Source: Two New Sciences, including centers of gravity and force of percussion (Discorsi e dimonstrazioni mathematiche, 1632), translated with introduction and notes by Stillman Drake, U of Wisconsin Press, 1974. Selections from the Fourth Day, pp. 217, 221-222.
We have considered properties existing in equable motion, and those in naturally accelerated motion over inclined planes of whatever slope. In the studies on which I now enter, I shall try to present certain leading essentials, and to establish them by firm demonstrations, bearing on a moveable when its motion is compounded from two movements; that is, when it is moved equably and is also naturally accelerated. Of this kind appear to be those which we speak of as projections, the origin of which I lay down as follows.
I mentally conceive of some moveable projected on a horizontal plane, all impediments being put aside. 2 Now it is evident from what has been said elsewhere at greater length that equable motion on this plane would be perpetual if the plane were of infinite extent 3 ; but if we assume it to be ended, and [situated] on high, the moveable (which I conceive of as being endowed with heaviness), driven to the end of this plane and going on further, adds on to its previous equable and indelible motion that downward tendency which it has from its own heaviness. Thus there emerges a certain motion, compounded from equable horizontal and from naturally accelerated downward [motion], which I call "projection." We shall demonstrate some of its properties [accidentia], of which the first is this:
PROPOSITION I. THEOREM I.
When a projectile is carried in motion compounded from equable horizontal and from naturally accelerated downward [motions], it describes a semiparabolic line in its movement. 4
Salv. This conclusion
is deduced from the converse of the first of the two lemmas given above.
For if the parabola is described through points B and H,
for example, and if either of the two [points], F or I, were
not in the parabolic line described, then it would lie either inside or
outside, and consequently line FG would be either less or greater
than that which would go to terminate in the parabolic line. Whence
the ratio that line LB has to BG, the square of HL
should have, not to the square of FG, but to [the square of] some
[line] greater or less [than FG]. But it [the square of HL]
does have [that ratio] to the square of FG. 8
F is on the parabola; and so on for all the others,
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