![Descartes.
[608] Harriott’s book is a thin folio of 180 pages, with very little besides
examples; for his principles are shortly and obscurely laid down.
Whoever is the author of the preface to this work cannot be said to
have suppressed or extenuated the merits of Vieta, or to have claimed
anything for Harriott but what he is allowed to have deserved.
Montucla justly observes, that Harriott very rarely makes an equation
equal to zero, by bringing all the quantities to one side of the equation.
18. Harriott introduced the use of small letters instead of capitals in
algebra; he employed vowels for unknown, consonants for known
quantities, and joined them to express their product.[609] There is
certainly not much in this; but its evident convenience renders it
wonderful that it should have been reserved for so late an era.
Wallis, in his History of Algebra, ascribes to Harriott a long list of
discoveries, which have been reclaimed for Cardan and Vieta, the
great founders of the higher algebra, by Cossali and Montucla.[610]
The latter of these writers has been charged, even by foreigners,
with similar injustice towards our countryman; and that he has been
provoked by what he thought the unfairness of Wallis to something
like a depreciation of Harriott, seems as clear as that he has himself
robbed Cardan of part of his due credit in swelling the account of
Vieta’s discoveries. From the general integrity, however, of
Montucla’s writings, I am much inclined to acquit him of any wilful
partiality.
[609] Oughtred, in his Clavis Mathematica, published in 1631, abbreviated
the rules of Vieta, though he still used capital letters. He also gives
succinctly the praxis of algebra, or the elementary rules we find in our
common books, which, though what are now first learned, were, from
the singular course of algebraical history, discovered late. They are,
however, given also by Harriott. Wallisii Algebra.
[610] These may be found in the article Harriott of the Biographia Britannica.
Wallis, however, does not suppress the honour due to Vieta quite as
much as is intimated by Montucla.
19. Harriott had shown what were the hidden laws of
algebra, as the science of symbolical notation. But one man, the
pride of France and wonder of his contemporaries, was destined to
flash light upon the labours of the analyst, and to point out what](https://image.slidesharecdn.com/4866-250322231313-357002d8/85/Project-Management-The-Managerial-Process-5th-Edition-Larson-Solutions-Manual-26-320.jpg)
![His application
of algebra to
curves.
those symbols, so darkly and painfully traced, and resulting
commonly in irrational or even impossible forms, might represent
and explain. The use of numbers, or of letters denoting numbers, for
lines and rectangles capable of division into aliquot parts, had long
been too obvious to be overlooked, and is only a compendious
abbreviation of geometrical proof. The next step made was the
perceiving that irrational numbers, as they are called, represent
incommensurable quantities; that is, if unity be taken for the side of
a square, the square-root of two will represent its diagonal.
Gradually the application of numerical and algebraical calculation to
the solution of problems respecting magnitude became more
frequent and refined.[611] It is certain, however, that no one before
Descartes had employed algebraic formulæ in the construction of
curves; that is, had taught the inverse process, not only how to
express diagrams by algebra, but how to turn algebra into diagrams.
The ancient geometers, he observes, were scrupulous about using
the language of arithmetic in geometry, which could only proceed
from their not perceiving the relation between the two; and this has
produced a great deal of obscurity and embarrassment in some of
their demonstrations.[612]
[611] See note in Vol. II., p. 445.
[612] Œuvres de Descartes, v. 323.
20. The principle which Descartes establishes is that
every curve, of those which are called geometrical, has
its fundamental equation expressing the constant
relation between the absciss and the ordinate. Thus, the rectangle
under the abscisses of a diameter of the circle is equal to the square
of the ordinate, and the other conic sections, as well as higher
curves, have each their leading property, which determines their
nature, and shows how they may be generated. A simple equation
can only express the relation of straight lines; the solution of a
quadratic must be found in one of the four conic sections; and the
higher powers of an unknown quantity lead to curves of a superior
order. The beautiful and extensive theory developed by Descartes in](https://image.slidesharecdn.com/4866-250322231313-357002d8/85/Project-Management-The-Managerial-Process-5th-Edition-Larson-Solutions-Manual-27-320.jpg)
![Suspected
plagiarism
from Harriott.
this short treatise displays a most consummate felicity of genius.
That such a man, endowed with faculties so original, should have
encroached on the just rights of others, is what we can only believe
with reluctance.
21. It must, however, be owned that independently of
the suspicions of an unacknowledged appropriation of
what others had thought before him, which
unfortunately hang over all the writings of Descartes, he has taken
to himself the whole theory of Harriott on the nature of equations in
a manner which, if it is not a remarkable case of simultaneous
invention, can only be reckoned a very unwarrantable plagiarism. For
not only he does not name Harriott, but he evidently introduces the
subject as an important discovery of his own, and in one of his
letters asserts his originality in the most positive language.[613]
Still it
is quite possible that, prepared as the way had been by Vieta, and
gifted as Descartes was with a wonderfully intuitive acuteness in all
mathematical reasoning, he may in this, as in other instances, have
struck out the whole theory by himself. Montucla extols the algebra
of Descartes, that is, so much of it as can be fairly claimed for him
without any precursor, very highly; and some of his inventions in the
treatment of equations have long been current in books on that
science. He was the first who showed what were called impossible or
imaginary roots, though he never assigns them, deeming them no
quantities at all. He was also perhaps the first who fully understood
negative roots, though he still retains the appellation, false roots,
which is not so good as Harriott’s epithet, privative. According to his
panegyrist, he first pointed out that in every equation (the terms
being all on one side) which has no imaginary roots, there are as
many changes of signs as positive roots, as many continuations of
them as negative.
[613] Tant s’en faut que les choses que j’ai écrites puissent être aisément
tirées de Viéte, qu’au contraire ce qui est cause que mon traité est
difficile à entendre, c’est que j’ai tâché à n’y rien mettre que ce que j’ai
crû n’avoir point été su ni par lui ni par aucun autre; comme on peut
voir si on confére ce que j’ai écrit du nombre des racines qui sont en](https://image.slidesharecdn.com/4866-250322231313-357002d8/85/Project-Management-The-Managerial-Process-5th-Edition-Larson-Solutions-Manual-28-320.jpg)
![Algebraic
geometry not
successful at
first.
who scrupled to follow him implicitly, and who might in any manner
be thought rivals of his fame. Yet it is this unhappy temper of
Descartes which ought to render us more unwilling to credit the
suspicions of his designed plagiarism from the discoveries of others;
since this, combined with his unwillingness to acknowledge their
merits, and affected ignorance of their writings, would form a
character we should not readily ascribe to a man of great genius,
and whose own writings give many apparent indications of sincerity
and virtue. But in fact there was in this age a great probability of
simultaneous invention in science, from developing principles that
had been partially brought to light. Thus Roberval discovered the
same method of indivisibles as Cavalieri, and Descartes must equally
have been led to this theory of tangents by that of Kepler. Fermat
also, who was in possession of his principal discoveries before the
geometry of Descartes saw the light, derived from Kepler his own
celebrated method, de maximis et minimis; a method of discovering
the greatest or least value of a variable quantity, such as the
ordinate of a curve. It depends on the same principle as that of
Kepler. From this he deduced a rule for drawing tangents to curves
different from that of Descartes. This led to a controversy between
the two geometers, carried on by Descartes, who yet is deemed to
have been in the wrong, with his usual quickness of resentment.
Several other discoveries, both in pure algebra and geometry,
illustrate the name of Fermat.[614]
[614] A good article on Fermat, by M. Maurice, will be found in the
Biographie Universelle.
23. The new geometry of Descartes was not received
with the universal admiration it deserved. Besides its
conciseness and the inroad it made on old prejudices as
to geometrical methods, the general boldness of the
author’s speculations in physical and metaphysical philosophy, as
well as his indiscreet temper, disinclined many who ought to have
appreciated it; and it was in his own country, where he had ceased
to reside, that Descartes had the fewest admirers. Roberval made
some objections to his rival’s algebra, but with little success. A](https://image.slidesharecdn.com/4866-250322231313-357002d8/85/Project-Management-The-Managerial-Process-5th-Edition-Larson-Solutions-Manual-30-320.jpg)
![Conjectures as
to comets.
Galileo’s
discovery of
Jupiter’s
satellites.
Cosmo graphicum, published in 1619; a work mingled up with many
strange effusions of a mind far more eccentric than any of the
planets with which it was engaged. In the Epitome Astronomiæ
Copernicanæ, printed the same year, he endeavours to deduce this
law from his theory of centrifugal forces. He had a very good insight
into the principles of universal gravitation, as an attribute of matter;
but several of his assumptions as to the laws of motion are not
consonant to truth. There seems indeed to have been a considerable
degree of good fortune in the discoveries of Kepler; yet, this may be
deemed the reward of his indefatigable laboriousness, and of the
ingenuousness with which he renounced any hypothesis that he
could not reconcile with his advancing knowledge of the phenomena.
25. The appearance of three comets in 1619 called once
more the astronomers of Europe to speculate on the
nature of those anomalous bodies. They still passed for harbingers
of worldly catastrophies; and those who feared them least could not
interpret their apparent irregularity. Galileo, though Tycho Brahe had
formed a juster notion, unfortunately took them for atmospheric
meteors. Kepler, though he brought them from the far regions of
space, did not suspect the nature of their orbits, and thought that,
moving in straight lines, they were finally dispersed and came to
nothing. But a Jesuit, Grassi, in a treatise, De Tribus Cometis, Rome,
1618, had the honour of explaining what had baffled Galileo, and
first held them to be planets moving in vast ellipses round the sun.
[615]
[615] The Biographie Universelle, art. Grassi, ascribes this opinion to Tycho.
26. But long before this time the name of Galileo had
become immortal by discoveries which, though they
would certainly have soon been made by some other,
perhaps far inferior, observer, were happily reserved for
the most philosophical genius of the age. Galileo assures us that,
having heard of the invention of an instrument in Holland which
enlarged the size of distant objects, but knowing nothing of its
construction, he began to study the theory of refractions till he](https://image.slidesharecdn.com/4866-250322231313-357002d8/85/Project-Management-The-Managerial-Process-5th-Edition-Larson-Solutions-Manual-32-320.jpg)
![found by experiment, that by means of a convex and concave glass
in a tube, he could magnify an object threefold. He was thus
encouraged to make another which magnified thirty times; and this
he exhibited in the autumn of 1609 to the inhabitants of Venice.
Having made a present of his first telescope to the senate, who
rewarded him with a pension, he soon constructed another; and in
one of the first nights of January, 1610, directing it towards the
moon, was astonished to see her surface and edges covered with
inequalities. These he considered to be mountains, and judged by a
sort of measurement that some of them must exceed those of the
earth. His next observation was of the milky way; and this he found
to derive its nebulous lustre from myriads of stars not
distinguishable through their remoteness, by the unassisted sight of
man. The nebulæ in the constellation Orion he perceived to be of
the same character. Before his delight at these discoveries could
have subsided, he turned his telescope to Jupiter, and was surprised
to remark three small stars, which, in a second night’s observation,
had changed there places. In the course of a few weeks, he was
able to determine by their revolutions, which are very rapid, that
these are secondary planets, the moons or satellites of Jupiter; and
he had added a fourth to their number. These marvellous revelations
of nature he hastened to announce in a work, aptly entitled Sidereus
Nuncius, published in March, 1610. In an age when the fascinating
science of astronomy had already so much excited the minds of
philosophers, it may be guessed with what eagerness this
intelligence from the heavens was circulated. A few, as usual,
through envy or prejudice, affected to contemn it. But wisdom was
justified of her children. Kepler, in his Narratio de observatis a se
Quatuor Jovis Satellitibus, 1610, confirmed the discoveries of Galileo.
Peiresc, an inferior name, no doubt, but deserving of every praise for
his zeal in the cause of knowledge, having with difficulty procured a
good telescope, saw the four satellites in November, 1610, and is
said by Gassendi to have conceived at that time the ingenious idea
that their occultations might be used to ascertain the longitude.[616]
[616] Gassendi Vita Peirescii, p. 77.](https://image.slidesharecdn.com/4866-250322231313-357002d8/85/Project-Management-The-Managerial-Process-5th-Edition-Larson-Solutions-Manual-33-320.jpg)
![Copernican
system held
by Galileo.
revealing at intervals the dark central mass of the sun’s body which
it envelopes.
29. Though it cannot be said, perhaps, that the
discoveries of Galileo would fully prove the Copernican
system of the world to those who were already
insensible to reasoning from its sufficiency to explain the
phenomena, and from the analogies of nature, they served to
familiarise the mind to it, and to break down the strong rampart of
prejudice which stood in its way. For eighty years, it has been said,
this theory of the earth’s motion had been maintained without
censure; and it could only be the greater boldness of Galileo in its
assertion which drew down upon him the notice of the church. But,
in these eighty years since the publication of the treatise of
Copernicus, his proselytes had been surprisingly few. They were now
becoming more numerous: several had written on that side; and
Galileo had begun to form a school of Copernicans who were
spreading over Italy. The Lincean society, one of the most useful and
renowned of Italian academies, founded at Rome by Frederic Cesi, a
young man of noble birth, in 1603, had, as a fundamental law, to
apply themselves to natural philosophy; and it was impossible that
so attractive and rational a system as that of Copernicus could fail of
pleasing an acute and ingenious nation strongly bent upon science.
The church, however, had taken alarm; the motion of the earth was
conceived to be as repugnant to Scripture as the existence of
antipodes had once been reckoned; and in 1616, Galileo, though
respected and in favour with the court of Rome, was compelled to
promise that he would not maintain that doctrine in any manner.
Some letters that he had published on the subject were put, with the
treatise of Copernicus and other works, into the Index
Expurgatorius, where, I believe, they still remain.[617]
[617] Drinkwater’s Life of Galileo. Fabroni, Vitæ Italorum, vol. i. The former
seems to be mistaken in supposing that Galileo did not endeavour to
prove his system compatible with Scripture. In a letter to Christina, the
Grand Duchess of Tuscany, the author (Brenna) of the Life in Fabroni’s
work, tells us, he argued very elaborately for that purpose. In ea](https://image.slidesharecdn.com/4866-250322231313-357002d8/85/Project-Management-The-Managerial-Process-5th-Edition-Larson-Solutions-Manual-35-320.jpg)
![His dialogues,
and
persecution.
videlicet epistolâ philosophus noster ita disserit, ut nihil etiam ab
hominibus, qui omnem in sacrarum literarum studio consumpsissent
ætatem, aut subtilius aut verius aut etiam accuratius explicatum
expectari potuerit, p. 118. It seems, in fact, to have been this over-
desire to prove his theory orthodox, which incensed the church against
it. See an extraordinary article on this subject in the eighth number of
the Dublin Review (1838). Many will tolerate propositions inconsistent
with orthodoxy, when they are not brought into immediate
juxtaposition with it.
30. He seems, notwithstanding this, to have flattered
himself that, after several years had elapsed, he might
elude the letter of this prohibition by throwing the
arguments in favour of the Ptolemaic and Copernican systems into
the form of a dialogue. This was published in 1632; and he might,
from various circumstances, not unreasonably hope for impunity. But
his expectations were deceived. It is well known that he was
compelled by the Inquisition at Rome, into whose hands he fell, to
retract, in the most solemn and explicit manner, the propositions he
had so well proved, and which he must have still believed. It is
unnecessary to give a circumstantial account, especially as it has
been so well done in a recent work, the Life of Galileo, by Mr.
Drinkwater Bethune. The papal court meant to humiliate Galileo, and
through him to strike an increasing class of philosophers with shame
and terror; but not otherwise to punish one, of whom even the
inquisitors must, as Italians, have been proud; his confinement,
though Montucla says it lasted for a year, was very short. He
continued, nevertheless, under some restraint for the rest of his life,
and though he lived at his own villa near Florence, was not
permitted to enter the city.[618]
[618] Fabroni. His Life is written in good Latin, with knowledge and spirit,
more than Tiraboschi has ventured to display.
It appears from some of Grotius’s Epistles, that Galileo had thought,
about 1635, of seeking the protection of the United Provinces. But on
account of his advanced age he gave this up: fessus senio constituit
manere in quibus est locis, et potius quæ ibi sunt incommoda perpeti,
quam malæ ætati migrandi onus, et novas parandi amicitias imponere.](https://image.slidesharecdn.com/4866-250322231313-357002d8/85/Project-Management-The-Managerial-Process-5th-Edition-Larson-Solutions-Manual-36-320.jpg)
![Descartes
alarmed by
this.
Progress of
Copernican
system.
The very idea shows that he must have deeply felt the restraint
imposed upon him in his country. Epist. Grot. 407, 446.
31. The church was not mistaken in supposing that she
should intimidate the Copernicans, but very much so in
expecting to suppress the theory. Descartes was so
astonished at hearing of the sentence on Galileo, that he was almost
disposed to burn his papers, or at least to let no one see them. “I
cannot collect,” he says, “that he who is an Italian, and a friend of
the pope, as I understand, has been criminated on any other
account than for having attempted to establish the motion of the
earth. I know that this opinion was formerly censured by some
cardinals; but I thought I had since heard that no objection was now
made to its being publicly taught even at Rome.”[619] It seems not at
all unlikely that Descartes was induced, on this account, to pretend a
greater degree of difference from Copernicus than he really felt, and
even to deny, in a certain sense of his own, the obnoxious tenet of
the earth’s motion.[620] He was not without danger of a sentence
against truth nearer at hand; Cardinal Richelieu having had the
intention of procuring a decree of the Sorbonne to the same effect,
which, by the good sense of some of that society, fell to the ground.
[621]
[619] Vol. vi., p. 239. He says here, of the motion of the earth, Je confesse
que s’il est faux, tous les fondemens de ma philosophie le sont aussi.
[620] Vol. vi., p. 50.
[621] Montucla, ii., p. 297.
32. The progress, however, of the Copernican theory in
Europe, if it may not actually be dated from its
condemnation at Rome, was certainly not at all slower
after that time. Gassendi rather cautiously took that side; the
Cartesians brought a powerful reinforcement; Bouillaud and several
other astronomers of note avowed themselves favourable to a
doctrine which, though in Italy it lay under the ban of the papal
power, was readily saved on this side of the Alps by some of the
salutary distinctions long in use to evade that authority.[622] But in](https://image.slidesharecdn.com/4866-250322231313-357002d8/85/Project-Management-The-Managerial-Process-5th-Edition-Larson-Solutions-Manual-37-320.jpg)
![Descartes
denies general
gravitation.
the middle of the seventeenth century, and long afterwards, there
were mathematicians of no small reputation, who struggled
staunchly for the immobility of the earth; and except so far as
Cartesian theories might have come in vogue, we have no reason to
believe that any persons unacquainted with astronomy, either in this
country or on the continent, had embraced the system of
Copernicus. Hume has censured Bacon for rejecting it; but if Bacon
had not done so, he would have anticipated the rest of his
countrymen by a full quarter of a century.
[622] Id., p. 50.
33. Descartes, in his new theory of the solar system,
aspired to explain the secret springs of nature, while
Kepler and Galileo had merely showed their effects. By
what force the heavenly bodies were impelled, by what law they
were guided, was certainly a very different question from that of the
orbit they described or the period of their revolution. Kepler had
evidently some notion of that universally mutual gravitation which
Hooke saw more clearly, and Newton established on the basis of his
geometry.[623] But Descartes rejected this with contempt. “For,” he
says “to conceive this we must not only suppose that every portion
of matter in the universe is animated, and animated by several
different souls which do not obstruct one another, but that those
souls are intelligent and even divine; that they may know what is
going on in the most remote places, without any messenger to give
them notice, and that they may exert their powers there.”[624] Kepler,
who took the world for a single animal, a leviathan that roared in
caverns and breathed in the ocean tides, might have found it difficult
to answer this, which would have seemed no objection at all to
Campanella. If Descartes himself had been more patient towards
opinions which he had not formed in his own mind, that constant
divine agency, to which he was, on other occasions, apt to resort,
could not but have suggested a sufficient explanation of the gravity
of matter, without endowing it with self-agency. He had, however,
fallen upon a complicated and original scheme; the most celebrated,](https://image.slidesharecdn.com/4866-250322231313-357002d8/85/Project-Management-The-Managerial-Process-5th-Edition-Larson-Solutions-Manual-38-320.jpg)
![Cartesian
theory of the
world.
perhaps, though not the most admirable, of the novelties which
Descartes brought into philosophy.
[623] “If the earth and moon,” he says, “were not retained in their orbits,
they would fall one on another, the moon moving about 33/34 of the
way, the earth the rest, supposing them equally dense.” By this
attraction of the moon he accounts for tides. He compares the
attraction of the planets towards the sun to that of heavy bodies
towards the earth.
[624] Vol.ix., p. 560.
34. In a letter to Mersenne, January 9th, 1639, he
shortly states that notion of the material universe, which
he afterwards published in the Principia Philosophiæ. “I
will tell you,” he says, “that I conceive, or rather I can demonstrate,
that besides the matter which composes terrestrial bodies, there are
two other kinds; one very subtle, of which the parts are round or
nearly round like grains of sand, and this not only occupies the pores
of terrestrial bodies, but constitutes the substance of all the
heavens; the other incomparably more subtle, the parts of which are
so small and move with such velocity, that they have no determinate
figure, but readily take at every instant that which is required to fill
all the little intervals which the other does not occupy.”[625] To this
hypothesis of a double æther he was driven by his aversion to admit
any vacuum in nature; the rotundity of the former corpuscles having
been produced, as he fancied, by their continual circular motions,
which had rubbed off their angles. This seems at present rather a
clumsy hypothesis, but it is literally that which Descartes presented
to the world.
[625] Vol. viii., p. 73.
35. After having thus filled the universe with different sorts of
matter, he supposes that the subtler particles, formed by the
perpetual rubbing off of the angles of the larger in their progress
towards sphericity, increased by degrees till there was a superfluity
that was not required to fill up the intervals; and this, flowing
towards the centre of the system, became the sun, a very subtle and](https://image.slidesharecdn.com/4866-250322231313-357002d8/85/Project-Management-The-Managerial-Process-5th-Edition-Larson-Solutions-Manual-39-320.jpg)
![liquid body, while in like manner, the fixed stars were formed in
other systems. Round these centres the whole mass is whirled in a
number of distinct vortices, each of which carries along with it a
planet. The centrifugal motion impels every particle in these vortices
of each instant to fly off from the sun in a straight line; but it is
retained by the pressure of those which have already escaped and
form a denser sphere beyond it. Light is no more than the effect of
particles seeking to escape from the centre, and pressing one on
another, though perhaps without actual motion.[626] The planetary
vortices contain sometimes smaller vortices, in which the satellites
are whirled round their principal.
[626] J’ai souvent averti que par la lumière je n’entendois pas tant le
mouvement que cette inclination ou propension que ces petits corps
ont à se mouvoir, et que ce que je dirois du mouvement, pour être plus
aisément entendu, se devoit rapporter à cette propension; d’où il est
manifeste qua selon moi l’on ne doit entendre autre chose par les
couleurs que les différentes variétés qui arrivent en ces propensions.
Vol. vii., p. 193.
36. Such, in a few words, is the famous Cartesian theory, which,
fallen in esteem as it now is, stood its ground on the continent of
Europe, for nearly a century, till the simplicity of the Newtonian
system, and, above all, its conformity to the reality of things, gained
an undisputed predominance. Besides the arbitrary suppositions of
Descartes, and the various objections that were raised against the
absolute plenum of space and other parts of his theory, it has been
urged that his vortices are not reconcilable, according to the laws of
motion in fluids, with the relation, ascertained by Kepler, between
the periods and distances of the planets; nor does it appear why the
sun should be in the focus, rather than in the centre of their orbits.
Yet, within a few years it has seemed not impossible, that a part of
his bold conjectures will enter once more with soberer steps into the
schools of philosophy. His doctrine as to the nature of light,
improved as it was by Huygens, is daily gaining ground over that of
Newton; that of a subtle æther pervading space, which in fact is
nearly the same thing, is becoming a favourite speculation, if we are](https://image.slidesharecdn.com/4866-250322231313-357002d8/85/Project-Management-The-Managerial-Process-5th-Edition-Larson-Solutions-Manual-40-320.jpg)
![numbers, 1, 3, 5, 7, and so forth; and that the whole space was half
what would have been traversed uniformly from the beginning with
the final velocity. These are the great laws of accelerated and
retarded motion, from which Galileo deduced most important
theorems. He showed that the time in which bodies roll down the
length of inclined planes is equal to that in which they would fall
down the height, and in different planes is proportionate to the
height; and that their acquired velocity is in the same ratios. In
some propositions he was deceived; but the science of dynamics
owes more to Galileo than to any one philosopher. The motion of
projectiles had never been understood; he showed it to be parabolic;
and in this he not only necessarily made use of a principle of vast
extent, that of compound motion, which, though it is clearly
mentioned in one passage by Aristotle[627] and may probably be
implied in the mechanical reasonings of others, does not seem to
have been explicitly laid down by modern writers, but must have
seen the principle of curvilinear deflection by forces acting in
infinitely small portions of time. The ratio between the times of
vibration in pendulums of unequal length, had early attracted
Galileo’s attention. But he did not reach the geometrical exactness of
which this subject is capable.[628] He developed a new principle as to
the resistance of solids to the fracture of their parts, which, though
Descartes as usual treated it with scorn, is now established in
philosophy. “One forms, however,” says Playfair, “a very imperfect
idea of this philosopher from considering the discoveries and
inventions, numerous and splendid as they are, of which he was the
undisputed author. It is by following his reasonings, and by pursuing
the train of his thoughts, in his own elegant, though somewhat
diffuse exposition of them, that we become acquainted with the
fertility of his genius, with the sagacity, penetration, and
comprehensiveness of his mind. The service which he rendered to
real knowledge is to be estimated not only from the truths which he
discovered, but from the errors which he detected; not merely from
the sound principles which he established, but from the pernicious
idols which he overthrew. Of all the writers who have lived in an age
which was yet only emerging from ignorance and barbarism, Galileo](https://image.slidesharecdn.com/4866-250322231313-357002d8/85/Project-Management-The-Managerial-Process-5th-Edition-Larson-Solutions-Manual-43-320.jpg)
![Mechanics of
Descartes.
has most entirely the tone of true philosophy, and is most free from
any contamination of the times, in taste, sentiment, and opinion.”[629]
[627] Drinkwater’s Life of Galileo, p. 80.
[628] Fabroni.
[629] Preliminary Dissertation to Encyclop. Britain.
40. Descartes, who left nothing in philosophy untouched,
turned his acute mind to the science of mechanics,
sometimes with signal credit, sometimes very unsuccessfully. He
reduced all statics to one principle, that it requires as much force to
raise a body to a given height, as to raise a body of double weight to
half the height. This is the theorem of virtual velocities in another
form. In many respects he displays a jealousy of Galileo, and an
unwillingness to acknowledge his discoveries, which puts himself
often in the wrong. “I believe,” he says, “that the velocity of very
heavy bodies which do not move very quickly in descending
increases nearly in a duplicate ratio; but I deny that this is exact,
and I believe that the contrary is the case when the movement is
very rapid.”[630]
This recourse to the air’s resistance, a circumstance
of which Galileo was well aware, in order to diminish the credit of a
mathematical theorem, is unworthy of Descartes; but it occurs more
than once in his letters. He maintained also, against the theory of
Galileo, that bodies do not begin to move with an infinitely small
velocity, but have a certain degree of motion at the first instance,
which is afterwards accelerated.[631] In this too, as he meant to
extend his theory to falling bodies, the consent of philosophers has
decided the question against him. It was a corollary from these
notions that he denies the increments of spaces to be according to
the progression of uneven numbers.[632] Nor would he allow that the
velocity of a body augments its force, though it is a concomitant.[633]
[630] Œuvres de Descartes, vol. viii., p. 24.
[631] Il faut savoir, quoique Galilée et quelques autres disent au contraire,
que les corps qui commencent à descendre, ou à se mouvoir en
quelque façon que ce soit, ne passent point par tous les degrés de](https://image.slidesharecdn.com/4866-250322231313-357002d8/85/Project-Management-The-Managerial-Process-5th-Edition-Larson-Solutions-Manual-44-320.jpg)
![Law of motion
laid down by
Descartes.
tardiveté; mais que des le premier moment ils ont certaine vitesse qui
s’augmente après de beaucoup, et c’est de cette augmentation que
vient la force de la percussion. viii., 181.
[632] Cette proportion d’augmentation selon les nombres impairs, 1, 3, 5, 7,
&c., qui est dans Galilée et que je crois vous avoir aussi écrite
autrefois, ne peut être vraie, qu’en supposant deux ou trois choses qui
sont très fausses, dont l’une est que le mouvement croisse par degrés
depuis le plus lent, ainsi que le songe Galilée, et l’autre que la
résistance de l’air n’empêche point. Vol. ix., p. 349.
[633] Je pense que la vitesse n’est pas la cause de l’augmentation de la
force, encore qu’elle l’accompagne toujours. Id. p. 356, See also vol.
viii., p. 14. He was probably perplexed by the metaphysical notion of
causation, which he knew not how to ascribe to mere velocity. The fact
that increased velocity is a condition or antecedent of augmented force
could not be doubted.
41. Descartes, however, is the first who laid down the
laws of motion; especially that all bodies persist in their
present state of rest or uniform rectilineal motion till
affected by some force. Many had thought, as the vulgar always do,
that a continuance of rest was natural to bodies, but did not
perceive that the same principle of inertia or inactivity was applicable
to them in rectilineal motion. Whether this is deducible from theory,
or depends wholly on experience, by which we ought to mean
experiment, is a question we need not discuss. The fact, however, is
equally certain; and hence Descartes inferred that every curvilinear
deflection is produced by some controlling force, from which the
body strives to escape in the direction of a tangent to the curve. The
most erroneous part of his mechanical philosophy is contained in
some propositions as to the collision of bodies, so palpably
incompatible with obvious experience that it seems truly wonderful
he could ever have adopted them. But he was led into these
paradoxes by one of the arbitrary hypotheses which always
governed him. He fancied it a necessary consequence from the
immutability of the divine nature that there should always be the
same quantity of motion in the universe; and rather than abandon
this singular assumption he did not hesitate to assert, that two hard](https://image.slidesharecdn.com/4866-250322231313-357002d8/85/Project-Management-The-Managerial-Process-5th-Edition-Larson-Solutions-Manual-45-320.jpg)
![Also those of
compound
forces.
Other
discoveries in
mechanics.
bodies striking each other in opposite directions would be reflected
with no loss of velocity; and, what is still more outrageously
paradoxical, that a smaller body is incapable of communicating
motion to a greater; for example, that the red billiard-ball cannot put
the white into motion. This manifest absurdity he endeavoured to
remove by the arbitrary supposition, that when we see, as we
constantly do, the reverse of his theorem take place, it is owing to
the air, which, according to him, renders bodies more susceptible of
motion, than they would naturally be.
42. Though Galileo, as well as others, must have been
acquainted with the laws of the composition of moving
forces, it does not appear that they had ever been so
distinctly enumerated as by Descartes, in a passage of his Dioptrics.
[634] That the doctrine was in some measure new may be inferred
from the objections of Fermat; and Clerselier, some years
afterwards, speaks of persons “not much versed in mathematics,
who cannot understand an argument taken from the nature of
compound motion.”[635]
[634] Vol. v., p. 18.
[635] Vol. vi., p. 508.
43. Roberval demonstrated what seems to have been
assumed by Galileo, that the forces on an oblique or
crooked lever balance each other, when they are
inversely as the perpendiculars drawn from the centre of motion to
their direction. Fermat, more versed in geometry than physics,
disputed this theorem which is now quite elementary. Descartes, in a
letter to Mersenne, ungraciously testifies his agreement with it.[636]
Torricelli, the most illustrious disciple of Galileo, established that
when weights balance each other in all positions, their common
centre of gravity does not ascend or descend, and conversely.
[636] Je suis de l’opinion, says Descartes, de ceux qui disent que pondera
sunt in æquilibrio quando sunt in ratione reciproca linearum
perpendicularium, &c., vol. xi., p. 357. He would not name Roberval;](https://image.slidesharecdn.com/4866-250322231313-357002d8/85/Project-Management-The-Managerial-Process-5th-Edition-Larson-Solutions-Manual-46-320.jpg)
Project Management The Managerial Process 5th Edition Larson Solutions Manual
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- 5. Chapter 06 - Developing A Project Plan 6-1 Chapter 6 DEVELOPING A PROJECT PLAN Chapter Outline 1. Developing the Project Network 2. From Work Package to Network 3. Constructing a Project Network A. Terminology B. Two Approaches C. Basic Rules to Follow in Developing Project Networks 4. Activity-on-Node (AON) Fundamentals 5. Network Computation Process A. Forward Pass—Earliest Times B. Backward Pass—Latest Times C. Determining Slack (or Float) D. Free Slack (Float) 6. Using the Forward and Backward Pass Information 7. Level of Detail for Activities 8. Practical Considerations A. Network Logic Errors B. Activity Numbering C. Use of Computer to Develop Networks D. Calendar Dates E. Multiple Starts and Multiple Projects 9. Extended Network Techniques to Come Closer to Reality A. Laddering B. Use of Lags C. An Example Using Lag Relationships—The Forward and Backward Pass
- 6. Chapter 06 - Developing A Project Plan 6-2 D. Hammock Activities 10. Summary 11. Key Terms 12. Review Questions 13. Exercises 14. Case: Advantage Energy Technology Data Center Migration 15. Case: Greendale Stadium Case 16. Appendix 6.1: Activity-on-Arrow Method A. Description B. Design of An AOA Project Network i. Forward Pass—Earliest Times ii. Backward Pass—Latest Times iii. Computer-Generated Networks C. Choice of Method—AON or AOA D. Summary 17. Appendix Review Questions 18. Appendix Exercises Chapter Objectives • To establish the linkage between the WBS and the project network • To diagram a project network using AON methods • To provide a process for computing early, late, and slack activity times and identify the critical path • To demonstrate understanding and application of “lags” in compressing projects or constraining the start or finish of an activity • To provide an overview framework for estimating times and costs • To suggest the importance of slack in scheduling projects. Review Questions 1. How does the WBS differ from the project network? a. The WBS is hierarchical while the project network is sequential. b. The network provides a project schedule by identifying sequential dependencies and timing of project activities. The network sets all project work, resource needs, and budgets into a sequential time frame; the WBS does not provide this information. c. The WBS is used to identify each project deliverable and the organization unit responsible for its accomplishment within budget and within a time duration. d. The WBS provides a framework for tracking costs to deliverables and organization units responsible. 2. How are WBS and project networks linked?
- 7. Chapter 06 - Developing A Project Plan 6-3 The network uses the time estimates found in the work packages of the WBS to develop the network. Remember, the time estimates, budgets, and resources required for a work package in the WBS are set in time frames, but without dates. The dates are computed after the network is developed. 3. Why bother creating a WBS? Why not go straight to a project network and forget the WBS? The WBS is designed to provide different information for decision making. For example, this database provides information for the following types of decisions: a. Link deliverables, organization units, and customer b. Provide for control c. Isolate problems to source d. Track schedule and cost variance. Network doesn’t. e. Assign responsibility and budgets f. Focus attention on deliverables g. Provide information for different levels in the organization. 4. Why is slack important to the project manager? Slack is important to the project manager because it represents the degree of flexibility the project manager will have in rearranging work and resources. A project network with several near critical paths and hence, little slack, gives the project manager little flexibility in changing resources or rearranging work. 5. What is the difference between free slack and total slack? Free slack usually occurs at the end of an activity chain—before a merge activity. It is the amount of time the activity can be delayed without affecting the early start of the activity immediately following it. Since free slack can be delayed without delaying following activities, it gives some resource flexibility to the project manager. Total slack is the amount of time an activity can be delayed before it becomes critical. Use of total slack prevents its use on a following activity. 6. Why are lags used in developing project networks? Two major reasons: a. To closer represent real situations found in projects b. To allow work to be accomplished in parallel when the finish-to-start relationship is too restrictive. 7. What is a hammock activity, and when is it used?
- 8. Chapter 06 - Developing A Project Plan 6-4 A hammock activity is a special purpose activity that exists over a segment of the life of the project. A hammock activity typically uses resources and is handled as an overhead cost—e.g., inspection. Hammock activities are used to identify overhead resources or costs tied directly to the project. The hammock duration is determined by the beginning of the first of a string of activities and the ending of the last activity in the string. Hammock activities are also used to aggregate sections of projects to avoid project detail—e.g., covering a whole subnetwork within a project. This approach gives top management an overview of the project by avoiding detail. Exercises Creating a Project Network 1.
- 9. Chapter 06 - Developing A Project Plan 6-5 Drawing AON Networks 2. Activity A is a burst activity. Activity D is a merge activity 3. Activity C is a burst activity. Activity G is a merge activity 4. Activity A is a burst activity. Activities D and H are merge activities.
- 10. Chapter 06 - Developing A Project Plan 6-6 5. Activity A is a burst activity. Activities F, H, and G are merge activities.
- 11. Chapter 06 - Developing A Project Plan 6-7 AON Network Times 6. The project will take 14 days 7. Air Control Company
- 12. Chapter 06 - Developing A Project Plan 6-8 8. We would expect to penalized for one day past the 15 day deadline 9. The project is expected to take 9 days. The project is very sensitive with 3 interrelated critical paths. None of the activities have slack.
- 13. Chapter 06 - Developing A Project Plan 6-9 11.
- 14. Chapter 06 - Developing A Project Plan 6-10 12. The project is expected to take 16 days. The project is not very sensitive with one dominant critical path. Activities B, D, E, G, H and J all have total slack of 7. Activities B, D, E, and G have zero free slack. Activities H and J have 7 days of free slack.
- 15. Chapter 06 - Developing A Project Plan 6-11 13. 1 2 3 4 5 6 7 8 9 10 11 12 Identify topic Research topic Draft paper Edit paper Create graphics References Final Draft
- 16. Chapter 06 - Developing A Project Plan 6-12 14.
- 17. Chapter 06 - Developing A Project Plan 6-13 Computer Exercises 15. The estimated completion date is 30 weeks which is well ahead of the 45 week deadline. 16. Whistler Ski Resort Project Assignment: 1. Identify the critical path on your network. 2. Can the project be completed by October 1? The critical path for this schedule is: Build road to site → Clear chair lift → Construct chair lift foundation → Install chair lift towers → Install chair lift cable → Install chairs. If the project starts on April 1, it should be completed by September 29 (based on a 2010 schedule)—1 day ahead of schedule. Hint: When assigning this exercise you should remind students to use an April 1 start date. Below is the MS Project Entry Table and Gantt Chart for the Whistler Ski Resort project.
- 18. Chapter 06 - Developing A Project Plan 6-14 MS Project files generated for this exercise can be found either on the teacher’s CD- Rom or at the Instructional Support Web Site. 17. Optical Disk Preinstallation Project Next is the MS Project Entry Table for the Optical Disk Preinstallation project. The estimated completion time is 44 weeks, so yes, the project can be completed in 45 weeks if everything goes as planned. Note: Prior to assigning this exercise you should announce to the students that they should assume that the project workweek will be 5 days (Monday - Friday) and that the project is scheduled to start January 1.
- 19. Chapter 06 - Developing A Project Plan 6-15 MS Project files generated for this exercise can be found either on the teacher’s CD- Rom or at the Instructional Support Web Site. A drawn network for the optical disk project is presented below detailing the critical path.
- 20. Chapter 06 - Developing A Project Plan 6-16 Lag Exercises 18.
- 21. Chapter 06 - Developing A Project Plan 6-17 19.
- 22. Chapter 06 - Developing A Project Plan 6-18 20. 21.
- 23. Chapter 06 - Developing A Project Plan 6-19 22. CyClon Project MS Project files generated for this exercise can be found either on the teacher’s CD- Rom or at the Instructional Support Web Site. The CyClon project team has started gathering information necessary to develop a project network-predecessor activities and activity time in days. The results of their meeting are found in the following table: Activity Description Duration Predecessor 1 CyClon Project 2 Design 10 3 Procure prototype parts 10 2 4 Fabricate parts 8 2 5 Assemble prototype 4 3,4 6 Laboratory test 7 5 7 Field test 10 6 8 Adjust design 6 7 9 Order stock components 10 8 10 Order custom components 15 8 11 Assemble test production unit 10 9,10 12 Test unit 5 11 13 Document results 3 12 Note: Prior to assigning this exercise you should announce to the students that they should assume that no work is completed on weekends and that the project is scheduled to start January 2, 2008. Part A. Create a network based on the above information. How long will the project take? What is the critical path? The project is scheduled to take 80 days. The critical path consists of activities: 2, 3, 5, 6, 7, 8, 10, 11, 12, 13. CyClon Project Entry Table and Gantt Chart Part A.
- 24. Chapter 06 - Developing A Project Plan 6-20 Part B. Upon further review the team recognizes that they missed three finish- to-start lags. Procure prototype parts will only involve 2 days of work but it will take 8 days for the parts to be delivered. Likewise, Order stock components will take 2 days of work and 8 days for delivery and Order custom components 2 days of work and 13 days for delivery. Reconfigure the CyClon schedule by entering the three finish-to-start lags. What impact did these lags have on the original schedule? On the amount of work required to complete the project? The schedule still takes 80 days to complete and there is no change in the critical path. However instead of taking 98 days of work to complete, the project will only take 69 days of work! We obtain 69 days by totaling the number of days for each lag. CyClon Project Entry Table and Gantt Chart Part B. Part C. Management is not happy with the schedule and wants the project completed as soon as possible. Unfortunately, they are not willing to approve additional resources. One team member pointed out that the network contained only finish-to-start relationships and that it might be possible to reduce project duration by creating start-to-start lags. After much deliberation the team concluded that the following relationships could be converted into start-to-start lags: • Procure prototype parts could start 6 days after the start of Design. • Fabricate parts could start 9 days after the start of Design. • Laboratory test could begin 1 day after the start of Assemble prototype. • Field test could start 5 days after the start of Laboratory test. • Adjust design could begin 7 days after the start of Field test. • Order stock and Order custom components could begin 5 days after Adjust design. • Test unit could begin 9 days after the start of Assemble test production unit. • Document results could start 3 days after the start of Test unit.
- 25. Another Random Scribd Document with Unrelated Content
- 26. Descartes. [608] Harriott’s book is a thin folio of 180 pages, with very little besides examples; for his principles are shortly and obscurely laid down. Whoever is the author of the preface to this work cannot be said to have suppressed or extenuated the merits of Vieta, or to have claimed anything for Harriott but what he is allowed to have deserved. Montucla justly observes, that Harriott very rarely makes an equation equal to zero, by bringing all the quantities to one side of the equation. 18. Harriott introduced the use of small letters instead of capitals in algebra; he employed vowels for unknown, consonants for known quantities, and joined them to express their product.[609] There is certainly not much in this; but its evident convenience renders it wonderful that it should have been reserved for so late an era. Wallis, in his History of Algebra, ascribes to Harriott a long list of discoveries, which have been reclaimed for Cardan and Vieta, the great founders of the higher algebra, by Cossali and Montucla.[610] The latter of these writers has been charged, even by foreigners, with similar injustice towards our countryman; and that he has been provoked by what he thought the unfairness of Wallis to something like a depreciation of Harriott, seems as clear as that he has himself robbed Cardan of part of his due credit in swelling the account of Vieta’s discoveries. From the general integrity, however, of Montucla’s writings, I am much inclined to acquit him of any wilful partiality. [609] Oughtred, in his Clavis Mathematica, published in 1631, abbreviated the rules of Vieta, though he still used capital letters. He also gives succinctly the praxis of algebra, or the elementary rules we find in our common books, which, though what are now first learned, were, from the singular course of algebraical history, discovered late. They are, however, given also by Harriott. Wallisii Algebra. [610] These may be found in the article Harriott of the Biographia Britannica. Wallis, however, does not suppress the honour due to Vieta quite as much as is intimated by Montucla. 19. Harriott had shown what were the hidden laws of algebra, as the science of symbolical notation. But one man, the pride of France and wonder of his contemporaries, was destined to flash light upon the labours of the analyst, and to point out what
- 27. His application of algebra to curves. those symbols, so darkly and painfully traced, and resulting commonly in irrational or even impossible forms, might represent and explain. The use of numbers, or of letters denoting numbers, for lines and rectangles capable of division into aliquot parts, had long been too obvious to be overlooked, and is only a compendious abbreviation of geometrical proof. The next step made was the perceiving that irrational numbers, as they are called, represent incommensurable quantities; that is, if unity be taken for the side of a square, the square-root of two will represent its diagonal. Gradually the application of numerical and algebraical calculation to the solution of problems respecting magnitude became more frequent and refined.[611] It is certain, however, that no one before Descartes had employed algebraic formulæ in the construction of curves; that is, had taught the inverse process, not only how to express diagrams by algebra, but how to turn algebra into diagrams. The ancient geometers, he observes, were scrupulous about using the language of arithmetic in geometry, which could only proceed from their not perceiving the relation between the two; and this has produced a great deal of obscurity and embarrassment in some of their demonstrations.[612] [611] See note in Vol. II., p. 445. [612] Œuvres de Descartes, v. 323. 20. The principle which Descartes establishes is that every curve, of those which are called geometrical, has its fundamental equation expressing the constant relation between the absciss and the ordinate. Thus, the rectangle under the abscisses of a diameter of the circle is equal to the square of the ordinate, and the other conic sections, as well as higher curves, have each their leading property, which determines their nature, and shows how they may be generated. A simple equation can only express the relation of straight lines; the solution of a quadratic must be found in one of the four conic sections; and the higher powers of an unknown quantity lead to curves of a superior order. The beautiful and extensive theory developed by Descartes in
- 28. Suspected plagiarism from Harriott. this short treatise displays a most consummate felicity of genius. That such a man, endowed with faculties so original, should have encroached on the just rights of others, is what we can only believe with reluctance. 21. It must, however, be owned that independently of the suspicions of an unacknowledged appropriation of what others had thought before him, which unfortunately hang over all the writings of Descartes, he has taken to himself the whole theory of Harriott on the nature of equations in a manner which, if it is not a remarkable case of simultaneous invention, can only be reckoned a very unwarrantable plagiarism. For not only he does not name Harriott, but he evidently introduces the subject as an important discovery of his own, and in one of his letters asserts his originality in the most positive language.[613] Still it is quite possible that, prepared as the way had been by Vieta, and gifted as Descartes was with a wonderfully intuitive acuteness in all mathematical reasoning, he may in this, as in other instances, have struck out the whole theory by himself. Montucla extols the algebra of Descartes, that is, so much of it as can be fairly claimed for him without any precursor, very highly; and some of his inventions in the treatment of equations have long been current in books on that science. He was the first who showed what were called impossible or imaginary roots, though he never assigns them, deeming them no quantities at all. He was also perhaps the first who fully understood negative roots, though he still retains the appellation, false roots, which is not so good as Harriott’s epithet, privative. According to his panegyrist, he first pointed out that in every equation (the terms being all on one side) which has no imaginary roots, there are as many changes of signs as positive roots, as many continuations of them as negative. [613] Tant s’en faut que les choses que j’ai écrites puissent être aisément tirées de Viéte, qu’au contraire ce qui est cause que mon traité est difficile à entendre, c’est que j’ai tâché à n’y rien mettre que ce que j’ai crû n’avoir point été su ni par lui ni par aucun autre; comme on peut voir si on confére ce que j’ai écrit du nombre des racines qui sont en
- 29. Fermat. chaque équation, dans la page 372, qui est l’endroit où je commence à donner les règles de mon algèbre, avec ce que Viéte en a écrit tout à la fin de son livre, De Emendatione Æquationum; car on verra que je le determine généralement en toutes équations, au lieu que lui n’en aiant donné que quelques exemples particuliers, dont il fait toutefois si grand état qu’il a voulu conclure son livre par là, il a montre qu’il ne le pouvoit déterminer en général. Et ainsi j’ai commencé où il avoit achevé, ce que j’ai fait toutefois sans y penser; car j’ai plus feuilleté Viéte depuis que j’ai reçu votre dernière que je n’avois jamais fait auparavant, l’ayant trouvé ici par hasard entre les mains d’un de mes amis; et entre nous, je ne trouve pas qu’il en ait tant su que je pensois, non obstant qu’il fût fort habile. This is in a letter to Mersenne in 1637. Œuvres de Descartes, vol. vi., p. 300. The charge of plagiarism from Harriott was brought against Descartes in his lifetime: Roberval, when an English gentleman showed him the Artis Analyticæ Praxis, exclaimed eagerly, Il l’a vu! il l’a vu! It is also a very suspicious circumstance, if true, as it appears to be, that Descartes was in England the year (1631) that Harriott’s work appeared. Carcavi, a friend of Roberval, in a letter to Descartes in 1649, plainly intimates to him that he has only copied Harriott as to the nature of equations Œvres des Descartes, vol. x., p. 373. To this accusation Descartes made no reply. See Biographia Britannica, art. Harriott. The Biographie Universelle unfairly suppresses all mention of this, and labours to depreciate Harriott. See Leibnitz’s catalogue of the supposed thefts of Descartes in Vol. III., p. 267, of this work. 22. The geometer next in genius to Descartes, and perhaps nearer to him than to any third, was Fermat, a man of various acquirements, of high rank in the parliament of Toulouse, and of a mind incapable of envy, forgiving of detraction, and delighting in truth, with almost too much indifference to praise. The works of Fermat were not published till long after his death in 1665; but his frequent discussions with Descartes, by the intervention of their common correspondent Mersenne, render this place more appropriate for the introduction of his name. In these controversies Descartes never behaved to Fermat with the respect due to his talents; in fact, no one was ever more jealous of his own pre- eminence, or more unwilling to acknowledge the claims of those
- 30. Algebraic geometry not successful at first. who scrupled to follow him implicitly, and who might in any manner be thought rivals of his fame. Yet it is this unhappy temper of Descartes which ought to render us more unwilling to credit the suspicions of his designed plagiarism from the discoveries of others; since this, combined with his unwillingness to acknowledge their merits, and affected ignorance of their writings, would form a character we should not readily ascribe to a man of great genius, and whose own writings give many apparent indications of sincerity and virtue. But in fact there was in this age a great probability of simultaneous invention in science, from developing principles that had been partially brought to light. Thus Roberval discovered the same method of indivisibles as Cavalieri, and Descartes must equally have been led to this theory of tangents by that of Kepler. Fermat also, who was in possession of his principal discoveries before the geometry of Descartes saw the light, derived from Kepler his own celebrated method, de maximis et minimis; a method of discovering the greatest or least value of a variable quantity, such as the ordinate of a curve. It depends on the same principle as that of Kepler. From this he deduced a rule for drawing tangents to curves different from that of Descartes. This led to a controversy between the two geometers, carried on by Descartes, who yet is deemed to have been in the wrong, with his usual quickness of resentment. Several other discoveries, both in pure algebra and geometry, illustrate the name of Fermat.[614] [614] A good article on Fermat, by M. Maurice, will be found in the Biographie Universelle. 23. The new geometry of Descartes was not received with the universal admiration it deserved. Besides its conciseness and the inroad it made on old prejudices as to geometrical methods, the general boldness of the author’s speculations in physical and metaphysical philosophy, as well as his indiscreet temper, disinclined many who ought to have appreciated it; and it was in his own country, where he had ceased to reside, that Descartes had the fewest admirers. Roberval made some objections to his rival’s algebra, but with little success. A
- 31. Astronomy.— Kepler. commentary on the treatise of Descartes by Schooten, professor of Geometry at Leyden, first appeared in 1649. 24. Among those who devoted themselves ardently and successfully to astronomical observations at the end of the sixteenth century, was John Kepler, a native of Wirtemburg, who had already shown that he was likely to inherit the mantle of Tycho Brahe. He published some astronomical treatises of comparatively small importance in the first years of the present period. But in 1609 he made an epoch in that science by his Astronomia Nova αιτιολογητος, or Commentaries on the Planet Mars. It had been always assumed that the heavenly bodies revolve in circular orbits round their centre, whether this were taken to be the sun or the earth. There was, however, an apparent eccentricity or deviation from this circular motion, which it had been very difficult to explain, and for this Ptolemy had devised his complex system of epicycles. No planet showed more of this eccentricity than Mars; and it was to Mars that Kepler turned his attention. After many laborious researches he was brought by degrees to the great discovery, that the motion of the planets, among which, having adopted the Copernican system, he reckoned the earth, is not performed in circular but in elliptical orbits, the sun not occupying the centre but one of the foci of the curve; and, secondly, that it is performed with such a varying velocity, that the areas described by the radius vector, or line which joins this focus to the revolving planet, are always proportional to the times. A planet, therefore, moves less rapidly as it becomes more distant from the sun. These are the first and second of the three great laws of Kepler. The third was not discovered by him till some years afterwards. He tells us himself that on the 8th May, 1618, after long toil in investigating the proportion of the periodic times of the planetary movements to their orbits, an idea struck his mind, which, chancing to make a mistake in the calculation, he soon rejected. But a week after, returning to the subject, he entirely established his grand discovery, that the squares of the times of revolution are as the cubes of the mean distances of the planets. This was first made known to the world in his Mysterium
- 32. Conjectures as to comets. Galileo’s discovery of Jupiter’s satellites. Cosmo graphicum, published in 1619; a work mingled up with many strange effusions of a mind far more eccentric than any of the planets with which it was engaged. In the Epitome Astronomiæ Copernicanæ, printed the same year, he endeavours to deduce this law from his theory of centrifugal forces. He had a very good insight into the principles of universal gravitation, as an attribute of matter; but several of his assumptions as to the laws of motion are not consonant to truth. There seems indeed to have been a considerable degree of good fortune in the discoveries of Kepler; yet, this may be deemed the reward of his indefatigable laboriousness, and of the ingenuousness with which he renounced any hypothesis that he could not reconcile with his advancing knowledge of the phenomena. 25. The appearance of three comets in 1619 called once more the astronomers of Europe to speculate on the nature of those anomalous bodies. They still passed for harbingers of worldly catastrophies; and those who feared them least could not interpret their apparent irregularity. Galileo, though Tycho Brahe had formed a juster notion, unfortunately took them for atmospheric meteors. Kepler, though he brought them from the far regions of space, did not suspect the nature of their orbits, and thought that, moving in straight lines, they were finally dispersed and came to nothing. But a Jesuit, Grassi, in a treatise, De Tribus Cometis, Rome, 1618, had the honour of explaining what had baffled Galileo, and first held them to be planets moving in vast ellipses round the sun. [615] [615] The Biographie Universelle, art. Grassi, ascribes this opinion to Tycho. 26. But long before this time the name of Galileo had become immortal by discoveries which, though they would certainly have soon been made by some other, perhaps far inferior, observer, were happily reserved for the most philosophical genius of the age. Galileo assures us that, having heard of the invention of an instrument in Holland which enlarged the size of distant objects, but knowing nothing of its construction, he began to study the theory of refractions till he
- 33. found by experiment, that by means of a convex and concave glass in a tube, he could magnify an object threefold. He was thus encouraged to make another which magnified thirty times; and this he exhibited in the autumn of 1609 to the inhabitants of Venice. Having made a present of his first telescope to the senate, who rewarded him with a pension, he soon constructed another; and in one of the first nights of January, 1610, directing it towards the moon, was astonished to see her surface and edges covered with inequalities. These he considered to be mountains, and judged by a sort of measurement that some of them must exceed those of the earth. His next observation was of the milky way; and this he found to derive its nebulous lustre from myriads of stars not distinguishable through their remoteness, by the unassisted sight of man. The nebulæ in the constellation Orion he perceived to be of the same character. Before his delight at these discoveries could have subsided, he turned his telescope to Jupiter, and was surprised to remark three small stars, which, in a second night’s observation, had changed there places. In the course of a few weeks, he was able to determine by their revolutions, which are very rapid, that these are secondary planets, the moons or satellites of Jupiter; and he had added a fourth to their number. These marvellous revelations of nature he hastened to announce in a work, aptly entitled Sidereus Nuncius, published in March, 1610. In an age when the fascinating science of astronomy had already so much excited the minds of philosophers, it may be guessed with what eagerness this intelligence from the heavens was circulated. A few, as usual, through envy or prejudice, affected to contemn it. But wisdom was justified of her children. Kepler, in his Narratio de observatis a se Quatuor Jovis Satellitibus, 1610, confirmed the discoveries of Galileo. Peiresc, an inferior name, no doubt, but deserving of every praise for his zeal in the cause of knowledge, having with difficulty procured a good telescope, saw the four satellites in November, 1610, and is said by Gassendi to have conceived at that time the ingenious idea that their occultations might be used to ascertain the longitude.[616] [616] Gassendi Vita Peirescii, p. 77.
- 34. Other discoveries by him. Spots of the sun discovered. 27. This is the greatest and most important of the discoveries of Galileo. But several others were of the deepest interest. He found that the planet Venus had phases, that is, periodical differences of apparent form like the moon; and that these are exactly such as would be produced by the variable reflection of the sun’s light on the Copernican hypothesis; ascribing also the faint light on that part of the moon which does not receive the rays of the sun, to the reflection from the earth, called by some late writers earth-shine; which, though it had been suggested by Mæstlin, and before him by Leonardo da Vinci, was not generally received among astronomers. Another striking phenomenon, though he did not see the means of explaining it, was the triple appearance of Saturn, as if smaller stars were conjoined as it were like wings to the planet. This, of course, was the ring. 28. Meantime the new auxiliary of vision which had revealed so many wonders could not lie unemployed in the hands of others. A publication, by John Fabricius, at Wittenberg, in July, 1611, De Maculis in Sole visis, announced a phenomenon in contradiction of common prejudice. The sun had passed for a body of liquid flame, or, if thought solid, still in a state of perfect ignition. Kepler had, some years before, observed a spot, which he unluckily mistook for the orb of Mercury in its passage over the solar orb. Fabricius was not permitted to claim this discovery as his own. Scheiner, a Jesuit, professor of mathematics at Ingolstadt, asserts in a letter, dated 12th of November, 1611, that he first saw the spots in the month of March in that year, but he seems to have paid little attention to them before that of October. Both Fabricius, however, and Scheiner may be put out of the question. We have evidence, that Harriott observed the spots on the sun as early as December 8th, 1610. The motion of the spots suggested the revolution of the sun round its axis, completed in twenty-four days, as it is now determined; and their frequent alterations of form, as well as occasional disappearance, could only be explained by the hypothesis of a luminous atmosphere in commotion, a sea of flame,
- 35. Copernican system held by Galileo. revealing at intervals the dark central mass of the sun’s body which it envelopes. 29. Though it cannot be said, perhaps, that the discoveries of Galileo would fully prove the Copernican system of the world to those who were already insensible to reasoning from its sufficiency to explain the phenomena, and from the analogies of nature, they served to familiarise the mind to it, and to break down the strong rampart of prejudice which stood in its way. For eighty years, it has been said, this theory of the earth’s motion had been maintained without censure; and it could only be the greater boldness of Galileo in its assertion which drew down upon him the notice of the church. But, in these eighty years since the publication of the treatise of Copernicus, his proselytes had been surprisingly few. They were now becoming more numerous: several had written on that side; and Galileo had begun to form a school of Copernicans who were spreading over Italy. The Lincean society, one of the most useful and renowned of Italian academies, founded at Rome by Frederic Cesi, a young man of noble birth, in 1603, had, as a fundamental law, to apply themselves to natural philosophy; and it was impossible that so attractive and rational a system as that of Copernicus could fail of pleasing an acute and ingenious nation strongly bent upon science. The church, however, had taken alarm; the motion of the earth was conceived to be as repugnant to Scripture as the existence of antipodes had once been reckoned; and in 1616, Galileo, though respected and in favour with the court of Rome, was compelled to promise that he would not maintain that doctrine in any manner. Some letters that he had published on the subject were put, with the treatise of Copernicus and other works, into the Index Expurgatorius, where, I believe, they still remain.[617] [617] Drinkwater’s Life of Galileo. Fabroni, Vitæ Italorum, vol. i. The former seems to be mistaken in supposing that Galileo did not endeavour to prove his system compatible with Scripture. In a letter to Christina, the Grand Duchess of Tuscany, the author (Brenna) of the Life in Fabroni’s work, tells us, he argued very elaborately for that purpose. In ea
- 36. His dialogues, and persecution. videlicet epistolâ philosophus noster ita disserit, ut nihil etiam ab hominibus, qui omnem in sacrarum literarum studio consumpsissent ætatem, aut subtilius aut verius aut etiam accuratius explicatum expectari potuerit, p. 118. It seems, in fact, to have been this over- desire to prove his theory orthodox, which incensed the church against it. See an extraordinary article on this subject in the eighth number of the Dublin Review (1838). Many will tolerate propositions inconsistent with orthodoxy, when they are not brought into immediate juxtaposition with it. 30. He seems, notwithstanding this, to have flattered himself that, after several years had elapsed, he might elude the letter of this prohibition by throwing the arguments in favour of the Ptolemaic and Copernican systems into the form of a dialogue. This was published in 1632; and he might, from various circumstances, not unreasonably hope for impunity. But his expectations were deceived. It is well known that he was compelled by the Inquisition at Rome, into whose hands he fell, to retract, in the most solemn and explicit manner, the propositions he had so well proved, and which he must have still believed. It is unnecessary to give a circumstantial account, especially as it has been so well done in a recent work, the Life of Galileo, by Mr. Drinkwater Bethune. The papal court meant to humiliate Galileo, and through him to strike an increasing class of philosophers with shame and terror; but not otherwise to punish one, of whom even the inquisitors must, as Italians, have been proud; his confinement, though Montucla says it lasted for a year, was very short. He continued, nevertheless, under some restraint for the rest of his life, and though he lived at his own villa near Florence, was not permitted to enter the city.[618] [618] Fabroni. His Life is written in good Latin, with knowledge and spirit, more than Tiraboschi has ventured to display. It appears from some of Grotius’s Epistles, that Galileo had thought, about 1635, of seeking the protection of the United Provinces. But on account of his advanced age he gave this up: fessus senio constituit manere in quibus est locis, et potius quæ ibi sunt incommoda perpeti, quam malæ ætati migrandi onus, et novas parandi amicitias imponere.
- 37. Descartes alarmed by this. Progress of Copernican system. The very idea shows that he must have deeply felt the restraint imposed upon him in his country. Epist. Grot. 407, 446. 31. The church was not mistaken in supposing that she should intimidate the Copernicans, but very much so in expecting to suppress the theory. Descartes was so astonished at hearing of the sentence on Galileo, that he was almost disposed to burn his papers, or at least to let no one see them. “I cannot collect,” he says, “that he who is an Italian, and a friend of the pope, as I understand, has been criminated on any other account than for having attempted to establish the motion of the earth. I know that this opinion was formerly censured by some cardinals; but I thought I had since heard that no objection was now made to its being publicly taught even at Rome.”[619] It seems not at all unlikely that Descartes was induced, on this account, to pretend a greater degree of difference from Copernicus than he really felt, and even to deny, in a certain sense of his own, the obnoxious tenet of the earth’s motion.[620] He was not without danger of a sentence against truth nearer at hand; Cardinal Richelieu having had the intention of procuring a decree of the Sorbonne to the same effect, which, by the good sense of some of that society, fell to the ground. [621] [619] Vol. vi., p. 239. He says here, of the motion of the earth, Je confesse que s’il est faux, tous les fondemens de ma philosophie le sont aussi. [620] Vol. vi., p. 50. [621] Montucla, ii., p. 297. 32. The progress, however, of the Copernican theory in Europe, if it may not actually be dated from its condemnation at Rome, was certainly not at all slower after that time. Gassendi rather cautiously took that side; the Cartesians brought a powerful reinforcement; Bouillaud and several other astronomers of note avowed themselves favourable to a doctrine which, though in Italy it lay under the ban of the papal power, was readily saved on this side of the Alps by some of the salutary distinctions long in use to evade that authority.[622] But in
- 38. Descartes denies general gravitation. the middle of the seventeenth century, and long afterwards, there were mathematicians of no small reputation, who struggled staunchly for the immobility of the earth; and except so far as Cartesian theories might have come in vogue, we have no reason to believe that any persons unacquainted with astronomy, either in this country or on the continent, had embraced the system of Copernicus. Hume has censured Bacon for rejecting it; but if Bacon had not done so, he would have anticipated the rest of his countrymen by a full quarter of a century. [622] Id., p. 50. 33. Descartes, in his new theory of the solar system, aspired to explain the secret springs of nature, while Kepler and Galileo had merely showed their effects. By what force the heavenly bodies were impelled, by what law they were guided, was certainly a very different question from that of the orbit they described or the period of their revolution. Kepler had evidently some notion of that universally mutual gravitation which Hooke saw more clearly, and Newton established on the basis of his geometry.[623] But Descartes rejected this with contempt. “For,” he says “to conceive this we must not only suppose that every portion of matter in the universe is animated, and animated by several different souls which do not obstruct one another, but that those souls are intelligent and even divine; that they may know what is going on in the most remote places, without any messenger to give them notice, and that they may exert their powers there.”[624] Kepler, who took the world for a single animal, a leviathan that roared in caverns and breathed in the ocean tides, might have found it difficult to answer this, which would have seemed no objection at all to Campanella. If Descartes himself had been more patient towards opinions which he had not formed in his own mind, that constant divine agency, to which he was, on other occasions, apt to resort, could not but have suggested a sufficient explanation of the gravity of matter, without endowing it with self-agency. He had, however, fallen upon a complicated and original scheme; the most celebrated,
- 39. Cartesian theory of the world. perhaps, though not the most admirable, of the novelties which Descartes brought into philosophy. [623] “If the earth and moon,” he says, “were not retained in their orbits, they would fall one on another, the moon moving about 33/34 of the way, the earth the rest, supposing them equally dense.” By this attraction of the moon he accounts for tides. He compares the attraction of the planets towards the sun to that of heavy bodies towards the earth. [624] Vol.ix., p. 560. 34. In a letter to Mersenne, January 9th, 1639, he shortly states that notion of the material universe, which he afterwards published in the Principia Philosophiæ. “I will tell you,” he says, “that I conceive, or rather I can demonstrate, that besides the matter which composes terrestrial bodies, there are two other kinds; one very subtle, of which the parts are round or nearly round like grains of sand, and this not only occupies the pores of terrestrial bodies, but constitutes the substance of all the heavens; the other incomparably more subtle, the parts of which are so small and move with such velocity, that they have no determinate figure, but readily take at every instant that which is required to fill all the little intervals which the other does not occupy.”[625] To this hypothesis of a double æther he was driven by his aversion to admit any vacuum in nature; the rotundity of the former corpuscles having been produced, as he fancied, by their continual circular motions, which had rubbed off their angles. This seems at present rather a clumsy hypothesis, but it is literally that which Descartes presented to the world. [625] Vol. viii., p. 73. 35. After having thus filled the universe with different sorts of matter, he supposes that the subtler particles, formed by the perpetual rubbing off of the angles of the larger in their progress towards sphericity, increased by degrees till there was a superfluity that was not required to fill up the intervals; and this, flowing towards the centre of the system, became the sun, a very subtle and
- 40. liquid body, while in like manner, the fixed stars were formed in other systems. Round these centres the whole mass is whirled in a number of distinct vortices, each of which carries along with it a planet. The centrifugal motion impels every particle in these vortices of each instant to fly off from the sun in a straight line; but it is retained by the pressure of those which have already escaped and form a denser sphere beyond it. Light is no more than the effect of particles seeking to escape from the centre, and pressing one on another, though perhaps without actual motion.[626] The planetary vortices contain sometimes smaller vortices, in which the satellites are whirled round their principal. [626] J’ai souvent averti que par la lumière je n’entendois pas tant le mouvement que cette inclination ou propension que ces petits corps ont à se mouvoir, et que ce que je dirois du mouvement, pour être plus aisément entendu, se devoit rapporter à cette propension; d’où il est manifeste qua selon moi l’on ne doit entendre autre chose par les couleurs que les différentes variétés qui arrivent en ces propensions. Vol. vii., p. 193. 36. Such, in a few words, is the famous Cartesian theory, which, fallen in esteem as it now is, stood its ground on the continent of Europe, for nearly a century, till the simplicity of the Newtonian system, and, above all, its conformity to the reality of things, gained an undisputed predominance. Besides the arbitrary suppositions of Descartes, and the various objections that were raised against the absolute plenum of space and other parts of his theory, it has been urged that his vortices are not reconcilable, according to the laws of motion in fluids, with the relation, ascertained by Kepler, between the periods and distances of the planets; nor does it appear why the sun should be in the focus, rather than in the centre of their orbits. Yet, within a few years it has seemed not impossible, that a part of his bold conjectures will enter once more with soberer steps into the schools of philosophy. His doctrine as to the nature of light, improved as it was by Huygens, is daily gaining ground over that of Newton; that of a subtle æther pervading space, which in fact is nearly the same thing, is becoming a favourite speculation, if we are
- 41. Transits of Mercury and Venus. Laws of Mechanics. Statics of Galileo. not yet to call it an established truth; and the affirmative of a problem, which an eminent writer has started, whether this æther has a vorticose motion round the sun, would not leave us very far from the philosophy it has been so long our custom to turn into ridicule. 37. The passage of Mercury over the sun was witnessed by Gassendi in 1631. This phenomenon, though it excited great interest in that age, from its having been previously announced, so as to furnish a test of astronomical accuracy, recurs too frequently to be now considered as of high importance. The transit of Venus is much more rare. It occurred on December 4, 1639, and was then only seen by Horrox, a young Englishman of extraordinary mathematical genius. There is reason to ascribe an invention of great importance, though not perhaps of extreme difficulty, that of the micrometer, to Horrox. 38. The satellites of Jupiter and the phases of Venus are not so glorious in the scutcheon of Galileo as his discovery of the true principles of mechanics. These, as we have seen in the former volume, were very imperfectly known till he appeared; nor had the additions to that science since the time of Archimedes been important. The treatise of Galileo, Della Scienza Mecanica, has been said, I know not on what authority, to have been written in 1592. It was not published, however, till 1634, and then only in a French translation by Mersenne, the original not appearing till 1649. This is chiefly confined to statics, or the doctrine of equilibrium; it was in his dialogues on motion, Della Nuova Scienza, published in 1638, that he developed his great principles of the science of dynamics, the moving forces of bodies. Galileo was induced to write his treatise on mechanics, as he tells us, in consequence of the fruitless attempts he witnessed in engineers to raise weights by a small force, “as if with their machines they could cheat nature, whose instinct as it were by fundamental law is that no resistance can be overcome except by a superior force.” But as one man may raise a weight to the height of
- 42. His Dynamics. a foot by dividing it into equal portions, commensurate to his power, which many men could not raise at once, so a weight, which raises another greater than itself, may be considered as doing so by successive instalments of force, during each of which it traverses as much space as a corresponding portion of the larger weight. Hence the velocity, of which space uniformly traversed in a given time is the measure, is inversely as the masses of the weights; and thus the equilibrium of the straight lever is maintained, when the weights are inversely as their distance from the fulcrum. As this equilibrium of unequal weights depends on the velocities they would have if set in motion, its law has been called the principle of virtual velocities. No theorem has been of more important utility to mankind. It is one of those great truths of science, which combating and conquering enemies from opposite quarters, prejudice and empiricism, justify the name of philosophy against both classes. The waste of labour and expense in machinery would have been incalculably greater in modern times, could we imagine this law of nature not to have been discovered; and as their misapplication prevents their employment in a proper direction, we owe in fact to Galileo the immense effect which a right application of it has produced. It is possible, that Galileo was ignorant of the demonstration given by Stevinus of the law of equilibrium in the inclined plane. His own is different; but he seems only to consider the case when the direction of the force is parallel to that of the plane. 39. Still less was known of the principles of dynamics than of those of statics, till Galileo came to investigate them. The acceleration of falling bodies, whether perpendicularly or on inclined planes, was evident; but in what ratio this took place, no one had succeeded in determining, though many had offered conjectures. He showed that the velocity acquired was proportional to the time from the commencement of falling. This might now be demonstrated from the laws of motion; but Galileo, who did not perhaps distinctly know them, made use of experiment. He then proved by reasoning that the spaces traversed in falling were as the squares of the times or velocities; that their increments in equal times were as the uneven
- 43. numbers, 1, 3, 5, 7, and so forth; and that the whole space was half what would have been traversed uniformly from the beginning with the final velocity. These are the great laws of accelerated and retarded motion, from which Galileo deduced most important theorems. He showed that the time in which bodies roll down the length of inclined planes is equal to that in which they would fall down the height, and in different planes is proportionate to the height; and that their acquired velocity is in the same ratios. In some propositions he was deceived; but the science of dynamics owes more to Galileo than to any one philosopher. The motion of projectiles had never been understood; he showed it to be parabolic; and in this he not only necessarily made use of a principle of vast extent, that of compound motion, which, though it is clearly mentioned in one passage by Aristotle[627] and may probably be implied in the mechanical reasonings of others, does not seem to have been explicitly laid down by modern writers, but must have seen the principle of curvilinear deflection by forces acting in infinitely small portions of time. The ratio between the times of vibration in pendulums of unequal length, had early attracted Galileo’s attention. But he did not reach the geometrical exactness of which this subject is capable.[628] He developed a new principle as to the resistance of solids to the fracture of their parts, which, though Descartes as usual treated it with scorn, is now established in philosophy. “One forms, however,” says Playfair, “a very imperfect idea of this philosopher from considering the discoveries and inventions, numerous and splendid as they are, of which he was the undisputed author. It is by following his reasonings, and by pursuing the train of his thoughts, in his own elegant, though somewhat diffuse exposition of them, that we become acquainted with the fertility of his genius, with the sagacity, penetration, and comprehensiveness of his mind. The service which he rendered to real knowledge is to be estimated not only from the truths which he discovered, but from the errors which he detected; not merely from the sound principles which he established, but from the pernicious idols which he overthrew. Of all the writers who have lived in an age which was yet only emerging from ignorance and barbarism, Galileo
- 44. Mechanics of Descartes. has most entirely the tone of true philosophy, and is most free from any contamination of the times, in taste, sentiment, and opinion.”[629] [627] Drinkwater’s Life of Galileo, p. 80. [628] Fabroni. [629] Preliminary Dissertation to Encyclop. Britain. 40. Descartes, who left nothing in philosophy untouched, turned his acute mind to the science of mechanics, sometimes with signal credit, sometimes very unsuccessfully. He reduced all statics to one principle, that it requires as much force to raise a body to a given height, as to raise a body of double weight to half the height. This is the theorem of virtual velocities in another form. In many respects he displays a jealousy of Galileo, and an unwillingness to acknowledge his discoveries, which puts himself often in the wrong. “I believe,” he says, “that the velocity of very heavy bodies which do not move very quickly in descending increases nearly in a duplicate ratio; but I deny that this is exact, and I believe that the contrary is the case when the movement is very rapid.”[630] This recourse to the air’s resistance, a circumstance of which Galileo was well aware, in order to diminish the credit of a mathematical theorem, is unworthy of Descartes; but it occurs more than once in his letters. He maintained also, against the theory of Galileo, that bodies do not begin to move with an infinitely small velocity, but have a certain degree of motion at the first instance, which is afterwards accelerated.[631] In this too, as he meant to extend his theory to falling bodies, the consent of philosophers has decided the question against him. It was a corollary from these notions that he denies the increments of spaces to be according to the progression of uneven numbers.[632] Nor would he allow that the velocity of a body augments its force, though it is a concomitant.[633] [630] Œuvres de Descartes, vol. viii., p. 24. [631] Il faut savoir, quoique Galilée et quelques autres disent au contraire, que les corps qui commencent à descendre, ou à se mouvoir en quelque façon que ce soit, ne passent point par tous les degrés de
- 45. Law of motion laid down by Descartes. tardiveté; mais que des le premier moment ils ont certaine vitesse qui s’augmente après de beaucoup, et c’est de cette augmentation que vient la force de la percussion. viii., 181. [632] Cette proportion d’augmentation selon les nombres impairs, 1, 3, 5, 7, &c., qui est dans Galilée et que je crois vous avoir aussi écrite autrefois, ne peut être vraie, qu’en supposant deux ou trois choses qui sont très fausses, dont l’une est que le mouvement croisse par degrés depuis le plus lent, ainsi que le songe Galilée, et l’autre que la résistance de l’air n’empêche point. Vol. ix., p. 349. [633] Je pense que la vitesse n’est pas la cause de l’augmentation de la force, encore qu’elle l’accompagne toujours. Id. p. 356, See also vol. viii., p. 14. He was probably perplexed by the metaphysical notion of causation, which he knew not how to ascribe to mere velocity. The fact that increased velocity is a condition or antecedent of augmented force could not be doubted. 41. Descartes, however, is the first who laid down the laws of motion; especially that all bodies persist in their present state of rest or uniform rectilineal motion till affected by some force. Many had thought, as the vulgar always do, that a continuance of rest was natural to bodies, but did not perceive that the same principle of inertia or inactivity was applicable to them in rectilineal motion. Whether this is deducible from theory, or depends wholly on experience, by which we ought to mean experiment, is a question we need not discuss. The fact, however, is equally certain; and hence Descartes inferred that every curvilinear deflection is produced by some controlling force, from which the body strives to escape in the direction of a tangent to the curve. The most erroneous part of his mechanical philosophy is contained in some propositions as to the collision of bodies, so palpably incompatible with obvious experience that it seems truly wonderful he could ever have adopted them. But he was led into these paradoxes by one of the arbitrary hypotheses which always governed him. He fancied it a necessary consequence from the immutability of the divine nature that there should always be the same quantity of motion in the universe; and rather than abandon this singular assumption he did not hesitate to assert, that two hard
- 46. Also those of compound forces. Other discoveries in mechanics. bodies striking each other in opposite directions would be reflected with no loss of velocity; and, what is still more outrageously paradoxical, that a smaller body is incapable of communicating motion to a greater; for example, that the red billiard-ball cannot put the white into motion. This manifest absurdity he endeavoured to remove by the arbitrary supposition, that when we see, as we constantly do, the reverse of his theorem take place, it is owing to the air, which, according to him, renders bodies more susceptible of motion, than they would naturally be. 42. Though Galileo, as well as others, must have been acquainted with the laws of the composition of moving forces, it does not appear that they had ever been so distinctly enumerated as by Descartes, in a passage of his Dioptrics. [634] That the doctrine was in some measure new may be inferred from the objections of Fermat; and Clerselier, some years afterwards, speaks of persons “not much versed in mathematics, who cannot understand an argument taken from the nature of compound motion.”[635] [634] Vol. v., p. 18. [635] Vol. vi., p. 508. 43. Roberval demonstrated what seems to have been assumed by Galileo, that the forces on an oblique or crooked lever balance each other, when they are inversely as the perpendiculars drawn from the centre of motion to their direction. Fermat, more versed in geometry than physics, disputed this theorem which is now quite elementary. Descartes, in a letter to Mersenne, ungraciously testifies his agreement with it.[636] Torricelli, the most illustrious disciple of Galileo, established that when weights balance each other in all positions, their common centre of gravity does not ascend or descend, and conversely. [636] Je suis de l’opinion, says Descartes, de ceux qui disent que pondera sunt in æquilibrio quando sunt in ratione reciproca linearum perpendicularium, &c., vol. xi., p. 357. He would not name Roberval;
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