[The following paper (also available in PDF) was reproduced
with the permission of the *Journal
of Royal Astronomical Society of Canada.* (J. Roy. Astron.
Soc. Can., Vol 83, No. 3, 1989:207-218). It
was written north of
the
70th parallel at the western entrance to the
Northwest Passage during the summer of 1988.

OF THE LAWS OF PLANETARY MOTION

Kepler's Third Law of planetary motion:T^{ 2}= R^{ 3}(T= period in years,R= mean distance in astronomical units) may be extended to include the inverse of the mean speedV(in units of the inverse of the Earth's mean orbital speed) such that:_{i}

R = V_{i }^{2}andT^{2}= R^{3}= V_{i }^{6}^{ }The first relationship - found in Galileo's last major work, theDialogues Concerning Two New Sciences(1638), - may also be restated and expanded to include relative speedV(in units of Earth's mean orbital speed_{r}k) and absolute speedV, thus:_{a}= kV_{r}

T = V_{i}^{ }^{3}

V_{i}= T/R

V_{r}= R/T

V_{a}= kR/T

V_{r}= kR^{ -}^{1/2}

V_{r}= kT^{-}^{1/3}

V_{a}= kR^{ -}^{1/2}

V_{a}= kT^{-}^{1/3}

This paper explains the context of Galileo's velocity expansions of the laws of planetary motion and applies these relationships to the parameters of the Solar System. A related "percussive origins" theory of planetary formation is also discussed.

The semi-parabola proves to be applicable to planetary motion as Galileo claimed, while the integral velocity variants of the laws of planetary motion and the implications of Galileo's application lead in turn to an examination of Galileo's "percussive origins" theory of planetary formation.

* 2. The
Parabola*.
The parabolas used by Galileo initially describe the paths followed by
projectiles in terrestrial applications. In this context Galileo elects
to standardize his procedures on the grounds that an infinite number of
uniform horizontal velocities may be compounded with the " naturally
accelerated"
velocity of a falling body. Accordingly, Galileo combines accelerated
velocity
on the vertical axis with a specific uniform velocity on the horizontal
axis to create a semi-parabola with the vertex at the
origin
and a distance of "four" units between the vertex and the directrix.
The
semi-parabola apparently has a second function, however, for following
its construction Galileo states in the dialogue

" ...the beautiful agreement between this thought of the Author (Galileo) and the views of Plato concerning the origin of the various uniform speeds with which the heavenly bodies revolve" (italics supplied).The relationship between the parabola, the "views of Plato," and planetary velocity is described in detail in the ensuing dialogue. At the conclusion Galileo states that he has applied the parabola to planetary motion and that:

From the last part of this passage it thus appears that Galileo successfully tested the new application on the parameters of the Solar System. Moreover, Galileo also asserts here that he as provided sufficient information for the reader to verify his results.he once made the computation and found a satisfactory correspondence with the observation. But he did not wish to speak of it, lest in view of the odium which his many new discoveries had already brought upon him, this might be adding fuel to the fire.But if anyone desires such information he can obtain it for himself from the theory set forth in the present treatment.(italics supplied)^{4}

Once alerted to Galileo's intention, it becomes clear that the inclusion of the " views of Plato " on the subject is a relevant and necessary device permitting the use of parameters and concepts common to both applications.

(1)R =V_{i}^{ 2}

while the further relationships:

(2)V_{r}= R/T

(3)V_{i }= T/R

(4)T = V_{i}^{ }^{3}

(5)V_{i}^{ 6}= R^{3}= T^{2}

where *T*
is the sidereal period in
years,
follow from Kepler's Third Law of planetary motion. Relation (**5**)
may also be expressed in exponents (i.e., [* V*_{i}^{
0}*,V*_{i}^{ 1}*,V*_{i}^{
2}*,V*_{i}^{ 3}*
*] )
and applied to the parabola in terms of the first three integer sets
which
illustrate the Third Law:^{7}

The parabola in the
standard context is
illustrated
in figure 1a, and in the astronomical context with (** V_{i},
R**)
as the subset of [

(2a)V_{a}= kR/T

where *k* is
the mean velocity of Earth.
With
*k* = 29.79 kilometers per second for this constant we obtain
the comparison with modern estimates (**Table 1**). The superior
planets
are shown on the inverted parabola in figure 2.

**
3. The
Parabola and Planetary
Origins**.
In spite of the limited treatment of the parabola in its astronomical
context,
it remains possible to hypothesize from material provided in

"among the decrees of the divine Architect was the thought of creating in the universe those globes which we behold continually revolving, and establishingwhile later, ina centre of their rotations in which the sun was located immovably. Next, suppose all the said globes to have beencreated in the same place, and their assigned tendencies of motion,descending towards the centreuntil they had acquired those degrees of velocity which originally seemed good to the Divine mind.These velocities being acquired, we lastly suppose that the globes were set in rotation, each retaining in its orbit its predetermined velocity. Now,at what altitude and distance from the sunwould have been the place where the said globes were first created, andcould they have been created in the same place?" (italics supplied)

"God, after having created the heavenly bodies, assigned them the proper and uniform speeds with which they were forever to revolve... (and)and finally asked with respect to the parabola:made them start from rest and move over definite distances under a natural and rectilinear acceleration such as governs the motion of terrestrial bodies...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......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 revolveswithout either receding from or approaching its desired goal." (italics supplied)

"whether or not aIn the initial passage Galileo poses two questions: firstly whether the planets originated in one place, and secondly, whether the place in question can be identified. From a heliocentric viewpoint, because relative velocity decreases with distance from the Sun, one can understand how Galileo may have come to conceive that the planets originated with "zero" velocity beyond the region of Saturn (the outer limit of the Solar System in Galileo's era), but this does not address the question of origins itself. In the second passage indefinite'sublimity' might be assigned to each planet, such that, if it were tostart 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 the orbit and its period of revolution would be those actually observed." (italics supplied)

Could
Galileo have extended his
treatment
of terrestrial projectile paths to embrace satellite orbits and also
have
expanded the idea one step further to include the planets as satellites
of the Sun? While acknowledging that there are dangers in attributing
to
Galileo modern or Newtonian concepts, it is necessary to recall that
the
initial discussion of the parabola concerned the path traced by a
projectile
with uniform horizontal velocity applied down the horizontal axis, and
"naturally accelerated" velocity applied down the vertical axis.
Visually,
a projectile launched almost horizontally will obviously gain very
little
height before falling back to ground when the initial velocity is
relatively
low.^{11} As the initial velocity
increases, however, some height
will be gained because of the curvature of the Earth, and although the
projectile may still fall to ground, with sufficient velocity, a
projectile
will finally "fall" into orbit around Earth itself.^{12}
Thus in
general, *by reversing matters, all objects in specific orbits may
be
treated in terms of a "percussive origins theory" with the parent body
the initial source. *The hypothesis may therefore be applied to the
planets and the Solar System with the *Sun* as the single
percussive
point of origin.^{13}

Could Galileo have taken this final step?
If he did, then undoubtedly criteria provided by Galileo in his
historical
aside becomes more significant than ever, i.e., if planetary origins
are
considered in terms of projectiles originating from the Sun, the
planets
would indeed "start with zero velocity" and "move through successive
speeds"
until their initial "rectilinear motion" changed into "circular motion"
(or orbital motion) as they "fell" into their respective orbitals
positions.
And once established, the planets would then "revolve without either
receding
from or approaching" their common point of origin, or deviating from
their
"final" positions in the Solar System.

Although no causal mechanism is associated
with this "percussive origins" (or "Small Bang") theory, the hypothesis
might possibly assume that the Sun was essentially formed at this
stage,
and for whatever reason, the planetary material was ejected from the
Sun
in one enormous explosion.^{14}

In this sense the hypothesis is a
variation
of catastrophe theory, with the exception that the source of the
catastrophe
is internal rather than external. The latter, involving collisions or
near
misses with double or triple stars, etc., are not generally well
supported
today, but the percussive elements of the basic hypothesis may perhaps
have some affinity with the massive explosion of the solar core (i.e.,
the "T Tauri winds") thought by some accretion theorists to be a
possible
explanation for the expulsion of unaccreted dust and gas from the Solar
System.

The relative paucity of direct information notwithstanding, it has proved feasible to apply Galileo's semi-parabolas in the given astronomical context and understand the application in terms of the relationship between mean inverse speed V

For a given mean distance R or a given sidereal period T the absolute mean speed V

## (

6)V_{r}= R^{-}^{1/2}## (

7)V_{r}= T^{-}^{1/3}## (

8)V_{a}= kR^{-}^{1/2}## (

9)V_{r }= kT^{-}^{1/3}

The
fundamental understanding and
application
of the semi-parabola in the astronomical context depends on the
heliocentric
concept, Kepler's Third Law for the mean distances, and relation (1).
Although
absolute confirmation may be lacking, it seems likely that Galileo -
the
originator of the material in its dual contexts - would have known, or
would have been able to derive (in one form or another) all the
velocity
expansions of the laws of planetary motion given here.

The "percussive origins" theory of planetary motion also credited in this work to Galileo may perhaps be open to alternative interpretations, but the transition from projectile trajectories to satellite orbits is nevertheless a logical one. In view of his pioneering researches in the former area, and his discoveries in the other (the four Jovian moons which bear his name) such a progression would seem in keeping with both his heliocentric orientation and the general directions of his research.

Finally, three and a half centuries have passed since Galileo published the*Dialogues Concerning The New
Science*.
Apart from Mersenne's negative assessment of related concepts,^{17}
scant
attention seems to have been paid to Galileo's oblique treatment of
planetary
velocities and planetary origins. Although his research may have been
overshadowed
by the works of Kepler and Newton, it seems that the obscure
methodology
forced on Galileo was if anything, only too successful. One cannot but
help admiring Galileo's tenacity, however, for *The New Sciences*
was written when he was in his seventies, with failing eyesight, and
under
the threat of most dire consequences should he ever attempt to discuss
the heliocentric hypothesis again. Galileo may or may not have claimed
at the conclusions of his trial that the Earth still moved, but it
appears
from the material in the *Dialogues Concerning The New Sciences*
that
he had the last word on the matter after all.

**APPENDIX**

The following dialogue between Salviati
and
Sagredo occurs in association with Galileo's "standard" parabola and an
unexpected expansion that includes Plato, planetary motion, and the
Solar
System. (Fourth Day, [282-283] pp.259-260).
*The Dialogues Concerning Two New
Sciences*,
Fourth Day, [282-283], translated by Henry Crew and Alphonso deSalvio,
1914, pp.259-260. The "sublimity" may be understood to correspond to
the
distance between the vertex and the directrix for the parabola in both
terrestrial and astronomical contexts.

**NOTES &
REFERENCES**

**1**. Galileo's obscure and
limited
treatment
of this subject may be explained by the fact that the
*Dialogues Concerning
Two New Sciences* was written after his trial for heresy for
supporting
the heliocentric concept in two previous treatises. Following his
conviction
by the Inquisition in 1633 he was forced to recant and expressly
forbidden
to discuss the heliocentric hypothesis again, or suffer the penalties
of
relapse.

The "percussive origins" theory of planetary motion also credited in this work to Galileo may perhaps be open to alternative interpretations, but the transition from projectile trajectories to satellite orbits is nevertheless a logical one. In view of his pioneering researches in the former area, and his discoveries in the other (the four Jovian moons which bear his name) such a progression would seem in keeping with both his heliocentric orientation and the general directions of his research.

Finally, three and a half centuries have passed since Galileo published the

SALVIATI. But before we proceed further, since this discussion is to 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 of both motions; and since from the innumerable uniform velocities one only, and 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. For the sake of clearness, draw the vertical lineto meet the horizontal lineac.bcis the height andAcis the amplitude of the semi-parabolabc, which is the resultant of the two motions, one that of a body falling from rest ataba,through the distance, with naturally accelerated motion, the other a uniform motion along the horizontalacThe speed acquired atad.cby a fall through the distanceis determined by the heightac; 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 heightacupwards tocajust as far as is necessary and will call this distanceethe "sublimity." Imagine a body to fall from rest ataeit is clear that we may make its terminal speed ate;athe same as that with which the same body travels along the horizontal line; this speed will be such that, in the time of descent alongad, it will describe a horizontal distance twice the length ofea.ea

This preliminary remark seems necessary. The reader is reminded that above I have called the horizontal linethe " amplitude " of the semi-parabolacb; the axisabof this parabola, I have called its " altitude "; but the lineacthe fall along which determines the horizontal speed I have called the " sublimity. " These matters having been explained, I proceed with the demonstration.ea

SAGREDO.. 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 its desired goal. ···Allow me, please, to interrupt in order that I may point outthe beautiful agreement between this thought of the Author and the views of Plato concerning the origin of the various uniform speeds with which the heavenly bodies revolve. In view of the fact that·This conception is truly worthy of Plato; and it is all the more highly prized since its undying 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, 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 the orbit and its period of revolution would be those actually observed.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

SALVIATI.(emphases supplied)I think I remember his having told me thathe once made the computation and found a satisfactory correspondence with the observation.But he did not wish to speak of it, lest in view of the odium which his many new discoveries had already brought upon him, this might be adding fuel to the fire.But if anyone desires such information he can obtain it for himself from the theory set forth in the present treatment.

**2**
Thus the ratio of the vertical
to the horizontal axis at the point of calibration is 2:1. For this
parabola
a uniform velocity of TWO on the horizontal axis corresponds to
distance
of FOUR on the vertical axis. The same parabola also appears to have
been
used by Galileo to illustrate his observation that "... a moving body
starting
from rest and acquiring velocity at a rate proportional to time, will
during
equal intervals of time, traverse distances which are related to each
other
as the odd numbers beginning with unity, 1, 3, 5; or considering the
total
space traversed, that covered in double time will be quadruple that
covered
during unit time in triple time, the space is nine times as great as in
unit time. And in general the spaces traversed are in the duplicate
ratio
of the times, i.e., in the ratio of the squares of the times." *The
Two
New Sciences*, Third Day [211-212]; also discussed in the Fourth Day
[272-273].

**3**
Galileo follows Plato in his
effective
use of the Dialectic Method. The passages dealing with the parabola and
the historical aside which follows are given here in the Appendix.

**4**
*The
Dialogues Concerning
Two
New Sciences*, Fourth Day [283-284].

**5**
For example, Galileo discusses
the sets: [1,2,4,8] and [1,3,9,27] with respect to squaring and cubing
in the *Two Dialogues Concerning Two New Sciences* (First Day
[83]).
The same sets are also mentioned by Plato in the__ __*Timaeus*
(35b and 43d) and the first set [1,2,4,8] is discussed again in the *Epinomis*
(991a-992a).

**6**
Matters are greatly simplified
if mean circular motion is assumed, i.e.,if the velocity of Earth is
expressed
in terms of the distance moved around the circumference divided by the
mean period of revolution: 2Pi1/1 = 2Pi then the ratios of the mean
velocities
of the planets with respect to that of Earth will also reduce to ratios
of mean distances divided by mean periods of revolution, i.e., 2PiR/T
divided
by 2Pi = R/T, and V_{a} = KR/T etc.

**7**
Also the ancient relationship
between a point, a line, an area, and a volume. See Galileo's
discussion
of the latter pair and the "sesquialteral ratio" between them in the *Two
New Sciences*, First Day, (134-135).

**8**.
The *Dialogue Concerning
the
Two Chief World Systems*, translated by Stillman Drake, 1967, p.29.

**9**.
The *Dialogues Concerning
The
New Sciences*, Fourth Day, (282-283), translated by Henry Crew and
Alphonso
de Salvio, 1914, pp. 259-260. The "sublimity" may be understood to
correspond
to the distance between the vertex and the directrix for the parabola
in
both terrestrial and astronomical contexts.

**10**.
I perhaps place too much
significance
on this point, but it does seem, in the last reference at least, that
Galileo
requires *a common, yet specific point of origin* with respect to
each of the planets *and* the parabola. It is relevant to note
here
that the rotation of Earth is not directly involved in this
application,
although Galileo's views on this subject are interest; for details see
Stillman Drake's "Galileo and the Projection Argument,"
*Annals of Science*,
**43**,
(1986), pp. 77-79.

**11**.
Galileo discusses
horizontal,
near-horizontal projectile trajectories, and the parabola near the end
of the *Dialogues Concerning The New Sciences in the Fourth Day*,
(309-321).

**12**.
But even though Galileo
*could*
have extended his work to this final conclusion, it should nevertheless
still be acknowledged that it is at odds with what is generally known
concerning
these aspects of Galileo's physics.

**13**.
At least from a theoretical
point of view or simple exercise, capture and accretion theories not
excluded;
to generalize, one might also include origins in other known satellite
systems, even perhaps those of Jupiter and Saturn.

**14**.
Or more than one single
explosion.

**15**.
Galileo seems to have
supplied
at least three alternative paths to reach this goal; once attained the
rest follows almost as a matter of course.

**16**.
The First and Second Laws of
planetary motion are found in Johannes Kepler's *New Astronomy*
published
in 1609; the materia containing the Third (or the Harmonic) Law occurs
in his Harmony of the Worlds published in 1618, some twenty years prior
to the publication of Galileo's *Dialogues Concerning The New
Sciences*.
Applying Kepler's Third law in Galileo's astronomical application of
the
semi-parabola therefore causes no difficulties historically. Galileo's
adherence to mean circular motion is also relevant in this same context.

**17**.
*Harmonie Universalle*,
second livre des mouvements, prop. 6. p.103, Paris, 1936. The
uncritical
acceptance of Mersenne's analysis appears to have unduly influenced
subsequent
commentators.

Galilei, G. *Dialogues
Concerning
the
Two Chief World Systems*, translated by Stillman Drake. 2nd Revised
Edition, Berkeley and Los Angeles, 1967.

Galilei, G. *Dialogues
Concerning
The
New Sciences*, translated by Henry Crew and Antonio de Salvio,
Dover,
New York, 1954.

Mersenne, M. *Harmonie
Universalle*,
Paris, 1636.

CITATION

Harris,
John N.
"PROJECTILES, PARABOLAS, AND VELOCITY
EXPANSIONS OF THE LAWS OF PLANETARY MOTION," Journal
of the Royal Astronomical Society
of Canada, Vol 83, No. 3,
1989:207-218.