part of the complete figure to the deficient figure.—10. The diameter of equiponderation of the complement of the half of any of the said deficient figures, divides that line which is drawn through the vertex parallel to the base, so that the part next the vertex is to the other part as the complete figure to the complement.—11. The centre of equiponderation of the half of any of the deficient figures in the first row of the table of art. 3, chap. XVII, may be found out by the numbers of the second row.—12. The centre of equiponderation of the half of any of the figures of the second row of the same table, may be found out by the numbers of the fourth row.—13. The centre of equiponderation of the half of any of the figures in the same table being known, the centre of the excess of the same figure above a triangle of the same altitude and base is also known.—14. The centre of equiponderation of a solid sector is in the axis so divided, that the part next the vertex be to the whole axis, wanting half the axis of the portion of the sphere, as 3 to 4. DEFINITIONS. [Sidenote: Definitions.] I. A _scale_ is a strait line, whose middle point is immovable, all the rest of its points being at liberty; and that part of the scale, which reaches from the centre to either of the weights, is called the _beam_. II. _Equiponderation_ is when the endeavour of one body, which presses one of the beams, resists the endeavour of another body pressing the other beam, so that neither of them is moved; and the bodies, when neither of them is moved, are said to be _equally poised_.