Theorem 3
Table of Contents
Theorem 3, Proposition 3
Let the motion take place along the line ab, starting from rest at a, and in this line choose any point c.
Let ac represent the time required for the body to fall through the space ac.
Let ac also represent the velocity [impetus seu momentum] at c acquired by a fall through the distance ac.
In the line ab select any other point b.
The problem now is to determine the velocity at b acquired by a body in falling through the distance ab and to express this in terms of the velocity at c, the measure of which is the length ac.
Take as a mean proportional between ac and ab. We shall prove that the speed at b is to that at c as the length as is to the length ac.
Draw the horizontal line cd, having twice the length of ac, and be, having twice the length of ba.
It follows, from the preceding theorems, that a body falling through the distance ac and turned so as to move along the horizontal cd with a uniform speed equal to that acquired on reaching c Fig 110 will traverse the distance cd in the same interval of time as that required to fall with accelerated motion from a to c.
Likewise, be will be traversed in the same time as ba. But the time of descent through ab is as; hence the horizontal distance be is also traversed in the time as.
Take a point l such that the time as is to the time ac as be is to bl; since the motion along be is uniform, the distance bl, if traversed with the speed [momentum celeritatis] acquired at b, will occupy the time ac; but in this same time-interval, ac, the distance cd is traversed with the speed acquired in c.
Two speeds are to each other as the distances traversed in equal intervals of time. Hence the speed at c is to the speed at b as cd is to bl.
But since dc is to be as their halves, namely, as ca is to ba, and since be is to bl as ba is to sa; it follows that dc is to bl as ca is to sa.
In other words, the speed at c is to that at b as ca is to sa, that is, as the time of fall through ab.
The method of measuring the speed of a body along the direction of its fall is thus clear; the speed is assumed to increase directly as the time.
We deal with the motion compounded of a uniform horizontal one and one accelerated vertically downwards – the path of a projectile, namely, a parabola – it is necessary that we define some common standard by which we may estimate the velocity, or momentum [velocitatem, impetum seu momentum] of both motions;
Since from the innumerable uniform velocities one only, and that not selected at random, is to be compounded with a velocity acquired by naturally accelerated motion, I can think of no simpler way of selecting and measuring this than to assume another of the same kind. *
Draw the vertical line ac to meet the horizontal line bc. Ac is the height and bc the amplitude of the semi-parabola ab, which is the resultant of the two motions, one that of a body falling [283] from rest at a, through the distance ac, with naturally accelerated motion, the other a uniform motion along the horizontal ad.
The speed acquired at c by a fall through the distance ac is determined by the height ac; for the speed of a body falling from the same elevation is always one and the same; but along the horizontal one may give a body an infinite number of uniform speeds.
However, in order that I may select one out of this multitude and separate it from the rest in a perfectly definite manner, I will extend the height ca upwards to e just as far as is necessary and will call this distance ae the “sublimity. "
Imagine a body to fall from rest at e; it is clear that we may make its terminal speed at a the same as that with which the same body travels along the horizontal line ad; this speed will be such that, in the time of descent along ea, it will describe a horizontal distance twice the length of ea. This preliminary remark seems necessary.
Fig 111
The reader is reminded that above I have called the horizontal line cb the “amplitude” of the semi-parabola ab; the axis ac of this parabola, I have called its “altitude”; but the line ea the fall along which determines the horizontal speed I have called the “sublimity. " These matters having been explained, I proceed with the demonstration. (261)
Galileo agrees with the views of Plato on the origin of the various uniform speeds with which the heavenly bodies revolve.
The latter chanced upon the idea that a body could not pass from rest to any given speed and maintain it uniformly except by passing through all the degrees of speed intermediate between the given speed and rest.
Plato thought that God, after having created the heavenly bodies, assigned them the proper and uniform speeds with which they were forever to revolve; and that He made them start from rest and move over definite distances under a natural and rectilinear acceleration such as governs the motion of terrestrial bodies.
He added that once these bodies had gained their proper and permanent speed, their rectilinear motion was converted into a circular one, the only motion capable of maintaining uniformity, a motion in which the body revolves without either receding from or approaching its desired goal. This conception is truly worthy of Plato.
It is to be all the more highly prized since its underlying principles remained hidden until discovered by our Author who removed from them the mask and poetical dress and set forth the idea in correct historical perspective.
In view of the fact that astronomical science furnishes us such complete information concerning the size of the planetary orbits, the distances of these bodies from their centers of revolution, and their velocities, I cannot help thinking that our Author (to whom this idea of Plato was not unknown) had some curiosity to discover whether or not a definite “sublimity” might be assigned to each planet, such that, if it were to start from rest at this particular height and to fall with naturally accelerated motion along a straight line, and were later to change the speed thus acquired into uniform motion, the size of its orbit and its period of revolution would be those actually observed.
PROBLEM 1, PROPOSITION 4
To determine the momentum of a projectile at each particular point in its given parabolic path.
Let bec be the semi-parabola whose amplitude is cd and whose height is db, which latter extended upwards cuts the tangent of the parabola ca in a.
Through the vertex draw the horizontal line bi parallel to cd.
If the amplitude cd is equal to the entire height da, then bi will be equal to ba and also to bd; and if we take ab as the measure of the time required for fall through the distance ab and also of the momentum acquired at b in consequence of its fall from rest at a, then if we turn into a horizontal direction the momentum acquired by fall through ab [impetum ab] the space traversed in the same interval of time will be represented by dc which is twice bi.
But a body which falls from rest at b along the line bd will during the same time-interval fall through the height of the parabola [285] bd.
Hence a body falling from rest at a, turned into a horizontal direction with the speed ab will traverse a space equal to dc.
If one superposes upon this motion a fall along bd, traversing the height bd while the parabola bc is described, then the momentum of the body at the terminal point c is the resultant of a uniform horizontal momentum, whose value is represented by ab, and of another momentum acquired by fall from b to the terminal point d or c; these two momenta are equal.
If, therefore, we take ab to be the measure of one of these momenta, say, the uniform horizontal one, then bi, which is equal to bd, will represent the momentum acquired at d or c; and ia will represent the resultant of these two momenta, that is, the total momentum with which the projectile, travelling along the parabola, strikes at c.
With this in mind let us take any point on the parabola, say e, and determine the momentum with which the projectile passes that point.
Draw the horizontal ef and take bg a mean proportional between bd and bf. Now since ab, or bd, is assumed to be the measure of the time and of the momentum [momentum velocitatis] acquired by falling from rest at b through the distance bd, it follows that bg will measure the time and also the momentum [impetus] acquired at f by fall from b.
If therefore we lay off bo, equal to bg, the diagonal line joining a and o will represent the momentum at the point e; because the length ab has been assumed to represent the momentum at b which, after diversion into a horizontal direction, remains constant; and because bo measures the momentum at f or e, acquired by fall, from rest at b, through the height bf. But the square of ao equals the sum of the squares of ab and bo. Hence the theorem sought.
Fig 112
The manner in which you compound these different momenta to obtain their resultant strikes me as so novel that my mind is left in no small confusion.
I do not refer to the composition of two uniform motions, even when unequal, and when one takes place along a horizontal, the other along a vertical direction; because in this case I am thoroughly convinced that the resultant is a motion whose square is equal to the sum of the squares of the two components. The confusion arises when one undertakes to compound a uniform horizontal motion with a vertical one which is naturally accelerated. I trust, therefore, we may pursue this discussion more at length. [286]
Even in the case of the 2 uniform motions, one horizontal, the other perpendicular, I wish to understand better the manner in which you obtain the resultant from the components.
