_State of the Science of Motion before Galileo._
IT is generally difficult to trace any branch of human knowledge up to
its origin, and more especially when, as in the case of mechanics, it is
very closely connected with the immediate wants of mankind. Little has
been told to us when we are informed that so soon as a man might wish to
remove a heavy stone, "he would be led, by natural instinct, to slide
under it the end of some long instrument, and that the same instinct
would teach him either to raise the further end, or to press it
downwards, so as to turn round upon some support placed as near to the
stone as possible."[115]
Montucla's history would have lost nothing in value, if, omitting "this
philosophical view of the birth of the art," he had contented himself
with his previous remark, that there can be little doubt that men were
familiar with the use of mechanical contrivances long before the idea
occurred of enumerating or describing them, or even of examining very
closely the nature and limits of the aid they are capable of affording.
The most careless observer indeed could scarcely overlook that the
weights heaved up with a lever, or rolled along a slope into their
intended places, reached them more slowly than those which the workmen
could lift directly in their hands; but it probably needed a much longer
time to enable them to see the exact relation which, in these and all
other machines, exists between the increase of the power to move, and
the decreasing swiftness of the thing moved.
In the preface to Galileo's Treatise on Mechanical Science, published in
1592, he is at some pains to set in a clear light the real advantages
belonging to the use of machines, "which (says he) I have thought it
necessary to do, because, if I mistake not, I see almost all mechanics
deceiving themselves in the belief that, by the help of a machine, they
can raise a greater weight than they are able to lift by the exertion of
the same force without it.—Now if we take any determinate weight, and
any force, and any distance whatever, it is beyond doubt that we can
move the weight to that distance by means of that force; because even
although the force may be exceedingly small, if we divide the weight
into a number of fragments, each of which is not too much for our force,
and carry these pieces one by one, at length we shall have removed the
whole weight; nor can we reasonably say at the end of our work, that
this great weight has been moved and carried away by a force less than
itself, unless we add that the force has passed several times over the
space through which the whole weight has gone but once. From which it
appears that the velocity of the force (understanding by velocity the
space gone through in a given time) has been as many times greater than
that of the weight, as the weight is greater than the force: nor can we
on that account say that a great force is overcome by a small one,
contrary to nature: then only might we say that nature is overcome when
a small force moves a great weight as swiftly as itself, which we assert
to be absolutely impossible with any machine either already or hereafter
to be contrived. But since it may occasionally happen that we have but a
small force, and want to move a great weight without dividing it into
pieces, then we must have recourse to a machine by means of which we
shall remove the given weight, with the given force, through the
required space. But nevertheless the force as before will have to travel
over that very same space as many times repeated as the weight surpasses
its power, so that, at the end of our work, we shall find that we have
derived no other benefit from our machine than that we have carried away
the same weight altogether, which if divided into pieces we could have
carried without the machine, by the same force, through the same space,
in the same time. This is one of the advantages of a machine, because it
often happens that we have a lack of force but abundance of time, and
that we wish to move great weights all at once."
This compensation of force and time has been fancifully personified by
saying that Nature cannot be cheated, and in scientific treatises on
mechanics, is called the "principle of virtual velocities," consisting
in the theorem that two weights will balance each other on any machine,
no matter how complicated or intricate the connecting contrivances may
be, when one weight bears to the other the same proportion that the
space through which the latter would be raised bears to that through
which the former would sink, in the first instant of their motion, if
the machine were stirred by a third force. The whole theory of machines
consists merely in generalizing and following out this principle into
its consequences; combined, when the machines are in a state of motion,
with another principle equally elementary, but to which our present
subject does not lead us to allude more particularly.
The credit of making known the principle of virtual velocities is
universally given to Galileo; and so far deservedly, that he undoubtedly
perceived the importance of it, and by introducing it everywhere into
his writings succeeded in recommending it to others; so that five and
twenty years after his death, Borelli, who had been one of Galileo's
pupils, calls it "that mechanical principle with which everybody is so
familiar[116]," and from that time to the present it has continued to be
taught as an elementary truth in most systems of mechanics. But although
Galileo had the merit in this, as in so many other cases, of
familiarizing and reconciling the world to the reception of truth, there
are remarkable traces before his time of the employment of this same
principle, some of which have been strangely disregarded. Lagrange
asserts[117] that the ancients were entirely ignorant of the principle
of virtual velocities, although Galileo, to whom he refers it,
distinctly mentions that he himself found it in the writings of
Aristotle. Montucla quotes a passage from Aristotle's Physics, in which
the law is stated generally, but adds that he did not perceive its
immediate application to the lever, and other machines. The passage to
which Galileo alludes is in Aristotle's Mechanics, where, in discussing
the properties of the lever, he says expressly, "the same force will
raise a greater weight, in proportion as the force is applied at a
greater distance from the fulcrum, and the reason, as I have already
said, is because it describes a greater circle; and a weight which is
farther removed from the centre is made to move through a greater
space."[118]
It is true, that in the last mentioned treatise, Aristotle has given
other reasons which belong to a very different kind of philosophy, and
which may lead us to doubt whether he fully saw the force of the one we
have just quoted. It appeared to him not wonderful that so many
mechanical paradoxes (as he called them) should be connected with
circular motion, since the circle itself seemed of so paradoxical a
nature. "For, in the first place, it is made up of an immoveable centre,
and a moveable radius, qualities which are contrary to each other. 2dly.
Its circumference is both convex and concave. 3dly. The motion by which
it is described is both forward and backward, for the describing radius
comes back to the place from which it started. 4thly. The radius is
_one_; but every point of it moves in describing the circle with a
different degree of swiftness."
Perhaps Aristotle may have borrowed the idea of virtual velocities,
contrasting so strongly with his other physical notions, from some older
writer; possibly from Archytas, who, we are told, was the first to
reduce the science of mechanics to methodical order;[119] and who by the
testimony of his countrymen was gifted with extraordinary talents,
although none of his works have come down to us. The other principles
and maxims of Aristotle's mechanical philosophy, which we shall have
occasion to cite, are scattered through his books on Mechanics, on the
Heavens, and in his Physical Lectures, and will therefore follow rather
unconnectedly, though we have endeavoured to arrange them with as much
regularity as possible.
After defining a body to be that which is divisible in every direction,
Aristotle proceeds to inquire how it happens that a body has only the
three dimensions of length, breadth, and thickness; and seems to think
he has given a reason in saying that, when we speak of two things, we do
not say "all," but "both," and three is the first number of which we say
"all."[120] When he comes to speak of motion, he says, "If motion is not
understood, we cannot but remain ignorant of Nature. Motion appears to
be of the nature of continuous quantities, and in continuous quantity
infinity first makes its appearance; so as to furnish some with a
definition who say that continuous quantity is that which is infinitely
divisible.—Moreover, unless there be time, space, and a vacuum, it is
impossible that there should be motion."[121]—Few propositions of
Aristotle's physical philosophy are more notorious than his assertion
that nature abhors a vacuum, on which account this last passage is the
more remarkable, as he certainly did not go so far as to deny the
existence of motion, and therefore asserts here the necessity of that of
which he afterwards attempts to show the absurdity.—"Motion is the
energy of what exists in power so far forth as so existing. It is that
act of a moveable which belongs to its power of moving."[122] After
struggling through such passages as the preceding we come at last to a
resting-place.—"It is difficult to understand what motion is."—When
the same question was once proposed to another Greek philosopher, he
walked away, saying, "I cannot tell you, but I will show you;" an answer
intrinsically worth more than all the subtleties of Aristotle, who was
not humble-minded enough to discover that he was tasking his genius
beyond the limits marked out for human comprehension.
He labours in the same manner and with the same success to vary the idea
of space. He begins the next book with declaring, that "those who say
there is a vacuum assert the existence of space; for a vacuum is space,
in which there is no substance;" and after a long and tedious reasoning
concludes that, "not only what space is, but also whether there be such
a thing, cannot but be doubted."[123] Of time he is content to say
merely, that "it is clear that time is not motion, but that without
motion there would be no time;"[124] and there is perhaps little fault
to be found with this remark, understanding motion in the general sense
in which Aristotle here applies it, of every description of change.
Proceeding after these remarks on the nature of motion in general to the
motion of bodies, we are told that "all local motion is either straight,
circular, or compounded of these two; for these two are the only simple
sorts of motion. Bodies are divided into simple and concrete; simple
bodies are those which have naturally a principle of motion, as fire and
earth, and their kinds. By simple motion is meant the motion of a simple
body."[125] By these expressions Aristotle did not mean that a simple
body cannot have what he calls a compound motion, but in that case he
called the motion violent or unnatural; this division of motion into
natural and violent runs through the whole of the mechanical philosophy
founded upon his principles. "Circular motion is the only one which can
be endless;"[126] the reason of which is given in another place: for
"that cannot be doing, which cannot be done; and therefore it cannot be
that a body should be moving towards a point (_i.e._ the end of an
infinite straight line) whither no motion is sufficient to bring
it."[127] Bacon seems to have had these passages in view when he
indulged in the reflections which we have quoted in page 14. "There are
four kinds of motion of one thing by another: Drawing, Pushing,
Carrying, Rolling. Of these, Carrying and Rolling may be referred to
Drawing and Pushing.[128]—The prime mover and the thing moved are
always in contact."
The principle of the composition of motions is stated very plainly:
"when a moveable is urged in two directions with motions bearing an
indefinitely small ratio to each other, it moves necessarily in a
straight line, which is the diameter of the figure formed by drawing the
two lines of direction in that ratio;"[129] and adds, in a singularly
curious passage, "but when it is urged for any time with two motions
which have an indefinitely small ratio one to another, the motion cannot
be straight, so that a body describes a curve, when it is urged by two
motions bearing an indefinitely small ratio one to another, and lasting
an indefinitely small time."[130]
He seemed on the point of discovering some of the real laws of motion,
when he was led to ask—"Why are bodies in motion more easily moved than
those which are at rest?—And why does the motion cease of things cast
into the air? Is it that the force has ceased which sent them forth, or
is there a struggle against the motion, or is it through the disposition
to fall, does it become stronger than the projectile force, or is it
foolish to entertain doubts on this question, when the body has quitted
the principle of its motion?" A commentator at the close of the
sixteenth century says on this passage: "They fall because every thing
recurs to its nature; for if you throw a stone a thousand times into the
air, it will never accustom itself to move upwards." Perhaps we shall
now find it difficult not to smile at the idea we may form of this
luckless experimentalist, teaching stones to fly; yet it may be useful
to remember that it is only because we have already collected an opinion
from the results of a vast number of observations in the daily
experience of life, that our ridicule would not be altogether misplaced,
and that we are totally unable to determine by any kind of reasoning,
unaccompanied by experiment, whether a stone thrown into the air would
fall again to the earth, or move for ever upwards, or in any other
conceivable manner and direction.
The opinion which Aristotle held, that motion must be caused by
something in contact with the body moved, led him to his famous theory
that falling bodies are accelerated by the air through which they pass.
We will show how it was attempted to explain this process when we come
to speak of more modern authors. He classed natural bodies into heavy
and light, remarking at the same time that it is clear that there are
some bodies possessing neither gravity nor levity."[131] By light bodies
he understood those which have a natural tendency to move from the
earth, observing that "that which is lighter is not always light."[132]
He maintained that the heavenly bodies were altogether devoid of
gravity; and we have already had occasion to mention his assertion, that
a large body falls faster than a small one in proportion to its
weight.[133] With this opinion may be classed another great mistake, in
maintaining that the same bodies fall through different mediums, as air
or water, with velocities reciprocally proportional to their densities.
By a singular inversion of experimental science, Cardan, relying on this
assertion, proposed in the sixteenth century to determine the densities
of air and water by observing the different times taken by a stone in
falling through them.[134] Galileo inquired afterwards why the
experiment should not be made with a cork, which pertinent question put
an end to the theory.
There are curious traces still preserved in the poem of Lucretius of a
mechanical philosophy, of which the credit is in general given to
Democritus, where many principles are inculcated strongly at variance
with Aristotle's notions. We find absolute levity denied, and not only
the assertion that in a vacuum all things would fall, but that they
would fall with the same velocity; and the inequalities which we observe
are attributed to the right cause, the impediment of the air, although
the error remains of believing the velocity of bodies falling through
the air to be proportional to their weight.[135] Such specimens of this
earlier philosophy may well indispose us towards Aristotle, who was as
successful in the science of motion as he was in astronomy in
suppressing the knowledge of a theory so much sounder than that which he
imposed so long upon the credulity of his blinded admirers.
An agreeable contrast to Aristotle's mystical sayings and fruitless
syllogisms is presented in Archimedes' book on Equilibrium, in which he
demonstrates very satisfactorily, though with greater cumbrousness of
apparatus than is now thought necessary, the principal properties of the
lever. This and the Treatise on the Equilibrium of Floating Bodies are
the only mechanical works which have reached us of this writer, who was
by common consent one of the most accomplished mathematicians of
antiquity. Ptolemy the astronomer wrote also a Treatise on Mechanics,
now lost, which probably contained much that would be interesting in the
history of mechanics; for Pappus says, in the Preface to the Eighth Book
of his Mathematical Collections: "There is no occasion for me to explain
what is meant by a heavy, and what by a light body, and why bodies are
carried up and down, and in what sense these very words 'up' and 'down'
are to be taken, and by what limits they are bounded; for all this is
declared in Ptolemy's Mechanics."[136] This book of Ptolemy's appears to
have been also known by Eutocius, a commentator of Archimedes, who lived
about the end of the fifth century of our era; he intimates that the
doctrines contained in it are grounded upon Aristotle's; if so, its loss
is less to be lamented. Pappus's own book deserves attention for the
enumeration which he makes of the mechanical powers, namely, the wheel
and axle, the lever, pullies, the wedge and the screw. He gives the
credit to Hero and Philo of having shown, in works which have not
reached us, that the theory of all these machines is the same. In Pappus
we also find the first attempt to discover the force necessary to
support a given weight on an inclined plane. This in fact is involved in
the theory of the screw; and the same vicious reasoning which Pappus
employs on this occasion was probably found in those treatises which he
quotes with so much approbation. Numerous as are the faults of his
pretended demonstration, it was received undoubtingly for a long period.
[Illustration: Chain.]
The credit of first giving the true theory of equilibrium on the
inclined plane is usually ascribed to Stevin, although, as we shall
presently show, with very little reason. Stevin supposed a chain to be
placed over two inclined planes, and to hang down in the manner
represented in the figure. He then urged that the chain would be in
equilibrium; for otherwise, it would incessantly continue in motion, if
there were any cause why it should begin to move. This being conceded,
he remarks further, that the parts AD and BD are also in equilibrium,
being exactly similar to each other; and therefore if they are taken
away, the remaining parts AC and BC will also be in equilibrium. The
weights of these parts are proportional to the lengths AC and BC; and
hence Stevin concluded that two weights would balance on two inclined
planes, which are to each other as the lengths of the planes included
between the same parallels to the horizon.[137] This conclusion is the
correct one, and there is certainly great ingenuity in this contrivance
to facilitate the demonstration; it must not however be mistaken for an
_à priori_ proof, as it sometimes seems to have been: we should remember
that the experiments which led to the principle of virtual velocities
are also necessary to show the absurdity of supposing a perpetual
motion, which is made the foundation of this theorem. That principle had
been applied directly to determine the same proportion in a work written
long before, where it has remained singularly concealed from the notice
of most who have written on this subject. The book bears the name of
Jordanus, who lived at Namur in the thirteenth century; but Commandine,
who refers to it in his Commentary on Pappus, considers it as the work
of an earlier period. The author takes the principle of virtual
velocities for the groundwork of his explanations, both of the lever and
inclined plane; the latter will not occupy much space, and in an
historical point of view is too curious to be omitted.
"_Quæst. 10._—If two weights descend by paths of different
obliquities, and the proportion be the same of the weights and the
inclinations taken in the same order, they will have the same descending
force. By the inclinations, I do not mean the angles, but the paths up
to the point in which both meet the same perpendicular.[138] Let,
therefore, _e_ be the weight upon _dc_, and _h_ upon _da_, and let _e_
be to _h_ as _dc_ to _da_. I say these weights, in this situation, are
equally effective. Take _dk_ equally inclined with _dc_, and upon it a
weight equal to _e_, which call 6. If possible let _e_ descend to _l_,
so as to raise _h_ to _m_, and take 6_n_ equal to _hm_ or _el_, and draw
the horizontal and perpendicular lines as in the figure.
[Illustration]
Then _nz_:_n_6 :: _db_:_dk_
and _mh_:_mx_ :: _da_:_db_
therefore _nz_:_mx_ :: _da_:_dk_ :: _h_:6, _and therefore since er is
not able to raise_ 6 _to n, neither will it be able to raise h to m_;
therefore they will remain as they are."[139] The passage in Italics
tacitly assumes the principle in question. Tartalea, who edited
Jordanus's book in 1565, has copied this theorem _verbatim_ into one of
his own treatises, and from that time it appears to have attracted no
further attention. The rest of the book is of an inferior description.
We find Aristotle's doctrine repeated, that the velocity of a falling
body is proportional to its weight; that the weight of a heavy body
changes with its form; and other similar opinions. The manner in which
falling bodies are accelerated by the air is given in detail. "By its
first motion the heavy body will drag after it what is behind, and move
what is just below it; and these when put in motion move what is next to
them, so that by being set in motion they less impede the falling body.
In this manner it has the effect of being heavier, and impels still more
those which give way before it, until at last they are no longer
impelled, but begin to drag. And thus it happens that its gravity is
increased by their attraction, and their motion by its gravity, whence
we see that its velocity is continually multiplied."
In this short review of the state of mechanical science before Galileo,
the name of Guido Ubaldi ought not to be omitted, although his works
contain little or nothing original. We have already mentioned Benedetti
as having successfully attacked some of Aristotle's statical doctrines,
but it is to be noticed that the laws of motion were little if at all
examined by any of these writers. There are a few theorems connected
with this latter subject in Cardan's extraordinary book "On
Proportions," but for the most part false and contradictory. In the
seventy-first proposition of his fifth book, he examines the force of
the screw in supporting a given weight, and determines it accurately on
the principle of virtual velocities; namely, that the power applied at
the end of the horizontal lever must make a complete circuit at that
distance from the centre, whilst the weight rises through the
perpendicular height of the thread. The very next proposition in the
same page is to find the same relation between the power and weight on
an inclined plane; and although the identity of principle in these two
mechanical aids was well known, yet Cardan declares the necessary
sustaining force to vary as the angle of inclination of the plane, for
no better reason than that such an expression will properly represent it
at the two limiting angles of inclination, since the force is nothing
when the plane is horizontal, and equal to the weight when
perpendicular. This again shows how cautious we should be in attributing
the full knowledge of general principles to these early writers, on
account of occasional indications of their having employed them.
FOOTNOTES:
[115] Histoire des Mathématiques, vol. i. p. 97.
[116] De vi Percussionis, Bononiæ, 1667.
[117] Mec. Analyt.
[118] Mechanica.
[119] Diog. Laert. In vit. Archyt.
[120] De Cœlo, lib. i. c. 1.
[121] Phys. lib. i. c. 3.
[122] Lib. iii. c. 2. The Aristotelians distinguished between things as
existing in act or energy (ενεργεια) and things in capacity or power
(δυναμις). For the advantage of those who may think the distinction
worth attending to, we give an illustration of Aristotle's meaning, from
a very acute and learned commentator:—"It (motion) is something more
than dead capacity; something less than perfect actuality; capacity
roused, and striving to quit its latent character; not the capable
brass, nor yet the actual statue, but the capacity in energy; that is to
say, the brass in fusion while it is becoming the statue and is not yet
become."—"The bow moves not because it may be bent, nor because it is
bent; but the motion lies between; lies in an imperfect and obscure
union of the two together; is the actuality (if I may so say) even of
capacity itself: imperfect and obscure, because such is capacity to
which it belongs."—Harris, Philosophical Arrangements.
[123] Lib. iv. c. 1.
[124] Lib. iv. c. 11.
[125] De Cœlo, lib. i. c. 2.
[126] Phys. lib. vii. c. 8.
[127] De Cœlo, lib. i. c. 6.
[128] Phys. lib. vii. c. 2.
[129] Mechanica.
[130] Εαν δε εν μηδενι λογῳ φερηται δυο φορας κατα μηδενα χρονον,
αδυνατον ευθειαν ειναι την φοραν. Εαν γαρ τινα λογον ενεχθῃ εν χρονῳ
τινι τουτον αναγκη τον χρονον ευθειαν ειναι φοραν δια τα προειρημενα,
ὡστε περιφερες γινεται δυο φερομενον φορας εν μηδενι λογῳ μηδενα
χρονον.—_i.e._ v = ds/dt
[131] De Cœlo, lib. i. c. 3.
[132] Lib. iv. c. 2.
[133] Phys., lib. iv. c. 8.
[134] De Proport. Basileæ, 1570.
[135]
"Nunc locus est, ut opinor, in his illud quoque rebus
Confirmare tibi, nullam rem posse suâ vi
Corpoream sursum ferri, sursumque meare.—
Nec quom subsiliunt ignes ad tecta domorum,
Et celeri flammâ degustant tigna trabeisque
Sponte suâ facere id sine vi subicente putandum est.
—Nonne vides etiam quantâ vi tigna trabeisque
Respuat humor aquæ? Nam quod magi' mersimus altum
Directâ et magnâ vi multi pressimus ægre:—
Tam cupide sursum revomit magis atque remittit
Plus ut parte foras emergant, exsiliantque:
—Nec tamen hæc, quantu'st in sedubitamus, opinor,
Quinvacuum per inane deorsum cuncta ferantur,
Sic igitur debent flammæ quoque posse per auras
Aeris expressæ sursum subsidere, quamquam
Pondera quantum in se est deorsum deducere pugnent.
—Quod si forte aliquis credit Graviora potesse
Corpora, quo citius rectum per Inane feruntur,
—Avius a verâ longe ratione recedit.
Nam per Aquas quæcunque cadunt atque Aera deorsum
Hæc pro ponderibus casus celerare necesse 'st
Propterea quia corpus Aquæ, naturaque tenuis
Aeris haud possunt æque rem quamque morari:
Sed citius cedunt Gravioribus exsuperata.
At contra nulli de nullâ parte, neque ullo
Tempore Inane potest Vacuum subsistere reii
Quin, sua quod natura petit, considere pergat:
Omnia quâ propter debent per Inane quietum
Æque ponderibus non æquis concita ferri."
De Rerum Natura, lib. ii, v. 184-239.
[136] Math. Coll. Pisani, 1662.
[137] Œuvres Mathématiques. Leyde, 1634.
[138] This is not a literal translation, but by what follows, is
evidently the Author's meaning. His words are, "Proportionem igitur
declinationum dico non angulorum, sed linearum usque ad æquidistantem
resecationem in quâ æqualiter sumunt de directo."
[139] Opusculum De Ponderositate. Venetiis, 1565.