## Galileo Galilei

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.

#### On the Motion of Projectiles1

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

Imagine a horizontal line or plane AB situated on high, upon which the moveable is carried from A to B in equable motion, but at B lacks support from the plane, whereupon there supervenes in the same moveable, from its own heaviness, a natural motion downward along the vertical BN.  Beyond the plane AB imagine the line BE, lying straight on, as if it were the flow or measure of time, on which there are noted any equal parts of time BC, CD, DE; and from points B, C, D, and E imagine lines drawn parallel to the vertical BN.  In the first of these, take some part CI; in the next, its quadruple DF; then its nonuple EH 5, and so on for the rest according to the rule of squares of CB, DB, and EB; or let us say, in the duplicate ratio of those lines. 6
If now to the moveable in equable movement beyond B toward C, we imagine to be added a motion of vertical descent according to the quantity CI, the moveable will be found after time BC to be situated at the point I.  Proceeding onwards, after time DB (that is, double BC), the distance of descent will be quadruple the first distance, CI; for it was demonstrated in the earlier treatise that the spaces run through by heavy things in naturally accelerated motion are in the squared ratio of the times. 7   And likewise the next space, EH, run through in time BE, will be as nine times [CI]; so that it manifestly appears that spaces EH, DF, and CI are to one another as the squares of lines EB, DB, and CB.  Now, from points I, F, and H, draw straight lines IO, FG, and HL parallel to EF; line by line, HL, FG, and IO will be equal to EB, DB, and CB respectively, and BO, BG, and BL will be equal to CI, DF, and EH.  And the square of HL will be to the square of FG as line LB is to BG, while the square of FG[will be] to the square of IO as GB is to BO; therefore points I, F, and H lie in one and the same parabolic line.
And it is similarly demonstrated, assuming any equal parts of time, of any size whatever, that the places of moveables carried in like compound motion will be found at those times in the same parabolic line.  Therefore the proposition is evident.

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   Therefore point F is on the parabola; and so on for all the others, etc.

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