35. If in a perpendicular Projection upwards; it ascends in the rate of the former, abated by that of the latter. Because here the impulse of Gravity is contrary to the Force impressed.
36. When therefore this latter (continually increasing) becomes equal to that former (continually decreasing) it then ceaseth to ascend; and doth thenceforth descend at the rate wherein the latter continually exceeds the former.
37. In an Horizontal, or Oblique Projection: If to a Tangent, whose Increments are as FL, LM, MN, &c. that is as 1⁄m f, &c. be fitted Ordinates (at a given Angle) whose Increments are as FL, FM, FN, &c. that is, as 1⁄m g, &c. The Curve answering to the Compound of these Motions, is that wherein the Project is to move.
38. This Curve (being hitherto without a Name) may be call'd Linea Projectorum; the Line of Projects, or things projected; which resembles a Parabola deform'd.
39. The Celerity and Tendency, as to each Point of this Line, is determined by a Tangent at that Point.
40. And that against which it makes the greatest Stroke or Percussion, is that which (at that Point) is at right Angles to that Tangent.
41. If the Projection (at ¶ 27) be not infinitely continued, but terminate (suppose) at N, so that the last Term in the first Column or Series erect be a; and consequently in the second, ma; in the third, mma, &c. (each Series having one Term fewer than that before it:) Then (for the same Reasons, as at ¶ 22) the Aggregates of the several Columns (or erect Series) will be 1-a / n, 1-ma / n, 1-mma / n, and so forth, till (the Multiple of a becoming = 1) the Progression expire.
42. Now all the Abatements here, a, ma, mma, &c. are the same with the Terms of the first Column taken backward. For a is the last, ma the next before it; and so of the rest.
43. And the Aggregate of all the Numerators is so many times 1, as is the Number of Terms (suppose t,) wanting the first Column; that is t - 1-a / n, or nt - 1 + a / n; and this again divided by the common Denominator n, becomes nt - 1 + a / nn. And therefore nt - 1 + a / nn g, is the Line of Descent by its own Gravity.
44. If therefore this be added to a projecting Force downward in a Perpendicular; or subducted from such projecting Force upward; that is, to or from 1-a / n f: The Descent in the first Case will be 1-a / n f + nt - 1 + a / nn g; and the Ascent in the other Case 1-a / n f - nt - 1 + a / nn g. And in this latter Case, when the ablative Part becomes equal to the positive Part, the Ascent is at the highest; and thenceforth (the ablative Part exceeding the positive) will descend.