Prop. III. The Velocity which a falling Body acquires in any Space of time, is double to that, wherewith it would have moved the Space, descended by an equable Motion, in the same time.
Demonstration. Draw the Line EC parallel to AB, and AE parallel to BC in the same Fig. 9. and compleat the Parallelogram ABCE, it is evident that the Area thereof may represent the Space, a Body moved equably with the Velocity BC would describe in the Time AB, and the Triangle ABC represents the Space describ'd by the Fall of a Body, in the same Time AB, by the second Proposition. Now the Triangle ABC is half of the Parallelogram ABCE, and consequently the Space described by the Fall, is half what would have been described by an equable Motion with the Velocity BC, in the same Time; wherefore the Velocity BC at the end of the Fall, is double to that Velocity, which in the Time AB, would have described the Space fallen, represented by the Triangle ABC with an equable Motion, Q. E. D.
Prop. IV. All Bodies on or near the Surface of the Earth, in their Fall, descend so, as at the end of the first Second of Time, they have described 16 Feet, 1 Inch, London Measure, and acquired the Velocity of 32 Feet, 2 Inches, in a Second.
This is made out from the 25th Proposition of the second Part of that excellent Treatise of Mr. Hugenius de Horologio Oscillatorio; wherein he demonstrates the time of the least Vibrations of a Pendulum, to be to the Time of the Fall of a Body, from the heighth of half the length of the Pendulum, as the Circumference of a Circle to its Diameter; whence, as a Corollary, it follows, That as the Square of the Diameter to the Square of the Circumference, so half the length of the Pendulum vibrating Seconds, to the Space described by the Fall of a Body in a Second of Time: And the Length of the Pendulum vibrating Seconds, being found 39, 125, or ⅛ Inches, the Descent in a Second will be found by the aforesaid Analogy 16 Foot and 1 Inch; and, by the third Proposition, the Velocity will be double thereto; and near to this it hath been found by several Experiments, which by reason of the swiftness of the Fall, cannot so exactly determine its Quantity. The Demonstration of Hugenius being the Conclusion of a long Train of Consequences, I shall for brevity sake omit; and refer you to his Book, where these things are more amply treated of.
From these Four Propositions, all Questions concerning the Perpendicular Fall of Bodies, are easily solved, and either Time, Height, or Velocity being assign'd, one may readily find the other two. From them likewise is the Doctrine of Projects deducible, assuming the two following Axioms; viz. That a Body set a moving, will move on continually in a right Line with an equable Motion, unless some other Force or Impediment intervene, whereby it is accelerated, or retarded, or deflected.
Secondly, That a Body being agitated by two Motions at a time, does by their compounded Forces pass through the same Points, as it would do, were the two Motions divided and acted successively. As for Instance, Suppose a Body moved in the Line GF, (Fig. 1. Tab. 5.) from G to R, and there stopping, by another Impulse, suppose it moved in a Space of Time equal to the former, from R towards K, to V. I say, the Body shall pass through the Point to V, though these two several Forces acted both in the same time.
Prop. V. The Motion of all Projects is in the Curve of a Parabola: Let the Line GRF (in Fig. 1.) be the Line in which the Project is directed, and in which by the first Axiom it would move equal Spaces in equal Times, were it not deflected downwards by the Force of Gravity. Let GB be the Horizontal Line, and GC a Perpendicular thereto. Then the Line GRF being divided into equal Parts, answering to equal Spaces of Time, let the Descents of the Project be laid down in Lines parallel to GC, proportioned as the Squares of the Lines GS, GR, GL, GF, or as the Squares of the Times, from S to T, from R to V, from L to X, and from F to B, and draw the Lines TH, VD, XY, BC parallel to GF; I say, the Points T, V, X, B, are Points in the Curve described by the Project, and that that Curve is a Parabola. By the second Axiom, they are Points in the Curve; and the Parts of the Descent GH, GD, GY, GC, = to ST, RV, LX, FB, being as the Squares of the Times (by the Second Proposition) that is, as the Squares of the Ordinates, HT, DU, YX, BC, equal to GS, GR, GL, GF, the Spaces measured in those Times; and there being no other Curve but the Parabola, whose Parts of the Diameter are as the Squares of the Ordinates, it follows that the Curve describ'd by a Project, can be no other than a Parabola: And saying, as RU the Descent in any time, to GR or UD the direct Motion in the same time, so is UD to a third proportional; that third will be the Line call'd by all Writers of Conicks, the Parameter of the Parabola to the Diameter GC, which is always the same in Projects cast with the same Velocity: And the Velocity being defined by the Number of Feet moved in a Second of Time, the Parameter will be found by dividing the Square of the Velocity, by 16 Feet, 1 Inch, the Fall of a Body in the same Time.
Lemma.
The Sine of the double of any Arch, is equal to twice the Sine of that Arch into its Co-sine, divided by Radius; and the versed Sine of the double of any Arch is equal to twice the Square of the Sine thereof divided by Radius.
Let the Arch BC (in Fig. 2. Tab. 5.) be double the Arch BF, and A the Center; draw the Radii AB, AF, AC, and the Chord BDC, and let fall BE perpendicular to AC, and the Angle EBC, will be equal to the Angle ABD, and the Triangle BCE, will be like to the Triangle BDA; wherefore it will be as AB to AD, so BC or twice BD, to BE; that is, as Radius to Co-sine, so twice Sine to Sine of the double Arch. And as AB to BD, so twice BD or BC to EC, that is, as Radius to Sine, so twice that Sine, to the Versed Sine of the double Arch; which two Analogies resolved into Equations, are the Propositions contained in the Lemma to be proved.