The Essential Galileo (63 page)

For imagine it were possible that this line cuts the parabola above or that its prolongation cuts it below; then through any point
g
on this line draw the straight line
fge
. Since the square of
fe
is greater than the square of
ge
, the square of
fe
will bear a greater ratio to the square of
bc
than the square of
ge
to that of
bc;
and since, by the preceding proposition, the square of
fe
is to that of
bc
as the line
ea
is to
ca;
it follows that the line
ea
will bear to the line
ca
a greater ratio than the square of
ge
to that of
bc
, or,
than the square of
ed
to that of
cd
(the sides of the triangles
deg
and
dcb
being proportional). But the line
ea
is to
ca
, or
da
, in the same ratio as four times the rectangle
ea-ad
is to four times the square of
ad
, or, what is the same, to the square of
cd
(since this is four times the square of
ad
). Hence four times the rectangle
ea-ad
bears to the square of
cd
a greater ratio than the square of
ed
to the square of
cd
. But that would make four times the rectangle
ea-ad
greater than the square of
ed
. This is false, the fact being just the opposite, because the two portions
ea
and
ad
of the line
ed
are not equal. Therefore the line
db
touches the parabola without cutting it. QED.

S
IMP.
    Your demonstration proceeds too rapidly and, it seems to me, you keep on assuming that all [272] of Euclid's theorems are as familiar and available to me as his first axioms, which is far from true. For example, you just sprang upon us that four times the rectangle
ea-ad
is less than the square of
ed
because the two portions
ea
and
ad
of the line
ed
are not equal; and this brings me little composure of mind, but rather leaves me in suspense.

S
ALV.
    Indeed, all real mathematicians assume on the part of the reader perfect familiarity with at least the
Elements
of Euclid. Here it is necessary in your case only to recall the proposition of Book II
³⁰
in which he proves that when a line is cut at two points into equal and unequal parts respectively, the rectangle formed on the unequal parts is less than that formed on the equal (i.e., less than the square on half the line), by an amount that is the square of the segment between the two cut points; from this it is clear that the square of the whole line, which is equal to four times the square of the half, is greater than four times the rectangle of the unequal parts. In order to understand the following portions of this treatise it will be necessary to keep in mind the two elementary theorems from conic sections which we have just demonstrated; these two theorems are indeed the only ones which the Author uses. We can now resume the reading of the text and see how he demonstrates his first proposition, in which he shows that a projectile undergoing motion compounded of uniform horizontal motion and naturally accelerated fall describes a semiparabola.

Let us imagine an elevated horizontal line or plane
ab
along which a body moves with uniform speed from
a
to
b
. Suppose this plane to end abruptly at
b;
then at this point the body will, on account of its weight, acquire also a natural motion downwards along the perpendicular
bn
. Draw the line
be
along the plane
ab
to represent the flow, or measure, of time; divide this line into a number of segments,
bc, cd
,
de
, representing equal intervals of time; and from the points
c, d, e
, let fall lines that are parallel to the perpendicular
bn
. On the first of these lay off any distance
ci;
[273] on the second a distance four times as long,
df;
on the third, one nine times as long,
eh;
and so on, in proportion to the squares of
cb, db, eb
, or, we may say, in the squared ratio of these same lines. Accordingly we see that while the body moves from
b
to
c
with uniform speed, it also falls perpendicularly through the distance
ci
, and at the end of the time interval
bc
it finds itself at the point
i
. In like manner at the end of the time interval
bd
, which is the double of
bc
, the vertical fall will be four times the first distance
ci;
for it has been shown in a previous discussion that the distance traversed by a freely falling body varies as the square of the time. In like manner the space
eh
traversed during the time
be
will be nine times
ci
. Thus it is evident that the distances
eh, df, ci
will be to one another as the squares of the lines
be, bd, bc
. Now, from the points
i, f, h
draw the straight lines
io
,
fg, hl
parallel to
be;
these lines
hl, fg, io
are equal to
eb, db
, and
cb
, respectively; so also are the lines
bo, bg, bl
respectively equal to
ci, df
, and
eh;
furthermore, the square of
hl
is to that of
fg
as the line
lb
is to
gb
, and the square of
fg
is to that of
io
as
gb
is to
ob;
therefore the points
i, f, h
lie on one and the same parabola. In like manner it may be shown that if we take equal time intervals of any size whatever, and if we imagine the body to be carried by a similar compound motion, its positions at the end of these time intervals will lie on one and the same parabola. QED.

This conclusion follows from the converse of the first of the two propositions given above. For, having drawn a parabola through the points
b
and
h
, any other two points,
f
and
i
, not falling on the parabola must lie either within or without; consequently the line
fg
is either longer or shorter than the line that terminates on the parabola. Therefore the square of
hl
will not bear to the square of
fg
the same ratio as the line
lb
to
gb
, but a greater or smaller. The fact is, however, that the square of
hl
does bear this same ratio to the square of
fg
. Hence the point
f
does lie on the parabola, and so do all the others.

S
AGR.
    One cannot deny that the argument is new, subtle, and conclusive. It also rests upon various assumptions, namely, that the horizontal motion remains uniform, that the vertical motion continues to be accelerated downwards in proportion to the square of the time, and that such motions and velocities as these combine without altering, disturbing, or hindering each other, so that as the motion proceeds the path of the projectile does not change into a different curve. But this, in my opinion, [274] is impossible. For the axis of the parabola along which we suppose the natural motion of a falling body to take place stands perpendicular to a horizontal surface and ends at the center of the earth; and since the parabola deviates more and more from its axis, no projectile can ever reach the center of the earth or, if it does, as seems necessary, then the path of the projectile must transform itself into some other curve very different from the parabola.

S
IMP.
    To these difficulties, I may add others. One of these is that we suppose the horizontal plane, which slopes neither up nor down, to be represented by a straight line as if each point on this line were equally distant from the center. This is not the case, for as one starts from the middle of the line and goes toward either end, one departs farther and farther from the center of the earth and so is constantly going uphill; whence it follows that the motion cannot remain uniform through any distance whatever, but must continually diminish. Besides, I do not see how it is possible to avoid the resistance of the medium, which must destroy the uniformity of the horizontal motion and change the law of acceleration of falling bodies. These various difficulties render it highly improbable that a result derived from such unreliable assumptions should hold true in practical experience.

S
ALV.
    All these difficulties and objections which you urge are so well founded that it is impossible to remove them; and as for me, I am ready to admit them all, which indeed I think our Author would also do. I grant that these conclusions proved in the abstract will be different when applied in the concrete and will be false to this extent, that neither will the horizontal motion be uniform, nor will the natural acceleration be in the ratio assumed, nor will the path of the projectile be a parabola, etc. But, on the other hand, I ask you not to begrudge our Author that which other eminent men have assumed, even if not strictly true.

The authority of Archimedes alone will satisfy everybody. In his works on mechanics and on the quadrature of the parabola, he takes for granted that the beam of a balance or steelyard is a straight line, every point of which is equidistant from the common center of all heavy bodies, and that the strings by which heavy bodies are suspended are parallel to each other. Some consider this assumption permissible because, in practice, our instruments and the distances [275] involved are so small in comparison with the enormous distance from the center of the earth that we may consider a minute of arc on a great circle as a straight line, and may regard the perpendiculars let fall from its two extremities as parallel. For if in actual practice one had to consider such small quantities, it would be necessary first of all to criticize the architects who presume, by the use of a plumb line, to erect high towers with parallel sides. I may add that, in all their discussions, Archimedes and the others considered themselves as located at an infinite distance from the center of the earth, in which case their assumptions were not false, and therefore their conclusions were absolutely correct. When we wish to apply our proven conclusions to distances which, though finite, are very large, it is necessary for us to infer, on the basis of demonstrated truth, what correction is to be made for the fact that our distance from the center of the earth is not really infinite, but merely very great in comparison with the small dimensions of our apparatus. The largest of these will be the range of our projectiles—and here we need consider only the artillery— which, however great, will never exceed four of those miles of which as many thousand separate us from the center of the earth; and since these paths terminate upon the surface of the earth, only very slight changes can take place in their parabolic shape, which, it is conceded, would be greatly altered if they terminated at the center of the earth.

As to the perturbation arising from the resistance of the medium, this is more considerable and does not, on account of its manifold forms, submit to fixed laws and exact description. Thus, if we consider only the resistance which the air offers to the motions studied by us, we shall see that it disturbs them all and disturbs them in an infinite variety of ways corresponding to the infinite variety in the form, weight, and velocity of the projectiles. As to velocity, the greater this is, the greater will be the resistance offered by the air; also resistance will be greater as the moving bodies become less dense. Thus, although the falling body ought to be accelerated in accordance with the rule of distance being proportional to the square of the duration of its motion, yet no matter how heavy the body is, if it falls from a very considerable height, the resistance of the air will be such as to eventually prevent any increase in [276] speed and render the motion uniform; and in proportion as the moving body is less dense, this uniformity will be attained more quickly and from smaller heights. Even horizontal motion, which would be uniform and constant if no impediment were offered, is altered by the resistance of the air and finally ceases; and here again, the less dense the body, the quicker the process.

Of such effects of weight, velocity, and also shape, which are infinite in number, it is not possible to give any exact description. Hence, in order to handle this matter in a scientific way, it is necessary to cut loose from these difficulties, to discover and demonstrate the theorems in the case of no impediments, and to use them and apply them with such limitations as experience will teach. The advantage of this method will not be small, for the material and shape of the projectile may be chosen as dense and round as possible, so that it will encounter the least resistance in the medium; and the spaces and velocities will be small enough for the most part that we shall be easily able to correct them with precision. Indeed, in the case of those projectiles we use, thrown from a sling or crossbow, and made of dense material and round in shape or of lighter material and cylindrical in shape (such as arrows), the deviation from an exact parabolic path is quite imperceptible. Furthermore, if you will allow me a little greater liberty, I can show you, by two experiments, that the dimensions of our apparatus are so small that these external and incidental resistances, among which that of the medium is the most considerable, are scarcely observable.