(1-_a_)/_m_ + (1-_a_)/_mm_ + (1-_a_)/_m_³ &c. = (1-_a_)/_n_.
And thus as to the Line of Projection, in which (secluding the
Resistance) the Motion is reputed uniform; dispatching equal Lengths in
equal Times. Consider we next the Line of Descent.
23. In the Descent of Heavy Bodies, it is supposed that to each Moment
of Time, there is superadded a new Impulse of Gravity to what was
before: And each of these, secluding the Consideration of the Air's
Resistance, to proceed equally (from their several beginnings) through
the succeeding Moments. As (in the erect Lines) 1 1 1 1, _&c._ 1 1 1,
_&c._ 1 1, _&c._ 1, _&c._ and so continually, as in the Line of
Projection.[16]
24. Hence ariseth (in the transverse Lines) for the first Moment 1, for
the second 1 + 1, for the third 1 + 1 + 1, and so forth, in Arithmetical
Progression: As are the Ordinates in a Triangle, at equal distance.
25. And such are the continual Increments of the Diameter, or of the
Ordinates in the exterior Parabola, answering to the interior Ordinates,
or Segments of the Tangent, equally increasing; as is known, and
commonly admitted.
26. If we take in the Consideration of the Air's Resistance; we are
then, for each of these equal Progressions, to substitute a decreasing
Progression Geometrical; in like manner (and for the same Reasons) as in
the Line of Projection.
27. Hence ariseth, for the first Moment 1/_m_; for the second 1/_m_ +
1/_m_²; for the third 1/_m_ + 1/_m_² + 1/_m_³, _&c._[17] And such is
therefore the Descent of a heavy Body falling by its own weight. The
several Impulses of Gravity being supposed equal.
28. That is (in the Figure of ¶ 12) as _FL_, _FM_, _FN_, &c. in the Line
of Descent, answering to _FL_, _LM_, _MN_, &c. in the Line of Projection.
29. But though the Progressions for the Line of Projection, are like to
each of those many in the Line of Descent; it is not to be thence
inferred, that therefore 1/_m_ in the one, is equal to 1/_m_ in the
other: But in the Line of Projection (suppose) (1/_m_)_f_ (such a Part
of the Force impressed, and a Celerity answerable:) in the Line of
Descent, (1/_m_)_g_ such a Part of the Impulse of Gravity.
30. Those for the Line of Descent (of the some Body) are all equal, each
to other: Because _g_ (the new Impulse of Gravity) in each Moment is
supposed to be the same.
31. But what is the Proportion of _f_ to _g_ (that of the Force
impressed, to the Impulse of Gravity in each Body) remains to be
inquired by Experiment.
32. This Proportion being found as to one known Force; the same is
thence known as to any other Force (whose Proportion to this is given)
in the same uniform _Medium_.
33. And this being known, as to one _Medium_; the same is thence known
as to any other _Medium_, the Proportion of whose Resistance to that of
this is known.
34. If a heavy Body be projected downward in a perpendicular Line; it
descends therefore at the Rate 1/_m_, 1/_mm_, 1/_m_³, _&c._ of _f_,
(the impressed Force) increased by 1/_m_, 1/_m_ + 1/_m_², 1/_m_ +
1/_m_² + 1/_m_³, _&c._ of _g_ the impulse of Gravity, (by ¶ 7, and ¶
27) Because both Forces are here united.
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.