YALE
UNIVERSITY
LIBRARY
THE
ENGLISH WORKS
OF
THOMAS HOBBES
OF MALMESBURY;
NOW FIRST COLLECTED AND EDITED
BY
SIR WILLIAM MOLESWORTH, BART.
VOL. I.
LONDON:
JOHN BOHN,
HENRIETTA STREET, COVENT GARDEN.
MDCCCXXXIX.
LONDON:
C. RICHARDS, PRINTER, ST. MARTIN'S LANE.
TO
GEORGE GROTE, ESQ.
M.P. FOR THE CITY OF LONDON.
Dear Grote,
I dedicate to you this edition of
the Works of Hobbes ; first, because I know
you will be well pleased to see a complete
collection of all the writings of an Author
for whom you have so high an admiration.
Secondly, because I am indebted to you for
my first acquaintance with the speculations of
one of the greatest and most original thinkers
in the English language, whose works, I have
often heard you regret, were so scarce, and so
much less read and studied than they deserved
to be. It now, therefore, gives me great satis-
dedication.
faction to be able to gratify a wish, you have fre¬
quently expressed, that some person, who had
time and due reverence for that illustrious
man, would undertake to edite his works, and
bring his views again before his countrymen,
who have so long and so unjustly neglected
him. And likewise, I am desirous, in some
way, to express the sincere regard and respect
that I feel for you, and the gratitude that I
owe you for the valuable instruction, that I have
obtained from your society, and from the
friendship with which you have honoured me,
during the many years we have been com¬
panions in political life.
Yours, truly,
William Molesworth.
February 25th, 1839.
79, Eaton Square, London.
ELEMENTS OF PHILOSOPHY.
THE FIRST SECTION,
CONCERNING BODY,
WRITTEN IN LATIN
BY
THOMAS HOBBES OF MALMESBURY,
AND
TRANSLATED INTO ENGLISH.
THE
TRANSLATOR TO THE READER,
If, when I had finished ray translation of this first section of
the Elements of Philosophy, I had presently committed the
same to the press, it might have come to your hands sooner
than now it doth. But as I undertook it with much diffidence
of my own ability to perform it well; so I thought fit, before
I published it, to pray Mr. Hobbes to view, correct, and order
it according to his own mind and pleasure. Wherefore, though
you find some places enlarged, others altered, and two chapters,
xvm and xx, almost wholly changed, you may nevertheless
remain assured, that as now I present it to you, it doth not at
all vary from the author's own sense and meaning. As for
the Six Lessons to the Savilian Professors at Oxford, they are
not of my translation, but were written, as here you have
them in English, by Mr. Hobbes himself; and are joined to
this book, because they are chiefly in defence of the same.*
* They will he published in a separate volume, with other works of a
similar description. W. M.
THE AUTHOR'S EPISTLE DEDICATORY,
TO THE
RIGHT HONORABLE, MY MOST HONORED LORD,
WILLIAM, EARL OF DEVONSHIRE.
This first section of the Elements of Philosophy, the
monument of my service and your Lordship's bounty,
though, after the Third Section published, long de¬
ferred, yet at last finished, I now present, my most
excellent Lord, and dedicate to your Lordship. A
little book, but full; and great enough, if men count
well for great; and to an attentive reader versed in
the demonstrations of mathematicians, that is, to
your Lordship, clear and easy to understand, and
almost new throughout, without any offensive novelty.
I know that that part of philosophy, wherein are
considered lines and figures, has been delivered to
us notably improved by the ancients; and withal a
most perfect pattern of the logic by which they were
enabled to find out and demonstrate such excellent
theorems as they have done. I know also that the
Vlll
EPISTLE DEDICATORY.
hypothesis of the earth's diurnal motion was the
invention of the ancients ; but that both it, and
astronomy, that is, celestial physics, springing up
together with it, were by succeeding philosophers
strangled with the snares of words. And therefore
the beginning of astronomy, except observations, I
think is not to be derived from farther time than from
Nicolaus Copernicus ; who in the age next preceding
the present revived the opinion of Pythagoras, Aris-
tarchus, and Philolaus. After him, the doctrine of
the motion of the earth being now received, and a dif¬
ficult question thereupon arising concerning the de¬
scent of heavy bodies, Galileus in our time, striving
with that difficulty, was the first that opened to us the
gate of natural philosophy universal, which is the
knowledgeof the nature of motion. So that neither can
the age of natural philosophy be reckoned higher than
to him. Lastly, the science of mans body, the most
profitable part of natural science, was first discovered
with admirable sagacity by our countryman Doctor
Harvey, principal Physician to King James and King
Charles, in his books of the Motion of the Blood,
and of the Generation of Living Creatures ; who is
the only man I know, that conquering envy, hath
established a new doctrine in his life-time. Before
these, there was nothing certain in natural philosophy
EPISTLE DEDICATORY.
ix
but every man's experiments to himself, and the
natural histories, if they may be called certain, that
are no certainer than civil histories. But since these,
astronomy and natural philosophy in general have,
for so little time, been extraordinarily advanced by
Joannes Keplerus, Petrus Gassendus, and Marinus
Mersennus; and the science of human bodies in
special by the wit and industry of physicians, the
only true natural philosophers, especially of our most
learned men of the College of Physicians in London.
Natural Philosophy is therefore but young; but
Civil Philosophy yet much younger, as being no older
(I say it provoked, and that my detractors may know
how little they have wrought upon me) than my own
book De Cive. But what ? were there no philoso¬
phers natural nor civil among the ancient Greeks ?
There were men so called; witness Lucian, by whom
they are derided; witness divers cities, from which
they have been often by public edicts banished. But
it follows not that there was philosophy. There
walked in old Greece a certain phantasm, for super¬
ficial gravity, though full within of fraud and filth, a
little like philosophy; which unwary men, thinking
to be it, adhered to the professors of it, some to one,
some to another, though they disagreed among them¬
selves, and with great salary put their children to
X EPISTLE DEDICATORY.
tliem to be taught, instead of wisdom, nothing but
to dispute, and, neglecting the laws, to determine
every question according to their own fancies. The
first doctors of the Church, next the Apostles, born
in those times, whilst they endeavoured to defend
the Christian faith against the Gentiles by natural
reason, began also to make use of philosophy, and
with the decrees of Holy Scripture to mingle the
sentences of heathen philosophers ; and first some
harmless ones of Plato, but afterwards also many
foolish and false ones out of the physics and meta¬
physics of Aristotle ; and bringing in the enemies,
betrayed unto them the citadel of Christianity. From
that time, instead of the worship of God, there entered
a thing called school divinity, walking on one foot
firmly, which is the Holy Scripture, but halted on
the other rotten foot, which the Apostle Paul called
vain, and might have called pernicious philosophy;
for it hath raised an infinite number of controversies
in the Christian world concerning religion, and from
those controversies, wars. It is like that Empusa in
the Athenian comic poet, which was taken in Athens
for a ghost that changed shapes, having one brazen
leg, but the other was the leg of an ass, and was sent,
as was believed, by Hecate, as a sign of some ap¬
proaching evil fortune. Against this Empusa I think
EPISTLE DEDICATORY.
xi
there cannot be invented a better exorcism, than to
distinguish between the rules of religion, that is, the
rules of honouring God, which we have from the
laws, and the rules of philosophy, that is, the opi¬
nions of private men; and to yield what is due to
religion to the Holy Scripture, and what is due to
philosophy to natural reason. And this I shall do,
if I but handle the Elements of Philosophy truly and
clearly, as I endeavour to do. Therefore having in
the Third Section, which I have published and dedi¬
cated to your Lordship, long since reduced all power
ecclesiastical and civil by strong arguments of reason,
without repugnance to God's word, to one and the
same sovereign authority ; I intend now, by putting
into a clear method the true foundations of natural
philosophy, to fright and drive away this metaphy¬
sical Empusa; not by skirmish, but by letting in the
light upon her. For I am confident, if any con¬
fidence of a writing can proceed from the writer's
fear, circumspection, and diffidence, that in the
three former parts of this book all that I have said
is sufficiently demonstrated from definitions ; and all
in the fourth part from suppositions not absurd.
But if there appear to your Lordship anything less
fully demonstrated than to satisfy every reader, the
cause was this, that I professed to write not all to
xii
EPISTLE DEDICATORY.
all, but some things to geometricians only. But that
your Lordship will be satisfied, I cannot doubt.
There remains the second section, which is con¬
cerning Man. That part thereof, where I handle the
Optics, containing six chapters, together with the
tables of the figures belonging to them, I have already
written and engraven lying by me above these six
years. The rest shall, as soon as I can, be added to
it; though by the contumelies and petty injuries of
some unskilful men, I know already, by experience,
how much greater thanks will be due than paid me,
for telling men the truth of what men are. But the
burthen I have taken 011 me I mean to carry through;
not striving to appease, but rather to revenge myself
of envy, by encreasing it. For it contents me that
I have your Lordship's favour, which, being all you
require, I acknowledge ; and for which, with my
prayers to Almighty God for your Lordship's safety,
I shall, to my power, be always thankful.
Your Lordship's most humble servant,
London,
April 23, 1655.
THOMAS HOBBES,
the
AUTHOR'S EPISTLE TO THE READER.
Think not, Courteous Header, that the philosophy, the
elements whereof I am going to set in order, is that which
makes philosophers' stones, nor that which is found in the
metaphysic codes; but that it is the natural reason of man,
busily flying up and down among the creatures, and bringing
back a true report of their order, causes and effects. Phi¬
losophy, therefore, the child of the world and your own mind,
is within yourself; perhaps not fashioned yet, but like the
world its father, as it was in the beginning, a thing confused.
Do, therefore, as the statuaries do, who, by hewing off that
which is superfluous, do not make but find the image. Or imi¬
tate the creation: if you will be a philosopher in good earnest,
let your reason move upon the deep of your own cogitations
and experience; those things that lie in confusion must be set
asunder, distinguished, and every one stamped with its own
name set in order; that is to say, your method must re¬
semble that of the creation. The order of the creation was,
light, distinction of day and niglit, the firmament, the lumi¬
naries, sensible creatures, man ; and, after the creation, the
commandment. Therefore the order of contemplation will
be, reason, definition, space, the stars, sensible quality,
man; and after man is grown up, subjection to command.
In the first part of this section, which is entitled Logic, I set
up the light of reason. In the second, which hath for title
author's epistle to the reader.
the Grounds of Philosophy, I distinguish the most common
notions by accurate definition, for the avoiding of confusion
and obscurity. The third part concerns the expansion of
space, that is Geometry. The fourth contains the Motion of
the Stars, together with the doctrine of sensible qualities.
In the second section, if it please God, shall be handled
Man. In the third section, the doctrine of Subjection is handled
already. This is the method I followed; and if it like you,
you may use the same; for I do but propound, not commend
to you anything of mine. But whatsoever shall be the
method you will like, I would very fain commend philosophy
to you, that is to say, the study of wisdom, for want of which
we have all suffered much damage lately. For even they, that
study wealth, do it out of love to wisdom; for their treasures
serve them but for a looking-glass, wherein to behold and
contemplate their own wisdom. Nor do they, that love to be
employed in public business, aim at anything but place
wherein to show their wisdom. Neither do voluptuous men
neglect philosophy, but only because they know not how great
a pleasure it is to the mind of man to be ravished in the
vigorous and perpetual embraces of the most beauteous world.
Lastly, though for nothing else, yet because the mind of man
is no less impatient of empty time than nature is of empty
place, to the end you be not forced for want of what to do, to
be troublesome to men that have business, or take hurt by
falling into idle company, but have somewhat of your own
wherewith to fill up your time, I recommend unto you to
study philosophy. Farewell.
T. H.
TITLES OF THE CHAPTERS.
PART FIRST,
OK LOGIC.
CHAP. PAGE.
1. Of Philosophy ........ 1
2. Of Names ....... 13
3. Of Proposition ........ 29
4. Of Syllogism ........ 44
5. Of Erring, Falsity, and Captions .... 55
6. Of Method 65
PART SECOND,
OR THE FIRST GROUNDS OP FHILOSOPHY.
7. Of Place and Time 91
8. Of Body and Accident 101
9. Of Cause and Effect . . • . . . .120
10. Of Power and Act . . . . . . .127
11. Of Identity and Difference ...... 132
12. Of Quantity. 138
13. Of Analogism, or the Same Proportion . . .144
14. Of Strait and Crooked, Angle and Figure . . . 176
TITLES OF THE CHAPTERS.
PART THIRD,
OF THE PROPORTIONS OF MOTIONS AND MAGNITUDES.
CHAP. PAGE.
15. Of the Nature, Properties, and divers Considerations of
Motion and Endeavour ...... 203
16. Of Motion Accelerated and Uniform, and of Motion by.
Concourse ........ 218
17. Of Figures Deficient ....... 246
18. Of the Equation of Strait Lines with the Crooked Lines
of Parabolas, and other Figures made in imitation of
Parabolas ....... 268
19. Of Angles of Incidence and Reflection, equal by Suppo¬
sition ......... 273
20. Of the Dimension of a Circle, and the Division of Angles
or Arches ........ 287
21. Of Circular Motion ....... 317
22. Of other Variety of Motions ..... 333
23. Of the Centre of Equiponderation of Bodies pressing
downwards in Strait Parallel Lines .... 350
24. Of Refraction and Reflection ..... 374
PART FOURTH,
OF PHYSICS, OR THE PHENOMENA OF NATURE.
25. Of Sense and Animal Motion ..... 387
26. Of the World and of the Stars 410
27. Of Light, Heat, and of Colours ..... 445
28. Of Cold, Wind, Hard, Ice, Restitution of Bodies bent,
Diaphanous, Lightning and Thunder, and of the
Heads of Rivers 466
29. Of Sound, Odour, Savour, and Touch . . . 485
30. Of Gravity 508
* I
-J
L.
\ t <4^ er^S^Cf
ocS^ j t«- J
; Is V^Uy -^y"^r^C.
S|N J^t^T ^M^t/yX'^c^' /rVf^^
Q-{^UaJjU /~rir^ 0K^^jf ^>^sUv~~ /£ a^)
f^As Ocuy^t^^^' & ^ US-£^vJ~1 / f^n C^ C6^Aj*^C
I*
( 4 . ^
"X N^
1^
\
COMPUTATION OR LOGIC.
CHAPTER I.
of philosophy.
1. The Introduction.—2. The Definition of Philosophy ex¬
plained.—3. Ratiocination of the Mind.—4. Properties, what
they are.—5. How Properties are known by Generation, and
contrarily.—6. The Scope of Philosophy.—7. The Utility of
it.—8. The Subject.—9. The Parts of it.—10. The Epilogue.
Philosophy seems to me to be amongst men now, part i.
in the same manner as corn and wine are said to —i -
have been in the world in ancient time. For from Introductlon-
the beginning there were vines and ears of corn
growing here and there in the fields; bnt no care
was taken for the planting and sowing of them.
Men lived therefore upon acorns ; or if any were
so bold as to venture upon the eating of those
unknown and doubtful fruits, they did it with dan¬
ger of their health. In like manner, every man
brought Philosophy, that is, Natural Reason, into
the world with him; for all men can reason to
some degree, and concerning some things: but
where there is need of a long series of reasons,
there most men wander out of the way, and fall
into error for want of method, as it were for want
vol. i. b
2
computation or logic.
part i. of sowing and planting, that is, of improving their
-—^—- reason. And from hence it comes to pass, that
introduction, they who content themselves with daily experience,
which may be likened to feeding upon acorns, and
either reject, or not much regard philosophy, are
commonly esteemed, and are, indeed, men of
sounder judgment than those who, from opinions,
though not vulgar, yet full of uncertainty, and
carelessly received, do nothing but dispute and
wrangle, like men that are not well in their wits.
I confess, indeed, that that part of philosophy by
which magnitudes and figures are computed, is
highly improved. But because I have not observed
the like advancement in the other parts of it, my
purpose is, as far forth as I am able, to lay open
the few and first Elements of Philosophy in gene¬
ral, as so many seeds from which pure and true
Philosophy may hereafter spring up by little and
little.
I am not ignorant how hard a thing it is to
weed out of men's minds such inveterate opinions
as have taken root there, and been confirmed in
them by the authority of most eloquent writers;
especially seeing true (that is, accurate) Philosophy
professedly rejects not only the paint and false
colours of language, but even the very ornaments
and graces of the same; and the first grounds of
all science are not only not beautiful, but poor,
arid, and, in appearance, deformed. Nevertheless,
there being certainly some men, though but few,
who are delighted with truth and strength of rea¬
son in all things, I thought I might do well to take
this pains for the sake even of those few. I proceed
therefore to the matter, and take my beginning
of philosophy.
3
from the very definition of philosophy, which is part i.
this. -—1;—-
2. Philosophy is such knowledge of effects or Definition of
7 ° J . . . Philosophy
appearances, as we acquire by true ratiocination explained.
from the knowledge we hare first of their causes
or generation: And again, of such causes or gene¬
rations as may be from knowing first their effects.
For the better understanding of which definition,
we must consider, first, that although Sense and
Memory of things, which are common to man and
all living creatures, be knowledge, yet because they
are given us immediately by nature, and not gotten
by ratiocination, they are not philosophy.
Secondly, seeing Experience is nothing but me¬
mory ; and Prudence, or prospect into the future
time, nothing but expectation of such things as
we have already had experience of, Prudence also
is not to be esteemed philosophy.
By ratiocination, I mean computation. Now
to compute, is either to collect the sum of many
things that are added together, or to know what
remains when one thing is taken out of another.
Ratiocination, therefore, is the same with addition
and substraction ; and if any man add multiplica¬
tion and division, I will not be against it, seeing
multiplication is nothing but addition of equals one
to another, and division nothing but a substraction
of equals one from another, as often as is possible.
So that all ratiocination is comprehended in these
two operations of the mind, addition and substrac¬
tion.
3. But how by the ratiocination of our mind, Ratiocination
i 1 • .i , , of the Mind.
we add and substract m our silent thoughts, with¬
out the use of words, it will be necessary for me
b 2
4
COMPUTATION OR LOGIC.
part i. to make intelligible by an example or two. If
—r—- therefore a man see something afar off and ob-
scurely, although no appellation had yet been given
to anything, he will, notwithstanding, have the
same idea of that thing for which now, by im¬
posing a name on it, we call it body. Again, when,
by coming nearer, he sees the same thing thus and
thus, now in one place and now in another, he
will have a new idea thereof, namely, that for
which we now call such a thing animated. Thirdly,
when standing nearer, he perceives the figure,
hears the voice, and sees other things which are
signs of a rational mind, he has a third idea,
though it have yet 110 appellation, namely, that for
which we now call anything rational. Lastly,
when, by looking fully and distinctly upon it, he
conceives all that he has seen as one thing, the
idea he has now is compounded of his former ideas,
which are put together in the mind in the same
order in which these three single names, body,
animated, rational, are in speech compounded into
this one name, body-animated-rational, or man.
In like manner, of the several conceptions of four
sides, equality of sides, and right angles, is com¬
pounded the conception of a square. For the
mind may conceive a figure of four sides without
any conception of their equality, and of that equa¬
lity without conceiving a right angle; and may
join together all these single conceptions into one
conception or one idea of a square. And thus we
see how the conceptions of the mind are com¬
pounded. Again, whosoever sees a man standing
near him, conceives the whole idea of that man;
and if, as he goes away, he follow him with his
of philosophy.
eyes only, he will lose the idea of those things part i.
which were signs of his being rational, whilst, -—i—■
nevertheless, the idea of a body-animated remains
still before his eyes, so that the idea of rational is
substracted from the whole idea of man, that is to
say, of body-animated-rational, and there remains
that of body-animated; and a while after, at a
greater distance, the idea of animated will be lost,
and that of body only will remain ; so that at last,
when nothing at all can be seen, the whole idea
will vanish out of sight. By which examples, I
think, it is manifest enough what is the internal
ratiocination of the mind without words.
We must not therefore think that computation,
that is, ratiocination, has place only in numbers,
as if man were distinguished from other living
creatures (which is said to have been the opinion
of Pythagoras) by nothing but the faculty of num¬
bering ; for magnitude, body, motion, time, degrees ^
of quality, action, conception, proportion, speech •
and names (in which all the kinds of philosophy
consist) are capable of addition and substraction.
Now such things as we add or substract, that is,
which we put into an account, we are said to con¬
sider, in Greek XoyilzaOai, in which language also
avWoylfeadai signifies to compute, reason, or reckon.
4. But effects and the appearances of things to Properties,
sense, are faculties or powers of bodies, whichwhat they are"
make us distinguish them from one another ; that
is to say, conceive one body to be equal or un¬
equal, like or unlike to another body ; as in the
example above, when by coming near enough to
any body, we perceive the motion and going of
the same, we distinguish it thereby from a tree, a
6
COMPUTATION OR LOGIC.
part i. column, and other fixed bodies ; and so that motion
—l'r—-- or going is the property thereof, as being proper
to living creatures, and a faculty by which they
make us distinguish them from other bodies.
i! ,wProperties 5. How the knowledge of any effect may be
g t- lie rati on^ gotten from the knowledge of the generation
",ul contranly- thereof, may easily be understood by the example
of a circle: for if there be set before us a plain
figure, having, as near as may be, the figure of a
circle, we cannot possibly perceive by sense whe¬
ther it be a true circle or no ; than which, never¬
theless, nothing is more easy to be known to him
that knows first the generation of the propounded
figure. For let it be known that the figure was
made by the circumduction of a body whereof one
end remained unmoved, and we may reason thus;
a body carried about, retaining always the same
length, applies itself first to one radius, then to
another, to a third, a fourth, and successively to
all; and, therefore, the same length, from the same
point, toucheth the circumference in every part
thereof, which is as much as to say, as all the radii
are equal. We know, therefore, that from such
generation proceeds a figure, from whose one
middle point all the extreme points are reached
unto by equal radii. And in like manner, by
knowing first what figure is set before us, we may
come by ratiocination to some generation of the
same, though perhaps not that by which it was
made, yet that by which it might have been made;
for he that knows that a circle has the property
above declared, will easily know whether a body
carried about, as is said, will generate a circle or
no.
of philosophy.
7
6. The end or scope of philosophy is, that we part i.
may make use to our benefit of effects formerly - L
seen; or that, by application of bodies to one scope of
another, we may produce the like effects of those Philosophy"
we conceive in our mind, as far forth as matter,
strength, and industry, will permit, for the com¬
modity of human life. For the inward glory and
triumph of mind that a man may have for the mas¬
tering of some difficult and doubtful matter, or for
the discovery of some hidden truth, is not worth
so much pains as the study of Philosophy requires ;
nor need any man care much to teach another
what he knows himself, if he think that will be the
only benefit of his labour. The end of knowledge
is power ; and the use of theorems (which, among
geometricians, serve for the finding out of proper¬
ties) is for the construction of problems; and,
lastly, the scope of all speculation is the perform¬
ing of some action, or thing to be done.
7. But what the utility of philosophy is, espe- utility of
cially of natural philosophy and geometry, will be 1
best understood by reckoning up the chief com¬
modities of which mankind is capable, and by
comparing the manner of life of such as enjoy
them, with that of others which want the same.
Now, the greatest commodities of mankind are the
arts ; namely, of measuring matter and motion ; of
moving ponderous bodies; of architecture; of
navigation; of making instruments for all uses ;
of calculating the celestial motions, the aspects of
the stars, and the parts of time ; of geography, &c.
By which sciences, how great benefits men receive
is more easily understood than expressed. These
benefits are enjoyed by almost all the people of
8
computation or logic.
part i. Europe, by most of those of Asia, and by some of
>—i—- Africa: but the Americans, and they that live near
Philosophy ^ie P°les> d° totally want them. But why ? Have
they sharper wits than these ? Have not all men
one kind of soul, and the same faculties of mind ?
What, then, makes this difference, except philo¬
sophy ? Philosophy, therefore, is the cause of all
these benefits. But the utility of moral and civil
philosophy is to be estimated, not so much by the
commodities we have by knowing these sciences,
as by the calamities we receive from not knowing
them. Now, all such calamities as may be avoided
by human industry, arise from war, but chiefly
from civil war ; for from this proceed slaughter,
solitude, and the want of all things. But the cause
of war is not that men are willing to have it; for
the will has nothing for object but good, at least
that which seemeth good. Nor is it from this,
that men know not that the effects of war are
evil; for who is there that thinks not poverty
and loss of life to be great evils ? The cause,
therefore, of civil war is, that men koow not the
causes neither of war nor peace, there being but
few in the world that have learned those duties
which unite and keep men in peace, that is to say,
that have learned the rules of civil life sufficiently.
Now, the knowledge of these rules is moral philo¬
sophy. But why have they not learned them,
unless for this reason, that none hitherto have
taught them in a clear and exact method ? For
what shall we say ? Could the ancient masters of
Greece, Egypt, Rome, and others, persuade the
unskilful multitude to their innumerable opinions
concerning the nature of their gods, which they
OF PHILOSOPHY.
9
themselves knew not whether they were true or part i.
false, and which were indeed manifestly false and - L ,
absurd; and could they not persuade the same utility of
multitude to civil duty, if they themselves had Phllosophy-
understood it? Or shall those few writings of
geometricians which are extant, be thought suffi¬
cient for the taking away of all controversy in the
matters they treat of, and shall those innumerable
and huge volumes of ethics be thought unsufficient,
if what they teach had been certain and well de¬
monstrated ? What, then, can be imagined to be
the cause that the writings of those men have
increased science, and the writings of these have
increased nothing but words, saving that the for¬
mer were written by men that knew, and the
latter by such as knew not, the doctrine they
taught only for ostentation of their wit and elo¬
quence ? Nevertheless, I deny not but the reading
of some such books is very delightful; for they
are most eloquently written, and contain many
clear, wholesome and choice sentences, which yet ^
are not universally true, though by them univer¬
sally pronounced. From whence it comes to pass,
that the circumstances of times, places, and per¬
sons being changed, they are no less frequently
made use of to confirm wicked men in their pur¬
poses, than to make them understand the precepts
of civil duties. Now that which is chiefly wanting
in them, is a true and certain rule of our actions,
by which we might know whether that we under¬
take be just or unjust. For it is to no purpose to
be bidden in every thing to do right, before there
be a certain rule and measure of right established,
which no man hitherto hath established. Seeing,
10
COMPUTATION OR LOGIC.
paiit i. therefore, from the not knowing of civil duties,
-—^—- that is, from the want of moral science, proceed
civil wars, and the greatest calamities of mankind,
we may very well attribute to such science the
production of the contrary commodities. And
thus much is sufficient, to say nothing of the praises
and other contentment proceeding from philosophy,
to let you see the utility of the same in every kind
thereof.
Subject of 8. The subject of Philosophy, or the matter it
Philosophy. treatg 0f? js every body of which we can conceive
any generation, and which we may, by any consi¬
deration thereof, compare with other bodies, or
which is capable of composition and resolution;
that is to say, every body of whose generation or
properties we can have any knowledge. And this
may be deduced from the definition of philosophy,
whose profession it is to search out the properties
of bodies from their generation, or their generation
from their properties ; and, therefore, where there
is no generation or property, there is no philo¬
sophy. Therefore it excludes Theology, I mean
the doctrine of God, eternal, ingenerable, incom¬
prehensible, and in whom there is nothing neither
to divide nor compound, nor any generation to be
conceived.
It excludes the doctrine of angels, and all such
things as are thought to be neither bodies nor
properties of bodies ; there being in them no place
neither for composition nor division, nor any capa¬
city of more and less, that is to say, no place for
ratiocination.
It excludes history, as well natural as political,
though most useful (nay necessary) to philosophy;
of philosophy.
11
because such knowledge is but experience, or part i.
authority, and not ratiocination. —^—-
It excludes all such knowledge as is acquired by
Divine inspiration, or revelation, as not derived to
us by reason, but by Divine grace in an instant,
and, as it were, by some sense supernatural.
It excludes not only all doctrines which are
false, but such also as are not well-grounded ; for
whatsoever we know by right ratiocination, can
neither be false nor doubtful; and, therefore, as¬
trology, as it is now held forth, and all such divi¬
nations rather than sciences, are excluded.
Lastly, the doctrine of God's worship is excluded
from philosophy, as being not to be known by
natural reason, but by the authority of the Church ;
and as being the object of faith, and not of know¬
ledge.
9. The principal parts of philosophy are two. Parts of
For two chief kinds of bodies, and very differentPhllosoi,hy-
from one another, offer themselves to such as
search after their generation and properties ; one
whereof being the work of nature, is called a natu¬
ral body, the other is called a commonwealth, and
is made by the wills and agreement of men. And
from these spring the two parts of philosophy,
called natural and civil. But seeing that, for the
knowledge of the properties of a commonwealth,
it is necessary first to know the dispositions, affec¬
tions, and manners of men, civil philosophy is again
commonly divided into two parts, whereof one,
which treats of men's dispositions and manners, is
called ethics; and the other, which takes cogni¬
zance of their civil duties, is called politics, or
simply civil philosophy. In the first place, there-
12
COMPUTATION OR LOGIC.
part i. fore (after I have set down such premises as ap-
- !' pertain to the nature of philosophy in general), I
will discourse of bodies natural; in the second,
of the dispositions and manners of men ; and in
the third, of the civil duties of subjects.
Epilogue. 10. To conclude; seeing there may be many
who will not like this my definition of philosophy,
and will say, that, from the liberty which a man
may take of so defining as seems best to himself,
he may conclude any thing from any thing (though
I think it no hard matter to demonstrate that this
definition of mine agrees with the sense of all men);
yet, lest in this point there should be any cause of
dispute betwixt me and them, I here undertake
no more than to deliver the elements of that science
by which the effects of anything may be found out
from the known generation of the same, or con-
trarily, the generation from the effects ; to the end
that they who search after other philosophy, may
be admonished to seek it from other principles.
OF NAMES.
13
CHAPTER II.
OF NAMES.
1. The necessity of sensible Moniments or Marks for the help
of Memory : a Mark defined.—2. The necessity of Marks for
the signification of the conceptions of the Mind.—3. Names
supply both those necessities.—4. The Definition of a Name.—
5. Names are Signs not of Things, but of our Cogitations.—
6. What it is we give Names to.—7. Names Positive and
Negative.—8. Contradictory Names.—9. A Common Name.—
10. Names of the First and Second Intention.—11. Universal,
Particular, Individual, and Indefinite Names.—12. Names
Univocal and Equivocal.—13. Absolute and Relative Names.—
14-. Simple and Compounded Names.—15. A Predicament
described.—16. Some things to be noted concerning Predica¬
ments.
1. How unconstant and fading men's thoughts part l
are, and how much the recovery of them depends —-•
upon chance, there is none but knows by infallible ^sensiSe
experience in himself. For no man is able to re- Moniments
L ... . or Marks
member quantities without sensible and present for the help
measures, nor colours without sensible and present °f Memoiy'
patterns, nor number without the names of num¬
bers disposed in order and learned by heart. So
that whatsoever a man has put together in his
mind by ratiocination without such helps, will
presently slip from him, and not be revocable but
by beginning his ratiocination anew. From which
it follows, that, for the acquiring of philosophy,
some sensible moniments are necessary, by which
our past thoughts may be not only reduced, but
14
computation or logic.
also registered every one in its own order. These
moniments I call marks, namely, sensible things
taken at pleasure, that, by the sense of them, such
thoughts may be recalled to our mind as are like
those thoughts for which we took them.
MarkTfVjr°the 2* Again, though some one man, of how excel-
signification of lent a wit soever, should spend all his time partly
the conceptions . . .. . . „
of the Mind, m reasoning, and partly m inventing marks for
the help of his memory, and advancing himself in
learning ; who sees not that the benefit he reaps to
himself will not be much, and to others none at
all ? For unless he communicate his notes with
others, his science will perish with him. But if
the same notes be made common to many, and so
one man's inventions be taught to others, sciences
will thereby be increased to the general good of
mankind. It is therefore necessary, for the ac¬
quiring of philosophy, that there be certain signs,
by which what one man finds out may be mani¬
fested and made known to others. Now, those
things we call signs are the antecedents of their
/ consequents, and the consequents of their antece¬
dents, as often as we observe them to go before
or follow after in the same manner. For example,
a thick cloud is a sign of rain to follow, and rain a
sign that a cloud has gone before, for this reason
only, that we seldom see clouds without the con¬
sequence of rain, nor rain at any time but when a
cloud has gone before. And of signs, some are
natural, whereof I have already given an example,
others are arbitrary, namely, those we make choice
of at our own pleasure, as a bush hung up, signi¬
fies that wine is to be sold there ; a stone set in
the ground signifies the bound of a field; and
of names.
15
words so and so connected, signify the cogitations part i.
and motions of our mind. The difference, there-
fore, betwixt marks and signs is this, that we
make those for our own use, but these for the use
of others.
3. Words so connected as that they become Names
signs of our thoughts, are called speech, of which both those
every part is a name. But seeing (as is said) both necessities-
marks and signs are necessary for the acquiring of
philosophy, (marks by which we may remember
our own thoughts, and signs by which we may
make our thoughts known to others), names do
both these offices ; but they serve for marks before
they be used as signs. For though a man were
alone in the world, they would be useful to him
in helping him to remember; but to teach others,
(unless there were some others to be taught) of
no use at all. Again, names, though standing
singly by themselves, are marks, because they serve
to recal our own thoughts to mind; but they can- y
not be signs, otherwise than by being disposed and
ordered in speech as parts of the same. For ex¬
ample, a man may begin with a word, whereby the
hearer may frame an idea of something in his
mind, which, nevertheless, he cannot conceive to
be the idea which was in the mind of him that
spake, but that he would say something which
began with that word, though perhaps not as by
itself, but as part of another word. So that the
nature of a name consists principally in this, that
it is a mark taken for memory's sake ; but it serves
also by accident to signify and make known to
others what we remember ourselves, and, therefore,
I will define it thus :
16
computation or logic.
part i. 4. A name is a word taken at pleasure to serve
- 2;—- for a mark, which may raise in our mind a thought
Definition jjj-e f0 some thought we had before, and which
of a Name. 7777
being pronounced to others, maij be to them a
sign of ivliat thought the speaker had, or had not
before in his mind. And it is for brevity's sake
that I suppose the original of names to be arbi¬
trary, judging it a thing that may be assumed as
unquestionable. For considering that new names
are daily made, and old ones laid aside; that
^ diverse nations use different names, and how im¬
possible it is either to observe similitude, or make
any comparison betwixt a name and a thing, how
can any man imagine that the names of things
were imposed from their natures ? For though
some names of living creatures and other things,
which our first parents used, were taught by God
himself; yet they were by him arbitrarily imposed,
and afterwards, both at the Tower of Babel, and
since, in process of time, growing everywhere out
of use, are quite forgotten, and in their room have
succeeded others, invented and received by men
at pleasure. Moreover, whatsoever the common
use of words be, yet philosophers, who were to
teach their knowledge to others, had always the
liberty, and sometimes they both had and will have
a necessity, of taking to themselves such names as
they please for the signifying of their meaning, if
they would have it understood. Nor had mathe¬
maticians need to ask leave of any but themselves
to name the figures they invented, parabolas, hij-
perboles, cissoeides, quadratices, &c. or to call
one magnitude A, another B.
OF NAMES.
]/
5. But seeing names ordered in speech (as is part i.
defined) are signs of our conceptions, it is mani- -—-r—-
fest they are not signs of the things themselves : Nam.es
0 ° 5 are signs
for that the sound of this word stone should be of things,
the sign of a stone, cannot be understood in any cognitions,
sense but this, that he that hears it collects that
he that pronounces it thinks of a stone. And,
therefore, that disputation, whether names signify
the matter or form, or something compounded of
both, and other like subtleties of the metaphysics,
is kept up by erring men, and such as understand
not the words they dispute about.
6. Nor, indeed, is it at all necessary that every what it is
name should be the name of something. For as we glv?
~ names to.
these, a man, a tree, a stone, are the names of the
things themselves, so the images of a man, of a
tree, and of a stone, which are represented to men
sleeping, have their names also, though they be
not things, but only fictions and phantasms of
things. For we can remember these ; and, there¬
fore, it is no less necessary that they have names
to mark and signify them, than the things them¬
selves. Also this word future is a name, but no
future thing has yet any being, nor do we know
whether that which we call future, shall ever have
a being or no. Nevertheless, seeing we use in our
mind to knit together things past with those that
are present, the name future serves to signify such
knitting together. Moreover, that which neither
is, nor has been, nor ever shall, or ever can be,
has a name, namely, that which neither is nor has
been, &c. ; or more briefly this, impossible. To
conclude ; this word nothing is a name, which yet
cannot be the name of any thing: for when, for
vol. i. c
18
COMPUTATION OR LOGIC.
Names Positive
and Negative.
example, we substract 2 and 3 from 5, and so
nothing remaining, we would call that substrac-
tion to mind, this speech nothing remains, and in
it the word nothing is not unuseful. And for the
same reason we say truly, less than nothing re¬
mains, when we substract more from less ; for the
mind feigns such remains as these for doctrine's
sake, and desires, as often as is necessary, to call
the same to memory. But seeing every name has
some relation to that which is named, though that
which we name be not always a thing that has a
being in nature, yet it is lawful for doctrine's sake
to apply the word thing to whatsoever we name ;
as if it were all one whether that thing be truly
existent, or be only feigned.
7. The first distinction of names is, that some
are positive, or affirmative, others negative, which
are also called privative and indefinite. Positive
are such as we impose for the likeness, equality,
or identity of the things we consider; negative,
for the diversity, unlikeness, or inequality of the
same. Examples of the former are, a man, a
philosopher; for a man denotes any one of a
multitude of men, and a philosopher, any one of
many philosophers, by reason of their similitude;
also, Socrates is a positive name, because it sig¬
nifies always one and the same man. Examples of
negatives are such positives as have the negative
particle not added to them, as not-man, not-
philosopher. But positives were before negatives;
for otherwise there could have been no use at all
of these. For when the name of white was
imposed upon certain things, and afterwards upon
other things, the names of black, blue, trans-
OF NAMES.
19
parent, 8?c. the infinite dissimilitudes of these part i.
with white could not be comprehended in any one r—
name, save that which had in it the negation of
white, that is to say, the name not-wliite, or some
other equivalent to it, in which the word white is
repeated, such as unlike to white, fyc. And by
these negative names, we take notice ourselves,
and signify to others what we have not thought of.
8. Positive and negative names are contra-Cmir^ci0}1
names.
dictory to one another, so that they cannot both
be the name of the same thing. Besides, of con¬
tradictory names, one is the name of anything
whatsoever ; for whatsoever is, is either man, or
not-man, white or not-white, and so of the rest.
And this is so manifest, that it needs no farther
proof or explication ; for they that say the same
thing cannot both be, ancl not he, speak obscurely;
but they that say, whatsoever is, either is, or is
not, speak also absurdly and ridiculously. The
certainty of this axiom, viz. of two contradictory
names, one is the name of anything whatsoever,
the other not, is the original and foundation of all
ratiocination, that is, of all philosophy; and
therefore it ought to be so exactly propounded,
that it may be of itself clear and perspicuous to
all men ; as indeed it is, saving to such, as
reading the long discourses made upon this sub¬
ject by the writers of metaphysics (which they
believe to be some egregious learning) think they
understand not, when they do.
9. Secondly, of names, some are common to A common
, name.
many things, as a man, a tree; others proper to
one thing, as he that writ the Iliad, Homer, this
man, that man. And a common name, being the
c '2
20
computation or logic.
part i. name of many things severally taken, but not
-—r—- collectively of all together (as man is not the name
of all mankind, but of every one, as of Peter,
John, and the rest severally) is therefore called an
universal name ; and therefore this word univer¬
sal is never the name of any thing existent in
nature, nor of any idea or phantasm formed in the
mind, but always the name of some word or
name; so that when a living creature, a stone, a
J spirit, or any other thing, is said to be universal,
it is not to be understood, that any man, stone,
&c. ever was or can be universal, but only that
these words, living creature, stone, $'c. are uni¬
versal names, that is, names common to many
things ; and the conceptions answering them in
our mind, are the images and phantasms of
several living creatures, or other things. And
therefore, for the understanding of the extent of
an universal name, we need no other faculty but
that of our imagination, by which we remember
that such names bring sometimes one thing, some¬
times another, into our mind. Also of common
names, some are more, some less common. More
common, is that which is the name of more
things; less common, the name of fewer things;
as living creature is more common than man, or
horse, or lion, because it comprehends them all:
and therefore a more common name, in respect of
a less common, is called the genus, or a general
name ; and this in respect of that, the species, or
a special name.
Names 10. And from hence proceeds the third distinc-
and second tion of names, which is, that some are called
intention, names of the first, others of the second intention.
of names.
21
Of the first intention are the names of things, tart i.
a man, stone, fyc. ; of the second are the names -—r—•
of names and speeches, as universal, particular,
genus, species, syllogism, and the like. But it
is hard to say why those are called names of the
first, and these of the second intention, unless
perhaps it was first intended by us to give names
to those things which are of daily use in this life,
and afterwards to such things as appertain to
science, that is, that our second intention was to
give names to names. But whatsoever the cause
hereof may be, yet this is manifest, that genus,
species, definition, 8?c. are names of words and
names only; and therefore to put genus and
species for things, and definition for the nature of
any thing, as the writers of metaphysics have
done, is not right, seeing they be only signifi¬
cations of what we think of the nature of things.
11. Fourthly, some names are of certain and Universal,
determined, others of uncertain and undetermined individual'
signification. Of determined and certain signifi- ^g^efinlte
cation is, first, that name which is given to any
one thing by itself, and is called an individual
name; as Homer, this tree, that living creature,
&c. Secondly that which has any of these words,
all, every, both, either, or the like added to it;
and it is therefore called an universal name,
because it signifies every one of those things to
which it is common ; and of certain signification
for this reason, that he which hears, conceives in
his mind the same thing that he which speaks
would have him conceive. Of indefinite significa¬
tion is, first, that name which has the word some,
or the like added to it, and is called a particular
22
COMPUTATION OR LOGIC.
part i. name; secondly, a common name set by itself
—■—" without any note either of universality or particu¬
larity, as man, stone, and is called an indefinite
name; but both particular and indefinite names
are of uncertain signification, because the hearer
knows not what thing it is the speaker would have
him conceive ; and therefore in speech, particular
and indefinite names are to be esteemed equivalent
to one another. But these words, all, every, some,
fyc. which denote universality and particularity,
are not names, but parts only of names ; so that
every man, and that man which the hearer con¬
ceives in his mind, are all one ; and some man,
and that man which the speaker thought of \ signify
the same. From whence it is evident, that the
use of signs of this kind, is not for a man's own
sake, or for his getting of knowledge by his own
private meditation (for every man has his own
thoughts sufficiently determined without such helps
as these) but for the sake of others ; that is, for
the teaching and signifying of our conceptions to
others ; nor were they invented only to make us
remember, but to make us able to discourse with
others.
Names 12. Fifthly, names are usually distinguished
univocal # J .
and equivocal, into univocal and equivocal. Univocal are those
which in the same train of discourse signify
always the same thing; but equivocal those which
mean sometimes one thing and sometimes another.
Thus, the name triangle is said to be univocal,
because it is always taken in the same sense; and
parabola to be equivocal, for the signification it
has sometimes of allegory or similitude, and some¬
times of a certain geometrical figure. Also every
OF NAMES.
23
metaphor is by profession equivocal. But tliis PAI^T L
distinction belongs not so much to names, as to • '<-—"
those that use names, for some use them properly
and accurately for the finding out of truth ; others
draw them from their proper sense, for ornament
or deceit.
13. Sixthly, of names, some are absolute, others Absolute
r> 7 i • and relative
relative. Relative are such as are imposed for names,
some comparison, as father, son, cause, effect,
lihe, unlike, equal, unequal, master, servant, fyc.
And those that signify no comparison at all are
absolute names. But, as it was noted above, that
universality is to be attributed to words and names
only, and not to things, so the same is to be said
of other distinctions of names ; for no things are
either univocal or equivocal, or relative or abso¬
lute. There is also another distinction of names
into concrete and abstract; but because abstract
names proceed from proposition, and can have no
place where there is no affirmation, I shall speak
of them hereafter.
14. Lastly, there are simple and compounded simple and
names. But here it is to be noted, that a name is names.undec
not taken in philosophy, as in grammar, for one
single word, but for any number of words put
together to signify one thing ; for among philoso¬
phers sentient animated body passes but for one
name, being the name of every living creature,
which yet, among grammarians, is accounted three
names. Also a simple name is not here distin¬
guished from a compounded name by a preposition,
as in grammar. But I call a simple name, that
which in every kind is the most common or most
universal; and that a compounded name, which
24
computation or logic.
part i. by the joining of another name to it, is made less
-—t— universal, and signifies that more conceptions than
compounded one were in the mind, for which that latter name
names. was added, ]?or example, in the conception of
man (as is shown in the former chapter.) First,
he is conceived to be something that has exten¬
sion, which is marked by the word body. Body,
therefore, is a simple name, being put for that
first single conception ; afterwards, upon the sight
of such and such motion, another conception
arises, for which he is called an animated body ;
and this I here call a compounded name, as I do
also the name animal, which is equivalent to an
animated body. And, in the same manner, an
animated rational body, as also a man, which is
equivalent to it, is a more compounded name.
And by this we see how the composition of con¬
ceptions in the mind is answerable to the compo¬
sition of names ; for, as in the mind one idea or
phantasm succeeds to another, and to this a
third; so to one name is added another and
another successively, and of them all is made one
compounded name. Nevertheless we must not
think bodies which are without the mind, are
compounded in the same manner, namely, that
there is in nature a body, or any other imaginable
thing existent, which at the first has no magnitude,
and then, by the addition of magnitude, comes
to have quantity, and by more or less quantity to
have density or rarity; and again, by the addition
of figure, to be figurate, and after this, by the
injection of light or colour, to become lucid or
coloured; though such has been the philosophy
of many.
of names.
25
15. The writers of logic have endeavoured to part i.
2
di gest the names of all the kinds of things into -—A—-
certain scales or degrees, by the continual subor- ^eP^ec^ment
dination of names less common, to names more
common, In the scale of bodies they put in the
first and highest place body simply, and in the
next place under it less common names, by which
it may be more limited and determined, namely
animated and inanimated, and so on till they
come to individuals. In like manner, in the
scale of qua?itities, they assign the first place to
quantity, and the next to line, superficies, and
solid, which are names of less latitude ; and these
orders or scales of names they usually call 'predi¬
caments and categories. And of this ordination
not only positive, but negative names also are
capable ; which may be exemplified by such forms
of the 'predicaments as follow :
The Form of the Predicament of Body.
Not-Body, or
Accident.
Body
| Not ani-
I mated.
{Animated
Not living
Creature.
Living
Creature
Not Man.
Man
Not Peter.
Peter.
Both Accident and Body]
are considered
(Quantity, or so much.
Absolutely, as \ ~ ,
1 I Quality, or such.
or
Comparatively, which is called
\ their Relation.
26
COMPUTATION OR LOGIC.
The Form of the Predicament of Quantity.
A predicament
described.
/Not continual,
as Number.
Quantity
/Of itself> as
^Continual
(Line.
Superficies.
Solid.
By accident, as-
'Time, by Line.
Motion, by Line and
Time.
Force, by Motion and
Solid.
Where, it is to be noted, that line, superficies,
arid solid, may be said to be of such and such
quantity, that is, to be originally and of their
own nature capable of equality and inequality;
but we cannot say there is either majority or
minority, or equality, or indeed any quantity at
all, in time, without the help of line and motion;
nor in motion, without line and time; nor in
force, otherwise than by motion and solid.
The Form of the Predicament of Quality.
Quality
Perception
by Sense
Sensible
Quality
Primary
Secondary
Seeing.
Hearing.
Smelling.
Tasting.
Touching.
(Imagination.
Affection
f pleasant.
{ unpleasant.
By Seeing, as Light and Colour.
By Hearing, as Sound.
By Smelling, as Odours.
By Tasting, as Savours.
By Touching, as Hardness, Heat,
Cold, &c.
OF NAMES.
27
Tiie Form of the Predicament of Relation.
Magnitudes, as Equality antl Inequality.
Qualities, as Likeness and. Unlikeness.
Relation of \
'Together
In Place.
In Time.
0rder (In Place j^mer'
Not together I I a ei.
Ti™ {Sr
16. Concerning which predicaments it is to be Somethings
noted, in the first place, that as the division is concerning
made in the first predicament into contradictory Predicaments-
names, so it might have been done in the rest.
For, as there, body is divided into animated and
not-animated, so, in the second predicament,
continual quantity may be divided into li?ie and
not-line, and again, not-line into superficies and
not-superficies, and so in the rest; but it was not
necessary.
Secondly, it is to be observed, that of positive
names the former comprehends the latter ; bnt of
negatives the former is comprehended by the
latter. For example, living-creature is the name
of every man, and therefore it comprehends the
name man ; bnt, on the contrary, not-man is the
name of everything which is not-living-creature,
and therefore the name not-living-creature, which
is put first, is comprehended by the latter name,
not-man.
Thirdly, we must take heed that we do not
think, that as names, so the diversities of things
themselves may be searched out and determined
by such distinctions as these; or that arguments
28
COMPUTATION OR LOGIC.
part i. may be taken from hence (as some have done
~ r—- ridiculously) to prove that the kinds of things are
Predicaments. no(. infinite
Fourthly, I would not have any man think I
deliver the forms above for a true and exact or¬
dination of names ; for this cannot be performed
as long as philosophy remains imperfect; nor that
by placing (for example) light in the predicament
of qualities, while another places the same in the
predicament of bodies, I pretend that either of
us ought for this to be drawn from his opinion;
for this is to be done only by arguments and
ratiocination, and not by disposing of words into
classes.
Lastly, I confess I have not yet seen any great
use of the predicaments in philosophy. I believe
Aristotle when he saw he could not digest the
things themselves into such orders, might never¬
theless desire out of his own authority to reduce
words to such forms, as I have done ; but I do it
only for this end, that it may be understood what
this ordination of words is, and not to have it
received for true, till it be demonstrated by good
reason to be so.
of proposition.
29
CHAPTER III.
of proposition.
1. Divers kinds of speech.—2. Proposition defined.—3. Subject,
predicate, and copula, what they are; and abstract and con¬
crete what. The use and abuse of names abstract.—5. Pro¬
position, universal and particular.—6. Affirmative and negative.
—7- True and false.—8. True and false belongs to speech,
and not to things.—9. Proposition, primary, not primary,
definition, axiom, petition.—10. Proposition, necessary and
contingent.— 11. Categorical and hypothetical.— 12. The
same proposition diversely pronounced.—13. Propositions that
may be reduced to the same categorical proposition, are equi¬
pollent.—14. Universal propositions converted by contradic¬
tory names, are equipollent.—15. Negative propositions are
the same, whether negation be before or after the copula.—
16. Particular propositions simply converted, are equipollent.
—17. What are subaltern, contrary, subcontrary, and con¬
tradictory propositions.—18. Consequence, what it is.—19.
Falsity cannot follow from truth.—20. How one proposition
is the cause of another.
1. From the connexion or contexture of names part i.
arise divers kinds of speech, whereof some signify —^—-
the desires and affections of men ; such are, first, Jflpeech!^8
interrogations, which denote the desire of know¬
ing : as, Who is a good man ? In which speech
there is one name expressed, and another desired
and expected from him of whom we ask the same.
Then prayers, which signify the desire of having
something ; promises, threats, wishes, commands,
complaints, and other significations of other
affections. Speech may also be absurd and in¬
significant ; as when there is a succession of
30
computation or logic.
part i. words, to which there can be no succession of
3.
-——- thoughts in mind to answer them ; and this hap¬
pens often to such, as, understanding nothing in
some subtle matter, do, nevertheless, to make
others believe they understand, speak of the same
incoherently; for the connection of incoherent
words, though it want the end of speech (which
is signification) yet it is speech; and is used by
writers of metaphysics almost as frequently as
speech significative. In philosophy, there is but
one kind of speech useful, which some call in Latin
dictum, others enunticitum et pronunciatum; but
most men call it 'proposition, and is the speech of
those that affirm or deny, and expresseth truth or
falsity.
Proposition 2. A proposition is a speech consisting of
defined.
two names copulated, by which he that speahetli
signifies lie conceives the latter name to be the
name of the same thing whereof the former is
the name; or (which is all one) that the former
name is comprehended, by the latter. For example,
this speech, man is a living creature, in which
two names are copulated by the verb is, is a pro¬
position, for this reason, that he that speaks it
conceives both living creature and man to be
names of the same thing, or that the former name,
man, is comprehended by the latter name, living
creature. Now the former name is commonly
called the subject, or antecedent, or the contained
name, and the latter the predicate, consequent,
or containing name. The sign of connection
amongst most nations is either some word, as the
word is in the proposition man is a living creature,
or some case or termination of a word, as in this
OF PROPOSITION.
31
proposition, man walketh (which i§ equivalent to part i.
this, man is walking) ; the termination by which it r—
is said he walheth, rather than he is walking,
signifieth that those two are understood to be
copulated, or to be names of the same thing.
But there are, or certainly may be, some nations
that have no word which answers to our verb is,
who nevertheless form propositions by the position
only of one name after another, as if instead of
man is a living creature, it should be said man
a living creature; for the very order of the
names may sufficiently show their connection ;
and they are as apt and useful in philosophy, as if
they were copulated by the verb is.
3. Wherefore, in every proposition three things Subject,
are to be considered, viz. the two names, which ^copula,
are the subject, and the predicate, and their ^Ibstract'6'
copulation ; both which names raise in our mind a»d concrete
the thought of one and the same thing ; but the
copulation makes us think of the cause for which
those names were imposed on that thing. As, for
example, when we say a body is moveable, though
we conceive the same thing to be designed by
both those names, yet our mind rests not there,
but searches farther what it is to be a body, or to
he moveable, that is, wherein consists the differ¬
ence betwixt these and other things, for which
these are so called, others are not so called.
They, therefore, that seek what it is to be any
thing, as to be moveable, to be hot, S?c. seek in
things the causes of their names.
And from hence arises that distinction of names
(touched in the last chapter) into concrete and
abstract. For concrete is the name of any thing
32
computation or logic.
part i. which we suppose to have a being, and is there-
*—-—- fore called the subject, in Latin suppositum, and
subject. -n Qreek vTTOKkifjLzvov ; as body, moveable, moved,
figurate, a cubit high, hot, cold, like, equal,
Appius, Lentulus, and the like ; and, abstract
is that which in any subject denotes the cause
of the concrete name, as to be a body, to be
moveable, to be moved, to be figurate, to be of
such quantity, to be hot, to be cold, to be like,
to be equal, to be Appius, to be Lentulus, $pc.
Or names equivalent to these, which are most
commonly called abstract names, as corporiety,
mobility, motion, figure, quantity, heat, cold,
likeness, equality, and (as Cicero has it) Appiety
and Lentulity. Of the same kind also are infini¬
tives ; for to live and to move are the same with.
life and motion, or to be living and to be moved.
But abstract names denote only the causes of
concrete names, and not the things themselves.
For example, when we see any thing, or conceive
in our mind any visible thing, that thing appears
to us, or is conceived by us, not in one point, but
as having parts distant from one another, that is,
as being extended and filling some space. Seeing
therefore we call the thing so conceived body,
the cause of that name is, that that thing is
exte7ided, or the extension or corporiety of it.
So when we see a thing appear sometimes here,
sometimes there, and call it moved or removed,
the cause of that name is that it is moved or the
motion of the same.
And these causes of names are the same witli
the causes of our conceptions, namely, some
power of action, or affection of the thing con-
OF PROPOSITION.
33
ceived, which some call the manner by which any part i.
thing works upon our senses, but by most men ^—
they are called accidents ; I say accidents, not in
that sense in which accident is opposed to
necessary; but so, as being neither the things
themselves, nor parts thereof, do nevertheless
accompany the things in such manner, that (saving
extension) they may all perish, and be destroyed,
but can never be abstracted.
4. There is also this difference betwixt concrete The use „
and abuse of
and abstract names,, that those were invented names abstract,
before propositions, but these after; for these
could have no being till there were propositions,
from whose copula they proceed. Now in all
matters that concern this life, but chiefly in philo¬
sophy, there is both great use and great abuse of
abstract names ; and the use consists in this, that
without them we cannot, for the most part, either
reason, or compute the properties of bodies ; for
when we would multiply, divide, add, or substract
heat, light, or motion, if we should double or add
them together by concrete names, saying (for
example) hot is double to hot, light double to
light, or moved double to moved, we should not
double the properties, but the bodies themselves
that are hot, light, moved, &c. which we would
not do. But the abuse proceeds from this, that
some men seeing they can consider, that is (as I
said before) bring into account the increasings
and decreasings of quantity, heat and other acci¬
dents, without considering their bodies or subjects
(which they call abstracting, or making to exist
apart by themselves) they speak of accidents, as
if they might be separated from all bodies. And
VOL. I. D
34 COMPUTATION Oil LOGIC.
part i. from hence proceed the gross errors of writers
-——- of metaphysics; for, because they can consider
thought without the consideration of body, they
infer there is no need of a thinking-body ; and
because quantity may be considered without con¬
sidering body, they think also that quantity may
be without body, and body without quantity ; and
that a body has quantity by the addition of quan¬
tity to it. From the same fountain spring those
insignificant words, abstract substance, separated
essence, and the like; as also that confusion of
words derived from the Latin verb est, as essence,
essentiality, entity, entitative ; besides reality,
aliquiddity, quiddity, &>c. which could never
have been heard of among such nations as do not
copulate their names by the verb is, but by
adjective verbs, as runneth, readeth, &c. or by
the mere placing of one name after another; and
yet seeing such nations compute and reason, it is
evident that philosophy has no need of those
words essence, entityf and other the like barbarous
terms.
Proposition, 5. There are many distinctions of propositions,
paJtTcuTar?nd whereof the first is, that some are universal,
others particular, others indefinite, and others
singular; and this is commonly called the dis¬
tinction of quantity. An universal proposition is
that whose subject is affected with the sign of an
universal name, as every man is a living creature.
Particular, that whose subject is affected with
the sign of a particular name, as some man is
learned. An indefinite proposition has for its
subject a common name, and put without any
sign, as man is a living creature, man is learned.
OF PROPOSITION.
35
And a singular proposition is that whose subject i'ART r-
is a singular name, as Socrates is a philosopher, -
this man is black.
(>. The second distinction is into affirmative Affirmative
and negative, and is called the distinction of
quality. An affirmative proposition is that whose
predicate is a positive name, as man is a living
creature. Negative, that whose predicate is a
negative name, as man is not a stone.
7. The third distinction is, that one is true, True & false,
another false. A true proposition is that, whose
predicate contains, or comprehends its subject, or
whose predicate is the name of every thing, of
which the subject is the name ; as man is a living
creature is therefore a true proposition, because
whatsoever is called man, the same is also called
living creature; and some man is sick, is true,
because sick is the name of some man. That
which is not true, or that whose predicate does
not contain its subject, is called a false proposi¬
tion, as man is a stone.
Now these words true, truth, and true 'propo¬
sition, are equivalent to one another; for truth
consists in speech, and not in the things spoken
of; and though true be sometimes opposed to
apparent or feigned, yet it is always to be referred
to the truth of proposition ; for the image of a
man in a glass, or a ghost, is therefore denied to
be a very man, because this proposition, a ghost
is a man, is not true ; for it cannot be denied but
that a ghost is a very ghost. And therefore truth
or verity is not any affection of the thing, but of
the proposition concerning it. As for that which
the writers of metaphysics say, that a thing, one
i) 2
36
computation or logic.
part i. thing, and a very thing, are equivalent to one
-—*—- another, it is but trifling and childish ; for who
does not know, that a man, one man, and a very
man, signify the same.
True & false 8. And from hence it is evident, that truth and
speech, and falsity have no place but amongst such living
not to things. crcatures as use speech. For though some brute
creatures, looking upon the image of a man in a
glass, may be affected with it, as if it were the
man himself, and for this reason fear it or fawn
upon it in vain ; yet they do not apprehend it as
true or false, but only as like ; and in this they are
not deceived. Wherefore, as men owe all their
true ratiocination to the right understanding of
speech ; so also they owe their errors to the mis¬
understanding of the same ; and as all the orna¬
ments of philosophy proceed only from man, so
from man also is derived the ugly absurdity of
false opinions. For speech has something in it
like to a spider's web, (as it was said of old of
Solon s laws) for by contexture of words tender
and delicate wits are ensnared and stopped;
but strong wits break easily through them.
From hence also this may be deduced, that the
first truths were arbitrarily made by those that
first of all imposed names upon things, or received
them from the imposition of others. For it is
true (for example) that man is a living creature,
\ but it is for this reason, that it pleased men to
impose both those names on the same thing.
Proposition, 9. Fourthly, propositions are distinguished into
primary, primary and not primary. Primaru is that
not primary, I J ...
definition, wherein the subject is explicated by a predicate of
axiom, . . T
petition. many names, as man is a body, animated,
OF PROPOSITION.
37
rational; for tliat which is comprehended in the part i.
name man, is more largely expressed in the names - <"—-
body, animated, and rational, joined together;
and it is called primary, because it is first in ratio¬
cination ; for nothing can be proved, without
understanding first the name of the thing in
question. Now primary propositions are nothing
but definitions, or parts of definitions, and these
only are the principles of demonstration, being
truths constituted arbitrarily by the inventors of
speech, and therefore not to be demonstrated.
To these propositions, some have added others,
which they call primary and principles, namely,
axioms, and common notions ; which, (though
they be so evident that they need no proof) yet,
because they may be proved, are not truly prin¬
ciples ; and the less to be received for such, in
regard propositions not intelligible, and some¬
times manifestly false, are thrust on us under the
name of principles by the clamour of men, who
obtrude for evident to others, all that they them¬
selves think true. Also certain petitions are com¬
monly received into the number of principles ; as,
for example, that a straight line may be drawn
between two points, and other petitions of the
writers of geometry ; and these are indeed the
principles of art or construction, but not of science
and demonstration.
10. Fifthly, propositions are distinguished into Proposition
necessary, that is, necessarily true ; and true, but contingent*
not necessarily, which they call contingent. A
necessary proposition is when nothing can at any
time be conceived or feigned, whereof the subject
is the name, but the predicate also is the name of
38
computation or logic.
part i. the same thing ; as man is a living creature is a
"—■—" necessary proposition, because at what time
ncces'saJy & soever we suppose the name man agrees with any
contingent, thing, at that time the name living-creature also
agrees with the same. But a contingent proposi¬
tion is that, which at one time may be true, at
another time false ; as every crow is hlach; which
may perhaps be true now, but false hereafter.
Again, in every necessary proposition, the predi¬
cate is either equivalent to the subject, as in this,
man is a rational living creature; or part of an
equivalent name, as in this, man is a living crea¬
ture, for the name rational-living-creature, or
man, is compounded of these two, rational and
living-creaturc. But in a contingent proposition
this cannot be ; for though this were true, every
man is a liar, yet because the word liar is no part
of a compounded name equivalent to the name
man, that proposition is not to be called necessary,
but contingent, though it should happen to be true
always. And therefore those propositions only
are necessary, which are of sempiternal truth, that
is, true at all times. From hence also it is mani¬
fest, that truth adheres not to things, but to
speech only, for some truths are eternal; for it
will be eternally true, if man, then living-crea¬
ture; but that any man, or living-creature, should
exist eternally, is not necessary.
Categorical & 11. A sixth distinction of propositions is into
hypothetical. cafeg0r\cal and hypothetical. A categorical
proposition is that which is simply or absolutely
pronounced, as every man is a living-creature,
no man is a tree; and hypothetical is that which
is pronounced conditionally, as, if any thing he a
OF PROPOSITION.
39
man, the same is also a living-creature, if any- paiit i.
thing be a man, the same is also not-a-stone. v—r—-
A categorical proposition, and an liypothe- hypothetical
tical answering it, do both signify tlie same, if the
propositions be necessary ; but not if they be con¬
tingent. For example, if this, every man is a
living-creature, be true, this also will be true, if
any thing be a man, the same is also a living-
creature ; but in contingent propositions, though
this be true, every crow is black, yet this, if any
thing be a crow, the same is black, is false. But
an hypothetical proposition is then rightly said
to be true, when the consequence is true, as every
man is a living-creature, is rightly said to be a
true proposition, because of whatsoever it is truly
said that is a, man, it cannot but be truly said also,
the same is a living creature. And therefore
whensoever an hypothetical proposition is true,
the categorical answering it, is not only true, but
also necessary ; which I thought worth the noting,
as an argument, that philosophers may in most
things reason more solidly by hypothetical than
categorical propositions.
12. But seeina; every proposition may be, andThosame
° . . n proposition
uses to be, pronounced and written m many forms, diversely
and we are obliged to speak in the same manner pionounce(i'
as most men speak, yet they that learn philosophy
from masters, had need to take heed they be not
deceived by the variety of expressions. And
therefore, whensoever they meet with any obscure
proposition, they ought to reduce it to its most
simple and categorical form ; in which the copu¬
lative word is must be expressed by itself, and not
mingled in any manner either with the subject or
40
COMPUTATION OR LOGIC.
part i. predicate, both which must be separated and
——- clearly distinguished one from another. For
example, if this proposition, man can not siny be
compared with this, man cannot sin, their dif¬
ference will easily appear if they be reduced to
these, man is able not to sin, and, man is not able
to sin, where the predicates are manifestly dif¬
ferent. But they ought to do this silently by
themselves, or betwixt them and their masters
only ; for it will be thought both ridiculous and
absurd, for a man to use such language publicly.
Being therefore to speak of equipollent proposi¬
tions, I put in the first place all those for equipol¬
lent, that may be reduced purely to one and the
same categorical proposition.
Propositions 13. Secondly, that which is categorical and
reduced^ to necessary, is equipollent to its hypothetical pro-
tegoricaipro- position; as this categorical, a right-lined tri-
position, are ancrle ]ms jfs three angles equal to two right
equipollent. 0 # . .
angles, to this hypothetical, if any figure be a
right-lined triangle, the three angles of it are
equal to two right angles.
universal 14. Also, any two universal propositions, of
con vert edTy which the terms of the one (that is, the subject
naliefar? an(l predicate) are contradictory to the terms of
equipollent, the other, and their order inverted, as these, every
man is a living creature, and every thing that is
not a living-creature is not a man, are equipollent.
For seeing every man is a living creature is a
true proposition, the name living creature con¬
tains the name man ; but they are both positive
names, and therefore (by the last article of the
precedent chapter) the negative name not man,
contains the negative name not living-creature;
OF PROPOSITION.
41
wherefore every thing that is not a living-crea- part i.
tnre, is not a man, is a true proposition. Likewise —-
these, no man is a tree, no tree is a man, are
equipollent. For if it be true that tree is not the
name of any man, then no one thing can be signi¬
fied by the two names man and tree, wherefore
no tree is a man is a true proposition. Also to
this, whatsoever is not a living-creature is not a
man, where both the terms are negative, this
other proposition is equipollent, only a living crea¬
ture is a man.
15. Fourthly, negative propositions, whether Negative
the particle of negation be set after the copula asSeUie'same,
some nations do, or before it, as it is in Latin and whet)!er ljlu
' 7 negation be
Greek, if the terms be the same, are equipollent: before or after
\ r . the copula.
as, ior example, man is not a tree, and, man is
not-a-tree, are equipollent, though Aristotle deny
it. Also these, every man is not a tree, and no
man is a tree, are equipollent, and that so mani¬
festly, as it needs not be demonstrated.
16. Lastly, all particular propositions that have Particular
their terms inverted, as these, some man is blind, Smpiy'con!
some blind thing is a man, are equipollent; for exponent,
either of the two names, is the name of some one
and the same man ; and therefore in which soever
of the two orders they be connected, they signify
the same truth.
17- Of propositions that have the same terms, what are sub-
and are placed in the same order, but varied either trlry^subcon-
by quantity or quality, some are called subaltern, ^XLy
others contrary, others subcontrary, and others propositions.
contradictory.
Subaltern, are universal and particular propo¬
sitions of the same quality; as, every man is a
This page was
missing from
the original text
at the time of
digitization.
- Yale University Library
This page was
missing from
the original text
at the time of
digitization.
- Yale University Library
44
COMPUTATION OR LOGIC.
part i. call it the formal-cause; whereas indeed it is no
—-<■—— cause at all; nor does the property of any figure
proposition the figure, but has its being at the same
is the cause time with it; only the knowledge of the figure
of another. J o cj
goes before the knowledge of the properties;
and one knowledge is truly the cause of another
knowledge, namely the efficient cause.
And thus much concerning proposition ; which
in the progress of philosophy is the first step,
like the moving towards of one foot. By the
due addition of another step I shall proceed to
syllogism, and make a complete pace. Of which
in the next chapter.
CHAPTER IV.
OF SYLLOGISM.
' 1. The definition of syllogism.—2. In a syllogism there are but
three terms.—3. Major, minor, and middle term ; also major
and minor proposition, what they are.—4. The middle term in
every syllogism ought to be determined in both the propositions
to one and the same thing.—5. From two particular propo¬
sitions nothing can be concluded.—6. A syllogism is the col¬
lection of two propositions into one sum.—7. The figure of a
' syllogism, what it is.—8. What is in the mind answering to a
syllogism.—9. The first indirect figure, how it is made.—
10. Tlife second indirect figure, how made.—11. How the third
indirect ijgure is made.—1 % There are many moods in every
figure, but most of them useless in philosophy.—13. An
hypothetical syllogism when equipollent to a categorical.
Definition 1. A speech, consisting of three propositions,
of syllogism. from two Qf ^rjneh the third follows, is called a
syllogism ; and that wrhich follows is called the
conclusion ; the other two premises. For example,
of syllogism.
45
this speech, every man is a Jiving creature, part i.
every living creature is a body, therefore, every -—r—
man is a body, is a syllogism, because the third
proposition follows from the two first ; that is, if
those be granted to be true, this must also be
granted to be true.
2. From two propositions which have not one in a syllogism
... „ ,, n there are but
term common, no conclusion can iollow ; and three terms,
therefore no syllogism can be made of them.
For let any two premises, a man is a living crea¬
ture, a tree is a plant, be both of them true, yet
because it cannot be collected from them that
plant is the name of a man, or man the name of
a plant, it is not necessary that this conclusion, a
man is a plant, should be true. Corollary : there¬
fore, in the premises of a syllogism there can be
but three terms.
Besides, there can be no term in the conclusion,
which was not in the premises. For let any two
premises be, a man is a living creature, a living
creature is a body, yet if any other term be put
in the conclusion, as man is two-footed; though
it be true, it cannot follow from the premises,
because from them it cannot be collected, that
the name two-footed belongs to a man; and
therefore, again, in every syllogism there can be
but three terms.
3. Of these terms, that which is the predicate Majoiy minor
in the conclusion, is commonly called the major; term; also
that which is the subject in the conclusion, proposition"01
the minor, and the other is the middle term ; as what the?are-
in this syllogism, a man is a living creature, a
living creature is a body, therefore, a man is a
body, body is the major, man the minor, and
46
COMPUTATION OR LOGIC.
PA^T I- ^lvinS creature the middle term. Also of the
1—.—- premises, that in which the major term is found,
is called the major proposition, and that which
has the minor term, the minor proposition.
The middle 4. if the middle term be not in both the pre-
erm in every . x
yiiogism to nnses determined to one and the same singular
in dSTo- thing, no conclusion will follow, nor syllogism be
oneantTthe madc. For let the minor term be man, the middle
same thing, term living creature, and the major term lion;
and let the premises be, man is a living creature,
some living creature is a lion, yet it will not fol¬
low that every or any man is a lion. By which
it is manifest, that in every syllogism, that propo¬
sition which has the middle term for its subject,
ought to be either universal or singular, but not
particular nor indefinite. For example, this syl¬
logism, every man is a living creature, some living
creature is four-footed, therefore some man is
four-footed, is therefore faulty, because the middle
term, living creature, is in the first of the premises
determined only to man, for there the name of
living creature is given to man only, but in the
latter premise it may be understood of some other
living creature besides man. But if the latter
premise had been universal, as here, every man is
a living creature, every living creature is a body,
therefore every man is a body, the syllogism had
been true; for it would have followed that body
had been the name of every living creature, that
is of man ; that is to say, the conclusion every man
is a body had been true. Likewise, when the
middle term is a singular name, a syllogism may
be made, I say a true syllogism, though useless in
philosophy, as this,«50w<? man is Socrates, Socrates
OF SYLLOGISM.
47
is a philosopher, therefore, some man is a philo- tart r.
sopher ; for the premises being granted, the con- -—'—-
elusion cannot be denied.
5. And therefore of two premises, in both From two
which the middle term is particular, a syllogism propoSdons
cannot be made ; for whether the middle term be bfco3Seci.
the subject in both the premises, or the predicate
in both, or the subject in one, and the predicate
in the other, it will not be necessarily determined
to the same thing. For let the premises be,
Some man is blind, ) In both which the middle
Some man is learned, ' terra is the subject,
it will not follow that blind is the name of any
learned man, or learned the name of any blind
man, seeing the name learned does not contain
the name blind, nor this that; and therefore it is
not necessary that both should be names of the
same man. So from these premises,
Every man is a living-creature, ) In both which the middle
Every horse is a living-creature, j term is the predicate,
nothing will follow. For seeing living creature
is in both of them indefinite, which is equivalent
to particular, and that man may be one kind of
living creature, and horse another kind, it is not
necessary that man should be the name of horse,
or horse of man. Or if the premises be,
Every man is a living- )Jn Qne rf which ,he mkldle_
creature,
Some living creature is
four-footed,
term is the subject, and in
the other the predicate,
the conclusion will not follow, because the name
48
computation or logic.
part i. living creature being not determined, it may in
r—- one of them be understood of man, in the other of
not-man.
a syllogism is g Now it is manifest from what has been said,
the collection # 7
of two propo- that a syllogism is nothing but a collection of the
sitions into „ J . . . . , , 1 ■,
one sum. sum 01 two propositions, joined together by a
common term, which is called the middle term.
And as proposition is the addition of two names,
so syllogism is the adding together of three.
the figure of 7. Syllogisms are usually distinguished according
Sufis!*1 to their diversity of figures, that is, by the diverse
position of the middle term. And again in
figure there is a distinction of certain moods,
which consist of the differences of propositions in
quantity and quality. The first figure is that, in
which the terms are placed one after another
according to their latitude of signification; in
which order the minor term is first, the middle
term next, and the major last; as, if the minor
term be man, the middle term, living creature,
and the major term, body, then, man is a living-
creature, is a body, will be a syllogism in the first
figure: in which, man is a living creature is the
minor proposition ; the major, living creature is
a body, and the conclusion, or sum of both, man is
a body. Now this figure is called direct, because
the terms stand in direct order ; and it is varied
by quantity and quality into four moods: of
which the first is that wherein all the terms are
positive, and the minor term universal, as every
man is a living creature, every living creature is
a body : in which all the propositions are affirma¬
tive, and universal. But if the major term be a
negative name, and the minor an universal name,
OF SYLLOGISM.
49
tlie figure will be in the second mood, as, evert) PAI4lT L
man is a living creature, every living creature is -—>—-
not a tree, in which the major proposition and
conclusion are both universal and negative. To
these two, are commonly added two more, by
making the minor term particular. Also it may
happen that both the major and middle terms
are negative terms, and then there arises another
mood, in which all the propositions are negative,
and yet the syllogism will be good; as, if the
minor term be man, the middle term not a stone,
and the major term not a flint, this syllogism,
no man is a stone, whatsoever is not a stone is
not a flint, therefore, no man is a flint, is true,
though it consist of three negatives. But in phi¬
losophy, the profession whereof is to establish
universal rules concerning the properties of things,
seeing the difference betwixt negatives and affirm¬
atives is only this, that in the former the subject
is affirmed by a negative name, and by a positive
in the latter, it is superfluous to consider any other
mood in direct figure, besides that, in which all
the propositions are both universal and affirm¬
ative.
8. The thoughts in the mind answering to a what is
direct syllogism, proceed in this manner; first, answering to
there is conceived a phantasm of the thing named,a syllosisni-
with that accident or quality thereof, for which it
is in the minor proposition called by that name
which is the subject; next, the mind has a phan¬
tasm of the same thing with that accident, or
quality, for which it hath the name, that in the
same proposition is the predicate ; thirdly, the
thought returns of the same thing as having that
VOL. i. E
50
COMPUTATION OR LOGTC,
part i. accident in it, for which it is called by the name,
■—r— that is the predicate of the major proposition ;
and lastly, remembering that all those are the acci¬
dents of one and the same thing, it concludes that
those three names are also names of one and the
same thing ; that is to say, the conclusion is true.
For example, when this syllogism is made, man is
a living creature, a living creature is a body,
therefore, man is a body, the mind conceives first
an image of a man speaking or discoursing, and
remembers that that, which so appears, is called
man; then it has the image of the same man
moving, and remembers that that, which appears
so, is called living creature; thirdly, it conceives
an image of the same man, as filling some place or
space, and remembers that what appears so is
called body ; and lastly, when it remembers that
that thing, which was extended, and moved and
spake, was one and the same thing, it concludes
that the three names, man, living creature, and
body, are names of the same thing, and that there¬
fore man is a living creature is a true proposition.
From whence it is manifest, that living creatures
that have not the use of speech, have no concep¬
tion or thought in the mind, answering to a syllo¬
gism made of universal propositions ; seeing it is
necessary to think not only of the thing, but also by
turns to remember the divers names, which for di¬
vers considerations thereof are applied to the same,
The first in- 9. The rest of the figures arise either from the
hoT m^dT inflexion, or inversion of the first or direct figure;
which is done by changing the major, or minor,
or both the propositions, into converted proposi¬
tions equipollent to them,
of syllogism.
51
From whence follow three other figures; of part t-
which, two are inflected, and the third inverted. "—-—-
The first of these three is made by the conversion aircct'l-ure
of the major proposition. For let the minor,bow made-
middle, and major terms stand in direct order,
thus, man is a living creature, is not a stone,
which is the first or direct figure ; the inflection
will be by converting the major proposition in this
manner, man is a living creature, a stone is not
a living creature ; and this is the second figure,
or the first of the indirect figures ; in which the
conclusion will be, man is not a stone. For
(having shown in the last chapter, art. 14, that
universal propositions, converted by contradiction
of the terms, are equipollent) both those syllogisms
conclude alike ; so that if the major be read (like
Hebrew) backwards, thus, a living creature is not
a stone, it will be direct again, as it was before.
In like manner this direct syllogism, man is not a
tree, is not a pear-tree, will be made indirect by
converting the major proposition (by contradiction
of the terms) into another equipollent to it, thus,
man is not a tree, a pear-tree is a tree; for the
same conclusion will follow, man is not a pear-tree.
But for the conversion of the direct figure into
the first indirect figure, the major term in the
direct figure ought to be negative. For though
this direct, man is a living creature, is a body, be
made indirect, by converting the major propo¬
sition, thus,
Man is a living creature,
Not a body is not a living creature,
Therefore, Every man is a, body;
Yet this conversion appears so obscure, that
e 2
computation or logic.
part i. this mood is of no use at all. By tlie conversion
-—r—' of tlie major proposition, it is manifest, that in this
figure, tlie middle term is always the predicate in
both the premises.
second indi- io. The second indirect figure is made by con-
rect figure . . . , ,
how made, verting the minor proposition, so as that the
middle term is the subject in both. But this
never concludes universally, and therefore is of no
use in philosophy. Nevertheless I will set down
an example of it; by which this direct
Every man is a living creature,
Every living creature is a body,
by conversion of the minor proposition, will stand
thus,
Some living creature is a man,
Every living creature is a body,
Therefore, Some man is a body.
For every man is a living creature cannot he
converted into this, every living creature is a
man: and therefore if this syllogism be restored
to its direct form, the minor proposition will be
some man is a living creature, and consequently
the conclusion will be some man is a body, seeing
the minor term man, which is the subject in the
conclusion, is a particular name,
in dire ct Vgu re 1 * • The third indirect or inverted figure, is
is made. made by the conversion of both the premises.
For example, this direct syllogism,
Every man is a living creature,
Every living creature is not a stone,
Therefore, Every man is not a stone,
being inverted, will stand thus,,
OF SYLLOGISM.
53
Every stone is not a living creature, TART I.
Whatsoever is not a living creature, is not a man, L ,
Therefore, Every stone is not a man ;
which conclusion is the converse of the direct
conclusion, and equipollent to the same.
The figures, therefore, of syllogisms, if they be
numbered by the diverse situation of the middle
term only, are but three ; in the first whereof, the
middle term has the middle place; in the second,
the last; and in the third, the first place. But if
they be numbered according to the situation of
the terms simply, they are four ; for the first may
be distinguished again into two, namely, into
direct and inverted. From whence it is evident,
that the controversy among logicians concerning
the fourth figure, is a mere Ao-yd^a^ia, or conten¬
tion about the name thereof; for, as for the thing
itself, it is plain that the situation of the terms
(not considering the quantity or quality by which
the moods are distinguished) makes four dif¬
ferences of syllogisms, which may be called
figures, or have any other name at pleasure.
/> i ^ i There are m a-
12. In every one or these figures there are ny moods in
many moods, which are made by varying the pre- but^moft^of
mises according to all the differences they are ^"T., uselfs
° _ J in philosophy.
capable of, by quantity and quality ; as namely,
in the direct figure there are six moods ; in the
first indirect figure, four ; in the second, fourteen;
and in the third, eighteen. But because from the
direct figure I rejected as superfluous all moods
besides that which consists of universal proposi¬
tions, and whose minor proposition is affirmative,
I do, together with it, reject the moods of the rest
54
COMPUTATION OR LOGIC.
part i. of the figures which are made by conversion of
v—r—- the premises in the direct figure.
Anhypotiieti- 13. Asit was showed before, that in necessary
cal syllogism . . . , . ,
when equipol- propositions a categorical and hypothetical propo-
goricaifCdte" sition are equipollent; so likewise it is manifest
that a categorical and hypothetical syllogism are
equivalent. For every categorical syllogism, as
this,
Every man is a living creature,
Every living creature is a body,
Therefore, Every man is a body,
is of equal force with this hypothetical syllogism:
If any thing be a man, the same is also a living creature,
If any thing be a living creature, the same is a body,
Therefore, If any thing be a man, the same is a body.
In like manner, this categorical syllogism in an
indirect figure,
No stone is a living creature,
Every man is a living creature,
Therefore, No man is a stone,
Or, No stone is a man,
is equivalent to this hypothetical syllogism:
If any thing be a man, the same is a living creature,
If any thing be a stone, the same is not a living creature,
Therefore, If any thing be a stone, the same is not a, man,
Or, If any thing be a man, the same is not a stone.
And thus much seems sufficient for the nature
of syllogisms ; (for the doctrine of moods and
figures is clearly delivered by others that have
written largely and profitably of the same). Nor
are precepts so necessary as practice for the
attaining of true ratiocination; and they that
study the demonstrations of mathematicians, will
OF SYLLOGISM.
55
sooner learn true logic, than they that spend time part t.
in reading the rules of syllogizing which logicians ■—r—-
have made; no otherwise than little children
learn to go, not by precepts, but by exercising
their feet. This, therefore, may serve for the first
pace in the way to Philosophy.
In the next place I shall speak of the faults and
errors into which men that reason unwarily are
apt to fall; and of their kinds and causes.
CHAPTER V.
OF EllUING, FALSITY, AND CAPTIONS.
1. Erring and falsity how they differ. Error of the mind by
itself without the use of words, how it happens.—2. A seven¬
fold incoherency of names, every one of which makes always
a false proposition.—3. Examples of the first manner of inco¬
herency.-—4. Of the second.—5. Of the third.—6. Of the
fourth.—7- Of the fifth.—8. Of the sixth.—9. Of the seventh.
10. Falsity of propositions detected by resolving the terms
with definitions continued till they come to simple names, or
names that are the most general of their kind.—11. Of the
fault of a syllogism consisting in the implication of the terms
with the copula.—-12. Of the fault which consists in equivo¬
cation.—13. Sophistical captions are oftener faulty in the
matter than in the form of syllogisms.
1. Men are subject to err not only in affirming and Erring & fai-
denying, but also in perception, and in silent dift'er. Error
cogitation. In affirming and denying, when they ^ei^wilhou't
call any thing by a name, which is not the name the, ufe of
J . . . words, how it
thereof; as if from seeing the sun first by reflec- happens,
tion in water, and afterwards again directly in the
56
COMPUTATION OR LOGIC.
part i. firmament, we should to both those appearances
-—r—- give the name of sun, and say there are two suns;
Si"/ hot which none but men can do, for no other living
they differ, creatures have the use of names. This kind of
error only deserves the name of falsity, as arising,
not from sense, nor from the things themselves,
but from pronouncing rashly ; for names have
their constitution, not from the species of things,
but from the will and consent of men. And hence
it comes to pass, that men pronounce falsely, by
their own negligence, in departing from such
appellations of things as are agreed upon, and are
not deceived neither by the things, nor by the
sense ; for they do not perceive that the thing
they see is called sun, but they give it that name
from their own will and agreement. Tacit
errors, or the errors of sense and cogitation, are
made, by passing from one imagination to the
imagination of another different thing; or by
feigning that to be past, or future, which never
was, nor ever shall be ; as when, by seeing the
image of the sun in water, we imagine the sun
itself to be there ; or by seeing swords, that there
has been or shall be fighting, because it uses to be
so for the most part; or when from promises we
feign the mind of the promiser to be such and such;
or lastly, when from any sign we vainly imagine
something to be signified, which is not. And
errors of this sort are common to all things that
have sense ; and yet the deception proceeds neither
from our senses, nor from the things we perceive;
but from ourselves while we feign such things as
are but mere images to be something more than
images. But neither things, nor imaginations of
of erring, falsity, etc.
57
things, can be said to be false, seeing they are part i.
truly what they are ; nor do they, as signs, pro- *—r—-
mise any thing which they do not perform ; for
they indeed do not promise at all, but we from
them ; nor do the clouds, but we, from seeing the
clouds, say it shall rain. The best way, therefore,
to free ourselves from such errors as arise from
natural signs, is first of all, before we begin to
reason concerning such conjectural things, to sup¬
pose ourselves ignorant, and then to make use of
our ratiocination ; for these errors proceed from
the want of ratiocination ; whereas, errors which
consist in affirmation and negation, (that is, the
falsity of propositions) proceed only from reasoning
amiss. Of these, therefore, as repugnant to phi¬
losophy, I will speak principally.
2. Errors which happen in reasoning, that is, a sevenfold
in syllogizing, consist either in the falsity of the names, all of
premises, or of the inference. In the first of these ^j.ls
cases, a syllogism is said to be faulty in the Pr°i)0sition-
matter of it; and in the second case, in the
form. I will first consider the matter, namely,
how many ways a proposition may be false ; and
next the form, and how it comes to pass, that
when the premises are true, the inference is, not¬
withstanding, false.
Seeing, therefore, that proposition only is true,
(chap, in, art. 7) in which are copulated two
names of one and the same thing; and that always
false, in which names of different things are copu¬
lated, look how many ways names of different
things may be copulated, and so many ways a
false proposition may be made.
Now, all things to which we give names, may be
58
COMPUTATION OF LOGIC.
reduced to these four kinds, namely, bodies, acci¬
dents, phantasms, and names themselves; and there¬
fore, in every true proposition, it is necessary that
the names copulated, be both of them names of
bodies, or both names of accidents, or both names of
phantasms, or both names of names. For names
otherwise copulated are incoherent, and constitute
a false proposition. It may happen, also, that the
name of a body, of an accident, or of a phantasm,
may be copulated with the name of a speech. So
that copulated names may be incoherent seven
manner of wrays.
1. If the name of a Body
c2. If the name of a Body
3. If the name of a Body
4. If the name of an Accident
5. If the name of an Accident
6. If the name of a Phantasm
7. If the name of a Body,
Accident, or Phantasm
the name of an Accident,
the name of a Phantasm.
►§ | the name of a Name.
the name of a Phantasm,
the name of a Name,
the name of a Name.
the name of a Speech.
c
f- £-1
Of all which I will give some examples,
of*he fir3* After the first of these ways propositions are
manner of false, when abstract names are copulated with
mcoherency. concre{.e names . as ('m Latin and Greek) esse est
ens, essentia est ens, to t'l vv aval (i.); quidditas
est ens, and many the like, which are found in
Aristotle's Metaphysics. Also, the understanding
worheth, the understanding understandeth, the
sight seeth; a body is magnitude, a body is
quantity, a body is extension; to be a man is a
man, whiteness is a white thing, &c.; wrhich is
as if one should say, the runner is the running,
or the walk ivalheth. Moreover, essence is sepa¬
rated, substance is abstracted: and others like
these, or derived from these, (with which common
OF ERRING, FALSITY, ETC.
59
philosophy abounds.) For seeing no subject of part i.
an accident (that is, no body) is an accident: no -—^—-
name of an accident ought to be given to a body,
nor of a body to an accident.
4. False, in the second manner, are such propo- The second,
sitions as these; a ghost is « body, or a spirit,
that is, a thin body; sensible species jly up and
down in the air, or are moved hither and thither,
which is proper to bodies; also, a shadoiv is
moved, or is a body ; light is moved, or is a
body; colour is the object of sight, sound of
hearing ; space or place is extended; and innu¬
merable others of this kind. For seeing ghosts,
sensible species, a shadow, light, colour, sound,
space, &c. appear to us no less sleeping than
waking, they cannot be things without us, but
only phantasms of the mind that imagines them ;
and therefore the names of these, copulated with
the names of bodies, cannot constitute a true
proposition.
5. False propositions of the third kind, are such Tlle tllird-
as these ; genus est ens, universale est ens, ens
de ente prcedicatur. For genus, and universale,
and predicare, are names of names, and not of
things. Also, number is infinite, is a false propo¬
sition ; for no number can be infinite, but only
the word number is then called an indefinite name
when there is no determined number answering to
it in the mind.
6. To the fourth kind belong such false propo- The fourtl1'
sitions as these, an object is of such magnitude or
figure as appears to the beholders ; colour, light,
sound, are in the object; and the like. For the
same object appears sometimes greater, sometimes
60 COMPUTATION OR LOGIC.
part i. lesser, sometimes square,, sometimes round, accor-
•—r—- ding to the diversity of the distance and medium;
but the true magnitude and figure of the thing
seen is always one and the same ; so that the
magnitude and figure which appears, is not the
true magnitude and figure of the object, nor any¬
thing but phantasm ; and therefore, in such pro¬
positions as these, the names of accidents are
copulated with the names of phantasms.
The fifth. 7. Propositions are false in the fifth manner,
when it is said that the definition is the essence of
a thing ; whiteness, or some other accident, is
the genus, or universal. For definition is not the
essence of any thing, but a speech signifying
what we conceive of the essence thereof; and so
also not whiteness itself, but the word whiteness,
is a genus, or an universal name.
The sixth. g. In the sixth manner they err, that say the
idea of anything is universal; as if there could
be in the mind an image of a man, which were
not the image of some one man, but a man simply,
which is impossible ; for every idea is one, and of
one thing; but they are deceived in this, that they
put the name of the thing for the idea thereof.
The seventh. 9. They err in the seventh manner, that make
this distinction between things that have being,
that some of them exist by themselves, others by
accident; namely, because Socrates is a man is
a necessary proposition, and Socrates is a musi¬
cian a contingent proposition, therefore they say
some things exist necessarily or by themselves,
others contingently or by accident ; whereby,
seeing necessary, contingent,by itself, by accident,
are not names of things, but of propositions, they
of erring, falsity, etc.
61
that say any thing that has bring, exists by acci- part r.
dent, copulate the name of a proposition with the -r—-
name of a thing. In the same manner also, they
err, which place some ideas in the understanding,
others in the fancy ; as if from the understanding
of this proposition, man is a living creature, we
had one idea or image of a man derived from
sense to the memory, and another to the under¬
standing ; wherein that which deceives them is
this, that they think one idea should be answerable
to a name, another to a proposition, which is
false; for proposition signifies only the order of
those things one after another, which wTe observe
in- the same idea of man ; so that this proposition,
man is a living creature raises but one idea in
us, though in that idea we consider that first, for
which he is called man, and next that, for which
he is called living creature. The falsities of pro¬
positions in all these several manners, is to be
discovered by the definitions of the copulated
names.
10. But when names of bodies are copulated Falsity ,of
with names of bodies, names of accidents with detected by
n • i , p -, i n resolving the
names oi accidents, names oi names with names of terms with
names, and names of phantasms with names ofdefin,tlons-
phantasms, if we, nevertheless, remain still doubt¬
ful whether such propositions are true, we ought
then in the first place to find out the definition of
both those names, and again the definitions of
such names as are in the former definition, and so
proceed by a continual resolution till we come to
a simple name, that is, to the most general or
most universal name of that kind ; and if after all
62
COMPUTATION OR LOGIC.
part i. this, tlie truth or falsity thereof be not evident,
-—~r~—' we must search it out by philosophy, and ratioci¬
nation, beginning from definitions. For every
proposition, universally true, is either a definition,
or part of a definition, or the evidence of it
% depends upon definitions,
of the fault 11. That fault of a syllogism which lies hid in
consisting in the form thereof, will always be found either in
Ih/tem^with implication of the copula with one of the
the copula, terms, or in the equivocation of some word ; and
in either of these ways there will be four terms,
which (as I have shewn) cannot stand in a true
syllogism. Now the implication of the copula
with either term, is easily detected by reducing
the propositions to plain and clear predication ;
as (for example) if any man should argue thus,
The hand toucheth the pen,
The 'pen toucheth the paper,
Therefore, The hand toucheth the paper ;
the fallacy will easily appear by reducing it, thus:
The hand, is, touching the fen,
The pen, is, touching the paper,
Therefore, The hand, is, touching the paper;
where there are manifestly these four terms, the
hand, touching tlie pen, the pen, and touching the
the paper. But the danger of being deceived by
sophisms of this kind, does not seem to be so
great, as that I need insist longer upon them,
of the 12. And though there may be fallacy in equi-
consisT^ill vocal terms, yet in those that be manifestly such,
equivocation, there is none at all; nor in metaphors, for they
profess the transferring of names from one thing
OF ERRING, FALSITY, ETC. G3
to another. Nevertheless, sometimes equivoeals tart r.
(and those not very obscure) may deceive; as in -—^—-
this argumentation :—It belongs to metaphysics
to treat of principles ; but the first principle of
all, is, that the same thing cannot both exist ancl
not exist at the same time ; and therefore it
belongs to metaphysics to treat whether the same
thing may both exist and not exist at the same
time ; where the fallacy lies in the equivocation
of the word principle ; for whereas Aristotle in
the beginning of his Metaphysics, says, that the
treating of principles belongs to primary science,
he understands by principles, causes of things,
and certain existences which he calls primary;
but where he says a primary proposition is a
principle, by principle, there, he means the
beginning and cause of knowledge, that is, the
understanding of words, which, if any man want,
he is incapable of learning.
13. But the captions of sophists and sceptics, Sophistical
by which they were wont, of old, to deride and are oftener
oppose truth, were faulty for the most part, not matter^than
in the form, but in the matter of syllogism; and in„ th,f f?rm
7 J ° ot syllogisms.
they deceived not others oftener than they were
themselves deceived. For the force of that famous
argument of Zeno against motion, consisted in
this proposition, whatsoever may be divided into
parts, infinite in number, the same is infinite;
which he, without doubt, thought to be true, yet
nevertheless is false. For to be divided into infi¬
nite parts, is nothing else but to be divided into
as many parts as any man will. But it is not
necessary that a line should have parts infinite in
64
COMPUTATION OR LOGIC.
part i. number, or be infinite, because I can divide and
vt" subdivide it as often as I please; for how many
parts soever I make, yet their number is finite;
but because he that says parts, simply, without
adding how many, does not limit any number, but
leaves it to the determination of the hearer, there¬
fore we say commonly, a line may be divided
conclusion, infinitely; which cannot be true in any other
sense.
And thus much may suffice concerning syllo¬
gism, which is, as it were, the first pace towards
philosophy; in which I have said as much as is
necessary to teach any man from whence all true
argumentation has its force. And to enlarge this
treatise with all that may be heaped together,
would be as superfluous, as if one should (as I
said before) give a young child precepts for the
teaching of him to go ; for the art of reasoning is
not so well learned by precepts as by practice, and
by the reading of those books in which the con¬
clusions are all made by severe demonstration.
And so I pass on to the way of philosophy, that is,
to the method of study.
OF METHOD.
65
CHAPTER VI.
of method.
1. Method and science defined.—2. It is more easily known
concerning singular, than universal things, that they are ; and
contrarily, it is more easily known concerning universal, than
singular things, why they are, or what are their causes.—•
3. What it is philosophers seek to know.—4. The first part,
by which principles are found out, is purely analytical.—5. The
highest causes, and most universal in every kind, are known
by themselves.—6. Method from principles found out, tending
to science simply, what it is.—7. That method of civil and
natural science, which proceeds from sense to principles, is
analytical; and again, that, which begins at principles, is
synthetical.—8. The method of searching out, whether any
thing propounded be matter or accident.—9. The method of
seeking whether any accident be in this, or in that subject.
10. The method of searching after the cause of any effect
propounded.—11. Words serve to invention, as marks; to
demonstration, as signs.—12. The method of demonstration
is synthetical.—13. Definitions only are primary and universal
propositions.—14. The nature and definition of a definition.
15. The properties of a definition.—16. The nature of a
demonstration.—17. The properties of a demonstration, and
order of things to be demonstrated.—18. The faults of a
demonstration.—19. Why the analytical method of geometri¬
cians cannot be treated of in this place.
1. For the understanding of method, it will be PA^T L
necessary for me to repeat the definition of philo- ->—-
sophy, delivered above (Chap, i, art. 2.) in this sc^ defied,
manner, Philosophy is the knowledge we acquire,
by true ratiocination, of appearances, or apparent
effects, from the knowledge we have of some pos¬
sible production or generation of the same ; and
vol. i. f
66
COMPUTATION OR LOGIC,
part i. 0f such production, as has been or may be, from
v T the knowledge we have of the effects. Method,
science defined therefore, 111 th.6 Study of philosophy, IS tll6
shortest way of finding out effects by their known
causes, or of causes by their known effects. But
we are then said to know any effect, when we
know that there be causes of the same, and in
what subject those causes are, and in what sub¬
ject they produce that effect, and in what manner
they ivork the same. And this is the science of
causes, or, as they call it, of the Sion. All other
science, which is called the on, is either percep¬
tion by sense, or the imagination, or memory
remaining after such perception.
The first beginnings, therefore, of knowledge,
are the phantasms of sense and imagination; and
that there be such phantasms we know well enough
by nature ; but to know why they be, or from
what causes they proceed, is the work of ratioci¬
nation ; which consists (as is said above, in the
1st Chapter, Art. 2) in composition, and division
or resolution. There is therefore no method, by
which we find out the causes of things, but is
/ either compositive or resolutive, or partly com¬
positive, and partly resolutive. And the resolutive
is commonly called analytical method, as the
compositive is called synthetical.
T . 2. It is common to all sorts of method, to pro-
It is easier 7 ±
known con- ceed from known things to unknown ; and this is
]ar than uni- manifest from the cited definition of philosophy.
tbatUicyaS But in knowledge by sense, the whole object is
and contrariiy more known, than any part thereof : as when we
it is easier •» j. ?
known con- see a man, the conception or whole idea of that
versailhansii- man is first or more known, than the particular
of method.
67
ideas of his being jig urate, animate, and rational; part i.
that is, we first see the whole man, and take —■—--
notice of his being, before we observe in him those wby diey
other particulars. And therefore in any know- °r ,what ave
x .... their causes.
ledge of the on, or that any thing is, the beginning
of our search is from the whole idea; and con-
trarily, in our knowledge of the Si on, or of the
causes of any thing, that is, in the sciences, we
have more knowledge of the causes of the parts
than of the whole. For the cause of the whole
is compounded of the causes of the parts ; but it
is necessary that we know the things that are to
be compounded, before we can know the whole
compound. Now, by parts, I do not here mean
parts of the thing itself, but parts of its nature ;
as, by the parts of man, I do not understand his
head, his shoulders, his arms, &c. but his figure,
quantity, motion, sense, reason, and the like;
which accidents being compounded or put together,
constitute the whole nature of man, but not the
man himself. And this is the meaning of that
common saying, namely, that some things are
more known to us, others more known to nature;
for I do not think that they, which so distinguish,
mean that something is known to nature, which
is known to no man ; and therefore, by those
things, that are more known to us, we are to
understand things we take notice of by our senses,
and, by more known to nature, those we acquire
the knowledge of by reason ; for in this sense it
is, that the whole, that is, those things that have
universal names, (which, for brevity's sake, I call
universal) are more known to us than the parts,
that is, such things as have names less universal,
f 2
68
COMPUTATION OR LOGIC.
(which I therefore call singular) ; and the causes
of the parts are more known to nature than the
cause of the whole; that is, universals than
singulars.
philosophers 3* In tlie stu(1Y of philosophy, men search after
seek to know, science either simply or indefinitely ; that is, to
know as much as they can, without propounding
to themselves any limited question; or they
enquire into the cause of some determined appear¬
ance, or endeavour to find out the certainty of
something in question, as what is the cause of
light, of heat, of gravity, of a figure propounded,
and the like ; or in what subject any propounded
accident is inherent; or what may conduce most
to the generation of some propounded effect from
many accidents ; or in what manner particular
causes ought to be compounded for the production
of some certain effect. Now, according to this
variety of things in question, sometimes the analy¬
tical method is to be used, and sometimes the
synthetical.
The first part, 4. But to those that search after science inde-
cfpils arePdis- finitely, which consists in the knowledge of the
covered, is causes of all things, as far forth as it may be
purely ana- . .
lyticai. attained, (and the causes of singular things are
compounded of the causes of universal or simple
things) it is necessary that they know the causes
of universal things, or of such accidents as are
common to all bodies, that is, to all matter, before
they can know the causes of singular things, that
is, of those accidents by which one thing is distin¬
guished from another. And, again, they must
know what those universal things are, before they
can know their causes. Moreover, seeing universal
OF METHOD.
69
things are contained in the nature of singular part i.
things, the knowledge of them is to be acquired -—4—-
by reason, that is, by resolution. For example, if
there be propounded a conception or idea of some
singular thing, as of a square, this square is to be
resolved into a plain, terminated with a certain
number of equal and straight lines and right
angles. For by this resolution we have these
things universal or agreeable to all matter, namely,
line, plain, (which contains superficies) termi¬
nated, angle, straightness, rectitude, and equality;
and if we can find out the causes of these, we may
compound them altogether into the cause of a
square. Again, if any man propound to himself
the conception of gold, he may, by resolving,
come to the ideas of solid, visible, heavy, (that is,
tending to the centre of the earth, or downwards)
and many other more universal than gold itself;
and these he may resolve again, till he come to
such things as are most universal. And in this
manner, by resolving continually, we may come to
know what those things are, whose causes being
first known severally, and afterwards compounded,
bring us to the knowledge of singular things.
I conclude, therefore, that the method of attaining
to the universal knowledge of things, is purely
analytical.
5. But the causes of universal things (of those, The highest
' * causes s,nd
at least, that have any cause) are manifest of most universal
themselves, or (as they say commonly) known to LreTinby
nature; so that they need 110 method at all; for themselves-
they have all but one universal cause, which is
motion. For the variety of all figures arises out
of the variety of those motions by which they are
70
COMPUTATION OR LOGIC.
part i. made ; and motion cannot be understood to have
-—v—- any other cause besides motion ; nor has the
variety of those things we perceive by sense, as of
colours, sounds, savours, &c. any other cause than
motion, residing partly in the objects that work
upon our senses, and partly in ourselves, in such
manner, as that it is manifestly some kind of
motion, though we cannot, without ratiocination,
come to know what kind. For though many
cannot understand till it be in some sort demon¬
strated to them, that all mutation consists in
motion; yet this happens not from any obscurity
in the thing itself, (for it is not intelligible that
anything can depart either from rest, or from the
motion it has, except by motion), but either by
having their natural discourse corrupted with
former opinions received from their masters, or
else for this, that they do not at all bend their
mind to the enquiring out of truth.
Method from By the knowledge therefore of universals,
principles p • i • •
found out, and of their causes (which are the first principles
science simply, by which we know the Sioti of things) we have in
what it is. pjace their definitions, (which are nothing
but the explication of our simple conceptions.)
For example, he that has a true conception of
place, cannot be ignorant of this definition, place
is that space which is possessed or filled ade¬
quately by some body ; and so, he that conceives
motion aright, cannot but know that motion is
the privation of one place, and the acquisition of
another. In the next place, we have their gene¬
rations or descriptions ; as (for example) that a
line is made by the motion of a point, superficies
by the motion of a line, and one motion by another
of method.
71
motion, &c. It remains, that we enquire what part r.
motion begets such and such effects ; as, what -—^—-
motion makes a straight line, and what a circular : Metho(]frora
° J ? principles
what motion thrusts, what draws, and by whatfound out>
i t i i tending to
way; what makes a thing which is seen or heard, science simply,
to be seen or heard sometimes in one manner, what u 1S'
sometimes in another. Now the method of this
kind of enquiry, is compositive. For first we are
to observe what effect a body moved produceth,
when we consider nothing in it besides its motion;
and we see presently that this makes a line, or
length ; next, what the motion of a long body
produces, which we find to be superficies; and so
forwards, till we see what the effects of simple
motion are; and then, in like manner, we are to
observe what proceeds from the addition, multipli¬
cation, subtraction, and division, of these motions,
and what effects, what figures, and what properties,
they produce; from which kind of contemplation
sprung that part of philosophy which is called
geometry.
From this consideration of what is produced by
simple motion, we are to pass to the consideration
of what effects one body moved worketh upon
another ; and because there may be motion in all
the several parts of a body, yet so as that the
whole body remain still in the same place, we
must enquire first, what motion causeth such and
such motion in the whole, that is, when one body
invades another body which is either at rest or in
motion, what way, and with what swiftness, the
invaded body shall move ; and, again, what motion
this second body will generate in a third, and so
forwards. From which contemplation shall be
72
computation or logic.
part i. drawn that part of philosophy which treats of
-—r—- motion.
Method from in the third place we must proceed to the
principles *
found out, enquiry of such effects as are made by the motion
science"simply, of the parts of any body, as, how it comes to
what it is. pass, that things when they are the same, yet
seem not to be the same, but changed. And here
the things we search after are sensible qualities,
such as light, colour, transparency, opacity,
sound, odour, savour, heat, cold, and the like ;
which because they cannot be known till we
know the causes of sense itself, therefore the
consideration of the causes of seeing, hearing,
smelling, tasting, and touching, belongs to this
third place; and all those qualities and changes,
above mentioned, are to be referred to the fourth
place; which two considerations comprehend
that part of philosophy which is called physics.
And in these four parts is contained whatsoever
in natural philosophy may be explicated by
demonstration, properly so called. For if a cause
were to be rendered of natural appearances in
special, as, what are the motions and influences of
the heavenly bodies, and of their parts, the reason
hereof must either be drawn from the parts of the
sciences above mentioned, or no reason at all will
be given, but all left to uncertain conjecture.
After physics we must come to moral philo¬
sophy ; in which we are to consider the motions
of the mind, namely, appetite, aversion, love,
benevolence, hope, fear, anger, emulation, envy,
cfc.; what causes they have, and of what they
be causes. / And the reason why these are to
be considered after physics is, that they have
of method.
73
their causes in sense and imagination, which are part i.
the subject of physical contemplation. Also the r—-
reason, why all these things are to be searched
after in the order above-said, is, that physics
cannot be understood, except we know first what
motions are in the smallest parts of bodies; nor
such motion of parts, till we know what it is that
makes another body move ; nor this, till we know
what simple motion will effect./ And because all
appearance of things to sense is determined, and
made to be of such and such quality and quantity
by compounded motions, every one of which has a
certain degree of velocity, and a certain and
determined way ; therefore, in the first place, we
we are to search out the ways of motion simply
(in which geometry consists) ; next the ways of
such generated motions as are manifest; and,
lastly, the ways of internal and invisible motions
(which is the enquiry of natural philosophers).
And, therefore, they that study natural philosophy,
study in vain, except they begin at geometry ;
and such writers or disputers thereof, as are l
ignorant of geometry, do but make their readers
and hearers lose their time.
7. Civil and moral philosophy do not so adhere That method
11 -a -| , , of civil and na-
to one another, but that they may be severed. tural science,
For the causes of the motions of the mind are j^msensf to
known, not only by ratiocination, but also by the is
experience of every man that takes the pains to and again,that
observe those motions within himself. And, at ' principles
therefore, not only they that have attained the1S synthetlcaL
knowledge of the passions and perturbations of
the mind, by the synthetical method, and from
the very first principles of philosophy, may by
7 4
COMPUTATION OR LOGIC.
part i. proceeding in the same way, come to the causes
•—r—- and necessity of constituting commonwealths, and
ofciviiTndn°ad to get the knowledge of what is natural right, and
turai science, what; are civii duties ; and, in every kind of
proceeding 7 J J
from sense government, what are the rights of the common-
is analytical; wealth, and all other knowledge appertaining to
whtclfTgh^ civil philosophy ; for this reason, that the princi-
is'synthetical' P^es the politics consist in the knowledge of
the motions of the mind, and the knowledge of
these motions from the knowledge of sense and
imagination; but even they also that have not
learned the first part of philosophy, namely,
geometry and physics, may, notwithstanding,
attain the principles of civil philosophy, by the
analytical method. For if a question be pro-
^ pounded, as, whether such an action be just or
unjust; if that unjust be resolved into fact against
law, and that notion law into the command of him
or them that have coercive power; and that
power be derived from the wills of men that con¬
stitute such power, to the end they may live in
peace, they may at last come to this, that the
appetites of men and the passions of their minds
are such, that, unless they be restrained by some
power, they will always be making war upon one
another; which may be known to be so by any
man's experience, that will but examine his own
mind. And, therefore, from hence he may pro¬
ceed, by compounding, to the determination of
the justice or injustice of any propounded action.
So that it is manifest, by what has been said, that
the method of philosophy, to such as seek science
simply, without propounding to themselves the
solution of any particular question, is partly
of method.
75
analytical, and partly synthetical; namely, that part i.
which proceeds from sense to the invention of —>—V
principles, analytical; and the rest synthetical.
8. To those that seek the cause of some certain The method
and propounded appearance or effect, it happens, out,Swhethe?
sometimes, that they know not whether the thing, *"^3^
whose cause is sought after, be matter or body,ormittcrorac-
f ' cident
some accident of a body. For though in geometry,
when the cause is sought of magnitude, or propor¬
tion, or figure, it be certainly known that these
things, namely magnitude, proportion, and figure,
are accidents ; yet in natural philosophy, where all
questions are concerning the causes of the phan¬
tasms of sensible things, it is not so easy to
discern between the things themselves, from which
those phantasms proceed, and the appearances of
those things to the sense; which have deceived
many, especially when the phantasms have been
made by light. For example, a man that looks
upon the sun, has a certain shining idea of the
magnitude of about a foot over, and this he calls
the sun, though he know the sun to be truly a
great deal bigger ; and, in like manner, the phan¬
tasm of the same thing appears sometimes round,
by being seen afar off, and sometimes square, by
being nearer. Whereupon it may well be doubted,
whether that phantasm be matter, or some body
natural, or only some accident of a body; in the ex¬
amination of which doubt we may use this method.
The properties of matter and accidents already
found out by us, by the synthetical method, from
their definitions, are to be compared with the idea
we have before us ; and if it agree with the pro¬
perties of matter or body, then it is a body; other-
76
COMPUTATION OR LOGIC.
part i. wise it is an accident. Seeing, therefore, matter
-—r— cannot by any endeavour of ours be either made or
destroyed, or increased, or diminished, or moved
out of its place, whereas that idea appears, vanishes,
is increased and diminished, and moved hither and
thither at pleasure; we may certainly conclude
that it is not a body, but an accident only. And
this method is synthetical.
The method 9 gut if there be a doubt made concerning the
01 seeking # °
whether any subject of anv known accident (for this mav be
accident be,,, " . , * .
in this or in doubted sometimes, as m the precedent example,
that subject, doubt may mac[e in wliat subject that splendour
and apparent magnitude of the sun is), then our
enquiry must proceed in this manner. First,
matter in general must be divided into parts,
as, into object, medium, and the sentient itself, or
such other parts as seem most conformable to the
thing propounded. Next, these parts are severally
to be examined how they agree with the definition
of the subject; and such of them as are not
capable of that accident are to be rejected. For
example, if by any trae ratiocination the sun be
found to be greater than its apparent magnitude,
then that magnitude is not in the sun ; if the sun
be in one determined straight line, and one deter¬
mined distance, and the magnitude and splendour
be seen in more lines and distances than one, as it
is in reflection or refraction, then neither that
splendour nor apparent magnitude are in the sun
itself, and, therefore, the body of the sun cannot
be the subject of that splendour and magnitude.
And for the same reasons the air and other parts
will be rejected, till at last nothing remain which
can be the subject of that splendour and mag-
of method.
77
nitude but the sentient itself. And this method, part i.
in regard the subject is divided into parts, is r—-
analytical; and in regard the properties, both of
the subject and accident, are compared with the
accident concerning whose subject the enquiry is
made, it is synthetical.
10. But when we seek after the cause of any Method of
propounded effect, we must in the first place get ^cause fof
into our mind an exact notion or idea of that
which we call cause, namely, that a cause is the
sum or aggregate of all such accidents, both in
the agents and the patient, as concur to the
producing of the effect propounded; all which
existing together, it cannot be understood but
that the effect existeth with them; or that
it can jjossibly exist if any one of them be
absent. This being known, in the next place we
must examine singly every accident that accom¬
panies or precedes the effect, as far forth as it
seems to conduce in any manner to the production
of the same, and see whether the propounded
effect may be conceived to exist, without the
existence of any of those accidents; and by this
means separate such accidents, as do not concur,
from such as concur to produce the said effect;
which being done, we are to put together the
concurring accidents, and consider whether we
can possibly conceive, that when these are all
present, the effect propounded will not follow;
and if it be evident that the effect will follow,
then that aggregate of accidents is the entire
cause, otherwise not; but we must still search out
and put together other accidents. For example,
if the cause of light be propounded to be sought
78
computation or logic.
part i. out. we examine things without us, and find
1^—- that whensoever light appears, there is some prin-
searching for cipal object, as it were the fountain of light,
Iry caieffec°tf without which we cannot have any pereeption of
propounded, light; and, therefore, the concurrence of that
object is necessary to the generation of light.
Next we consider the medium, and find, that
unless it be disposed in a certain manner, namely,
that it be transparent, though the object remain
the same, yet the effect will not follow; and,
therefore, the concurrence of transparency is also
necessary to the generation of light. Thirdly, we
observe our own body, and find that by the indis¬
position of the eyes, the brain, the nerves, and the
heart, that is, by obstructions, stupidity, and
debility, we are deprived of light, so that a fitting
disposition of the organs to receive impressions
from without is likewise a necessary part of the
cause of light. Again, of all the accidents inherent
in the object, there is none that can conduce to
the effecting of light, but only action (or a certain
motion), which cannot be conceived to be wanting,
whensoever the effect is present; for, that anything
may shine, it is not requisite that it be of such or
such magnitude or figure, or that the whole
body of it be moved out of the place it is in (unless
it may perhaps be said, that in the sun, or other
body, that which causes light is the light it hath
in itself; which yet is but a trifling exception,
seeing nothing is meant thereby but the cause of
light; as if any man should say that the cause of
light is that in the sun which produceth it); it
remains, therefore, that the action, by which light
is generated, is motion only in the parts of the
of method.
79
object. Which being understood, we may easily tart i.
conceive what it is the medium contributes, ^—-
namely, the continuation of that motion to the
eye ; and, lastly, what the eye and the rest of the
organs of the sentient contribute, namely, the
continuation of the same motion to the last organ
of sense, the heart. And in this manner the cause
of light may be made up of motion continued
from the original of the same motion, to the
original of vital motion, light being nothing but
the alteration of vital motion, made by the impres¬
sion upon it of motion continued from the object.
But I give this only for an example, for I shall
speak more at large of light, and the generation of
it, in its proper place. In the mean time it is
manifest, that in the searching out of causes, there
is need partly of the analytical, and partly of the
synthetical method; of the analytical, to con¬
ceive how circumstances conduce severally to the
production of effects; and of the synthetical, for
the adding together and compounding of what they
can effect singly by themselves. And thus much
may serve for the method of invention. It remains
that I speak of the method of teaching, that is, of
demonstration, and of the means by which we
demonstrate.
11. In the method of invention, the use oft^0^sve™
words consists in this, that they may serve for as marks; to
_ demonstration
marks, by which, whatsoever we have round out as signs,
may be recalled to memory ; for without this all
our inventions perish, nor will it be possible for
us to go on from principles beyond a syllogism
or two, by reason of the weakness of memory.
For example, if any man, by considering a triangle
80
computation or logic.
part i. set before him, should find that all its angles
^— together taken are equal to two right angles, and
that by thinking of the same tacitly, without any
use of words either understood or expressed; and
it should happen afterwards that another triangle,
unlike the former, or the same in different situa¬
tion, should be offered to his consideration, he
would not know readily whether the same pro¬
perty were in this last or no, but would be forced,
as often as a different triangle were brought before
him (and the difference of triangles is infinite) to
begin his contemplation anew ; which he would
have 110 need to do if he had the use of names,
for every universal name denotes the conceptions
we have of infinite singular things. Nevertheless,
as I said above, they serve as marks for the help
of our memory, whereby we register to ourselves
our own inventions ; but not as signs by which
we declare the same to others ; so that a man may
be a philosopher alone by himself, without any
master; Adam had this capacity. But to teach,
that is, to demonstrate, supposes two at the least,
and syllogistical speech.
The method of 12. And seeing teaching is nothing but leading
^synthetfcaT the mind of him we teach, to the knowledge of
our inventions, in that track by which we attained
the same with our own mind; therefore, the same
method that served for our invention, will serve
also for demonstration to others, saving that we
omit the first part of method which proceeded
from the sense of things to universal principles,
which, because they are principles, cannot be
demonstrated ; and seeing they are known by
nature, (as was said above in the 5th article) they
of method.
81
need no demonstration, though they need expli- tart i.
cation. The whole method, therefore, of demon- -—■A—-
stration, is synthetical, consisting of that order of
speech which begins from primary or most
universal propositions, which are manifest of
themselves, and proceeds by a perpetual com¬
position of propositions into syllogisms, till at
last the learner understand the truth of the
conclusion sought after.
13. Now, such principles are nothing but defi- Definitions
± _M.— a— only are
nitions, whereof there are two sorts; one of primary,
11 , • • j? i & universal
names, that signily such things as have some con- propositions,
ceivable cause, and another of such names as
signify things of which we can conceive no cause
at all. Names of the former kind are, body, or
matter, quantity, or extension, motion, and what¬
soever is common to all matter. Of the second
kind, are such a body, such and so great motion,
so great magnitude, such figure, and whatsoever
we can distinguish one body from another by.
And names of the former kind are Avell enough
defined, when, by speech as short as may be, we
raise in the mind of the hearer perfect and clear
ideas or conceptions of the things named, as when
we define motion to be the leaving of one place,
and the acquiring of another continually ; for
though no thing moved, nor any cause of motion
be in that definition, yet, at the hearing of that
speech, there will come into the mind of the
hearer an idea of motion clear enough. But
definitions of things, which may be understood to
have some cause, must consist of such names as
express the cause or manner of their generation,
as when we define a circle to be a figure made by
vol. i. g
82
COMPUTATION OR LOGIC.
part i. the circumduction of a straight line in a plane, &c.
1Besides definitions, there is no other proposition
mfiy are°lls that ought to be called primary, or (according
primary, seVere truth) be received into the number of
& universal '
propositions, principles. For those axioms of Euclid, seeing
they may be demonstrated, are no principles of
demonstration, though they have by the consent of
all men gotten the authority of principles, because
they need not be demonstrated. Also, those
petitions, or postulata, (as they call them) though
they be principles, yet they are not principles of
demonstration, but of construction only ; that is,
not of science, but of power; or (which is all one)
not of theorems, which are speculations, but of
problems, which belong to practice, or the doing
of something. But as for those common received
opinions. Nature abhors vacuity, Nature doth
nothing in vain, and the like, which are neither
evident in themselves, nor at all to be demon¬
strated, and which are oftener false than true,
they are much less to be acknowledged for
principles.
To return, therefore, to definitions; the reason
why I say that the cause and generation of such
things, as have any cause or generation, ought to
enter into their definitions, is this. The end of
science is the demonstration of the causes and
generations of things ; which if they be not in the
definitions, they cannot be found in the conclusion
of the first syllogism, that is made from those
definitions; and if they be not in the first con¬
clusion, they will not be found in any further
conclusion deduced from that; and, therefore, by
proceeding in this manner, we shall never come to
of method.
83
science ; which is against the scope and intention part r.
of demonstration. r—-
14. Now, seems; definitions (as I have said) are The nature
. ■, . ... & definition
principles, or primary propositions, they are there- of a definition,
fore speeches; and seeing they are used for the
raising of an idea of some thing in the mind of
the learner, whensoever that thing has a name,
the definition of it can be nothing but the expli¬
cation of that name by speech ; and if that name
be given it for some compounded conception, the
definition is nothing but a resolution of that name
into its most universal parts. As when we define
man, saying man is a body animated', sentientL,
rational, those names, body animated, cfc. are
parts of that whole name man; so that definitions \
of this kind always consist of genus and difference; t
the former names being all, till the last, general; |
and the last of all, difference. But if any name
be the most universal in its kind, then the defini¬
tion of it cannot consist of genus and difference,
but is to be made by such circumlocution, as best
explicateth the force of that name. Again, it is
possible, and happens often, that the genus and
difference are put together, and yet make no
definition ; as these words, a straight line, contain
both the genus and difference; but are not a
definition, unless we should think a straight line
may be thus defined, a straight line is a straight
line: and yet if there were added another name,
consisting of different words, but signifying the
same thing which these signify, then these might
be the definition of that name. From what has
been said, it may be understood how a defini¬
tion ought to be defined, namely, that it is a
g 2
84
computation or logic.
part i. proposition, whose predicate resolves the subject,
" —' when it may ; and when it may not, it exemplifies
the same.
Properties of 15. The properties of a definition are :
a definition. . . .
First, that it takes away equivocation, as also
all that multitude of distinctions, which are used
by such as think they may learn philosophy by
disputation. For the nature of a definition is to
define, that is, to determine the signification of
the defined name, and to pare from it all other
signification besides what is contained in the
definition itself; and therefore one definition does
as much, as all the distinctions (how many soever)
that can be used about the name defined.
Secondly, that it gives an universal notion of
the thing defined, representing a certain universal
picture thereof, not to the eye, but to the mind.
For as when one paints a man, he paints the image
of some man ; so he, that defines the name man,
makes a representation of some man to the mind.
Thirdly, that, it is not necessary to dispute
whether definitions are to be admitted or no. For
when a master is instructing his scholar, if the
scholar understand all the parts of the thing
defined, which are resolved in the definition, and
yet will not admit of the definition, there needs no
further controversy betwixt them, it being all one
as if he refused to be taught. But if he under¬
stand nothing, then certainly the definition is
faulty ; for the nature of a definition consists in
this, that it exhibit a clear idea of the thing defined;
and principles are either known by themselves, or
else they are not principles.
Fourthly, that, in philosophy, definitions are
OF METHOD.
85
before defined names. For in teaching philosophy, PA^T T-
the first beginning is from definitions ; and all pro- -—A—-
gression in the same, till we come to the knowledge aEk?on°f
of the thing compounded, is compositive. Seeing,
therefore, definition is the explication of a com¬
pounded name by resolution, and the progression
is from the parts to the compound, definitions
must be understood before compounded names ;
nay, when the names of the parts of any speech
be explicated, it is not necessary that the definition
should be a name compounded of them. For
example, when these names, equilateral, quadri¬
lateral, right-angled, are sufficiently understood,
it is not necessary in geometry that there should
be at all such a name as square; for defined
names are received in philosophy for brevity's
sake only.
Fifthly, that compounded names, which are de¬
fined one way in some one part of philosophy,
may in another part of the same be otherwise
defined; as a parabola and an hyperbole have
one definition in geometry, and another in rhetoric;
for definitions are instituted and serve for the
understanding of the doctrine which is treated of.
And, therefore, as in one part of philosophy, a
definition may have in it some one fit name for
the more brief explanation of some proposition in
geometry ; so it may have the same liberty in
other parts of philosophy ; for the use of names is
particular (even where many agree to the settling
of them) and arbitrary.
Sixthly, that no name can be defined by any
one word ; because no one word is sufficient for
the resolving of one or more words.
86
computation or logic.
part i. Seventhly, that a defined name ought not to be
—r—- repeated in the definition. For a defined name is
the whole compound, and a definition is the reso¬
lution of that compound into parts ; but no total
can be part of itself.
nature of a 16. Any two definitions, that may be com-
dumonstration. p0Unded into a syllogism, produce a conclusion;
which, because it is derived from principles, that
is, from definitions, is said to be demonstrated;
and the derivation or composition itself is called a
demonstration. In like manner, if a syllogism be
made of two propositions, whereof one is a defi¬
nition, the other a demonstrated conclusion, or
neither of them is a definition, but both formerly
demonstrated, that syllogism is also called a de¬
monstration, and so successively. The definition
therefore of a demonstration is this, a demonstra¬
tion is a syllogism, or series of syllogisms,
derived and continued, from the definitions of
names, to the last conclusion. And from hence it
may be understood, that all true ratiocination,
which taketh its beginning from true principles,
produceth science, and is true demonstration.
For as for the original of the name, although that,
which the Greeks called avoSk^g, and the Latins
demonstration was understood by them for that
sort only of ratiocination, in which, by the de¬
scribing of certain lines and figures, they placed
the thing they were to prove, as it were before
men's eyes, which is properly (itto^ikvvhv, or to
shew by the figure ; yet they seem to have done it
for this reason, that unless it were in geometry,
(in which only there is place for such figures)
there was no ratiocination certain, and ending in
OF METHOD.
87
science, their doctrines concerning all other things part i.
being nothing but controversy and clamour; —-
which, nevertheless, happened, not because the
truth to which they pretended could not be made
evident without figures, but because they wanted
true principles, from which they might derive
their ratiocination ; and, therefore, there is no
reason but that if true definitions were premised
in all sorts of doctrines, the demonstrations also
would be true.
17- It is proper to methodical demonstration, Properties of a
First, that there be a true succession of one
reason to another, according to the rules of svllo- t1hings t0 be
0 J demonstrated.
gizing delivered above.
Secondly, that the premises of all syllogisms be
demonstrated from the first definitions.
Thirdly, that after definitions, he that teaches
or demonstrates any thing, proceed in the same
method by which he found it out; namely, that
in the first place those things be demonstrated,
which immediately succeed to universal definitions
(in which is contained that part of philosophy
which is called plulosophia prima). Next, those
things wrhich may be demonstrated by simple
motion (in which geometry consists). After
geometry, such tilings as may be taught or shewed
by manifest action, that is, by thrusting from, or
pulling towards. And after these, the motion 01*
mutation of the invisible parts of things, and the
doctrine of sense and imaginations, and of the
internal passions, especially those of men, in which
are comprehended the grounds of civil duties, or
civil philosophy ; which takes up the last place.
And that this method ought to be kept in all sorts of
philosophy, is evident from hence, that such things
88
COMPUTATION OR LOGIC.
part i. as I have said are to be taught last, cannot be de-
r— monstrated, till such as are propounded to be first
treated of, be fully understood. Of which method
no other example can be given, but that treatise
of the elements of philosophy, which I shall begin
in the next chapter, and continue to the end of
the work.
Faults of a 18. Besides those paralogisms, whose fault lies
demonstration, the falsity of the premises, or the want
of true composition, of which I have spoken in
the precedent chapter, there are two more, which
are frequent in demonstration ; one whereof is
commonly called petit to principii; the other is
the supposing of a false cause ; and these do not
only deceive unskilful learners, but sometimes
masters themselves, by making them take that for
well demonstrated, which is not demonstrated at
all. Petitio principii is, when the conclusion to
be proved is disguised in other words, and put
for the definition or principle from whence it is
to be demonstrated; and thus, by putting for the
cause of the thing sought, either the thing itself or
some effect of it, they make a circle in their
demonstration. As for example, he that would
demonstrate that the earth stands still in the
centre of the world, and should suppose the earth's
gravity to be the cause thereof, and define gravity
to be a quality by which every heavy body tends
towards the centre of the world, would lose his
labour; for the question is, what is the cause of
that quality in the earth ? and, therefore, he that
supposes gravity to be the cause, puts the thing
itself for its own cause.
Of a false cause I find this example in a cer¬
tain treatise where the thing to be demonstrated
OF METHOD.
89
is the motion of the earth. He begins, therefore, PAI^'r L
with this, that seeing the earth and the sun are -—-—'
not always in the same situation, it must needs be
that one of them be locally moved, which is true ;
next, he affirms that the vapours, which the sun
raises from the earth and sea, are, by reason of
this motion, necessarily moved, which also is true;
from whence he infers the winds are made, and
this may pass for granted ; and by these winds he
says, the waters of the sea are moved, and by
their motion the bottom of the sea, as if it were
beaten forwards, moves round; and let this also
be granted; wherefore, he concludes, the earth is
moved ; which is, nevertheless, a paralogism. For,
if that wind were the cause why the earth was,
from the beginning, moved round, and the motion
either of the sun or the earth' were the cause of
that wind, then the motion of the sun or the earth
was before the wind itself; and if the earth were
moved, before the wind was made, then the wind
could not be the cause of the earth's revolution ;
but, if the sun were moved, and the earth stand
still, then it is manifest the earth might remain
unmoved, notwithstanding that wind ; and there¬
fore that motion was not made by the cause which
he allegeth. But paralogisms of this kind are
very frequent among the writers of physics,
though none can be more elaborate than this in
the example given.
19. It may to some men seem pertinent to treat w^y ,the auna-
J L lytical method
in this place of that art of the geometricians, of geometri-
which tliey call logistica, that is, the art, by be treated of
which, from supposing the thing in question to be in thls place*
true, they proceed by ratiocination, till either they
come to something known, by which they may
90
COMPUTATION OR LOGIC.
part i. demonstrate the truth of the thing sought for; or
-—A—- to something which is impossible, from whence
Micailnctw collect that to be false, which they supposed
of geometri- true. But this art cannot be explicated here, for
cians cannot . .
be treated of this reason, that the method of it can neither be
m tins place. pracj.jge(^ nor understood, unless by such as are
well versed in geometry ; and among geometri¬
cians themselves, they, that have most theorems in
readiness, are the most ready in the use of this
logistic a ; so that, indeed, it is not a distinct
thing from geometry itself; for there are, in the
method of it, three parts ; the first whereof con¬
sists in the finding out of equality betwixt known
and unknown things, which they call equation;
and this equation cannot be found out, but by such
as know perfectly the nature, properties, and
transpositions of proportion, as also the addition,
subtraction, multiplication, and division of lines
and superficies, and the extraction of roots ; which
are the parts of no mean geometrician. The
second is, when an equation is found, to be able to
judge whether the truth or falsity of the question
may be deduced from it, or no ; which yet requires
greater knowledge. And the third is, when such
an equation is found, as is fit for the solution of
the question, to know how to resolve the same in
such manner, that the truth or falsity may there¬
by manifestly appear; which, in hard questions,
cannot be done without the knowledge of the
nature of crooked-lined figures; but he that un¬
derstands readily the nature and properties of
these, is a complete geometrician. It happens
besides, that for the finding out of equations, there
is no certain method, but he is best able to do it,
that has the best natural wit.
PART II.
the
FIRST GROUNDS OF PHILOSOPHY.
CHAPTER VII.
of place and time.
I. Things that have no existence, may nevertheless be under¬
stood and computed.—2. What is Space.—3. Time.—4% Part.
5. Division. — 6. One.—7. Number. — 8. Composition.—
9. The whole.—10. Spaces and times contiguous, and con¬
tinual.—11. Beginning, end, way, finite, infinite.—12. What
is infinite in power. Nothing infinite can be truly said to be
either whole, or one; nor infinite spaces or times, many.—
13. Division proceeds not to the least.
1. In the teaching of natural philosophy, I cannot part ii.
begin better (as I have already shewn) than from -—•
privation ; that is, from feigning the world to beJ^^jt
annihilated. But, if such annihilation of all things >stence, may
nevertheless
be supposed, it may perhaps be asked, what would be understood
/. / -i it i computed.
remain for any man (whom only I except from
this universal annihilation of things) to consider
as the subject of philosophy, or at all to reason
upon; or what to give names unto for ratiocina¬
tion's sake.
92
philosophy.
part ii. j say5 therefore, there would remain to that man
-—'—- ideas of the world, and of all such bodies as he
have nVeS before their annihilation, seen with his eyes,
istence, may or perceived by any other sense; that is to say,
nevertheless r ... .
be understood the memory and imagination of magnitudes,
compute . mot*onS) colours, &e. as also of their order
and parts. All which things, though they be
nothing but ideas and phantasms, happening in¬
ternally to him that imagineth; yet they will
appear as if they were external, and not at all
depending upon any power of the mind. And
these are the things to which he would give
names, and subtract them from, and compound
them with one another. For seeing, that after the
destruction of all other things, I suppose man
still remaining, and namely that he thinks, ima¬
gines, and remembers, there can be nothing for
him to think of but what is past; nay, if we do
but observe diligently what it is we do when we
consider and reason, we shall find, that though
all things be still remaining in the world, yet we
compute nothing but our own phantasms. For
when we calculate the magnitude and motions of
heaven or earth, we do not ascend into heaven
that we may divide it into parts, or measure the
motions thereof, but we do it sitting still in our
closets or in the dark. Now things may be con¬
sidered, that is, be brought into account, either as
internal accidents of our mind, in which manner
we consider them when the question is about
some faculty of the mind ; or as species of external
things, not as really existing, but appearing only
to exist, or to have a being without us. And in
this manner we are now to consider them.
OF PLACE AND TIME.
93
2. If therefore we remember, or have a phantasm part ir.
of any thing that was in the world before the — -
supposed annihilation of the same ; and consider, ^space
not that the thing was such or such, but only that
it had a being without the mind, we have pre¬
sently a conception of that we call space : an
imaginary space indeed, because a mere phantasm,
yet that very thing which all men call so. For no
man calls it space for being already filled, but
because it may be filled; nor does any man
think bodies carry their places away with them,
but that the same space contains sometimes one,
sometimes another body; which could not be if
space should always accompany the body which is
once in it. And this is of itself so manifest, that
I should not think it needed any explaining at all,
but that I find space to be falsely defined by
certain philosophers, who infer from thence, one,
that the world is infinite (for taking space to be
the extension of bodies, and thinking extension
may encrease continually, he infers that bodies
may be infinitely extended); and, another, from the
same definition, concludes rashly, that it is im¬
possible even to God himself to create more
worlds than one ; for, if another world were to be
created, he says, that seeing there is nothing
without this world, and therefore (according to his
definition) no space, that new world must be
placed in nothing; but in nothing nothing can be
placed; which he affirms only, without showing
any reason for the same; whereas the contrary is
the truth : for more cannot be put into a place
already filled, so much is empty space fitter than
that, which is full, for the receiving of new bodies.
94
philosophy.
part ii. Having therefore spoken thus much for these
- men's sakes, and for theirs that assent to them,
I return to my purpose, and define space thus:
space is the phantasm of a thing existing without
the mind simply ; that is to say, that phantasm,
in which we consider 110 other accident, but only
that it appears without us.
Time. 3, As a body leaves a phantasm of its magnitude
in the mind, so also a moved body leaves a
phantasm of its motion, namely, an idea of that
body passing out of one space into another by
continual succession. And this idea, or phantasm,
is that, which (without receding much from the
common opinion, or from Aristotle's definition)
I call Time. For seeing all men confess a year
to be time, and yet do not think a year to be
the accident or affection of any body, they must
needs confess it to be, not in the things without
us, but only in the thought of the mind. So
when they speak of the times of their predecessors,
they do not think after their predecessors are
gone, that their times can be any where else than
in the memory of those that remember them.
And as for those that say, days, years, and months
are the motions of the sun and moon, seeing it is
all one to say, motion past and motion destroyed,
and that future motion is the same with motion
which is not yet begun, they say that, which they
do not mean, that there neither is, nor has been,
nor shall be any time: for of whatsoever it may
be said, it has been or it shall be, of the same
also it might have been said heretofore, or may
be said hereafter, it is. What then can days,
months, and years, be, but the names of such
OF PLACE AND TIME.
95
computations made in our mind ? Time therefore part ii.
is a phantasm, but a phantasm of motion, for if --
we would know by what moments time passes Time'
away, we make use of some motion or other, as
of the sun, of a clock, of the sand in an hour¬
glass, or we mark some line upon which we
imagine something to be moved, there being no
other means by which we can take notice of any
time at all. And yet, when I say time is a phan¬
tasm of motion, I do not say this is sufficient to
define it by; for this word time comprehends the
notion of former and latter, or of succession
in the motion of a body, in as much as it is first
here then there. Wherefore a complete definition
of time is such as this, time is the phantasm of
before and after in motion; which agrees with
this definition of Aristotle, time is the number of
motion according to former and latter ; for that
numbering is an act of the mind ; and therefore
it is all one to say, time is the number of motion
according to former and latter ; and time is a
phantasm of motion numbered. But that other
definition, time is the measure of motion, is not
so exact, for we measure time by motion and
not motion by time.
4. One space is called part of another space, Part,
and one time part of another time, when this
contains that and something besides. From
whence it may be collected, that nothing can
rightly be called a part, but that which is com¬
pared with something that contains it.
5. And therefore to make parts, or to part or Division.
divide space or time, is nothing else but to con¬
sider one and another within the same ; so that
96
philosophy.
part ii. if any man iUvide space or time, tlie diverse
v—■—- conceptions he has are more, by one, than the
Division. parts makes . for conception is of that
which is to be divided, then of some part of it,
and again of some other part of it, and so
forwards as long as he goes on in dividing.
But it is to be noted, that here, by division, I
do not mean the severing or pulling asunder of
one space or time from another (for does any
man think that one hemisphere may be separated
from the other hemisphere, or the first hour from
the second ?) but diversity of consideration ; so
that division is not made by the operation of the
hands but of the mind,
one. g When space or time is considered among
other spaces or times, it is said to be one, namely
one of them; for except one space might be
added to another, and subtracted from another
space, and so of time, it would be sufficient to
say space or time simply, and superfluous to say
one space or one time, if it could not be conceived
that there were another. The common definition
of one, namely, that one is that which is undivided,
is obnoxious to an absurd consequence ; for it may
thence be inferred, that whatsoever is divided is
many things, that is, that every divided thing, is
divided things, which is insignificant.
Number. 7. Number is one and one, or one one and one,
and so forwards ; namely, one and one make the
number two, and one one and one the number
three ; so are all other numbers made; which is
all one as if we should say, number is unities.
composition, g# To compound space of spaces, or time of
times, is first to consider them one after another,
of place and time.
9/
and then altogether as one ; as if one should part ii.
reckon first the head, the feet, the arms, and the •——-
body, severally, and then for the account of them
all together put man. And that which is so put
for all the severals of which it consists, is called
the whole ; and those severals, when by the
division of the whole they come again to be
considered singly, are parts thereof; and therefore
the whole and all the parts taken together are
the same thing. And as I noted above, that in
division it is not necessary to pull the parts
asunder; so in composition, it is to be understood,
that for the making up of a whole there is 110
need of putting the parts together, so as to make
them touch one another, but only of collecting
them into one sum in the mind. For thus all men,
being considered together, make up the whole of
mankind, though never so much dispersed by time
and place ; and twelve hours, though the hours of
several days, may be compounded into one number
of twelve.
9. This being well understood, it is manifest, The whole-
that nothing can rightly be called a whole, that is
not conceived to be compounded of parts, and that
it may be divided into parts ; so that if we deny
that a thing has parts, we deny the same to be a
whole. For example, if we say the soul can have
no parts, we affirm that 110 soul can be a whole
soul. Also it is manifest, that nothing has parts
till it be divided ; and when a thing is divided,
the parts are only so many as the division makes
them. Again, that a part of a part is a part of
the whole ; and thus any part of the number four,
as two, is a part of the number eight; for four is
vol. 1. 15
98
philosophy.
part ii. made of two and two; but eight is compounded
-—^—- of two, two, and four, and therefore two, which
is a part of the part four, is also a part of the
whole eight.
spaces and 10. Two spaces are said to be contiguous,
tiguTus°and when there is 110 other space betwixt them. But
continual. ^wo times, betwixt which there is no other time, are
called immediate, as A B, B C. ^ B C
And any two spaces, as well as — —
times, are said to be continual, when they have
one common part, as A C, 1> L), ^ g ^ ^
where the part B C is common; —-——
and more spaces and times are continual, when
every two which are next one another are
continual.
Beginning, n T}iat part which is between two other parts,
end, way, 1 .
finite, infinite, is called a mean ; and that which is not between
two other parts, an extreme. And of extremes,
that which is first reckoned is the beginning,
and that which last, the end; and all the means
together taken are the way. Also, extreme parts
and limits are the same thing. And from hence
it is manifest, that beginning and end depend
upon the order in which we number them; and
that to terminate or limit space and time, is the
same thing with imagining their beginning and
end; as also that every thing is finite or infi¬
nite, according as we imagine or not imagine it
limited or terminated every wTay ; and that the
limits of any number are unities, and of these,
that which is the first in our numbering is the
beginning, and that which we number last, is the
end. When we say number is infinite, we mean
only that no number is expressed; for when we
of place and time.
09
speak of the numbers two, three, a thousand, &c. part n-
they are always finite. But when no more is said *——-
but this, number is infinite, it is to be understood
as if it were said, this name number is an indefi¬
nite name.
12. Space or time is said to be finite in power, What is infi-
or terminable, when there may be assigned a NothingTfi-
number of finite spaces or times, as of paces or JJJJ c™d
hours, than which there can be no greater number be either whole
° _ or one; nor m-
of the same measure in that space or time; and spaces
infinite in power is that space or time, in which 1 es'many'
a greater number of the said paces or hours may
be assigned, than any number that can be given.
But we must note, that, although in that space or
time which is infinite in power, there may be
numbered more paces or hours than any number
th at can be assigned, yet their number will always
be finite ; for every number is finite. And there¬
fore his ratiocination was not good, that under¬
taking to prove the world to be finite, reasoned
thus; If the world he infinite, then there may be
taken in it some part which is distant f rom us an
infinite number of paces : but no such part can
be taken; wherefore the world is not infinite;
because that consequence of the major proposition
is false; for in an infinite space, whatsoever we take
or design in our mind, the distance of the same
from us is a finite space ; for in the very designing
of the place thereof, we put an end to that space,
of which we ourselves are the beginning ; and
whatsoever any man with his mind cuts off both
ways from infinite, he determines the same, that
is, he makes it finite.
Of infinite space or time, it cannot be said that
h 2
100
philosophy.
part ii. it is a whole or one: not a whole, because not
7.
-—■—' compounded of parts ; for seeing parts, how many
soever they be, are severally finite, they will also,
when they are all put together, make a whole
finite : nor one, because nothing can be said to be
one, except there be another to compare it with;
but it cannot be conceived that there are two
spaces, or two times, infinite. Lastly, when we
make question whether the world be finite or
infinite, we have nothing in our mind answering
to the name world; for whatsoever we imagine,
is therefore finite, though our computation reach
the fixed stars, or the ninth or tenth, nay, the
thousandth sphere. The meaning of the question
is this only, whether God has actually made so
great an addition of body to body, as we are able
to make of space to space.
Division 13. And, therefore, that which is commonly
rrlr sa^> *^at sPace and t^me lliay divided infinitely,
is not to be so understood, as if there might be
any infinite or eternal division; but rather to be
taken in this sense, whatsoever is divided, is
divided into such parts as may again be divided;
or thus, the least divisible thing is not to be
given; or, as geometricians have it, no quantity
is so small, but a less may be taken ; which may
easily be demonstrated in this manner. Let any
space or time, that which was thought to be the
least divisible, be divided into two equal parts, A
and B. I say either of them, as A, may be
divided again. For suppose the part A to be
contiguous to the part B of one side, and of the
other side to some other space equal to B. This
whole space, therefore, being greater than the
of place and time.
101
space given, is divisible. Wherefore, if it be i'art ii.
divided into two equal parts, the part in the -—-r—-
middle, which is A, will be also divided into two
equal parts ; and therefore A was divisible.
CHAPTER VIII.
of body and accident.
1. Body defined.—2. Accident defined.—3. How an accident
may be understood to be in its subject.—4. Magnitude, what
it is.—5. Place, what it is, and that it is immovable. •—
6. What is full and empty.-—7. Here, there, somewhere, what
they signify.—8. Many bodies cannot be in one place, nor
one body in many places.—9. Contiguous and continual, what
they are. —10. The definition of motion. No motion intelli¬
gible but with time.—11. What it is to be at rest, to have
been moved, and to be moved. No motion to be conceived,
without the conception of past and future.—12. A point, a
line, superficies and solid, what they are.—13. Equal, greater,
and less in bodies and magnitudes, what they are.—14. One
and the same body has always one and the same magnitude.
15. Velocity, what it is.—16. Equal, greater, and less in times,
what they are.—17* Equal, greater, and less, in velocity, what.
18. Equal, greater, and less, in motion, what.—19. That
which is at rest, will always be at rest, except it be moved by
some external thing; and that which is moved, will always be
moved, unless it be hindered by some external thing.—
20. Accidents are generated and destroyed, but bodies not so.
21. An accident cannot depart from its subject.—22. Nor be
moved. — 23. Essence, form, and matter, what they are.
24. First matter, what.—25. That the whole is greater than
any part thereof, why demonstrated.
1. Having understood what imaginary space is, Body defined,
in which we supposed nothing remaining without
us, but all those things to be destroyed, that, by
102
philosophy.
part ii. existing heretofore, left images of themselves in
r— our minds ; let us now suppose some one of those
things to be placed again in the world, or created
anew. It is necessary, therefore, that this new-
created or replaced thing do not only fill some
part of the space above mentioned, or be coinci¬
dent and 'coextended with it, but also that it have
no dependance upon our thought. And this is
that which, for the extension of it, we commonly
call body; and because it depends not upon our
thought, we say is a thing subsisting of itself;
as also existing, because without us; and, lastly,
it is called the subject, because it is so placed in
and subjected to imaginary space, that it may be
understood by reason, as well as perceived by
sense. The definition, therefore, of body may be
this, a body is that, which having no dependance
upon our thought, is coincident or coextended
with some part of space.
Accident 2. But what an accident is cannot so easily be
explained by any definition, as by examples. Let
us imagine, therefore, that a body fills any space,
or is coextended with it; that coextension is not
the coextended body: and, in like manner, let us
imagine that the same body is removed out of its
place ; that removing is not the removed body : or
let us think the same not removed; that not
removing or rest is not the resting body. What,
then, are these things ? They are accidents of
that body. But the thing in question is, what is
an accident ? which is an enquiry after that which
we know already, and not that which we should
enquire after. For who does not always and in
the same manner understand him that says any
of body and accident.
103
tiling is extended, or moved, or not moved ? But part ir.
most men will have it be said that an accident is -—-
something, namely, some part of a natural thing, ^ecf^nt
when, indeed, it is no part of the same. To satisfy
these men, as well as may be, they answer best
that define an accident to be the manner by which
any body is conceived; which is all one as if they
should say, an accident is that faculty of any
body, by which it works in lis a conception of
itself. Which definition, though it be not an
answer to the question propounded, yet it is an
answer to that question which should have been
propounded, namely, whence does it happen that
one part of any body appears here, another
there ? For this is well answered thus : it happens
from the extension of that body. Or, how comes
it to pass that the whole body, by succession, is
seen now here, now there ? and the answer will be,
by reason of its motion. Or, lastly, whence is it
that any body possesseth the same space for
sometime ? and the answer will be, because it is
not moved. For if concerning the name of a
body, that is, concerning a concrete name, it be
asked, ivhat is it ? the answer must be made by
definition ; for the question is concerning the
signification of the name. But if it be asked
concerning an abstract name, what is it ? the
cause is demanded why a thing appears so or so.
As if it be asked, what is hard? The answer
will be, hard is that, whereof no part gives place,
but when the whole gives place. But if it be
demanded, what is hardness ? a cause must be
shewn why a part does not give place, except the
104
PHILOSOPHY.
part n. Miole give place. Wherefore, I define an accident
t -- to be the manner of our conception of body.
Howanaeci- 3. When an accident is said to be in a body, it
dent may be .
understood is not so to be understood, as it any thing were
subject? tS contained in that body ; as if, for example, redness
were in blood, in the same manner, as blood is in
a bloody cloth, that is, as a part in the whole;
for so, an accident would be a body also. But, as
magnitude, or rest, or motion, is in that which is
great, or which resteth, or which is moved, (which,
how it is to be understood, every man understands)
so also, it is to be understood, that every other
accident is in its subject. And this, also, is
explicated by Aristotle 110 otherwise than nega¬
tively, namely, that an accident is in its subject,
not as any part thereof * but so as that it may be
away, the subject still remaining; which is right,
saving that there are certain accidents which can
never perish except the body perish also; for no
body can be conceived to be without extension, or
without figure. All other accidents, which are
not common to all bodies, but peculiar to some
only, as to be at rest, to be moved, colour,
hardness, and the like, do perish continually, and
are succeeded by others; yet so, as that the body
never perisheth. And as for the opinion that some
may have, that all other accidents are not in their
bodies in the same manner that extension, motion,
rest, or figure, are in the same; for example, that
colour, heat, odour, virtue, vice, and the like, are
otherwise in them, and, as they say, inherent;
I desire they would suspend their judgment for
the present, and expect a little, till it be found out
OF BODY AND ACCIDENT.
105
by ratiocination, whether these very accidents are part ii.
not also certain motions either of the mind of the -—r—-
perceiver, or of the bodies themselves which are
perceived; for in the search of this, a great part
of natural philosophy consists.
4. The extension of a body, is the same thing Magnitude,
with the magnitude of it, or that which some call
real space. But this magnitude does not depend
upon our cogitation, as imaginary space doth ; for
this is an effect of our imagination, but magnitude
is the cause of it; this is an accident of the mind,
that of a body existing out of the mind.
5. That space, by which word I here understand Plaee> what it
. . . is, and that it
imaginary space, which is coincident with the is immovable,
magnitude of any body, is called the place of that
body; and the body itself is that which we call
the thing placed. Now place, and the magnitude
of the thing placed, differ. First in this, that a
body keeps always the same magnitude, both
when it is at rest, and when it is moved; but when
it is moved, it does not keep the same place.
Secondly in this, that place is a phantasm of any
body of such and such quantity and figure ; but
magnitude is the peculiar accident of every body ;
for one body may at several times have several
places, but has always one and the same magnitude.
Thirdly in this, that place is nothing out of the
mind, nor magnitude any thing within it. And
lastly, place is feigned extension, but magnitude
true extension; and a placed body is not extension,
but a thing extended. Besides, place is immovable;
for, seeing that which is moved, is understood to
be carried from place to place, if place were
moved, it would also be carried from place to
106
philosophy.
part it. place, so that one place must have another place,
~'—' and that place another place, and so on infinitely,
il an'cuhat !t which is ridiculous. And as for those, that, by
is immovable. making place to be of the same nature with real
space, would from thence maintain it to be
immovable, they also make place, though they do
not perceive they make it so, to be a mere phan¬
tasm. For whilst one affirms that place is therefore
said to be immovable, because space in general is
considered there; if he had remembered that
nothing is general or universal besides names or
signs, he would easily have seen that that space,
which he says is considered in general, is nothing
but a phantasm, in the mind or the memory, of a
body of such magnitude and such figure. And
whilst another says: real space is made immovable
by the understanding; as when, under the super¬
ficies of running water, we imagine other and
other water to come by continual succession, that
superficies fixed there by the understanding, is the
immovable place of the river : what else does he
make it to be but a phantasm, though he do it
obscurely and in perplexed words ? Lastly, the
nature of place does not consist in the superficies
of the ambient, but in solid space; for the whole
placed body is coextended with its whole place,
and every part of it with every answering part of
the same place ; but seeing every placed body is a
solid thing, it cannot be understood to be coex¬
tended with superficies. Besides, how can any
whole body be moved, unless all its parts be
moved together with it ? Or how can the internal
parts of it be moved, but by leaving their place ?
But the internal parts of a body cannot leave the
of body and accident.
107
superficies of an external part contiguous to it; part ii.
and, therefore, it follows, that if place be the ——r—-
superficies of the ambient, then the parts of a
body moved, that is, bodies moved, are not moved.
6. Space, or place, that is possessed by a body, ^[jate^p^u
is called full, and that which is not so possessed,
is called empty.
7. Here, there, in the country, in the city, and Hcre> therc>
. J somewhere,
other the like names, by which answer is made to what they
the question where is it ? are not properly names Slgmfy'
of place, nor do they of themselves bring into the
mind the place that is sought; for here and there
signify nothing, unless the thing be shewn at the
same time with the finger or something else ; but
when the eye of him that seeks, is, by pointing or
some other sign, directed to the thing sought, the
place of it is not hereby defined by him that
answers, but found out by him that asks the ques¬
tion. Now such she wings as are made by words
only, as when we say, in the country, or in the
city, are some of greater latitude than others, as
when we say, in the country, in the city, in such a
street, in a house, in the chamber, in bed, &c.
For these do, by little and little, direct the seeker
nearer to the proper place; and yet they do not
determine the same, but only restrain it to a lesser
space, and signify 110 more, than that the place of
the thing is within a certain space designed by
those words, as a part is in the whole. And all
such names, by which answer is made to the ques¬
tion where ? have, for their highest genus, the
name somewhere. From whence it may be under¬
stood, that whatsoever is somewhere, is in some
place properly so called, which place is part of
108
PHILOSOPHY.
part ii. that greater space that is signified by some of these
v—-■r—- names, in the country, in the city, or the like.
cannot be^in ^ ^ody, anc^ the magnitude, and the place
one place, nor thereof, are divided by one and the same act of
mMy^piace^ the mind ; for, to divide an extended body, and the
extension thereof, and the idea of that extension,
which is place, is the same with dividing any one
of them ; because they are coincident, and it
cannot be done but by the mind, that is by the
division of space. From whence it is manifest,
that neither two bodies can be together in the
same place, nor one body be in two places at the
same time. Not two bodies in the same place;
because when a body that fills its whole place is
divided into two, the place itself is divided into
two also, so that there will be two places. Not
one body in two places ; for the place that a body
fills being divided into two, the placed body will
be also divided into two ; for, as I said, a place
and the body that fills that place, are divided both
together; and so there will be two bodies.
Contiguous 9. Two bodies are said to be contiguous to one
and continual, , , 7 . ,
what they are. another, and continual, m the same manner as
spaces are ; namely, those are contiguous, between
which there is no space. Now, by space I under¬
stand, here as formerly, an idea or phantasm of a
body. Wherefore, though between two bodies
there be put no other body, and consequently no
magnitude, or, as they call it, real space, yet if
another body may be put between them, that is, if
there intercede any imagined space which may
receive another body, then those bodies are not
contiguous. And this is so easy to be understood,
that I should wonder at some men, who being
OF BODY AND ACCIDENT.
109
otherwise skilful enough in philosophy, are of a part ii.
different opinion, but that I find that most of those -—A—
that affect metaphysical subtleties wander from
truth, as if they were led out of their way by an
ignis fatuus. For can any man that has his
natural senses, think that two bodies must
therefore necessarily touch one another, because
no other body is between them ? Or that there
can be no vacuum, because vacuum is nothing, or
as they call it, non ens ? Which is as childish, as
if one should reason thus; no man can fast,
because to fast is to eat nothing ; but nothing
cannot be eaten. Continual, are any two bodies
that have a common part ; and more than two are
continual, when every two, that are next to one
another, are continual.
10. Motion is a continual relinquishing of The definition
7 7 . . n ,7 , of motion. No
one place, arid acquiring oj another ; and that motion inteiii-
place which is relinquished is commonly called the but Wlth
terminus a quo, as that wThich is acquired is called
the terminus ad quem ; I say a continual relin¬
quishing, because no body, how little soever, can
totally and at once go out of its former place into
another, so, but that some part of it will be in a
part of a place which is common to both, namely,
to the relinquished and the acquired places. For
example, let any body be in the :
place A C B D; the same body can¬
not come into the place B D E F,
but it must first be in G H I K,
, , n Ilinr . CHDKF
whose part G H B 1) is common to
both the places A C B D, and G H I K, and
whose part B D I K, is common to both the places
G H I K, and B D E F. Now it cannot be con-
110
philosophy.
part ii. ceived that any thing can be moved without time;
-——• for time is, by the definition of it, a phantasm, that
is, a conception of motion ; and, therefore, to con¬
ceive that any thing may be moved without time,
were to conceive motion without motion, which is
impossible.
what it is i| That is said to be at rest, which, during
to be at rest, 7 ' o
to have been any time, is in one place ; and that to be moved,
to be moved, or to have been moved, which, whether it be now
be°conceived° resI or moved, was formerly in another place
without the than that which it is now in. From which defini-
conception ot
past and future, tions it may be inferred, first, that whatsoever is
moved, has been moved; for if it be still in the
same place in which it was formerly, it is at rest,
that is, it is not moved, by the definition of rest;
but if it be in another place, it has been moved,
by the definition of moved. Secondly, that what
is moved, will yet be moved; for that which is
moved, leaveth the place where it is, and therefore
will be in another place, and consequently will
be moved still. Thirdly, that whatsoever is
moved, is not in one place during any time, how
little soever that time be ; for by the definition of
rest, that which is in one place during any time,
is at rest.
There is a certain sophism against motion, which
seems to spring from the not understanding of
this last proposition. For they say, that, if any
body be moved, it is moved either in the place
where it is, or in the place where it is not; both
which are false ; and therefore nothing is moved.
But the falsity lies in the major proposition; for
that which is moved, is neither moved in the place
where it is, nor in the place where is not; but
OF BODY AND ACCIDENT.
Ill
from the place where it is, to the place where it is PART IL
not. Indeed it cannot be denied but that what- '——'
soever is moved, is moved somewhere, that is,
within some space; but then the place of that
body is not that whole space, but a part of it, as
is said above in the seventh article. From what
is above demonstrated, namely, that whatsoever is
moved, has also been moved, and will be moved,
this also may be collected, that there can be no
conception of motion, without conceiving past
and future time.
12. Though there be no body which has not A point, a line,
some magnitude, yet if, when any body is moved, and^on^'
the magnitude of it be not at all considered, the what tliey ar0,
way it makes is called a line, or one single
dimension ; and the space, through which it
passeth, is called length; and the body itself, a
'point; in which sense the earth is called a point,
and the way of its yearly revolution, the ecliptic
line. But if a body, which is moved, be considered
as long, and be supposed to be so moved, as that
all the several parts of it be understood to make
several lines, then the way of every part of that
body is called breadth, and the space which is
made is called superficies, consisting of two
dimensions, one whereof to every several part of
the other is applied whole. Again, if a body be
considered as having superficies, and be under¬
stood to be so moved, that all the several parts of
it describe several lines, then the way of every
part of that body is called thickness or depth,
and the space which is made is called solid,
consisting of three dimensions, any two whereof
are applied whole to every several part of the
third,
112
PHILOSOPHY.
part ii. But if a body be considered as solid, then it is
-—'■ not possible tliat all the several parts of it should
describe several lines; for what way soever it be
moved, the way of the following part will fall into
the way of the part before it, so that the same
solid will still be made which the foremost super¬
ficies would have made by itself. And therefore
there can be no other dimension in any body, as
ifc is a body, than the three which I have now
describedthough, as it shall be shewed hereafter,
velocity, which is motion according to length,
may, by being applied to all the parts of a solid,
make a magnitude of motion, consisting of four
dimensions; as the goodness of gold, computed
in all the parts of it, makes the price and value
thereof.
Equal, great, 13. Bodies, how many soever they be, that
greater, and . .- . „
less, in bodies can nil every one the place or every one, are said
tudes^what to be equal every one to every other. Now, one
they are. body may fill the same place which another body
filleth, though it be not of the same figure with
that other body, if so be that it may be understood
to be reducible to the same figure, either by
flexion or transposition of the parts. And one
body is greater than another body, when a part
of that is equal to all this ; and less, when all
that is equal to a part of this. Also, magnitudes
are equal, or greater, or lesser, than one another,
for the same consideration, namely, when the
bodies, of which they are the magnitudes, are
either equal, or greater, or less, &c.
One and the 14. One and the same body is always of one
haTaiways one and the same magnitude. For seeing a body
and the same | magnitude and place thereof cannot be
magnitude. ~ 1
comprehended in the mind otherwise than as they
of body and accident.
113
are coincident, if any body be understood to be at part ir.
rest, that is, to remain in the same place during •—~—-
some time, and the magnitude thereof be in one
part of that time greater, and in another part less,
that body's place, which is one and the same, will
be coincident sometimes with greater, sometimes
with less magnitude, that is, the same place will be
greater and less than itself, which is impossible.
But there would be no need at all of demonstrating
a thing that is in itself so manifest, if there were
not some, whose opinion concerning bodies and
their magnitudes is, that a body may exist separated
from its magnitude, and have greater or less mag¬
nitude bestowed upon it, making use of this
principle for the explication of the nature of varum
and densum.
15. Motion, in as much as a certain length may
in a certain time be transmitted by it, is called
velocity or swiftness : &c. For though swift
be very often understood with relation to slower
or less swift, as great is in respect of less, yet
nevertheless, as magnitude is by philosophers taken
absolutely for extension, so also velocity or swift¬
ness may be put absolutely for motion according to
length.
16. Many motions are said to be made in equal Eciua1'
J 1 greater, and
times, when every one of them begins and ends less, in times,
together with some other motion, or if it hadw at they are'
begun together, would also have ended together
with the same. For time, which is a phantasm of
motion, cannot be reckoned but by some exposed
motion; as in dials by the motion of the sun or of
the hand ; and if two or more motions begin and
end with this motion, they are said to be made in
vol. i. i
114
philosophy.
part ii. equal times ; from whence also it is easy to under-
v—r—- stand what it is to be moved in greater or longer
time, and in less time or not so long ; namely,
that that is longer moved, which beginning with
another, ends later; or ending together, began
sooner.
equal, greater, 17. Motions are said to be equally swift, when
and less, in ve- . ,
locity, what, equal lengths are transmitted m equal times; and
greater swiftness is that, wherein greater length is
passed in eqnal time, or equal length in less time.
Also that swiftness by which equal lengths are
passed in equal parts of time, is called uniform
swiftness or motion ; and of motions not uniform,
such as become swifter or slower by equal in-
creasings or decreasings in equal parts of time, are
said to be accelerated or retarded uniformly.
equal, greater, 18. But motion is said to be greater, less, and
modon^what. equal, not only in regard of the length which is
transmitted in a certain time, that is, in regard of
swiftness only, but of swiftness applied to every
smallest particle of magnitude; for when any
body is moved, every part of it is also moved ; and
supposing the parts to be halves, the motions of
those halves have their swiftness equal to one
another, and severally equal to that of the whole;
but the motion of the whole is equal to those two
motions, either of which is of equal swiftness with
it; and therefore it is one thing for two motions
to be equal to one another, and another thing for
them to be equally swift. And this is manifest in
two horses that draw abreast, where the motion of
both the horses together is of equal swiftness with
the motion of either of them singly; but the
motion of both is greater than the motion of one
OF BODY AND ACCIDENT.
115
of them, namely, double. Wherefore motions are part ii.
said to be simply equal to one another, when the • r—-
swiftness of one, computed in every part of its
magnitude, is equal to the swiftness of the other
computed also in every part of its magnitude:
and greater than one another, when the swiftness
of one computed as above, is greater than the
swiftness of the other so computed; and less,
when less. Besides, the magnitude of motion
computed in this manner is that which is commonly
called force.
19. Whatsoever is at rest, ivill always be at That which
rest, unless there be some other body besides it, aiways^^at
which, by endeavouring to get into its place by ^movTdby
motion, suffers it no longer to remain at rest, some external
. thing.
For suppose that some finite body exist and be at
rest, and that all space besides be empty ; if now
this body begin to be moved, it will certainly be
moved some way ; seeing therefore there was
nothing in that body which did not dispose it to
rest, the reason why it is moved this way is in
something out of it; and in like manner, if it had
been moved any other way, the reason of motion
that way had also been in something out of it; but
seeing it was supposed that nothing is out of it,
the reason of its motion one way would be the
same with the reason of its motion every other
way, wherefore it would be moved alike all ways
at once ; which is impossible.
In like manner, whatsoever is moved, willThat wllich is
77 777 m0Ve(* al"
always be mo ved, except there be some other body ways be moved,
besides it, which causeth it to rest. For if we dercT some
suppose nothing to be without it, there will be no external thing*
reason why it should rest now, rather than at
I 2
PHILOSOPHY.
part ii. another time ; wherefore its motion would cease
o.
v—■—' 111 every particle of time alike; which is not
intelligible-
Accidents are /20. When we say a living creature, a tree, or any
generated cind i • i •
destroyed, but other specified body is generated or destroijed,
o ies no so. -g nQj. gQ underst00d as if there were made
a body of that which is not-body, or not a body of
a body, but of a living creature not a living crea¬
ture, of a tree not a tree, &c. that is, that those
accidents for which we call one thing a living
creature, another thing a tree, and another by
some other name, are generated and destroyed;
and that therefore the same names are not to be
given to them now, which were given them before.
But that magnitude for which we give to any
thing the name of body is neither generated nor
destroyed.^/For though we may feign in our mind
that a point may swell to a huge bulk, and that
this may again contract itself to a point; that is,
though we may imagine something to arise where
before was nothing, and nothing to be there where
before was something, yet we cannot comprehend
in our mind how this may possibly be done in
nature. And therefore philosophers, who tie
themselves to natural reason, suppose that a body
can neither be generated nor destroyed, but only
that it may appear otherwise than it did to us,
that is, under different species, and consequently
be called by other and other names; so that that
which is now called man, may at another time
have the name of not-man ; but that which is once
called body, can never be called not-body. But it
is manifest, that all other accidents besides magni¬
tude or extension may be generated and destroyed;
of body and accident.
117
as when a white thing is made black, the whiteness part ii.
that was in it perislieth, and the blackness that -- »■' -
was not in it is now generated; and therefore
bodies, and the accidents under which they appear
diversely, have this difference, that bodies are
things, and not generated; accidents are generated,
and not things.
21. And therefore, when any thing appearsAn accident
J o x x cannot depart
otherwise than it did by reason of other and other from its subjeci
accidents, it is not to be thought that an accident
goes out of one subject into another, (for they are
not, as I said above, in their subjects as a part in
the whole, or as a contained thing in that which
contains it, or as a master of a family in his house,)
but that one accident perisheth, and another is
generated. For example, when the hand, being
moved, moves the pen, motion does not go out of
the hand into the pen; for so the writing might be
continued though the hand stood still; but a new
motion is generated in the pen, and is the pen's
motion.
22. And therefore also it is improper to say, an Nor be moved,
accident is moved; as when, instead of saying,
figure is an accident of a body carried away, we
say, a body carries away its figure.
23. Now that accident for which we give a Essence, form,
certain name to any body, or the accident which "hat Syak
denominates its subject, is commonly called the
essence thereof; as rationality is the essence of
a man ; whiteness, of any white thing, and exten¬
sion the essence of a body. And the same essence,
in as much as it is generated, is called the form.
Again, a body, in respect of any accident, is called
the subject, and in respect of the form it is
called the matter.
118
PHILOSOPHY.
part ii. Also, the production or perishing of any accident
*—r—' makes its subject be said to be changed; only the
production or perishing of form makes it be said it
is generated or destroyed; but in all generation
and mutation, the name of matter still remains.
For a table made of wood is not only wooden, but
wood; and a statue of brass is brass as well as
brazen ; though Aristotle, in his Metaphysics, says,
that whatsoever is made of any thing ought not to
be called e/c«vo, but uhvlvov; as that which is made
of wood, not £;v\ov, but &\ivov, that is, not wood,
but wooden.
First mat- 24. And as for that matter which is common to
ter, what.
all things, and which philosophers, following Aris¬
totle, usually call materia prima, that is, first
matter, it is not any body distinct from all other
bodies, nor is it one of them. What then is it ?
A mere name ; yet a name which is not of vain
use ; for it signifies a conception of body without
the consideration of any form or other accident
except only magnitude or extension, and aptness
to receive form and other accident. So that when¬
soever we have use of the name body in general,
if we use that of materia prima, we do well. For
as when a man not knowing which was first,
water or ice, would find out which of the two were
the matter of both, he would be fain to suppose
some third matter which were neither of these
two ; so he that would find out what is the matter
of all things, ought to suppose such as is not the
matter of anything that exists. Wherefore materia
prima is nothing ; and therefore they do not
attribute to it either form or any other accident
besides quantity ; whereas all singular things have
their forms and accidents certain.
OF BODY AND ACCIDENT,
119
Materia 'prima, therefore, is body in general, PAR8T IL
that is, body considered universally, not as having 1—■—'
neither form nor any accident, but in which no
form nor any other accident bnt quantity are at all
considered, that is, they are not drawn into argu¬
mentation.
25. From what has been said, those axioms may ?hat thfc wJole
J J is greater than
be demonstrated, which are assumed by Euclid in any Part there-
. p 1 , of, why denion-
the beginning ot his first element, about the equa- strated.,
lity and inequality of magnitudes ; of which,
omitting the rest, I will here demonstrate only
this one, the whole is greater than any part
thereof; to the end that the reader may know that
those axioms are not indemonstrable, and therefore
not principles of demonstration ; and from hence
learn to be wary how he admits any thing for a
principle, which is not at least as evident as these
are. Greater is defined to be that, whose part is
equal to the whole of another. Now if we suppose
any whole to be A, and a part of it to be B ;
seeing the whole B is equal to itself, and the same
B is a part of A; therefore a part of A will be
equal to the whole B. Wherefore, by the definition
above, A is greater than B; which was to be proved.
120
PHILOSOPHY.
CHAPTER IX.
of cause and effect.
1. Action and passion, what they are.—2. Action and passion
mediate and immediate. — 3. Cause simply taken. Cause
without which no effect follows, or cause, necessary by sup¬
position.— 4-. Cause efficient and material.—5. An entire
cause is always sufficient to produce its effect. At the same
instant that the cause is entire, the effect is produced. Every
effect has a necessary cause.—6. The generation of effects is
continual. What is the beginning in causation.—7. No cause
of motion but in a body contiguous and moved.—8. The same
agents and patients, if alike disposed, produce like effects
though at different times. — 9. All mutation is motion.
10. Contingent accidents, what they are.
tart ii. l. A body is said to work upon or act, tiiat is to
-—^—- say, do something to another body, when it either
tm\'passion, generates or destroys some accident in it: and the
what they are. |)0(ly in which an accident is generated or destroyed
is said to suffer, that is, to have something done to
it by another body ; as when one body by putting
forwards another body generates motion in it, it is
called the agent ; and the body in which motion
is so generated, is called the patient ; so fire that
warms the hand is the agent, and the hand, which
is warmed, is the patient. That accident, which
is generated in the patient, is called the effect.
Action and 2. When an agent and patient are contiguous to
passion, me- _ , . .
diate, and one another, their action and passion are then said
immediate. ^ |)(1 immediate, otherwise, mediate; and when
another body, lying betwixt the agent and patient,
is contiguous to them both, it is then itself both an
of cause and effect.
121
agent and a patient; an agent in respect of the part ii.
body next after it, upon which it works, and a -——-
patient in respect of the body next before it, from
which it suffers. Also, if many bodies be so
ordered that every two which are next to one
another be contiguous, then all those that are
betwixt the first and the last are both agents and
patients, and the first is an agent only, and the last
a patient only.
3. An agent is understood to produce its deter- simply
mined or certain effect in the patient, according to
some certain accident or accidents, with which
both it and the patient are affected ; that is to say,
the agent hath its effect precisely such, not because
it is a body, but because such a body, or so moved.
For otherwise all agents, seeing they are all bodies
alike, would produce like effects in all patients.
And therefore the fire, for example, does not warm,
because it is a body, but because it is hot; nor
does one body put forward another body because it
is a body, but because it is moved into the place
of that other body. The cause, therefore, of all
effects consists in certain accidents both in the
agents and in the patients ; wThich when they are
all present, the effect is produced ; but if any one
of them be wanting, it is not produced; and that Cause without
accident either of the agent or patient, without JbnowS"oieffect
which the effect cannot be produced, is called £ausenecessary
r 3 by supposition,
causa sine qua non, or cause necessary by sup¬
position, as also the cause requisite for the pro¬
duction of the effect. But a cause simply, or an
entire cause, is the aggregate of all the accidents
both of the agents how many soever they be, and
of the patient, put together ; which when they
PHILOSOPHY.
part ii. are ajj supposed to be present, it cannot, be under -
■ stood but that the effect is produced at the same
instant ; and if any one of them be wanting, it
cannot be understood but that the effect is not
produced.
fnTmateria?' 4' aggregate of accidents in the agent or
agents, requisite for the production of the effect,
the effect being produced, is called the efficient
cause thereof; and the aggregate of accidents in
the patient, the effect being produced, is usually
called the material cause ; I say the effect being
produced ; for where there is no effect, there can
be no cause ; for nothing can be called a cause,
where there is nothing that can be called an
effect. But the efficient and material causes are
both but partial causes, or parts of that cause, which
in the next precedent article I called an entire
cause. And from hence it is manifest, that the
effect we expect, though the agents be not defective
on their part, may nevertheless be frustrated by a
defect in the patient; and when the patient is
sufficient., by a defect in the agents.
An entire cause 5. An entire cause is always sufficient for the
is always suf- . _ J
ficient to pro- production of its effect, if the effect be at all
ciuce its effect, p0SS^e> por ^ any effect whatsoever be pro¬
pounded to be produced; if the same be produced,
it is manifest that the cause which produced it was
a sufficient cause ; but if it be not produced, and
yet be possible, it is evident that something was
wanting either in some agent, or in the patient,
without which it could not be produced ; that is,
that some accident was wanting which was requi¬
site for its production; and therefore, that cause was
not entire, which is contrary to what was supposed.
of cause and effect.
123
It follows also from lience, that in whatsoever part it.
9.
instant the cause is entire, in the same instant the -—r—-
effect is produced. For if it be not produced,
something is still wanting, which is requisite for cause is entire,
• tlie effect is pro-
the production of it; and therefore the cause was duced.
not entire, as was supposed.
And seeing a necessary cause is defined to be Every effect
that, which being supposed, the effect cannot but s^r/cause!"
follow ; this also may be collected, that whatsoever
effect is produced at any time, the same is produced
by a necessary cause. For whatsoever is produced,
in as much as it is produced, had an entire cause,
that is, had all those things, which being supposed,
it cannot be understood but that the effect fol¬
lows ; that is, it had a necessary cause. And in the
same manner it may be shewn, that whatsoever
effects are hereafter to be produced, shall have a
necessary cause; so that all the effects that have
been, or shall be produced, have their necessity in
things antecedent.
6. And from this, that whensoever the cause is The genera-
entire, the effect is produced in the same instant, i^ntiuud.3
it is manifest that causation and the production ,What i8 the
L beginning in
of effects consist in a certain continual progress; causation,
so that as there is a continual mutation in the
agent or agents, by the working of other agents
upon them, so also the patient, upon which they
work, is continually altered and changed. For
example: as the heat of the fire increases more
and more, so also the effects thereof, namely, the
heat of such bodies as are next to it, and again, of
such other bodies as are next to them, increase
more and more accordingly; which is already no
little argument that all mutation consists in motion
124
philosophy.
part ii. only; the truth whereof shall be further demon-
^ strated in the ninth article. But in this progress
of causation, that is, of action and passion, if any
man comprehend in his imagination a part thereof,
and divide the same into parts, the first part or
beginning of it cannot be considered otherwise
than as action or cause; for, if it should be consi¬
dered as effect or passion, then it would be neces¬
sary to consider something before it, for its cause
or action; which cannot be, for nothing can be
before the beginning. And in like manner, the
last part is considered only as effect; for it cannot
be called cause, if nothing follow it; but after the
last, nothing follows. And from hence it is, that in
all action the beginning and cause are taken for
the same thing. But every one of the intermediate
parts are both action and passion, and cause and
effect, according as they are compared with the
antecedent or subsequent part.
No cause of 7. There can be no cause of motion, except in a
a bod°y contigu-body contiguous and moved. For let there be
ous and moved. any two which are not contiguous, and be¬
twixt which the intermediate space is empty, or, if
filled, filled with another body which is at rest;
and let one of the propounded bodies be supposed
to be at rest; I say it shall always be at rest. For
if it shall be moved, the cause of that motion, by
the 8th chapter, article 19, will be some external
body; and, therefore, if between it and that ex¬
ternal body there be nothing but empty space,
then whatsoever the disposition be of that external
body or of the patient itself, yet if it be supposed
to be now at rest, we may conceive it will con¬
tinue so till it be touched by some other body.
of cause and effect.
125
But seeing cause, by the definition, is the aggre- part ir.
gate of all such accidents, which being supposed -—r—'
to be present, it cannot be conceived but that the
effect will follow, those accidents, which are either
in external bodies, or in the patient itself, cannot
be the cause of future motion. And in like manner,
seeing we may conceive that whatsoever is at rest
will still be at rest, though it be touched by some
other body, except that other body be moved;
therefore in a contiguous body, Avhich is at rest,
there can be no cause of motion. Wherefore there
is no cause of motion in any body, except it be
contiguous and moved.
The same reason may serve to prove that what¬
soever is moved, will always be moved on in the
same way and with the same velocity, except it
be hindered by some other contiguous and moved
body; and consequently that no bodies, either
when they are at rest, or when there is an inter¬
position of vacuum, can generate or extinguish or
lessen motion in other bodies. There is one that
has written that things moved are more resisted
by things at rest, than by things contrarily moved;
for this reason, that he conceived motion not to be
so contrary to motion as rest. That which deceived
him was, that the words rest and motion are but
contradictory names; whereas motion, indeed, is
not resisted by rest, but by contrary motion.
8. But if a body work upon another body at one The same
time, and afterwards the same body work upon the jSis"#
same body at another time, so that both the agent sed^oduc'e
and patient, and all their parts, be in all things as |jke e®"ecj^.f
they were ; and there be no difference, except only ferent times,
in time, that is, that one action be former, the
126
PHILOSOPHY.
part ir. other later in time; it is manifest of itself, that the
-——- effects will be equal and like, as not differing in
anything besides time. And as effects themselves
proceed from their causes, so the diversity of them,
depends upon the diversity of their causes also.
ah mutation 9. This being true, it is necessary that mutation
is motion. -i-ii • /• i
can be nothing else but motion of the parts of that
body which is changed. For first, we do not say
anything is changed, but that which appears to our
senses otherwise than it appeared formerly. Se¬
condly, both those appearances are effects pro¬
duced in the sentient; and, therefore, if they be
different, it is necessary, by the preceding article,
that either some part of the agent, which was for¬
merly at rest, is now moved, and so the mutation
consists in this motion; or some part, which was
formerly moved, is now otherwise moved, and so
also the mutation consists in this new motion; or
which, being formerly moved, is now at rest,
which, as I have shewn above, cannot come to
pass without motion; and so again, mutation is
motion ; or lastly, it happens in some of these
manners to the patient, or some of its parts; so
that mutation, howsoever it be made, will consist
in the motion of the parts, either of the body
which is perceived, or of the sentient body, or of
both. Mutation therefore is motion, namely, of
the parts either of the agent or of the patient;
which was to be demonstrated. And to this it is
consequent, that rest cannot be the cause of any¬
thing, nor can any action proceed from it; seeing
neither motion nor mutation can be caused by it,
contingent 10. Accidents, in respect of other accidents
accidents. precec[e them, or are before them in time,
of power and act.
127
and upon which they do not depend as upon their part ii.
causes, are called contingent accidents; I say, in -—*—'
respect of those accidents by which they are not
generated; for, in respect of their causes, all things
come to pass with equal necessity; for otherwise
they would have no causes at all; which, of things
generated, is not intelligible.
CHAPTER X.
of power and act.
1. Power and cause are the same thing.—2. An act is produced
at the same instant in which the power is plenary.—3. Active
and passive power are parts only of plenary power.—4. An
act, when said to be possible.—5. An act necessary and con¬
tingent, what.—6. Active power consists in motion.-—7. Cause,
formal and final, what they are.
1. Correspondent to cause and effect, are power and
t cause are the
power and act; nay, those and these are thesamething,
same things; though, for divers considerations,
they have divers names. For whensoever any
agent has all those accidents which are necessarily
requisite for the production of some eifect in the
patient, then we say that agent has power to pro¬
duce that effect, if it be applied to a patient. But,
as I have shewn in the precedent chapter, those
accidents constitute the efficient cause; and there¬
fore the same accidents, which constitute the
efficient cause, constitute also the power of the
agent. Wherefore the power of the agent and
the efficient cause are the same thing. But they
are considered with this difference, that cause is
128
philosophy.
part ii. go called in respect of the effect already produced,
-—r—' and power in respect of the same effect to be pro¬
duced hereafter; so that cause respects the past,
'power the future time. Also, the power of the
agent is that which is commonly called active
power.
In like manner, whensoever any patient has all
those accidents which it is requisite it should have,
for the production of some effect in it, we say it is
in the power of that patient to produce that effect,
if it be applied to a fitting agent. But those acci¬
dents, as is defined in the precedent chapter, con¬
stitute the material cause ; and therefore the power
of the patient, commonly called passive power,
and material cause, are the same thing; but with
this different consideration, that in cause the past
time, and in power the future, is respected.
Wherefore the power of the agent and patient
together, which may be called entire or plenary
power, is the same thing with entire cause; for
they both consist in the sum or aggregate of all
the accidents, as well in the agent as in the patient,
which are requisite for the production of the effect.
Lastly, as the accident produced is, in respect of
the cause, called an effect, so in respect of the
power, it is called an act.
An act is pro- 2. As therefore the effect is produced in the
same instant same instant in which the cause is entire, so also
power is pie- eyeiT act that may be produced, is produced in the
nai7- same instant in which the power is plenary. And
as there can be 110 effect but from a sufficient and
necessary cause, so also 110 act can be produced but
by sufficient power, or that power by which it
could not but be produced.
OF POWER AND ACT.
129
3. And as it is manifest, as I liave shewn, that part it.
10.
the efficient and material causes are severally and -—^—-
by themselves parts only of an entire cause, and power
cannot produce any effect but by beina- ioinedar® Parts only
x J . . of plenary
together, so also power, active and passive, are power,
parts only of plenary and entire power ; nor, except
they be joined, can any act proceed from them;
and therefore these powers, as I said in the first
article, are but conditional, namely, the agent has
power, if it be applied to a patient; and the
patient has power, if it be applied to an agent;
otherwise neither of them have power, nor can the
accidents, which are in them severally, be properly
called powers; nor any action be said to be pos¬
sible for the power of the agent alone or of the
patient alone.
4. For that is an impossible act, for the pro due- An act, when
tion of which there is no power plenary. For possible,
seeing plenary power is that in which all things
concur, which are requisite for the production of
an act, if the power shall never be plenary, there
will always be wanting some of those things, with¬
out which the act cannot be produced ; wherefore
that act shall never be produced ; that is, that act
is impossible : and every act, which is not impos¬
sible, is possible. Every act, therefore, which is
possible, shall at some time be produced ; for if it
shall never be produced, then those things shall
never concur which are requisite for the produc¬
tion of it; wherefore that act is impossible, by the
definition ; which is contrary to what was sup¬
posed.
5. A necessary act is that, the production An act neces-
whereof it is impossible to hinder ; and therefore tingent, wi°a.
vol. i. k
130
philosophy.
part ii. every act, that shall be produced, shall necessarily
-—A—- be produced; for, that it shall not he produced, is
sar^fnd econ- impossible; because, as is already demonstrated,
tingent, what, every possible act shall at some time be produced;
nay, this proposition, what .shall be, shall be, is as
necessary a proposition as this, a man is a man.
But here, perhaps, some man may ask whether
those future things, which are commonly called
contingents, are necessary. I say, therefore, that
generally all contingents have their necessary
causes, as is shewn in the preceding chapter; but
are called contingents in respect of other events,
upon which they do not depend; as the rain, which
shall be tomorrow, shall be necessary, that is,
from necessary causes ; but we think and say it
happens by chance, because we do not yet perceive
the causes thereof, though they exist now ; for men
commonly call that casual or contingent, whereof
they do not perceive the necessary cause; and in
the same manner they used to speak of things past,
when not knowing whether a thing be done or 110,
they say it is possible it never was done.
Wherefore, all propositions concerning future
things, contingent or not contingent, as this, it
will rain to]i$orrow, or this, tomorrow the sun
will rise, are either necessarily true, or necessarily
false; but we call them contingent, because we do
not yet know whether they be true or false;
whereas their verity depends not upon our know¬
ledge, but upon the foregoing of their causes. But
there are some, who though they confess this whole
proposition, tomorrow it will either rain, or not
rain, to be true, yet they will not acknowledge the
parts of it, as, tomorrow it will rain, or, tomorrow
of power and act.
131
it will not rain, to be either of them true by itself; PAI^ Ir-
because they say neither this nor that is true deter- --—'
minutely. But what is this deter minutely true, but
true upon our knowledge, or evidently true ? And
therefore they say 110 more but that it is not yet
known whether it be true or no ; but they say it
more obscurely, and darken the evidence of the
truth with the same words, with which they endea¬
vour to hide their own ignorance.
G. In the 9tli article of the preceding chapter, I Active power
, ™ . ? ,, . consists in
have shewn that the efficient cause of all motion motion,
and mutation consists in the motion of the agent,
or agents ; and in the first article of this chapter,
that the power of the agent is the same thing with
the efficient cause. From whence it may be under¬
stood, that all active power consists in motion also;
and that power is not a certain accident, which
differs from all acts, but is, indeed, an act, namely,
motion, which is therefore called power, because
another act shall be produced by it afterwards.
For example, if of three bodies the first put
forward the second, and this the third, the motion
of the second, in respect of the first which pro-
duceth it, is the act of the second body; but, in
respect of the third, it is the active power of the
same second body.
7. The writers of metaphysics reckon up two cause, formal
other causes besides the efficient and material, what they are.
namely, the essence, which some call the formal
cause, and the end, or final cause; both which
are nevertheless efficient causes. For when it is
said the essence of a thing is the cause thereof, as
to he rational is the cause of man, it is not intel¬
ligible ; for it is all one, as if it were said, to be a
k 2
132
philosophy.
part ii. man is the cause of man ; which is not well said.
10.
-—r—' And yet the knowledge of the essence of anything,
is the cause of the knowledge of the thing itself;
for, if I first know that a thing is rational, I know
from thence, that the same is man; but this is no
other than an efficient cause. A final cause has no
place but in such things as have sense and will;
and this also I shall prove hereafter to be an effi¬
cient cause.
CHAPTER XL
of identity and difference.
1. What it is for one thing to differ from another.—2. To differ
in number, magnitude, species, and genus, what.—3. What is
relation, proportion, and relatives.—4. Proportionals, what.—
5. The proportion of magnitudes to one another, wherein it
consists.—6. Relation is no new accident, but one of those
that were in the relative before the relation or comparison was
made. Also the causes of accidents in the correlatives, are the
cause of relation.—7. Of the beginning of individuation.
what it is l. Hitherto I have spoken of body simply, and
to differ from accidents common to all bodies., as magnitude,
motion, rest, action, passion, power, 'possible, fyc.;
and I should now descend to those accidents by
which one body is distinguished from another, but
that it is first to be declared what it is to be dis¬
tinct and not distinct, namely, what are the same
and different ; for this also is common to all
bodies, that they may be distinguished and differ¬
enced from one another. Now, two bodies are
said to differ from one another, when something
may be said of one of them, which cannot be said
of the other at the same time.
of identity and difference.
133
2. And, first of all, it is manifest that no two part n.
11.
bodies are the ,same ; for seeing they are two, they -—>—-
are in two places at the same time ; as that, which is ^0ndJmber,
the same, is at the same time in one and the same
species, and
place. All bodies therefore differ from one another genus, what,
in number, namely, as one and another; so that
the same and different in number, are names
opposed to one another by contradiction.
In magnitude bodies differ when one is greater
than another, as a cubit long, and two cubits long,
of two pound weight, and of three pound weight.
And to these, equals are opposed.
Bodies, which differ more than in magnitude, are
called unlike; and those, which differ only in mag¬
nitude, like. Also, of unlike bodies, some are said
to differ in the species, others in the genus ; in the
species, when their difference is perceived by one
and the same sense, as white and black; and in the
genus, when their difference is not perceived but
by divers senses, as white and hot.
3. And the likeness, or unlikeness, equality, or What is
inequality of one body to another, is called their proportion,
relation ; and the bodies themselves relatives orand relatives.
correlatives ; Aristotle calls them rd rrpur A ; the
first whereof is usually named the antecedent, and
the second the consequent ; and the relation of the
antecedent to the consequent, according to mag¬
nitude, namely, the equality, the excess or defect
thereof, is called the proportion of the ante¬
cedent to the consequent; so that proportion is
nothing but the equality or inequality of the mag¬
nitude of the antecedent compared to the magni¬
tude of the consequent by their difference only,
or compared also with their difference. For ex-
134
philosophy.
part ii. ample, the proportion of three to two consists
—r——^ only in this, that three exceeds two by unity ; and
the proportion of two to five in this, that two,
compared with five, is deficient of it by three,
either simply, or compared with the numbers dif¬
ferent ; and therefore in the proportion of unequals,
the proportion of the less to the greater, is called de¬
fect ; and that of the greater to the less, excess.
Proportion- 4. Besides, of unequals, some are more, some
' ' less, and some equally unequal; so that there is
proportion of proportions, as well as of magni¬
tudes ; namely, where two unequals have relation
to two other unequals; as, when the inequality
which is between 2 and 3, is compared with the
inequality which is between 4 and 5. In which
comparison there are always four magnitudes ; or,
which is all one, if there be but three, the middle¬
most is twice numbered; and if the proportion of
the first to the second, be equal to the proportion
of the third to the fourth, then the four are said
to be proportionals ; otherwise they are not pro¬
portionals.
The propor- 5. The proportion of the antecedent to the con-
niTudfs"!)^" sequent consists in their difference, not only
whereiuU61"' simply taken, but also as compared with one of
consists. the relatives; that is, either in that part of the
greater, by which it exceeds the less, or in the re¬
mainder, after the less is taken out of the greater;
as the proportion of two to five consists in the
three by which five exceeds two, not in three
simply only, but also as compared with five or two.
For though there be the same difference between
two and five, which is between nine and twelve,
namely three, yet there is not the same inequality;
OF IDENTITY AND DIFFERENCE. 135
and therefore the proportion of two to five is not PART IL
in all relation the same with that of nine to twelve, *—-
but only in that which is called arithmetical.
(5. But we must not so think of relation, as if it Relation is no
were an accident differing from all the other acci- ^Ton^o'Sose
dents of the relative; but one of them, namely, S^o ^cfore
that by which the comparison is made. For ex-the relation or
, P . , 7 comparison was
ample, the likeness or one white to another white, made. Also the
vi , 7 / 7 • . i •!, causes of acci-
or its unlikeness to black, is the same accident dents in con-e-
with its whiteness ; and equality and inequality, ^us^o? reia!.0
the same accident with the magnitude of the thing tion*
compared, though under another name: for that
which is called white or great, when it is not com¬
pared with something else, the same when it is
compared, is called like or unlike, equal or un¬
equal. And from this it follows that the causes
of the accidents, which are in- relatives, are the
causes also of likeness, unlikeness, equality and
inequality ; namely, that he, that makes two unequal
bodies, makes also their inequality; and he, that
makes a rule and an action, makes also, if the
action be congruous to the rule, their congruity;
if incongruous, their incongruity. And thus much
concerning comparison of one body with another.
7- But the same body may at different times be ofthebegin-
compared with itself. And from hence springs a Suatfo'n dl~
great controversy among philosophers about the
beginning of individuation, namely, in what sense
it may be conceived that a body is at one time the
same, at another time not the same it wras formerly.
For example, whether a man grown old be the
same man he was whilst he was young, or another
man; or whether a city be in different ages the
same, or another city. Some place individuity in
136
PHILOSOPHY.
part ii. the unity of matter; others, in the unity ofform;
v—<— and one says it consists in the unity of the aggre-
nin^onndi-* gut? °f nil the accidents together. For matter,
valuation. it is pleaded that a lump of wax, whether it be
spherical or cubical, is the same wax, because the
same matter. For form, that when a man is grown
from an infant to be an old man, though his matter
be changed, yet he is still the same numerical
man ; for that identity, which cannot be attributed
to the matter, ought probably to be ascribed to the
form. For the aggregate of accidents, no instance
can be made ; but because, when any new accident
is generated, a new name is commonly imposed on
the thing, therefore he, that assigned this cause of
individuity, thought the thing itself also was
become another thing. According to the first
opinion, he that -sins, and he that is punished,
should not be the same man, by reason of the per¬
petual flux and change of man's body ; nor should
the city, which makes laws in one age and abro¬
gates them in another, be the same city; which
were to confound all civil rights. According to
the second opinion, two bodies existing both at
once, would be one and the same numerical body.
For if, for example, that ship of Theseus, concern¬
ing the difference whereof made by continual re¬
paration in taking out the old planks and putting
in new, the sophisters of Athens were wont to dis¬
pute, were, after all the planks were changed, the
same numerical ship it was at the beginning; and
if some man had kept the old planks as they were
taken out, and by putting them afterwards together
in the same order, had again made a ship of them,
this, without doubt, had also been the same nume-
OF IDENTITY AND DIFFERENCE.
137
rical ship with that which was at the beginning; part n.
and so there would have been two ships numerically -—^—-
the same, which is absurd. But, according to the 01 thc,be?i,m'
7 ^ o liingotnuii-
third opinion, nothing; would be the same it viduation.
was ; so that a man standing would not be the same
he was sitting; nor the water, which is in the vessel,
the same with that which is poured out of it.
Wherefore the beginning of individuation is not
always to be taken either from matter alone, or
from form alone.
But we must consider by what name anything
is called, when we inquire concerning the identity
of it. For it is one thing to ask concerning Socrates,
whether he be the same man, and another to ask
whether he be the same body; for his body, when
he is old, cannot be the same it was when he was
an infant, by reason of the difference of magnitude ;
for one body has always one and the same magni¬
tude ; yet, nevertheless, he may be the same man.
And therefore, whensoever the name, by which it
is asked whether a thing be the same it was, is
given it for the matter only, then, if the matter be
the same, the thing also is individually the same;
as the water, which was in the sea, is the same
which is afterwards in the cloud; and any body is
the same, whether the parts of it be put together,
or dispersed; or whether it be congealed, or dis¬
solved. Also, if the name be given for such form
as is the beginning of motion, then, as long as that
motion remains, it will be the same individual
thing ; as that man will be always the same, whose
actions and thoughts proceed all from the same
beginning of motion, namely, that which was in
his generation; and that will be the same river
138
philosophy.
part ii. which flows from one and the same fountain,
-——- whether the same water, or other water, or some-
nin-'ofhfdi-" tiling else than water, flow from thence; and that
viduation. the same city, whose acts proceed continually from
the same institution, whether the men be the same
or 110. Lastly, if the name be given for some
accident, then the identity of the thing will depend
upon the matter; for, by the taking away and
supplying of matter, the accidents that were, are
destroyed, and other new ones are generated,
which cannot be the same numerically ; so that a
ship, which signifies matter so figured, will be the
same as long as the matter remains the same; but
if no part of the matter be the same, then it is
numerically another ship ; and if part of the matter
remain and part be changed, then the ship will
be partly the same, and partly not the same.
CHAPTER XII.
of quantity.
1. The definition of quantity.—2. Tlic exposition of quantity,
what it is.—13. How line, superficies, and solid, arc exposed.
4. How time is exposed.—5. How number is exposed.—6. How
velocity is exposed.—7. How weight is exposed.—8. How the
proportion of magnitudes is exposed.—9. How the proportion
of times and velocities is exposed.
definition ]. What and how manifold dimension is, has
quantity. ^eil saj(j \n |}1(. nth chapter, namely, that there are
three dimensions, line or length, superficies, and
solid ; every one of which, if it be determined, that
is, if the limits of it be made known, is commonly
called quantity; for by quantity all men under-
of quantity.
139
stand that which is signified by that word, by part it-
which answer is made to the question, How much *—^—-
is it? Whensoever, therefore, it is asked, for
example, How long is the journey ? it is not
answered indefinitely, length; nor, when it is
asked, How big is the field? is it answered inde¬
finitely, superficies ; nor, if a man ask, How great
is the bulk? indefinitely, solid: but it is answered
determinately, the journey is a hundred miles ; the
field is a hundred acres; the bulk is a hundred
cubical feet; or at least in some such manner, that
the magnitude of the thing enquired after may
by certain limits be comprehended in the mind.
Quantity, therefore, cannot otherwise be defined,
than to be a dimension determined, or a dimen¬
sion, whose limits are set out, either by their
'place, or by some comparison.
2. And quantity is determined two ways ; one, The exposi-
by the sense, when some sensible object is set ti'ty, wiiatTt'L
before it; as when a line, a superficies or solid,
of a foot or cubit, marked out in some matter, is
objected to the eyes; which way of determining,
is called exposition, and the quantity so known
is called exposed quantity ; the other by memory,
that is, by comparison with some exposed quan¬
tity. In the first manner, when it is asked of what
quantity a thing is, it is answered, of such quantity
as you see exposed. In the second manner, answer
cannot be made but by comparison with some
exposed quantity ; for if it be asked, how long is
the way ? the answer is, so many thousand paces ;
that is, by comparing the way with a pace, or some
other measure, determined and known by exposi¬
tion ; or the quantity of it is to some other quan-
140
philosophy.
part ii. tity known by exposition, as the diameter of a
'—-— square is to the side of the same, or by some
Srn °f other the like means. But it is to be understood,
( Udlll 11V, J
what it is. that the quantity exposed must be some standing
or permanent thing, such as is marked out in
consistent or durable matter ; or at least something
which is revocable to sense ; for otherwise no com¬
parison can be made by it. Seeing, therefore, by
what has been said in the next preceding chapter,
comparison of one magnitude writh another is the
same thing with proportion; it is manifest, that
quantity determined in the second manner is
nothing else but the proportion of a dimension not
exposed to another which is exposed; that is, the
comparison of the equality or inequality thereof
with an exposed quantity.
How line, 3 Lines, superficies, and solids, are exposed
superficies, . . 5
and solids, first, by motion, in such manner as in the 8th
aie exposeu. j ]iave sa}(;[ they are generated; but so as
that the marks of such motion be permanent; as
when they are designed upon some matter, as a
line upon paper; or graven in some durable
matter. Secondly, by apposition; as when one
line or length is applied to another line or length,
one breadth to another breadth, and one thickness
to another thickness ; which is as much as to
describe a line by points, a superficies by lines,
and a solid by superficies; saving that by points
in this place are to be understood very short
lines; and, by superficies, very thin solids.
Thirdly, lines and superficies may be exposed by
section, namely, a line may be made by cutting
an exposed superficies ; and a superficies, by the
cutting of an exposed solid.
OF QUANTITY.
141
4. Time is exposed, not only by the exposition PAT]l9T IL
of a line, but also of some moveable tiling, which -—•—-
is moved uniformly upon that line, or at least is **™Posed
supposed so to be moved. For, seeing time is an
idea of motion, in which we consider former and
latter, that is succession, it is not sufficient for the
exposition of time that a line be described; but
we must also have in our mind an imagination of
some moveable thing passing over that line ; and
the motion of it must be uniform, that time may
be divided and compounded as often as there shall
be need. And, therefore, when philosophers, in
their demonstrations, draw a line, and say, Let
that line be time, it is to be understood as if they
said, Let the conception of uniform motion upon
that line, he time. For though the circles in dials
be lines, yet they are not of themselves sufficient
to note time by, except also there be, or be sup¬
posed to be, a motion of the shadow or the hand.
5. Number is exposed, either by the exposition How number
of points, or of the names of number, one, two, ls expose •
three, &?c.; and those points must not be conti¬
guous, so as that they cannot be distinguished by
notes, but they must be so placed that they may
be discerned one from another ; for, from this it
is, that number is called discreet quantity,
whereas all quantity, which is designed by motion,
is called continual quantity. But that number
may be exposed by the names of number, it is
necessary that they be recited by heart and in
order, as one, two, three, &c. ; for by saying one,
one, one, and so forward, we know not what
number we are at beyond two or three; which
also appear to us in this manner, not as number,
but as figure.
142
PHILOSOPHY.
PART II.
12.
How velocity
is exposed.
How weight
is exposed.
IIovv the pro¬
portion of
magnitudes
is exposed.
6. For the exposition of velocity, which, by the
definition thereof, is a motion which, in a certain
time, passeth over a certain space, it is requisite,
not only that time be exposed, but that there be
also exposed that space which is transmitted by
the body, whose velocity Ave would determine;
and that a body be understood to be moved in
that space also ; so that there must be exposed two
lines, upon one of which uniform motion must be
understood to be made, that the time may be de¬
termined ; and, upon the other, the
velocity is to be computed. As if A ?
we would expose the velocity of the C D
body A, we draw two lines A B
and C 1), and place a body in C also; which done,
we say the velocity of the body A is so great,
that it passeth over the line A B in the same time
in which the body C passeth over the line C D
with uniform motion.
7. Weight is exposed by any heavy body, of
what matter soever, so it be always alike heavy.
8. The proportion of two magnitudes is then
exposed, when the magnitudes themselves are ex¬
posed, namely, the proportion of equality, when
the magnitudes are equal; and of inequality, when
they are unequal. For seeing, by the 5th article
of the preceding chapter, the proportion of two
unequal magnitudes consists in their difference,
compared with either of them; and when two un¬
equal magnitudes are exposed, their difference is
also exposed; it follows, that when magnitudes,
which have proportion to one another, are ex¬
posed, their proportion also is exposed with them;
and, in like manner, the proportion of equals,
OF QUANTITY.
143
which consists in this, that there is 110 difference fart Tr-
. 12-
of magnitude betwixt them, is exposed at the -—<—-
same time when the equal magnitudes themselves
are exposed. For example, if the exposed lines
A B and C D be equal, the propor- ^
tion of equality is exposed in them ;
and if the exposed lines, E F and E G C D
be unequal, the proportion which g Q ]?
E F has to E G, and that which E G
has to E F are also exposed in. them; for not only
the lines themselves, but also their difference, G F,
is exposed. The proportion of unequals is quan¬
tity ; for the difference, G F, in which it consists,
is quantity. But the proportion of equality is not
quantity; because, between equals, there is 110
difference ; nor is one equality greater than another,
as one inequality is greater than another inequality.
9. The proportion of two times, or of two uni- iiowthepro-
forni velocities, is then exposed, when two lines times and
are exposed by which two bodies are understood yelocitles
i J is exposed.
to be moved uniformly; and therefore the same
two lines serve to exhibit both their own propor¬
tion, and that of the times and velocities, accord¬
ing as they are considered to be exposed for the
magnitudes themselves, or for the times or velo¬
cities. For let the two lines A and B be ex- ^
posed; their proportion therefore (by the
last foregoing article) is exposed; and if ^
they be considered as drawn with equal
and uniform velocity, then, seeing their times are
greater, or equal, or less, according as the same
spaces are transmitted in greater, or equal, or
less time, the lines A and B will exhibit the
equality or inequality, that is, the proportion
144
philosophy.
tart it. of the times. To conclude, if the same lines, A
-—'—- and B, he consideTed as drawn in the same time,
then, seeing' their velocities are greater, or equal,
or less, according as they pass over in the same
time longer, or equal, or shorter lines, the same
lines, A and B, will exhibit the equality, or in¬
equality, that is, the proportion of their velocities.
CHAPTER XIII.
OF ANATOCISM, Oil THE SAME PROPORTION.
1, 2, 3, 4. The nature and definition of proportion, arithmetical
and geometrical.—5. The definition, and some properties of
the same arithmetical proportion.—6, 7. The definition and
transmutations of analogisin, or the same geometrical propor¬
tion.—S, 9. The definitions of hyperlogism and hypologism,
that is, of greater and less proportion, and their transmuta¬
tions.—10, 1], 12. Comparison of analogical quantities, ac¬
cording to magnitude.—13,14,15. Composition of proportions.
16, 17, 18, 19, 20, 21, 22, 23, 24, 25. The definition and
properties of continual proportion.—26, 27, 2S, 29. Compa¬
rison of arithmetical and geometrical proportions.
[Note, that in this chapter the sign + signifies that the quantities betwixt
which it is put, are added together; and this sign— the remainder after
the latter quantity is taken out of the former. So that A-j-13 is equal to
both A and 15 together; and where you see A—13, there A is the whole,
B the part taken out of it, and A—13 the remainder. Also, two letters, set
together without any sign, signify, unless they belong to a figure, that one
of ihe quantities is multiplied by the other ; as A 13 signifies the product of
A multiplied by 13.]
the nature }. Great and little are not intelligible, but by com-
aud definition . _T . . . . . ,
of proportion, parison. i\ ow that, to which they are compared,
&'geometrical. *s something exposed; that is, some magnitude
either perceived by sense, or so defined by words,
that it may be comprehended by the mind. Also
that, to which any magnitude is compared, is either
OF ANALOG1SM.
145
greater or less, or equal to it. And therefore pro- part ii.
portion (which, as I have shewn, is the estimation -—^—-
or comprehension of magnitudes by comparison,)
is threefold, namely, proportion of equality, that
is, of equal to equal; or of excess, which is of the
greater to the less ; or of defect, which is the pro¬
portion of the less to the greater.
Again, every one of these proportions is two¬
fold ; for if it be asked concerning any magnitude
given, how great it is, the answer may be made
by comparing it two ways; first, by saying it is
greater or less than another magnitude, by so
much; as seven is less than ten, by three unities ;
and this is called arithmetical proportion. Se¬
condly, by saying it is greater or less than another
magnitude, by such a part or parts thereof; as
seven is less than ten, by three tenth parts of the
same ten. And though this proportion be not
always explicable by number, yet it is a deter¬
minate proportion, and of a different kind from
the former, and called geometrical proportion,
and most commonly proportion simply.
2. Proportion, whether it be arithmetical or The nature
geometrical, cannot be exposed but in two magni-
tion of pro-
tudes, (of which the former is commonly called the Portlon> &c-
antecedent, and the latter the consequent of the
proportion) as I have shewn in the 8th article of
the preceding chapter. And, therefore, if two
proportions be to be compared, there must be four
magnitudes exposed, namely, two antecedents and
two consequents ; for though it happen sometimes
that the consequent of the former proportion be
the same with the antecedent of the latter, yet in
that double comparison it must of necessity be
VOL. i. L
146 philosophy.
part ii. twice numbered ; so that there will be always four
-—r*~ ' terms.
The nature 3. Of two proportions, whether tliey be arith-
and denm- . x x /
tion of pro- metical or geometrical, when the magnitudes com-
portion, &c. pare(j *n both (which Euclid, in the fifth definition
of his sixth book, calls the quantities of propor¬
tions,) are equal, then one of the proportions
cannot be either greater or less than the other;
for one equality is neither greater nor less than
another equality. But of two proportions of in¬
equality, whether they be proportions of excess or
of defect, one of them may be either greater or less
than the other, or they may both be equal; for
though there be propounded two magnitudes that
are unequal to one another, yet there may be
other two more unequal, and other two equally
unequal, and other two less unequal than the two
which were propounded. And from hence it may
be understood, that the proportions of excess and
defect are quantity, being capable of more and
less; but the proportion of equality is not quan¬
tity, because not capable neither of more, nor of
less. And therefore proportions of inequality may
be added together, or subtracted from one another,
or be multiplied or divided by one another, or by
number; but proportions of equality not so.
4. Two equal proportions are commonly called
the same proportion; and, it is said, that the
proportion of the first antecedent to the first
consequent is the same with that of the second
antecedent to the second consequent. And when
four magnitudes are thus to one another in geo¬
metrical proportion, they are called proportionals;
and by some, more briefly, analogism. And greater
of analogism.
147
proportion is the proportion of a greater ante- PA^T IL
cedent to the same consequent, or of the same -—A--
antecedent to a less consequent; and when the
proportion of the first antecedent to the first con¬
sequent is greater than that of the second ante¬
cedent to the second consequent, the four magni¬
tudes, which are so to one another, may be called
hyperlogism.
Less proportion is the proportion of a less ante¬
cedent to the same consequent, or of the same
antecedent to a greater consequent; and when the
proportion of the first antecedent to the first conse¬
quent is less than that of the second to the second,
the four magnitudes may be called hypologism.
5. One arithmetical proportion is the same with The definition
another arithmetical proportion, when one of the perties of the
antecedents exceeds its consequent, or is exceeded ™caTp^o-
by it, as much as the other antecedent exceeds its Portion-
consequent, or is exceeded by it. And therefore,
in four magnitudes, arithmetically proportional,
the sum of the extremes is equal to the sum of the
means. For if A. B :: C. D be arithmetically pro¬
portional, and the difference on both sides be the
same excess, or the same defect, E, then B + C (if
A be greater than B) will be equal to A—E + C;
and A+D will be equal to A + C—E ; but A—E + C
and A + C—E are equal. Or if A be less than B,
then B + C will be equal to A+E+C; and A+D
will be equal to A + C + E ; but A+E + C and A + C
+ E are equal.
Also, if there be never so many magnitudes,
arithmetically proportional, the sum of them all
will be equal to the product of half the number of
the terms multiplied by the sum of the extremes.
l 2
148
philosophy.
tart ii. For if A. B : : C. D : : E. F be arithmetically pro-
vportional, the couples A + F, B + E, C + D will be
Ind some'pro" equal to one another ; and their sum will be equal
perties of, See. £0 A + F, multiplied by the number of their combi¬
nations, that is, by half the number of the terms.
If, of four unequal magnitudes, any two, together
taken, be equal to the other two together taken,
then the greatest and the least of them will be in
the same combination. Let the unequal magni¬
tudes be A, B, C, D; and let A + B be equal to
C + D ; and let A be the greatest of them all; I say
B will be the least. For, if it may be, let any of
the rest, as D, be the least. Seeing therefore A
is greater than C, and B than D, A + B will be
greater than C + D; which is contrary to what was
supposed.
If there be any four magnitudes, the sum of the
greatest and least, the sum of the means, the
difference of the two greatest, and the difference
of the two least, will be arithmetically propor¬
tional. For, let there be four magnitudes, whereof
A is the greatest, D the least, and B and C the
means; I say A+D. B + C:: A—B. C—D are
arithmetically proportional. For the difference
between the first antecedent and its consequent is
this, A + D—B—C ; and the difference between
the second antecedent and its consequent this,
A—B—C + D ; but these two differences are equal;
and therefore, by this 5th article, A + D. B + C::
A—B. C—D are arithmetically proportional.
If, of four magnitudes, two be equal to the other
two, they will be in reciprocal arithmetical pro¬
portion. For let A + B be equal to C+D, I say
A. C : : D. B are arithmetically proportional. For
OF ANALOGISM.
149
if tliey be not, let A. C :: D. E (supposing E to be IL
greater or less than B) be arithmetically propor- ——r—'
tional, and then A+E will be equal to C + D ;
wherefore A + B and C + D are not equal; which is
contrary to what was supposed.
(). One geometrical proportion is the same with The definition
. i , ,, and transmu-
another geometrical proportion; when the same tations of ana-
cause, producing equal effects in equal times, de-laSe geom*0
termines both the proportions. tricaipro-.
. . portion.
If a point uniformly moved describe two lines,
either with the same, or different velocity, all the
parts of them which are contemporary, that is,
which are described in the same time, will be two
to two, in geometrical proportion, whether the
antecedents be taken in the same line, or not.
For, from the point A (in the 10th figure at the
end of the 14th chapter) let the two lines, AD,
A G, be described with uniform motion; and let
there be taken in them two parts A B, AE, and
again, two other parts, AC, AF; in such man¬
ner, that A B, A E, be contemporary, and likewise
A C, A F contemporary. I say first (taking the
antecedents A B, AC in the line A D, and the con-
quents AE, AF in the line A G) that AB. AC::
A E. A F are proportionals. For seeing (by the
8th chap, and the 15tli art.) velocity is motion
considered as determined by a certain length or
line, in a certain time transmitted by it, the quan¬
tity of the line A B will be determined by the
velocity and time by which the same A B is de¬
scribed ; and for the same reason, the quantity of
the line A C will be determined by the velocity
and time, by which the same A C is described;
and therefore the proportion of A B to A C, whe-
150
philosophy.
part ii. ther it be proportion of equality, or of excess or
-—A— defect, is determined by the velocities and times
andtransmuTa-which A B, AC are described ; but seeing the
tions of anaio- motion of the point A upon A B and A C is uni-
gism, &c. 1 . i i •
form, they are both described with equal velocity ;
and therefore whether one of them have to the
other the proportion of majority or of minority,
the sole cause of that proportion is the difference
of their times ; and by the same reason it is evi¬
dent, that the proportion of A E to A F is deter¬
mined by the difference of their times only. Seeing
therefore A B, A E, as also A C, A F are contem¬
porary, the difference of the times in which A B
and A C are described, is the same with that in
which A E and A F are described. Wherefore the
proportion of A B to AC, and the proportion of
A E to A F are both determined by the same cause.
But the cause, which so determines the proportion
of both, works equally in equal times, for it is uni¬
form motion ; and therefore (by the last precedent
definition) the proportion of A B to A C is the same
with that of A E to A F; and consequently A B.
AC : : A E. A F are proportionals; which is the
first.
Secondly, (taking the antecedents in different
lines) I say, A B. A E :: A C. A F are proportion¬
als ; for seeing A B, A E are described in the same
time, the difference of the velocities in which they
are described is the sole cause of the proportion
they have to one another. And the same may be
said of the proportion of A C to A F. But seeing
both the lines A D and A G are passed over by
uniform motion, the difference of the velocities in
which A B, A E are described, will be the same
OF ANALOGISM.
151
with the difference of the velocities, in which A C, taut ir.
13
A F are described. Wherefore the cause which 1—^—■-
determines the proportion of A B to A E, is the Thf.definition
1 r ^ 5 and trailsmuta-
same with that which determines the proportion of 'i.01"of anaio-
A C to AF; and therefore A B. AE :: AC. AF, glsm'&c'
are proportionals ; which remained to be proved.
Coroll. i. If four magnitudes be in geometrical
proportion, they will also be proportionals by per¬
mutation., that is, by transposing the middle terms.
For I have shown, that not only A B. A C :: A E.
A F, but also that, by permutation, A B. A E : :
A C. A F are proportionals.
Coroll. ii. If there be four proportionals, they
will also be proportionals by inversion or conver¬
sion,, that is, by turning the antecedents into con¬
sequents. For if in the last analogism, I had for
A B, A C, put by inversion A C, A B, and in like
manner converted A E, A F into A F, A E, yet the
same demonstration had served. For as well A C,
A B, as A B, A C are of equal velocity; and A C,
A F, as well as A F, A C are contemporary.
Coroll. hi. If proportionals be added to propor¬
tionals, or taken from them, the aggregates, or
remainders, will be proportionals. For contempo¬
raries, whether they be added to contemporaries,
or taken from them, make the aggregates or re¬
mainders contemporary, though the addition or
subtraction be of all the terms, or of the antece¬
dents alone, or of the consequents alone.
Coroll. iv. If both the antecedents of four pro¬
portionals, or both the consequents, or all the
terms, be multiplied or divided by the same num¬
ber or quantity, the products or quotients will be
proportionals. For the multiplication and division
152
PHILOSOPHY.
pa^t IL of proportionals, is the same with the addition and
v—subtraction of them,
an'd transmuta- Coroll. v. If there be four proportionals, they
°Lanal°"a^so proportionals by composition, that is,
by compounding: an antecedent of the antecedent
and consequent put together, and by taking for
consequent either the consequent singly, or the
antecedent singly. For this composition is nothing
but addition of proportionals, namely, of conse¬
quents to their own antecedents, which by suppo¬
sition are proportionals.
Coroll. yi . In like manner, if the antecedent
singly, or consequent singly, be put for antecedent,
and the consequent be made of both put together,
these also will be proportionals. For it is the in¬
version of proportion by composition,
Coroll. vii. If there be four proportionals, they
will also be proportionals by division, that is, by
taking the remainder after the consequent is sub¬
tracted from the antecedent, or the difference
between the antecedent and consequent for ante¬
cedent, and either the whole or the subtracted for
consequent; as if A. B :: C. D be proportionals,
they will by division be A—B. B :: C—D. D, and
A—B. A : : C—D. C ; and when the consequent is
greater than the antecedent, B—A. A : : D—C. C,
and B—A. B :: D—C. D. For in all these divisions,
proportionals are, by the very supposition of the
analogism A. B : : C. D, taken from A and B, and
from C and D.
Coroll. viii. If there be four proportionals, they
will also be proportionals by the conversion of
proportion, that is, by inverting the divided pro¬
portion, or by taking the whole for antecedent,
and the difference or remainder for consequent.
OF ANALOGISM.
153
As, if A. B :: C. D be proportionals, then A. A—B part ii.
:: C. C—D, as also B. A—B :: D. C—D will be —.
, • i i • , -i -1 The definition
proportionals, r or seeing these inverted be pro-and transmuta-
portionals, they are also themselves proportionals. ^ &c®nal°"
Coroll. ix. If there be two analogisms which
have their quantities equal, the second to the se¬
cond, and the fourth to the fourth, then either the
sum or difference of the first quantities will be to
the second, as the sum or difference of the third
quantities is to the fourth. Let A. B :: C. D and
E. B :: F. D be analogisms ; I say A +E. B : : C + F.D
are proportionals. For the said analogisms will
by permutation be A. C :: B. D, and E. F :: B. D ;
and therefore A.C::E. F will be proportionals,
for they have both the proportion of B to D com¬
mon. Wherefore, if in the permutation of the
first analogism, there be added E and F to A and
C, which E and F are proportional to A and C,
then (by the third coroll.) A + E. B : : C + F. D will
be proportionals ; which was to be proved.
Also in the same manner it may be shown, that
A—E. B : : C—F. D are proportionals.
7. If there be two analogisms, where four an¬
tecedents make an analogism, their consequents
also shall make an analogism ; as also the sums of
their antecedents will be proportional to the sums
of their consequents. For if A. B : : 0. D and
E. F : : G. H be two analogisms, and A. : : C. G be
proportionals, then by permutation A. C : : E. G,
and E. G : : F. H, and A. C : : B. D will be propor¬
tionals ; wherefore B.D : : E.G, that is, B.D : : F.H,
and by permutation B. F :: D. H are proportionals;
which is the first. Secondly, I say A + E. B + F: :
C + G. D + H will be proportionals. For seeing
154
PHILOSOPHY.
PA1JJ IT- A. E : : C. G are proportionals, A + E. E :: C + G. G
-—■—' will also by composition be proportionals, and by
permutation A + E. C + G::E. G will be propor¬
tionals; wherefore, also A+E. C + G : : F. H will
be proportionals. Again, seeing, as is shown above,
B. F: : D. H are proportionals, B + F. F : : D + H.H
will also by composition be proportionals ; and by
permutation B + F. D + H : : F. H will also be pro¬
portionals ; wherefore A+E. C + G: : B + F. D + H
are proportionals ; which remained to be proved.
Coroll. By the same reason, if there be never so
many analogisms, and the antecedents be propor¬
tional to the antecedents, it may be demonstrated
also that the consequents will be proportional to
the consequents, as also the sum of the antece¬
dents to the sum of the consequents.
The definition 8. In an liyperlogism, that is, where the pro-
an^hypo°o-Ism portion of the first antecedent to its consequent
ofSgreattrVnd *s greater than the proportion of the second ante-
lessproportion, cedent to its consequent, the permutation of the
and their trans- . , i t • f • i
mutations. proportionals, and the addition of proportionals to
proportionals, and substraction of them from one
another, as also their composition and division, and
their multiplication and division by the same num¬
ber, produce always an liyperlogism. For suppose
A. B : : C.l) and A.C :: E. F be analogisms, A+E. B
: : C + F. D will also be an analogism; but A + E.
B : : C. D will be an liyperlogism; wherefore by
permutation, A + E. C :: B. D is an liyperlogism,
because A. B :: C. I) is an analogism. Secondly, if
to the liyperlogism A + E. B::C. D the propor¬
tionals G and H be added, A + E + G. B :: C + H. D
will be an liyperlogism, by reason A+E+G.
B :: C + F + H. D is an analogism. Also, if G and
OF ANALOGISM.
155
H be taken away, A + E—G. B :: C—H. D will be paut il
an hyperlogism ; for A + E—G. B : : C + F—H. D -—-
is an analogism. Thirdly, by composition A + E defimUons
° J7 j i 1 oi hyperlogism,
+ B. B:: C + D. D will be an hyperlogism, because hypoiogism&c.
A+E + B. B : : C + F + D. D is an analogism, and so
it will be in all the varieties of composition.
Fourthly, by division, A+E—B. B : : C—D. D will
by an hyperlogism, by reason A E—B. B :: C+F
—D. D is an analogism. Also A + E—B. A+E: :
C—D. C is an hyperlogism ; for A + E—B. A + E ::
C + F—D. C is an analogism. Fifthly, by multipli¬
cation 4 A + 4 E. B : : 4 C. D is an hyperlogism, be¬
cause 4 A. B : : 4 C. D is an analogism ; and by
division J A + JE. B :: ] C. D is an hyperlogism,
because A. B :: 7] C. I) is an analogism.
9. But if A+E. B: :C.D be an hyperlogism,
then by inversion B. A+E : : D. C will be an hy-
pologism, because B. A: : D. C being an analo¬
gism, the first consequent will be too great. Also,
by conversion of proportion, A + E. A+E—B :: C.
C—1) is an hypologism, because the inversion of
it, namely A + E—B. A + E : : C—D. C is an hyper¬
logism, as I have shown but now. So also B. A +
E—B :: D. C—D is an hypologism, because, as I
have newly shown, the inversion of it, namely
A+E—B. B : : C—D. D is an hyperlogism. Note
that this hypologism A + E. A + E—B : : C. C—D is
commonly thus expressed; if the proportion of
the whole, (A + E) to that which is taken out of it
(B), be greater than the proportion of the whole
(C) to that which is taken out of it (D), then the
proportion of the whole (A+E) to the remainder
(A + E—B) will be less than the proportion of the
whole (C) to the remainder (C—D).
156
philosophy.
part ii. 10. if there be four proportionals, the difference
v——- of the two first, to the difference of the two last,
oAnalogical will be as the first antecedent is to the second
according' antecedent, or as the first consequent to the second
to magnitude, consequent. For if A. B :: C. D be proportionals,
then by division A—B. B : : C—D. D will be pro¬
portionals ; and by permutation A—B. C—D ::
B. 1) ; that is, the differences are proportional to
the consequents, and therefore they are so also to
the antecedents.
11. Of four proportionals, if the first be greater
than the second, the third also shall be greater
than the fourth. For seeing the first is greater
than the second, the proportion of the first to the
second is the proportion of excess; but the pro¬
portion of the third to the fourth is the same with
that of the first to the second ; and therefore also
the proportion of the third to the fourth is the
proportion of excess ; wherefore the third is greater
than the fourth. In the same maimer it may be
proved, that whensoever the first is less than the
second, the third also is less than the fourth; and
when those are equal, that these also are equal.
12. If there be four proportionals whatsoever,
A.B : : C.D, and the first and third be multiplied by
any one number, as by 2 ; and again the second and
fourth be multiplied by any one number, as by 3 ;
and the product of the first 2 A, be greater than
the product of the second 3 B ; the product also
of the third 2 C, will be greater than the product
of the fourth 3 D. But if the product of the first
be less than the product of the second, then the
product of the third will be less than that of the
fourth. And lastly, if the products of the first
OF ANALOGISM.
15/
and second be equal, the products of the third and PA^T n*
fourth shall also be equal. Now this theorem -—r—-
is all one with Euclid's definition of the .same
proportion; and it may be demonstrated thus.
Seeing A. B :: C. D are proportionals, by permu¬
tation also (art. 6, coroll. i.) A. C : : B. D will be
proportionals ; wherefore (by coroll. iv. art. 6) 2 A.
2 C: : 3 B. 3 I) will be proportionals; and again,
by permutation, 2 A. 3 B : : 2 C. 3D will be pro¬
portionals ; and therefore, by the last article, if
2 A be greater than 3 B, then 2 C will be greater
than 3 D ; if less, less ; and if equal, equal; which
was to be demonstrated.
13. If any three magnitudes be propounded, or Composition
three things whatsoever that have any proportion ° proportloIls'
one to another, as three numbers, three times,
three degrees, &c. ; the proportions of the first to
the second, and of the second to the third, together
taken, are equal to the proportion of the first to
the third. Let there be three lines, for any pro¬
portion may be reduced to the proportion of lines,
A B, A C, A D ; and in the first place, let the pro¬
portion as well of the first A B to the second A C,
^ ^ ^ as of the second A C to the
third AD, be the proportion
of defect, or of less to greater ; I say the propor¬
tions together taken of A B to A C, and of A C to
A D, are equal to the proportion of A B to A D.
Suppose the point A to be moved over the whole
line A D with uniform motion ; then the propor¬
tions as well of A B to A C, as of A C to A D, are
determined by the difference of the times in which
they are described ; that is, A B has to A C such
proportion as is determined by the different times
of their description ; and A C to AD such propor-
158
philosophy.
part ii. tion as is determined by their times. But the
13. J
■—r—' proportion of A B to A D is such as is determined
of°proportions. difference of the times in which A B and
A D are described; and the difference of the times
in which AB and A C are described, together with
the difference of the times in which A C and A D
are described, is the same with the difference of
the times in which A B and A D are described.
And therefore, the same cause which determines
the two proportions of A B to A C and of A C to
A D, determines also the proportion of AB to
A D. Wherefore, by the definition of the same
proportion, delivered above in the 6th article, the
proportion of A B to AC together with the pro¬
portion of AC to AD, is the same with the pro¬
portion of A B to A D.
In the second place, let A D be the first, A C
the second, and A B the third, and let their pro¬
portion be the proportion of excess, or the greater
to less ; then, as before, the proportions of A D to
A C, and of A C to A B, and of A D to A B, will be
determined by the difference of their times ; which
in the description of A D and A C, and of A C and
A B together taken, is the same with the differ¬
ence of the times in the description of AD and
A B. Wherefore the proportion of A D to A B is
equal to the two proportions of A D to A C and of
A C to A B.
In the last place. If one of the proportions,
namely of A D to A B, be the proportion of excess,
and another of them, as of A B to A C be the pro¬
portion of defect, thus also the proportion of A D
to A C will be equal to the two proportions toge¬
ther taken of A D to A B, and of A B to A C. For
the difference of the times in which A D and AB
of analogism.
159
are described, is excess of time; for there goes part ir.
more time to the description of A D than of A B ; • ^—-
and the difference of the times in which A B and of° proportions.
A C are described, is defect of time, for less time
goes to the description of A B than of A C ; but
this excess and defect being added together, make
D B—B C, which is equal to D C, by which the
first A D exceeds the third A C ; and therefore the
proportions of the first AD to the second A B,
and of the second A B to the third A C, are deter¬
mined by the same cause which determines the
proportion of the first A D to the third A C.
Wherefore, if any three magnitudes, &c.
Coroll. i. If there be never so many magnitudes
having proportion to one another, the proportion
of the first to the last is compounded of the pro¬
portions of the first to the second, of the second
to the third, and so on till you come to the last;
or, the proportion of the first to the last is the
same with the sum of all the intermediate propor¬
tions. For any number of magnitudes having pro¬
portion to one another, as A, B, G, D, E being
propounded, the proportion of A to E, as is newly
shown, is compounded of the proportions of A to D
and of D to E ; and again, the proportion of A to
D, of the proportions of A to C, and of C to D ;
and lastly, the proportion of A to C, of the pro¬
portions of A to B, and of B to C.
Coroll. ii. From hence it may be understood
how any two proportions may be compounded. For
if the proportions of A to B, and of C to D, be
propounded to be added together, let B have to
something else, as to E, the same proportion which
C has to D, and let them be set in this order,
160
philosophy.
part it. A, B, E ; for so the proportion of A to E will evi-
-—r—- dently be the sum of the two proportions of A to B,
ofpropoSns.and of B to E, that is, of C to D. Or let it be as
D to C, so A to something else, as to E, and let
them be ordered thus, E, A, B ; for the proportion
of E to B will be compounded of the proportions
of E to A, that is, of C to D, and of A to B. Also,
it may be understood how one proportion may be
taken out of another. For if the proportion of C
to D be to be subtracted out of the proportion of
A to B, let it be as C to D, so A to something else,
as E, and setting them in this order, A, E, B, and
taking away the proportion of A to E, that is, of
C to D, there will remain the proportion of E to B.
Coroll. in. If there be two orders of magnitudes
which have proportion to one another, and the
several proportions of the first order be the same
and equal in number with the proportions of the
second order; then, whether the proportions in
both orders be successively answerable to one ano¬
ther, which is called ordinate proportion, or not
successively answerable, which is called perturbed
proportion, the first and the last in both will be pro¬
portionals. For the proportion of the first to the
last is equal to all the intermediate proportions;
which being in both orders the same, and equal in
number, the aggregates of those proportions will
also be equal to one another ; but to their aggre¬
gates, the proportions of the first to the last are
equal; and therefore the proportion of the first to
the last in one order, is the same with the propor¬
tion of the first to the last in the other order.
Wherefore the first and the last in both are pro¬
portionals.
of analogism.
161
14. If any two quantities be made of the mutual part it.
multiplication of many quantities, which have pro- «—r—-
portion to one another, and the efficient quantities Composition
r ' 1 of proportions.
on both sides be equal in number, the proportion
of the products will be compounded of the several
proportions, which the efficient quantities have to
one another.
First, let the two products be A B and C D,
whereof one is made of the multiplication of A
into B, and the other of the multiplication of C
into D. I say the proportion of AB to CD is
compounded of the proportions of the efficient A
to the efficient C, and of the efficient B to the
efficient D. For let A B, C B and C D be set in
order; and as B is to D, so let C be to another
quantity as E ; and let A, C, E be
set also in order. Then (by
coroll. iv. of the 6th art.) it will
be as A B the first quantity to CB
the second quantity in the first order, so A to C in
the second order ; and again, as C B to C D in the
first order, so B to D, that is, by construction,
so C to E in the second order; and therefore (by
the last corollary) A B. C D : : A. E will be pro¬
portionals. But the proportion of A to E is com¬
pounded of the proportions of A to C, and of B to
D; wherefore also the proportion of A B to C D
is compounded of the same.
Secondly, let the two products be ABF, and
C D G, each of them made of three efficients, the
first of A, B and F, and the second of C, D and
G; I say, the proportion of ABF to C D G is
compounded of the proportions of A to C, of B to
D, and of F to G. For let them be set in order as
VOL. I. M
AB. A.
C B. C.
CD. E.
162
philosophy.
part ii. before; and as B is to D, so let C be to another
13.
-—A— quantity E ; and again, as F is to G, so let E be to
ofproponicns an°ther, H ; and let the first order stand thus,
ABF, CBF, CDF and CDG; A p A
and the second order thus, CBF C*
A, C, E, H. Then the propor- CDF E
tion of A B F to C B F in the nr\n u
i mi i * JJ Ijr. II.
first order, will be as A to C m
the second ; and the proportion of C B F to C D F
in the first order, as B to D, that is, as C to E (by
construction) in the second order ; and the pro¬
portion of CDF to C D G in the first, as F to G,
that is, as E to H (by construction) in the second
order; and therefore ABF. C D G :: A. H will be
proportionals. But the proportion of A to H is
compounded of the proportions of A to C, B to D,
and F to G. Wherefore the proportion of the
product A B F to C D G is also compounded of the
same. And this operation serves, how many soever
the efficients be that make the quantities given.
From hence ariseth another way of compounding
many proportions into one, namely, that which is
supposed in the 5th definition of the 6th book of
Euclid; which is, by multiplying all the antece¬
dents of the proportions into one another, and in
like manner all the consequents into one another.
And from hence also it is evident, in the first
place, that the cause why parallelograms, which
are made by the duction of two straight lines into
one another, and all solids which are equal to
figures so made, have their proportions compounded
of the proportions of the efficients; and in the
second place, why the multiplication of two or
more fractions into one another is the same thing
of analogism.
163
with the composition of the proportions of their part it.
several numerators to their several denominators. -—r-^—■
For example, if these fractions L I, i be to be compositi?n
r# 1 ^ a of proportions.
multiplied into one another, the numerators 1, 2, 3,
are first to be multiplied into one another, which
make 6 ; and next the denominators 2, 3, 4, which
make 24 ; and these two products make the frac¬
tion yt. In like manner, if the proportions of 1
to 2, of 2 to 3, and of 3 to 4, be to be compounded,
by working as I have shown above, the same pro¬
portion of 6 to 24 will be produced.
15. If any proportion be compounded with itself
inverted, the compound will be the proportion of
equality. For let any proportion be given, as of
A to B, and let the inverse of it be that of C to D ;
and as C to D, so let B be to another quantity ;
for thus they will be compounded (by the second
coroll. of the 12th art.) Now seeing the propor¬
tion of C to D is the inverse of the proportion of
A to B, it will be as C to D, so B to A ; and there¬
fore if they be placed in order, A, B, A, the propor¬
tion compounded of the proportions of A to B, and
of C to D, will be the proportion of A to A, that
is, the proportion of equality. And from hence
the cause is evident why two equal products have
their efficients reciprocally proportional. For, for
the making of two products equal, the proportions
of their efficients must be such, as being com¬
pounded may make the proportion of equality,
which cannot be except one be the inverse of the
other ; for if betwixt A and A any other quantity,
as C, be interposed, their order will be A,C, A, and
the later proportion of C to A will be the inverse
of the former proportion of A to C.
m 2
164
philosophy.
part ii. i(j. A proportion is said to be multiplied by a
-——' number, when it is so often taken as there be
L^propTnies unities in that number ; and if the proportion be
of continual 0f greater to the less, then shall also the
proportion. °
quantity of the proportion be increased by the
multiplication ; but when the proportion is of the
less to the greater, then as the number increaseth,
the quantity of the proportion diminisheth ; as in
these three numbers, 4, 2, 1, the proportion of 4 to
1 is not only the duplicate of 4 to 2, but also twice
as great; but inverting the order of those numbers
thus, 1, 2, 4, the proportion of 1 to 2 is greater
than that of 1 to 4 ; and therefore though the
proportion of 1 to 4 be the duplicate of 1 to 2, yet
it is not twice so great as that of 1 to 2, but con-
trarily the half of it. In like manner, a proportion
is said to be divided, when between two quantities
are interposed one or more means in continual
proportion, and then the proportion of the first to
the second is said to be subduplicate of that of the
first to the third, and subtriplicate of that of the
first to the fourth, &c.
This mixture of proportions, where some are
proportions of excess, others of defect, as in a
merchant's account of debtor and creditor, is not
so easily reckoned as some think; but maketh the
composition of proportions sometimes to be addi¬
tion, sometimes substraction ; which soundeth
absurdly to such as have always by composition
understood addition, and by diminution substrac¬
tion. Therefore to make this account a little
clearer, we are to consider (that which is com¬
monly assumed, and truly) that if there be never
so many quantities, the proportion of the first to
of analogism.
165
the last is compounded of the proportions of the part it.
first to the second, and of the second to the third, ■ ^—-
and so on to the last, without regarding their ThfdefiriltIon
7 . and properties
equality, excess, or defect; so that if two propor- of continual
tions, one of inequality, the other of equality, be proportlon'
added together, the proportion is not thereby made
greater nor less ; as for example, if the proportions
of A to B and of B to B be compounded, the pro¬
portion of the first to the second is as much as the
sum of both, because proportion of equality, being
not quantity, neither augmenteth quantity nor
lesseneth it. But if there be three quantities,
A, B, C, unequal, and the first be the greatest, the
last least, then the proportion of B to C is an ad¬
dition to that of A to B, and makes it greater;
and on the contrary, if A be the least, and C the
greatest quantity, then doth the addition of the
proportion of B to C make the compounded pro¬
portion of A to C less than the proportion of A to
B, that is, the whole less than the part. The com¬
position therefore of proportions is not in this case
the augmentation of them, but the diminution ;
for the same quantity (Euclid v. 8) compared with
two other quantities, hath a greater proportion to
the lesser of them than to the greater. Likewise,
when the proportions compounded are one of
excess, the other of defect, if the first be of excess,
as in these numbers, 8, 6, 9, the proportion com¬
pounded, namely, of 8 to 9, is less than the pro¬
portion of one of the parts of it, namely, of 8
to 6; but if the proportion of the first to the
second be of defect, and that of the second to
the third be of excess, as in these numbers, 6, 8, 4,
then shall the proportion of the first to the third
166
philosophy.
part ii. be greater than that of the first to the second, as
r—- 6 hath a greater proportion to 4 than to 8; the
properties reason whereof is manifestly this, that the less any
of continual quantity is deficient of another, or the more one
proportion. J \
exceedeth another, the proportion of it to that
other is the greater.
Suppose now three quantities in continual pro¬
portion, A B 4, A C 6, AD 9. Because therefore
AD is greater than AC, but not greater than AD,
the proportion of AD to AC will be (by Euclid,
v. 8) greater than that of A D to A D ; and like¬
wise, because the proportions of A D to A C, and
of A C to A B are the same, the proportions of A D
to A C and of A C to A B, being both proportions
of excess, make the whole proportion of AD to
A B, or of 9 to 4, not only the duplicate of AD to
AC, that is, of 9 to 6, but also the double, or
twice so great. On the other side, because the
proportion of A D to A D, or 9 to 9, being propor¬
tion of equality, is no quantity, and yet greater
than that of AC to AD, or 6 to 9, it will be as 0—9
to 0—6, so A C to AD, and again, as 0—9 to 0—6,
so 0—6 to 0—4 ; but 0—4, 0—6, 0—9 are in con¬
tinual proportion ; and because 0—4 is greater
than 0—6, the proportion of 0—4 to 0—6 will be
double to the proportion of 0—4 to 0—9, double I
say, and yet not duplicate, but subduplicate.
If any be unsatisfied with this ratiocination, let
him first consider that (by Euclid v. 8) the propor¬
tion of A B to A C is greater than that of A B to
AD, wheresoever D be
placed in the line AC BCD
prolonged; and the A E
further off the point
OF ANALOGISM.
167
D is from C, so much the greater is the proportion PART IL
of A B to A C than that of A B to AD. There is v—
therefore some point (which suppose be E) in such Sproperties
distance from C, as that the proportion of A B to ploponlm?1
A C will be twice as great as that of A B to A E.
That considered, let him determine the length of
the line AE, and demonstrate, if he can, that AE
is greater or less than A D.
By the same method, if there be more quantities
than three, as A, B, C, D, in continual proportion,
and A be the least, it may be made appear that
the proportion of A to B is triple magnitude,
though subtriple in multitude, to the proportion of
A to D.
17- If there be never so many quantities, the
number whereof is odd, and their order such, that
from the middlemost quantity both ways they
proceed in continual proportion, the proportion of
the two which are next 011 either side to the mid¬
dlemost is subduplicate to the proportion of the
two which are next to these on both sides, and
subtriplicate of the proportion of the two which
are yet one place more remote, &c. For let the
magnitudes be C, B, A, D, E, and let A, B, C, as
also A, D, E be in continual proportion; I say
the proportion of D to B is subduplicate of the
proportion of E to C. For the proportion of D to
B is compounded of the proportions of D to A, and
of A to B once taken; but the proportion of E to
C is compounded of the same twice taken ; and
therefore the proportion of D to B is subduplicate
of the proportion of E to C. And in the same
manner, if there were three terms 011 either side.,
it might be demonstrated that the proportion of
168
PHILOSOPHY.
PAI}J TL ^ to ® would be subtriplicate of that of the ex-
—»—' tremes, &c.
and properties If there be never so many continual propor-
of continual tionals, as the first, second, third. &c. their differ-
proportion. y
ences will be proportional to them, tor the second,
third, &c. are severally consequents of the preceding,
and antecedents of the following proportion. But
(by art. x.) the difference of the first antecedent
and consequent, to difference of the second antece¬
dent and consequent, is as the first antecedent to
the second antecedent, that is, as the first term to
the second, or as the second to the third, &c. in
continual proportionals.
19. If there be three continual proportionals,
the sum of the extremes, together with the mean
twice taken, the sum of the mean and either of
the extremes, and the same extreme, are conti¬
nual proportionals. For let A. B. C be continual
proportionals. Seeing, therefore, A. B :: B. C are
proportionals, by composition also A +B. B :: B+C.
C will be proportionals; and by permutation A+B.
B + C : : B. C will also be proportionals ; and again,
by composition A + 2B+C. B+C:: B + C. C; which
was to be proved.
20. In four continual proportionals, the greatest
and the least put together is a greater quantity
than the other two put together. Let A. B : : C. D
be continual proportionals ; whereof let the great¬
est be A, and the least be D ; I say A+D is greater
than B+C. For by art. 10, A—B. C—D :: A. C
are proportionals ; and therefore A—B is, by art.
11, greater than C—D. Add B on both sides, and
A will be greater than C + B—D. And again, add
D on both sides, and A+D will be greater than
B + C ; which was to be proved.
of analogism.
169
21. If there be four proportionals, the extremes part ii.
multiplied into one another, and the means multi- v—A-'
plied into one another, will make equal products. topTrties
Let A. B : : C. D be proportionals ; I say A D IS of continual
equal to B C. For the proportion of AD to B Cproportlon'
is compounded, by art. 13, of the proportions of
A to B, and D to C, that is, its inverse B to A;
and therefore, by art. 14, this compounded pro¬
portion is the proportion of equality ; and there¬
fore also, the proportion of A D to B C is the pro¬
portion of equality. Wherefore they are equal.
22. If there be four quantities, and the propor¬
tion of the first to the second be duplicate of the
proportion of the third to the fourth, the product
of the extremes to the product of the means, will
be as the third to the fourth. Let the four quan¬
tities be A, B, C and D ; and let the proportion of
A to B be duplicate of the proportion of C to D,
I say A D, that is, the product of A into D is to
B C, that is, to the product of the means, as C to D.
For seeing the proportion of A to B is duplicate of
the proportion of C to D, if it be as C to D, so D
to another, E, then A. B : : C. E will be propor¬
tionals ; for the proportion of A to B is by suppo¬
sition duplicate of the proportion of C to D ; and
C to E duplicate also of that of C to D by the defi¬
nition, art. 15. Wherefore, by the last article, A E
or A into E is equal to B C or B into C ; but, by
coroll. iv. art. 6, AD is to AE as D to E, that is,
as C to D ; and therefore A D is to B C, which as
I have shown is equal to A E, as C to D ; which
was to be proved.
Moreover, if the proportion of the first A to
the second B be triplicate of the proportion of
170
philosophy.
part ii. the third C to the fourth D, the product of th(
s 777 extremes to the product of the means will b(
and properties duplicate the proportion of the third to the
proportion^ fourth- For if it be as C to D so D to E, and
again, as D to E so E to another, F, then the
proportion of C to F will be triplicate of the pro¬
portion of C to D ; and consequently, A. B :: C. F
will be proportionals, and A F equal to B C. But
as A D to A F, so is D to F; and therefore, also,
as A D to B C, so D to F, that is, so C to E ; but
the proportion of C to E is duplicate of the pro¬
portion of C to D ; wherefore, also, the proportion
of A D to B C is duplicate of that of C to D, as was
propounded.
23. If there be four proportionals, and a mean
be interposed betwixt the first and second, and
another betwixt the third and fourth, the first of
these means will be to the second, as the first of
the proportionals is to the third, or as the second
of them is to the fourth. For let A. B :: C. D be
proportionals, and let E be a mean betwixt A and
B, and F a mean betwixt C and D ; I say A. C: :
E. F are proportionals. For the proportion of A
to E is subduplicate of the proportion of A to 13,
or of C to D. Also, the proportion of C to F is
subduplicate of that of C to D; and therefore
A. E :: C. F are proportionals ; and by permutation
A. C : : E. F are also proportionals ; which was to
be proved.
24. Any thing is said to be divided into extreme
and mean proportion, when the whole and the
parts are in continual proportion. As for example,
when A+ B. A. B are continual proportionals; or
when the straight line A C is so divided in B, that
OF ANALOGISM.
171
A C. A B. B C are in continual proportion. And if PA1jg n*
the same line A C be again divided ^ B C "—'—'
in D, so as that AC. CD. AD be 1 1
continual proportionals; then also ^
AC. A B. AD will be continual proportionals;
and in like manner, though in contrary order,
CA. CD. CB will be continual proportionals;
which cannot happen in any line otherwise
divided.
25. If there be three continual proportionals, and
again, three other continual proportions, which
have the same middle term, their extremes will be
in reciprocal proportion. For let A. B. C and
D. 13. E be continual proportionals, I say A. D ::
E. C shall be proportionals. For the proportion of
A to D is compounded of the proportions of A to B,
and of B to D; and the proportion of E to C is
compounded of those of E to B, that is, of B to D,
and of B to C, that is, of A to B. Wherefore, by
equality, A. D : : E. C are proportionals.
26. If any two unequal quantities be made ex- comparison of
tremes, and there be interposed betwixt them any and^ometri-
number of means in geometrical proportion, and cal ProPortlon-
the same number of means in arithmetical propor¬
tion, the several means in geometrical proportion
will be less than the several means in arithmetical
proportion. For betwixt A the lesser, and E the
greater extreme, let there be interposed three
means, B, C, D, in geometrical proportion, and as
many more, F, G, H, in arithmetical proportion ;
I say B will be less than F, C than G, and D than
H. For first, the difference between A and F is the
same with that between F and G, and with that
between G and H, by the definition of arithme-
172
PHILOSOPHY.
part ii. tical proportion ; and therefore., the difference of
—A- the proportionals which stand next to one another,
arithmetical°f to difference of the extremes, is, when there is
and geometri- but one mean, half their difference ; when two, a
cal proportions.
third part of it; when three, a quarter, &c.; so that
in this example it is a quarter. But the difference
between D and E, by art. 17, is more than a
quarter of the difference be-
tween the extremes, because _
the proportion is geometrical, B F
and therefore the difference q ~q~
between A and D is less than —
three quarters of the same ^
difference of the extremes. In E E
like manner, if the difference
between A and D be understood to be divided
into three equal parts, it may be proved, that the
difference between A and C is less than two quar¬
ters of the difference of the extremes A and E.
And lastly, if the difference between A and C be
divided into two equal parts, that the difference
between A and B is less than a quarter of the
difference of the extremes A and E.
From the consideration hereof, it is manifest,
that B, that is A together with something else
which is less than a fourth part of the difference of
the extremes A and E, is less than F, that is, than
the same A with something else which is equal to
the said fourth part. Also, that C, that is A with
something else which is less than two fourth parts
of the said difference, is less than G, that is, than
A together with the said two-fourths. And lastly,
that D, which exceeds A by less than three-fourths
of the said difference, is less than H, which ex-
of analogism.
173
ceeds the same A by three entire fourths of the part ii.
said difference. And in the same manner it would —•
be if there were four means, saving that instead aruhmetfcai °f
of fourths of the difference of the extremes we are and geometri-
„ . . cal proportions.
to take filth parts ; and so on.
27- Lemma. If a quantity being given, first one
quantity be both added to it and subtracted from
it, and then another greater or less, the propor¬
tion of the remainder to the aggregate, is greater
where the less quantity is added and substracted,
than where the greater quantity is added and sub¬
stracted. Let B be added to and substracted from
the quantity A ; so that A—B be the remainder,
and A + B the aggregate ; and again, let C, a
greater quantity than B, be added to and sub¬
stracted from the same A, so that A—C be the
remainder and A + C the aggregate ; I say A—B.
A + B :: A—C. A + C will be an hyperlogism. For
A—B. A :: A—C. A is an hyperlogism of a greater
antecedent to the same consequent; and therefore
A—B. A + B :: A—C. A + C is a much greater hy¬
perlogism, being made of a greater antecedent to
a less consequent.
28. If unequal parts be taken from two equal
quantities, and betwixt the whole and the part of
each there be interposed two means, one in geome¬
trical, the other in arithmetical proportion; the
difference betwixt the two means will be greatest,
where the difference betwixt the whole and its part
is greatest. For let A B and A B be two equal quan¬
tities, from which let two unequal parts be taken,
namely, A E the less, and A F the greater; and
betwixt A B and A E let A G be a mean in geo¬
metrical proportion, and A H a mean in arithme-
174
philosophy.
part it. tical proportion. Also
- 13'-" betwixt A B and A F let ± , , ,
S=S;fAI be a mean in geo- J >. 'j
and geometn- metrical proportion, and
cal proportions. L L ?
A K a mean in arithmetical proportion; I say
H G is greater than K I.
For in the first place we have
this analogism * " . . A B. A G : : B G. G E, by
article 18.
Then by composition we have
this AB + AG. AB::B G+GE
that is, B E. B G.
And by taking the halves of
the antecedents this third . i A B-f I A G. A B::{BG +
\ G E, that is, B H. B G.
And by conversion a fourth . A B. i A B + ^ A G :: B G. B H.
And by division this fifth . i A B—J AG. \ K B+J A G
::HG. BH.
And by doubling the first an¬
tecedent and the first con¬
sequent AB—AG. AB + AG::HG.BH.
Also by the same method may
be found out this analogism A B—A I. A B-J- AI:: K I. B K.
Now seeing the proportion of A B to A E is
greater than that of A B to A F, the proportion of
A B to A G, which is half the greater proportion,
is greater than the proportion of A B to A I the
half of the less proportion ; and therefore AI is
greater than A G. Wherefore the proportion of
A B—A G to A B + A G, by the precedent lemma,
will be greater than the proportion of A B—A I to
AB + AI ; and therefore also the proportion of
H G to B H will be greater than that of KI to BK,
and much greater than the proportion of K I to
B H, which is greater than BK; for B H is the
half of B E, as B K is the half of B F, which, by
OF ANALOGISM.
175
supposition, is less than B E. Wherefore H G is PAI^ IL
greater than K I; which was to be proved. —'
Coroll. It is manifest from hence, that if any arithmetic^ °f
quantity be supposed to be divided into equal an,d seome.tn~
a J r r a cal proportions.
parts infinite in number, the difference between
the arithmetical and geometrical means will be
infinitely little, that is, none at all. And upon
this foundation, chiefly, the art of making those
numbers, which are called Logarithms, seems to
have been built.
29. If any number of quantities be propounded,
whether they be unequal, or equal to ione ano¬
ther ; and there be another quantity, which multi¬
plied by the number of the propounded quantities,
is equal to them all; that other quantity is a mean
in arithmetical proportion to all those propounded
quantities.
1/6
PHILOSOPHY.
CHAP. XIV.
OF STRAIT AND CROOKED, ANGLE AND
FIGURE.
1. The definition and properties of a strait line.—2. The defini¬
tion and properties of a plane superficies.—3. Several sorts of
crooked lines.—4*. The definition and properties of a circular
line. — 5. The properties of a strait line taken in a plane.
6. The definition of tangent lines. — 7. The definition of an
angle, and the kinds thereof.—8. In concentric circles, arches
of the same angle are to one another, as the whole circumfer¬
ences are.—9. The quantity of an angle, in what it consists.
10. The distinction of angles, simply so called.—11. Of strait
lines from the centre of a circle to a tangent of the same.
12. The general definition of parallels, and the properties of
strait parallels.—13. The circumferences of circles are to
one another, as their diameters are.—14. In triangles, strait
lines parallel to the bases are to one another, as the parts of
the sides which they cut off from the vertex.—15. By what
fraction of a strait line the circumference of a circle is made.
16. That an angle of contingence is quantity, but of a differ¬
ent kind from that of an angle simply so called; and that it
can neither add nor take away any thing from the same.
17. That the inclination of planes is angle simply so called.
18. A solid angle what it is.—19. What is the nature of
asymptotes. — 20. Situation, by what it is determined.—
21. What is like situation ; what is figure; and what are like
figures.
part ii. 1. Between two points given, the shortest line is
^—- that, whose extreme points cannot be drawn far-
Ind properties ther asunder without altering the quantity, that is,
of a strait line. without altering the proportion of that line to any
other line given. For the magnitude of a line is
computed by the greatest distance which may be
OF STRAIT AND CROOKED.
1 77
between its extreme points ; so that any one line, PAI^J IL
whether it be extended or bowed, has always one ^—-
and the same length, because it can have but one Ildp^opTrties
greatest distance between its extreme points. of a strait line.
And seeing the action, by which a strait line is
made crooked, or contrarily a crooked line is made
strait, is nothing but the bringing of its extreme
points nearer to one another, or the setting of
them further asunder, a croolied line may rightly
be defined to be that, whose extreme points may
be understood to be drawn further asunder;
and a strait line to be that, whose extreme
points cannot be drawn further asunder; and
comparatively, a more crooked, to be that line
whose extreme points are nearer to one another
than those of the other, supposing both the lines
to be of equal length. Now, howsoever a line
be bowed, it makes always a sinus or cavity, some¬
times on one side, sometimes on another ; so that
the same crooked line may either have its whole
cavity on one side only, or it may have it part 011
one side and part on the other side. Which
being well understood, it will be easy to under¬
stand the following comparisons of strait and
crooked lines.
First, if a strait and a crooked line have their
extreme points common, the crooked line is longer
than the strait line. For if the extreme points of
the crooked line be drawn out to their greatest
distance, it will be made a strait line, of which
that, which was a strait line from the beginning,
will be but a part; and therefore the strait line
was shorter than the crooked line, which had the
same extreme points. And for the same reason,
VOL. I. N
178
philosophy.
pari ii. if tWo crooked lines have their extreme points
common, and both of them have all their cavity on
JLne definition - J
and properties one and the same side, the outermost of the two
ofa strait line. wiU be ^ ^
Secondly, a strait line and a perpetually crook¬
ed line cannot be coincident, 110, not in the least
part. For if they should, then not only some
strait line would have its extreme points common
with some crooked line, but also they would, by
reason of their coincidence, be equal to one ano¬
ther ; which, as I have newly shown, cannot be.
Thirdly, between two points given, there can
be understood but one strait line ; because there
cannot be more than one least interval or length
between the same points. For if there may be
two, they will either be coincident, and so both of
them will be one strait line; or if they be not
coincident, then the application of one to the other
by extension will make the extended line have its
extreme points at greater distance than the other;
and consequently, it was crooked from the begin¬
ning.
Fourthly, from this last it follows, that two
strait lines cannot include a superficies. For if
they have both their extreme points common, they
are coincident; and if they have but one or neither
of them common, then at one or both ends the
extreme points will be disjoined, and include no
superficies, but leave all open and undetermined.
Fifthly, every part of a strait line is a strait
line. For seeing every part of a strait line is the
least that can be drawn between its own extreme
points, if all the parts should not constitute a strait
OF STRATT AND CROOKED.
1/9
line, tliey would altogether be longer than the part ii.
whole line. >—
2. A plane or a plane superficies, is that which The definition
is described by a strait line so moved, that all 7lpianc'su-
the several points thereof describe several strait Perficies*
lines. A strait line, therefore, is necessarily all of
it in the same plane which it describes. Also the
strait lines, which are made by the points that
describe a plane, are all of them in the same plane.
Moreover, if any line whatsoever be moved in a
plane, the lines, which are described by it, are all
of them in the same plane.
All other superficies, which are not plane, are
crooked, that is, are either concave or convex.
And the same comparisons, which were made of
strait and crooked lines, may also be made of plane
and crooked superficies.
For, first, if a plane and crooked superficies be
terminated with the same lines, the crooked super¬
ficies is greater than the plane superficies. For if
the lines, of which the crooked superficies con¬
sists, be extended, they will be found to be longer
than those of which the plane superficies consists,
which cannot be extended, because they are strait.
Secondly, two superficies, whereof the one is
plane, and the other continually crooked, cannot
be coincident, no, not in the least part. For if they
were coincident, they would be equal; nay, the
same superficies would be both plane and crooked,
which is impossible.
Thirdly, within the same terminating lines
there can be 110 more than one plane superficies;
because there can be but one least superficies
within the same.
N 2
180
PHILOSOPHY.
rAI^4 IT* Fourthly, 110 number of plane superficies can
s—' include a solid, unless more than two of them end
in a common vertex. For if two planes have both
the same terminating lines, they are coincident,
that is, they are but one superficies; and if their
terminating lines be not the same, they leave one
or more sides open.
Fifthly, every part of a plane superficies is a
plane superficies. For seeing the whole plane
superficies is the least of all those, that have the
same terminating lines ; and also every part of the
same superficies is the least of all those, that are
terminated with the same lines; if every part
should not constitute a plane superficies, all the
parts put together would not be equal to the
whole.
Several sorts of 3. Of straitness, whether it be in lines or in
crooked lines. SUpergcies^ there is but one kind ; but of crooked¬
ness there are many kinds ; for of crooked magni¬
tudes, some are congruous, that is, are coincident
when they are applied to one other ; others are
incongruous. Again, some are o/noiofxepug or uni¬
form, that is, have their parts, howsoever taken,
congruous to one another ; others are dvopoioptpug
or of several forms. Moreover, of such as are
crooked, some are continually crooked, others have
parts which are not crooked.
Definition and 4. If a strait line be moved in a plane, in such
circular line.& manner, that while one end of it stands still, the
whole line be carried round about till it come
again into the same place from whence it was first
moved, it will describe a plane superficies, which
will be terminated every way by that crooked line,
which is made by that end of the strait line which
OF STRAIT AND CROOKED.
181
was carried round. Now this superficies is called part ir.
a circle ; and of this circle, the unmoved point is -—r-^
the centre; the crooked line which terminates it, Definiti.on a°d
' properties ot a
the perimeter; and every part of that crooked circular line,
line, a circumference or arch; the strait line, which
generated the circle, is the semidiameter or ra¬
dius ; and any strait line, which passeth through
the centre and is terminated on both sides in the
circumference, is called the diameter. Moreover,
every point of the radius, which describes the
circle, describes in the same time its own peri¬
meter, terminating its own circle, which is said to
be concentric to all the other circles, because this
and all those have one common centre.
Wherefore in every circle, all strait lines from
the centre to the circumference are equal. For
they are all coincident with the radius which
generates the circle.
Also the diameter divides both the perimeter
and the circle itself into two equal parts. For if
those two parts be applied to one another, and the
semiperimeters be coincident, then, seeing they
have one common diameter, they will be equal;
and the semicircles will be equal also; for these
also will be coincident. But if the semiperimeters
be not coincident, then some one strait line, which
passes through the centre, which centre is in the
diameter, will be cut by them in two points.
Wherefore, seeing all the strait lines from the
centre to the circumference are equal, a part of
the same strait line will be equal to the whole;
which is impossible.
For the same reason the perimeter of a circle
182
philosophy.
part ii. will be uniform, that is, any one part of it will be
-—r^—• coincident with any other equal part of the same.
Ues ofa'strait From hence may be collected this property
line taken in 0f a strait line, namely, that it is all contained in
that plane which contains both its extreme points.
For seeing both its extreme points are in the
plane, that strait line, which describes the plane,
will pass through them both; and if one of them
be made a centre, and at the distance between
both a circumference be described, whose radius
is the strait line which describes the plane, that
circumference will pass through the other point.
Wherefore between the two propounded points,
there is one strait line, by the definition of a circle,
contained wholly in the propounded plane ; and
therefore if another strait line might be drawn
between the same points, and yet not be contained
in the same plane, it would follow, that between
two points two strait lines may be drawn; which
has been demonstrated to be impossible.
It may also be collected, that if two planes cut
one another, their common section will be a strait
line. For the two extreme points of the inter¬
section are in both the intersecting planes; and
between those points a strait line may be drawn;
but a strait line between any two points is in the
same plane, in which the points are; and seeing
these are in both the planes, the strait line which
connects them will also be in both the same planes,
and therefore it is the common section of both.
And every other line, that can be drawn between
those points, will be either coincident with that
line, that is, it will be the same line ; or it will not
OF STRAIT AND CROOKED.
183
be coincident, and then it will be in neither, or part ir.
but in one of those planes. . I4' -
As a strait line may be understood to be
moved round about whilst one end thereof remains
fixed; as the centre ; so in like manner it is easy to
understand, that a plane may be circumduced
about a strait line, whilst the strait line remains
still in one and the same place, as the axis of that
motion. Now from hence it is manifest, that any
three points are in some one plane. For as any
two points, if they be connected by a strait line,
are understood to be in the same plane in which
the strait line is ; so, if that plane be circumduced
about the same strait line, it will in its revolution
take in any third point, howsoever it be situate;
and then the three points will be all in that plane ;
and consequently the three strait lines which con¬
nect those points, will also be in the same plane.
6. Two lines are said to touch one another, Definition of
which being both drawn to one and the same
point, will not cut one another, though they be
produced, produced, I say, in the same manner in
which they were generated. And therefore if two
strait lines touch one another in any one point,
they will be contiguous through their whole length.
Also two lines continually crooked will do the
same, if they be congruous and be applied to one
another according to their congruity ; otherwise,
if they be incongruously applied, they will, as all
other crooked lines, touch one another, where they
touch, but in one point only. Which is manifest
from this, that there can be 110 congruity between
a strait line and a line that is continually crooked ;
for otherwise the same line might be both strait
184 PHILOSOPHY.
part ii. and crooked. Besides, when a strait line touches
1 i-
r—- a crooked line, if the strait line be never so little
moved about upon the point of contact, it will cut
the crooked line; for seeing it touches it but in
one point, if it incline any way, it will do more
than touch it; that is, it will either be congruous
to it, or it will cut it; but it cannot be congruous
to it; and therefore it will cut it.
The definition j an^le, according to the most general
ol an angle, .
and the kinds acceptation of the word, may be thus defined;
when two lines, or many superficies, concur in one
sole point, and diverge every where else, the
quantity of that divergence is an angle. And an
angle is of two sorts; for, first, it may be made
by the concurrence of lines, and then it is a super¬
ficial angle ; or by the concurrence of superficies,
and then it is called a solid angle.
Again, from the two ways by which two lines
may diverge from one another, superficial angles
are divided into two kinds. For two strait lines,
which are applied to one another, and are con¬
tiguous in their whole length, may be separated or
pulled open in such manner, that their concur¬
rence in one point will still remain; and this
separation or opening may be either by circular
motion, the centre whereof is their point of con¬
currence, and the lines will still retain their strait-
ness, the quantity of which separation or divergence
is an angle simply so called; or they may be
separated by continual flexion or curvation in every
imaginable point; and the quantity of this sepa¬
ration is that, which is called an angle of con-
tingence.
Besides, of superficial angles simply so called,.
OF STIIAIT AND CROOKED.
185
those, which are in a plane superficies, are plane ; part ti.
and those^ which are not plane, are denominated r-—-
from the superficies in which they are.
Lastly, those are strait-lined angles, which are
made by strait lines ; as those which are made by
crooked lines are crooked-lined; and those which
are made both of strait and crooked lines, are
mixed angles.
8. Two arches intercepted between two radii ofTn concentric
circles, arches
concentric circles, have the same proportion to one of the same
another, which their whole perimeters have to one Jllf'another,0
another. For let the point A (in the first figure) cLumfcinces
be the centre of the two circles BCD and E F G,are-
in which the radii AEB and AFC intercept the
arches B C and E F; I say the proportion of the
arch B C to the arch E F is the same with that of
the perimeter BCD to the perimeter EFG. For
if the radius A F C be understood to be moved
about the centre A with circular and uniform
motion, that is, with equal swiftness everywhere,
the point C will in a certain time describe the
perimeter BCD, and in a part of that time the
arch B C ; and because the velocities are equal by
which both the arch and the whole perimeter are
described, the proportion of the magnitude of the
perimeter B C D to the magnitude of the arch BC
is determined by nothing but the difference of the
times in which the perimeter and the arch are
described. But both the perimeters are described
in one and the same time, and both the arches in
one and the same time ; and therefore the propor¬
tions of the perimeter BCD to the arch B C, and
of the perimeter EFG to the arch E F, are
both determined by the same cause. Wherefore
18(3
PHILOSOPHY.
PARi4 TL BCD. BC::EFG. EF are proportionals (by the
^—- 6th art. of the last chapter), and by permutation
BCD.EFG::BC. EF will also be proportionals ;
which was to be demonstrated.
ofh an "angie^ Nothing is contributed towards the quantity
consists 1 an angle5 neither by the length, nor by the
equality, nor by the inequality of the lines which
comprehend it. For the lines A B and A C com¬
prehend the same angle which is comprehended by
the lines A E and A F, or AB and A F. Nor is an
angle either increased or diminished by the abso¬
lute quantity of the arch, which subtends the
same ; for both the greater arch B C and the
lesser arch E F are subtended to the same angle.
But the quantity of an angle is estimated by the
quantity of the subtending arch compared with the
quantity of the whole perimeter. And therefore
the quantity of an angle simply so called may be
thus defined : the quantity of an angle is an arch
or circumference of a circle, determined by its
proportion to the whole perimeter. So that when
an arch is intercepted between two strait lines
drawn from the centre, look how great a portion
that arch is of the whole perimeter, so great is the
angle. From whence it may be understood, that
when the lines which contain an angle are strait
lines, the quantity of that angle may be taken at
any distance from the centre. But if one or both
of the containing lines be crooked, then the quan¬
tity of the angle is to be taken in the least distance
from the centre, or from their concurrence ; for
the least distance is to be considered as a strait
line, seeing no crooked line can be imagined so
little, but that there may be a less strait line. And
OF STRAIT AND CROOKED.
187
although the least strait line cannot be given, PAR£ IL
because the least given line may still be divided, t—
yet we may come to a part so small, as is not at all
considerable; which we call a point. And this
point may be understood to be in a strait line
which touches a crooked line; for an angle is
generated by separating, by circular motion, one
strait line from another which touches it, as has
been said above in the 7th article. Wherefore an
angle, which two crooked lines make, is the same
with that which is made by two strait lines which
touch them.
10. From hence it follows, that vertical angles, The distinction
such as are ABC, DBF in the second figure, are piysfcaiied.1"
equal to one another. For if, from the two semi-
perimeters D A C, FDA, which are equal to one
another, the common arch D A be taken away, the
remaining arches A C, D F will be equal to one
another.
Another distinction of angles is into right and
oblique. A right angle is that, whose quantity is
the fourth part of the perimeter. And the lines,
which make a right angle, are said to be perpen¬
dicular to one another. Also, of oblique angles,
that which is greater than a right, is called an
obtuse angle; and that which is less, an acute
angle. From whence it follows, that all the angles
that can possibly be made at one and the same
point, together taken, are equal to four right
angles ; because the quantities of them all put
together make the whole perimeter. Also, that
all the angles, which are made on one side of a
strait line, from any one point taken in the same,
are equal to two right angles ; for if that point be
188
PHILOSOPHY.
tart ii. made the eentre, that strait line will be the dia-
14.
■ ^—- meter of a circle, by whose circumference the
quantity of an angle is determined; and that dia¬
meter will divide the perimeter into two equal
parts.
of strait lines j \ jf a tangent be made the diameter of a
from the ceil- ° . „
ti e of a circle circle, whose centre is the point ot contact, a
SIS strait line drawn from the centre of the former
circle to the centre of the latter circle, will make
two angles with the tangent, that is, with the dia¬
meter of the latter circle, equal to two right angles,
by the last article. And because, by the 6th article,
the tangent has on both sides equal inclination to
the circle, each of them will be a right angle; as
also the semidiameter will be perpendicular to the
same tangent. Moreover, the semidiameter, inas¬
much as it is the semidiameter, is the least
strait line which can be drawn from the centre
to the tangent; and every other strait line, that
reaches the tangent, will pass out of the circle,
and will therefore be greater than the semidia¬
meter. In like manner, of all the strait lines,
which may be drawn from the centre to the tan¬
gent, that is the greatest which makes the greatest
angle with the perpendicular ; which will be mani¬
fest, if about the same centre another circle be
described, whose semidiameter is a strait line
taken nearer to the perpendicular, and there be
drawn a perpendicular, that is, a tangent, to the
same.
From whence it is also manifest, that if two
strait lines, which make equal angles on either
side of the perpendicular, be produced to the tan¬
gent, they will be equal.
of strait and crooked.
189
12. There is in Euclid a definition of strait- n.
lined parallels ; but I do not find that parallels in —
general are anywhere defined; and therefore for SnftioTof
an universal definition of them, I say that any two parallels;
. the proper-
lines whatsoever, strait or crooked, as also any ties of strait
• 7,7 parallels.
two superficies, are parallel ; when two equal
strait lines, wheresoever they fall upon them,
make always equal angles with each of them.
From which definition it follows; first, that any
two strait lines, not inclined opposite ways, falling
upon two other strait lines, which are parallel, and
intercepting equal parts in both of them, are them¬
selves also equal and parallel. As if A B and C D
(in the third figure), inclined both the same way,
fall upon the parallels A C and B D, and A C and
B D be equal, A B and C D will also be equal and
parallel. For the perpendiculars B E and D F
being drawn, the right angles E B D and F D H
will be equal. Wherefore, seeing E F and B D are
parallel, the angles E B A and F D C will be equal.
Now if D C be not equal to B A, let any other
strait line equal to B A be drawn from the point D ;
which, seeing it cannot fall upon the point C, let
it fall upon G. Wherefore AG will be either
greater or less than B D ; and therefore the angles
E B A and F D C are not equal, as was supposed.
Wherefore A B and C D are equal; which is the
first.
Again, because they make equal angles with the
perpendiculars B E and 1) F ; therefore the angle
C D H will be equal to the angle A B D, and, by
the definition of parallels, A B and C D will be
parallel; which is the second.
That plane, which is included both ways
within parallel lines,is called a parallelogram.
190
philosophy.
part it. Coroll. i. From this last it follows, that the
14.
*—r—- angles ABD and CDH are equal, that is, that
Sio^ofp?-" a strait line> as B H, falling upon two parallels, as
raiieis, &c. A B and C D, makes the internal angle ABD
equal to the external and opposite angle CDH.
Coroll. ii. And from hence again it follows, that
a strait line falling upon two parallels, makes the
alternate angles equal, that is, the angle A G F, in
the fourth figure, equal to the angle G F D. For
seeing G F D is equal to the external opposite
angle E G B, it will be also equal to its vertical
angle AGF, which is alternate to G F D.
Coroll. hi. That the internal angles on the
same side of the line F G are equal to two right
angles. For the angles at F, namely, G F C and
G F D, are equal to two right angles. But G F D
is equal to its alternate angle AGF. Wherefore
both the angles G F C and AGF, which are in¬
ternal 011 the same side of the line F G, are equal
to two right angles.
Coroll. iv. That the three angles of a strait-
lined plain triangle are equal to two right angles;
and any side being produced, the external angle
will be equal to the two opposite internal angles.
For if there be drawn by the vertex of the plain
triangle ABC (fig. 5) a parallel to any of the
sides, as to A B, the angles A and B will be equal
to their alternate angles E and F, and the angle C
is common. But, by the 10th article, the three
angles E, C and F, are equal to two right angles;
and therefore the three angles of the triangle are
equal to the same ; which is the first. Again,
the two angles B and D are equal to two right
angles, by the 10th article. Wherefore taking
of strait and crooked.
191
away B, there will remain tlie angles A and C, PARJ IL
equal to the angle D ; which is the second. -—^—-
Coroll. v. If the angles A and B be equal, the nit^ofpa-"
sides A C and C B will also be equal, because A B rallels>&c-
and E F are parallel; and, on the contrary, if the
sides A C and C B be equal, the angles A and B
will also be equal. For if they be not equal, let
the angles B and G be equal. Wherefore, seeing
G B and E F are parallels, and the angles G and B
equal, the sides G C and C B will also be equal;
and because C B and A C are equal by supposi¬
tion, C G and C A will also be equal; which cannot
he, by the 11 th article.
Coroll. yt. From hence it is manifest, that if
two radii of a circle be connected by a strait line,
the angles they make with that connecting line
will be equal to one another; and if there be
added that segment of the circle, which is sub¬
tended by the same line which connects the radii,
then the angles, which those radii make with the
circumference, will also be equal to one another.
For a strait line, which subtends any arch, makes
equal angles with the same; because, if the arch
and the subtense be divided in the middle, the two
halves of the segment will be congruous to one
another, by reason of the uniformity both of the
circumference of the circle, and of the strait line.
13. Perimeters of circles are to one another, as The circumfe-
i ences of cir-
their semidiameters are. For let there be any twro
cles are to one
circles, as, in the first figure, B C D the greater,
and E F G the lesser, having their common centre
at A; and let their semidiameters be A C and A E.
I say, A C has the same proportion to A E, which
the perimeter BCD has to the perimeter E F G.
192
philosophy.
part ii. for the magnitude of tlie semidiameters A C and
14. . .
*■—»—' A E is determined by the distance of the points
C and E from the centre A; and the same dis¬
tances are acquired by the uniform motion of a
point from A to C, in such manner, that in equal
times the distances acquired be equal. But the
perimeters B C D and E F G are also determined
by the same distances of the points C and E from
the centre A ; and therefore the perimeters BCD
and E F G, as well as the semidiameters A C and
A E, have their magnitudes determined by the
same cause, which cause makes, in equal times,
equal spaces. Wherefore, by the 13th chapter and
6th article, the perimeters of circles and their
semidiameters are proportionals; which was to be
proved.
in tnangics 14 jf two strait lines, which constitute an angle,
strait lines pa- ' o '
railei to the be cut "by strait-lined parallels, the intercepted pa-
bases are to one * A i • i
another, as the rallels will be to one another, as the parts which
sides which they cut off from the vertex. Let the strait lines
from STc vertex. A B alld A C' in the 6th figure, make an angle at
A, and be cut by the two strait-lined parallels B C
and D E, so that the parts cut off from the vertex
in either of those lines, as in A B, may be A B
and AD. I say, the parallels B C and D E are to
one another, as the parts A B and AD. For let
A B be divided into any number of equal parts, as
into A F, F D, D B ; and by the points F and D,
let F G and D E be drawn parallel to the base B C,
and cut A C in G and E ; and again, by the points
G and E, let other strait lines be drawn parallel
to A B, and cut B C in H and I. If now the point
A be understood to be moved uniformly over A B,
and in the same time B be moved to C, and all the
OF STRAIT AND CROOKED.
193
points F, D, and B be moved uniformly and with part 11.
equal swiftness over F G, D E, and B C ; then shall —■—-
B pass over B H, equal to F G, in the same time that
A passes over A F; and A F and F G will be to one
another, as their velocities are; and when A is in
F, D will be in K ; when A is in D, D will be in E ;
and in what manner the point A passes by the
points F, D, and B, in the same manner the point
B will pass by the points H, I, and C; and the
strait lines F G, D K, K E, B H, H I, and I C, are
equal, by reason of their parallelism ; and therefore,
as the velocity in A B is to the velocity in B C, so
is A D to D E; but as the velocity in A B is to
the velocity in B C, so is A B to B C; that is to say,
all the parallels will be severally to all the parts
cut off from the vertex, as A F is to F G. Where¬
fore, A F. G F: : A D. D E : : A B. B C are propor¬
tionals.
The subtenses of equal angles in different circles,
as the strait lines B C and FE (in fig. 1), are to
one another as the arches which they subtend.
For (by art. 8) the arches of equal angles are to
one another as their perimeters are ; and (by art.
13) the perimeters as their semidiameters ; but the
subtenses B C and F E are parallel to one another
by reason of the equality of the angles which they
make with the semidiameters ; and therefore the
same subtenses, by the last precedent article, will
be proportional to the semidiameters, that is, to
the perimeters, that is, to the arches which they
subtend.
15. If in a circle any number of equal subtenses tBy wl3at frac"
J *- _ tion of a strait
be placed immediately after one another, and strait line the circum-
lines be drawn from the extreme point of the first cle is made.
VOL. I. O
194
PHILOSOPHY.
PAfiT XI" Su^tense to the extreme points of all the rest, the
•—*—- first subtense being produced will make with the
Sn oft strait second subtense an external angle double to that,
linethecircum-which is made by the same first subtense, and a
ference of a cir- Jm
cie is made, tangent to the circle touching it in the extreme
points thereof; and if a strait line which subtends
two of those arches be produced, it will make an
external angle with the third subtense, triple to
the angle which is made by the tangent with the
first subtense ; and so continually. For with the
radius AB (in fig. 7) let a circle be described, and
in it let any number of equal subtenses, B C, C D,
and D E, be placed ; also let B D and B E be drawn;
and by producing B C, B D and B E to any dis¬
tance in G, H and I, let them make angles with
the subtenses which succeed one another, namely,
the external angles GCD, and H D E. Lastly, let
the tangent K B be drawn, making with the first
subtense the angle K B C. I say the angle G C D
is double to the angle K B C, and the angle H D E
triple to the same angle K B C. For if A C be
drawn cutting B D in M, and from the point C
there be drawn L C perpendicular to the same A C,
then C L and M D will be parallel, by reason of
the right angles at C and M; and therefore the
alterne angles LCD and B D C will be equal: as
also the angles B D C and C B D will be equal,
because of the equality of the strait lines B C and
C D. Wherefore the angle G C D is double to
either of the angles C B D or CDB ; and there¬
fore also the angle G C D is double to the angle
LCD, that is, to the angle K BC. Again, CD is
parallel to BE, by reason of the equality of the
angles C B E and DEB, and of the strait lines
of strait and crooked.
195
C B and D E ; and therefore the angles G C D and part ii.
GBE are equal; and consequently GBE, as also -—-
D E B is double to the angle K B C. But the ex- tion^a stndt
ternal angle HDE is equal to the two internal fe"ence ofaUck-
D E B and DBE; and therefore the angle HDE cle is made,
is triple to the angle K B C, &c.; which was to be
proved.
Coroll. i. From hence it is manifest, that the
angles K B C and C B D, as also, that all the angles
that are comprehended by two strait lines meeting
in the circumference of a circle and insisting upon
equal arches, are equal to one another.
Coroll. 11. If the tangent B K be moved in the
circumference with uniform motion about the
centre B, it will in equal times cut off equal arches ;
and will pass over the whole perimeter in the same
time in which itself describes a semiperimeter about
the centre B.
Coroll. hi. From hence also we may under¬
stand, what it is that determines the bending or
curvation of a strait line into the circumference of
a circle ; namely, that it is fraction continually in¬
creasing in the same manner, as numbers, from
one upwards, increase by the continual addition of
unity. For the indefinite strait line K B being
broken in B according to any angle, as that of
K B C, and again in C according to a double angle,
and in D according to an angle which is triple, •
and in E according to an angle which is quadru¬
ple to the first angle, and so continually, there will
be described a figure which will indeed be recti¬
lineal, if the broken parts be considered as having
magnitude ; but if they be understood to be the
least that can be, that is, as so many points, then
o 2
196
PHILOSOPHY.
part ii. the figure described will not be rectilineal, but a
>—■ circle, whose circumference will be the broken
line.
Coroll. iv. From what has been said in this pre¬
sent article, it may also be demonstrated, that an
angle in the centre is double to an angle in the
circumference of the same circle, if the intercepted
arches be equal. For seeing that strait line, by
whose motion an angle is determined, passes over
equal arches in equal times, as well from the centre
as from the circumference; and while that, which
is from the circumference, is passing over half its
own perimeter, it passes in the same time over the
whole perimeter of that which is from the centre,
the arches, which it cuts off in the perimeter whose
centre is A, will be double to those, wThich it makes
in its own semiperimeter, whose centre is B. But
in equal circles, as arches are to one another, so
also are angles.
It may also be demonstrated, that the external
angle made by a subtense produced and the next
equal subtense is equal to an angle from the centre
insisting upon the same arch ; as in the last dia¬
gram, the angle G C D is equal to the angle CAD;
for the external angle G C D is double to the angle
C B D ; and the angle CAD insisting upon the
same arch C D is also double to the same angle
C B D or K B C.
That an angle 16. An angle of contingence, if it be compared
is^quantity^but with an angle simply so called, how little soever,
kind* from that ^as suc^ ProPortion to it as a point has to a line;
pfynsT8caiied' *s' no ProPortion at all, nor any quantity. For
and that it can first, an angle of contingence is made by continual
neither add nor n • ,i , • ~ .. . ^
flexion so that 111 the generation of it there is 110
OF STRAIT AND CROOKED.
197
circular motion at all, in which consists the nature partii.
14.
of an angle simply so called ; and therefore it can- •——-
not be compared with it according to quantity, any Eg
Secondly, seeing the external angle made by afrom the samc'
subtense produced and the next subtense is equal
to an angle from the centre insisting upon the
same arcli, as in the last figure the angle G C D is
equal to the angle C A D, the angle of contingence
will be equal to that angle from the centre, which
is made by A B and the same A B ; for 110 part of
a tangent can subtend any arch ; but as the point
of contact is to be taken for the subtense, so the
angle of contingence is to be accounted for the
external angle, and equal to that angle whose arch
is the same point B.
Now, seeing an angle in general is defined to be
the opening or divergence of two lines, which con¬
cur in one sole point; and seeing one opening is
greater than another, it cannot be denied, but that
by the very generation of it, an angle of contin¬
gence is quantity ; for wheresoever there is greater
and less, there is also quantity ; but this quantity
consists in greater and less flexion ; for how much
the greater a circle is, so much the nearer comes
the circumference of it to the nature of a strait
line ; for the circumference of a circle being made
by the curvation of a strait line, the less that strait
line is, the greater is the curvation ; and therefore,
when one strait line is a tangent to many circles,
the angle of contingence, which it makes with a
less circle, is greater than that which it makes
with a greater circle.
Nothing therefore is added to or taken from an
angle simply so called, by the addition to it or
198
PHILOSOPHY.
part ii. taking from it of never so many angles of contin-
■—r1—- genee. And as an angle of one sort can never be
equal to an angle of the other sort, so they cannot
be either greater or less than one another.
From whence it follows, that an angle of a seg¬
ment, that is, the angle, which any strait line
makes with any arch, is equal to the angle which
is made by the same strait line, and another which
touches the circle in the point of their concur¬
rence ; as in the last figure, the angle which is
made between G B and B K is equal to that which
is made between G B and the arch B C.
That the incli- 17- An angle, which is made by two planes, is
ilfangfimpty commonly called the inclination of those planes •
so called. an(j kecause planes have equal inclination in all
their parts, instead of their inclination an angle is
taken, which is made by two strait lines, one of
which is in one, the other in the other of those
planes, but both perpendicular to the common
section.
A solid angle 18. A solid angle may be conceived two ways.
First, for the aggregate of all the angles, which are
made by the motion of a strait line, while one ex¬
treme point thereof remaining fixed, it is carried
about any plain figure, in which the fixed point of
the strait line is not contained. And in this sense,
it seems to be understood by Euclid. Now it is
manifest, that the quantity of a solid angle so con¬
ceived is no other, than the aggregate of all the
angles in a superficies so described, that is, in the
superficies of a pyramidal solid. Secondly, when
a pyramis or cone has its vertex in the centre of a
sphere, a solid angle may be understood to be the
proportion of a spherical superficies subtending
OF STRAIT AND CROOKED.
199
that vertex to the whole superficies of the sphere, partii.
In which sense, solid angles are to one another as s—-
the spherical bases of solids, which have their ver¬
tex in the centre of the same sphere.
19. All the ways, by which two lines respect one What is the
nature of
another, or all the variety of their position, may asymptotes,
be comprehended under four kinds; for any two
lines whatsoever are either parallels, or being pro¬
duced, if need be, or moved one of them to the
other parallelly to itself, they make an angle; or
else, by the like production and motion, they touch
one another; or lastly, they are asymptotes. The
nature of parallels, angles, and tangents, has been
already declared. It remains that I speak briefly
of the nature of asymptotes.
Asvmptosy depends upon this, that quantity is
infinitely divisible. And from hence it follows, that
any line being given, and a body supposed to be
moved from one extreme thereof towards the other,
it is possible, by taking degrees of velocity always
less and less, in such proportion as the parts of the
line are made less by continual division, that the
same body may be always moved forwards in that
line, and yet never reach the end of it. For it is
manifest, that if any strait line, as A F, (in the 8th
figure) be cut anywhere in B, and again B F be cut
in C, and C F in D, and D F in E, and so eternally,
and there be drawn from the point F, the strait
line F F at any angle AFF; and lastly, if the strait
lines A F, B F, C F, D F, E F, &c., having the same
proportion to one another with the segments of
the line A F, be set in order and parallel to the
same A F, the crooked line ABCD E, and the
strait line F F, will be asymptotes, that is, they
200
philosophy.
PA^J II- will always come nearer and nearer together, but
v—■—' never touch one another. Now, because any line
may be cut eternally according to the proportions
which the segments have to one another, therefore
the divers kinds of asymptotes are infinite in num¬
ber, and not necessary to be further spoken of in
this place. In the nature of asymptotes in general
there is no more, than that they come still nearer
and nearer, but never touch. But in special in the
asymptosy of hyperbolic lines, it is understood
they should approach to a distance less than any
given quantity,
what it°is by 20. Situation is the relation of one place to
determined, another; and where there are many places, their
situation is determined by four things; by their
distances from one another; by several distances
from a place assigned; by the order of strait
lines drawn from a place assigned to the places
of them all; and by the angles which are made
by the lines so drawn. For if their distances,
order, and angles, be given, that is, be certainly
known, their several places will also be so certainly
known, as that they can be no other,
what is 21. Points, how many soever they be, have like
like situation; 7 J .
what is figure; situation with an equal number of other points,
like figures'.6 when all the strait lines, that are drawn from some
one point to all these, have severally the same
proportion to those, that are drawn in the same
order and at equal angles from some one point to
all those. For let there be any number of points
as A, B, and C, (in the 9th figure) to which from
some one point D let the strait lines D A, D B, and
D C be drawn ; and let there be an equal number
of other points, as E, F, and G, and from some
OF STRAIT AND CROOKED.
201
point H let the strait lines H E, H F, and H G be partit.
drawn, so that the angles A D B and B D C be -—A-'
severally and in the same order equal to the angles Nation
E H F and F H G, and the strait lines D A, D B,is fisi,re; &c*
and D C proportional to the strait lines H E, H F,
and H G ; I say, the three points A, B, and C, have
like situation with the three points E, F, and G, or
are placed alike. For if H E be understood to be
laid upon D A, so that the point H be in D, the
point F will be in the strait line D B, by reason of
the equality of the angles A D B and EHF; and
the point G will be in the strait line D C, by reason
of the equality of the angles B D C and FHG;
and the strait lines A B and E F, as also B C and
F G, will be parallel, because A D. E H : : B D.
F H :: CD. G H are proportionals by construction ;
and therefore the distances between the points A
and B, and the points B and C, will be propor¬
tional to the distances between the points E and F,
and the points F and G. Wherefore, in the situa¬
tion of the points A, B, and C, and the situation
of the points E, F and G, the angles in the same
order are equal; so that their situations differ in
nothing but the inequality of their distances from
one another, and of their distances from the points
D and H. Now, in both the orders of points, those
inequalities are equal; for A B. B C : : E F. F G,
which are their distances from one another, as
also DA. DB. DC :: HE. H F. H G, which are
their distances from the assumed points D and
H, are proportionals. Their difference, therefore,
consists solely in the magnitude of their distances.
But, by the definition of like, (chapter i. article 2)
those things, which differ only in magnitude, are
like. Wherefore the points A, B, and C, have to
202
philosophy.
PAi4T 1L one an°ther like situation with the points E, F,
v—^—- and G, or are placed alike ; which was to be proved,
situation;'what Figure is quantity, determined by the situation
is %ure, &c. or placing of all its extreme points. Now I call
those points extreme, which are contiguous to the
place wrhich is without the figure. In lines there¬
fore and superficies, all points may be called ex¬
treme ; but in solids only those which are in the
superficies that includes them.
Like figures are those, whose extreme points in
one of them are all placed like all the extreme
points in the other; for such figures differ in
nothing but magnitude.
And like figures are alike placed', when in both
of them the homologal strait lines, that is, the strait
lines which connect the points which answer one
another, are parallel, and have their proportional
sides inclined the same way.
And seeing every strait line is like every other
strait line, and every plane like every other plane,
when nothing but planeness is considered; if the
lines, which include planes, or the superficies,
which include solids, have their proportions known,
it will not be hard to know whether any figure
be like or unlike to another propounded figure.
And thus much concerning the first grounds of
philosophy. The next place belongs to geometry;
in which the quantities of figures are sought out
from the proportions of lines and angles. Where¬
fore it is necessary for him, that would study geo¬
metry, to know first what is the nature of quantity,
proportion, angle and figure. Having therefore
explained these in the three last chapters, I
thought fit to add them to this part; and so pass
to the next.
PART III.
PROPORTIONS OF MOTIONS
AND MAGNITUDES.
CHAPTER XV.
of the nature, properties, and divers
considerations of motion and
endeavour.
1. Repetition of some principles of the doctrine of motion
formerly set down. — 2. Other principles added to them.
3. Certain theorems concerning the nature of motion.—4.
Divers considerations of motion.—5. The way by which the
first endeavour of bodies moved tendeth.—6. In motion which
is made by concourse, one of the movents ceasing, the endea¬
vour is made by the way by which the rest tend.—7. All endea¬
vour is propagated in infinitum.—8. How much greater the
velocity or magnitude is of a movent, so much the greater is
the efficacy thereof upon any other body in its way.
1. The next things in order to be treated of are partiii.
motion and magnitude, which are the most —
common accidents of all bodies. This place there- Repetition
A or some prin-
fore most properly belongs to the elements of cipies ofthe
geometry. But because this part of philosophy, moYoTfor-
having been improved by the best wits of all ages, merlysetdown-
has afforded greater plenty of matter than can well
204
MOTIONS AND MAGNITUDES.
part iii. be thrust together within the narrow limits of this
15. °
— discourse, I thought fit to admonish the reader,
ofsome'011 that before he proceed further, he take into his
principles, &c. hands the works of Euclid, Archimedes, Apollo-
nius, and other as well ancient as modern writers.
For to what end is it, to do over again that which
is already done ? The little therefore that I shall
say concerning geometry in some of the following
chapters, shall be such only as is new, and con¬
ducing to natural philosophy.
I have already delivered some of the principles
of this doctrine in the eighth and ninth chapters;
which I shall briefly put together here, that the
reader in going on may have their light nearer at
hand.
First, therefore, in chap. viii. art. 10, motion is
defined to be the continual privation of one place,
and acquisition of another.
Secondly, it is there shown, that whatsoever is
moved is moved in time.
Thirdly, in the same chapter, art. 11, I have
defined rest to be when a body remains for some
time in one place.
Fourthly, it is there shown, that whatsoever is
moved is not in any determined place; as also
that the same has been moved, is still moved, and
will yet be moved; so that in every part of that
space, in which motion is made, we may consider
three times, namely, the past, the present, and
the future time.
Fifthly, in art. 15 of the same chapter, I have
defined velocity or swiftness to be motion con¬
sidered as power, namely, that power by ivkich a
body moved may in a certain time transmit a
motion and endeavour.
205
certain length ; which also may more briefly be part hi.
enunciated thus, velocity is the quantity of motion —'
determined by time and line. Repetition
*j oi soiTi0
Sixthly, in the same chapter, art. 16, I have priuciplesi&c-
shown that motion is the measure of time.
Seventhly, in the same chapter, art. 17, I have
defined motions to be equally swift, when in equal
times equal lengths are transmitted by them.
Eighthly, in art. 18 of the same chapter, motions
are defined to be equal, when the swiftness of one
moved body, computed in every part of its mag¬
nitude., is equal to the swiftness of another, com¬
puted also in every part of its magnitude. From
whence it is to be noted, that motions equal to
one cmother, and motions equally swift, do not
signify the same thing; for when two horses draw
abreast, the motion of both is greater than the
motion of either of them singly ; but the swiftness
of both together is but equal to that of either.
Ninthly, in art. 19 of the same chapter, I have
shown, that whatsoever is at rest will always be
at rest, unless there be some other body besides
it, which by getting into its place suffers it no
longer to remain at rest. And that whatsoever is
moved, will always be moved, unless there be some
other body besides it, which hinders its motion.
Tenthly, in chap. ix. art. 7,1 have demonstrated,
that when any body is moved which was formerly
at rest, the immediate efficient cause of that motion
is in some other moved and contiguous body.
Eleventhly, I have shown in the same place, that
whatsoever is moved, will always be moved in the
same way, and with the same swiftness, if it be
206
motions and magnitudes.
part iii. not hindered by some other moved and contiguous
— body.
other prin- 2. To which principles I shall here add those
ciples added x 1
to them. that follow. First, I define endeavour to be
motion made in less space and time than can be
given; that is, less than can be determined or
assigned by exposition or number ; that is, motion
made through the length of a point, and in an
instant or point of time. > For the explaining of
which definition it must be remembered, that by a
point is not to be understood that which has no
quantity, or which cannot by any means be
divided: for there is no such thing in nature;
but that, whose quantity is not at all considered,
that is, whereof neither quantity nor any part is
computed in demonstration; so that a point is not
to be taken for an indivisible, but for an undivided
thing; as also an instant is to be taken for an
undivided, and not for an indivisible time.
In like manner, endeavour is to be conceived as
motion; but so as that neither the quantity of the
time in which, nor of the line in which it is made,
may in demonstration be at all brought into com¬
parison with the quantity of that time, or of that
line of which it is a part. And yet, as a point may
be compared with a point, so one endeavour may
be compared with another endeavour, and one
may be found to be greater or less than another.
For if the vertical points of two angles be com¬
pared, they will be equal or unequal in the same
proportion which the angles themselves have to
one another. Or if a strait line cut many circum¬
ferences of concentric circles, the inequality of the
points of intersection will be in the same propor-
motion and endeavour.
207
tion which the perimeters have to one another, part hi.
And in the same manner, if two motions begin ——.—"
and end both together, their endeavours will be Sodded
equal or unequal, according to the proportion of t0 them*
their velocities ; as we see a bullet of lead descend
with greater endeavour than a ball of wool.
Secondly, I define impetus, or quickness of
motion, to be the swiftness or velocity of the body
moved, but considered in the severed points of
that time in which it is moved. In which sense
impetus is nothing else but the quantity or velocity
of endeavour. But considered with the whole
time, it is the whole velocity of the body moved
taken together throughout all the time, and equal
to the product of a line representing the time,
multiplied into a line representing the arith¬
metically mean impetus or quickness. Which
arithmetical mean, what it is, is defined in the 29th
article of chapter xm.
And because in equal times the ways that are
passed are as the velocities, and the impetus is the
velocity they go withal, reckoned in all the several
points of the times, it folioweth that during any
time whatsoever, howsoever the impetus be in¬
creased or decreased, the length of the way passed
over shall be increased or decreased in the same
proportion; and the same line shall represent
both the way of the body moved, and the several
impetus or degrees of swiftness wherewith the way
is passed over.
And if the body moved be not a point, but a
strait line moved so as that every point thereof
make a several strait line, the plane described by
its motion, whether uniform, accelerated, or re-
208
motions and magnitudes.
part i n. tarded, shall be greater or less, the time being the
'—' same, in the same proportion with that of the
dpi"added impetus reckoned in one motion to the impetus
to them. reckoned in the other. For the reason is the same
in parallelograms and their sides.
For the same cause also, if the body moved be a
plane, the solid described shall be still greater or
less in the proportions of the several impetus or
quicknesses reckoned through one line, to the
several impetus reckoned through another.
This understood, let A B C D, (in figure 1, chap,
xvn.) be a parallelogram ; in which suppose the
side A B to be moved parallelly to the opposite side
C D, decreasing all the way till it vanish in the
point C, and so describing the figure A B E F C;
the point B, as A B decreaseth, will therefore de¬
scribe the line BEFC; and suppose the time of
this motion designed by the line C D ; and in the
same time C D, suppose the side A C to be moved
parallel and uniformly to B D. From the point 0
taken at adventure in the line C D, draw 0 R pa¬
rallel to B D, cutting the line BEFC in E, and
the side A B in R. And again, from the point Q,
taken also at adventure in the line C D, draw Q S
parallel to B D, cutting the line B E F C in F, and
the side A B in S ; and draw E G and F H parallel
to CD, cutting A C in G and H. Lastly, suppose
the same construction done in all the points possi¬
ble of the line BEFC. I say, that as the propor¬
tions of the swiftness wrherewith QF, 0 E, D B,
and all the rest supposed to be drawn parallel to
D B and terminated in the line BEFC, are to
the proportions of their several times designed by
the several parallels H F, G E, A B, and all the
MOTION AND ENDEAVOUR.
209
rest supposed to be drawn parallel to the line of tart iii.
time C D and terminated in the line B E F C, the -—A—
aggregate to the aggregate, so is the area or plane
D B E F C to the area or plane A C F E B. Fort0 them-
as A B decreasing continually by the line B E F C
vanisheth in the time C D into the point C, so in
the same time the line D C continually decreasing
vanisheth by the same line C F E B into the point
B ; and the point D describeth in that decreasing
motion the line D B equal to the line A C described
by the point A in the decreasing motion of A B;
and their swiftnesses are therefore equal. Again,
because in the time G E the point 0 describeth the
line 0 E, and in the same time the point S de¬
scribeth the line S E, the line O E shall be to the
line S E, as the swiftness wherewith O E is de¬
scribed to the swiftness wherewith SE is described.
In like manner, because in the same time H F the
point Q, describeth the line Q F, and the point R
the line R F, it shall be as the swiftness by which
Q F is described to the swiftness by which R F is
described, so the line itself Q, F to the line itself
R F; and so in all the lines that can possibly be
drawn parallel to B D in the points where they
cut the line B E F C. But all the parallels to B D,
as S E, R F, A C, and the rest that can possibly be
drawn from the line A B to the line B E F C, make
the area of the plane ABEFC; and all the paral¬
lels to the same B D, as Q-F, O E, DB and the
rest drawn to the points where they cut the same
line B E F C, make the area of the plane BE F C D.
As therefore the aggregate of the swiftnesses
wherewith the plane B E F C D is described, is
to the aggregate of the swiftnesses wherewith
VOL. i. P
210
motions and magnitudes.
part hi. the plane A C F E B is described, so is the plane
-—- itself B E F C D to the plane itself A C F E B. But
cipies added the aggregate of the times represented by the pa-
to them. rallels AB, G E, H F and the rest, maketh also
the area A C F E B. And therefore, as the aggre¬
gate of all the lines Q, F, 0 E, D B and all the rest
of the lines parallel to B D and terminated in the
line BEFC, is to the aggregate of all the lines
H F, G E, A B and all the rest of the lines pa¬
rallel to C D and terminated in the same line
BEFC; that is, as the aggregate of the lines
of swiftness to the aggregate of the lines of time,
or as the whole swiftness in the parallels to D B to
the whole time in the parallels to C D, so is the
plane B E F C D to the plane A C F E B. And the
proportions of Q, F to F H, and of O E to E G, and
of D B to B A, and so of all the rest taken toge¬
ther, are the proportions of the plane D B E F C
to the plane A B E F C. But the lines Q F, 0 E,
D B and the rest are the lines that design the swift¬
ness ; and the lines H F, G E, A B and the rest are
the lines that design the times of the motions;
and therefore the proportion of the plane DBEFC
to the plane A B E F C is the proportion of all the
velocities taken together to all the times taken
together. Wherefore, as the proportions of the
swiftnesses, &c.; which was to be demonstrated.
The same holds also in the diminution of the
circles, whereof the lines of time are the semidia-
meters, as may easily be conceived by imagining
the whole plane A B C D turned round upon the
axis B D ; for the line BEFC will be everywhere
in the superficies so made, and the lines H F, G E,
A B, which are here parallelograms, will be there
motion and endeavour.
211
cylinders, the diameters of whose bases are the part iii.
lines H F, G E, A B, &c. and the altitude a point, —^—-
that is to say, a quantity less than any quantity added
that can possibly be named; and the lines QF, O E, t0 them-
D B, &c. small solids whose lengths and breadths
are less than any quantity that can be named.
But this is to be noted, that unless the propor¬
tion of the sum of the swiftnesses to the proportion
of the sum of the times be determined, the propor¬
tion of the figure DBEFC to the figure ABEFC
cannot be determined.
Thirdly,[I define resistance to be the endea¬
vour of one moved body either wholly or in part
contrary to the endeavour of another moved body,
which toucheth the same. J I say, wholly contrary,
when the endeavour of two bodies proceeds in the
same strait line from the opposite extremes, and
contrary in part, when two bodies have their en¬
deavour in two lines, which, proceeding from the
extreme points of a strait line, meet without the
same.
Fourthly, that I may define what it is to press,
I say, that of two moved bodies one presses the
other, when with its endeavour it makes either all
or part of the other body to go out of its place.
Fifthly, a body, which is pressed and not
wholly removed, is said to restore itself when,
the pressing body being taken away, the parts
which were moved do, by reason of the internal
constitution of the pressed body, return every one
into its own place. And this we may observe in
springs, in blown bladders, and in many other
bodies, whose parts yield more or less to the en¬
deavour which the pressing body makes at the
p 2
212
motions and magnitudes.
PARi5m' ^rst arr^va^; but afterwards, when the pressing
v' body is removed, they do, by some force within
them, restore themselves, and give their wThole
body the same figure it had before.
Sixthly, I define force to be the impetus or
quickness of motion multiplied either into itself \
or into the magnitude of the movent, by means
whereof the said movent works more or less upon
the body that resists it.
Certain theo- 3. Having premised thus much, I shall now
rems concern- or 3
ing the nature demonstrate, first, thatlif a point moved come to
touch another point which is at rest, how little
soever the impetus or quickness of its motion be,
it shall move that other point. For if by that
impetus it do not at all move it out of its place,
neither shall it move it with double the same
impetus. For nothing doubled is still nothing;
and for the same reason it shall never move it with
that impetus, how many times soever it be multi¬
plied, because nothing, however it be multiplied,
will for ever be nothing. Wherefore, when a
point is at rest, if it do not yield to the least
impetus, it will yield to none ; and consequently
it will be impossible that that, which is at rest,
should ever be moved.
Secondly, that when a point moved, how little
soever the impetus thereof be, falls upon a point of
any body at rest, how7 hard soever that body be, it
will at the first touch make it yield a little. For if
it do not yield to the impetus which is in that
point, neither will it yield to the impetus of never
so many points, which have all their impetus seve¬
rally equal to the impetus of that point. For seeing
all those points together work equally, if any one
MOTION AND ENDEAVOUR.
213
of them have no effect, the .aggregate of them all part iii.
together shall have no effect as many times told as v—^—-
there are points in the whole body, that is, still no
effect at all; and by consequent there would be
some bodies so hard that it would be impossible to
break them ; that is, a finite hardness, or a finite
force, would not yield to that which is infinite;
which is absurd.
Coroll. It is therefore manifest, that rest does
nothing at all, nor is of any efficacy; and that
nothing but motion gives motion to such things
as be at rest, and takes it from things moved.
Thirdly, that cessation in the movent does not
cause cessation in that which was moved by it.
For (by number 11 of art. 1 of this chapter) what¬
soever is moved perseveres in the same way and
with the same swiftness, as long as it is not hin¬
dered by something that is moved against it. Now
it is manifest, that cessation is not contrary mo¬
tion ; and therefore it follows that the standing
still of the movent does not make it necessary that
the thing moved should also stand still.
Coroll. They are therefore deceived, that reckon
the taking away of the impediment or resistance
for one of the causes of motion.
4. Motion is brought into account for divers ^1nvseird8erations
respects; first, as in a body undivided, that is, of motions,
considered as a point; or, as in a divided body.
In an undivided body, when we suppose the way,
by which the motion is made, to be a line; and in
a divided body, when we compute the motion of
the several parts of that body, as of parts.
Secondly, from the diversity of the regulation
of motion, it is in body, considered as undivided.
214 MOTIONS AND MAGNITUDES.
part hi. sometimes uniform and sometimes multiform. Uni-
v^ form is that by which equal lines are always
considerations transmitted in equal times ; and [multiform, when
of motion. [n one ^me more? jn another time less space is
transmitted. Again, of multiform motions, there
are some in which the degrees of acceleration and
retardation proceed in the same proportions, which
the spaces transmitted have, whether duplicate, or
triplicate, or by whatsoever number multiplied;
and others in which it is otherwise.
Thirdly, from the number of the movents; that
is, one motion is made by one movent only, and
another by the concourse of many movents.
Fourthly, from the position of that line in which
a body is moved, in respect of some other line;
and from hence one motion is called perpendicular,
another oblique, another parallel.
Fifthly, from the position of the movent in re¬
spect of the moved body; from whence one motion
is pulsion or driving, another traction or drawing.
Pulsion, when the movent makes the moved body
go before it; and traction, when it makes it follow.
Again, there are two sorts of pulsion ; one, when
the motions of the movent and moved body begin
both together, which may be called trusion or
thrusting and vection ; the other, when the movent
is first moved, and afterwards the moved body,
which motion is called percussion or stroke.
Sixthly, motion is considered sometimes from
the effect only which the movent works in the
moved body, which is usually called moment. Now
moment is the excess of motion which the movent
has above the motion or endeavour of the resisting
body.
motion and endeavour.
215
Seventhly, it may be considered from the diver- part hi.
sity of the medium ; as one motion may be made 1—A-'
in vacuity or empty place; another in a fluid;
another in a consistent medium, that is, a medium
whose parts are by some power so consistent and
cohering, that no part of the same will yield to the
movent, unless the whole yield also.
Eighthly, when a moved body is considered as
having parts, there arises another distinction of
motion into simple and compound. Simple, when
all the several parts describe several equal lines;
compounded, when the lines described are unequal.
5. All endeavour tends towards that part, that is The way by
to say, in that way which is determined by the endeavour ofSt
motion of the movent, if the movent be but one; ^odfsumoved
7 ' tendeth.
or, if there be many movents, in that way which
their concourse determines. For example, if a
moved body have direct motion, its first endeavour
will be in a strait line; if it have circular motion,
its first endeavour will be in the circumference of a
circle.
6. And whatsoever the line be, in which a in motion,
i i i •, , • r« , i c; j which is made
body has its motion from the concourse ot two concourse,
movents, as soon as in any point thereof the force ventsfceas™g0"
of one of the movents ceases, there immediately pie endeavour
is made bv the
the former endeavour of that body will be changed way by which
into an endeavour in the line of the other movent.the lest ttnd'
Wherefore, when any body is carried on by
the concourse of two winds, one of those winds
ceasing, the endeavour and motion of that body
will be in that line, in which it would have been
carried by that wind alone which blows still. And
in the describing of a circle, where that which is
moved has its motion determined by a movent in a
216 motions and magnitudes.
part hi. tangent, and by the radius which keeps it in a cer-
•—>—- tain. distance from the centre, if the retention of
the radius cease, that endeavour, which was in the
circumference of the circle, will now be in the tan¬
gent, that is, in a strait line. For, seeing endea¬
vour is computed in a less part of the circum¬
ference than can be given, that is, in a point, the
way by which a body is moved in the circumference
is compounded of innumerable strait lines, of which
every one is less than can be given; which are
therefore called points. Wherefore when any body,
which is moved in the circumference of a circle, is
freed from the retention of the radius, it will pro¬
ceed in one of those strait lines, that is, in a
tangent.
ah endeavour ^ endeavour, whether strong; or weak, is
is propagated # # J # 7
in infinitum, propagated to infinite distance ; for it is motion. If
therefore the first endeavour of a body be made in
space which is empty, it will always proceed with
the same velocity; for it cannot be supposed that
it can receive any resistance at all from empty
space; and therefore, (by art. 7? chap, ix) it will
always proceed in the same way and with the
same swiftness. And if its endeavour be in space
which is filled, yet, seeing endeavour is motion,
that which stands next in its way shall be removed,
and endeavour further, and again remove that
which stands next, and so infinitely. Wherefore
the propagation of endeavour, from one part of full
space to another, proceeds infinitely. Besides, it
reaches in any instant to any distance, how great
soever. For in the same instant in which the first
part of the full medium removes that which is next
it, the second also rempves that part which is next
motion and endeavour.
217
to it; and therefore all endeavour, whether it be in part hi.
15
empty or in full space, proceeds not only to any ■- .'
distance, how great soever, but also in any time,
how little soever, that is, in an instant. Nor makes
it any matter, that endeavour, by proceeding, grows
weaker and weaker, till at last it can no longer be
perceived by sense; for motion may be insensible ;
and I do not here examine things by sense and ex¬
perience, but by reason.
8. When two movents are of equal magnitude, How much
the swifter of them works with greater force than velocity or
the slower, upon a body that resists their motion. 5? mown"
Also, if two movents have equal velocity, thes0 mucl!
7 A J J greater is the
greater of them works with more force than the efficacy there-
less. For where the magnitude is equal, the movent other body
of greater velocity makes the greater impressionin lts way'
upon that body upon which it falls ; and where the
velocity is equal, the movent of greater magnitude
falling upon the same point, or an equal part of
another body, loses less of its velocity, because the
resisting body works only upon that part of the
movent which it touches, and therefore abates the
impetus of that part only; whereas in the mean
time the parts, which are not touched, proceed,
and retain their whole force, till they also come to
be touched; and their force has some effect.
Wherefore, for example, in batteries a longer than
a shorter piece of timber of the same thickness and
velocity, and a thicker than a slenderer piece of
the same length and velocity, work a greater
effect upon the wall.
218
motions and magnitudes.
CHAPTER XVI.
of motion accelerated and uniform, and
of motion by concourse.
1. The velocity of any body, in what time soever it be computed,
is that which is made of the multiplication of the impetus, or
quickness of its motion into the time.—2-5. In all motion,
the lengths which are passed through are to one another, as the
products made by the impetus multiplied into the time.—6. If
two bodies be moved with uniform motion through two lengths,
the proportion of those lengths to one another will be com¬
pounded of the proportions of time to time, and impetus to im¬
petus, directly taken.—7. If two bodies pass through two lengths
with uniform motion, the proportion of their times to one
another will be compounded of the proportions of length to
length, and impetus to impetus reciprocally taken; also the
proportion of their impetus to one another will be compounded
of the proportions of length to length, and time to time reci¬
procally taken.—8. If a body be carried on with uniform motion
by two movents together, which meet in an angle, the line by
which it passes will be a strait line, subtending the comple¬
ment of that angle to two right angles.—9, &c. If a body be
carried by two movents together, one of them being moved
with uniform, the other with accelerated motion, and the pro¬
portion of their lengths to their times being explicable in
numbers, how to find out what line that body describes.
part nr. l. The velocity of any body, in whatsoever time it
•—r—- be moved, has its quantity determined by the sum
any bod"hi°f the several quicknesses or impetus, which it
what time so- hath in the several points of the time of the body's
ever it be com-
puted, is that motion. For seeing velocity, (by the definition of
If the muUi-e it, chap, viii, art. 15) is that power by which a
d'/e Impetus body can in a certain time pass through a certain
or quickness length; and quickness of motion or impetus, (by
accelerated and uniform motion. 219
chap, xv, art. 2, num. 2) is velocity taken in one part hi.
point of time only, all the impetus, together taken -—^—-
in all the points of time, will be the same thing
with the mean impetus multiplied into the whole
time, or which is all one, will be the velocity of the
whole motion.
Coroll. If the impetus be the same in every
point, any strait line representing it may be taken
for the measure of time: and the quicknesses or
impetus applied ordinately to any strait line
making an angle with it, and representing the way
of the body's motion, will design a parallelogram
which shall represent the velocity of the whole
motion. But if the impetus or quickness of mo¬
tion begin from rest and increase uniformly, that
is, in the same proportion continually with the
times which are passed, the whole velocity of the
motion shall be represented by a triangle, one side
whereof is the whole time, and the other the
greatest impetus acquired in that time; or else by
a parallelogram, one of whose sides is the whole
time of motion, and the other, half the greatest
impetus ; or lastly, by a parallelogram having for
one side a mean proportional between the whole
time and the half of that time, and for the other
side the half of the greatest impetus. For both
these parallelograms are equal to one another, and
severally equal to the triangle which is made of
the whole line of time, and of the greatest ac¬
quired impetus; as is demonstrated in the ele¬
ments of geometry.
2. In all uniform motions the lengths which are in all motion,
transmitted are to one another, as the product of SiSS'pass,
the mean impetus multiplied into its time, to the cd throu«h are
220 MOTIONS AND MAGNITUDES.
part hi. product of the mean impetus multiplied also into
—A—- its time.
Is the products For let A B (in fig. 1) be the time, and A C the
™XhLn- by which any body passes with uniform
plied into time, motion through the length D E ; and in any part of
the time A B, as in the time A F, let another body
be moved with uniform motion, first, with the same
impetus A C. This body, therefore, in the time
A F with the impetus A C will pass through the
length A F. Seeing, therefore, when bodies are
moved in the same time, and with the same velo¬
city and impetus in every part of their motion, the
proportion of one length transmitted to another
length transmitted, is the same with that of time
to time, it followeth, that the length transmitted in
the time A B with the impetus A C will be to the
length transmitted in the time A F with the same
impetus A C, as A B itself is to A F, that is, as the
parallelogram AI is to the parallelogram A H,
that is, as the product of the time A B into the
mean impetus AC is to the product of the time
AF into the same impetus A C. Again, let it be
supposed that a body be moved in the time A F,
not with the same but with some other uniform
impetus, as A L. Seeing therefore, one of the
bodies has in all the parts of its motion the impetus
AC, and the other in like manner the impetus
A L, the length transmitted by the body moved
with the impetus A C will be to the length trans¬
mitted by the body moved with the impetus A L,
as A C itself is to A L, that is, as the parallelogram
A H is to the parallelogram F L. Wherefore, by
ordinate proportion it will be, as the parallelogram
A 1 to the parallelogram F L, that is, as the pro-
accelerated and uniform motton. 221
duct of the mean impetus into the time is to the PA^ IIL
product of the mean impetus into the time, so the 1'
length transmitted in the time A B with the impe- j£e
tus A C, to the length transmitted in the time A F
with the impetus A L; which was to be demon¬
strated.
Coroll. Seeing, therefore, in uniform motion, as
has been shown, the lengths transmitted are to
one another as the parallelograms which are made
by the multiplication of the mean impetus into the
times, that is, by reason of the equality of the im¬
petus all the way, as the times themselves, it will
also be, by permutation, as time to length, so time
to length; and in general, to this place are appli¬
cable all the properties and transmutations of ana-
logisms, which I have set down and demonstrated
in chapter xiii.
3. In motion begun from rest and uniformly
accelerated, that is, where the impetus increaseth
continually according to the proportion of the
times, it will also be, as one product made by the
mean impetus multiplied into the time, to another
product made likewise by the mean impetus multi¬
plied into the time, so the length transmitted in
the one time to the length transmitted in the other
time.
For let A B (in fig. 1) represent a time ; in the
beginning of which time A, let the impetus be as
the point A; but as the time goes on, so let the
impetus increase uniformly, till in the last point of
that time A B, namely in B, the impetus acquired
be B I. Again, let A F represent another time, in
whose beginning A, let the impetus be as the point
itself A; but as the time proceeds, so let the im-
222
motions and magnitudes.
part in. petus increase uniformly, till in the last point F of
-—1^_- the time A F the impetus acquired be F K; and
the lengths &c' ^ E be the length passed through in the time
A B with impetus uniformly increased. I say, the
length D E is to the length transmitted in the time
A F, as the time A B multiplied into the mean of
the impetus increasing through the time A B, is to
the time A F multiplied into the mean of the im¬
petus increasing through the time A F.
For seeing the triangle A B I is the whole velo¬
city of the body moved in the time A B, till the
impetus acquired be B I; and the triangle A F K
the whole velocity of the body moved in the time
A F with impetus increasing till there be acquired
the impetus F K ; the length D E to the length
acquired in the time A F with impetus increasing
from rest in A till there be acquired the impetus
F K, will be as the triangle A B I to the triangle
A F K, that is, if the triangles A B I and A F K be
like, in duplicate proportion of the time A B to the
time A F; but if unlike, in the proportion com¬
pounded of the proportions of A B to A F and of
BI to F K. Wherefore, as A BI is to A F K, so
let D E be to DP; for so, the length transmitted
in the time A B with impetus increasing to B I,
will be to the length transmitted in the time A F
with impetus increasing to F K, as the triangle
A B I is to the triangle A F K ; but the triangle
A B I is made by the multiplication of the time
A B into the mean of the impetus increasing to
B I; and the triangle A F K is made by the multi¬
plication of the time A F into the mean of the
impetus increasing to F K; and therefore the
length D E which is transmitted in the time A B
ACCELERATED AND UNIFORM MOTION. 223
with impetus increasing to B I, to the length D P part iii.
which is transmitted in the time A F with impetus •—r-^
increasing to F K, is as the product which is made ^
of the time A B multiplied into its mean impetus,
to the product of the time A F multiplied also into
its mean impetus ; which was to be proved.
Coroll. i. In motion uniformly accelerated, the
proportion of the lengths transmitted to that of
their times, is compounded of the proportions of
their times to their times, and impetus to impetus.
Coroll. ii. In motion uniformly accelerated, the
lengths transmitted in equal times, taken in conti¬
nual succession from the beginning of motion, are
as the differences of square numbers beginning
from unity, namely, as 3, 5, 7, &c. For if in the
first time the length transmitted be as 1, in the
first and second times the length transmitted will
be as 4, which is the square of 2, and in the three
first times it will be as 9, which is the square of 3,
and in the four first times as 16, and so on. Now
the differences of these squares are 3, 5, 7, &c.
Coroll. hi. In motion uniformly accelerated from
rest, the length transmitted is to another length
transmitted uniformly in the same time, but with
such impetus as was acquired by the accelerated
motion in the last point of that time, as a triangle
to a parallelogram, which have their altitude and
base common. For seeing the length D E (in fig. 1)
is passed through with velocity as the triangle
A B I, it is necessary that for the passing through
of a length which is double to D E, the velocity be
as the parallelogram AI; for the parallelogram AI
is double to the triangle A B I.
4. In motion, which beginning from rest is so ac-
224 motions and magnitudes.
PARTiir. lerated, that the impetus thereof increases conti-
v—<—' uually in proportion duplicate to the proportion of
the fengths'&c! the times in which it is made, a length transmitted
in one time will be to a length transmitted in ano¬
ther time, as the product made by the mean impetus
multiplied into the time of one of those motions, to
the product of the mean impetus multiplied into
the time of the other motion.
For let A B (in fig. 2) represent a time, in whose
first instant A let the impetus be as the point A;
but as the time proceeds, so let the impetus in¬
crease continually in duplicate proportion to that
of the times, till in the last point of time B the
impetus acquired be B I; then taking the point F
anywhere in the time A B, let the impetus F K
acquired in the time A F be ordinately applied to
that point F. Seeing therefore the proportion of
F K to BI is supposed to be duplicate to that of
A F to A B, the proportion of A F to A B will be
subduplicate to that of F K to BI; and that of
AB to AF will be (by chap. xiii. art 16) dupli¬
cate to that of B I to F K ; and consequently the
point K will be in a parabolical line, whose dia¬
meter is A B and base B I; and for the same
reason, to what point soever of the time A B the
impetus acquired in that time be ordinately ap¬
plied, the strait line designing that impetus will be
in the same parabolical line A K I. Wherefore the
mean impetus multiplied into the whole time A B
will be the parabola A K I B, equal to the paralle¬
logram A M, which parallelogram has for one side
the line of time A B and for the other the line of
the impetus A L, which is two-thirds of the im¬
petus B I; for every parabola is equal to two-
ACCELERATED AND UNIFORM MOTTON. 225
thirds of that parallelogram with which it has its PAR1gm*
altitude and base common. Wherefore the whole -—.—-
velocity in A B will be the parallelogram A M, as thefeiigth^&c,
being made by the multiplication of the impetus
A L into the time A B. And in like manner, if
F N be taken, which is two-thirds of the impetus
F K, and the parallelogram F 0 be completed, F 0
will be the whole velocity in the time A F, as being
made by the uniform impetus AO or FN multi¬
plied into the time A F. Let now the length
transmitted in the time A B and with the velocity
A M be the strait line D E ; and lastly, let the
length transmitted in the time A F with the velo¬
city A N be D P ; I say that as A M is to AN, or as
the parabola A K I B to the parabola A K F, so is
D E to D P. For as A M is to F L, that is, as A B
is to A F, so let D E be to D G. Nowr the propor¬
tion of A M to A N is compounded of the propor¬
tions of A M to F L, and of F L to A N. But as
AM to FL, so by construction is D E to DG;
and as F L is to A N (seeing the time in both is the
same, namely, A F), so is the length D G to the
length D P; for lengths transmitted in the same
time are to one another as their velocities are.
Wherefore by ordinate proportion, as A M is to
A N, that is, as the mean impetus A L multiplied
into its time A B, is to the mean impetus A O
multiplied into A F, so is D E to D P; which was
to be proved.
Coroll. 1. Lengths transmitted with motion so
accelerated, that the impetus increase continually
in duplicate proportion to that of their times, if
the base represent the impetus, are in triplicate
proportion of their impetus acquired in the last
VOL. i. Q
220
motions and magnitudes.
part nr. point of their times. For as the length D E is to
■ i'• the length DP, so is the parallelogram AM to the
the^engtiis^&c. parallelogram A N, and so the parabola A K I B
to the parabola A K F. But the proportion of the
parabola A KIB to the parabola A K F is triplicate
to the proportion which the base B I has to the
base FK. Wherefore also the proportion of DE
to D P is triplicate to that of B I to F K.
Coroll. ii. Lengths transmitted in equal times
succeeding one another from the beginning, by
motion so accelerated, that the proportion of the
impetus be duplicate to the proportion of the
times, are to one another as the differences of cubic
numbers beginning at unity, that is as 7, 19, 37, &c.
For if in the first time the length transmitted be as
1, the length at the end of the second time will be
as 8, at the end of the third time as 27, and at the
end of the fourth time as 64, &c.; which are cubic
numbers, whose differences are 7, 19, 37, &c.
Coroll. hi. In motion so accelerated, as that the
length transmitted be always to the length trans¬
mitted in duplicate proportion to their times, the
length uniformly transmitted in the whole time,
and with impetus all the way equal to that which
is last acquired, is as a parabola to a parallelogram
of the same altitude and base, that is, as 2 to 3.
For the parabola AKIB is the impetus increasing
in the time AB ; and the parallelogram A I is the
greatest uniform impetus multiplied into the same
time AB. Wherefore the lengths transmitted will
be as a parabola to a parallelogram, &c., that is,
as 2 to 3.
5. If I should proceed to the explication of such
motions as are made by impetus increasing in pro-
ACCELERATED AND UNIFORM MOTION. 22/
portion triplicate, quadruplicate, quintuplicate, &c., part tit
to that of their times, it would be a labour infinite •—^—-
and unnecessary. For by the same method by
which I have computed such lengths, as are trans¬
mitted with impetus increasing in single and dupli¬
cate proportion, any man may compute such as are
transmitted with impetus increasing in triplicate,
quadruplicate, or what other proportion he pleases.
In making which computation he shall find, that
where the impetus increase in proportion triplicate
to that of the times, there the whole velocity will
be designed by the first parabolaster (of which see
the next chapter); and the lengths transmitted
will be in proportion quadruplicate to that of
the times. And in like manner, where the im¬
petus increase in quadruplicate proportion to that
of the times, that there the whole velocity will be
designed by the second parabolaster, and the
lengths transmitted will be in quintuplicate pro¬
portion to that of the times ; and so on continually.
6. If two bodies with uniform motion transmit If two bodies
two lengths, each with its own impetus and time, unifm^mo-th
the proportion of the lengths transmitted will be
compounded of the proportions of time to time, the proportion
_ . . tit of those lengths
and impetus to impetus, directly taken. to one another,
Let two bodies be moved uniformly (as in fig. 3), poLde/of'the
one in the time A B with the impetus A C, the proportions of
1 time to tune,
other in the time AD with the impetus AE. I and impetus
- .to impetus,
say the lengths transmitted have tlieir proportion directly taken,
to one another compounded of the proportions of
AB to AD, and of AC to AE. For let any
length whatsoever, as Z, be transmitted by one of
the bodies in the time A B with the impetus A C ;
and any other length, as X, be transmitted by the
Q 2
228
MOTIONS AND MAGNITUDES.
PARi6m* °^er body in the time A D with the impetus A E ;
v—r—- and let the parallelograms A F and A G be com-
blmoid^&c. pleted. Seeing now Z is to X (by art. 2) as the
impetus A C multiplied into the time A B is to the
impetus A E multiplied into the time A D, that is,
as A F to AG; the proportion of Z to X will be
compounded of the same proportions, of which the
proportion of A F to A G is compounded ; but the
proportion of AF to AG is compounded of the
proportions of the side AB to the side AD, and of
the side A C to the side A E (as is evident by the
Elements of Euclid), that is, of the proportions of
the time A B to the time A D, and of the imnetus
A C to the impetus A E. Wherefore also the
proportion of Z to X is compounded of the same
proportions of the time A B to the time A D, and
of the impetus AC to the impetus AE ; which was
to be demonstrated.
Coroll. i. When two bodies are moved with
uniform motion, if the times and impetus be in
reciprocal proportion, the lengths transmitted shall
be equal. For if it were as A B to A D (in the
same fig. 3) so reciprocally A E to AC, the pro¬
portion of AF to AG would be compounded of
the proportions of A B to AD, and of A C to AE,
that is, of the proportions of AB to AD, and of
A D to A B. Wherefore, A F would be to AG as
AB to AB, that is, equal; and so the two products
made by the multiplication of impetus into time
would be equal; and by consequent, Z would be
equal to X.
Coroll. ii. If two bodies be moved in the same
time, but with different impetus, the lengths trans¬
mitted will be as impetus to impetus. For if the
accelerated and uniform motion. 229
time of both of them be AD, and their different partiii.
impetus be A E and A C, the proportion of AG to -—r—
D C will be compounded of the proportions of A E
to A C and of AD to AD, that is, of the propor¬
tions of A E to A C and of A C to AC; and so
the proportion of A G to D C, that is, the propor¬
tion of length to length, will be as A E to A C, that
is, as that of impetus to impetus. In like manner,
if two bodies be moved uniformly, and both of
them with the same impetus, but in different
times, the proportion of the lengths transmitted by
them will be as that of their times. For if they
have both the same impetus A C, and their dif¬
ferent times be AB and AD, the proportion of AF
to D C will be compounded of the proportions of
AB to AD and of AC to AC; that is, of the
proportions of AB to AD and of AD to AD;
and therefore the proportion of A F to D C, that is,
of length to length, will be the same with that of
AB to A D, which is the proportion of time to time.
7. If two bodies pass through two lengths with if two bodies
uniform motion, the proportion of the times in fwoMngth?
which they are moved will be compounded of the modon"1?™
proportions of length to length and impetus to proportion of
f 1 o o r t]ieir times t0
impetus reciprocally taken. one another,
For let any two lengths be given, as (in the same poundedTf'the
fig. 3) Z and X, and let one of them be transmitted ot
with the impetus A C, the other with the impetus len§'th>and hn-
x ... petus to impe-
AE. I say the proportion of the times in which tus recipro-
they are transmitted, will be compounded of the aShep^por-
proportions of Z to X, and of A E, which is the ^pC°t[jhte0nor)e
impetus with which X is transmitted, to A C, the anotber>wil1 be
. . . ■ compounded of
impetus with which Z is transmitted, h or seeing the proportions
A F is the product of the impetus A C multiplied Llgtifand
230
motions and magnitudes.
part hi. into the time A B, the time of motion through Z
16.. .
-—.—' will be a line, which is made by the application of
reciprocally8 the parallelogram AF to the strait line AC, which
taken. ^lie A B; and therefore A B is the time of
motion through Z. In like manner, seeing A G is
the product of the impetus AE multiplied into the
time A D, the time of motion through X will be a
line which is made by the application of AG to the
strait line A D; but A D is the time of motion
through X. Now the proportion of A B to A D
is compounded of the proportions of the parallelo¬
gram A F to the parallelogram A G, and of the
impetus A E to the impetus A C ; which may be
demonstrated thus. Put the parallelograms in
order A F, AG, DC, and it will be manifest that
the proportion of AF to DC is compounded of the
proportions of A F to A G and of AG to D C; but
A F is to D C as A B to A D ; wherefore also the
proportion of A B to AD is compounded of the
proportions of A F to AG and of AG to DC.
And because the length Z is to the length X as
A F is to A G, and the impetus A E to the impetus
A C as AG to DC, therefore the proportion of
A B to A D will be compounded of the proportions
of the length Z to the length X, and of the impetus
A E to the impetus A C ; which was to be demon¬
strated.
In the same manner it may be proved, that in
two uniform motions the proportion of the impetus
is compounded of the proportions of length to
length and of time to time reciprocally taken.
For if we suppose A C (in the same fig. 3) to be
the time, and A B the impetus with which the
length Z is passed through ; and A E to be the-
ACCELERATED AND UNIFORM MOTION. 231
time, and A D the impetus with which the length X pARJ IIL
is passed through, the demonstration will proceed ^—'
as in the last article.
8. If a body be carried by two movents toe;e- If a bo(1y be,
J . _ ° carried on with
ther, which move with strait and uniform motion,, uniform motion
and concur in any given angle, the line by which SLts"together,
that body passes will be a strait line. rn'ang'iT^he
Let the movent AB (in fie;. 4) have strait andlinebywhichit
N ° 7 passes will be a
uniform motion, and be moved till it come into the strait lire, sub-
place CD; and let another movent AC, havingcompliantof
likewise strait and uniform motion, and making ^^1tlta^ets('2
with the movent A B any given angle CAB, be
understood to be moved in the same time to D B;
and let the body be placed in the point of their
concourse, A. I say the line which that body de¬
scribes with its motion is a strait line. For let the
parallelogram A B D C be completed, and its dia¬
gonal A D be drawn; and in the strait line A B
let any point E be taken; and from it let E F be
drawn parallel to the strait lines A C and B D,
cutting A D in G ; and through the point G let H I
be drawn parallel to the strait lines A B and C D ;
and lastly, let the measure of the time be A C.
Seeing therefore both the motions are made in the
same time, when A B is in C D, the body also
will be in C D ; and in like manner, when A C is
in B D, the body will be in B D. But A B is in
C D at the same time when A C is in B D ; and
therefore the body will be in C D and B D at the
same time ; wherefore it will be in the common
point D. Again, seeing the motion from A C to
B D is uniform, that is, the spaces transmitted by
it are in proportion to one another as the times
in which they are transmitted, when AC is in E F,
232
MOTIONS AND MAGNITUDES.
part hi. the proportion of A B to A E will be the same with
-—•.—- that of E F to E G, that is, of the time A C to the
time AH. Wherefore A B will be in HI in the
same time in which A C is in E F, so that the body
will at the same time be in E F and H I, and there¬
fore in their common point G. And in the same
manner it will be, wheresoever the point E be
taken between A and B. Wherefore the body will
always be in the diagonal A D ; which was to be
demonstrated.
Coroll. From hence it is manifest, that the body
will be carried through the same strait line A D,
though the motion be not uniform, provided it
have like acceleration; for the proportion of A B
to A E will always be the same with that of A C
to AH.
if a body be 9. If a body be carried by two movents toge-
carried by two i • i , • i i
movents toge- tlier, which meet m any given angle, and are
them beingf moved, the one uniformly, the other with motion
moved with uniformly accelerated from rest, that is, that the
uniform, the # .
other with ac- proportion of their impetus be as that of their
tion^and "the times, that is, that the proportion of their lengths
tS°iengths°to duplicate to that of the lines of their times, till
their times be-the line of greatest impetus acquired by accelera-
ing explicable . ° ■* A . n
in numbers, tion be equal to that of the line of time of the uni-
whltHne^hat' form motion ; the line in which the body is carried
body describes. ^ie crooked line of a semiparabola, whose
base is the impetus last acquired, and vertex the
point of rest.
Let the straight line A B (in fig. 5) be under¬
stood to be moved with uniform motion to C D;
and let another movent in the strait line A C be
supposed to be moved in the same time to BD,
but with motion uniformly accelerated, that is,
ACCELERATED AND UNIFORM MOTION. 233
with such motion, that the proportion of the PARjJ IIL
spaces which are transmitted be always duplicate v—■—-
to that of the times, till the impetus acquired be ^rrieT^cT
B D equal to the strait line A C ; and let the
semiparabola A G D B be described. I say that by
the concourse of those two movents, the body will
be carried through the semiparabolical crooked
line A G D. For let the parallelogram A B D C be
completed; and from the point E, taken anywhere
in the strait line A B, let E F be drawn parallel to
A C and cutting the crooked line in G ; and lastly,
through the point G let H I be drawn parallel to
the strait lines A B and C D. Seeing therefore
the proportion of A B to A E is by supposition
duplicate to the proportion of E F to E G, that is,
of the time A C to the time A H, at the same time
when A C is in E F, A B will be in H I ; and there¬
fore the moved body will be in the common point
G. And so it will always be, in what part soever
of A B the point E be taken. Wherefore the moved
body will always be found in the parabolical line
A G D ; which was to be demonstrated.
10. If a body be carried by two movents toge¬
ther, which meet in any given angle, and are
moved the one uniformly, the other with impetus
increasing from rest, till it be equal to that of the
uniform motion, and with such acceleration, that
the proportion of the lengths transmitted be every
where triplicate to that of the times in which they
are transmitted; the line, in which that body is
moved, will be the crooked line of the first semi-
parabolaster of two means, whose base is the im¬
petus last acquired.
Let the strait line AB (in the (ithfigure) beinoved
234
motions and magnitudes.
part hi. uniformly to CD; and let another movent A C be
-—r~~ ^ moved at the same time to B D with motion so
carriedC'&e.e accelerated, that the proportion of the lengths
transmitted be everywhere triplicate to the pro¬
portion of their times ; and let the impetus acquired
in the end of that motion be BD, equal to the
strait line A C; and lastly, let A G D be the crooked
line of the first semiparabolaster of two means. I
say, that by the concourse of the two movents
together, the body will be always in that crooked
line A G D. For let the parallelogram A B D C be
completed; and from the point E, taken anywhere
in the strait line A B, let E F be drawn parallel to
A C, and cutting the crooked line in G; and
through the point G let HI be drawn parallel to
the strait lines A B and C D. Seeing therefore the
proportion of A B to A E is, by supposition, tripli¬
cate to the proportion of E F to E G, that is, of the
time A C to the time A H, at the same time when
AC is in E F, A B will be in H I; and therefore
the moved body will be in the common point G.
And so it will always be, in what part soever of
A B the point E be taken; and by consequent, the
body will always be in the crooked line A G D;
which was to be demonstrated.
11. By the same method it may be shown, what
line it is that is made by the motion of a body
carried by the concourse of any two movents,
which are moved one of them uniformly, the other
with acceleration, but in such proportions of spaces
and times as are explicable by numbers, as dupli¬
cate, triplicate, &c., or such as may be designed
by any broken number whatsoever. For which
this is the rule. Let the two numbers of the length
ACCELERATED AND UNIFORM MOTION. 235
and time be added together ; and let their sum be rArjJ m-
the denominator of a fraction, whose numerator v—-
must be the number of the length. Seek this frac- eL-riclf&c6
tion in the table of the third article of the xvnth
chapter; and the line sought will be that, which
denominates the three-sided figure noted 011 the
left hand; and the kind of it will be that, which is
numbered above over the fraction. For example,
let there be a concourse of two movents, whereof
one is moved uniformly, the other with motion so
accelerated, that the spaces are to the times as 5
to 3. Let a fraction be made whose denominator
is the sum of 5 and 3, and the numerator 5, namely
the fraction f. Seek in the table, and you will
find f to be the third in that row, which belongs
to the three-sided figure of four means. Wherefore
the line of motion made by the concourse of two
such movents, as are last of all described, will be
the crooked line of the third parabolaster of four
means.
12. If motion be made by the concourse of two
movents, whereof one is moved uniformly, the
other beginning from rest in the angle of concourse
with any acceleration whatsoever; the movent,
which is moved uniformly, shall put forward the
moved body in the several parallel spaces, less
than if both the movents had uniform motion ; and
still less and less, as the motion of the other
movent is more and more accelerated.
Let the body be placed in A, (in the 7th figure)
and be moved by two movents, by one with uni -
form motion from the strait line A B to the strait
line C D parallel to it; and by the other with any
acceleration, from the strait line A C to the strait
236
motions and magnitudes.
part in. line B D parallel to it; and in the parallelogram
-—.—- A B D C let a space be taken between any two pa-
cLhc^&c! rallels E F and G H. I say, that whilst the movent
A C passes through the latitude which is between
E F and G H, the body is less moved forwards from
A B towards C D, than it would have been, if the
motion from A C to B D had been uniform.
For suppose that whilst the body is made to
descend to the parallel E F by the power of the
movent from A C towards B D, the same body in
the same time is moved forwards to any point F
in the line E F, by the power of the movent from
AB towards CD; and let the strait line AFbe
drawn and produced indeterminately, cutting GII
in H. Seeing therefore, it is as A E to A G, so E F
to GH ; if AC should descend towards BD with
uniform motion, the body in the time GII, (for I
make A C and its parallels the measure of time,)
would be found in the point H. But because A C
is supposed to be moved towards B D with motion
continually accelerated, that is, in greater propor¬
tion of space to space, than of time to time, in the
time G H the body will be in some parallel beyond
it, as between G H and B D. Suppose now that in
the end of the time G H it be in the parallel IK,
and in I K let I L be taken equal to G H. When
therefore the body is in the parallel I K, it will be
in the point L. Wherefore when it was in the
parallel G H, it was in some point between G and
H, as in the point M ; but if both the motions had
been uniform, it had been in the point H; and
therefore whilst the movent A C passes over the
latitude which is between E F and G H, the body is
less moved forwards from A B towards C D, than
ACCELERATED AND UNIFORM MOTTON. 23/
it would have been, if both the motions had been PA^ Iir-
uniform ; which w as to be demonstrated. —,—-
13. Any length being given, which is passed Jaiiedf&c!
through in a given time with uniform motion, to
find out what length shall be passed through in the
same time with motion uniformly accelerated, that
is, with such motion that the proportion of the
lengths passed through be continually duplicate to
that of their times, and that the line of the impetus
last acquired be equal to the line of the wrhole time
of the motion.
Let A B (in the 8th figure) be a length, trans¬
mitted with uniform motion in the time A C; and
let it be required to find another length, which
shall be transmitted in the same time with motion
uniformly accelerated, so that the line of the im¬
petus last acquired be equal to the strait line A C.
Let the parallelogram ABDC be completed;
and let B D be divided in the middle at E ; and
between B E and B D let B F be a mean propor¬
tional ; and let A F be draw^n and produced till it
meet w7ith C D produced in G; and lastly, let the
parallelogram A C G H be completed. I say, A H
is the length required.
For as duplicate proportion is to single propor¬
tion, so let A H be to A I, that is, let A I be the
half of A H ; and let I K be drawn parallel to the
strait line A C, and cutting the diagonal A D in K,
and the strait line A G in L. Seeing therefore A I
is the half of A H, I L will also be the half of B D,
that is, equal to BE; and I K equal to B F; for
B D, that is, G H, B F, and B E, that is, I L, being
continual proportionals, AH, A B and A I will
also be continual proportionals. But as A P* is to
238
motions and magnitudes,
part hi. A I, that is, as A H is to A B, so is BD to I K, and
-—r"—- so also is G II, that is, B D to B F; and therefore
ifa bodybe £ p an(j I are eqUa}. Now the proportion of
carried, etc. ^
A H to A I is duplicate to the proportion of A B
to A I, that is, to that of B D to I K, or of G H to
I K. Wherefore the point K will be in a parabola,
whose diameter is A H, and base G H, which G H
is equal to A C. The body therefore proceeding
from rest in A, with motion uniformly accelerated
in the time A C, when it has passed through the
length A H, will acquire the impetus G H equal to
the time A C, that is, such impetus, as that with it
the body will pass through the length A C in the
time A C. Wherefore any length being given, &c.,
which was propounded to be done.
14. Any length being given, which in a given
time is transmitted with uniform motion, to find
out what length shall be transmitted in the same
time with motion so accelerated, that the lengths
transmitted be continually in triplicate proportion
to that of their times, and the line of the impetus
last of all acquired be equal to the line of time
given.
Let the given length A B (in the 9th figure) be
transmitted with uniform motion in the time A C;
and let it be required to find what length shall be
transmitted in the same time with motion so acce¬
lerated, that the lengths transmitted be continually
in triplicate proportion to that of their times, and
the impetus last acquired be equal to the time
given.
Let the parallelogram A B D C be completed;
and let B D be so divided in E, that B E be a third
part of the whole B D ; and let B F be a mean pro-
ACCELERATED AND UNIFORM MOTION. 239
portional between B I) and B E ; and let, A F be part in¬
drawn and produced till it meet the strait line C D r—'
in G; and lastly, let the parallelogram A C G II be carried^&c.C
completed. I say, A H is the length required.
For as triplicate proportion is to single propor¬
tion, so let A H be to another line, A I, that is,
make Ala third part of the whole A H ; and let
I K be drawn parallel to the strait line A C, cutting
the diagonal A D in K, and the strait line A G in
L ; then, as A B is to A I, so let AI be to another,
A N ; and from the point N let N Q, be drawn pa¬
rallel to A C, cutting A G, A D, and F K produced
in P, M, and O ; and last of all, let F 0 and L M
be drawn, which will be equal and parallel to the
strait lines B N and I N. By this construction, the
lengths transmitted A H, A B, A I, and A N, will
be continual proportionals; and, in like manner,
the times G H, B F, I L and N P, that is, N Q,
N 0, N M and N P, will be continual proportionals,
and in the same proportion with A H, A B, A I
and A N. Wherefore the proportion of A H to
A N is the same with that of B D, that is, of N Q,
to N P; and the proportion of N Q to N P tripli¬
cate to that of N Q to N O, that is, triplicate to
that of B D to I K ; wherefore also the length A H
is to the length A N in triplicate proportion to that
of the time B D, to the time I K; and therefore
the crooked line of the first three-sided figure of
two means whose diameter is A H, and base G H
equal to A C, shall pass through the point O ; and
consequently, A H shall be transmitted in the time
A C, and shall have its last acquired impetus G H
equal to A C, and the proportions of the lengths
acquired in any of the times triplicate to the pro-
240
motions and magnitudes.
r-irt in. portions of the times themselves. Wherefore A H
-—t is the length required to be found out.
Lwied^&c! By the same method, if a length be given which
is transmitted with uniform motion in any given
time, another length may be found out which shall
be transmitted in the same time with motion so
accelerated, that the lengths transmitted shall be
to the times in which they are transmitted, in pro¬
portion quadruplicate, quintuplicate, and so on
infinitely. For if B D be divided in E, so that B D
be to B E as 4 to 1 : and there be taken between
B D and B E a mean proportional F B; and as
A H is to A B, so A B be made to a third, and
again so that third to a fourth, and that fourth to
a fifth, A N, so that the proportion of A H to A N
be quadruplicate to that of AH to A B, and the
parallelogram N B F 0 be completed, the crooked
line of the first three-sided figure of three means
will pass through the point O ; and consequently,
the body moved will acquire the impetus G H
equal to A C in the time A C. And so of the rest.
15. Also, if the proportion of the lengths trans¬
mitted be to that of their times, as any number to
any number, the same method serves for the find¬
ing out of the length transmitted with such
impetus, and in such time.
For let A C (in the 10th figure) be the time in
which a body is transmitted with uniform motion
from A to B ; and the parallelogram A B D C being
completed, let it be required to find out a length
in which that body may be moved in the same time
A C from A, with motion so accelerated, that the
proportion of the lengths transmitted to that of
the times be continually as 3 to 2.
Let B D be so divided in E, that B D be to B E
accelerated and uniform motion.
241
as 3 to 2 ; and between B D and B E let B F be a part hi
mean proportional; and let A F be drawn and pro- -—^—-
duced till it meet with C D produced in G; andIf a.b?d{ be
carried, occ«
making A M a mean proportional between A H
and A B, let it be as A M to A B, so A B to AI;
and so the proportion of A H to A I will be to that
of A H to A B as 3 to 2 ; for of the proportions, of
which that of A H to A M is one, that of A H to
A B is two, and that of A H to A I is three ; and
consequently, as 3 to 2 to that of G H to B F, and
(F K being drawn parallel to B I and cutting A D
in K) so likewise to that of G H or B D to I K.
Wherefore the proportion of the length A H to A I
is to the proportion of the time B D to I K as 3 to
2; and therefore if in the time A C the body be
moved with accelerated motion, as was pro¬
pounded, till it acquire the impetus H G equal to
A C, the length transmitted in the same time will
be AH.
16. But if the proportion of the lengths to that
of the times had been as 4 to 3, there should then
have been taken two mean proportionals between
A H and A B, and their proportion should have
been continued one term further, so that A H to
A B might have three of the same proportions,
of which AH to AI has four ; and all things else
should have been done as is already shown. Now
the way how to interpose any number of means
between two lines given, is not yet found out.
Nevertheless this may stand for a general rule ; if
there be a time given, and a length be transmitted
in that time with uniform motion ; as for example,
if the time be A C, and the length A B, the strait
line A G, which determines the length C G or A H,
vol. i. r
242
motions and magnitudes.
part in. transmitted in the same time A C with any acce-
——r~—- lerated motion, shall so cut B D in F, that B F
carried^&c6 shall be a mean proportional between B D and
BE, BE being so taken in B D, that the propor¬
tion of length to length be everywhere to the pro¬
portion of time to time, as the whole B D is to its
part B E.
17- If in a given time two lengths be trans¬
mitted, one with uniform motion, the other with
motion accelerated in any proportion of the lengths
to the times ; and again, in part of the same time,
parts of the same lengths be transmitted with the
same motions, the whole length will exceed the
other length in the same proportion in which one
part exceeds the other part.
For example, let A B (in the 8th figure) be a
length transmitted in the time A C, with uniform
motion ; and let A H be another length transmitted
in the same time with motion uniformly accele¬
rated, so that the impetus last acquired be G H
equal to A C ; and in A H let any part AI be taken,
and transmitted in part of the time A C with uni¬
form motion ; and let another part A B be taken
and transmitted in the same part of the time A C
with motion uniformly accelerated ; I say, that as
A H is to A B, so will A B be to A I.
Let B D be drawn parallel and equal to H G,
and divided in the midst at E, and between B D and
BE let a mean proportional be taken as BF;
and the strait line A G, by the demonstration of
art. 13, shall pass through F. And dividing AH
in the midst at I, A B shall be a mean proportional
between A H and A I. Again, because AI and A B
are described by the same motions, if I K be
accelerated and uniform motion. 243
drawn parallel and equal to B F or AM, and pa(^t in-
divided in the midst at N, and between IK and v
I N be taken the mean proportional I L, the strait Ifab°dJ'be
r r # J carried, otc.
line A F will, by the demonstration of the same
art. 13, pass through L. And dividing A B in the
midst at O, the line AI will be a mean proportional
between A B and A O. Where A B is divided in
I and 0, in like manner as A H is divided in B and
I; and as AH to AB, so is A B to A I. Which
was to be proved.
Coroll. Also as A H to A B, so is H B to B I;
and so also B I to I O.
And as this, where one of the motions is uni¬
formly accelerated, is proved out of the demonstra¬
tion of art. 13 ; so, when the accelerations are
in double proportion to the times, the same may be
proved by the demonstration of art. 14 ; and by
the same method in all other accelerations, whose
proportions to the times are explicable in numbers.
18. If two sides, which contain an angle in any
parallelogram, be moved in the same time to the
sides opposite to them, one of them with uniform
motion, the other with motion uniformly accele¬
rated ; that side, which is moved uniformly, will
affect as much with its concourse through the
whole length transmitted, as it would do if the
other motion were also uniform, and the length
transmitted by it in the same time were a mean
proportional between the whole and the half.
Let the side A B of the parallelogram A B D C,
(in the 11th figure) be understood to be moved with
uniform motion till it be coincident with C D ; and
let the time of that motion be A C or B D. Also
in the same time let the side A C be understood to
r 2
244
motions and magnitudes.
part hi. be moved with motion uniformly accelerated, till
-—A—■ it be coincident with B D ; then dividing A B in
carried,d&c.e the middle in E, let A F be made a mean propor¬
tional between A B and A E ; and drawing F G
parallel to A C, let the side A C be understood to
be moved in the same time A C with uniform
motion till it be coincident with F G. I say, the
whole A B confers as much to the velocity of the
body placed in A, when the motion of A C is uni¬
formly accelerated till it comes to B D, as the part
A F confers to the same, when the side A C is
moved uniformly and in the same time to F G.
For seeing A F is a mean proportional between
the whole A B and its half A E, B D will (by the
] 3tli article) be the last impetus acquired by A C,
with motion uniformly accelerated till it come to
the same B D; and consequently, the strait line
F B will be the excess, by which the length, trans¬
mitted by A C with motion uniformly accelerated,
will exceed the length transmitted by the same
A C in the same time with uniform motion, and
with impetus every where equal to B D. Where¬
fore, if the whole A B be moved uniformly to C D
in the same time in which A C is moved uniformly
to F G, the part F B, seeing it concurs not at all
with the motion of the side A C which is supposed
to be moved only to F G, will confer nothing to its
motion. Again, supposing the side AC to be
moved to B D with motion uniformly accelerated,
the side A B with its uniform motion to C D will
less pat forwards the body when it is accelerated
in all the parallels, than when it is not at all acce¬
lerated ; and by how much the greater the accele¬
ration is, by so much the less it will put it for-
ACCELERATED AND UNIFORM MOTION. 245
wards, as is shown in the 12th article. When PA^ IIL
therefore AC is in FG with accelerated motion, < ^—-
the body will not be in the side C D at the point G, *Lriedd&ce
but at the point D ; so that G D will be the excess,
by which the length transmitted with accelerated
motion to BD exceeds the length transmitted with
uniform motion to FG; so that the body by its
acceleration avoids the action of the part AF, and
comes to the side C D in the time A C, and makes
the length C D, which is equal to the length A B.
Wherefore uniform motion from A B to C D in the
time A C, works no more in the whole length A B
upon the body uniformly accelerated from A C to
B D, than if A C were moved in the same time
with uniform motion to FG; the difference con¬
sisting only in this, that when A B works upon the
body uniformly moved from AC to FG, that, by
which the accelerated motion exceeds the uniform
motion, is altogether in F B or G D; but when the
same A B works upon the body accelerated, that,
by which the accelerated motion exceeds the uni¬
form motion, is dispersed through the whole length
A B or C D, yet, so that if it were collected and
put together, it would be equal to the same F B or
GD. Wherefore, if two sides which contain an
angle, &e.; which was to be demonstrated.
19. If two transmitted lengths have to their
times any other proportion explicable by number,
and the side A B be so divided in E, that A B be
to A E in the same proportion which the lengths
transmitted have to the times in which they are
transmitted, and between A B and A E there be
taken a mean proportional A F; it may be shown
by the same method, that the side, which is moved
246 motions and magnitudes.
I>A^6r m- with uniform motion, works as much with its con-
-—.—- course through the whole length A B, as it would
do if the other motion were also uniform, and the
length transmitted in the same time A C were that
mean proportional A F.
And thus much concerning motion by concourse.
CHAP. XVII.
of figures deficient.
1. Definitions of a deficient figure ; of a complete figure; of the
complement of a deficient figure; and of proportions which
are proportional and commensurable to one another.— 2. The
proportion of a deficient figure to its complement.—3. The
proportions of deficient figures to the parallelograms in which
they are described, set forth in a table.—4. The description
and production of the same figures.—5. The drawing of tan¬
gents to them.—6. In what proportion the same figures exceed a
strait-lined triangle of the same altitude and base.—7. A table
of solid deficient figures described in a cylinder.—S. In what
proportion the same figures exceed a cone of the same altitude
and base.—9. How a plain deficient figure may be described
in a parallelogram, so that it be to a triangle of the same base
and altitude, as another deficient figure, plain or solid, twice
taken, is to the same deficient figure, together with the com¬
plete figure in which it is described.—10. The transferring of
certain properties of deficient figures described in a parallelo¬
gram to the proportions of the spaces transmitted with several
degrees of velocity.—11. Of deficient figures described in
a circle.—12. The proposition demonstrated in art. 2 confirmed
from the elements of philosophy.—13. An unusual way of
reasoning concerning the equality between the superficies of a
portion of a sphere and a circle.—14. How from the descrip¬
tion of deficient figures in a parallelogram, any number of mean
proportionals may be found out between two given strait lines.
Definition of a 1. I call those deficient figures which may be
deficientfiguie. un(ierstood be generated by the uniform motion
OF FIGURES DEFICIENT.
247
of some quantity, which decreases continually, till PAR^ 111 •
at last it have 110 magnitude at all. '—*—
And I call that a complete figure, answering to ^mpietefigS
a deficient figure, which is generated with theof the p001?1®-
0 / ment of a dea-
same motion and in the same time, by a quantity cient figure;
which retains always its whole magnitude. tionswiHaJe
The complement of a deficient figure is that which commensurabt
being added to the deficient figure makes it com-t0 one another,
plete.
Four proportions are said to be proportional,
when the first of them is to the second as the third
is to the fourth. For example, if the first propor¬
tion be duplicate to the second, and again, the
third be duplicate to the fourth, those proportions
are said to be proportional.
And commensurable proportions are those, which
are to one another as number to number. As
when to a proportion given, one proportion is
duplicate, another triplicate, the duplicate propor¬
tion will be to the triplicate proportion as 2 to 3 ;
but to the given proportion it will be as 2 to 1 ;
and therefore I call those three proportions com-
mensurable.
2. A deficient figure, which is made by a quantity The proportion
.. , . , . of a deficient
continually decreasing to nothing by proportions figure to its
everywhere proportional and commensurable, is toconip ement'
its complement, as the proportion of the whole
altitude to an altitude diminished in any time is
to the proportion of the whole quantity, which
describes the figure, to the same quantity dimi¬
nished in the same time.
Let the quantity AB (in fig. 1), by its motion
through the altitude AC, describe the complete
figure A D; and again, let the same quantity, by
248 motions and magnitudes.
part hi. decreasing continually to nothing in C, describe
- ■' ' the deficient figure A B E F C, whose complement
ot'addkicnt011 the figure B D CFE. Now let AB be
figure to its supposed to be moved till it lie in GK, so that the
complement.
altitude diminished be G C, and A B diminished
be G E ; and let the proportion of the whole alti¬
tude A C to the diminished altitude G C, be, for
example, triplicate to the proportion of the whole
quantity A B or G K to the diminished quantity
GE. And in like manner, let HI be taken equal
to G E, and let it be diminished to H F; and let
the proportion of G C to H C be triplicate to that
of HI to H F; and let the same be done in as
many parts of the strait line AC as is possible;
and a line be drawn through the points B, E, F
and C. I say the deficient figure A B E F C is to
its complement B D C F E as 3 to 1, or as the pro¬
portion of A C to G C is to the proportion of A B,
that is, of G K to G E.
For (by art. 2, chapter xv.) the proportion of
the complement B E F C D to the deficient figure
A B E F C is all the proportions of D B to B A,
O E to EG, Q F to F H, and of all the lines
parallel to D B terminated in the line B E F C, to
all the parallels to A B terminated in the same
points of the line B E F C. And seeing the pro¬
portions of D B to O E, and of D B to CI F &c.
are everywhere triplicate of the proportions of AB
to G E, and of A B to H F &c. the proportions of
H F to AB, and of GE to A B &c. (by art. 16,
chap, xiii.), are]'"triplicateJof the proportions of
Q,F to 1) B, and of OE to DB &c. and therefore
the deficient figure A B E F C, which is the aggre-
of figures deficient.
249
gate of all the lines H F, GE, A B, &e. is triple part til
to the complement 13 EFCD made of all the lines •—A-
QF, OE,DB, &c.; which was to be proved. ^dcfiSent0"
It follows from hence, that the same complement %ure t0 its
. »n complement.
B E H U is } oi the whole parallelogram. And
by the same method may be calculated in all other
deficient figures, generated as above declared, the
proportion of the parallelogram to either of its
parts; as that when the parallels increase from a
point in the same proportion, the parallelogram
will be divided into two equal triangles ; when
one increase is double to the other, it will be
divided into a semiparabola and its complement,
or into 2 and 1.
The same construction standing, the same con¬
clusion may otherwise be demonstrated thus.
Let the strait line C B be drawn cutting G K in
L, and through L let MN be drawn parallel to the
strait line A C ; wherefore the parallelograms GM
and L D will be equal. Then let L K be divided
into three equal parts, so that it may be to one of
those parts in the same proportion which the pro¬
portion of A C to G C, or of G K to G L, hath to
the proportion of G K to GE. Therefore LK will
be to one of those three parts as the arithmetical
proportion between G K and G L is to the arith¬
metical proportion between GK and the same GK
wanting the third part of L K ; and K E will be
somewhat greater than a third of L K. Seeing
now the altitude AG or M L is, by reason of the
continual decrease, to be supposed less than any
quantity that can be given; L K, which is inter¬
cepted between the diagonal BC and the side BD,
250
MOTIONS AND MAGNITUDES.
tart111. wiu be also less than any quantity tliat can be
v—>—given; and consequently, if G be put so near to A
oVadeEnt°nin g, as that the difference between Cg- and C A
complement. ^ess than any quantity that can be assigned,
the difference also between C I (removing L to I)
and CB, will be less than any quantity that can be
assigned; and the line g I being drawn and pro¬
duced to the line B D in k, cutting the crooked
line in e, the proportion of G k to GI will still be
triplicate to the proportion of G k to G e, and the
difference between k and e, the third part of h I,
will be less than any quantity that can be given ;
and therefore the parallelogram e D will differ
from a third part of the parallelogram A e by a
less difference than any quantity that can be
assigned. Again, let H I be drawn parallel and
equal to G E, cutting C B in P, the crooked line in
F, and OE in I, and the proportion of Cg* to C H
will be triplicate to the proportion of H F to HP,
and IF will be greater than the third part of P I.
But again, setting H in h so near to g, as that the
difference between C li and C g may be but as a
point, the point P will also in p be so near to I,
as that the difference between and CI will be
but as a point; and drawing hp till it meet with
B D in i, cutting the crooked line in f, and having
drawn e o parallel to B D, cutting D C in o, the pa¬
rallelogram f o will differ less from the third part of
the parallelogram g f\ than by any quantity that
can be given. And so it will be in all other spaces
generated in the same manner. Wherefore the
differences of the arithmetical and geometrical
means, which are but as so many points B,e,f, &c.
of figures deficient.
251
(seeing the whole figure is made up of so many part iit.
indivisible spaces) will constitute a certain line, v—>—•
such as is the line B E F C, which will divide the
complete figure AD into two parts, whereof one,
namely, A B E F C, which I call a deficient figure,
is triple to the other, namely, B D C F E, which I
call the complement thereof. And whereas the
proportion of the altitudes to one another is in
this case everywhere triplicate to that of the
decreasing quantities to one another; in the same
manner, if the proportion of the altitudes had
been everywhere quadruplicate to that of the de¬
creasing quantities, it might have been demon¬
strated that the deficient figure had been quadruple
to its complement; and so in any other proportion.
Wherefore, a deficient figure, which is made, &c.
which wras to be demonstrated.
The same rule holdeth also in the diminution of
the bases of cylinders, as is demonstrated in the
second article of chapter xv.
3. By this proposition, the magnitudes of all The proportion
deficient figures, when the proportions by which gUr^o the pa-
their bases decrease continually are proportional ^hichStheySare
to those by which their altitudes decrease, may be described, set
J . . forth in a table
compared with the magnitudes of their comple¬
ments ; and consequently, with the magnitudes of
their complete figures. And they will be found to
be, as I have set them down in the following
tables; in which I compare a parallelogram with
three-sided figures; and first, with a strait-lined
triangle, made by the base of the parallelogram
continually decreasing in such manner, that the
altitudes be always in proportion to one another
252
motions and magnitudes.
part hi. as the bases are, and so the triangle will be equal
•—'—- to its complement; or the proportions of the alti-
ofdefident'1011 tudes and bases will be as 1 to 1, and then the
figures, &c. triangle will be half the parallelogram. Secondly,
with that three-sided figure which is made by the
continual decreasing of the bases in subduplicate
proportion to that of the altitudes; and so the
deficient figure will be double to its complement,
and to the parallelogram as 2 to 3. Then, with
that where the proportion of the altitudes is tripli¬
cate to that of the bases ; and then the deficient
figure will be triple to its complement, and to the
parallelogram as 3 to 4. Also the proportion of
the altitudes to that of the bases may be as 3 to 2 ;
and then the deficient figure will be to its comple¬
ment as 3 to 2, and to the parallelogram as 3 to 5 ;
and so forwards, according as more mean propor¬
tionals are taken, or as the proportions are more
multiplied, as may be seen in the following table.
For example, if the bases decrease so, that the
proportion of the altitudes to that of the bases be
always as 5 to 2, and it be demanded what pro¬
portion the figure made has to the parallelogram,
which is supposed to be unity; then, seeing that
where the proportion is taken five times, there
must be four means; look in the table amongst
the three-sided figures of four means, and seeing
the proportion was as 5 to 2, look in the upper¬
most row for the number 2, and descending in
the second column till you meet with that three-
sided figure, you will find f ; which shows that the
deficient figure is to the parallelogram as f to 1,
or as 5 to 7-
OF FIGURES DEFICIENT.
253
Parallelogram
Strait-sided triangle
Three-sided figure of 1 mean
Three-sided figure of 2 means .
Three-sided figure of 3 means .
Three-sided figure of 4 means .
Three-sided figure of 5 means .
Three-sided figure of 6 means .
Three-sided figure of 7 means .
4. Now for the better understanding of the Description &
nature of these three-sided figures, I will show Sesame"
how they may be described by points ; and first, fisures-
those which are in the first column of the table.
Any parallelogram being described, as A B C D
(in figure 2) let the diagonal B D be drawn;
and the strait-lined triangle BCD will be half the
parallelogram ; then let any number of lines, as
E F, be drawn parallel to the side B C, and cutting
the diagonal B D in G; and let it be everywhere,
as E F to E G, so E G to another, E H; and through
all the points H let the line BHHD be drawn;
and the figure B H I I1) C will be that which I call
a three-sided figure of one mean, because in three
proportionals, as E F, EG and E H, there is but
one mean, namely, E G; and this three-sided
figure will be § of the parallelogram, and is called
a parabola. Again, let it be as E G to E H, so E H
to another, E I, and let the line BITD be drawn,
making the three-sided figure B 11 D C; and this
will be f of the parallelogram, and is by many
called a cubic parabola. In like manner, if the
MOTIONS AND MAGNITUDES.
tart in. proportions be further continued in E F, there will
"—' be made the rest of the three-sided figures of the
Description & „ i-iti - t
production of nrst column ; which I thus demonstrate. Let there
figures™6 be drawn strait lines, as H K and G L, parallel to
the base D C. Seeing therefore the proportion of
E F to E H is duplicate to that of E F to E G, or of
B C to B L, that is, of C D to L G, or of K M (pro¬
ducing K H to A D in M) to K H, the proportion
of B C to B K will be duplicate to that of K M to
K H ; but as B C is to B K, so is DC or KM to
K N, and therefore the proportion of K M to K N
is duplicate to that of K M to K H ; and so it will
be wheresoever the parallel KM be placed. Where¬
fore the figure B H H D C is double to its comple¬
ment B H H D A, and consequently § of the whole
parallelogram. In the same manner, if through I
be drawn 0 PI Q parallel and equal to C D, it may
be demonstrated that the proportion of 0 Q, to
0 P, that is, of B C to B O, is triplicate that of
0 Q to O I, and therefore that the figure B 11D C
is triple to its complement B 11 D A, and conse¬
quently f of the whole parallelogram, &c.
Secondly, such three-sided figures as are in any
of the transverse rows, may be thus described.
Let A B C D (in fig. 3) be a parallelogram, whose
diagonal is B D. I would describe in it such
figures, as in the preceding table I call three-sided
figures of three means. Parallel to DC, I draw
E F as often as is necessary, cutting BD in G ; and
between EF and EG, I take three proportionals
EH, EI and E K. If now there be drawn lines
through all the points H, I and K, that through all
the points H will make the figure B H D C, which
is the first of those three-sided figures ; and that
OF FIGURES DEFICIENT.
255
through all the points I, will make the figure PAIj^Iir*
13 I D C, which is the second ; and that which is -—■—'
drawn through all the points K, will make the ^oSiiTof
figure B K D C the third of those three-sided *he same
o figures.
figures. The first of these, seeing the proportion
of E F to E G is quadruplicate of that E F to E H,
will be to its complement as 4 to 1, and to the
parallelogram as 4 to 5. The second, seeing the
proportion of E F to E G is to that of E F to E I as
4 to 2, will be double to its complement, and ^ or
f of the parallelogram. The third, seeing the pro-
proportion of E F to EG is that of E F to E K as
4 to 3, will be to its complement as 4 to 3, and to
the parallelogram as 4 to 7-
Any of these figures being described may be
produced at pleasure, thus ; let A B C D (in fig. 4)
be a parallelogram, and in it let the figure BKDC
be described, namely, the third three-sided figure
of three means. Let B D be produced indefinitely
to E, and let E F be made parallel to the base D C,
cutting A D produced in G, and B C produced in
F; and in G E let the point H be so taken, that the
proportion of F E to F G may be quadruplicate to
that of F E to F H, which may be done by making
FH the greatest of three proportionals between
F E and F G ; the crooked line B K D produced,
will pass through the point H. For if the strait
line BH be drawn, cutting CD in I, and HL be
drawn parallel to G D, and meeting C D produced
in L ; it will be as F E to F G, so C L to C I, that
is, in quadruplicate proportion to that of F E to
F H, or of C D to C I. Wherefore if the line B K D
be produced according to its generation, it will
fall upon the point H.
256
motions and magnitudes.
tart hi. 5. A strait line may be drawn so as to touch
s—■—the crooked line of the said figure in any point, in
of tangents this manner. Let it be required to draw a tangent
to them. tjie j-ne g d h (in fig. 4) in the point D. Let
the points B and D be connected, and drawing
D A equal and parallel to B C, let B and A be con¬
nected ; and because this figure is by construction
the third of three means, let there be taken in A B
three points, so, that by them the same A B be
divided into four equal parts ; of which take three,
namely, A M, so that A B may be to A M, as the
figure B K D C is to its complement. I say, the
strait line M D will touch the figure in the point
given D. For let there be drawn anywhere be¬
tween A B and D C a parallel, as R Q-, cutting the
strait line B D, the crooked line B K D, the strait
line M D, and the strait line A D, in the points
P, K, 0 and Q,. R K will therefore, by construc¬
tion, be the least of three means in geometrical
proportion between R Q, and R P. Wherefore (by
coroll. of art. 28, chapter xiii.) R K will be less
than R 0 ; and therefore M D will fall without the
figure. Now if M D be produced to N, FN will
be the greatest of three means in arithmetical pro¬
portion betwreen F E and F G; and F H will be the
greatest of three means in geometrical proportion
between the same F E and F G. Wherefore (by
the same coroll. of art. 28, chapter xiii.) FH will
be less than F N ; and therefore D N will fall with¬
out the figure, and the strait line M N will touch
the same figure only in the point D.
in whatpropor- The proportion of a deficient figure to its
tion the same . .
figures exceed complement being known, it may also be known
OF FIGURES DEFICIENT. 257
what proportion a strait-lined triangle has to the part iii.
excess of the deficient figure above the same tri- —A--
angle ; and these proportions I have set down in jungle oftfie
the following table ; where if you seek, for ex- ^,eba^tltude
ample, how much the fourth three-sided figure of
five means exceeds a triangle of the same altitude
and base, you will find in the concourse of the
fourth column with the three-sided figures of five
means to- ; by which is signified, that that three-
sided figure exceeds the triangle by two-tenths or
by one-fifth part of the same triangle.
12 3 4 5 6 7
The triangle
o
.
1
r A three-sided fi<>\ of 1
mean .
1
A three-sided fig-, of c2
means
*
A
A three-sided of 3
means
3
'>
0"
1.
<[ A three-sided fig. of 4
means
4
G
~7~
"5"
_
1
9
A three-sided fig. of o
means
4
8
"if
"To
1
1 1
A three-sided fi<>\ of (5
O
means
6
8
«r
4
1 0
3
1 1
TIT
1 1
l 1
. A three-sided fig. of 7
means
7
9
6
TT5"
5
1 1
4
1 2
3
1 3
2 1
14 15 1
7. In the next table are set down the proportion a. table of solid
of a cone and the solids of the said three-sided described lu ^
figures, namely, the proportions between them andcyhndcr"
a cylinder. As for example, in the concourse of
the second column with the three-sided figures of
four means, you have 1; which gives you to un¬
derstand, that the solid of the second three-sided
figure of four means is to the cylinder as 5 to 1, or
as 5 to 9.
VOL. I.
s
258
MOTIONS AND MAGNITUDES.
A cylinder
A cone
'o
co
O
pi
rA three-sided fig. of 1 mean
A three-sided fig. of 2 means
A three-sided fig. of 3 means
A three-sided fig. of 4 means
A three-sided fig. of 5 means
A three-sided fig. of 6 means
. A three-sided fig. of 7 means
1
1
2
3
4
5
c
7
1 1
1 H*®
co[o |
3
1 "^1° J
4
8
-i
To
5
«o|» |
i'r
5
13
50|x> I
6
10
6
i 2
6
1 i
«
3 (i
7
9
7
11
7
i 3
7
15
7
1 7
7
1 9
TV
s
1 -J
s
1 4:
_a_
1 6
1 8
A
In what propor¬
tion the same
figures exceed
a cone of the
same altitude
and base.
8. Lastly, the excess of the solids of the said
three-sided figures above a cone of the same alti¬
tude and base, are set down in the table which
follows :
1 2 3 4- 5 6 7
The Cone
•5 8
Of the solid of a three-sided j
figure of 1 mean
Ditto ditto, 2 means
■)
Ditto ditto, 3 means
H Ditto ditto, 4 means
Ditto ditto, 5 means
Ditto ditto, 6 means
'-Ditto ditto, 7 means
•3 V,
£ 3
O tD
How a plain
deficient figure
may be describ¬
ed in a paralle¬
logram, so that
it be to a tri¬
angle of the
same base and
altitude, as an¬
other deficient
figure, plain or
solid, twice ta¬
ken, is to the
same deficient
9. If any of these deficient figures, of which I
have now spoken, as ABCD (in the 5th figure) be
inscribed within the complete figure B E, having
ADCE for its complement; and there be made
upon C B produced the triangle A B I; and the
parallelogram A B I K be completed; and there be
drawn parallel to the strait line C I, any number
of lines, as M F, cutting every one of them the
OF FIGURES DEFICIENT.
259
crooked line of the deficient figure in D, and the 1>ART m-
° 17.
strait lines AC, A B and AI in H, G, and L; and 1—■—-
as G F is to G D, so G L be made to another, G N ; wf"hrethegcom-
and through all the points N there be drawn the Plete %ure'in
which it is cIg"-
line A N I : there will be a deficient figure A N I B, scribed,
whose complement will be A N I K. I say, the
figure A NIB is to the triangle A B I, as the de¬
ficient figure A B C D twice taken is to the same
deficient figure together with the complete figure
BE.
For as the proportion of A B to A G, that is, of
G M to G L, is to the proportion of G M to G N,
so is the magnitude of the figure A NIB to that
of its complement A NI K, by the second article
of this chapter.
But, by the same article, as the proportion of
A B to A G, that is, of G M to G L, is to the pro¬
portion of GFtoGD, that is, by construction, of
GL to GN, so is the figure ABCD to its comple¬
ment A D C E.
And by composition, as the proportion of G M
to G L, together with that of G L to G N, is to the
proportion of G M to G L, so is the complete figure
B E to the deficient figure ABCD.
And by conversion, as the proportion of G M to
G L is to both the proportions of G M to G L and
of G L to G N, that is, to the proportion of G M to
G N, which is the proportion compounded of both,
so is the deficient figure A B C D to the complete
figure B E.
But it was, as the proportion of G M to G L to
that of G M to G N, so the figure A NI B to its
complement AN IK. And therefore, ABCD. BE
:: A NIB. AN IK are proportionals. And by com-
s 2
260
MOTIONS AND MAGNITUDES.
PART III. position, ABCD + BE. ABCD : : BK. ANIB
-—r~—- are proportionals.
And by doubling the consequents, ABCD +
BE. 2 A B C D : : B K. 2 ANIB are proportionals.
And by taking the halves of the third and the
fourth, ABCD + BE. 2 ABCD: : ABI. ANIB
are also proportionals; which was to be proved.
The transfer- 10. From what has been said of deficient figures
properties 111 described in a parallelogram, may be found out
figures^des- what proportions spaces, transmitted with accele-
cribed in a pa- rated motion in determined times, have to the
rallelogram to
the propor- times themselves, according as the moved body is
tions of spaces , , . , ,
transmitted accelerated m the several times with one or more
degreeesVeofal degrees of velocity.
velocity. ;por ]et the parallelogram ABCD, in the 6th
figure, and in it the three-sided figure D E B C be
described; and let F G be drawn anywhere parallel
to the base, cutting the diagonal B D in H, and
the crooked line B E D in E ; and let the propor¬
tion of B C to B F be, for example, triplicate to
that of F G to F E ; whereupon the figure D E B C
will be triple to its complement BEDA; and in
like manner, IE being drawn parallel to B C, the
three-sided figure EKBF will be triple to its com¬
plement B K E I. Wherefore the parts of the de¬
ficient figure cut off from the vertex by strait lines
parallel to the base, namely, D E B C and EKBF,
will be to one another as the parallelograms A C
and I F ; that is, in proportion compounded of the
proportions of the altitudes and bases. Seeing
therefore the proportion of the altitude B C to the
altitude B F is triplicate to the proportion of the
base D C to the base F E, the figure D E B C to the
of figures deficient.
261
figure EKBF will be quadruplicate to the propor- part iii.
tion of the same DC to FE. And by the same •—
method, may be found out what proportion any of o^certair
the said three-sided figures has to any part of the properties
c . of deficient
same, cut oft from the vertex by a strait line pa-figures, &c.
rallel to the base.
Now as the said figures are understood to be
described by the continual decreasing of the base,
as of C D, for example, till it end in a point, as in
B ; so also they may be understood to be described
by the continual increasing of a point, as of B, till
it acquire any magnitude, as that of C D.
Suppose now the figure B E D C to be described
by the increasing of the point B to the magnitude
C D. Seeing therefore the proportion of B C to
B F is triplicate to that of C D to F E, the propor¬
tion of FE to CD will, by conversion, as I shall
presently demonstrate, be triplicate to that B F to
B C. Wherefore if the strait line B C be taken for
the measure of the time in which the point B is
moved, the figure EKBF will represent the sum
of all the increasing velocities in the time B F ; and
the figure D E B C will in like manner represent
the sum of all the increasing velocities in the time
B C. Seeing therefore the proportion of the figure
EKBF to the figure D E B C is compounded of
the proportions of altitude to altitude, and base to
base ; and seeing the proportion of F E to C D is
triplicate to that of B F to B C; the proportion of
the figure EKBF to the figure D E B C will be
quadruplicate to that of B F to B C ; that is, the
proportion of the sum of the velocities in the time
B F, to the sum of the velocities in the time B C,
will be quadruplicate to the proportion of B F to
262 motions and magnitudes.
part hi. ]3 C. Wherefore if a body be moved from B with
17* 1
■ <—' velocity so increasing, that the velocity acquired
ring of c«tain'm the time BF be to the velocity acquired in the
properties ^me g q triplicate proportion to that of the
of deficient # jt ± r
figures, &c. times themselves B F to B C, and the body be
carried to F in the time B F ; the same body in the
time B C will be carried through a line equal to
the fifth proportional from B F in the continual
proportion of B F to B C. And by the same
manner of working, we may determine what spaces
are transmitted by velocities increasing according
to any other proportions.
It remains that I demonstrate the proportion of
F E to C D to be triplicate to that of B F to B C.
Seeing therefore the proportion of C D, that is, of
F G to F E is subtriplicate to that of B C to B F ;
the proportion of F G to F E will also be subtripli¬
cate to that of F G to F H. Wherefore the propor¬
tion of F G to F H is triplicate to that of F G, that
is, of C D to F E. But in four continual propor¬
tionals, of which the least is the first, the propor¬
tion of the first to the fourth, (by the 16th article
of chapter xiii), is subtriplicate to the proportion
of the third to the same fourth. Wherefore the
proportion of F FI to G F is subtriplicate to that of
F E to CD; and therefore the proportion of F E
to C D is triplicate to that of F H to F G, that is, of
B F to B C; which was to be proved.
It may from hence be collected, that when the
velocity of a body is increased in the same propor¬
tion with that of the times, the degrees of velocity
above one another proceed as numbers do in im¬
mediate succession from unity, namely, as 1, 2, 3, 4,
&rc. And when the velocity is increased in pro-
of figures deficient.
263
portion duplicate to that of the times, the degrees tart nr.
proceed as numbers from unity, skipping one, as -—*—'
1, 3, 5, 7, &c. Lastly, when the proportions of
the velocities are triplicate to those of the times,
the progression of the degrees is as that of num¬
bers from unity, skipping two in every place, as
1, 4, 7, 10, &c., and so of other proportions. For
geometrical proportionals, when they are taken in
every point, are the same with arithmetical pro¬
portionals.
] 1. Moreover, it is to be noted that as in quan- of deficient &.
tities, which are made by any magnitudes decreas-
ing, the proportions of the figures to one another
are as the proportions of the altitudes to those of
the bases; so also it is in those, which are made
with motion decreasing, which motion is nothing
else but that power by which the described figures
are greater or less. And therefore in the descrip¬
tion of Archimedes spiral, which is done by the
continual diminution of the semidiameter of a
circle in the same proportion in which the circum¬
ference is diminished, the space, which is con¬
tained within the semidiameter and the spiral
line, is a third part of the whole circle. For the
semidiameters of circles, inasmuch as circles are
understood to be made up of the aggregate of
them, are so many sectors; and therefore in the
description of a spiral, the sector which describes
it is diminished in duplicate proportions to the
diminutions of the circumference of the circle in
which it is inscribed; so that the complement of
the spiral, that is, that space in the circle which
is without the spiral line, is double to the space
within the spiral line. In the same manner, if
2(54 motions and magnitudes.
part in. there be taken a mean proportional everywhere
v—.—- between the semidiameter of the circle, which
contains the spiral, and that part of the semidia¬
meter which is within the same, there will be
made another figure, which will be half the circle.
And to conclude, this rule serves for all such
spaces as may be described by a line or superficies
decreasing either in magnitude or power; so that
if the proportions, in which they decrease, be
commensurable to the proportions of the times in
which they decrease, the magnitudes of the figures
they describe will be known.
The proposi- j2. The truth of that proposition, which I de-
tion demon- # 1 _ x
strated in art. monstrated in art. 2, which is the foundation of all
from the eie- that has been said concerning deficient figures,
iosophy.f plu" maY derived from the elements of philosophy,
as having its original in this; that all equality
and inequality between two effects, that is, all
proportion, proceeds from, and is determined by,
the equal and unequal causes of those effects, or
from the proportion which the causes, concurring
to one effect, have to the causes which concur to
the producing of the other effect; and that there¬
fore the proportions of quantities are the same
with the proportions of their causes. Seeing,
therefore, two deficient figures, of which one is
the complement of the other, are made, one by
motion decreasing in a certain time and proportion,
the other by the loss of motion in the same time ;
the causes, which make and determine the quanti¬
ties of both the figures, so that they can be 110
other than they are, differ only in this, that the
proportions by which the quantity which generates
the figure proceeds in describing of the same, that
OF FIGURES DEFICIENT.
265
is, the proportions of the remainders of all the part hi.
times and altitudes, may be other proportions than ^—-
those by which the same generating quantity de¬
creases in making the complement of that figure,
that is, the proportions of the quantity which gene¬
rates the figure continually diminished. Wherefore,
as the proportion of the times in which motion is
lost, is to that of the decreasing quantities by
which the deficient figure is generated, so will the
defect or complement be to the figure itself which
is generated.
13. There are also other quantities which are An unusual
determinable from the knowledge of their causes, ing concerning
namely, from the comparison of the motions by biSnthe
which they are made; and that more easily than s,'perficies
J « of a portion
from the common elements of geometry. Forof a sPhere
example, that the superficies of any portion of a'
sphere is equal to that circle, whose radius is a
strait line drawn from the pole of the portion to
the circumference of its base, I may demonstrate
in this manner. Let BAG (in fig. 7) be a portion
of a sphere, whose axis is A E, and whose base is
B C; and let A B be the strait line drawn from
the pole A to the base in B; and let AD, equal to
A B, touch the great circle B A C in the pole A.
It is to be proved that the circle made by the
radius AD is equal to the superficies of the portion
BAG. Let the plain AEBD be understood to
make a revolution about the axis A E; and it is
manifest that by the strait line A D a circle will be
described ; and by the arch A B the superficies of
a portion of a sphere ; and lastly, by the subtense
A B the superficies of a right cone. Now seeing
both the strait line A B and the arch A B make
266
motions and magnitudes.
part in. one and the same revolution, and both of them
-—r~"—- have the same extreme points A and B, the cause
w a yUq f1 r e Is on- W^Y the spherical superficies, which is made by
ing, &c. the arch, is greater than the conical superficies,
which is made by the subtense, is, that A B the
arch is greater than A B the subtense; and the
cause why it is greater consists in this, that
although they be both drawn from A to B, yet the
subtense is drawn strait, but the arch angularly,
namely, according to that angle which the arch
makes with the subtense, which angle is equal to
the angle D A B (for an angle of contingence adds
nothing to an angle of a segment, as has been shown
in chapter xiv, article 16.) Wherefore the mag¬
nitude of the angle D A B is the cause why the
superficies of the portion, described by the arch
AB, is greater than the superficies of the right
cone described by the subtense A B.
Again, the cause why the circle described by
the tangent A D is greater than the superficies of
the right cone described by the subtense A B (not¬
withstanding that the tangent and the subtense
are equal, and both moved round .in the same
time) is this, that A D stands at right angles to
the axis, but A B obliquely ; which obliquity con¬
sists in the same angle DAB. Seeing therefore
the quantity of the angle DAB is that which
makes the excess both of the superficies of the
portion, and of the circle made by the radius A D,
above the superficies of the right cone described
by the subtense A B ; it follows, that both the
superficies of the portion and that of the circle
do equally exceed the superficies of the cone.
Wherefore the circle made by A D or A B, and
OF FIGURES DEFICIENT.
2 67
the spherical superficies made by the arch A B, are rAIJTI1L
equal to one another; which was to be proved. 1—>—'
14. If these deficient figures, which I have de- How from the
scribed in a parallelogram, were capable of exact JeSufigui^s
description, then any number of mean propor- in a i)arallel°"
1 J . . gram,any num-
tionals might be found out between two strait lines ber of mean
given. For example, in the parallelogram A BCD, be^fomui
"(in figure 8) let the three-sided figure of two means twogfven^rait
be described (which many call a cubical parabola); lilies-
and let R and S be two given strait lines; between
which, if it be required to find two mean propor¬
tionals, it may be done thus. Let it be as R to S,
so B C to B F ; and let F E be drawn parallel to
B A, and cut the crooked line in E ; then through
E let G H be drawn parallel and equal to the strait
line A D, and cut the diagonal B D in I; for thus
we have G I the greatest of two means between
G H and G E, as appears by the description of the
figure in article 4. Wherefore, if it be as G H to
G I, so R to another line, T, that T will be the
greatest of two means between R and S. And
therefore if it it be again as R to T, so T to ano¬
ther line, X, that will be done which was required.
In the same manner, four mean proportionals
may be found out, by the description of a three-
sided figure of four means ; and so any other num¬
ber of means, &c.
268
motions and magnitudes.
CHAPTER XVIII.
of the equation of strait lines with the
crooked lines of parabolas and other
figures made in imitation of parabolas.
1. To find the strait line equal to the crooked line of a semipa-
rabola.—2. To find a strait line equal to the crooked line of
the first semiparabolaster, or to the crooked line of any other
of the deficient figures of the table of the 3d article of the
precedent chapter.
part in. y A parabola being given, to find a strait line
v—■—' equal to the crooked line of the semiparabola.
line equal to the Let the parabolical line given be A B C (in
aTcmijLraboia!'figure 1)? and the diameter found be AD, and the
base drawn D C ; and the parallelogram A D C E
being completed, draw the strait line A C. Then
dividing A D into two equal parts in F, draw F H
equal and parallel to D C, cutting A C in K, and
the parabolical line in 0; and between F H and
F O take a mean proportional F P, and draw A 0,
AP and PC. I say that the two lines AP and
P C, taken together as one line, are equal to the
parabolical line A B 0 C.
For the line A B 0 C being a parabolical line, is
generated by the concourse of two motions, one
uniform from A to E, the other in the same time
uniformly accelerated from rest in A to D. And
because the motion from A to E is uniform, A E
may represent the times of both those motions
from the beginning to the end. Let therefore
A E be the time ; and consequently the lines ordi-
OF EQUATION OF STUAIT LINES, ETC. 269
nately applied in the semiparabola will design tlie PAiy£ IIL
parts of time wherein the body, that describeth ^
the line ABO C, is in every point of the same ; so nLf&c.'1 stralt
that as at the end of the time AE or DC it is in C,
so at the end of the time F O it will be in O. And
because the velocity in A D is increased uniformly,
that is, in the same proportion with the time, the
same lines ordinately applied in the semiparabola
will design also the continual augmentation of the
impetus, till it be at the greatest, designed by the
base D C. Therefore supposing uniform motion
in the line A F, in the time F K the body in A by
the concourse of the two uniform motions in A F
and F K will be moved uniformly in the line A K ;
and KO will be the increase of the impetus or
swiftness gained in the time F K; and the line
A 0 will be uniformly described by the concourse
of the two uniform motions in A F and F O in the
time F 0. From 0 draw 0 L parallel to EC,
cutting A C in L ; and draw L N parallel to DC,
cutting E C in N, and the parabolical line in M;
and produce it on the other side to A D in I; and
IN, I M and I L will be, by the construction of a
parabola, in continual proportion, and equal to
the three lines F Ii, F P and F O; and a strait
line parallel to E C passing through M will fall on
P; and therefore 0 P will be the increase of im¬
petus gained in the time F O or IL. Lastly, pro¬
duce PM to CD in Gi; and Q-C or MN or PH will
be the increase of impetus proportional to the time
F P or I M or D Q-. Suppose now uniform motion
from H to C in the time P H. Seeing therefore in
the time F P with uniform motion and the impetus
increased in proportion to the times, is described
2/0 motions and magnitudes.
part nr. the straight line A P; and in the rest of the time
18
-—A— and impetus, namely, P H, is described the line
C P uniformly; it followeth that the whole line
A P C is described with the whole impetus, and in
the same time wherewith is described the parabo¬
lical line ABC; and therefore the line A P C,
made of the two strait lines A P and P C, is equal
to the parabolical line ABC; which was to be
proved.
To find a strait 2. To find a strait line equal to the crooked line
line equal to _
the crookediine oi the first semiparabolaster.
mi para bokis t e r Let ABC be the crooked line of the first semi-
or/°the <:rook-parabolaster ; AD the diameter; DC the base ;
ed line ot any r 7
other of the de- and let the parallelogram completed be ADCE,
S"theSieof whose diagonal is A C. Divide the diameter into
cedVng'chapter!two equal parts in F, and draw F H equal and
parallel to D C, cutting AC in K, the crooked line
in O, and EC in H. Then draw O L parallel to
E C, cutting AC in L; and draw L N parallel
to the base ,D C, cutting the crooked line in M,
and the strait line EC in N; and produce it on
the other side to AD in I. Lastly, through the
point M draw P M Q, parallel and equal to H C,
cutting F H in P; and join CP, AP and AO.
I say, the two strait lines A P and P C are equal to
the crooked line A B O C.
For the line A B O C, being the crooked line of
the first semiparabolaster, is generated by the
concourse of two motions, one uniform from A to
E, the other in the same time accelerated from
rest in A to D, so as that the impetus increaseth
in proportion perpetually triplicate to that of the
increase of the time, or which is all one, the
lengths transmitted are in proportion triplicate to
of equation of strait lines, etc. 2/1
that of the times of their transmission ; for as the part hi.
18
impetus or quicknesses increase, so the lengths -—re¬
transmitted increase also. And because the mo-strdit
tion from A to E is uniform, the line AE may
serve to represent the time, and consequently the
lines, ordinately drawn in the semiparabolaster,
will design the parts of time wherein the body,
beginning from rest in A, describetli by its
motion the crooked line A B O C. And because
D C, which represents the greatest acquired im¬
petus, is equal to A E, the same ordinate lines will
represent the several augmentations of the impetus
increasing from rest in A. Therefore, supposing
uniform motion from A to F, in the time F K there
will be described, by the concourse of the two
uniform motions A F and F K, the line A K uni¬
formly, and K O will be the increase of impetus iiv
the time F K; and by the concourse of the two
uniform motions in A F and F O will be described
the line AO uniformly. Through the point L
draw the strait line L M N parallel to D C, cutting
the strait line A D in I, the crooked line ABC in
M, and the strait line E C in N ; and through the
point M the strait line P M Q parallel and equal to
H C, cutting D C in Q, and F H in P. By the
concourse therefore of the two uniform motions in
A F and F P in the time F P will be uniformly
described the strait line A P; and L M or O P
will be the increase of impetus to be added for the
time F O. And because the proportion of I N to
I L is triplicate to the proportion of I N to I M,
the proportion of FH to F 0 will also be tripli¬
cate to the proportion of F H to F P; and the
proportional impetus gained in the time F P is P H.
2 72
MOTIONS AND MAGNITUDES.
PAI|i8 IIL ^ ^ equal to D C, which designed
"—1—" the whole impetus acquired by the acceleration,
To find a strait „ . c • , i_ i
line, &c. tnei e is no more increase ot impetus to be com¬
puted. Now in the time P H suppose an uniform
motion from H to C ; and by the two uniform mo¬
tions in C H and H P will be described uniformly
the strait line P C. Seeing therefore the two strait
lines AP and PC are described in the time AE
with the same increase of impetus, wherewith the
crooked line A B O C is described in the same
time A E, that is, seeing the line A P C and the
line A B O C are transmitted by the same body in
the same time and with equal velocities, the lines
themselves are equal; which was to be demon¬
strated.
By the same method (if any of the semipara-
bolasters in the table of art. 3 of the precedent
chapter be exhibited) may be found a strait line
equal to the crooked line thereof, namely, by
dividing the diameter into two equal parts, and
proceeding as before. Yet no man hitherto hath
compared any crooked with any strait line, though
many geometricians of every age have endeavoured
it. But the cause, why they have not done it,
may be this, that there being in Euclid no defini¬
tion of equality, nor any mark by which to judge
of it besides congruity (which is the 8th axiom of
the first Book of his Elements) a thing of no use
at all in the comparing of strait and crooked; and
others after Euclid (except Archimedes and Apol-
lonius, and in our time Bonaventura) thinking the
industry of the ancients had reached to all that
was to be done in geometry, thought also, that
all that could be propounded was either to be
of angles of incidence, etc. 273
deduced from what they had written, or else that part hi.
it was not at all to be done : it was therefore dis- -—re¬
puted by some of those ancients themselves, wlie- J°efi^dca strait
ther there might be any equality at all betwreen
crooked and strait lines; which question Archi¬
medes, who assumed that some strait line was
equal to the circumference of a circle, seems to
have despised, as he had reason. And there is a
late writer that granteth that between a strait
and a crooked line there is equality; but nowT,
says he, since the fall of Adam, without the spe¬
cial assistance of Divine Grace it is not to be
found.
CHAPTER XIX.
of angles of incidence and reflection,
equal by supposition.
1. If two strait lines falling upon another strait line be parallel,
the lines reflected from them shall also be parallel.—2. If two
strait lines drawn from one point fall upon another strait line,
the lines reflected from them, if they be drawn out the other
way, will meet in an angle equal to the angle made by the lines
of incidence.—3. If two strait parallel lines, drawn not oppo¬
sitely, but from the same parts, fall upon the circumference of
a circle, the lines reflected from them, if produced they meet
within the circle, will make an angle double to that which is
made by two strait lines drawn from the centre to the points of
incidence.—4. If two strait lines drawn from the same point
without a circle fall upon the circumference, and the lines
reflected from them being produced meet within the circle,
they will make an angle equal to twice that angle, which is
made by two strait lines drawn from the centre to the points of
incidence, together with the angle which the incident lines
themselves make.—5. If two strait lines drawn from one point
vol. i. t
274 motions and magnitudes.
fall upon the concave circumference of a circle, and the angle
they make be less than twice the angle at the centre, the lines
reflected from them and meeting within the circle will make an
angle, which being added to the angle of the incident lines will
be equal to twice the angle at the centre.—G. If through any
one point two unequal chords be drawn cutting one another,
and the centre of the circle be not placed between them, and
the lines reflected from them concur wheresoever, there can¬
not through the point, through which the two former lines
were drawn, be drawn any other strait line whose reflected
line shall pass through the common point of the two former
lines reflected.—7. In equal chords the same is not true.
8. Two points being given in the circumference of a circle, to
draw two strait lines to them, so that their reflected lines may-
contain any angle given.—9. If a strait line falling upon the
circumference of a circle be produced till it reach the semi-
diameter, and that part of it, which is intercepted between
the circumference and the semidiameter, be equal to that part
of the semidiameter which is between the point of concourse
and the centre, the reflected line will be parallel to the semi¬
diameter.—10. If from a point within a circle, two strait lines
be drawn to the circumference, and their reflected lines meet
in the circumference of the same circle, the angle made by the
reflected lines will be a third part of the angle made by the in¬
cident lines.
Whether a body falling upon the superficies of
another body and being reflected from it, do make
equal angles at that superficies, it belongs not to
and reflection, this place to dispute, being a knowledge which
depends upon the natural causes of reflection; of
which hitherto nothing has been said, but shall be
spoken of hereafter.
In this place, therefore, let it be supposed that
the angle of incidence is equal to the angle of
reflection; that our present search may be ap¬
plied, not to the finding out of the causes, but
some consequences of the same.
I call an angle of incidence, that which is made
PART III.
19.
Angles of
incidence
of angles of incidence, etc. 275
oetween a strait line and another line, strait or part iii.
19.
crooked, upon which it falls, and which I call the ——t—-
line reflecting ; and an angle of reflection equal
to it, that which is made at the same point between
the strait line which is reflected and the line
reflecting.
1. If two strait lines, which fall upon another if two strait
strait line, be parallel, their reflected lines shall be upon another
also parallel. _ ^ _ pneUhe
Let the two strait lines A B and C D (in fig. 1), lines «flected
v . from them
which fall upon the strait line E F, at the points shall also be
B and 1), be parallel; and let the lines reflectedparalle1'
from them be B G and D H. I say, B G and D H
are also parallel.
For the angles ABE and C D E are equal by
reason of the parallelism of A B and CD ; and the
angles G B F and H DF are equal to them by sup¬
position ; for the lines B G and D H are reflected
from the lines A B and C D. Wherefore B G and
D H are parallel.
2. If two strait lines drawn from the sameIf tw0 strait
r „ -a ,. it ^nes drawn
point tall upon another strait line, the lines re- from one point
fleeted from them, if they be drawn out the other £ strait finej
way, will meet in an angle equal to the angle of the flectedfrom
incident lines. them>if they
t, - . „ v , , . be drawn out
rrom the point A (m fig. 2) let the two strait the other way,
lines A B and A D be drawn; and let them fall Mgi™eqiS To
upon the strait line E K at the points B and D ; {,y0theSnnesa of
and let the lines B I and D G be reflected from incidence,
them. I say, IB and GD do converge, and that if
they be produced on the other side of the line E K,
they shall meet, as in F ; and that the angle B F D
shall be equal to the angle BAD.
For the angle of reflection IB K is equal to the
t 2
2/6
motions and magnitudes.
part iii. angle of incidence ABE; and to the angle I B K
-—^—- its vertical angle E B F is equal; and therefore
the angle ABE is equal to the angle E B F.
Again, the angle A D E is equal to the angle of
reflection G D K, that is, to its vertical angle
E D F; and therefore the two angles A B D and
A D B of the triangle A B D are one by one equal
to the two angles F B D and F D B of the triangle
F B D; wherefore also the third angle B A D is
equal to the third angle B F D ; which was to be
proved.
Coroll. i. If the strait line AF be drawn, it will
be perpendicular to the strait line E K. For both
the angles at E will be equal, by reason of the
equality of the two angles ABE and F B E, and
of the two sides A B and F B.
Coroll. ii. If upon any point between B and D
there fall a strait line, as AC, whose reflected line is
CH, this also produced beyond C, will fall upon F;
which is evident by the demonstration above,
if two strait 3. if from two points taken without a circle,
parallel lines, ± 7
drawn not op- two strait parallel lines, drawn not oppositely, but
from the same from the same parts, fall upon the circumference;
S?circumfe-n the lines reflected from them, if produced they
rence °f a «r- meet within the circle, will make an angle double
cle, the lines 7 #°
reflected from to that which is made by two strait lines drawn
duced they° from the centre to the points of incidence.
Let the two strait parallels A B and D C (in
make an angle fio- 3) fall upon the circumference B C at the
double to that ' 1
which is made points B and C; and let the centre of the circle be
lines drawn E ; and let A B reflected be B F, and D C reflected
SpS^eCG; and let the lines FB and GC produced
incidence. meet within the circle in H ; and let E B and E C
OF ANGLES OF INCIDENCE, ETC. 277
be connected. I say the angle F H G is double to rA^J ITL
tlie angle 13 E C. -—.—-
For seeing A B and D C are parallels, and E B parXriin^,
cuts A B in B, the same E B produced will cut drawn>&c*
D C somewhere ; let it cut it in D; and let D C
be produced howsoever to I, and let the intersec¬
tion of D C and B F be at K. The angle therefore
IC H, being external to the triangle C K II, will
be equal to the two opposite angles CKH and
CHK. Again, ICE being external to the triangle
C D E, is equal to the two angles at D and E.
Wherefore the angle I C H, being double to the
angle ICE, is equal to the angles at D and E
twice taken ; and therefore the two angles CKH
and CHK are equal to the two angles at D and E
twice taken. But the angle C K H is equal to the
angles D and A B D, that is, D twice taken ; for
A B and D C being parallels, the altern angles D
and A B D are equal. Wherefore C H K, that is
the angle F H G is also equal to the angle at E
twice taken ; which was to be proved.
Coroll. If from two points taken within a circle
two strait parallels fall upon the circumference,
the lines reflected from them shall meet in an
angle, double to that which is made by two strait
lines drawn from the centre to the points of
incidence. For the parallels A B and IC falling
upon the points B and C, are reflected in the lines
B H and C H, and make the angle at H double to
the angle at E, as was but now demonstrated.
4. If two strait lines drawn from the same point if two strait
without a circle fall upon the circumference, and fr0m the^same
the lines reflected from them being produced meet cirdefaUupon
within the circle, they will make an angle equal tothe circumfe-
278
motions and magnitudes.
part hi. twice that angle, which is made by two strait lines
•—r——*' drawn from the centre to the points of incidence,
HneT'refTectld6 together with the angle which the incident lines
from them themselves make.
being produced
meet within Let the two strait lines A B and A C (in fig. 4)
will make an be drawn from the point A to the circumference
twfce that'In- °f the circle, whose centre is D ; and let the lines
gie which is reflected from them be B E and C G, and, being;
made by two _ # ' ~
strait lines produced, make within the circle the angle H;
centre to°theh6 also let the two strait lines D B and D C be drawn
dence, to-ether from the centre D to the points of incidence B
with the angle an(j (j. j say the angle H is equal to twice the
which the 1 • i i
incident angle at D together with the angle at A.
seives^ake. For let AC be produced howsoever to I. There¬
fore the angle I C H, which is external to the
triangle C K H, will be equal to the two angles
CI(H and 0 H K. Again, the angle I CD, which
is external to the triangle C L D, will be equal to
the two angles C L D and C D L. But the angle
I C H is double to the angle I CD, and is therefore
equal to the angles C L D and C D L twice taken.
Wherefore the angles CKH and C H K are equal
to the angles C L D and C D L twice taken. But
the angle CLD, being external to the triangle
A L B, is equal to the two angles LAB and LB A;
and consequently CLD twice taken is equal to
LAB and L B A twice taken. Wherefore CKH
and C H K are equal to the angle C D L together
with LAB and LB A twice taken. Also the
angle C K H is equal to the angle LAB once and
ABK, that is, L B A twice taken. Wherefore
the angle C H K is equal to the remaining angle
C D L, that is, to the angle at D, twice taken, and
OF ANGLES OF INCIDENCE, ETC. 279
the angle LAB, that is, the angle at A, once PAIJ£UI*
taken ; which was to be proved. v—■—'
Coroll. If two strait converging lines, as IC and
M B, fall upon the concave circumference of a
circle, their reflected lines, as C H and B H, will
meet in the angle H, equal to twice the angle D,
together with the angle at A made by the incident
lines produced. Or, if the incident lines be H B
and IC, whose reflected lines C H and B M meet
in the point N, the angle C N B will be equal to
twice the angle D, together with the angle C K H
made by the lines of incidence. For the angle
C N B is equal to the angle H, that is, to twice
the angle D, together with the two angles A, and
NBH, that is, K B A. But the angles I(B A
and A are equal to the angle C K H. Wherefore
the angle C N B is equal to twice the angle D,
together with the angle C K H made by the lines
of incidence I C and H B produced to K.
5. If two strait lines drawn from one point fall two strait
. . lines drawn
upon the concave circumference 01 a circle, and from one point
the angle they make be less than twice the angle concave"?-6
at the centre, the lines reflected from them and cu™ffence! of
7 a circle, and
meeting within the circle will make an angle, the^angie they
which being added to the angle of the incident than twice the
lines, will be equal to twice the angle at the centre,
Let the two lines AB and AC (in fig. 5), drawn l'nesyfflected
\ o J? irom them and
from the point A, fall upon the concave circum- meeting within
the circle will
ference of the circle whose centre is D ; and let make an angle,
their reflected lines B E and C E meet in the point added ?o the
E; also let the angle A be less than twice the ansle °f the m-
o cioent lines will
angle D. I sav, the angles A and E togetherbe eq«al to
_ 1-1 1 twk'e tHe anSle
taken are equal to twrice the angle D. at the centre.
For let the strait lines A B and E C cut the
280
motions and magnitudes.
tart hi. strait lines D C and D B in the points G and H ;
and the angle BHC will be equal to the two
if two strait angles EBH and E; also the same angle BHC
lines drawn
from one, &c. will be equal to the two angles D and DCH; and
in like manner the angle B G C will be equal to
the two angles A C D and A. and the same angle
B G C will be also equal to the two angles DBG
and D. Wherefore the four angles E B H, E,
A CD and A, are equal to the four angles D, DCH,
DBG and D. If, therefore, equals be taken away
011 both sides, namely, on one side A CD and
E B H, and 011 the other side DCH and DBG,
(for the angle EBH is equal to the angle DBG,
and the angle A C D equal to the angle D C H),
the remainders 011 both sides will be equal, namely,
on one side the angles A and E, and on the other
the angle D twice taken. Wherefore the angles
A and E are equal to twice the angle D.
Coroll. If the angle A be greater than twice the
angle D, their reflected lines will diverge. For, by
the corollary of the third proposition, if the angle
A be equal to twice the angle D, the reflected lines
B E and C E will be parallel; and if it be less,
they will concur, as has now been demonstrated.
And therefore, if it be greater, the reflected lines
B E and C E will diverge, and consequently, if
they be produced the other way, they will concur
and make an angle equal to the excess of the angle
A above twice the angle D ; as is evident by art. 4.
if through any 6. If through any one point twTo unequal chords
one point two - . - . . . . . ,
unequal chords be drawn cutting one another, either withm the
thigdl'onT ano" circle, or, if they be produced, without it, and the
ther, and the centre of the circle be not placed between them,
centre of the -r
circle be not and the lines reflected from them concur where-
of angles of incidence, etc.
281
soever; there cannot, through the point through hi.
which the former lines were drawn, be drawn v-—.—•
another strait line, whose reflected line shall pass
through the point where the two former reflected lines reflected
from them con-
lines concur. cur whereso-
Let any two unequal chords, as B K and C H cLTn'ofthrough
(in fig. 6), be drawn through the point A in the JJ^0J°j1nt^vhic]l
circle B C ; and let their reflected lines B D and the two former
C E meet in F; and let the centre not be between drawee6
A B and A C ; and from the point A let any other oSeTsSiine
strait line, as A G, be drawn to the circumference vvhose reflected
„ . line shall pass
between B and C. 1 say, b JN, which passes through the
through the point F, where the reflected lines B D o™°twoPS-
and C E meet, will not be the reflected line of A G. fleecrtJlnes re~
For let the arch B L be taken equal to the arch
B G, and the strait line B M equal to the strait
line B A ; and LM being drawn, let it be produced
to the circumference in O. Seeing therefore B A
and B M are equal, and the arch B L equal to the
arch B G, and the angle MBL equal to the angle
A B G, A G and M L will also be equal, and, pro¬
ducing G A to the circumference in I, the whole
lines LO and GI will in like manner be equal.
But L 0 is greater than G F N, as shall presently
be demonstrated ; and therefore also GI is greater
than GN. Wherefore the angles NGC and IGB
are not equal. Wherefore the line GFN is not
reflected from the line of incidence A G, and con¬
sequently no other strait line, besides A B and
A C, which is drawn through the point A, and
falls upon the circumference B C, can be reflected
to the point F ; which was to be demonstrated.
It remains that I prove L O to be greater than
G N; which I shall do in this manner. L 0 and
282
motions and magnitudes.
[Part hi. G N cut one another in P ; and P L is greater than
1—*— P G. Seeing now L P. P G :: P N. P O are propor¬
tionals, therefore the two extremes L P and P 0
together taken, that is L O, are greater than P G
and P N together taken, that is, G N; which re¬
mained to be proved.
in equal chords ^ if two equal chords be drawn through one
the same is not 1 °
true. point within a circle, and the lines reflected from
them meet in another point, then another strait line
may be drawn between them through the former
point, whose reflected line shall pass through the
latter point.
Let the two equal chords B C and E D (in the
7th figure) cut one another in the point A within
the circle BCD; and let their reflected lines C H
and D I meet in the point F. Then dividing the
arch C D equally in G, let the two chords G K and
G L be drawn through the points A and F. I say,
G L will be the line reflected from the chord Iv G.
For the four chords B C, C H, E D and D I are by
supposition all equal to one another; and therefore
the arch B C H is equal to the arch EDI; as also
the angle BCH to the angle EDI; and the angle
AM C to its verticle angle FMD; and the strait
line D M to the strait line G M ; and, in like man¬
ner, the strait line A C to the strait line F D ; and
the chords C G and G D being drawn, will also be
equal; and also the angles F D G and A C G, in the
equal segments GDI and G C B. Wherefore the
strait lines F G and A G are equal; and, therefore,
the angle F G D is equal to the angle A G C, that
is, the angle of incidence equal to the angle of re¬
flection. Wherefore the line G L is reflected from
the incident line C G; which was to be proved.
OF ANGLES OF INCIDENCE, ETC.
283
Coroll. By the very sight of the figure it is mani- pAI^m-
fest, that if G be not the middle point between C ■—-—-
and D, the reflected line G L will not pass through
the point F.
8. Two points in the circumference of a circle Two points be-
1 # # ing given in the
being given to draw two strait lines to them, so as circumference
that their reflected lines may be parallel, or con- draw two" strait
a. ' „v. lines to them so
tain any angle given. _ as thattheir re.
In the circumference of the circle, whose centre flected lines
may contain
is A, (in the 8th figure) let the two points B and any angle given
C be given ; and let it be required to draw to them
from two points taken without the circle two inci¬
dent lines, so that their reflected lines may, first,
be parallel.
Let A B and A C be drawn ; as also any incident
line D C, with its reflected line C F ; and let the
angle E C D be made double to the angle A ; and
let H B be drawn parallel to E C, and produced
till it meet with D C produced in I. Lastly, pro¬
ducing A B indefinitely to K, let G B be drawn so
that the angle G B K may be equal to the angle
H B K, and then G B will be the reflected line of
the incident line H B. I say, DC and H B are two
incident lines, whose reflected lines C F and B G
are parallel.
For seeing the angle E C D is double to the angle
B AC, the angle H I C is also, by reason of the
parallels E C and H I, double to the same B A C ;
therefore also F C and G B, namely, the lines re¬
flected from the incident lines D C and H B, are
parallel. Wherefore the first thing required is
done.
Secondly, let it be required to draw to the points
B and C two strait lines of incidence, so that the
284 motions and magnitudes.
part i it. lines reflected from them may contain the given
-——- angle Z.
ing given in the To the angle ECD made at the point C, let there
circumference ke added on one side the ansrle D C L eqnal to half
of a circle, &.c. # 0 1
Z, and on the other side the angle ECM equal to
the angle DCL; and let the strait line BN be
drawn parallel to the strait line C M; and let the
angle K B O be made equal to the angle NBK;
which being done, B 0 will be the line of reflection
from the line of incidence N B. Lastly, from the
incident line L C, let the reflected line C 0 be
drawn, cutting B O at O, and making the angle
COB. I say, the angle C O B is equal to the
angle Z.
Let N B be produced till it meet with the strait
line LC produced in P. Seeing, therefore, the
angle L C M is, by construction, equal to twice the
angle B AC, together with the angle Z ; the angle
N P L, which is equal to L C M by reason of the
parallels N P and M C, will also be equal to twice
the same angle BAG, together with the angle Z.
And seeing the two strait lines O C and 0 B fall
from the point O upon the points C and B; and
their reflected lines L C and N B meet in the point
P; the angle NPL will be equal to twice the angle
B A C together with the angle COB. But I have
already proved the angle NPL to be equal to twice
the angle BAG together with the angle Z. There¬
fore the angle COB is equal to the angle Z; where¬
fore, two points in the circumference of a circle
being given, Ihave drawn,&c.; which was tobe done.
But if it be required to draw the incident lines
from a point within the circle, so that the lines re¬
flected from them may contain an angle equal to
OF ANGLES OF INCIDENCE, ETC. 285
the angle Z, the same method is to be used, saving PARj^ IIT#
that in this case the angle Z is not to be added to 1—»—-
twice the angle B AC, but to be taken from it.
9. If a strait line, falling upon the circumference J*»str£lit ]i"e
. . . falling upon the
of a circle, be produced till it reach the semidia- circumference
meter, and that part of it which is intercepted be- produced till it
tween the circumference and the semidiameter be J^mVter, se™id
equal to that part of the semidiameter which is th1a.t Pan ,of il'
1 \ winch is mter-
between the point of concourse and the centre, the eepted between
reflected line will be parallel to the semidiameter. enceand the se-
Let any line AR (in the 9th figure) be the semi- ^ua^tlTthat
diameter of the circle whose centre is A : and upon of the se"
x midiameter
the circumference B D let the strait line C D fall, which is be-
and be produced till it cut A B in E, so that E D If^oncourse"1
and E A may be equal; and from the incident line reflected6'
C D let the line D F be reflected. I say, A B and ]i"f f'11 \e Pa-
J 7 rallel to the se-
D F will be parallel. midiameter.
Let A G be drawn through the point D. Seeing,
therefore, E D and E A are equal, the angles E DA
and E AD will also be equal. But the angles FDG
and EDA are equal; for each of them is half the
angle EDH or FDC. Wherefore the angles FDG
and EAD are equal; and consequently DF and
A B are parallel; which was to be proved.
Coroll. If E A be greater then E D, then D F
and A B being produced will concur ; but if E A
be less than E D, then B A and D H being produced
will concur.
10. If from a point within a circle two straitIf a point
, , . „ i -1 • within a circle
lines be drawn to the circumference, and their re- two strait lines
fleeted lines meet in the circumference of the same circumference,6
circle, the angle made by the lines of reflection will
be a third part of the angle made by the lines of meet in the cir-
incidence.
286
MOTIONS AND MAGNITUDES.
part hi. From the point B (in the 10th figure) taken
-—within the circle whose centre is A, let the two
conference of strait lines B C and B D be drawn to the circum-
the same circle,
the angle made ference; and let their reflected lines C E and D E
by the reflected 1 . ,
lines will be a meet m the circumterence ot the same circle at the
angiePmade*by point E. I say, the angle C E D will be a third
lines1'101'16111 Pai t angle C B D.
Let A C and A D be drawn. Seeing, therefore,
the angles C E D and C B D together taken are
equal to twice the angle CAD (as has been de¬
monstrated in the 5th article) ; and the angle
C A D twice taken is quadruple to the angle C E D;
the angles C E D and C B D together taken will
also be equal to the angle C E D four times taken;
and therefore if the angle C E D be taken away on
both sides, there will remain the angle C B D on
one side, equal to the angle C E D thrice taken on
the other side ; which was to be demonstrated.
Coroll. Therefore a point being given within a
circle, there may be drawn two lines from it to the
circumference, so as their reflected lines may meet
in the circumference. For it is but trisecting the
angle C B D, which how it may be done shall be
shown in the following chapter.
dimension of a circle, etc.
287
CHAPTER XX.
of the dimension of a circle, and the
division of angles or arches.
1. The dimension of a circle never determined in numbers by
Archimedes and others.—2. The first attempt for the finding out
of the dimension of a circle by lines.—3. The second attempt
for the finding out of the dimension of a circle from the
consideration of the nature of crookedness.—4. The third
attempt; and some things propounded to be further searched
into.—5. The equation of the spiral of Archimedes with a
strait line.—6. Of the analysis of geometricians by the powers
of lines.
1. In tlie comparing of an arch of a circle with a PAll2^IIL
strait line, many and great geometricians, even —• '
,» . •, , . • -1 t , i The dimension
trom the most ancient times, have exercised their 0fa circle never
wits; and more had done the same, if they had nuS'Trsby1"
not seen their pains, though undertaken for the Archimedes
-A, f and others.
common good, it not brought to perfection, vilified
by those that envy the praises of other men.
Amongst those ancient writers whose works are
come to our hands, Archimedes was the first that
brought the length of the perimeter of a circle
within the limits of numbers very little differing
from the truth ; demonstrating the same to be
less than three diameters and a seventh part, but
greater than three diameters and ten seventy-one
parts of the diameter. So that supposing the
radius to consist of 10,000,000 equal parts, the
arch of a quadrant will be between 15,714,285
and 15,704,225 of the same parts. In our times,
Ludovicus Van Cullen and Willebrordus Snellius,
288
MOTIONS AND MAGNITUDES.
PAIooIIL w^h joint endeavour, have come yet nearer to
v—.—- the truth ; and pronounced from true principles,
ofact'X'never t^iat the arch of a quadrant, putting, as before,
determined in io 000,000 for radius, differs not one whole unity
numbers by ' # J
Archimedes from the number 15,707,963 ; which, if they had
exhibited their arithmetical operations, and no
man had discovered any error in that long work
of theirs, had been demonstrated by them. This
is the furthest progress that has been made by the
way of numbers; and they that have proceeded
thus far deserve the praise of industry. Never¬
theless, if we consider the benefit, which is the
scope at which all speculation should aim, the
improvement they have made has been little or
none. For any ordinary man may much sooner
and more accurately find a strait line equal to the
perimeter of a circle, and consequently square the
circle, by winding a small thread about a given
cylinder, than any geometrician shall do the same
by dividing the radius into 10,000,000 equal parts.
But though the length of the circumference were
exactly set out, either by numbers, or mechanically,
or only by chance, yet this would contribute no
help at all towards the section of angles, unless
happily these two problems, to divide a given
angle according to any proportion assigned, and
to find a strait line equal to the arcli of a circle,
were reciprocal, and followed one another. Seeing
therefore the benefit proceeding from the know¬
ledge of the length of the arch of a quadrant
consists in this, that we may thereby divide an
angle according to any proportion, either accu¬
rately, or at least accurately enough for common
use; and seeing this cannot be done by arithmetic, I
dimension of a circle, etc.
289
thought fit to attempt the same by geometry, part nr.
and in this chapter to make trial whether it might -—r-^
not be performed by the drawing of strait and
circular lines.
2. Let the square A B C D (in the first figure) The first
be described; and with the radii A B, B C, and ^finding
D C, the three arches B D, C A, and AC ; of which ^mendonof a
let the two B D and C A cut one another in E, andcircIe by lines.
the two B D and A C in F. The diagonals there¬
fore BD and AC being drawn will cut one another
in the centre of the square G, and the two arches
B D and C A in two equal parts in H and Y; and
the arch BHD will be trisected in F and E.
Through the centre G let the two strait lines K G L
and M G N be drawn parallel and equal to the
sides of the square A B and A D, cutting the four
sides of the same square in the points K, L, M,
and N ; which being done, K L will pass through
F, and M N through E. Then let 0 P be drawn
parallel and equal to the side B C, cutting the
arch B F D in F, and the sides A B and D C in 0
and P. Therefore OF will be the sine of the arch
B F, which is an arch of 30 degrees; and the
same OF will be equal to half the radius. Lastly,
dividing the arch BF in the middle in Q, let RQ,
the sine of the arch B Q, be drawn and produced
to S, so that QS be equal to RQ, and consequently
R S be equal to the chord of the arch B F; and
let F S be drawn and produced to T in the side
BC. I say, the strait line BT is equal to the
arch BF ; and consequently that BY, the triple of
BT, is equal to the arch of the quadrant B FE D.
Let T F be produced till it meet the side B A
produced in X ; and dividing 0 F in the middle
vol. i. u
290 MOTIONS AND MAGNITUDES.
PAI2om' *n ^ drawn and produced till it meet
v—*—with the side B A produced. Seeing therefore the
attempffor strait lines R S and 0 F are parallel, and divided
out of'theg *n ^ie midst in Q, and Z, Q, Z produced will fall
dimension of a upon X, and XZQ produced to the side B C will
circle by lines, . .
cut B I m the midst m a.
Upon the strait line F Z, the fourth part of the
radius A B, let the equilateral triangle a Z F be
constituted ; and upon the centre a, with the
radius a Z, let the arch Z F be drawn ; which arch
Z F will therefore be equal to the arch Gt F, the
half of the arch B F. Again, let the strait line
Z 0 be cut in the midst in b, and the strait line
b O in the midst in c ; and let the bisection be
continued in this manner till the last part 0 c be
the least that can possibly be taken ; and upon it,
and all the rest of the parts equal to it into which
the strait line 0 F may be cut, let so many equi¬
lateral triangles be understood to be constituted ;
of which let the last be d 0 c. If, therefore, upon
the centre d, with the radius d 0, be drawn the
arch O c, and upon the rest of the equal parts
of the strait line O F be drawn in like manner so
many equal arches, all those arches together taken
will be equal to the whole arch B F, and the half
of them, namely, those that are comprehended
between 0 and Z, or between Z and F, will be
equal to the arch B Q or Q, F, and in sum,
what part soever the strait line 0 c be of the
strait line 0 F, the same part will the arch 0 c be
of the arch B F, though both the arch and the
chord be infinitely bisected. Now seeing the
arch 0 c is more crooked than that part of the
arch B F which is equal to it; and seeing also
DIMENSION OF A CTRCLE, ETC. 291
that the more the strait line Xc is produced, the PAR^ 11L
more it diverges from the strait line X O, if the —-—'
points O and c be understood to be moved for-
wards with strait motion in X O and X c, the the fl'1(3ins
7 out or the
arch O c will thereby be extended by little and dimension of a
little, till at the last it come somewhere to have CUC L by
the same crookedness with that part of the arch. B F
which is equal to it. In like manner, if the strait
line X b be drawn, and the point b be understood
to be moved forwards at the same time, the arch
c b will also by little and little be extended, till
its crookedness come to be equal to the crooked¬
ness of that part of the arch B F which is equal
to it. And the same will happen in all those
small equal arches which are described upon so
many equal parts of the strait line 0 F. It is also
manifest, that by strait motion in X 0 and X Z
all those small arches will lie in the arch BF,
in the points B, Q, and F. And though the same
small equal arches should not be coincident with
the equal parts of the arch B F in all the other
points thereof, yet certainly they will constitute
two crooked lines, not only equal to the two
arches B Q, and Q F, and equally crooked, but
also having their cavity towards the same parts ;
which how it should be, unless all those small
arches should be coincident with the arch B F in
all its points, is not imaginable. They are there¬
fore coincident, and all the strait lines drawn
from X, and passing through the points of division
of the strait line O F, will also divide the arch
BF into the same proportions into which 0 F is
divided.
Now seeing X b cuts off from the point B the
u 2
292 MOTIONS AND MAGNITUDES.
part In- fourth part of the arch B F, let that fourth part
1be B e ; and let the sine thereof, f e, be produced
Juemptfor to F T in g, for so f e will be the fourth part of
out of th"S stra^ ^ne f because as O b is to 0 F, so is
dimension of a f e to f g. But B T is greater than f g; and
circle by lmes. same B T is greater than four sines
of the fourth part of the arch B F. And in like
manner, if the arch B F be subdivided into any
number of equal parts whatsoever, it may be
proved that the strait line B T is greater than the
sine of one of those small arches, so many times
taken as there be parts made of the whole arch
B F. Wherefore the strait line B T is not less
than the arch B F. But neither can it be greater,
because if any strait line whatsoever, less than
B T, be drawn below B T, parallel to it, and ter¬
minated in the strait lines X B and X T, it would
cut the arch B F; and so the sine of some one_of the
parts of the arch B F, taken so often as that small
arch is found in the whole arch B F, would be
greater than so many of the same arches ; which
is absurd. Wherefore the strait line B T is equal
to the arch B F ; and the strait line B V equal to
the arch of the quadrant B F D ; and B V four
times taken, equal to the perimeter of the circle
described with the radius A B. Also the arch
B F and the strait line B T are everywhere divided
into the same proportions ; and consequently any
given angle, whether greater or less than B A F,
may be divided into any proportion given.
But the strait line BY, though its magnitude
fall within the terms assigned by Archimedes, is
found, if computed by the canon of signs, to be
somewhat greater than that which is exhibited by
DIMENSION OF A CIRCLE, ETC.
293
the Rudolpliine numbers. Nevertheless, if in the tart iii.
place of B T, another strait line, though never so —-
little less, be substituted, the division of angles is attempt^
immediately lost, as may by any man be demon- *huet
strated by this very scheme. dimension of a
J i#i i • • circle by lines.
Howsoever^ if any man think this my strait line
B V to be too great, yet, seeing the arch and all the
parallels are everywhere so exactly divided, and
B V comes so near to the truth, I desire he would
search out the reason, why, granting BY to be
precisely true, the arches cut off should not be
equal.
But some man may yet ask the reason why the
strait lines, drawn from X through the equal parts
of the arch B F, should cut off in the tangent B V
so many strait lines equal to them, seeing the con¬
nected straight line XV passes not through the
point D, but cuts the strait line A D produced in I;
and consequently require some determination of
this problem. Concerning which, I will say what
I think to be the reason, namely, that whilst the
magnitude of the arch doth not exceed the magni¬
tude of the radius, that is, the magnitude of the
tangent B C, both the arch and the tangent are cut
alike by the strait lines drawn from X; otherwise
not. For A Y being connected, cutting the arch
BHD in I, if X C being drawn should cut the
same arch in the same point I, it would be as true
that the arch B I is equal to the radius B C, as it is
true that the arch B F is equal to the strait line BT;
and drawing X K it would cut the arch BI in the
midst in i; also drawing A i and producing it to the
tangent B C in k, the strait line B k will be the
294
motions and magnitudes.
part nr. tangent of the arch B i, (which arch is equal to
V—- half the radius) and the same strait line B k will
be equal to the strait line k I. I say all this is true,
if the preceding demonstration be true; and con¬
sequently the proportional section of the arch and
its tangent proceeds hitherto. But it is manifest
by the golden rule, that taking B h double to B T,
the line X h shall not cut off the arch B E, which
is double to the arch B F, but a much greater. For
the magnitude of the straight lines X M, X B, and
M E, being known (in numbers), the magnitude of
the strait line cut off in the tangent by the strait
line XE produced to the tangent, may also be
known ; and it will be found to be less than B h;
Wherefore the strait line X h being drawn, will cut
off a part of the arch of the quadrant greater than
the arch BE. But I shall speak more fully in
the next article concerning the magnitude of the
arch B I.
And let this be the first attempt for the finding
out of the dimension of a circle by the section of
the arch B F.
The second 3. I shall now attempt the same by arguments
attempt for the r JO
finding out o f drawn from the nature of the crookedness of the
of a d'rciefrom circle itself; but I shall first set down some pre-
tion<of1 tile°r'l" mises necessary for this speculation ; and
nature of First, if a strait line be bowed into an arch of
crookedness.
a circle equal to it, as when a stretched thread,
which toucheth a right cylinder, is so bowed in
every point, that it be everywhere coincident with
the perimeter of the base of the cylinder, the
flexion of that line will be equal, in all its points ;
and consequently the crookedness of the arch of a
DIMENSION OF A CIRCLE, ETC.
295
circle is everywhere uniform ; which needs no other m*
demonstration than this, that the perimeter of a •—<—'
circle is an uniform line. _ Sem^tfo?the
Secondly, and consequently: if two unequal findins> &c*
arches of the same circle be made by the bowing of
two strait lines equal to them, the flexion of the
longer line, whilst it is bowed into the greater
arch, is greater than the flexion of the shorter line,
whilst it is bowed into the lesser arch, according
to the proportion of the arches themselves; and
consequently, the crookedness of the greater arch
is to the crookedness of the lesser arch, as the
greater arch is to the lesser arch.
Thirdly : if two unequal circles and a strait line
touch one another in the same point, the crooked¬
ness of any arch taken in the lesser circle, wrill be
greater than the crookedness of an arch equal to it
taken in the greater circle, in reciprocal proportion
to that of the radii with which the circles are
described; or, which is all one, any strait line
being drawn from the point of contact till it cut
both the circumferences, as the part of that strait
line cut off by the circumference of the greater
circle to that part which is cut off by the circum¬
ference of the lesser circle.
For let A B and A C (in the second figure) be
two circles, touching one another, and the strait
line A D in the point A; and let their centres be
E and F; and let it be supposed, that as A E is to
A F, so is the arch A B to the arch AH. I say the
crookedness of the arch A C is to the crookedness
of the arch AH, as AE is to AF. For let the
strait line A D be supposed to be equal to the arch
A B, and the strait line A G to the arch A C ; and
296
MOTIONS AND MAGNITUDES.
rAI2o m' ^ ^or example? be double to A G. There-
vfore, by reason of the likeness of the arches A B
Attempt for the aRd AC, the strait line AB will be double to the
finding, &C. strait line AC, and the radius AE double to the ra¬
dius A F, and the arch A B double to the arch A H.
And because the strait line AD is so bowed to be
coincident with the arch A B equal to it, as the
strait line A G is bowed to be coincident with the
arch A C equal also to it, the flexion of the strait
line A G into the crooked line A C will be equal to
the flexion of the strait line A D into the crooked
line A B. But the flexion of the strait line A D
into the crooked line A B is double to the flexion
of the strait line A G into the crooked line A H ;
and therefore the flexion of the strait line A G into
the crooked line A C is double to the flexion of
the same strait line AG into the crooked line
A H. Wherefore, as the arch A B is to the arch
A C or A H ; or as the radius A E is to the radius
A F; or as the chord A B is to the chord A C ; so
reciprocally is the flexion or uniform crookedness
of the arch A C, to the flexion or uniform crooked¬
ness of the arch A H, namely, here double. And
this may by the same method be demonstrated in
circles whose perimeters are to one another triple,
quadruple, or in whatsoever given proportion. The
crookedness therefore of two equal arches taken in
several circles are in proportion reciprocal to that
of their radii, or like arches, or like chords ; which
was to be demonstrated.
Let the square A B C D be again described (in
the third figure), and in it the quadrants A B D,
B C A and DAC; and dividing each side of the
square A B C D in the midst in E, F, G and H, let
dimension of a circle, etc. 29/
E G and F H be connected, which will cnt one an- pai^ hi.
other in the centre of the square at I, and divide -—r—-
the arch of the quadrant A B D into three equal ^empt°for the
parts in K and L. Also the diagonals A C and fillding> &c-
13 D being drawn will cut one another in I, and
divide the arches BKD and C L A into two equal
parts in M and N. Then with the radius B F let
the arch F E be drawn, cutting the diagonal B D
in 0 ; and dividing the arch B M in the midst in P,
let the strait line E a equal to the chord B P be set
off from the point E in the arch E F, and let the
arch a b be taken equal to the arch 0 a, and let
B a and B b be drawn and produced to the arch
AN in c and d; and lastly, let the strait line Ad
be drawn. I say the strait line A d is equal to the
arch AN or BM.
I have proved in the preceding article, that the
arch E 0 is twice as crooked as the arch B P, that
is to say, that the arch E 0 is so much more
crooked than the arch B P, as the arch B P is more
crooked than the strait line E a. The crookedness
therefore of the chord E «, of the arch B P, and of
the arch EO, are as 0, 1, 2. Also the difference
between the arches E O and E O, the difference
between the arches E O and E a, and the difference
between the arches E O and E b, are as 0, 1, 2. So
also the difference between the arches AN and
A N, the difference between the arches A N and
Ac, and the difference between the arches AN
and A d, are as 0, 1, 2 ; and the strait line Ac is
double to the chord B P or E a, and the strait line
A d double to the chord E b.
Again, let the strait line B F be divided in the
midst in Q,, and the arch B P in the midst in R ;
298 MOTIONS AND MAGNITUDES.
PAI2o IIL anc* describing the quadrant B Q S (whose arch
1—r—- Q S is a fourth part of the arch of the quadrant
JmpTfoTthu1 B M D, as the arch B R is a fourth part of the arch
finding, &c. B M, which is the arch of the semiquadrant ABM)
let the chord S e equal to the chord BR be set oif
from the point S in the arch S Q,; and let B e be
drawn and produced to the arch AN in f; which
being done, the strait line Ay will be quadruple to
the chord BR or S e. And seeing the crooked¬
ness of the arch S e, or of the arch A c, is double
to the crookedness of the arch B R, the excess of
the crookedness of the arch Af above the crook¬
edness of the arch A c will be subduple to the
excess of the crookedness of the arch A c above the
crookedness of the arch A N ; and therefore the
arch N c will be double to the arch cf. Wherefore
the arch c d is divided in the midst in J\ and the
arch N / is f of the arch N d. And in like manner
if the arch B R be bisected in V, and the strait
line BQ in X, and the quadrant B X Y be de¬
scribed, and the strait line Y g equal to the chord
B Y be set off from the point Y in the arch Y X,
it may be demonstrated that the strait line B"1
being drawn and produced to the arch A N, will
cut the arch f d into two equal parts, and that a
strait line drawn from A to the point of that sec¬
tion, will be equal to eight chords of the arch B V,
and so on perpetually ; and consequently, that the
strait line A d is equal to so many equal chords of
equal parts of the arch B M, as may be made by
infinite bisections. Wherefore the strait line A d
is equal to the arch B M or A N, that is, to half
the arch of the quadrant A B D or B C A.
Coroll. An arch being given not greater than
DIMENSION OF A CIRCLE., ETC. 299
the arch of a quadrant (for being made greater, it PAI^ in¬
comes again towards the radius B A produced, -—•—-
from which it receded before) if a strait line double ^pTfor^iie
to the chord of half the given arch be adapted finding, &c.
from the beginning of the arch, and by how much
the arch that is subtended by it is greater than the
given arch, by so much a greater arch be sub¬
tended by another strait line, this strait line shall
be equal to the first given arch.
Supposing the strait line BV (in fig. 1) be equal
to the arch of the quadrant BHD, and AV be
oonnected cutting the arch BHD in I, it may be
asked what proportion the arch BI has to the
arch I D. Let therefore the arch A Y be divided
in the midst in o, and in the strait line A D let
Ap be taken equal, and A q double to the drawn
chord A o. Then upon the centre A, with the
radius A q, let an arch of a circle be drawn cutting
the arch A Y in r, and let the arch Y r be doubled
at t; which being done, the drawn strait line A t
(by what has been last demonstrated) will be
equal to the arch AY. Again, upon the centre A
with the radius A t let the arch t u be drawn
cutting A D in u ; and the strait line A u will be
equal to the arch A Y. From the point u let the
strait line u s be drawn equal and parallel to the
strait line A B, cutting M N in x, and bisected by
MN in the same point x. Therefore the strait
line A x being drawn and produced till it meet
with B C produced in Y, it w ill cut off B Y double
to Bs, that is, equal to the arch BHD. Now let
the point, where the strait line A V cuts the arch
B H D, be I ; and let the arch D I be divided in
the midst in y; and in the strait line D C, let 1) £
300 MOTIONS AND MAGNITUDES.
rA12o 1IL k0 ta^ei1 equal, and D 8 double to the drawn chord
^—- D y ; and upon the centre D with the radius D $■
^mpTTo^the ^et an arch of a circle be drawn cutting the arch
finding, &c. BHD in the point n ; and let the arch n m be
taken equal to the arch I n ; which being done,
the strait line D m will (by the last foregoing
corollary) be equal to the arch D I. If now the
strait lines D m and C Y be equal, the arch BI
will be equal to the radius A B or B C ; and con¬
sequently X C being drawn, will pass through the
point I. Moreover, if the semicircle BHDS being
completed, the strait lines 61 and B I be drawn,
making a right angle (in the semicircle) at I, and
the arch B I be divided in the midst at i, it will
follow that A i being connected will be parallel to
the strait line 61, and being produced to B C in h,
will cut off the strait line B It equal to the strait
line k I, and equal also to the strait line A y cut
off in A D by the strait line 61. All which is
manifest, supposing the arch B I and the radius
B C to be equal.
But that the arch BI and the radius B C are
precisely equal, cannot (how true soever it be) be
demonstrated, unless that be first proved which is
contained in art. 1, namely, that the strait lines
drawn from X through the equal parts of 0 F
(produced to a certain length) cut off so many
parts also in the tangent B C severally equal to
the several arches cut off; which they do most
exactly as far as B C in the tangent, and BI in the
arch B E; insomuch that no inequality between
the arch BI and the radius B C can be discovered
either by the hand or by ratiocination. It is
therefore to be further enquired, whether the
dimension of a circle, etc.
301
strait line A V cut the arch of the quadrant in I part hi.
. . 20.
in the same proportion as the point C divides the -—^—-
strait line B V, which is equal to the arch of the
quadrant. But however this be, it has been de¬
monstrated that the strait line B Y is equal to the
arch BHD.
4. I shall now attempt the same dimension of a The thirdat-
L _ tempt; and.
circle another way, assuming the two following some things
.. propounded
lemmas. to be further
Lemma i. If to the arch of a quadrant, and the searchedint0-
radius, there be taken in continual proportion a
third line Z ; then the arch of the semiquadrant,
half the chord of the quadrant, and Z, will also be
in continual proportion.
For seeing the radius is a mean proportional
between the chord of a quadrant and its semi-
chord, and the same radius a mean proportional
between the arch of the quadrant and Z, the
square of the radius will be equal as well to the
rectangle made of the chord and semichord of the
quadrant, as to the rectangle made of the arch of
the quadrant and Z; and these two rectangles
will be equal to one another. Wherefore, as the
arch of a quadrant is to its chord, so reciprocally
is half the chord of the quadrant to Z. But as the
arch of the quadrant is to its chord, so is half the
arch of the quadrant to half the chord of the
quadrant. Wherefore, as half the arch of the
quadrant is to half the chord of the quadrant (or
to the sine of 45 degrees), so is half the chord of
the quadrant to Z ; which was to be proved.
Lemma n. The radius, the arch of the semi-
quadrant, the sine of 45 degrees, and the semi-
radius, are proportional.
302 MOTIONS AND MAGNITUDES.
PAI2o 111 ^or see^nS the sine °f 45 degrees is a mean
s—r—- proportional between the radius and the semi-
attempt,d&c. radius; and the same sine of 45 degrees is also a
mean proportional (by the precedent lemma) be¬
tween the arch of 45 degrees and Z; the square
of the sine of 45 degrees will be equal as well to
the rectangle made of the radius and semiradius,
as to the rectangle made of the arch of 45 degrees
and Z. Wherefore, as the radius is to the arch of
45 degrees, so reciprocally is Z to the semiradius;
which wras to be demonstrated.
Let now AB CD (in fig. 4) be a square ; and
with the radii AB, BC and DA, let the three
quadrants ABD, B C A and D A C, be described ;
and let the strait lines E F and G H, drawn parallel
to the sides BC and AB, divide the square A BCD
into four equal squares. They will therefore cut
the arch of the quadrant ABD into three equal
parts in I and K, and the arch of the quadrant
BCA into three equal parts in K and L. Also let
the diagonals A C and B D be drawn, cutting the
arches BID and A L C in M and N. Then upon
the centre H with the radius H F equal to half
the chord of the arch B M D, or to the sine of 45
degrees, let the arch FO be drawn cutting the
arch C K in O; and let A O be drawn and pro¬
duced till it meet with B C produced in P; also
let it cut the arch B M D in Q, and the strait line
D C in R. If now the strait line H Q, be equal to
the strait line D R, and being produced to D C in
S, cut off D S equal to half the strait line B P ; I
say then the strait line B P will be equal to the
arch BMD.
For seeing P B A and A D R are like triangles,
DIMENSION OF A CIRCLE, ETC. 303
it will be as P B to the radius B A or A D, so A D PAR^ IIL
to D II; and therefore as well P B, AD and D R, ——•—'
as P B, A D (or A Q) and Q, H are in continual ^empt'&c.
proportion ; and producing HO to D C in T, DT
will be equal to the sine of 45 degrees, as shall by
and by be demonstrated. Now D S, D T and D R
are in continual proportion by the first lemma;
and by the second lemma D C. D S :: D R. DF are
proportionals. And thus it will be, wdiether B P
be equal or not equal to the arch of the quadrant
B M D. But if they be equal, it will then be, as
that part of the arch B M D which is equal to the
radius, is to the remainder of the same arch BMD;
so A a to H a, or so B C to C P. And then w ill
B P and the arch B M D be equal. But it is not
demonstrated that the strait lines H Q, and D R
are equal; though if from the point B there be
drawn (by the construction of fig. 1) a strait line
equal to the arch BMD, then D R to HQ-, and
also the half of the strait line B P to D S, will
always be so equal, that no inequality can be dis¬
covered between them. I will therefore leave this
to be further searched into. For though it be
almost out of doubt, that the strait line B P and
the arch BMD are equal, yet that may not be
received without demonstration; and means of
demonstration the circular line admitteth none
that is not grounded upon the nature of flexion, or
of angles. But by that way I have already exhi¬
bited a strait line equal to the arch of a quadrant
in the first and second aggression.
It remains that I prove D T to be equal to the
sine of 45 degrees.
304
MOTIONS AND MAGNITUDES.
PAI2o IIL In B A produced let A V be taken equal to the
v—■■—- sine of 45 degrees ; and drawing and producing
attempt'&c. V H, it will cut the arch of the quadrant C N A in
the midst in N, and the same arch again in O, and
the strait line D C in T, so that D T will be equal
to the sine of 45 degrees, or to the strait line A V;
also the strait line Y H will be equal to the strait
line H I, or the sine of 60 degrees.
For the square of A V is equal to two squares of
the semiradius ; and consequently the square of
Y H is equal to three squares of the semiradius.
But HI is a mean proportional between the semi¬
radius and three semiradii; and, therefore, the
square of H I is equal to three squares of the semi¬
radius. Wherefore HI is equal to H V. But
because A D is cut in the midst in H, therefore Y H
and H T are equal; and, therefore, also D T is
equal to the sine of 45 degrees. In the radius
B A let B X be taken equal to the sine of 45 de¬
grees ; for so Y X will be equal to the radius; and
it will be as Y A to A H the semiradius, so Y X the
radius to X N the sine of 45 degrees. Wherefore
Y H produced passes through. N. Lastly, upon the
centre Y with the radius Y A let the arch of a circle
be drawn cutting V H in Y; which being done,
Y Y will be equal to H O (for H O is, by construc¬
tion, equal to the sine of 45 degrees) and YH will
be equal to OT; and, therefore, VT passes through
O. All which was to be demonstrated.
I will here add certain problems, of which if
any analyst can make the construction, he will
thereby be able to judge clearly of what I have now
said concerning the dimension of a circle. Now
DIMENSION OF A CIRCLE, ETC.
305
these problems are nothing else (at least to sense) part iit.
but certain symptoms accompanying the construe- -——-
tion of the first and third figure of this chapter. attemptrd&c
Describing, therefore, again, the square A B C D
(in fig. 5) and the three quadrants A B D, B C A
and DAC, let the diagonals AC and BD be drawn,
cutting the arches BHD and CI A in the middle
in H and I; and the strait lines E F and G L, di¬
viding the square A B C D into four equal squares,
and trisecting the arches BHD and CIA, namely,
B H D in K and M, and CIA in M and 0. Then
dividing the arch B K in the midst in P, let Q, P
the sine of the arch B P, be drawn and produced to
R, so that Q, R be double to Gl P; and, connecting
K R, let it be produced one way to B C in S, and
the other way to B A produced in T. Also let B Y
be made triple to B S, and consequently ^ (by the
second article of this chapter) equal to the arch
B D. This construction is the same with that of
the first figure., which I thought fit to renew dis¬
charged of all lines but such as are necessary for my
present purpose.
In the first place, therefore, if A V be drawn,
cutting the arch B H D in X, and the side D C in
Z, I desire some analyst would, if he can, give a
reason why the strait lines T E and T C should cut
the arch B D, the one in Y, the other in X, so as
to make the arch B Y equal to the arch Y X ; or if
they be not equal, that he would determine their
difference.
Secondly, if in the side D A, the strait line D a
be taken equal to D Z, and V a be drawn; why
V a and Y B should be equal; or if they be not
equal, what is the difference.
VOL. I. x
306
MOTIONS AND MAGNITUDES.
PAR2o In* Thirdly, drawing Z b parallel and equal to the
v—y—' side C B, cutting the arch B H D in c, and draw-
Jttempt'r&c. ing the strait line A c, and producing it to B Y in
d; why A d should be equal and parallel to the
strait line a V, and consequently equal also to the
arch BD.
Fourthly, drawing e K the sine of the arch B K,
and taking (in e A produced) ef equal to the dia¬
gonal A C, and connecting f C ; why f C should
pass through a (which point being given, the length
of the arch B H D is also given) and c; and why
f e and f c should be equal; or if not, why un¬
equal.
Fifthly, drawing jfZ, I desire he would show,
why it is equal to B V, or to the arch BD; or if
they be not equal, what is their difference.
Sixthly, granting f Z to be equal to the arch
B D, I desire he would determine whether it fall
all without the arch B C A, or cut the same, or
touch it, and in what point.
Seventhly, the semicircle BD^ being completed,
why g I being drawn and produced, should pass
through X, by which point X the length of the
arch B D is determined. And the same g I being
yet further produced to D C in h, why A d, which
is equal to the arch B D, should pass through that
point h.
Eighthly, upon the centre of the square A BCD,
which let be k, the arch of the quadrant E i L being
drawn, cutting e K produced in i, why the drawn
strait line i X should be parallel to the side C D.
Ninthly, in the sides B A and B C taking g I
and B m severally equal to half B V, or to the arch
B H, and drawing m n parallel and equal to the
DIMENSION OF A CIRCLE, ETC.
307
side B A, cutting the arch B D in o, why the strait part nr.
line which connects V I should pass through the ——-
noint o The third
PUllltW* _ _ attempt, &c.
Tenthly, I would know of him why the strait
line which connects a H should be equal to B m ;
or if not, how much it differs from it.
The analyst that can solve these problems with¬
out knowing first the length of the arch B D, or
using any other known method than that which
proceeds by perpetual bisection of an angle, or is
drawn from the consideration of the nature of
flexion, shall do more than ordinary geometry is
able to perform. But if the dimension of a circle
cannot be found by any other method, then I have
either found it> or it is not at all to be found.
From the known length of the arch of a quad¬
rant, and from the proportional division of the arch
and of the tangent B C, may be deduced the sec¬
tion of an angle into any given proportion ; as also
the squaring of the circle, the squaring of a given
sector, and many the like propositions, which it is
not necessary here to demonstrate. I will, there¬
fore, only exhibit a strait line equal to the spiral of
Archimides, and so dismiss this speculation.
5. The length of the perimeter of a circle being ^eSp-1r^t0i^°f
found, that strait line is also found, which touches chimedes with
a spiral at the end of its first conversion. For upona strait
the centre A (in fig. 6) let the circle B C D E be de¬
scribed ; and in it let Archimedes' spiral A F G H B
be drawn, beginning at A and ending at B. Through
the centre A let the strait line C E be drawn, cut¬
ting the diameter B D at right angles; and let it be
produced to I, so that A I be equal to the perimeter
B C D E B. Therefore I B being drawn will touch
x 2
308
motions and magnitudes.
part hi. the spiral A F G H B in B; which is demonstrated
v^— by Archimedes in his book De Spiralibus.
JAhe^phcTof And for a strait line eqnal to the given spiral
Archimedes A F G H B, it may be found thus.
mthastraitline . .
Let the strait line A I, which is equal to the pe¬
rimeter B C D E, be bisected in K; and taking K L
equal to the radius A B, let the rectangle I L be
completed. Let M L be understood to be the axis,
and K L the base of a parabola, and let M K be
the crooked line thereof. Now if the point M be
conceived to be so moved by the concourse of two
movents, the one from I M to K L with velocity
encreasing continually in the same proportion with
the times, the other from ML to I K uniformly,
that both those motions begin together in M and
end in K; Galilseus has demonstrated that by such
motion of the point M, the crooked line of a para¬
bola will be described. Again, if the point A be
conceived to be moved uniformly in the strait line
A B, and in the same time to be carried round
upon the centre A by the circular motion of all the
points between A and B ; Archimedes has demon¬
strated that by such motion will be described a
spiral line. And seeing the circles of all these mo¬
tions are concentric in A ; and the interior circle
is always less than the exterior in the proportion
of the times in which A B is passed over with uni¬
form motion ; the velocity also of the circular mo¬
tion of the point A will continually increase pro¬
portionally to the times. And thus far the gene¬
rations of the parabolical line M K, and of the spiral
line A F G H B, are like. But the uniform motion
in A B concurring with circular motion in the peri¬
meters of all the concentric circles, describes that
dimension of a circle., etc.
309
circle, whose centre is A, and perimeter B C D E ; pa12q iil
and, therefore, that circle is (by the coroll. of art. •—r—-
1, chap, xyi) the aggregate of all the velocities to- JfhtheqspirLinof
o-ether taken of the point A whilst it describes the Archimedes
~ r with a strait line
spiral A F G H B. Also the rectangle I K L M is
the aggregate of all the velocities together taken
of the point M, whilst it describes the crooked line
M K. And, therefore the whole velocity by which
the parabolical line M K is described, is to the
whole velocity with which the spiral line AFGHB
is described in the same time, as the rectangle
IK L M is to the circle B C DE, that is to the
triangle A I B. But because AI is bisected in K,
and the strait lines I M and A B are equal, there¬
fore the rectangle I K L M and the triangle A I B
are also equal. Wherefore the spiral line AFGHB,
and the parabolical line M K, being described with
equal velocity and in equal times, are equal to one
another. Now, in the first article of chap, xviii, a
strait line is found out equal to any parabolical
line. Wherefore also a strait line is found out equal
to a given spiral line of the first revolution described
by Archimedes ; which was to be done.
6. In the sixth chapter, which is of Method, 0f the analysis
that which I should there have spoken of the ana- cians by the
lytics of geometricians I thought fit to defer, be-povversofhnes'
cause I could not there have been understood, as
not having then so much as named lines, superfi¬
cies, solids, equal and unequal, 8pc. Wherefore I
will in this place set down my thoughts concern¬
ing it.
Analysis is continual reasoning from the defini¬
tions of the terms of a proposition we suppose
true, and again from the definitions of the terms of
310
motions and magnitudes.
part iii. those definitions, and so on, till we come to some
20.
—r—- tilings known, the composition whereof is the
«?/ geometn-S1S demonstration of the truth or falsity of the first
dans by the supposition ; and tliis composition or demonstration
powers of lines. 1 1 1
is that we call Synthesis. Analytica, therefore, is
that art, by which onr reason proceeds from some¬
thing supposed, to principles, that is, to prime
propositions, or to such as are known by these, till
we have so many known propositions as are suffi¬
cient for the demonstration of the truth or falsity
of the thing supposed. Synthetica is the art itself
of demonstration. Synthesis, therefore, and ana¬
lysis, differ in nothing, but in proceeding forwards
or backwards; and Logistica comprehends both.
So that in the analysis or synthesis of any question,
that is to say, of any problem, the terms of all the
propositions ought to be convertible ; or if they be
enunciated hypothetically, the truth of the conse¬
quent ought not only to follow out of the truth of
its antecedent, but contrarily also the truth of the
antecedent must necessarily be inferred from the
truth of the consequent. For otherwise, when by
resolution we are arrived at principles, we cannot
by composition return directly back to the thing
sought for. For those terms which are the first in
analysis, will be the last in synthesis; as for ex¬
ample, when in resolving, we say, these two
rectangles are equal, and therefore their sides are
reciprocally proportional, we must necessarily in
compounding say, the sides of these rectangles are
reciprocally proportional, and therefore the rect¬
angles themselves are equal; wThich we could not
say, unless rectangles have their sides recipro-
DIMENSION OF A CIRCLE, ETC. 311
call// proportional, and rectangles are equal, part ht.
were terms convertible. -—r—-
Now in every analysis, that which is sought is
the proportion of two quantities; by which pro-ciansby<-he
11 . -ii i • powers of lines.
portion, a figure being described, the quantity
sought for may be exposed to sense. And this
exposition is the end and solution of the question,
or the construction of the problem.
And seeing analysis is reasoning from something
supposed, till we come to principles, that is, to
definitions, or to theorems formerly known; and
seeing the same reasoning tends in the last place to
some equation, we can therefore make no end of re¬
solving, till we come at last to the causes themselves
of equality and inequality, or to theorems formerly
demonstrated from those causes; and so have a
sufficient number of those theorems for the demon¬
stration of the thing sought for.
And seeing also, that the end of the analytics is
either the construction of such a problem as is pos¬
sible, or the detection of the impossibility thereof;
whensoever the problem may be solved, the analyst
must not stay, till he come to those things which
contain the efficient cause of that whereof he is to
make construction. But he must of necessity stay,
when he comes to prime propositions; and these
are definitions. These definitions therefore must
contain the efficient cause of his construction; I
say of his construction, not of the conclusion which
he demonstrates; for the cause of the conclusion
is contained in the premised propositions ; that is
to say, the truth of the proposition he proves is
drawn from the propositions which prove the same.
312
motions and magnitudes.
part hi. But the cause of his construction is in the things
-—v—' themselves, and consists in motion., or in the con-
o^/geometri-S13 course of motions. Wherefore those propositions,
da'18 bythe in which analysis ends, are definitions, but such as
powers of lines. ^ J *
signify in what manner the construction or gene¬
ration of the thing proceeds. For otherwise, when
he goes back by synthesis to the proof of his
problem, he will come to no demonstration at all;
there being no true demonstration but such as is
scientifical; and no demonstration is scientifical,
but that which proceeds from the knowledge of the
causes from which the construction of the problem
is drawn. To collect therefore what has been said
into few words; analysis is ratiocination from
the supposed construction or generation of a thing
to the efficient cause or coefficient causes of that
which is constructed or generated. And syn¬
thesis is ratiocination from the first causes of
the construction, continued through all the middle
causes till we come to the thing itself which is
constructed or generated.
But because there are many means by which the
same thing may be generated, or the same problem
be constructed, therefore neither do all geometri¬
cians, nor doth the same geometrician always, use
one and the same method. For, if to a certain
quantity given, it be required to construct another
quantity equal, there may be some that will inquire
whether this may not be done by means of some
motion. For there are quantities, whose equality
and inequality may be argued from motion and
time, as well as from congruence; and there is
motion, by which two quantities, whether lines or
superficies, though one of them be crooked, the
DIMENSION OF A CIRCLE, ETC.
313
other strait, maybe made congruous or coincident. PAR2£m*
And this method Archimedes made use of in his -—r—-
book De Spircilibus. Also the equality or inequa- ®/g^oen^tar^sls
litv of two quantities may be found out and ciansbythe
J A • i • r» • l powers oi lines.
demonstrated from the consideration of weight, as
the same Archimedes did in his quadrature of the
parabola. Besides, equality and inequality are found
out often by the division of the two quantities into
parts which are considered as indivisable; as
Cavallerius Bonaventura has done in our time, and
Archimedes often. Lastly, the same is performed
by the consideration of the powers of lines, or the
roots of those powers, and by the multiplication,
division, addition, and subtraction, as also by the
extraction of the roots of those powers^ or by find¬
ing where strait lines of the same proportion
terminate. For example, when any number of
strait lines, how many soever, are drawn from a
strait line and pass all through the same point,
look what proportion they have, and if their parts
continued from the point retain everywhere the
same proportion, they shall all terminate in a strait
line. And the same happens if the point be taken
between two circles. So that the places of all their
points of termination make either strait lines, or
circumferences of circles, and are called plane
places. So also when strait parallel lines are
applied to one strait line, if the parts of the strait
line to which they are applied be to one another in
proportion duplicate to that of the contiguous
applied lines, they will all terminate in a conical
section ; which section, being the place of their
termination, is called a solid place ^ because it
serves for the finding out of the quantity of any
314
MOTIONS AND MAGNITUDES.
PA2o'111' e(lua^on which consists of three dimensions. There
-—v—' are therefore three ways of finding out the cause of
of'geo m e tr'i^ equality or inequality between two given quantities;
cians by the namely, first, by the computation of motions ; for
powers of lines, J . .
by equal motion, and equal time, equal spaces are
described; and ponderation is motion. Secondly,
by indivisibles: because all the parts together
taken are equal to the whole. And thirdly, by the
powers: for when they are equal, their roots also
are equal; and contrarily, the powers are equal,
when their roots are equal. But if the question
be much complicated, there cannot by any of these
ways be constituted a certain rule, from the sup¬
position of which of the unknown quantities the
analysis may best begin; nor out of the variety of
equations, that at first appear, which we were
best to choose; but the success will depend upon
dexterity, upon formerly acquired science, and
many times upon fortune.
For no man can ever be a good analyst without
being first a good geometrician ; nor do the rules
of analysis make a geometrician, as synthesis doth;
which begins at the very elements, and proceeds
by a logical use of the same. For the true teaching
of geometry is by synthesis, according to Euclid's
method ; and he that hath Euclid for his master,
may be a geometrician without Yieta, though Yieta
was a most admirable geometrician; but he that
has Yieta for his master, not so, without Euclid.
And as for that part of analysis which works by
the powers, though it be esteemed by some geo¬
metricians, not the chiefest, to be the best way of
solving all problems, yet it is a thing of no great
extent; it being all contained in the doctrine of
DIMENSION OF A CIRCLE, ETC. 315
rectangles, and rectangled solids. So that although PA^ IIL
they come to an equation which determines the v—■—'
quantity sought, yet they cannot sometimes by ^/geometri-81"
art exhibit that quantity in a plane, but in some cians
conic section; that is, as geometricians say, not
geometrically, but mechanically. Now such pro¬
blems as these, they call solid; and when they
cannot exhibit the quantity sought for with the
help of a conic section, they call it a lineary pro¬
blem. And therefore in the quantities of angles,
and of the arches of circles, there is no use at all
of the analytics which proceed by the powers ; so
that the ancients pronounced it impossible to ex¬
hibit in a plane the division of angles, except
bisection, and the bisection of the bisected parts,
otherwise than mechanically. For Pappus, (before
the 31st proposition of his fourth book) distin¬
guishing and defining the several kinds of pro¬
blems, says that " some are plane, others solid,
and others lineary. Those, therefore, which may
be solved by strait lines and the circumferences of
circles, (that is, which may be described with the
rule and compass, without any other instrument),
are fitly called plane; for the lines, by which
such problems are found out, have their generation
in a plane. But those which are solved by the
using of some one or more conic sections in their
construction, are called solid, because their con¬
struction cannot be made without using the super¬
ficies of solid figures, namely, of cones. There
remains the third kind, which is called lineary,
because other lines besides those already mentioned
are made use of in their construction, &c." And a
316
MOTIONS AND MAGNITUDES.
PAR2o IIL a^er he says, " of this kind are the spiral
-—t—' lines, the qua dratr ices, the conchoeides, and the
llveonfetri-'818 cissoeides. And geometricians think it no small
dans by the fault, when for the fin ding out of a plane problem
powers ei lines. • v " at
any man makes use of conies, or new lines. Now
he ranks the trisection of an angle among solid
problems, and the quinquesection among lineary.
But what! are the ancient geometricians to be
blamed, who made use of the quadratrix for the
finding out of a strait line equal to the arch of a
circle ? And Pappus himself, was he faulty, when
he found out the trisection of an angle by the
help of an hyperbole ? Or am I in the wrong,
who think I have found out the construction of
both these problems by the rule and compass only?
Neither they, nor I. For the ancients made use
of this analysis which proceeds by the powers;
and with them it was a fault to do that by a more
remote power, which might be done by a nearer;
as being an argument that they did not sufficiently
understand the nature of the thing.
The virtue of this kind of analysis consists in the
changing and turning and tossing of rectangles and
analogisms; and the skill of analysts is mere logic, by
which they are able methodically to find out whatso¬
ever lies hid either in the subject or predicate of the
conclusion sought for. But this doth not properly
belong to algebra, or the analytics specious, sym¬
bolical, or cossick ; which are, as I may say, the
brachygraphy of the analytics, and an art neither
of teaching nor learning geometry, but of register¬
ing with brevity and celerity the inventions of
geometricians. For though it be easy to discourse
of circular motion.
31/
by symbols of very remote propositions; yet PA^T IIL
whether such discourse deserve to be thought very -—^—■"
profitable, when it is made without any ideas of
the things themselves, I know not.
CHAPTER XXL
of circular motion.
]. In simple motion, every strait line taken in the body moved
is so carried, that it is always parallel to the places in which it
formerly was.-—2. If circular motion be made about a resting
centre, and in that circle there be an epicycle, whose revolution
is made the contrary way, in such manner that in equal
times it make equal angles, every strait line taken in that
epicycle will be so carried, that it will always be parallel to the
places in which it formerly was.—3. The properties of simple
motion.—4. If a fluid be moved with simple circular motion,
all the points taken in it will describe their circles in times
proportional to the distances from the centre.—5. Simple
motion dissipates heterogeneous and congregates homogeneous
bodies.—6. If a circle made by a movent moved with simple
motion be commensurable to another circle made by a point
which is carried about by the same movent, all the points of
both the circles will at some time return to the same situation.
7. If a sphere have simple motion, its motion will more
dissipate heterogeneous bodies by how much it is more remote
from the poles.—8. If the simple circular motion of a fluid
body be hindered by a body which is not fluid, the fluid body
will spread itself upon the superficies of that body.—9. Cir¬
cular motion about a fixed centre casteth off by the tangent
such things as lie upon the circumference and stick not to it.
10. Such things, as are moved with simple circular motion,
beget simple circular motion.—11. If that which is so moved
have one side hard and the other side fluid, its motion will not
be perfectly circular.
1. I have already defined simple motion to beinsimple
,i . i-i-i i • i ln°tion, &c,
that, m which the several points taken m a moved
318
MOTIONS AND MAGNITUDES.
PA 21ITI* body 'm several equal times describe several
—■—' equal arches. And therefore in simple circular
moti'onfevery motion it is necessary that every strait line taken
taken h" the the moved body be always carried parallel to
body moved itself; which I thus demonstrate.
is so carried,
that it is always First, let A B (in the first figure) be any strait
theTpiaces line taken in any solid body ; and let A D be any
formerly was. arc^ drawn upon any centre C and radius CA.
Let the point B be understood to describe towards
the same parts the arch B E, like and equal to the
arch AD. Now in the same time in which the
point A transmits the arch AD, the point B,
which by reason of its simple motion is supposed
to be carried with a velocity equal to that of A,
will transmit the arch B E ; and at the end of
the same time the whole A B will be in D E ; and
therefore A B and D E are equal. And seeing the
arches AD and BE are like and equal, their subtend¬
ing strait lines AD and BE will also be equal; and
therefore the four-sided figure A B D E will be a
parallelogram. Wherefore A B is carried parallel
to itself. And the same may be proved by the
same method, if any other strait line be taken in
the same moved body in which the strait line A B
was taken. So that all strait lines, taken in a
body moved with simple circular motion, will be
carried parallel to themselves.
Coroll. i. It is manifest that the same will
also happen in any body which hath simple motion,
though not circular. For all the points of any
strait line whatsoever will describe lines, though
not circular, yet equal; so that though the crooked
lines A D and B E were not arches of circles, but
of parabolas, ellipses, or of any other figures,
OF CIRCULAR MOTION.
319
yet both they, and their subtenses, and the strait PAR^IIL
lines which join them, would be equal and parallel. ->■—'
Coroll. ii. It is also manifest, that the radii
of the equal circles A D and B E, or the axis of a
sphere, will be so carried, as to be always parallel
to the places in which they formerly were. For
the strait line B F drawn to the centre of the arch
B E being equal to the radius A C, will also be
equal to the strait line FE or CD ; and the angle
B F E will be equal to the angle A C D. Now the
intersection of the strait lines C A and B E being
at G, the angle C G E (seeing B E and A D are
parallel) will be equal to the angle DAC. But
the angle E B F is equal to the same angle DAC;
and therefore the angles C G E and E B F are also
equal. Wherefore A C and B F are parallel;
which was to be demonstrated.
2. Let there be a circle given (in the second If circular
figure) whose centre is A, and radius A B ; and made about a
upon the centre B and any radius B C let the aSTn tha?10'
epicycle C D E be described. Let the centre B circle.the?'e be
1 J an epicycle
be understood to be carried about the centre A, whose revoiu-
and the whole epicycle with it till it be coincident the contrary
with the circle FGH, whose centre is I; and let Sne" that in
B AI be any angle given. But in the time that equ1al time?il
J # ° ° make equal
the centre B is moved to I, let the epicycle C D E angles, every
1 , . . strait line
nave a contrary revolution upon its own centre, taken in that
namely from E by D to C, according to the samebfsocarded,
proportions ; that is, in such manner, that in both
the circles, equal angles be made in equal times. parallel to
I say E C, the axis of the epicycle, will be always in which it
carried parallel to itself. Let the angle FIG be former'y was*
made equal to the angle B AI ; IF and A B will
320
MOTIONS AND MAGNITUDES.
par2Ti m* therefore be parallel; and how much the axis
—*—' A G has departed from its former place A C (the
motion" &c. measure of which progression is the angle CAG,
or C B D, which I suppose equal to it) so much in
the same time has the axis IG, the same with B C,
departed from its own former situation. Where¬
fore, in what time B C comes to I G by the motion
from B to I upon the centre A, in the same time
G will come to F by the contrary motion of the
epicycle ; that is, it will be turned backwards to
F, and I G will lie in IF. But the angles FIG
and G A C are equal; and therefore A C, that is,
B C, and I F, (that is the axis, though in different
places) will be parallel. Wherefore, the axis of
the epicycle E D C will be carried always parallel
to itself; which was to be proved.
Coroll. From hence it is manifest, that those
two annual motions which Copernicus ascribes
to the earth, are reducible to this one circular
simple motion, by which all the points of the
moved body are carried always with equal velocity,
that is, in equal times they make equal revolutions
uniformly.
This, as it is the most simple, so it is the most
frequent of all circular motions ; being the same
which is used by all men when they turn anything
round with their arms, as they do in grinding or
sifting. For all the points of the thing moved
describe lines which are like and equal to one
another. So that if a man had a ruler, in which
many pens' points of equal length were fastened,
he might with this one motion write many lines
at once.
OF CIRCULAR MOTION.
321
3. Having shown what simple motion is, I will part nr.
• 21
here also set down some properties of the same. v—^—-
First, when a body is moved with simple motion Properties of
. simple motion:
in a fluid medium which hath no vacuity, it changes
the situation of all the parts of the fluid ambient
which resist its motion ; I say there are no parts
so small of the fluid ambient, how far soever it be
continued, but do change their situation in such
manner, as that they leave their places continually
to other small parts that come into the same.
For (in the same second figure) let any body,
as KLM N, be understood to be moved with
simple circular motion; and let the circle, which
every point thereof describes, have any deter¬
mined quantity, suppose that of the same K L M N.
Wherefore the centre A and every other point,
and consequently the moved body itself, will be
carried sometimes towards the side where is K,
and sometimes towards the other side where
is M. When therefore it is carried to K, the
parts of the fluid medium on that side will go
back ; and, supposing all space to be full, others
on the other side will succeed. And so it will be
when the body is carried to the side M, and to N,
and every way. Now when the nearest parts of
the fluid medium go back, it is necessary that the
parts next to those nearest parts go back also ;
and supposing still all space to be full, other parts
will come into their places with succession perpe¬
tual and infinite. Wlierefore all, even the least
parts of the fluid medium, change their places, &c.
Which was to be proved.
It is evident from hence, that simple motion,
whether circular or not circular, of bodies which
VOL. I. Y
322 motions and magnitudes.
part nr. make perpetual returns to their former places,
v- liatli greater or less force to dissipate the parts of
resisting bodies, as it is more or less swift, and as
the lines described have greater or less magnitude.
Now the greatest velocity that can be, may be
understood to be in the least circuit, and the least
in the greatest; and may be so supposed, when
there is need.
if a fluid be 4, Secondly, supposing: the same simple motion
moved with . ° .
simple circular in the air, water, or other fluid medium ; the
po^ntTt'aken'in parts of the medium, which adhere to the moved
it will describe ])0({y ])(. carried about with the same motion
their circles in J J
times propor- and velocity, so that in what time soever any point
tional to the _ -
distances from oi the movent finishes its circle, in the same time
the centre. every part of the medium, which adheres to the
movent, shall also describe such a part of its
circle, as is equal to the whole circle of the
movent; I say, it shall describe a part, and not
the whole circle, because all its parts receive their
motion from an interior concentric movent, and of
concentric circles the exterior are always greater
than the interior ; nor can the motion imprinted
by any movent be of greater velocity than that of
the movent itself. From whence it follows, that
the more remote parts of the fluid ambient shall
finish their circles in times, which have to one
another the same proportion with their distances
from the movent. For every point of the fluid
ambient, as long as it toucheth the body which
carries it about, is carried about with it, and would
make the same circle, but that it is left behind so
much as the exterior circle exceeds the interior.
So that if we suppose some thing, which is not fluid,
to float in that part of the fluid ambient which is
of circular motion.
323
nearest to the movent, it will together with the partiii.
21.
movent be carried about. Now that part of the —r—-
fluid ambient, which is not the nearest but almost
the nearest, receiving its degree of velocity from
the nearest, which degree cannot be greater than
it was in the giver, doth therefore in the same
time make a circular line, not a whole circle, yet
equal to the whole circle of the nearest. There¬
fore in the same time that the movent describes its
circle, that which doth not touch it shall not
describe its circle ; yet it shall describe such a part
of it, as is equal to the whole circle of the movent.
And after the same manner, the more remote parts
of the ambient will describe in the same time such
parts of their circles, as shall be severally equal to
the whole circle of the movent; and, by consequent,
they shall finish their whole circles in times pro¬
portional to their distances from the movent;
which was to be proved.
5. Thirdly, the same simple motion of a body simple motion
i -i. n*i i" i dissipates hete-
placed m a fluid medium, congregates or gathers rogeneous and
into one place such things as naturally float in that homogfifeous
medium, if they be homogeneous ; and if they be bodies-
heterogeneous, it separates and dissipates them.
But if such things as be heterogeneous do not
float, but settle, then the same motion stirs and
mingles them disorderly together. For seeing
bodies, which are unlike to one another, that is,
heterogeneous bodies, are not unlike in that they
are bodies; for bodies, as bodies, have no differ¬
ence ; but only from some special cause, that is,
trom some internal motion, or motions of their
smallest parts (for I have shown in chap, ix, art. 9,
that all mutation is such motion), it remains that
y 2
324
motions and magnitudes.
partiii. heterogeneous bodies have their unlikeness or
v—*—' difference from one another from their internal or
dissip'ateThe'te- specifical motions. Now bodies which have such
rogeneous, &c. difference receive unlike and different motions
from the same external common movent; and
therefore they will not be moved together, that is
to say, they will be dissipated. And being dissi¬
pated they will necessarily at some time or other
meet with bodies like themselves, and be moved
alike and together with them; and afterwards
meeting with more bodies like themselves, they
will unite and become greater bodies. Wherefore
homogeneous bodies are congregated, and hetero¬
geneous dissipated by simple motion in a medium
where they naturally float. Again, such as being
in a fluid medium do not float, but sink, if the
motion of the fluid medium be strong enough,
will be stirred up and carried away by that motion,
and consequently they will be hindered from re¬
turning to that place to which they sink naturally,
and in which only they would unite, and out of
which they are promiscuously carried; that is,
they are disorderly mingled.
Now this motion, by which homogeneous bodies
are congregated and heterogeneous are scattered,
is that which is commonly called fermentation,
from the Latin fewer e ; as the Greeks have their
Zup?, which signifies the same, from Ztw ferveo.
For seething makes all the parts of the water
change their places ; and the parts of any thing,
that is thrown into it, will go several ways ac¬
cording to their several natures. And yet all
fervour or seething is not caused by fire; for new
wine and many other things have also their fer-
OF CIRCULAR MOTION.
325
mentation and fervour, to which fire contributes PAI^Iir*
little, and sometimes nothing. But when in fer- 1——'
mentation we find heat, it is made by the fer¬
mentation.
0. Fourthly, in what time soever the movent, [fa'cirdemade
7 _ by a movent
whose centre is A (in fig. 2) moved in K L N, shall, moved with
by any number of revolutions, that is, when the J1"
commensu-
perimeters BI and K L N be commensurable, have
described a line equal to the circle which passes made bya point
x 9 • n which is car-
tlirough the points B and I; in the same time all ried about by
the points of the floating body, whose centre is B, ven^aTAhT0"
shall return to have the same situation in respect
of the movent, from which they departed. Forat some tir?e
J *- return to the
seeing it is as the distance B A, that is, as the same situation,
radius of the circle which passes through BI is to
the perimeter itself B I, so the radius of the circle
K L N is to the perimeter K L N ; and seeing the
velocities of the points B and K are equal, the
time also of the revolution in I B to the time of
one revolution in K L N, will be as the perimeter
BI to the perimeter KLN; and therefore so
many revolutions in K L N, as together taken are
equal to the perimeter B I, will be finished in the
same time in which the whole perimeter B I is
finished ; and therefore also the points L, N, F
and H, or any of the rest, will in the same time
return to the same situation from which they de¬
parted ; and this may be demonstrated, whatsoever
be the points considered. Wherefore all the points
shall in that time return to the same situation ;
which was to be proved.
From hence it follows, that if the perimeters BI
and L K N be not commensurable, then all the
326
MOTIONS AND MAGNITUDES.
m- points will never return to have the same situation
v—r—— or configuration in respect of one another,
w sample 7. In simple motion, if the body moved be of a
motion, its mo- spherical figure, it liatli less force towards its poles
tion will more . . . •,
dissipate hete- than towards its middle to dissipate heteroge-
rogeneous bo- , i v
dies by how neons, or to congregate homogeneous bodies.
™mtteVrom°ie Let there he a sphere (as in the third figure)
the poles. whose centre is A and diameter B C ; and let it be
conceived to be moved with simple circular motion;
of which motion let the axis be the strait line D E,
cutting the diameter B C at right angles in A. Let
now the circle, which is described by any point B
of the sphere, have B F for its diameter ; and taking
F G equal to B C, and dividing it in the middle in
H, the centre of the sphere A will, when half a
revolution is finished, lie in H. And seeing HF
and A B are equal, a circle described upon the
centre H with the radius HF or IIG, will be equal
to the circle whose centre is A and radius AB.
And if the same motion be continued, the point B
will at the end of another half revolution return to
the place from whence it began to be moved; and
therefore at the end of half a revolution, the point B
will be carried to F, and the whole hemisphere D BE
into that hemisphere in which are the points L, K
and F. Wherefore that part of the fluid medium,
which is contiguous to the point F, will in the same
time go back the length of the strait line B F; and
in the return of the point F to B, that is, of G to C,
the fluid medium will go back as much in a strait
line from the point C. And this is the effect of
simple motion in the middle of the sphere, where
the distance from the poles is greatest. Let now
the point I be taken in the same sphere nearer to
OF CIRCULAR MOTION.
32 7
the pole E, and through it let the strait line IK be part m-
drawn parallel to the strait line B F, cutting the -—■—'
arch F L in K, and the axis H L in M ; then con¬
necting H K, upon H F let the perpendicular Iv N
be drawn. In the same time therefore that B
comes to F the point I will come to Iv, B F and
I Iv being equal and described with the same velo¬
city. Now the motion in I K to the fluid medium
upon which it works, namely, to that part of the
medium which is contiguous to the point K, is
oblique, whereas if it proceeded in the strait line
H K it would be perpendicular ; and therefore
the motion which proceeds in I K has less power
than that which proceeds in H K with the same
velocity. But the motions in H K and H F do
equally thrust back the medium: and therefore
the part of the sphere at K moves the medium
less than the part at F, namely, so much less as
K N is less than H F. Wherefore also the same
motion hatli less power to disperse heterogeneous,
and to congregate homogeneous bodies, when it is
nearer, than when it is more remote from the
poles; which was to be proved.
Coroll. It is also necessary, that in planes which
are perpendicular to the axis, and more remote
than the pole itself from the middle of the sphere,
this simple motion have no effect. For the axis
D E with simple motion describes the superficies of
a cylinder; and towards the bases of the cylinder
there is in this motion 110 endeavour at all.
8. If in a fluid medium moved about, as hath J.f a ®imPIe.
. . . circular motion
been said, with simple motion, there be conceived °f a fluid body
to float some other spherical body which is not fluid, a body'wEcm!
the parts of the medium, which are stopped by that not fluid'the
328 MOTIONS AND MAGNITUDES.
PAI^FIIT* body, will endeavour to spread themselves every
—-—' way upon tlie superficies of it. And this is manifest
fpnidTtteifVl11 enough by experience, namely, by the spreading
perfici'cs6 of1" water poured out upon a pavement. But the
that body. reason of it may be this. Seeing the sphere A (in
fig. 3) is moved towards B, the medium also in
which it is moved will have the same motion. But
because in this motion it falls upon a body not
liquid, as G, so that it cannot go on ; and seeing
the small parts of the medium cannot go forwards,
nor can they go directly backwards against the
force of the movent; it remains, therefore, that
they diffuse themselves upon the superficies of that
body, as towards 0 and P; which was to be
proved.
Circular mo- g Compounded circular motion, in which all the
tion about a 1 7
fixed centre parts of the moved body do at once describe cir-
casteth oft' by
the tangent cumierences, some greater, others less, according
Suponthecir-to the proportion of their several distances from
eumference & f]10 common centre, carries a.bout with it such
stick not to it. 3
bodies, as being not fluid, adhere to the body so
moved; and such as do not adhere, it casteth for¬
wards in a strait line which is a tangent to the
point from which they are cast off.
For let there be a circle whose radius is AB (in
fig. 4) ; and let a body be placed in the circumfe¬
rence in B, which if it be fixed there, will neces¬
sarily be carried about with it, as is manifest of
itself. But whilst the motion proceeds, let us sup¬
pose that body to be unfixed in B. I say, the body
will continue its motion in the tangent B C. For
let both the radius A B and the sphere B be con¬
ceived to consist of hard matter; and let us sup¬
pose the radius A B to be stricken in the point B
OF CIRCULAR MOTION.
329
by some other body which falls upon it in the tan- part nr.
gent D B. Now, therefore, there will be a motion •—-r—-
made by the concourse of two things, the one, en¬
deavour towards C in the strait line D B produced,
in which the body B would proceed, if it were not
retained by the radius A B ; the other, the reten¬
tion itself. But the retention alone causeth no
endeavour towards the centre ; and, therefore, the
retention being taken away, which is done by the
unfixing of B, there will remain but one endeavour
in B, namely, that in the tangent B C. Wherefore
the motion of the body B unfixed will proceed in
the tangent B C ; which was to be proved.
By this demonstration it is manifest, that cir¬
cular motion about an unmoved axis shakes off and
puts further from the centre of its motion such
things as touch, but do not stick fast to its super¬
ficies ; and the more, by how much the distance is
greater from the poles of the circular motion ; and
so much the more also, by how much the things,
that are shaken off, are less driven towards the
centre by the fluid ambient, for other causes.
10. If in a fluid medium a spherical body be Such thinss aR
. . . . x J are moved with
moved with simple circular motion, and m the same simple circular
medium there float another sphere whose matter is simpiecircuS
not fluid, this sphere also shall be moved with sim- motlon-
pie circular motion.
Let BCD (in fig. 5) be a circle, whose centre is
A, and in whose circumference there is a sphere
so moved, that it describes with simple motion the
the perimeter BCD. Let also E F G be another
sphere of consistent matter, whose semidiameter is
E H, and centre H ; and with the radius A H let
the circle H I be described. I say, the sphere
330 motions and magnitudes.
part hi. E F G will, by the motion of the body in B C D,
v ' be moved in the circumference H I with simple
Such tilings as . •
are moved, &.c. HlOtioil.
For seeing the motion in 15 CD (by art. 4 of this
chapter) makes all the points of the fluid medium
describe in the same time circular lines equal to
one another, the points E, H and G of the strait
line EHG will in the same time describe with equal
radii equal circles. Let E B be drawn equal and
parallel to the strait line A H ; and let A B be con¬
nected, which will therefore be equal and parallel
to E H ; and therefore also, if upon the centre B
and radius B E the arch E K be drawn equal to the
arch H I, and the strait lines A I, B K and I K be
drawn, B K and AI will be equal; and they will
also be parallel, because the two arches E K and
III, that is, the two angles KBE and I A H are
equal; and, consequently, the strait lines A B
and K I, which connect them, will also be equal
and parallel. Wherefore KI and E H are parallel.
Seeing, therefore, E and H are carried in the same
time to K and I, the whole strait line I K will be
parallel to E H, from whence it departed. And,
therefore, seeing the sphere E F G is supposed to
be of consistent matter, so as all its points keep
always the same situation, it is necessary that every
other strait line, taken in the same sphere, be car¬
ried always parallel to the places in which it for¬
merly was. Wherefore the sphere E F G is moved
with simple circular motion; which was to be
demonstrated.
If 11);;t which is 1 ]. If in a fluid medium, whose parts are stirred
one" side hard by a body moved with simple motion., there float
and the other another body, which hath its superficies either
OF CIRCULAR MOTION.
331
wholly hard, or wholly fluid, the parts of this body par2T W-
shall approach the centre equally on all sides ; that '—^
is to say, the motion of the body shall be circular, Motion wfi'i not
and concentric with the motion of the movent.1,e perfectly
circular.
But if it have one side hard, and the other side
fluid, then both those motions shall not have the
same centre, nor shall the floating body be moved
in the circumference of a perfect circle.
Let a body be moved in the circumference of the
circle K LMN (infig 2.) wThose centre is A. And
let there be another body at I, whose superficies is
either all hard or all fluid. Also let the medium, in
which both these bodies are placed, be fluid. I
say, the body at I will be moved in the circle I B
about the centre A. For this has been demonstrated
in the last article.
Wherefore let the superficies of the body at I be
fluid 011 one side, and hard on the other. And
first, let the fluid side be towards the centre. See¬
ing, therefore, the motion of the medium is such,
as that its parts do continually change their places,
(as hath been shown in art 5); if this change of
place be considered in those parts of the medium
which are contiguous to the fluid superficies, it must
needs be that the small parts of that superficies
enter into the places of the small parts of the me¬
dium which are contiguous to them; and the like
change of place will be made with the next conti¬
guous parts towards A. And if the fluid parts of
the body at I have any degree at all of tenacity (for
there are degrees of tenacity, as in the air and
water) the whole fluid side will be lifted up a little,
but so much the less, as its parts have less tena¬
city ; whereas the hard part of the superficies.
332
MOTIONS AND MAGNITUDES.
part in. which is contiguous to the fluid part, has 110 cause
—-—- at all of elevation, that is to say, no endeavour
Jo'mov'Iij &c! towards A.
Secondly, let the hard superficies of the body at
I be towards A. By reason, therefore, of the said
change of place of the parts which are contiguous
to it, the hard superficies must, of necessity, seeing
by supposition there is 110 empty space, either come
nearer to A, or else its smallest parts must supply
the contiguous places of the medium, which other¬
wise would be empty. But this cannot be, by rea¬
son of the supposed hardness ; and, therefore, the
other must needs be, namely, that the body come
nearer to A. Wherefore the body at I has greater
endeavour towards the centre A, when its hard
side is next it, than when it is averted from it.
But the body in I, while it is moving in the circum¬
ference of the circle I B, has sometimes one side,
sometimes another, turned towards the centre; and,
therefore, it is sometimes nearer, sometimes fur¬
ther off from the centre A. Wherefore the body
at I is not carried in the circumference of a perfect
circle ; which was to be demonstrated.
of other variety of motion.
333
CHAPTER XXII.
of other variety of motion.
i. Endeavour and pressure how they differ.—2. Two kinds of
mediums in which bodies are moved.—3. Propagation of mo¬
tion, what it is.—4. What motion bodies have, when they press
one another.—5. Fluid bodies, when they are pressed together,
penetrate one another.—6. When one body presseth another
and dotli not penetrate it, the action of the pressing body is
perpendicular to the superficies of the body pressed.—7. When
a hard body, pressing another body, penetrates the same, it
doth not penetrate it perpendicularly, unless it fall perpendicu¬
larly upon it.—8. Motion sometimes opposite to that of the
movent.—9. In a full medium, motion is propagated to any
distance.—10. Dilatation and contraction what they are.
11. Dilatation and contraction suppose mutation of the smallest
parts in respect of their situation.-—12. All traction is pulsion.
13. Such things as being pressed or bent restore themselves,
have motion in their internal parts.—14. Though that which
carrieth another be stopped, the body carried will proceed.
15, 16. The effects of percussion not to be compared with
those of weight.—17, 18. Motion cannot begin first in the
internal parts of a body.—19. Action and reaction proceed in
the same line.—20. Habit, what it is.
1. I have already (chapter xv. art. 2) defined rAI^IIL
endeavour to be motion through some length, v—1—'
though not considered as length, but as a point, pressure how'd
Whether, therefore, there be resistance or no re- they dlffer*
sistance, the endeavour will still be the same. For
simply to endeavour is to go. But when two bodies,
having opposite endeavours, press one another, then
the endeavour of either of them is that which we
call pressure, and is mutual when their pressures
are opposite.
334
MOTIONS AND MAGNITUDES.
part hi. 2. Bodies moved, and also the mediums in which.
22.
—tliey are moved, are of two kinds. For either they
median's In °f have their parts coherent in such manner, as no
which bodies part of the moved body will easily yield to the
are moved. A J # J J
movent, except the whole body yield also, and such
are the things we call hard: or else their parts,
while the whole remains unmoved, will easily yield
to the movent, and these we call fluid or .soft
bodies. For the words fluid, soft, tough, and hard,
in the same manner as great and little, are used
only comparatively; and are not different kinds,
but different degrees of quality.
Propagation 3 rf0 (J0 an(j to suffer, is to move and to be
of motion, m ,I,J
what it is. moved; and nothing is moved but by that which
touelieth it and is also moved, as has been formerly
shown. And how great soever the distance be,
we say the first movent moveth the last moved
body, but mediately; namely so, as that the first
moveth the second, the second the third, and so
011, till the last of all be touched. When there¬
fore one body, having opposite endeavour to an¬
other body, moveth the same, and that moveth a
third, and so on, I call that action propagation of
motion.
bodies have°n 4. When two fluid bodies, which are in a free
when they press an(j 0r)en space, press one another, their parts will
one another. r r 7 r 7
endeavour, or be moved, towards the sides; not
only those parts which are there where the mutual
contact is, but all the other parts. For in the first
contact, the parts, which are pressed by both the
endeavouring bodies, have no place either forwards
or backwards in which they can be moved; and
therefore they are pressed out towards the sides.
And this expressure, when the forces are equal, is
OF OTHER VARIETY OF MOTION. 335
in a line perpendicular to the bodies pressing. But part hi.
whensoever the foremost parts of both the bodies —-
are pressed, the hindermost also must be pressed
at the same time ; for the motion of the hinder-
most parts cannot in an instant be stopped by the
resistance of the foremost parts, but proceeds for
some time; and therefore, seeing they must have
some place in which they may be moved, and that
there is 110 place at all for them forwards, it is neces¬
sary that they be moved into the places which are
towards the sides every way. And this effect fol¬
lows of necessity, not only in fluid, but in consistent
and hard bodies, though it be not always manifest
to sense. For though from the compression of
two stones we cannot with our eyes discern any
swelling outwards towards the sides, as we per¬
ceive 111 two bodies of wax; yet we know well
enough by reason, that some tumour must needs be
there, though it be but little.
5. But when the space is enclosed, and both the Fluid bodies,
bodies be fluid, they will, if they be pressed toge- pressed'Toge-6
tlier, penetrate one another, though differently,
according to their different endeavours. For sup¬
pose a hollow cylinder of hard matter, well
stopped at both ends, but filled first, below with
some heavy fluid body, as quicksilver, and above
with water or air. If now the bottom of the
cylinder be turned upwards, the heaviest fluid
body, which is now at the top, having the greatest
endeavour downwards, and being by the hard
sides of the vessel hindered from extending itself
sideways, must of necessity either be received by
the lighter body, that it may sink through it, or
else it must open a passage through itself, by
336
MOTIONS AND MAGNITUDES.
rA^9I1L which the lighter body may ascend. For of the
-—-■—' two bodies, that, whose parts are most easily sepa-
when they6 are rated, will be the first divided ; which being done,
pressed toge- js not necessary that the parts of the other suffer
thcr, penetrate _ J 1
one another. any separation at all. And therefore when two
liquors, which are enclosed in the same vessel,
change their places, there is no need that their
smallest parts should be mingled with one another;
for a way being opened through one of them, the
parts of the other need not be separated.
Now if a fluid body, which is not enclosed, press
a hard body, its endeavour will indeed be towards
the internal parts of that hard body; but being
excluded by the resistance of it, the parts of the
fluid body will be moved every way according to
the superficies of the hard body, and that equally,
if the pressure be perpendicular; for when all the
parts of the cause are equal, the effects will be
equal also. But if the pressure be not perpendi¬
cular, then the angles of the incidence being un¬
equal, the expansion also will be unequal, namely,
greater 011 that side where the angle is greater,
because that motion is most direct which proceeds
by the directest line,
when one body 6. If a body, pressing another body, do not
presseth an- J " L o J ■>
other and doth penetrate it, it will nevertheless give to the part it
not penetrate _ -111 1
it, the action of presseth an endeavour to yield, and recede m a
bodyis^erpen- strait line perpendicular to its superficies in that
dicular to the p0jjQt \n wllicll it is pressed.
superficies of 1 .
the body Let A BCD (in fig. 1) be a hard body, and let
another body, falling upon it in the strait line E A,
with any inclination or without inclination, press
it in the point A. I say the body so pressing, and
not penetrating it, will give to the part A an
of other variety of motion. 337
endeavour to yield or recede in a strait line per- part iii
pendicular to the line A D.
For let A B be perpendicular to A I), and let
B A be produced to F. If therefore A. F be coin¬
cident with AE, it is of itself manifest that the
motion in E A will make A to endeavour in the
line A B. Let now E A be oblique to A D, and
from the point E let the strait line E C be drawn,
cutting A D at right angles in D, and let the
rectangles A B C D and A D E F be completed. I
have shown (in the 8th article of chapter xvi)
that the body will be carried from E to A by the
concourse of two uniform motions, the one in E F
and its parallels, the other in E D and its parallels.
But the motion in E F and its parallels, whereof
D A is one, contributes nothing to the body in A
to make it endeavour or press towards B; and
therefore the whole endeavour, which the body
hath in the inclined line E A to pass or press the
strait line A D, it hath it all from the perpendicular
motion or endeavour in F A. Wherefore the body
E, after it is in A, will have only that perpendicu¬
lar endeavour which proceeds from the motion in
F A, that is, in A B; which wTas to be proved.
7. If a hard body falling upon or pressing an- when a hard
other body penetrate the same, its endeavour
after its first penetration will be neither in the Penetrates the
r _ same, it dotli
inclined line produced, nor in the perpendicular, not penetrate it
but sometimes betwixt both, sometimes without ly, unless'it fall
ti perpendicular-
ly upon it.
Let E A G (in the same fig. 1) be the inclined
line produced; and first, let the passage through
the medium, in which E A is, be easier than the
passage through the medium in which AG is. As
vol. 1. z
338 MOTIONS AND MAGNITUDES.
part iii. soon therefore as the body is within the medium
22. .
-—r—' in which is A G, it will find greater resistance to
bodyn&c.'ard motion in D A and its parallels, than it did
whilst it was above A D ; and therefore below A D
it will proceed with slower motion in the parallels
of D A, than above it. Wherefore the motion
which is compounded of the two motions in E F
and E D will be slower below A D than above it;
and therefore also, the body will not proceed from
A in E A produced, but below it. Seeing, there¬
fore, the endeavour in A B is generated by the
endeavour in F A; if to the endeavour in F A there
be added the endeavour in D A, which is not all
taken away by the immersion of the point A into
the lower medium, the body will not proceed from
A in the perpendicular A B, but beyond it; namely,
in some strait line between A B and A G, as in the
line A H.
Secondly, let the passage through the medium E A
be less easy than that through A G. The motion,
therefore, which is made by the concourse of the
motions in E F and F B, is slower above A D than
below it; and consequently, the endeavour will
not proceed from A in E A produced, but beyond
it, as in A I. Wherefore, if a hard body falling,
&c.; which was to be proved.
This divergency of the strait line A H from the
strait line A G is that which, the writers of optics
commonly called refraction, which, when the pas¬
sage is easier in the first than in the second
medium, is made by diverging from the line of
inclination towards the perpendicular; and con-
trarily, when the passage is not so easy in the
OF OTHER VARIETY OF MOTION. 339
first medium, by departing farther from the per- rAR2^ Iir-
pendicular. •—A-*
8. By the 6th theorem it is manifest, that the Motion some-
„ r , , ill times opposite
iorce ot the movent may be so placed, as that to that of the
the body moved by it may proceed in a way almost movent*
directly contrary to that of the movent, as we see
in the motion of ships.
For let A B (in fig. 2) represent a ship, whose
length from the prow to the poop is A B, and let
the wind lie upon it in the strait parallel lines C B,
D E and F G; and let D E and F G be cut in E and
and G by a strait line drawn from B perpendicular
to A B ; also let B E and E G be equal, and the
angle ABC any angle how small soever. Then
between B C and B A let the strait line BI be
drawn; and let the sail be conceived to be spread
in the same line B I, and the wind to fall upon it
in the points L, M and B; from which points, per¬
pendicular to B I, let BK, MQ, and L P be drawn.
Lastly, let E N and G 0 be drawn perpendicular to
B G, and cutting B K in H and K; and let H N
and K 0 be made equal to one another, and seve¬
rally equal to B A. I say, the ship BA, by the
wind falling upon it in C B, D E, F G, and other
lines parallel to them, will be carried forwards
almost opposite to the wind, that is to say, in a
way almost contrary to the way of the movent.
For the wind that blows in the line C B will (as
hath been shown in art. 6) give to the point B an en¬
deavour to proceed in a strait line perpendicular to
the strait line B I, that is, in the strait line B K ;
and to the points M and L an endeavour to pro¬
ceed in the strait lines M Q and L P, which are
parallel to B K. Let now the measure of the time
z 2
340
motions and magnitudes.
part hi. be B G, which is divided in the middle in E ; and
v—.—' let the point B be carried to H in the time B E.
dmes*'opposite th0- same time, therefore, by the wind blowing
movent1 °f the i*1 ^ M and F L, and as many other lines as may
be drawn parallel to them, the whole ship will be
applied to the strait line H N. Also at the end of
the second time E G, it will be applied to the strait
line K O. Wherefore the ship will always go for¬
ward ; and the angle it makes with the wind will
be equal to the angle ABC, how small soever that
angle be; and the way it makes will in every time be
equal to the strait line EH. I say, thus it would
be, if the ship might be moved with as great
celerity sideways from B A towards K 0, as it may
be moved forwards in the line B A. But this is
impossible, by reason of the resistance made by the
great quantity of water which presseth the side, much
exceeding the resistance made by the much smaller
quantity which presseth the prow of the ship; so
that the way the ship makes sideways is scarce
sensible; and, therefore, the point B will proceed
almost in the very line B A, making with the wind
the angle ABC, how acute soever; that is to say,
it will proceed almost in the strait line B C, that
is, in a way almost contrary to the way of the
movent; which was to be demonstrated.
But the sail in B I must be so stretched as that
there be left in it 110 bosom at all; for otherwise
the strait lines LP, Mft and B K will not be per¬
pendicular to the plane of the sail, but falling below
P, Q, and K, will drive the ship backwards. But
by making use of a small board for a sail, a little
waggon with wheels for the ship, and of a smooth
pavement for the sea, I have by experience found
OF OTHER VARIETY OF MOTION. 341
this to be so true, that I could scarce oppose the m-
board to the wind in any obliquity, though never ^—»—-
so small, but the waggon was carried forwards Ss^opposhe
iky jj- to that of the
. movent.
By the same 6th theorem it may be found, how
much a stroke, which falls obliquely, is weaker than
a stroke falling perpendicularly, they being like
and equal in all other respects.
Let a stroke fall upon the wall A B obliquely, as
for example, in the strait line CA (in fig. 3.) Let
C E be drawn parallel to A B, and D A perpendi¬
cular to the same A B and equal to C A ; and let
both the velocity and time of the motion in C A be
equal to the velocity and time of the motion in
DA. I say, the stroke in C A will be weaker than
that in D A, in the proportion of E A to D A. For
producing D A howsoever to F, the endeavour of
both the strokes will (by art. 6) proceed from A
in the perpendicular A F. But the stroke in CA is
made by the concourse of two motions in C E and
E A, of which that in C E contributes nothing to
the stroke in A, because CE and B A are parallels;
and, therefore, the stroke in C A is made by the
motion which is in E A only. But the velocity or
force of the perpendicular stroke in E A, to the
velocity or force of the stroke in D A, is as E A to
DA. Wherefore the oblique stroke in C A is weaker
than the perpendicular stroke in D A, in the pro¬
portion of E A to D A or C A; which was to be
proved.
9. In a full medium, all endeavour proceeds as in a full me-
far as the medium itself reacheth; that is to say, if iS propagated
the medium be infinite, the endeavour will proceedtoaiiydl8tance*
infinitely.
342
MOTIONS AND MAGNITUDES.
PAR22IIT' ^or whosoever endeavoureth is moved, and
therefore whatsoever standeth in its way it maketh
dhim, motion" it yield, at least a little, namely, so far as the movent
to any distance. ^self is moved forwards. But that which yieldeth
is also moved, and consequently maketh that to yield
which is in its way, and so 011 successively as long
as the medium is full; that is to say, infinitely, if
the full medium be infinite ; which was to be
proved.
Now although endeavour thus perpetually pro¬
pagated do not always appear to the senses as
motion, yet it appears as action, or as the efficient
cause of some mutation. For if there be placed
before our eyes some very little object, as for
example, a small grain of sand, which at a certain
distance is visible; it is manifest that it maybe re¬
moved to such a distance as not to be any longer
seen, though by its action it still work upon the
organs of sight, as is manifest from that wThich was
last proved, that all endeavour proceeds infinitely.
Let it be conceived therefore to be removed from
our eyes to any distance how great soever, and a
sufficient number of other grains of sand of the
same bigness added to it; it is evident that the
aggregate of all those sands will be visible ; and
though none of them can be seen when it is single
and severed from the rest, yet the whole heap or
hill which they make will manifestly appear to the
sight; which would be impossible, if some action
did not proceed from each several part of the whole
heap.
Dilatation jq Between the degrees of hard and soft are
& contraction 0
what they are. those things which we call tough, tough being that
which may be bent without being altered from
OF OTHER VARIETY OF MOTION. 343
what it was; and the bending of a line is either PA^2 IIL
the adduction or diduction of the extreme parts, ■—r—'
that is, a motion from straitness to crookedness,
or contrarily, whilst the line remains still the same
it was; for by drawing out the extreme points of
a line to their greatest distance, the line is made
strait, which otherwise is crooked. So also the
bending of a superficies is the diduction or adduc¬
tion of its extreme lines, that is, their dilatation and
contraction.
11. Dilatation and contraction,as also all flexion, Pllatat,on
° 3 & contraction
supposes necessarily that the internal parts of the suppose mu-
, , , , . . , , tation of the
body bowed do either come nearer to the external smallest parts
parts, or go further from them. For though flexion Ihe^Ttuation.
be considered only in the length of a body, yet
when that body is bowed, the line which is made
on one side will be convex, and the line on the
other side will be concave ; of which the concave,
being the interior line, will, unless something be
taken from it and added to the convex line, be the
more crooked, that is, the greater of the twro.
But they are equal; and, therefore, in flexion there
is an accession made from the interior to the ex¬
terior parts ; and, on the contrary, in tension, from
the exterior to the interior parts. And as for those
things which do not easily suffer such transposition
of their parts, they are called brittle ; and the
great force they require to make them yield, makes
them also with sudden motion to leap asunder, and
break in pieces.
12. Also motion is distinguished into pulsion ah traction
and traction. And pulsion, as I have already de-1S pulsion-
fined it, is when that which is moved goes before
that which moveth it. But contrarily, in traction
344 MOTIONS AND MAGNITUDES.
part m* the movent goes before that which is moved. Never-
<—r—- theless, considering it with greater attention, it
seemeth to be the same with pulsion. For of two
parts of a hard body, when that which is foremost
drives before it the medium in which the motion is
made, at the same time that which is thrust for¬
wards thrusteth the next, and this again the next,
and so 011 successively. In which action, if we sup¬
pose that there is no place void, it must needs be,
that by continual pulsion, namely, when that action
has gone round, the movent will be behind that
part, which at the first seemed not to be thrust
forwards, but to be drawn ; so that now the body,
which was drawn, goes before the body which
gives it motion ; and its motion is 110 longer trac¬
tion, but pulsion.
Such things as 13. Such things as are removed from their
being pressed , , „ . , . . ,
or bent restore places by forcible compression or extension, and, as
hlvTmotion in soon as the force is taken away, do presently return
their internal an(j restore themselves to their former situation,
parts.
have the beginning of their restitution within them¬
selves, namely, a certain motion in their internal
parts, which was there, when, before the taking
away of the force, they were compressed, or ex¬
tended. For that restitution is motion, and that
which is at rest cannot be moved, but by a moved
and a contiguous movent. Nor doth the cause of
their restitution proceed from the taking away of
the force by which they were compressed or ex¬
tended ; for the removing of impediments hath not
the efficacy of a cause, as has been shown at the
end of the 3rd article of chap. xv. The cause
therefore of their restitution is some motion either
of the parts of the ambient, or of the parts of the
OF OTHER VARIETY OF MOTION. 345
body compressed or extended. But the parts of PARrm.
the ambient have no endeavour which contributes «—*—-
to their compression or extension, nor to the set¬
ting of them at liberty, or restitution. It remains
therefore that from the time of their compression or
extension there be left some endeavour or motion,
by which, the impediment being removed, every
part resumes its former place; that is to say, the
whole restores itself.
14. In the carriage of bodies, if that body, which Though that
... . which carrietli
carries another, hit upon any obstacle, or be by another be
any means suddenly stopped, and that which is blTy ca'nSd
carried be not stopped, it will go on, till its motion wiU Proceed-
be by some external impediment taken away.
For I have demonstrated (chap, vm, art. 19)
that motion, unless it be hindered by some external
resistance, will be continued eternally with the
same celerity; and in the 7th article of chap, ix,
that the action of an external agent is of no effect
without contact. When therefore that, which car-
rieth another thing, is stopped, that stop doth not
presently take away the motion of that which is
carried. It will therefore proceed, till its motion
be by little and little extinguished by some external
resistance : which was to be proved; though expe¬
rience alone had been sufficient to prove this.
In like manner, if that body which carrieth
another be put from rest into sudden motion, that
which is carried will not be moved forwards toge¬
ther with it, but will be left behind. For the con¬
tiguous part of the body carried hath almost the
same motion with the body which carries it; and
the remote parts will receive different velocities
according to their different distances from the body
346 MOTIONS AND MAGNITUDES.
fart hi. that carries them; namely, the more remote the
r—- parts are, the less will be their degrees of velocity.
It is necessary, therefore, that the body, which is
carried, be left accordingly more or less behind.
And this also is manifest by experience, when at
the starting forward of the horse the rider falleth
backwards.
The effects of jg. jn percussion, therefore, when one hard
percussion not ■* . . ,
to be compared body is in some small part of it stricken by another
weight. °se ° with great force, it is not necessary that the whole
body should yield to the stroke with the same
celerity with which the stricken part yields. For
the rest of the parts receive their motion from the
motion of the part stricken and yielding, which
motion is less propagated every way towards the
sides, than it is directly forwards. And hence it
is, that sometimes very hard bodies, which being
erected can hardly be made to stand, are more
easily broken than thrown down by a violent
stroke ; when, nevertheless, if all their parts toge¬
ther were by any weak motion thrust forwards,
they would easily be cast down.
16. Though the difference between trusion and
percussion consist only in this, that in trusion the
motion both of the movent and moved body begin
both together in their very contact; and in percus¬
sion the striking body is first moved, and after¬
wards the body stricken; yet their effects are so
different, that it seems scarce possible to compare
their forces with one another. I say, any effect of
percussion being propounded, as for example, the
stroke of a beetle of any weight assigned, by
which a pile of any given length is to be driven
OF OTHER VARIETY OF MOTION. 347
into earth of any tenacity given, it seems to me PA22 IIL
very hard, if not impossible, to define with what <—r—'
weight, or with what stroke, and in what time, the
same pile may be driven to a depth assigned into
the same earth. The cause of which difficulty is
this, that the velocity of the percutient is to be
compared with the magnitude of the ponderant.
Now velocity, seeing it is computed by the length
of space transmitted, is to be accounted but as one
dimension; but weight is as a solid thing, being
measured by the dimension of the whole body.
And there is no comparison to be made of a solid
body with a length, that is, with a line.
1/. If the internal parts of a body be at rest, or Motion cannot
; begin first in
retain the same situation with one another tor any the internal
time how little soever, there cannot in those parts paitsofabody'
be generated any new motion or endeavour, whereof
the efficient cause is not without the body of which
they are parts. For if any small part, which is
comprehended within the superficies of the whole
body, be supposed to be now at rest, and by and
by to be moved, that part must of necessity receive
its motion from some moved and contiguous body.
But by supposition, there is 110 such moved and
contiguous part within the body. Wherefore, if
there be any endeavour or motion or change of
situation in the internal parts of that body, it must
needs arise from some efficient cause that is
without the body which contains them ; which was
to be proved.
18. I11 hard bodies, therefore, which are com¬
pressed or extended, if, that which compresseth or
extendetli them being taken away, they restore
348 MOTIONS AND MAGNITUDES.
part iii. themselves to their former place or situation, it
v—t——' must needs be that that endeavour or motion of
their internal parts, by which they were able to
recover their former places or situations, was not
extinguished when the force by which they were
compressed or extended was taken away. There¬
fore, when the lath of a cross-bow bent doth, as
soon as it is at liberty, restore itself, though to him,
that judges by sense, both it and all its parts seem
to be at rest; yet he, that judging by reason doth
not account the taking away of impediment for an
efficient cause, nor conceives that without an effi¬
cient cause any thing can pass from rest to motion,
will conclude that the parts were already in motion
before they began to restore themselves,
action* proceed 19- Action and reaction proceed in the same
line 6 same ^ne' from 0PP0Site terms. For seeing reaction is
nothing but endeavour in the patient to restore itself
to that situation from which it was forced by the
agent; the endeavour or motion both of the agent
and patient or reagent will be propagated between
the same terms ; yet so, as that in action the term,
from which, is in reaction the term to which. And
seeing all action proceeds in this manner, not only
between the opposite terms of the whole line in
which it is propagated, but also in all the parts of
that line, the terms from which and to which, both
of the action and reaction, will be in the same line.
Wherefore action and reaction proceed in the same
line, &c.
Habit, 20. To what has been said of motion, I will add
what it is. wjiat i ]iaYe to say concerning habit. Habit,
therefore, is a generation of motion, not of motion
of other variety of motion. 349
simply, but an easy conducting of the moved body PA^ hi.
in a certain and designed way. And seeing it is *—'
attained by the weakening of such endeavours as is<
divert its motion, therefore such endeavours are
to be weakened by little and little. But this cannot
be done but by the long continuance of action, or
by actions often repeated; and therefore custom
begets that facility, which is commonly and rightly
called habit; and it may be defined thus: habit
is motion made more easy and ready by custom;
that is to say, by perpetual endeavour, or by
iterated endeavours in a way differing from that
in which the motion proceeded from the beginning,
and opposing such endeavours as resist. And to
make this more perspicuous by example, we may
observe, that when one that has no skill in music
first puts his hand to an instrument, he cannot
after the first stroke carry his hand to the place
where he would make the second stroke, without
taking it back by a new endeavour, and, as it were
beginning again, pass from the first to the second.
Nor will he be able to go on to the third place
without another new endeavour; but he will be
forced to draw back his hand again, and so suc¬
cessively, by renewing his endeavour at every
stroke; till at the last, by doing this often, and by
compounding many interrupted motions or endea¬
vours into one equal endeavour, he be able to make
his hand go readily on from stroke to stroke in
that order and way which was at the first designed.
Nor are habits to be observed in living creatures
only, but also in bodies inanimate. For we find
that when the lath of a cross-bow is strongly bent,
350
MOTIONS AND MAGNITUDES.
PAI22IIL an(^ wou^if ^ie impediment were removed return
v—*—- again with great force ; if it remain a long time
St it is. bent, it will get such a habit, that when it is loosed
and left to its own freedom, it will not only not
restore itself, but will require as much force for
the bringing of it back to its first posture, as it did
for the bending of it at the first.
CHAP. XXIII.
OF THE CENTRE OF EQUIPONDERATION ; OF
BODIES PRESSING DOWNWARDS IN STRAIT
PARALLEL LINES.
1. Definitions and suppositions.— 2. Two planes of eqnipondera-
tion are not parallel.—3. The centre of equiponderation is in
every plane of equiponderation.—4. The moments of equal
ponderants are to one another as their distances from the
centre of the scale.— 5, 6. The moments of unequal pon¬
derants have their proportion to one another compounded
of the proportions of their weights and distances from the
centre of the scale.—7. If two ponderants have their weights
and distances from the centre of the scale in reciprocal pro¬
portion, they are equally poised ; and contrarily.—8. If the
parts of any ponderant press the beams of the scale every
where equally, all the parts cut off', reckoned from the centre
of the scale, will have their moments in the same proportion
with that of the parts of a triangle cut off from the vertex by
strait lines parallel to the base.—9. The diameter of equipon¬
deration of figures, which are deficient according to commen¬
surable proportions of their altitudes and bases, divides the
axis, so that the part taken next the vertex is to the other part
of the complete figure to the deficient figure.—10. The dia¬
meter of equiponderation of the complement of the half of any
of the said deficient figures, divides that line which is drawn
through the vertex parallel to the base, so that the part next
the vertex is to the other part as the complete figure to the
CENTRE OF EQUIPONDERATION. 351
complement.—11. The centre of equiponderation of the lialf PART III.
of any of the deficient figures in the first row of the table of . 23' .
art. 3, chap, xvir, may be found out by the numbers of the
second row.—12. The centre of equiponderation of the half
of any of the figures of the second row of the same table, may
be found out by the numbers of the fourth row.—13. The
centre of equiponderation of the half of any of the figures in
the same table being known, the centre of the excess of the
same figure above a triangle of the same altitude and base is
also known.—14. The centre of equiponderation of a solid
sector is in the axis so divided, that the part next the vertex
be to the whole axis, wanting half the axis of the portion of the
sphere, as 3 to 4.
DEFINITIONS.
I. A scale is a strait line, whose middle point Definitions,
is immovable, all the rest of its points being at
liberty ; and that part of the scale, which reaches
from the centre to either of the weights, is called
the beam.
II. Equiponderation is when the endeavour of
one body, which presses one of the beams, resists
the endeavour of another body pressing the other
beam, so that neither of them is moved; and the
bodies, when neither of them is moved, are said to
be equally poised.
hi. Weight is the aggregate of all the endea¬
vours, by which all the points of that body, which
presses the beam, tend downwards in lines parallel
to one another ; and the body which presses is
called the ponder ant.
iv. Moment is the power which the ponderant
lias to move the beam, by reason of a determined
situation.
v. The plane of equiponderation is that by
which the ponderant is so divided, that the mo¬
ments on both sides remain equal.
352 motions and magnitudes.
vi. The diameter of equiponderation is the
common section of the two planes of equipondera¬
tion, and is in the strait line by which the weight
is hanged.
vii. The centre of equiponderation is the com¬
mon point of the two diameters of equiponderation.
suppositions.
Suppositions. i. When two bodies are equally poised, if weight
be added to one of them and not to the other,
their equiponderation ceases.
ii. When two ponderants of equal magnitude,
and of the same species or matter, press the beam
011 both sides at equal distances from the centre of
the scale, their moments are equal. Also when
two bodies endeavour at equal distances from the
centre of the scale, if they be of equal magnitude
and of the same species, their moments are equal.
Two planes 2. No two planes of equiponderation are parallel.
of equiponde- T .
ration are Let A 13 C D (in ng. 1) be any ponderant what¬
not parallel. soeyer. anc[ jn ^ \eE p be a plane of equipon¬
deration ; parallel to which, let any other plane
be drawn, as G H. I say, G H is not a plane of
equiponderation. For seeing the parts AEFD
and E B C F of the ponderant A B C D are equally
poised ; and the weight EGHF is added to the
part AEFD, and nothing is added to the part
E B C F, but the weight EGHF is taken from
it; therefore, by the first supposition, the parts
A G H D and G B C H will not be equally poised;
and consequently G H is not a plane of equiponde¬
ration. Wherefore, 110 two planes of equiponde¬
ration are parallel; which was to be proved.
PART in.
23.
Definitions.
CENTRE OF EQUIPONDERATION. 353
3. The centre of equiponderation is in every tart nr.
plane of equiponderation. >—•
For if another plane of equiponderation be Jqhueipondl7a-0f
taken, it will not, by the last article, be parallel to d,on is ir* evei7
J A plane ot equi-
the former plane; and therefore both those planes ponderation.
will cut one another. Now that section (by the
6th definition) is the diameter of equiponderation.
Again, if another diameter of equiponderation be
taken, it will cut that former diameter; and in
that section (by the 7th definition) is the centre of
equiponderation. Wherefore the centre of equi¬
ponderation is in that diameter which lies in the
said plane of equiponderation.
4. The moment of anyponderant applied to one The moments
point of the beam, to the moment of the same or rants are to one
an equal ponderant applied to any other point of winces8from
the beam, is as the distance of the former point t!ie ce,ntre of
1 the scale.
from the centre of the scale, to the distance of the
latter point from the same centre. Or thus, those
moments are to one another, as the arches of
circles which are made upon the centre of the
scale through those points, in the same time. Or
lastly thus., they are as the parallel bases of two
triangles, which have a common angle at the
centre of the scale.
Let A (in fig. 2) be the centre of the scale; and
let the equal ponderants D and E press the beam
A B in the points B and C; also let the strait lines
B D and C E be diameters of equiponderation;
and the points D and E in the ponderants D and E
be their centres of equiponderation. Let AGF be
drawn howsoever, cutting D B produced in F, and
E C in G; and lastly, upon the common centre A,
let the two arches B H and CI be described, cut-
yol. i. a a
354 MOTIONS AND MAGNITUDES.
PA^,ITTL A G F in H and I. I say, the moment of the
-—•—' ponderant D to the moment of the ponderant E
is as AB to AC, or as BH to CI, or as BF to CG.
For the effect of the ponderant D, in the point B,
is circular motion in the arch B H ; and the effect
of the ponderant E, in the point C, circular motion
in the arch C I; and by reason of the equality of
the ponderants D and E, these motions are to one
another as the quicknesses or velocities with which
the points B and C describe the arches B H and
C I, that is, as the arches themselves B H and C I,
or as the strait parallels B F and C G, or as the
parts of the beam A B and AC; for A B. A C ::
B F. CG :: B H. CI. are proportionals ; and there¬
fore the effects, that is, by the 4th definition, the
moments of the equal ponderants applied to several
points of the beam, are to one another as A B and
AC; or as the distances of those points from the
centre of the scale ; or as the parallel bases of the
triangles which have a common angle at A ; or as
the concentric arches B H and CI; which was to
be demonstrated.
The moments 5 Unequal ponderants, when they are applied
of unequal pon- . .
derants have to several points of the beam, and hang at liberty,
tioTto'Tne" that is, so as the line by which they hang be the
pounded of'the diameter of equiponderation, whatsoever be the
proportions offjp-ure 0f the ponderant, have their moments to
their weights 0 . .
and distances one another 111 proportion compounded of the
from the centre ,. r ,1 • n. . n
of the scale, proportions ot their distances irom the centre 01
the scale, and of their weights.
Let A (in fig. 3) be the centre of the scale, and
A B the beam; to which let the two ponderants
C and D be applied at the points B and E. I say,
the proportion of the moment of the ponderant C
CENTRE OF EQUIPONDERATION. 355
to the moment of the ponderant D, is compounded of PA^IIL
the proportions of A B to A E, and of the weight •—-
C to the weight D ; or, if C and D be of the same ^fhuen™3epon-
species, of the magnitude C to the magnitude D. derants, &c.
Let either of them, as C, be supposed to be
bigger than the other, D. If, therefore, by the
addition of F, F and D together be as one body-
equal to C, the moment of C to the moment of
F + D will be (by the last article) as BG is to EH.
Now as F + D is to D, so let E H be to another
EI; and the moment of F + D, that is of C, to the
moment of D, will be as B G to E I. But the pro¬
portion of B G to EI is compounded of the propor¬
tions of B G to E H, that is, of A B to A E, and of
E H to E I, that is, of the weight C to the weight
D. Wherefore unequal ponderants, when they
are applied, &c. Which was to be proved.
G. The same figure remaining, if I K be drawn
parallel to the beam AB, and cutting AG in K;
and K L be drawn parallel to B G, cutting A B in
L, the distances A B and AL from the centre will
be proportional to the moments of C and D. For
the moment of C is B G, and the moment of D is
E I, to which K L is equal. But as the distance
A B from the centre is to the distance A L from
the centre, so is B G, the moment of tlie ponderant
C, to L K, or E I the moment of the ponderant D.
7. If two ponderants have their weights and if two ponde-
V , .t , • i , ■ rants have their
distances irom the centre m reciprocal proportion, weights and
and the centre of the scale be between the points ^ce^tre^S
to which the ponderants are applied, they will be the.scale in
n -i * i «1 . ~ 1 - ,, reciprocal pro-
equally poised. And contranly, if they be equally portion, they
poised, their weights and distances from the centre pSsedTand
of the scale will be in reciprocal proportion. contranly.
A A 2
356 MOTIONS AND MAGNITUDES.
PAR23ni" cen^re °f the scale (in the same third
v—^ figure) be A, the beam A B ; and let any ponderant
lLls,\l°.ndc' C, having B G for its moment, be applied to the
point B; also let any other ponderant D, whose
moment is E I, be applied to the point E. Through
the point I let I K be drawn parallel to the beam
AB, cutting AG in K; also let KL be drawn
parallel to B G, KL will then be the moment of the
ponderant D ; and by the last article, it will be as
B G, the moment of the ponderant C in the point
B, to L K the moment of the ponderant D in the
point E, so A B to A L. On the other side of the
centre of the scale, let A N be taken equal to A L;
and to the point N let there be applied the ponde¬
rant 0, having to the ponderant C the proportion
of A B to A N. I say, the ponderants in B and N
will be equally poised. For the proportion of the
moment of the ponderant O, in the point N, to the
moment of the ponderant C in the point B, is by
the 5th article, compounded of the proportions of
the weight 0 to the weight C, and of the distance
from the centre of the scale AN or AL to the
distance from the centre of the scale A B. But
seeing we have supposed, that the distance AB
to the distance AN is in reciprocal propor¬
tion of the weight 0 to the weight C, the propor¬
tion of the moment of the ponderant 0, in the point
N, to the moment of the ponderant C, in the point
B, will be compounded of the proportions of A B
to A N, and of A N to A B. Wherefore, setting in
order AB, AN, AB, the moment of O to the mo¬
ment of C will be as the first to the last, that is, as
A B to A B. Their moments therefore are equal;
and consequently the plane which passes through
CENTRE OF EQUIPONDERATION.
35/
A will (by tlie fifth definition) be a plane of equi- PAI^ Iir*
ponderation. Wherefore they will be equally ■—'
poised; as was to be proved.
Now the converse of this is manifest. For if
there be equiponderation and the proportion of the
weights and distances be not reciprocal, then both
the weights will always have the same moments,
although one of them have more weight added to
it or its distance changed.
Coroll. When ponderants are of the same species,
and their moments be equal; their magnitudes and
distances from the centre of the scale will be reci¬
procally proportional. For in homogeneous bodies,
it is as weight to weight, so magnitude to mag¬
nitude.
8. If to the whole length of the beam there be If the p*rts of
applied a parallelogram, or a parallelopipedum, or p?L the beams
a prisma, or a cylinder, or the superficies of a gJery where
cylinder, or of a sphere, or of any portion of ajJjtaJffthe
sphere or prisma; the parts of any of them cut reckoned from
• -i centre of tlie
off with planes parallel to the base will have their scale, will have
moments in the same proportion with the parts of ^ the same pro-
a triangle, which has its vertex in the centre of the p°rtion with
° 3 _ that of the parts
scale, and for one of its sides the beam itself, which ofa triangle cut
parts are cut off by planes parallel to the base. vertex by strait
First, let the rectangled parallelogram A B C D IhTbLT'1611°
(in figure 4) be applied to the whole length of
the beam A B ; and producing C B howsoever to E,
let the triangle A B E be described. Let now any
part of the parallelogram, as*A F, be cut off by the
plane F G, parallel to the base C B ; and let F G be
produced to AE in the point H. I say, the mo¬
ment of the whole A B C D to the moment of its
358 MOTIONS AND MAGNITUDES.
PAI23 ni* Pai ^ ^ *s as ^ie tr^an^e ABE to the triangle
s—r—- AGH, that is, in proportion duplicate to that of
anytpmfdcran°t!'distances from the centre of the scale.
&c- For, the parallelogram A B C D being divided
into equal parts, infinite in number, by strait lines
drawn parallel to the base; and supposing the
moment of the strait line CB to be BE, the mo¬
ment of the strait line F G will (by the 7th article)
be G H ; and the moments of all the strait lines of
that parallelogram will be so many strait lines in
the triangle ABE drawn parallel to the base B E;
all which parallels together taken are the moment
of the whole parallelogram ABCD; and the same
parallels do also constitute the superficies of the
triangle ABE. Wherefore the moment of the
parallelogram A B C D is the triangle ABE; and
for the same reason, the moment of the parallelo¬
gram A F is the triangle AGH; and therefore the
moment of the whole parallelogram to the moment
of a parallelogram which is part of the same, is as
the triangle A B E to the triangle A G H, or in
proportion duplicate to that of the beams to which
they are applied. And what is here demonstrated
in the case of a parallelogram may be understood
to serve for that of a cylinder, and of a prisma,
and their superficies; as also for the superficies of
a sphere, of an hemisphere, or any portion of a
sphere. For the parts of the superficies of a sphere
have the same proportion with that of the parts of
the axis cut off by the same parallels, by which the
parts of the superficies are cut off, as Archimedes
has demonstrated; and therefore when the parts
of any of these figures are equal and at equal
CENTRE OF EQUIPONDERATION. 359
distances from the centre of the scale, their mo- part hi.
23.
ments also are equal, in the same manner as they —.—•
are in parallelograms.
Secondly, let the parallelogram A KIB not be
rectangled; the strait line IB will nevertheless
press the point B perpendicularly in the strait line
B E ; and the strait line L G will press the point
G perpendicularly in the strait line G H ; and all
the rest of the strait lines which are parallel to IB
will do the like. Whatsoever therefore the moment
be which is assigned to the strait line IB, as here,
for example, it is supposed to be BE, if A E be
drawn, the moment of the whole parallelogram AI
will be the triangle ABE; and the moment of the
part A L will be the triangle AGH. Wherefore
the moment of any ponderant, which has its sides
equally applied to the beam, whether they be
applied perpendicularly or obliquely, will be always
to the moment of a part of the same in such pro¬
portion as the whole triangle has to a part of the
same cut off by a plane which is parallel to the base.
9. The centre of equiponderation of any figure, The diameter of
which is deficient according to commensurable STfigures
proportions of the altitude and base diminished, which are ^
1 1 _ cient according
and whose complete figure is either a parallelogram to commensu-
or a cylinder, or a parallelopipedum, divides the tions of their
axis, so, that the part next the vertex, to the other blsesfdividfs
part, is as the complete figure to the deficient JJ®
figure. next the vertex
For let CIA P E (in fig. 5) be a deficient figure, paJtas^hecom-
whose axis is AB, and whose complete figure is Jhe^Slt fi-
C D F E ; and let the axis A B be so divided in Z, §ure-
that A Z be to Z B as C D F E is to CIA P E. I
360 motions and magnitudes.
part in. Say, the centre of equiponderation of the figure
v—^—' CIA P E will be in the point Z.
of'equSponde- First, that the centre of equiponderation of the
ration, &c. flgure CIA P E is somewhere in the axis A B is
manifest of itself; and therefore A B is a diameter
of equiponderation. Let AE be drawn, and let
B E be put for the moment of the strait line C E ;
the triangle ABE will therefore (by the third
article) be the moment of the complete figure
C D F E. Let the axis A B be equally divided in
L, and let G L H be drawn parallel and equal to
the strait line C E, cutting the crooked line
CIA P E in I and P, and the strait lines A C and
A E in Iv and M. Moreover, let Z O be drawn
parallel to the same C E ; and let it be, as L G to
LI, so LM to another, LN ; and let the same be
done in all the rest of the strait lines possible,
parallel to the base ; and through all the points N,
let the line A N E be drawn ; the three-sided figure
A N E B will therefore be the moment of the figure
CI APE. Now the triangle ABE is (by the
9th article of chapter xvii) to the three-sided
figure A N E B, as A B C D + AIC B is to AIC B
twice taken, that is, as CDFE+CIAPE is to
CIA P E twice taken. But as CIA P E is to
C DFE, that is, as the weight of the deficient
figure is to the weight of the complete figure, so is
CIA P E twice taken to C D F E twice taken.
Wherefore, setting in order CDFE + CIAPE.
2 CIAPE. 2CDFE; the proportion of CDFE +
CIAPE to CDFE twice taken will be com¬
pounded of the proportion of CDFE + CIAPE
to CIAPE twice taken, that is, of the proportion
centre of equiponderat1 on. 361
of the triangle ABE to the three-sided figure part hi.
A N E B, that is, of the moment of the complete -—*—-
figure to the moment of the deficient figure, and of ofheeq„SpITnde-
the proportion of CIA P E twice taken to C D F E ration» &c-
twice taken, that is, to the proportion reciprocally
taken of the weight of the deficient figure to the
weight of the complete figure.
Again, seeing by supposition AZ.ZB::CDFE.
CIA P E are proportionals ; AB. AZ::CDFE +
CIAPE.CDFE wTill also, by compounding, be
proportionals. And seeing A L is the half of A B,
AL. AZ::CDFE + CI APE. 2 C D F E will also
be proportionals. But the proportion of CDFE +
CIAPEto2CDFE is compounded, as was but
now shown, of the proportions of moment to mo¬
ment, &c., and therefore the proportion of A L to
AZ is compounded of the proportion of the mo¬
ment of the complete figure CDFEto the moment
of the deficient figure CI APE, and of the pro¬
portion of the wreight of the deficient figure CIAPE
to the weight of the complete figure CD FE ; but
the proportion of A L to A Z is compounded of the
proportions of AL to B Z and of BZ to AZ. Now
the proportion of B Z to A Z is the proportion of
the weights reciprocally taken, that is to say, of the
weight CIAPE to the weight C D F E. There¬
fore the remaining proportion of A L to B Z, that
is, of L B to B Z, is the proportion of the moment
of the weight CDFE to the moment of the wreight
CIAPE. But the proportion of A L to B Z is
compounded of the proportions of A L to A Z and
of A Z to Z B ; of which proportions that of A Z to
Z B is the proportion of the weight C D F E to the
weight CIAPE. Wherefore (by art. 5 of this
362 motions and magnitudes.
part hi. chapter) the remaining proportion of AL to AZ is
-—- the proportion of the distances of the points Z and
The diameter ^ from the centre of the scale, which is A. And,
of equiponde- _ y 7
ration, &c. therefore, (by art. 6) the weight CI APE shall hang
from O in the strait line O Z. So that O Z is one
diameter of eqniponderation of the weight CI APE.
But the strait line AB is the other diameter of equi-
ponderation of the same weight CI APE. Where¬
fore (by the 7th definition) the point Z is the centre
of the same eqniponderation; which point, by con¬
struction, divides the axis so, that the part AZ,
which is the part next the vertex, is to the other
part Z B, as the complete figure C D F E is to the
deficient figure CI APE ; which is that which was
to be demonstrated.
Coroll. i. The centre of equiponderation of any of
those plane three-sided figures, which are compared
with their complete figures in the table of art. 3,
chap, xvn, is to be found in the same table, by
taking the denominator of the fraction for the part
of the axis cut off next the vertex, and the nume¬
rator for the other part next the base. For example,
if it be required to find the centre of equipondera¬
tion of the second three-sided figure of four means,
there is in the concourse of the second column
with the row of three-sided figures of four means
this fraction f, which signifies that that figure is
to its parallelogram or complete figure as f to
unity, that is, as f to 7r, or as 5 to 7 ; and, there¬
fore the centre of equiponderation of that figure
divides the axis, so that the part next the vertex
is to the other part as 7 to 5.
Coroll. ii. The centre of equiponderation of any
of the solids of those figures., which are contained
centre of equiponderation. 363
in the table of art. 7 of the same chap, xvii, is PA IIL
exhibited in the same table. For example, if the 1 *—'
centre of equiponderation of a cone be sought for,
the cone will be found to be i of its cylinder ; and,
therefore, the centre of its equiponderation will so
divide the axis, that the part next the vertex to
the other part will be as 3 to 1. Also the solid of
a three-sided figure of one mean, that is, a para¬
bolical solid, seeing it is t, that is } of its cylinder,
will have its centre of equiponderation in that
point, which divides the axis, so that the part
towards the vertex be double to the part towards
the base.
10. The diameter of equiponderation of the com- The diameter
plement of the half of any of those figures which ration of the
are contained in the table of art. 3, chap, xvii, hsftflny
divides that line which is drawn through the ver- °f.tlle ®aid de_
° iicient figures,
tex parallel and equal to the base, so that the part divides that
next the vertex will be to the other part, as the drawn "hroigh
complete figure to the complement. niiiei'tolhe1^1"
For let A I C B (in the same fig. 5) be the half b,ase> so that
° the part next
of a parabola, or of any other of those three-sided the vertex is to
figures which are in the table of art. 3, chap, xvii, lthfcompus
whose axis is A B, and base B C, having A D complement,
drawn from the vertex, equal and parallel to the
base B C, and whose complete figure is the pa¬
rallelogram A B C D. Let I Q, be drawn at any
distance from the side C D, but parallel to it; and
let AD be the altitude of the complement AICD,
and Q, I a line ordinately applied in it. Wherefore
the altitude A L in the deficient figure AI C B is
equal to Q, I the line ordinately applied in its com¬
plement ; and contrarily, LI the line ordinately
applied in the figure AICB is equal to the altitude
364 motions and magnitudes.
PA^J IIT* ^ Q in its complement; and so in all the rest of
-—■—- the ordinate lines and altitudes the mutation is
o^equiponde- suc^j that that line, which is ordinately applied in
ration, &c. the figure, is the altitude of its complement. And,
therefore, the proportion of the altitudes decreas¬
ing to that of the ordinate lines decreasing, being
multiplicate according to any number in the defi¬
cient figure, is submultiplicate according to the
same number in its complement. For example, if
A I C B be a parabola, seeing the proportion of
A B to A L is duplicate to that of B C to L I, the
proportion of AD to AO, in the complement AICD,
which is the same with that of B C to L I, will be
subduplicate to that of CD to Q,I, which is the
same with that of A B to A L ; and consequently,
in a parabola., the complement will be to the paral¬
lelogram as 1 to 3 ; in a three-sided figure of
two means, as 1 to 4 ; in a three-sided figure of
three means, as 1 to 5, &c. But all the ordinate
lines together in AICD are its moment; and all
the ordinate lines in AICB are its moment. Where¬
fore the moments of the complements of the halves
of deficient figures in the table of art. 3 of chap,
xvn, being compared, are as the deficient figures
themselves ; and, therefore, the diameter of equi-
ponderation will divide the strait line A D in such
proportion, that the part next the vertex be to the
other part, as the complete figure A B C D is to
the complement AICD.
Coroll. The diameter of equiponderation of these
halves may be found by the table of art. 3 of chap.
xyii, in this manner. Let there be propounded
any deficient figure, namely, the second three-sided
figure of two means. This figure is to the com_
CENTRE OF EQUIPONDERATION. 365
plete figure as I to 1, that is 3 to 5. Wherefore part nr.
the complement to the same complete figure is as -—
2 to 5; and, therefore, the diameter of equipon-
deration of this complement will cut the strait
line drawn from the vertex parallel to the base, so
that the part next the vertex will be to the other
part as 5 to 2. And, in like manner, any other of
the said three-sided figures being propounded, if
the numerator of its fraction found out in the table
be taken from the denominator, the strait line
drawn from the vertex is to be divided, so that the
part next the vertex be to the other part, as the
denominator is to the remainder which that sub¬
traction leaves.
11. The centre of equiponderation of the half of Thecen1-re of
^ , . . equipondera-
any of those crooked-lined figures, which are in tion of the half
the first row of the table of art. 3 of chap, xvn, is deficient fi-6
in that strait line which, being parallel to the axis, |"srtesro^ JJ®he
divides the base according to the numbers of the table of art 3>
n . , •, . . , -i -l chapter xvn,
traction next below it m the second row, so that may be found
the numerator be answerable to that part which is numYer^ofthe
towards the axis. second row.
For example, let the first figure of three means
be taken, whose half is A B C D (in fig. 6), and let
the rectangle ABED be completed. The com¬
plement therefore will be B C D E. And seeing
A B E D is to the figure A B C D (by the table) as
5 to 4, the same ABED will be to the comple¬
ment B C D E as 5 to 1. Wherefore, if F G be
drawn parallel to the base D A, cutting the axis so
that A G be to G B as 4 to 5, the centre of equi¬
ponderation of the figure A B C D will, by the pre¬
cedent article, be somewhere in the same F G.
Again, seeing, by the same article, the complete
366
MOTIONS AND MAGNITUDES.
PAIo3 Iir* figure ABED, is to the complement B C D E as
s—-—- 5 to 1, therefore if B E and A D be divided in I
equlpondara-" an(l H as 5 to 1, the centre of eqniponderation of
tion, &c. complement B C D E will be somewhere in the
strait line which connects H and I. Let now the
strait line L K be drawn through M the centre of
the complete figure, parallel to the base ; and the
strait line N O through the same centre M, perpen¬
dicular to it; and let the strait lines LK and FG cut
the strait line HI in P and Q,. Let P R be taken
quadruple to PQ ; and let RM be drawn and pro¬
duced to F G in S. R M therefore will be to M S
as 4 to 1, that is, as the figure A B C D to its com¬
plement B C D E. Wherefore, seeing M is the
centre of the complete figure ABED, and the dis¬
tances of R and S from the centre M be in propor¬
tion reciprocal to that of the weight of the com¬
plement BCDE to the weight of the figure ABCD,
R and S will either be the centres of equiponderation
of their own figures, or those centres will be in some
other points of the diameters of equiponderation
HI and FG. But this last is impossible. For no other
strait line can be drawn through the point M ter¬
minating in the strait lines HI and FG, and retain¬
ing the proportion of MR to M S, that is, of the
figure ABCD to its complement BCDE. The
centre, therefore, of equiponderation of the figure
A B C D is in the point S. Now, seeing PM hath
the same proportion to Q S which R P hath to RQ,
Q S will be 5 of those parts of which P M is four,
that is, of which IN is four. But IN or P M is 2
of those parts of which EB or FG is 6 ; and, there¬
fore, if it be as 4 to 5, so 2 to a fourth, that fourth
CENTRE OF E QUI POND E RATI ON. 367
will be 2-2-. Wherefore QS is 2£ of those parts rAR<J111,
of whieh F G is 6. But FQ, is 1 ; and, therefore, —'
FS is 3}. Wherefore the remaining part GS is 21.
So that F G is so divided in S, that the part to¬
wards the axis is in proportion to the other part,
as to 3i, that is as 5 to 7; which answereth to
the fraction f in the second row, next under the
fraction i in the first row. Wherefore drawing
S T parallel to the axis, the base will be divided in
like manner.
By this method it is manifest, that the base of a
semiparabola will be divided into 3 and 5 ; and the
base of the first three-sided figure of two means,
into 4 and 6 ; and of the first three-sided figure of
four means, into 6 and 8. The fractions, there¬
fore, of the second row denote the proportions,
into which the bases of the figures of the first row
are divided by the diameters of equiponderation.
But the first row begins one place higher than the
second row.
12. The centre of equiponderation of the half of The centre of
any of the figures in the second row of the same tion of the half
table of art. 3, chap, xvn, is in a strait line parallel °0ff the
to the axis, and dividing the base according to the ^he°"aine°tlb"e
numbers of the fraction in the fourth row, two may he found
. out by the
places lower, so as that the numerator be answer-numbers of the
able to that part which is next the axis.
Let the half of the second three-sided figure of
two means be taken; and let it be A B C D (in
fig. 7); whose complement is BCDE, and the
rectangle completed ABED. Let this rectangle
be divided by the two strait lines L K and N O,
cutting one another in the centre M at right
angles ; and because A B E D is to A B C D as 5 to
368 motions and magnitudes.
part hi. 3 let AB be divided in G, so that AG to BG be as
23
^—- 3 to 5 ; and let F G be drawn parallel to the base,
^iponto,-'Also because ABED is (by art. 9) to BCDE
tion, &c. as 5 to 2, let B E be divided in the point I, so that
B I be to IE as 5 to 2 ; and let I H be drawn
parallel to the axis, cutting LK and FG in P and Q,.
Let now PR be so taken, that it be to P Q, as 3 to
2, and let RM be drawn and produced to FG in S.
Seeing, therefore, R P is to P Q, that is, R M to
MS, as A B C D is to its complement BCDE,
and the centres of equiponderation of A B C D and
BCDE are in the strait lines F G and H I, and
the centre of equiponderation of them both toge¬
ther in the point M ; R will be the centre of the
complement BCDE, and S the centre of the
figure A B C D. And seeing P M, that is IN, is
to Q S, as R P is to R Q ; and IN or P M is 3 of
those parts, of which B E, that is F G, is 14 ; there¬
fore Q S is 5 of the same parts ; and E I, that is
FQ,, 4; and F S, 9 ; and G S, 5. Wherefore the
strait line S T being drawn parallel to the axis,
will divide the base A D into 5 and 9. But the
fraction f is found in the fourth row of the table,
two places below the fraction f in the second row.
By the same method, if in the same second row
there be taken the second three-sided figure of
three means, the centre of equiponderation of the
half of it will be found to be in a strait line parallel
to the axis, dividing the base according to the
numbers of the fraction tV, two places below in
the fourth row. And the same way serves for all
the rest of the figures in the second row. In like
manner, the centre of equiponderation of the third
three-sided figure of three means will be found to
centre of e qui pon deration. 369
be in a strait line parallel to the axis, dividing the part tii.
base, so that the part next the axis be to the other •—*—'
part as 7 to 13, &e.
Coroll. The centres of equiponderation of the
halves of the said figures are known, seeing they
are in the intersection of the strait lines S T and
F G, which are both known.
13. The centre of equiponderation of the half of The centre of
, . , . , ,, r. equipondera-
any of the figures, which (m the table of art. 3, tion of the half
chap, xvii) are compared with their parallelo- figures in the"
grams, being known; the centre of equiponderation ® n
of the excess of the same figure above its triangle l}ie centre of
° ° the excess of
is also known. the same figure
For example, let the semiparabola A B C D (in jfe° of the^ame
fig. 8) be taken, whose axis is A B; whose com-
plete figure is ABED; and whose excess above knowu-
its triangle is B C D B. Its centre of equiponde¬
ration may be found out in this manner. Let FG
be drawn parallel to the base, so that A F be a
third part of the axis ; and let HI be drawn pa¬
rallel to the axis, so that A H be a third part of
the base. This being done, the centre of equi¬
ponderation of the triangle ABD will be I. Again,
let K L be drawn parallel to the base, so that
A K be to A B as 2 to 5 ; and M N parallel to the
axis, so that A M be to A D as 3 to 8; and let
MN terminate in the strait line KL. The centre,
therefore, of equiponderation of the parabola
A BCD is N; and therefore we have the centres
of equiponderation of the semiparabola A B C D,
and of its part the triangle ABD. That we may
now find the centre of equiponderation of the
remaining part B CDB, let IN be drawn and
produced to 0, so that N 0 be triple to I N ; and
vol. i. b b
370 MOTIONS AND MAGNITUDES.
PA^3 111 ^ cen^re sought for. For seeing the
v—t—' weight of A B D to the weight of B C D B is in
Jqif;pon°dera-f proportion reciprocal to that of the strait line N 0
half °&che s^ra^ ^ne I 5 aild N is the centre of the
whole, and I the centre of the triangle ABD; O
will be the centre of the remaining part, namely,
of the figure BDCB; which was to be found.
Coroll. The centre of equiponderation of the
figure BDCB is in the concourse of two strait
lines, whereof one is parallel to the base, and
divides the axis, so that the part next the base be
f or of the whole axis; the other is parallel to
the axis, and so divides the base, that the part
towards the axis be i, or J-f of the whole base.
For drawing 0 P parallel to the base, it will be as
I N to N 0, so F K to K P, that is, so 1 to 3, or
5 to 15. But A F is tV, or | of the whole AB ;
and A K is T6T, or f; and FI( n; and K P tV;
and therefore A P is tV of the axis A B. Also AH
is i, or ^4 ; and A M §■, or of the whole base;
and therefore 0 Q, being drawn parallel to the
axis, M Q, which is triple to H M, will be -A.
Wherefore A Q is H, or k of the base A D.
The excesses of the rest of the three-sided
figures in the first row of the table of art. 3, chap,
xvn, have their centres of equiponderation in two
strait lines, which divide the axis and base accord¬
ing to those fractions, which add 4 to the nume¬
rators of the fractions of a parabola x-5- and if; and
6 to the denominators, in this manner:—
In a parabola, the axis the base
In the first three-sided figure, the axis M, the base It
In the second three-sided figure, the axis the base II, &c.
And by the same method, any man, if it be
CENTRE OF EQUIPONDERATION. 371
wortli the pains, may find out the centres of equi- part IIL
ponderation of the excesses above their triangles «—r—-
of the rest of the figures in the second and third
row, &c.
14. The centre of equiponderation of the sector The centre of
of a sphere, that is, of a figure compounded of a S^oTa^oiid
right cone, whose vertex is the centre of the aSfSo divide^
sphere, and the portion of the sphere whose base that ll)e Part
x . . . next the vertex
is the same with that of the cone, divides the strait be to the whole
line which is made of the axis of the cone and half ha]fthenaxisgof
the axis of the portion together taken, so that the sphere) as
part next the vertex be triple to the other part, or310 4-
to the whole strait line as 3 to 4.
For let A B 0 (in fig. 9) be the sector of a
sphere, whose vertex is the centre of the sphere A;
whose axis is A D ; and the circle upon B C is the
common base of the portion of the sphere and of
the cone whose vertex is A; the axis of which
portion is E D, and the half thereof F D ; and the
axis of the cone, A E. Lastly, let A G be f- of the
strait line A F. I say, G is the centre of equipon¬
deration of the sector ABC.
Let the strait line F H be drawn of any length,
making right angles with A F at F ; and drawing
the strait line AH, let the triangle AFH be made-
Then upon the same centre A let any arch I K be
drawn, cutting A D in L; and its chord, cutting
AD in M ; and dividing ML equally in N, let NO
be drawn parallel to the strait line F H, and meet¬
ing with the strait line A H in 0.
Seeing now B D C is the spherical superficies of
the portion cut off with a plane passing through
B C, and cutting the axis at right angles; and
seeing F H divides E D, the axis of the portion,
15 B 2
372
MOTIONS AND MAGNITUDES.
PAI23m' *n*:o ^W0 e(lua^ parts in F ; the centre of equipon-
v—<■— deration of the superficies B D C will be in F (by
ponderation "of art. 8); and for the same reason the centre of
a solid, &c. eqniponderation of the superficies ILK, K being
in the strait line A C, will be in N. And in like
manner, if there were drawn, between the centre
of the sphere A and the outermost spherical super¬
ficies of the sector, arches infinite in number, the
centres of equiponderation of the spherical super¬
ficies, in which those arches are, would be found
to be in that part of the axis, which is intercepted
between the superficies itself and a plane passing
along by the chord of the arch, and cutting the
axis in the middle at right angles.
Let it now be supposed that the moment of the
outermost spherical superficies BDC is FH. See¬
ing therefore the superficies B D C is to the super¬
ficies ILK in proportion duplicate to that of the
arch BD C to the arch ILK, that is, of BE to
IM, that is, of F H to NO; let it be as F H to
N 0, so N 0 to another N P; and again, as N 0 to
N P, so N P to another N Q; and let this be done
in all the strait lines parallel to the base F H that
that can possibly be drawn between the base and
the vertex of the triangle A F H. If then through
all the points Q there be drawn the crooked line
A Q, H, the figure A F H Q, A will be the comple¬
ment of the first three-sided figure of two means;
and the same will also be the moment of all the
spherical superficies, of which the solid sector
A B C D is compounded ; and by consequent, the
moment of the sector itself. Let now F H be un¬
derstood to be the semidiameter of the base of a
right cone, whose side is AH, and axis AF
CENTRE OF EQUIPONDERATION. 373
Wherefore, seeing the bases of the cones, which PAll2^IIL
pass through F and N and the rest of the points v—-—'
of the axis, are in proportion duplicate to that of p^derafon^of
the strait lines F H and N O, &c., the moment ofasolid'&c-
all the bases together, that is, of the whole cone,
will be the figure itself AFH QA; and therefore
the centre of equiponder ation of the cone A F H is
the same with that of the solid sector. Wherefore,
seeing A G is f of the axis A F, the centre of equi-
ponderation of the cone A F H is in G; and there¬
fore the centre of the solid sector is in G also, and
divides the part A F of the axis so that A G is
triple to G F; that is, A G is to A F as 3 to 4;
which was to be demonstrated.
Note, that when the sector is a hemisphere, the
axis of the cone vanisheth into that point which
is the centre of the sphere; and therefore it
addetli nothing to half the axis of the portion.
Wherefore, if in the axis of the hemisphere there
be taken from the centre | of half the axis, that is,
f of the semidiameter of the sphere, there will be
the centre of equiponderation of the hemisphere.
3/4 motions and magnitudes.
CHAPTER XXIV.
of refraction and reflection.
1. Definitions.—2. In perpendicular motion there is no refrac¬
tion.—3. Things thrown out of a thinner into a thicker me¬
dium are so refracted that the angle refracted is greater than
the angle of inclination.—4. Endeavour, which from one
point tendeth every way, will be so refracted, as that the sine
of the angle refracted will be to the sine of the angle of incli¬
nation, as the density of the first medium is to the density of
the second medium, reciprocally taken.—5. The sine of the
refracted angle in one inclination is to the sine of the refracted
angle in another inclination, as the sine of the angle of that
inclination is to the sine of the angle of this inclination.—6. If
two lines of incidence, having equal inclination, be the one in
a thinner, the other in a thicker medium, the sine of the angle
of inclination will be a mean proportional between the two
sines of the refracted angles.—7- If the angle of inclination
be semirect, and the line of inclination be in the thicker me¬
dium, and the proportion of their densities be the same with
that of the diagonal to the side of a square, and the separating
superficies be plane, the refracted line will be in the separating
superficies.—8. If a body be carried in a strait line upon
another body, and do not penetrate the same, but be reflected
from it, the angle of reflection will be equal to the angle of
incidence.—9. The same happens in the generation of motion
in the line of incidence.
PART III.
24.
Definitions.
definitions.
i. Refraction is the breaking of that strait
line, in which a body is moved or its action would
proceed in one and the same medium, into two
strait lines, by reason of the different natures of
the two mediums.
ii. The former of these is called the line of
incidence; the latter the refracted line.
OF REFRACTION and REFLECTION. 375
hi. The point of refraction is the common partiil
point of the line of incidence, and of the refracted -—.—'
Jjjjg Definitions.
iy. The refracting superficies, which also is
the separating superficies of the two mediums, is
that in which is the point of refraction.
v. The angle refracted is that, which the re¬
fracted line makes in the point of refraction with
that line, which from the same point is drawn per¬
pendicular to the separating superficies in a diffe¬
rent medium.
yi. The angle of refraction is that which the
refracted line makes with the line of incidence
produced.
vii. The angle of inclination is that which the
line of incidence makes with that line, which from
the point of refraction is drawn perpendicular to
the separating superficies.
viii. The angle of incidence is the complement
to a right angle of the angle of inclination.
And so, (in fig. 1) the refraction is made in
A B F. The refracted line is B F. The line of
incidence is A B. The point of incidence and of
refraction is B. The refracting or separating su¬
perficies is D B E. The line of incidence produced
directly is A B C. The perpendicular to the sepa¬
rating superficies is B H. The angle of refraction
is C B F. The angle refracted is H B F. The
angle of inclination is A B G or H B C. The angle
of incidence is A B D.
ix. Moreover the thinner medium is understood
to be that in which there is less resistance to mo¬
tion, or to the generation of motion; and the
thicker that wherein there is greater resistance.
3/6 MOTIONS AND MAGNITUDES.
PA]^4 iit* x* medium in which there is equal re-
—r—' sistance everywhere, is a homogeneous medium.
All other mediums are heterogeneous.
In perpendi- 2. If a body pass, or there be generation of mo-
there Ts no'1 tion from one medium to another of different
refraction, density, in a line perpendicular to the separating
superficies, there will be no refraction.
For seeing on every side of the perpendicular
all things in the mediums are supposed to be like
and equal, if the motion itself be supposed to be
perpendicular, the inclinations also will be equal,
or rather none at all; and therefore there can be
110 cause from which refraction may be inferred to
be on one side of the perpendicular, which will
not conclude the same refraction to be on the
other side. Which being so, refraction on one
side will destroy refraction on the other side ; and
consequently either the refracted line will be
everywhere, which is absurd, or there will be no
refracted line at all; which was to be demonstrated.
Coroll. It is manifest from hence, that the cause
of refraction consisteth only in the obliquity of the
line of incidence, whether the incident body pene¬
trate both the mediums, or without penetrating,
propagate motion by pressure only.
Tilings thrown 3. If a body, without any change of situation of
thicker internal parts, as a stone, be moved obliquely
refracted81 that out thinner medium, and proceed penetrating
the angle re- the thicker medium, and the thicker medium be
greater than such, as that its internal parts being moved restore
iilciinafioiK themselves to their former situation ; the angle
refracted will be greater than the angle of incli¬
nation.
of refraction and reflection. 377
For let DBE (in the same first figure) be tlie part hi.
separating superficies of two mediums ; and let a -——-
body, as a stone thrown, be understood to be t^™
moved as is supposed in the strait line ABC; and int° a thicker
# medium, &.c»
let A B be m the thinner medium, as m the air ;
and B C in the thicker, as in the water. I say the
stone, which being thrown, is moved in the line
A B, will not proceed in the line B C, but in some
other line, namely, that, with which the perpendi¬
cular B H makes the refracted angle H B F greater
than the angle of inclination H B C.
For seeing the stone coming from A, and falling
upon B, makes that which is at B proceed towards
H, and that the like is done in all the strait lines
which are parallel to B H ; and seeing the parts
moved restore themselves by contrary motion in
the same line; there will be contrary motion gene¬
rated in H B, and in all the strait lines which are
parallel to it. Wherefore, the motion of the stone
will be made by the concourse of the motions in
A G, that is, in D B, and in G B, that is, in B H,
and lastly, in H B, that is, by the concourse of
three motions. But by the concourse of the mo¬
tions in A G and B H, the stone will be carried to
C ; and therefore by adding the motion in H B, it
will be carried higher in some other line, as in
B F, and make the angle H B F greater than the
angle HBC.
And from hence may be derived the cause, why
bodies which are thrown in a very oblique line, if
either they be any thing fiat, or be thrown with
great force, will, when they fall upon the water, be
cast up again from the water into the air.
For let A B (in fig. 2) be the superficies of the
3/8
MOTIONS AND MAGNITUDES.
Pai^4 m' water; into which, from the point C, let a stone be
-—•—' thrown in the strait line C A, making with the line
B A produced a very little angle CAD; and pro¬
ducing B A indefinitely to D, let C D be drawn per¬
pendicular to it, and A E parallel to CD. The stone
therefore will be moved in C A by the concourse
of two motions in C D and D A, whose velocities
are as the lines themselves C D and D A. And from
the motion in C D and all its parallels downwards,
as soon as the stone falls upon A, there will be
reaction upwards, because the water restores itself
to its former situation. If now the stone be thrown
with sufficient obliquity, that is, if the strait line
C D be short enough, that is, if the endeavour of
the stone downwards be less than the reaction of
the water upwards, that is, less than the endeavour
it hath from its own gravity (for that may be), the
stone will by reason of the excess of the endeavour
which the water hath to restore itself, above that
which the stone hath downwards, be raised again
above the superficies A B, and be carried higher,
being reflected in a line which goes higher, as the
line A G.
Endeavour, 4. If from a point, whatsoever the medium be, en-
whicli from one A .
point tendeth deavour be propagated every way into all the parts
belrfrefra'cted, of that medium ; and to the same endeavour there
ofth^an^iere- obliquely opposed another medium of a different
fracted will be nature, that is, either thinner or thicker; that
to the sine of
the angle of in- endeavour will be so refracted, that the sine or the
elination.as the i t _p*i*
density of the angle retracted, to the sine or the angle or mclina-
t::::;;; tion, will be as the density of the first medium to
of the second the density of the second medium, reciprocally
medium, reei- i
procally taken, taken.
First, let a body be in the thinner medium in A
of refraction and reflection. 379
(fig. 3), and let it be understood to have endeavour PAI^ hi.
every way, and consequently, that its endeavour *—-
proceed in the lines A B and A b ; to which let
B b the superficies of the thicker medium be
obliquely opposed in B and b, so that A B and A b
be equal; and let the strait line B b be produced
both ways. From the points B and b, let the per¬
pendiculars B C and b c be drawn ; and upon the
centres B and b, and at the equal distances B A and
b A, let the circles AC and A c be described, cutting
B C and b c in C and c, and the same C B and c b
produced in D and d, as also A B and A b produced
in E and e. Then from the point A to the strait
lines B C and b c let the perpendiculars A F and A f
be drawn. A F therefore will be the sine of the
angle of inclination of the strait line A B, and Af
the sine of the angle of inclination of the strait
line A h, which two inclinations are by construc¬
tion made equal. I say, as the density of the
medium in which are B C and b c is to the density
of the medium in which are B D and b d, so is the
sine of the angle refracted, to the sine of the angle
of inclination.
Let the strait line F G be drawn parallel to the
strait line A B, meeting with the strait line b B
produced in G.
Seeing therefore A F and B G are also parallels,
they will be equal; and consequently, the endea¬
vour in AF is propagated in the same time, in
which the endeavour in B G would be propagated
if the medium were of the same density. But
because B G is in a thicker medium, that is, in a
medium which resists the endeavour more than the
Medium in which AF is, the endeavour will be
380
MOTIONS AND MAGNITUDES.
PAI^>4 m* ProPagated less in B G than in A F, according to
the proportion which the density of the medium, in
which*'&c.r' which A F is, hath to the density of the medium in
which B G is. Let therefore the density of the
medium, in which B G is, be to the density of the
medium, in which A F is, as B G is to BH; and
let the measure of the time be the radius of the
circle. Let H I be drawn parallel to B D, meeting
with the circumference in I; and from the point
I let I K be drawn perpendicular to BD; which
being done, B H and I K will be equal; and IK
will be to A F, as the density of the medium in
which is A F is to the density of the medium in
which is IK. Seeing therefore in the time AB,
which is the radius of the circle, the endeavour is
propagated in A F in the thinner medium, it will
be propagated in the same time, that is, in the
time BI in the thicker medium from K to I.
Therefore, B I is the refracted line of the line of
incidence A B; and I K is the sine of the angle
refracted : and A F the sine of the angle of incli¬
nation. Wherefore, seeing IK is to AF, as the
density of the medium in which is A F to the
density of the medium in which is IK; it will be
as the density of the medium in which is A F or.
B C to the density of the medium in which is
IK or B D, so the sine of the angle refracted to
the sine of the angle of inclination. And by the
same reason it may be shown, that as the density
of the thinner medium is to the density of the
thicker medium, so will KI the sine of the angle
refracted be to A F the sine of the angle of incli¬
nation.
Secondly, let the body, which endeavoureth every
OF REFRACTION AND REFLECTION. 381
way, be in the thicker medium at I. If, therefore, part iii.
both the mediums were of the same density, the *—
endeavour of the body in I B would tend directly
to L; and the sine of the angle of inclination L M
would be equal to I K or B II. But because the
density of the medium, in which is I Iv, to the
density of the medium, in which is L M, is as B H
to B G, that is, to A F, the endeavour will be pro¬
pagated further in the medium in which L M is,
than in the medium in which I K is, in the propor¬
tion of density to density, that is, of M L to A F.
Wherefore, B A being drawn, the angle refracted
will be C B A, and its sine A F. But L M is the
sine of the angle of inclination; and therefore
again, as the density of one medium is to the
density of the different medium, so reciprocally
is the sine of the angle refracted to the sine of
the angle of inclination; which was to be demon¬
strated.
In this demonstration, I have made the sepa¬
rating superficies B b plane by construction. But
though it were concave or convex, the theorem
would nevertheless be true. For the refraction
being made in the point B of the plane separating
superficies, if a crooked line, as P Q, be drawn,
touching the separating line in the point B ; neither
the refracted line B I, nor the perpendicular B D,
will be altered ; and the refracted angle K B I, as
also its sine K I, will be still the same they were.
5. The sine of the angle refracted in one incli- The sine of the
• • • « • refracted tingle
nation is to the sine of the angle refracted in in one inciina-
another inclination, as the sine of the angle of that j,1™, 0'fs t\°e ^
inclination to the sine of the ane:le of this incli- footed angle in
° another inch-
nation.
382
motions and magnitudes.
part hi. For seeing the sine of the refracted angle is to
v—r—- the sine of the angle of inclination, whatsoever
sine "of tiTe an- inclination be, as the density of one medium
gie of that in-t0 ^ie density of the other medium ; the propor-
chnation is to J A _ 1
the sine of the tion of the sine of the refracted angle, to the sine of
inclination. I1S the angle of inclination, will be compounded of the
proportions of density to density, and of the sine
of the angle of one inclination to the sine of the
angle of the other inclination. But the propor¬
tions of the densities in the same homogeneous
body are supposed to be the same. Wherefore
refracted angles in different inclinations are as the
sines of the angles of those inclinations; which
was to be demonstrated,
if two lines o jf two lines of incidence, having; equal inch-
of incidence, # # .
having equal nation, be the one in a thinner, the other in a
oneina thinner thicker medium, the sine of the angle of their in-
thiekerme-11 a clination will be a mean proportional between the
dium, the sine two sines of their angles refracted.
or the angle or ...
inclination will For let the strait line A13 (in fig. 3) have its in-
portimiai be-r°" clination in the thinner medium, and be refracted
SfeTofthere- ^ie thicker medium in BI; and let EB have as
fracted angles, much inclination in the thicker medium, and be
refracted in the thinner medium in B S; and let
R S, the sine of the angle refracted, be drawn. I
say, the strait lines R S, A F, and I K are in con¬
tinual proportion. For it is, as the density of the
thicker medium to the density of the thinner me¬
dium, so R S to A F. But it is also as the den¬
sity of the same thicker medium to that of the
same thinner medium, so A F to IK. Wherefore
R S. A F :: A F. IK are proportionals; that is, R S,
A F, and I K are in continual proportion, and A F
is the mean proportional; which was to be proved.
of refraction and reflection.
383
7. If the allele of inclination be semirect, and part hi.
. . . . 21-.
the line of inclination be in the thicker medium, *—'
and the proportion of the densities be as that of a ScUnaX'6 be
diagonal to the side of its square, and the sepa- tiT" n^'of^n-
ratine: superficies be plain, the refracted line will cimation be in
, . . the thicker me-
be m that separating superficies. dium, and the
For in the circle A C (fig. 4) let the angle of in- fhei^densiticl
clination ABC be an angle of 45 degrees. Let^^hUiatoHhe
C B be produced to the circumference in D ; and diagonal to the
, . i side ofa square,
let C Jli, the sine or the angle LBC, be drawn, to and the sePa-
which let B F be taken equal in the separating Sfbe pE"
lineBG. BCEF will therefore be a parallelo- ]he ref^cted.
a line will be m
gram, and F E and B C, that is F E and B G equal. the separating
# superficies*
Let A G be drawn, namely the diagonal of the
square whose side is B G, and it will be, as A G to
E F so B G to B F; and so, by supposition, the
density of the medium, in which C is, to the den¬
sity of the medium in which D is; and so also the
sine of the angle refracted to the sine of the angle
of inclination. Drawing therefore F D, and from
D the line D H perpendicular to A B produced,
D H will be the sine of the angle of inclination.
And seeing the sine of the angle refracted is to
the sine of the angle of inclination, as the density
of the medium, in which is C, is to the density of
the medium in which is D, that is, by supposition,
as A G is to F E, that is as B G is to D H; and
seeing D H is the sine of the angle of inclination,
BG will therefore be the sine of the angle re¬
fracted. Wherefore B G will be the refracted line,
and lye in the plain separating superficies ; which
was to be demonstrated.
Coroll. It is therefore manifest, that when the
inclination is greater than 45 degrees, as also
384 MOTIONS AND MAGNITUDES.
part nr. when it is less, provided the density be greater, it
-—r—' may happen that the refraction will not enter the
thinner medium at all.
if a body be 8. If a body fall in a strait line upon another
stvakHneaponhody, and do not penetrate it, but be reflected
another body, fr0m it, the angle of reflection will be equal to
and do not pe- ' . 9 1
netrateit,butbe the angle of incidence.
it, the allele of Let there be a body at A (in fig. 5), which fall-
be'equai" o 'tile strait motion in the line A C upon another
d'ence °f body passeth no further, but is reflected; and
let the angle of incidence be any angle, as A C D.
Let the strait line C E be drawn, making with D C
produced the angle E C F equal to the angle
A C D ; and let A D be drawn perpendicular to
the strait line D F. Also in the same strait line
D F let C G be taken equal to C D; and let the
perpendicular G E be raised, cutting C E in E.
This being done, the triangles A C D and E C G
will be equal and like. Let C H be drawn equal
and parallel to the strait line A D ; and let H C be
produced indefinitely to I. Lastly let E A be
drawn, which will pass through H, and be parallel
and equal to GD. I say the motion from A to C,
in the strait line of incidence A C, will be reflected
in the strait line CE.
For the motion from A to C is made by two co¬
efficient or concurrent motions, the one in A H
parallel to D G, the other in A D perpendicular to
the same D G; of which two motions that in A H
works nothing upon the body A after it has been
moved as far as C, because, by supposition, it doth
not pass the strait line D G; whereas the endea¬
vour in A D, that is in H C, worketh further to¬
wards I. But seeing it doth only press and not
OF REFRACTION AND REFLECTION. 385
penetrate, there will be reaction in H, which PA^IIL
causeth motion from C towards H ; and in the -—-—'
meantime the motion in H E remains the same it
was in A H ; and therefore the body will now be
moved by the concourse of two motions in C H
and H E, which are equal to the two motions it
had formerly in A H and H C. Wherefore it will
be carried on in CE. The angle therefore of re¬
flection will be E C G, equal, by construction, to
the angle ACD; which was to be demonstrated.
Now when the body is considered but as a point,
it is all one whether the superficies or line in
which the reflection is made be strait or crooked;
for the point of incidence and reflection C is as
well in the crooked line which toucheth D G in C,
as in D G itself.
9. But if we suppose that not a body be moved, The samthap"
r r J J pens in the
but some endeavour only be propagated from A to generation of
C, the demonstration will nevertheless be the line of inci-
same. For all endeavour is motion ; and when itdence'
hath reached the solid body in C, it presseth it,
and endeavoureth further in CI. Wherefore the
reaction will proceed in C H ; and the endeavour
in C H concurring with the endeavour in H E,
will generate the endeavour in C E, in the same
manner as in the repercussion of bodies moved.
If therefore endeavour be propagated from any
point to the concave superficies of a spherical body,
the reflected line with the circumference of a great
circle in the same sphere will make an angle equal
to the angle of incidence.
For if endeavour be propagated from A (in fig.
0) to the circumference in B, and the centre of
the sphere be C, and the line C B be drawn, as
VOL. 1. c c
386 MOTIONS AND MAGNITUDES.
PAR2^ IIJ. also the tangent D BE; and lastly if the angle
t— FB D,be made equal to the angle ABE, the re¬
flection will be made in the line B F, as hath been
newly shown. Wherefore the angles, which the
strait lines A B and F B make with the circumfe¬
rence, will also be eqnal. But it is here to be
noted, that if CB be produced howsoever to G,
the endeavour in the line G B C will proceed only
from the perpendicular reaction in G B ; and that
therefore there will be 110 other endeavour in the
point B towards the parts which are within the
sphere, besides that which tends towards the
centre.
And here I put an end to the third part of this
discourse; in which I have considered motion and
magnitude by themselves in the abstract. The
fourth and last part, concerning the phenomena of
nature, that is to say, concerning the motions and
magnitudes of the bodies which are parts of the
world, real and existent, is that which follows.
PART IV.
PHYSICS,
OR THE PHENOMENA OF NATURE.
CHAPTER XXV.
OF SENSE AND ANIMAL MOTION.
1. The connexion of what hath been said with that which fol-
lovveth.—2. The investigation of the nature of sense, and the
definition of sense.—3. The subject and object of sense.
4. The organs of sense.-—5. All bodies are not indued with
sense.—6. But one phantasm at one and the same time.
7. Imagination the remains of past sense, which also is memory.
Of sleep.—8. How phantasms succeed one another.—9.
Dreams, whence they proceed.—10. Of the senses, their kinds,
their organs, and phantasms proper and common.—11. The
magnitude of images, how and by what it is determined.
12. Pleasure, pain, appetite and aversion, what they are.
13. Deliberation and will, what.
1. I have, in the first chapter, defined philosophy part iv.
to be knowledge of effects acquired by true ratio- -—r—
cination, from knowledge first had of their causes ofhewh™nhathn
and generation; and of such causes or genera- sa|4 ™ith
Hons as may be, from former knowledge of their foiioweth.
effects or appearances. There are, therefore,
two methods of philosophy; one, from the gene¬
ration of things to their possible effects; and the
c c 2
388
physics.
part iv. other, from their effects or appearances to some
•—^—• possible generation of the same. In the former
^e-™°f these the truth of the first principles of our
been said with ratiocination, namely definitions, is made and con-
that which fol- s -J-. i i „ ,
loweth. stituted by ourselves, whilst we consent and agree
about the appellations of things. And this part I
have finished in the foregoing chapters; in which,
if I am not deceived, I have affirmed nothing,
saving the definitions themselves, which hath not
good coherence with the definitions I have given ;
that is to say, which is not sufficiently demonstrated
to all those, that agree with me in the use of words
and appellations; for whose sake only I have
written the same. I now enter upon the other
part; which is the finding out by the appearances
or effects of nature, which we know by sense, some
ways and means by which they may be, I do not
say they are, generated. The principles, therefore,
upon which the following discourse depends, are not
such as we ourselves make and pronounce in gene¬
ral terms, as definitions ; but such, as being placed
in the things themselves by the Author of Nature,
are by us observed in them ; and we make use of
them in single and particular, not universal propo¬
sitions. Nor do they impose upon us any necessity
of constituting theorems ; their use being only,
though not without such general propositions as
have been already demonstrated, to show us the
possibility of some production or generation. See¬
ing, therefore, the science, which is here taught,
hath its principles in the appearances of nature,
and endeth in the attaining of some knowledge of
natural causes, I have given to this part the title
of Physics, or the Phenomena of Nature. Now
OF SENSE AND ANIMAL MOTION. 389
such things as appear, or are shown to us by na- PARJiy-
ture, we call phenomena or appearances. v—*—'
Of all the phenomena or appearances which are ^VhaT^ath
near us, the most admirable is apparition itself,
to (palvtaOai; namely, that some natural bodies have loweth.
in themselves the patterns almost of all things, and
others of none at all. So that if the appearances
be the principles by which we know all other
things, we must needs acknowledge sense to be the
principle by which we know those principles, and
that all the knowledge we have is derived from it.
And as for the causes of sense, we cannot begin
our search of them from any other phenomenon
than that of sense itself. But you will say, by what
sense shall we take notice of sense ? I answer, by
sense itself, namely, by the memory which for some
time remains in us of things sensible, though they
themselves pass away. For he that perceives that
he hath perceived, remembers.
In the first place, therefore, the causes of our
perception, that is, the causes of those ideas and
phantasms which are perpetually generated within us
whilst we makeuse of our senses, are to be enquired
into ; and in what manner their generation pro¬
ceeds. To help which inquisition, we may observ
first of all, that our phantasms or ideas are not
always the same; but that new ones appear to us,
and old ones vanish, according as we apply our
organs of sense, now to one object, now to another.
Wherefore they are generated, and perish. And
from hence it is manifest, that they are some
change or mutation in the sentient.
2. Now that all mutation or alteration is mo¬
tion or endeavour (and endeavour also is motion)
390
PHYSICS.
PAI25IV' *n in^ernal parts of the thing that is altered,
-—r—- hath been proved (in art. 9, chap, viii) from this,
don o"tiTe'na- whilst even the least parts of any body remain
•mdthe defini'^n same situation in respect of one another, it
tion of sense, cannot be said that any alteration, nnless perhaps
that the whole body together hath been moved, hath
happened to it; but that it both appeareth and is
the same it appeared and was before. Sense,
therefore, in the sentient, can be nothing else but
motion in some of the internal parts of the sentient;
and the parts so moved are parts of the organs of
sense. For the parts of our body, by which we
perceive any thing, are those we commonly call
the organs of sense. And so we find what is the
subject of our sense, namely, that in which are the
phantasms; and partly also we have discovered
the nature of sense, namely, that it is some in¬
ternal motion in the sentient.
I have shown besides (in chap, ix, art. 7) that
no motion is generated but by a body contiguous
and moved: from whence it is manifest, that the
immediate cause of sense or perception consists in
this, that the first organ of sense is touched and
pressed. For when the uttermost part of the organ
is pressed, it no sooner yields, but the part next
within it is pressed also ; and, in this manner, the
pressure or motion is propagated through all the
parts of the organ to the innermost. And thus
also the pressure of the uttermost part proceeds
from the pressure of some more remote body, and
so continually, till we come to that from which,
as from its fountain, we derive the phantasm or idea
that is made in us by our sense. And this, what¬
soever it be, is that we commonly call the object,
of sense and animal motion. 391
Sense, therefore, is some internal motion in the pAIlT IV-
25.
sentient, generated by some internal motion of the v—*—-
parts of the object, and propagated through all the
media to the innermost part of the organ. By
which words I have almost defined what sense is.
Moreover, I have shown (art. 2, chap, xv) that
all resistance is endeavour opposite to another en¬
deavour, that is to say, reaction. Seeing, there¬
fore, there is in the whole organ, by reason of its
own internal natural motion, some resistance or
reaction against the motion which is propagated
from the object to the innermost part of the organ,
there is also in the same organ an endeavour oppo¬
site to the endeavour which proceeds from the
object; so that when that endeavour inwards is
the last action in the act of sense, then from the
reaction, how little soever the duration of it be, a
phantasm or idea hath its being ; which, by reason
that the endeavour is now outwards, doth always
appear as something situate without the organ.
So that now I shall give you the whole definition
of sense, as it is draw n from the explication of the
causes thereof and the order of its generation, thus:
sense is a phantasm, made by the reaction and
endeavour outwards in the organ of sense, caused
by an endeavour inwards from the object, remain¬
ing for some time more or less.
3. The subject of sense is the sentient itself, Xllesubjectand
namely, some living creature ; and we speak more object of sense-
correctly, when we say a living creature seeth,
than when we say the eye seeth. The object is the
thing received; and it is more accurately said,
that we see the sun, than that we see the light.
For light and colour, and heat and sound, and
392
PHYSICS.
™25 IV ot^er qualities which are commonly called sensible,
1—' are not objects, bnt phantasms in the sentients.
and object'of For a phantasm is the act of sense, and differs 110
sense. otherwise from sense than fieri, that is, being a
doing, differs from factum esse, that is, being
done; which difference, in things that are done in
an instant, is none at all; and a phantasm is made in
an instant. For in all motion which proceeds by
perpetual propagation, the first part being moved
moves the second, the second the third, and so 011
to the last, and that to any distance, how great
soever. And in what point of time the first or
foremost part proceeded to the place of the second,
which is thrust on, in the same point of time the
last save one proceeded into the place of the last
yielding part; which by reaction, in the same
instant, if the reaction be strong enough, makes a
phantasm ; and a phantasm being made, perception
is made together with it.
The organs of 4. The organs of sense, which are in the sen¬
tient, are such parts thereof, that if they be hurt,
the very generation of phantasms is thereby de¬
stroyed, though all the rest of the parts remain
entire. Now these parts in the most of living
creatures are found to be certain spirits and mem¬
branes, which, proceeding from the pi a mater,
involve the brain and all the nerves ; also the
brain itself, and the arteries which are in the
brain; and such other parts, as being stirred, the
heart also, which is the fountain of all sense, is
stirred together with them. For whensoever the
action of the object reacheth the body of the
sentient, that action is by some nerve propagated
to the brain ; and if the nerve leading thither be
OF SENSE AND ANIMAL MOTION.
393
so hurt or obstructed, tliat the motion can be PAI^IV-
2d*
propagated no further, 110 sense follows. Also if " ■—
the motion be intercepted between the brain and
the heart by the defect of the organ by which the
action is propagated, there will be no perception
of the object.
5. But though all sense, as I have said, be made A1| bo<jies are
~ ' 7 not endued
by reaction, nevertheless it is not necessary that with sense,
every thing that reacteth should have sense. I
know there have been philosophers, and those
learned men, who have maintained that all bodies
are endued with sense. Nor do I see how they
can be refuted, if the nature of sense be placed in
reaction only. And, though by the reaction of
bodies inanimate a phantasm might be made, it
would nevertheless cease, as soon as ever the
object were removed. For unless those bodies
had organs, as living creatures have, fit for the
retaining of such motion as is made in them, their
sense would be such, as that they should never
remember the same. And therefore this hath
nothing to do with that sense which is the subject
of my discourse. For by sense, we commonly
understand the judgment we make of objects by
their phantasms; namely, by comparing and dis¬
tinguishing those phantasms; which we could
never do, if that motion in the organ, by which
the phantasm is made, did not remain there for
some time, and make the same phantasm return.
Wherefore sense, as I here understand it, and
which is commonly so called, hath necessarily
some memory adhering to it, by which former and
later phantasms may be compared together, and
distinguished from one another.
394
PHYSICS.
partiv. Sense, therefore, properly so called, must ne-
-—r—' cessarily have in it a perpetual variety of phan-
not endue/16 tasms, that they may be discerned one from
with sense, another. For if we should suppose a man to be
made with clear eyes, and all the rest of his organs
of sight well disposed, but endued with no other
sense; and that he should look only upon one
thing, which is always of the same colour and
figure, without the least appearance of variety,
he would seem to me, whatsoever others may say,
to see, no more than I seem to myself to feel the
bones of my own limbs by my organs of feeling;
and yet those bones are always and on all sides
touched by a most sensible membrane. I might
perhaps say he were astonished, and looked upon
it; but I should not say he saw it; it being almost
all one for a man to be always sensible of one and
the same thing, and not to be sensible at all of
any thing.
But one phan- 6. And yet such is the nature of sense, that it
tasm at one and , . , .
the same time. CiOeS not permit a man to discern many things at
once. For seeing the nature of sense consists in
motion ; as long as the organs are employed about
one object, they cannot be so moved by another at
the same time, as to make by both their motions
one sincere phantasm of each of them at once.
And therefore two several phantasms will not be
made by two objects working together, but only
one phantasm compounded from the action of both.
Besides, as when we divide a body, we divide
its place; and when we reckon many bodies, we
must necessarily reckon as many places ; and con-
trarily, as I have shown in the seventh chapter; so
what number soever we say there be of times, we
of sense and animal motion.
395
must understand the same number of motions part iv.
25.
also ; and as oft as we count many motions, so oft 1—*—^
we reckon many times. For though the object we ®^°a"e0Ijfaa"d
look upon be of divers colours, yet with those the same time-
divers colours it is but one varied object, and not
variety of objects.
Moreover, whilst those organs which are com¬
mon to all the senses, such as are those parts of
every organ which proceed in men from the root
of the nerves to the heart, are vehemently stirred
by a strong action from some one object, they are,
by reason of the contumacy which the motion,
they have already, gives them against the reception
of all other motion, made the less fit to receive
any other impression from whatsoever other ob¬
jects, to what sense soever those objects belong.
And hence it is, that an earnest studying of one
object, takes away the sense of all other objects for
the present. For study is nothing else but a pos¬
session of the mind, that is to say, a vehement
motion made by some one object in the organs
of sense, which are stupid to all other motions as
long as this lasteth ; according to what was said
by Terence, " Populus studio stupidus in funam-
bulo animurn occuparat." For what is stupor but
that which the Greeks call dvcti<jOii<jia, that is, a
cessation from the sense of other things r Where¬
fore at one and the same time, we cannot by sense
perceive more than one single object; as in read¬
ing, we see the letters successively one by one,
and not all together, though the whole page be
presented to our eye ; and though every several
letter be distinctly written there, yet when we look
upon the whole page at once, we read nothing.
396
physics.
25 ' From hence it is manifest, that every endeavour
v T ' of the organ outwards, is not to be called sense,
but that only, which at several times is by vehe¬
mence made stronger and more predominant than
the rest; which deprives us of the sense of other
phantasms, no otherwise than the sun deprives
the rest of the stars of light, not by hindering their
action, but by obscuring and hiding them with his
excess of brightness.
Semains'of /• But the motion of the organ, by which a
SifchTho' Phantasm is made, is not commonly called sense,
is memory, except the object be present. And the phantasm
remaining after the object is removed or past by,
is called fancy, and in Latin imciginatio; which
word, because all phantasms are not images, doth
not fully answer the signification of the word fancy
in its general acceptation. Nevertheless I may
use it safely enough, by understanding it for the
Greek QavTcurla.
Imagination therefore is nothing else but sense
decaying, or weakened, by the absence of the
object. But what may be the cause of this decay
or weakening ? Is the motion the weaker, because
the object is taken away ? If it were, then phan¬
tasms would always and necessarily be less clear
in the imagination, than they are in sense; which
is not true. For in dreams, which are the imagina¬
tions of those that sleep, they are no less clear
than in sense itself. But the reason why in men
waking the phantasms of things past are more
obscure than those of things present, is this, that
their organs being at the same time moved by
other present objects, those phantasms are the less
predominant. Whereas in sleep, the passages
OF SENSE AND ANIMAL MOTTON. 397
being shut up, external action dotli not at all parti v.
disturb or hinder internal motion. —<—-
If this be true, the next thing to be considered, 0fsleeP-
will be, whether any cause may be found out, from
the supposition whereof it will follow, that the pas¬
sage is shut up from the external objects of sense
to the internal organ. I suppose, therefore, that
by the continual action of objects, to which a re¬
action of the organ, and more especially of the
spirits, is necessarily consequent, the organ is
wearied, that is, its parts are no longer moved by
the spirits without some pain; and consequently
the nerves being abandoned and grown slack, they
retire to their fountain, which is the cavity either
of the brain or of the heart; by which means the
action which proceeded by the nerves is necessarily
intercepted. For action upon a patient, that re¬
tires from it, makes but little impression at the
first; and at last, when the nerves are by little
and little slackened, none at all. And therefore
there is no more reaction, that is, no more sense,
till the organ being refreshed by rest, and by a
supply of new spirits recovering strength and
motion, the sentient awaketh. And thus it seems
to be always, unless some other preternatural
cause intervene; as heat in the internal parts
from lassitude, or from some disease stirring the
spirits and other parts of the organ in some extra¬
ordinary manner.
8. Now it is not without cause, nor so casual a Howphan-
, i • ■, ,i . .. .t , ■. , • tasms succeed
thing as many perhaps think it, that phantasms m one another,
this their great variety proceed from one another;
and that the same phantasms sometimes bring into
the mind other phantasms like themselves, and at
398
PHYSICS.
part Iv- other times extremely unlike. For in the motion
—t—' of any continued body, one part follows another by
tasms?succeea cohesion ; and therefore, whilst we turn our eyes
one another. an(j o^er organs successively to many objects, the
motion which was made by every one of them re¬
maining, the phantasms are renewed as often as
any one of those motions comes to be predominant
above the rest; and they become predominant in
the same order in which at any time formerly they
were generated by sense. So that when by length
of time very many phantasms have been generated
within us by sense, then almost any thought may
arise from any other thought; insomuch that it
may seem to be a thing indifferent and casual,
which thought shall follow which. But for the
most part this is not so uncertain a thing to waking
as to sleeping men. For the thought or phantasm
of the desired end brings in all the phantasms,
that are means conducing to that end, and that in
order backwards from the last to the first, and
again forwards from the beginning to the end.
But this supposes both appetite, and judgment to
discern what means conduce to the end, which is
gotten by experience; and experience is store of
phantasms, arising from the sense of very many
things. For (pavralecrOai and meminisse, fancy and
memory, differ only in this, that memory supposeth
the time past, which fancy doth not. In memory,
the phantasms we consider are as if they were worn
out with time ; but in our fancy we consider them
as they are ; which distinction is not of the things
themselves, but of the considerations of the sen¬
tient. For there is in memory something like that
which happens in looking upon things at a great
OF SENSE AND ANIMAL MOTION. 399
distance ; in which as the small parts of the object PA^J Iy-
are not discerned, by reason of their remoteness; so •—«—'
in memory, many accidents and places and parts
of things, which were formerly perceived by sense,
are by length of time decayed and lost.
The perpetual arising of phantasms, both in
sense and imagination, is that which we commonly
call discourse of the mind, and is common to men
with other living creatures. For he that thinketh,
compareth the phantasms that pass, that is, taketli
notice of their likeness or unlikeness to one an¬
other. And as he that observes readily the like¬
nesses of things of different natures, or that are
very remote from one another, is said to have a
good fancy; so he is said to have a good judgment,
that finds out the unlikenesses or differences of
things that are like one another. Now this obser¬
vation of differences is not perception made by a
common organ of sense, distinct from sense or
perception properly so called, but is memory of the
differences of particular phantasms remaining for
some time; as the distinction between hot and
lucid, is nothing else but the memory both of a
heating, and of an enlightening object.
9. The phantasms of men that sleep, are dreams. Dreams,
1 whence they
Concerning which we are taught by experience proceed,
these five things. First, that for the most part
there is neither order nor coherence in them.
Secondly, that we dream of nothing but what is
compounded and made up of the phantasms of
sense past. Thirdly, that sometimes they proceed,
as in those that are drowsy, from the interruption
of their phantasms by little and little, broken and
altered through sleepiness; and sometimes also
400
PHYSICS.
part iv. they begin in the midst of sleep. Fourthly, that
-—-r—' they are clearer than the imaginations of waking
whence'they men> except such as are made by sense itself, to
proceed. which they are equal in clearness. Fifthly, that
when we dream, we admire neither the places nor
the looks of the things that appear to ns. Now
from what hath been said, it is not hard to show
what may be the causes of these phenomena. For
as for the first, seeing all order and coherence
proceeds from frequent looking back to the end,
that is, from consultation; it must needs be, that
seeing in sleep we lose all thought of the end, our
phantasms succeed one another, not in that order
which tends to any end, but as it happeneth, and in
such manner, as objects present themselves to our
eyes when we look indifferently upon all things
before us, and see them, not because we would see
them, but because we do not shut our eyes; for
then they appear to us without any order at all.
The second proceeds from this, that in the silence
of sense there is no new motion from the objects,
and therefore no new phantasm, unless we call that
new, which is compounded of old ones, as a chi¬
mera, a golden mountain, and the like. As for the
third, why a dream is sometimes as it were the
continuation of sense, made up of broken phan¬
tasms, as in men distempered with sickness, the
reason is manifestly this, that in some of the organs
sense remains, and in others it faileth. But how
some phantasms may be revived, when all the
exterior organs are benumbed with sleep, is not
so easily shown. Nevertheless that, which hath
already been said, contains the reason of this also.
For whatsoever strikes the pia mater, reviveth
OF SENSE AND ANIMAL MOTION. 401
some of those phantasms that are still in motion PAI^T IV-
in the brain ; and when any internal motion of the *-—-—
heart reacheth that membrane, then the predomi- whence'they
nant motion in the brain makes the phantasm. Proceed-
Now the motions of the heart are appetites and
aversions, of which I shall presently speak farther.
And as appetites and aversions are generated by
phantasms, so reciprocally phantasms are gene¬
rated by appetites and aversions. For example,
heat in the heart proceeds from anger and fight¬
ing ; and again, from heat in the heart, whatsoever
be the cause of it, is generated anger and the
image of an enemy, in sleep. And as love and
beauty stir up heat in certain organs ; so heat in
the same organs, from whatsoever it proceeds,
often causetli desire and the image of an unresist¬
ing beauty. Lastly, cold doth in the same manner
generate fear in those that sleep, and causeth them
to dream of ghosts, and to have phantasms of
horror and danger; as fear also causeth cold in
those that wake. So reciprocal are the motions
of the heart and brain. The fourth, namely, that
the things we seem to see and feel in sleep, are as
clear as in sense itself, proceeds from two causes;
one, that having then no sense of things without
us, that internal motion which makes the phan¬
tasm, in the absence of all other impressions, is
predominant; and the other, that the parts of our
phantasms which are decayed and worn out by
time, are made up with other fictitious parts. To
conclude, when we dream, we do not wonder at
strange places and the appearances of things un¬
known to us, because admiration requires that the
things appearing be new and unusual, which can
VOL. T. D D
402
PHYSICS.
PAR25IV* haPPen to none those that remember former
s—■—' appearances; whereas in sleep, all things appear
as present.
But it is here to be observed, that certain dreams,
especially such as some men have when they are
between sleeping and wTaking, and such as happen
to those that have 110 knowledge of the nature of
dreams and are withal superstitious, were not
heretofore nor are now accounted dreams. For
the apparitions men thought they saw, and the
voices they thought they heard in sleep, were not
believed to be phantasms, but things subsisting of
themselves, and objects without those that dreamed.
For to some men, as well sleeping as waking, but
especially to guilty men, and in the night, and in
hallowed places, fear alone, helped a little with the
stories of such apparitions, hath raised in their
minds terrible phantasms, which have been and
are still deceitfully received for things really true,
under the names of ghosts and incorporeal sub¬
stances.
or the. scnMs, most living creatures there are observed
their kinds, o
their organs five kinds of senses, which are distinguished by their
and phantasms, °
proper and organs, and by their diirerent kinds 01 phantasms;
namely, sight, hearing, smell, taste, and touch;
and these have their organs partly peculiar to each
of them severally, and partly common to them all.
The organ of sight is partly animate, and partly
inanimate. The inanimate parts are the three
humours ; namely, the watery humour, which by
the interposition of the membrane called uvea, the
perforation whereof is called the apple of the eye,
is contained on one side by the first concave super¬
ficies of the eye, and 011 the other side by the
OF SENSE AND ANIMAL MOTION. 403
ciliary processes, and the coat of the crystalline PA JrT IY-
humour; the crystalline, which, hanging in the -—r—
midst between the ciliary processes, and being t°eifkVd?S
almost of spherical figure, and of a thick con¬
sistence, is enclosed 011 all sides with its own trans¬
parent coat; and the vitreous or glassy humour,
which filletli all the rest of the cavity of the eye,
and is somewhat thicker then the watery humour,
but thinner than the crystalline. The animate part
of the organ is, first, the membrane clioroeides,
which is a part of the pia mater, saving that it is
covered with a coat derived from the marrow of
the optic nerve, which is called the retina; and
this clioroeides, seeing it is part of the pia mater, is
continued to the beginning of the medulla spinalis
within the scull, in which all the nerves which are
within the head have their roots. Wherefore all
the animal spirits that the nerves receive, enter
into them there; for it is not imaginable that they
can enter into them anywhere else. Seeing there¬
fore sense is nothing else but the action of objects
propagated to the furthest part of the organ ; and
seeing also that animal spirits are nothing but vital
spirits purified by the heart, and carried from it by
the arteries; it follows necessarily, that the action
is derived from the heart by some of the arteries
to the roots of the nerves which are in the head,
whether those arteries be the plexus retiformis, or
whether they be other arteries which are inserted
into the substance of the brain. And, therefore,
those arteries are the complement or the remain¬
ing part of the whole organ of sight. And this
last part is a common organ to all the senses;
whereas, that which reaelieth from the eye to the
d u 2
404
PHYSICS.
pa^t IV- roots of the nerves is proper only to sight. The
-—proper organ of hearing is the tympanum of the
theiSnds^&c' ear and its own nerve; from which to the heart
the organ is common. So the proper organs of
smell and taste are nervous membranes, in the
palate and tongue for the taste, and in the nostrils
for the smell; and from the roots of those nerves
to the heart all is common. Lastly, the proper
organ of touch are nerves and membranes dispersed
through the whole body; which membranes are
derived from the root of the nerves. And all
things else belonging alike to all the senses seem
to be administered by the arteries, and not by the
nerves.
The proper phantasm of sight is light; and
under this name of light, colour also, which is
nothing but perturbed light, is comprehended.
Wherefore the phantasm of a lucid body is light;
and of a coloured body, colour. But the object of
sight, properly so called, is neither light nor colour,
but the body itself which is lucid, or enlightened,
or coloured. For light and colour, being phan¬
tasms of the sentient, cannot be accidents of the
object. Which is manifest enough from this, that
visible things appear oftentimes in places in which
we know assuredly they are not, and that ill dif¬
ferent places they are of different colours, and
may at one and the same time appear in divers
places. Motion, rest, magnitude, and figure, are
common both to the sight and touch; and the
whole appearance together of figure, and light or
colour, is by the Greeks commonly called aSoc, and
uSoAov, and « ; and by the Latins, species and
OF SENSE AND ANIMAL MOTION. 405
imago; all which names signify no more but ap- PA^7IV*
pearance. -—^—-
The phantasm, which is made by hearing, is their kinds" &c!
sound; by smell, odour; by taste, savour; and by
touch, hardness and softness, heat and cold, wet¬
ness, oiliness, and many more, which are easier to
be distinguished by sense than words. Smooth¬
ness, roughness, rarity, and density, refer to figure,
and are therefore common both to touch and sight.
And as for the objects of hearing, smell, taste, and
touch, they are not sound, odour, savour, hard¬
ness, &c., but the bodies themselves from which
sound, odour, savour, hardness, &c. proceed; of
the causes of which, and of the manner how they
are produced, I shall speak hereafter.
But these phantasms, though they be effects in
the sentient, as subject, produced by objects work¬
ing upon the organs; yet there are also other
effects besides these, produced by the same objects
in the same organs ; namely certain motions pro¬
ceeding from sense, which are called animal
motions. For seeing in all sense of external things
there is mutual action and reaction, that is, two
endeavours opposing one another, it is manifest
that the motion of both of them together will be
continued every way, especially to the confines of
both the bodies. And when this happens in the
internal organ, the endeavour outwards will pro¬
ceed in a solid angle, which will be greater, and
consequently the idea greater, than it would have
been if the impression had been weaker.
11. From hence the natural cause is manifest, Themagnit.ide
first, why those filings seem to be greater, which, aLn>ywh;!t°it
cceteris paribus, are seen in a greater angle :ls
406
physics.
tart iy. secondly, why in a serene cold night, when the
-—>—- moon doth not shine, more of the fixed stars ap¬
pear than at another time. For their action is less
hindered by the serenity of the air, and not ob¬
scured by the greater light of the moon, which is
then absent; and the cold, making the air more
pressing, helpeth or strengthened the action of the
stars upon our eyes; in so much as stars may then
be seen which are seen at no other time. And this
may suffice to be said in general concerning sense
made by the reaction of the organ. For, as for
the place of the image, the deceptions of sight, and
other things of which we have experience in our¬
selves by sense, seeing they depend for the most
part upon the fabric itself of the eye of man, I shall
speak of them then when I come to speak of man.
pleasure,pain, 12. But there is another kind of sense, of which
aversion', what I will say something in this place, namely, the
they are. sense of pleasure and pain, proceeding not from
the reaction of the heart outwards, but from con¬
tinual action from the outermost part of the organ
towards the heart. For the original of life being
in the heart, that motion in the sentient, which is
propagated to the heart, must necessarily make
some alteration or diversion of vital motion, namely,
by quickening or slackening, helping or hindering
the same. Now when it helpeth, it is pleasure;
and when it hindereth, it is pain, trouble, grief,
&c. And as phantasms seem to be without, by
reason of the endeavour outwards, so pleasure and
pain, by reason of the endeavour of the organ in¬
wards, seem to be within ; namely, there where the
first cause of the pleasure or pain is ; as when the
OF SENSE AND ANIMAL MOTION. 407
pain proceeds from a wound, we think the pain IV*
and the wound are both in the same place. -—^—-
Now vital motion is the motion of the blood, 3^"^' a,T'
perpetually circulating (as hath been shown from aversion, what
many infallible signs and marks by Doctor Harvey,
the first observer of it) in the veins and arteries.
Which motion, when it is hindered by some other
motion made by the action of sensible objects, may
be restored again either by bending or setting
strait the parts of the body ; which is done when
the spirits are carried now into these, now into
other nerves, till the pain, as far as is possible, be
quite taken away. Bat if vital motion be helped
by motion made by sense, then the parts of the
organ will be disposed to guide the spirits in such
manner as conduceth most to the preservation and
augmentation of that motion, by the help of the
nerves. And in animal motion this is the very
first endeavour, and found even in the embryo;
which while it is in the womb, moveth its limbs
with voluntary motion, for the avoiding of what¬
soever troubleth it, or for the pursuing of what
pleaseth it. And this first endeavour, when it
tends towards such things as are known by expe¬
rience to be pleasant, is called appetite, that is, an
approaching; and when it shuns what is trouble¬
some, aversion, or flying from it. And little in¬
fants, at the beginning and as soon as they are
born, have appetite to very few things, as also they
avoid very few, by reason of their want of experi¬
ence and memory ; and therefore they have not so
great a variety of animal motion as we see in those
that are more grown. For it is not possible, with-
408
physics,
part iv. out sucli knowledge as is derived from sense, that
*—■—' is, without experience and memory, to know what
will prove pleasant or hurtful; only there is some
place for conjecture from the looks or aspects of
things. And hence it is, that though they do not
know what may do them good or harm, yet some¬
times they approach and sometimes retire from
the same thing, as their doubt prompts them. But
afterwards, by accustoming themselves by little
and little, they come to know readily what is to be
pursued and what to be avoided ; and also to have
a ready use of their nerves and other organs, in
the pursuing and avoiding of good and bad.
Wherefore appetite and aversion are the first en¬
deavours of animal motion.
Consequent to this first endeavour, is the impul¬
sion into the nerves and retraction again of animal
spirits, of which it is necessary there be some recep¬
tacle or place near the original of the nerves ; and
this motion or endeavour is followed by a swelling
and relaxation of the muscles; and lastly, these
are followed by contraction and extension of the
limbs, which is animal motion.
Deliberation 13 rpjjg considerations of appetites and aver-
and will, what. <
sions are divers. For seeing living creatures have
sometimes appetite and sometimes aversion to the
same thing, as they think it will either be for their
good or their hurt; while that vicissitude of appe¬
tites and aversions remains in them, they have that
series of thoughts which is called deliberation;
which lasteth as long as they have it in their power
to obtain that which pleasetli, or to avoid that
which displeasetli them. Appetite, therefore, and
aversion are simply so called as long as they follow
of sense and animal motion. 409
not deliberation. Bnt if deliberation have gone part iv.
before, tlien the last act of it, if it be appetite, is •——'
called will; if aversion, unwillingness. So that ^wuCwha^
the same thing is called both will and appetite;
but the consideration of them, namely, before and
after deliberation, is divers. Nor is that which is
done within a man whilst he willeth any thing,
different from that which is done in other living
creatures, whilst, deliberation having preceded,
they have appetite.
Neither is the freedom of willing or not willing,
greater in man, than in other living creatures. For
where there is appetite, the entire cause of appetite
hath preceded; and, consequently, the act of
appetite could not choose but follow, that is, hath
of necessity followed (as is shown in chapter ix,
article 5). And therefore such a liberty as is free
from necessity, is not to be found in the will either
of men or beasts. But if by liberty we understand
the faculty or power, not of willing, but of doing
what they will, then certainly that liberty is to be
allowed to both, and both may equally have it,
whensoever it is to be had.
Again, when appetite and aversion do with cele¬
rity succeed one another, the whole series made by
them hath its name sometimes from one, some¬
times from the other. For the same deliberation,
whilst it inclines sometimes to one, sometimes to
the other, is from appetite called hope, and from
aversion, fear. For where there is no hope, it is
not to be called fear, but hate; and where 110 fear,
not hope, but desire. To conclude, all the passions,
called passions of the mind, consist of appetite and
aversion, except pure pleasure and pain, which are
410
physics.
PA^FIV- a certain fruition of good or evil ; as anger is aver-
^—r—' sion from some imminent evil, but such as is joined
and will, what.
with appetite of avoiding that evil by force. But
because the passions and perturbations of the mind
are innumerable, and many of them not to be
discerned in any creatures besides men; I will
speak of them more at large in that section which
is concerning man. As for those objects, if there
be any such, which do not at all stir the mind, we
are said to contemn them.
And thus much of sense in general. In the next
place I shall speak of sensible objects.
CHAPTER XXVI.
of the world and of the stars.
1. The magnitude and duration of the world, inscrutable.—2. No
place in the world empty.—i3. The arguments of Lucretius for
vacuum, invalid.—4. Other arguments for the establishing of
vacuum, invalid.—5. Six suppositions for the salving of the
phenomena of nature.—6. Possible causes of the motions
annual and diurnal; and of the apparent direction, station, and
retrogradation of the planets.—7. The supposition of simple
motion, why likely.—8. The cause of the eccentricity of the
annual motion of the earth.— 9. The cause why the moon hath
always one and the same face turned towards the earth.
10. The cause of the tides of the ocean.—11. The cause of the
precession of the equinoxes.
The magnitude 1. Consequent to the contemplation of sense is
the world,0Iin- the contemplation of bodies, which are the efficient
scrutabie. causes or objects of sense. Now every object is
either a part of the whole world, or an aggregate
of parts. The greatest of all bodies, or sensible
objects, is the world itself; which we behold when
TIIE WORLD AND THE STARS.
411
we look round about us from this point of tlie same PAIIV-
which we call the earth. Concerning the world., s—*—'
as it is one aggregate of'many parts, the things
that fall under inquiry are but few ; and those we the wor,ld> in-
scrutable.
can determine, none. Or the whole world we may
inquire what is its magnitude, what its duration,
and how many there be, but nothing else, For as
for place and time, that is to say, magnitude and
duration, they are only our own fancy of a body
simply so called, that is to say, of a body indefi¬
nitely taken, as I have shown before in chapter vn.
All other phantasms are of bodies or objects, as
they are distinguished from one another ; as colour,
the phantasm of coloured bodies ; sound, of bodies
that move the sense of hearing, &c. The questions
concerning the magnitude of the world are whether
it be finite or infinite, full or not full; concerning
its duration, whether it had a beginning, or be
eternal; and concerning the number, whether there
be one or many; though as concerning the num¬
ber, if it were of infinite magnitude, there could
be 110 controversy at all. Also if it had a begin¬
ning, then by what cause and of what matter it was
made ; and again, from whence that cause and
that matter had their being, will be new questions ;
till at last we come to one or many eternal cause
or causes. And the determination of all these
things belongeth to him that professeth the uni¬
versal doctrine of philosophy, in case as much
could be known as can be sought. But the know¬
ledge of what is infinite can never be attained by a
finite inquirer. Whatsoever we know that are men,
we learn it from our phantasms; and of infinite,
whether magnitude or time, there is no phantasm
412
PHYSICS.
PAI26IV* at ' so ^at ^ *s imP0SSi^»le either for a man or
v—■—' any other creature to have any conception of infi-
and dn rati on of nite* And though a man may from some effect
tbe world, in- proceed to the immediate cause thereof, and from
scrutable.
that to a more remote cause, and so ascend conti¬
nually by right ratiocination from cause to cause ;
yet he will not be able to proceed eternally, but
wearied will at last give over, without knowing
whether it were possible for him to proceed to an
end or not. But whether we suppose the world to
be finite or infinite, no absurdity will follow. For
the same things which now appear, might appear,
whether the Creator had pleased it should be finite
or infinite. Besides, though from this, that nothing
can move itself, it may rightly be inferred that
there was some first eternal movent; yet it can
never be inferred, though some used to make such
inference, that that movent was eternally immove¬
able, but rather eternally moved. For as it is true,
that nothing is moved by itself; so it is true also
that nothing is moved but by that which is already
moved. The questions therefore about the mag¬
nitude and beginning of the world, are not to be
determined by philosophers, but by those that are
lawfully authorized to order the worship of God.
For as Almighty God, when he had brought his
people into Judaea, allowed the priests the first
fruits reserved to himself; so when he had delivered
up the world to the disputations of men, it was his
pleasure that all opinions concerning the nature of
infinite and eternal, known only to himself, should,
as the first fruits of wisdom, be judged by those
whose ministry he meant to use in the ordering of
religion, I cannot therefore commend those that
THE WOULD AND THE STARS.
413
boast they have demonstrated, by reasons drawn PA^ IV'
from natural things, that the world had a beginning. s—■—'
They are contemned by idiots, because they under- ^deNation
stand them not; and by the learned, because they t,ie w°''ld> in"
J scrutable.
understand them; by both deservedly. For who
can commend him that demonstrates thus ? "If the
world be eternal, then an infinite number of days,
or other measures of time, preceded the birth of
Abraham. But the birth of Abraham preceded
the birth of Isaac; and therefore one infinite is
greater than another infinite, or one eternal than
another eternal; which," he says, "is absurd." This
demonstration is like his, who from this, that the
number of even numbers is infinite, would con¬
clude that there are as many even numbers as there
are numbers simply, that is to say, the even num¬
bers are as many as all the even and odd together.
They, which in this manner take away eternity
from the world, do they not by the same means
take away eternity from the Creator of the world ?
From this absurdity therefore they run into another,
being forced to call eternity nunc stems, a standing
still of the present time, or an abiding now; and,
which is much more absurd, to give to the infinite
number of numbers the name of unity. But why
should eternity be called an abiding now, rather
than an abiding then ? Wherefore there must
either be many eternities, or now and then must
signify the same. With such demonstrators as
these, that speak in another language, it is im¬
possible to enter into disputation. And the men,
that reason thus absurdly, are not idiots, but,
which makes the absurdity unpardonable, geome¬
tricians, and such as take upon them to be judges,
414
PHYSICS.
PA?JIV- impertinent, but severe judges of other men's
•—t—- demonstrations. The reason is this, that as soon
as they are entangled in the words infinite and
eternal, of which we have in our mind 110 idea, but
that of our own insufficiency to comprehend them,
they are forced either to speak something absurd,
or, which they love worse, to hold their peace. For
geometry hath in it somewhat like wine, which,
when new, is windy; but afterwards though less
pleasant, yet more wholesome. Whatsoever there¬
fore is true, young geometricians think demonstra¬
ble ; but elder not. Wherefore I purposely pass
over the questions of infinite and eternal; content¬
ing myself with that doctrine concerning the
beginning and magnitude of the world, which I
have been persuaded to by the holy Scriptures and
fame of the miracles which confirm them; and by
the custom of my country, and reverence due to
the laws. And so I pass on to such things as it is
not unlawful to dispute of.
No place in the 2. Concerning the world it is further questioned,
world empty. .
whether the parts thereot be contiguous to one
another, in such manner as not to admit of the least
empty space between ; and the disputation both for
and against it is carried on with probability enough.
For the taking away of vacuum, I will instance in
only one experiment, a common one, but I think
unanswerable.
Let A B (in fig. 1) represent a vessel, such as
gardeners use to water their gardens withal; whose
bottom B is full of little holes ; and whose mouth
A may be stopped with one's finger, when there
shall be need. If now this vessel be filled with
water, the hole at the top A being stopped, the
THE WORLD AND THE STARS.
415
water will not flow out at any of the holes in the tart iv.
• 26.
bottom B. But if the finger be removed to let in ■—-
the air above, it will run out at them all; and as ^rfdTmpty!16
soon as the finger is applied to it again, the water
will suddenly and totally be stayed again from
running out. The cause whereof seems to be no
other but this, that the water cannot by its natural
endeavour to descend drive down the air below it,
because there is no place for it to go into, unless
either by thrusting away the next contiguous air,
it proceed by continual endeavour to the hole A,
where it may enter and succeed into the place of
the water that floweth out, or else, by resisting the
endeavour of the water downwards, penetrate the
same and pass up through it. By the first of these
ways, while the hole at A remains stopped, there
is no possible passage ; nor by the second, unless
the holes be so great that the water, flowing out
at them, can by its own weight force the air at the
same time to ascend into the vessel by the same
holes : as we see it does in a vessel whose mouth
is wide enough, when we turn suddenly the bottom
upwards to pour out the water ; for then the air
being forced by the weight of the water, enters, as
is evident by the sobbing and resistance of the
water, at the sides or circumference of the orifice.
And this I take for a sign that all space is full;
for without this, the natural motion of the water,
which is a heavy body, downwards, would not be
hindered.
3. On the contrary, for the establishing of va- The arguments
J \ _ of Lucretius
cuum, many and specious arguments and experi- for vacuum
ments have been brought. Nevertheless thereinvalld>
seems to be something wanting in all of them to
416
PHYSICS.
PAI2o IV' conc^U(^e ifc firmly. Tliese arguments for vacuum
v——- are partly made by tlie followers of the doctrine
ofhLucfetTusntS Epicurus ; who taught that the world consists
invIndUUm veiT small spaces not filled by any body, and of
very small bodies that have within them no empty
space, which by reason of their hardness he calls
atoms ; and that these small bodies and spaces are
every where intermingled. Their arguments are
thus delivered by Lucretius.
And first he says, that unless it were so, there
could be no motion. For the office and property
of bodies is to withstand and hinder motion. If,
therefore, the universe were filled with body,
motion would everywhere be hindered, so as to
have 110 beginning anywhere ; and consequently
there would be no motion at all. It is true that in
whatsoever is full and at rest in all its parts, it is
not possible motion should have beginning. But
nothing is drawn from hence for the proving of
vacuum. For though it should be granted that
there is vacuum, yet if the bodies which are inter¬
mingled with it, should all at once and together
be at rest, they would never be moved again.
For it has been demonstrated above, in chap, ix,
art. 7, that nothing can be moved but by that
which is contiguous and already moved. But
supposing that all things are at rest together, there
can be nothing contiguous and moved, and there¬
fore no beginning of motion. Now the denying
of the beginning of motion, doth not take away
present motion, unless beginning be taken away
from body also. For motion may be either co-
eternal, or concreated with body. Nor doth it
seem more necessary that bodies were first at rest,
THE WORLD AND THE STARS. 417
and afterwards moved, than that they were first part iv-
26.
moved, and rested, if ever they rested at all, after- •—*—-
wards. Neither doth there appear any cause, why 0rfh[uacrSfs
the matter of the world should, for the admission for va™um
• • i invalid.
of motion, be intermingled with empty spaces
rather than full; I say full, but withal fluid. Nor,
lastly, is there any reason why those hard atoms
may not also, by the motion of intermingled fluid
matter, be congregated and brought together into
compounded bodies of such bigness as we see.
Wherefore nothing can by this argument be con¬
cluded, but that motion was either coeternal, or of
the same duration with that which is moved;
neither of which conclusions consisteth with the
doctrine of Epicurus, who allows neither to the
world nor to motion any beginning at all. The
necessity, therefore, of vacuum is not hitherto de¬
monstrated. And the cause, as far as I understand
from them that have discoursed with me of vacuum,
is this, that whilst they contemplate the nature of
fluid, they conceive it to consist, as it were, of
small grains of hard matter, in such manner as
meal is fluid, made so by grinding of the corn;
when nevertheless it is possible to conceive fluid
to be of its own nature as homogeneous as either
an atom, or as vacuum itself.
The second of their arguments is taken from
weight, and is contained in these verses of Lu¬
cretius :
Corporis officium est quoniam premere omnia deorsum;
Contra autem natura manet sine pondere inanis;
Ergo, quod magnum est aeque, leviusque videtur,
Nimirum plus esse sibi declarat inanis.—I. 363-66.
That is to say, seeing the office and property of
VOL. 1. EE
418
physics.
PAfJIV body is to press all things downwards; and on
-—v—' the contrary, seeing the nature of vacuum is to
Sl2ST have 710 weight at oil ! therefore when of two
invandUUm bodies of equal magnitude, one is lighter than the
other, it is manifest that the lighter body hatli in
it more vacuum than the other.
To say nothing of the assumption concerning
the endeavour of bodies downwards, which is not
rightly assumed, because the world hath nothing
to do with downwards, which is a mere fiction of
ours ; nor of this, that if all things tended to the
same lowest part of the world, either there would
be no coalescence at all of bodies, or they would
all be gathered together into the same place: this
only is sufficient to take away the force of the
argument, that air, intermingled with those his
atoms, had served as well for his purpose as his
intermingled vacuum.
The third argument is drawn from this, that
lightning, sound, heat and cold, do penetrate all
bodies, except atoms, how solid soever they be.
But this reason, except it be first demonstrated
that the same things cannot happen without va¬
cuum by perpetual generation of motion, is alto¬
gether invalid. But that all the same things may
so happen, shall in due place be demonstrated.
Lastly, the fourth argument is set down by the
same Lucretius in these verses :
Duo de concursu corpora lata
Si cita dissiliant, nempe aer omne necesse est,
Inter corpora quod fuerat, possidat inane.
Is porro quamvis circum celerantibus auris
Confluat, haud poterit tamen uno tempore totum
Compleri spatium; nam primum quemque necesse est
Occupet ille locum, deinde omnia possideantur.—I. 385-91.
the world and the stars. 419
That is, if two fiat bodies be suddenly pulled PA^r lv-
asunder, of necessity the air must come between v—•—-
them to fill up the space they left empty. But Jf LuSh^
with what celerity soever the air flow in, yet it for vacuum
cannot in one instan t of time fill the whole space,
but first one part of it, then successively all.
Which nevertheless is more repugnant to the opi¬
nion of Epicurus, than of those that deny vacuum.
For though it be true, that if two bodies were of
infinite hardness, and were joined together by their
superficies which were most exactly plane, it would
be impossible to pull them asunder, in regard it
could not be done but by motion in an instant;
yet, if as the greatest of all magnitudes cannot be
given, nor the swiftest of all motions, so neither
the hardest of all bodies ; it might be, that by the
application of very great force, there might be
place made for a successive flowing in of the air,
namely, by separating the parts of the joined
bodies by succession, beginning at the outermost
and ending at the innermost part. He ought,
therefore, first to have proved, that there are some
bodies extremely hard, not relatively as compared
with softer bodies, but absolutely, that is to say,
infinitely hard ; which is not true. But if we sup¬
pose, as Epicurus doth, that atoms are indivisible,
and yet have small superficies of their own ; then
if two bodies should be joined together by many,
or but one only small superficies of either of them,
then I say this argument of Lucretius would be a
firm demonstration, that no two bodies made up
of atoms, as he supposes, could ever possibly be
pulled asunder by any force whatsoever. But this
is repugnant to daily experience.
e e 2
420
PHYSICS.
part iv. 4. And thus much of the arguments of Lucretius.
26.
■—v—- Let us now consider the arguments which are
mints fofthe drawn from the experiments of later writers,
establishing of 1 ^he experiment is this : that if a hollow
vacuum,invalid A #
vessel be thrust into water with the bottom up-
wards, the water will ascend into it; which they
say it could not do, unless the air within were thrust
together into a narrower place ; and that this were
also impossible, except there were little empty
places in the air. Also, that when the air is com¬
pressed to a certain degree, it can receive no further
compression, its small particles not suffering them¬
selves to be pent into less room. This reason, if
the air could not pass through the water as it
ascends within the vessel, might seem valid. But
it is sufficiently known, that air will penetrate
water by the application of a force equal to the
gravity of the water. If therefore the force, by
which the vessel is thrust down, be greater or
equal to the endeavour by which the water natu¬
rally tendeth downwards, the air will go out that
way where the resistance is made, namely, towards
the edges of the vessel. For, by how much the
deeper is the water which is to be penetrated, so
much greater must be the depressing force. But
after the vessel is quite under water, the force by
which it is depressed, that is to say, the force by
which the water riseth up, is no longer increased.
There is therefore such an equilibration between
them, as that the natural endeavour of the water
downwards is equal to the endeavour by which
the same water is to be penetrated to the increased
depth.
ii. The second experiment is, that if a concave
THE WORLD AND THE STARS. 421
cylinder of sufficient length, made of glass, that PA^g IV*
the experiment may be the better seen, having v—>—'
one end open and the other close shut, be filled ^ents SbfVsta-
with quicksilver, and the open end being stopped c^nfn^Hvda"
with one's finger, be together with the finger
dipped into a dish or other vessel, in which
also there is quicksilver, and the cylinder be set
upright, we shall, the finger being taken away to
make way for the descent of the quicksilver, see it
descend into the vessel under it, till there be only
so much remaining within the cylinder as may fill
about twenty-six inches of the same; and thus it
will always happen whatsoever be the cylinder,
provided that the length be not less than twenty-
six inches. From whence they conclude that the
cavity of the cylinder above the quicksilver remains
empty of all body. But in this experiment I find
no necessity at all of vacuum. For when the
quicksilver which is in the cylinder descends, the
vessel under it must needs be filled to a greater
height, and consequently so much of the conti¬
guous air must be thrust away as may make place
for the quicksilver which is descended. Now if it
be asked whither that air goes, what can be an¬
swered but this, that it thrusteth away the next
air, and that the next, and so successively, till
there be a return to the place where the propulsion
first began. And there, the last air thus thrust
on will press the quicksilver in the vessel with the
same force with which the first air was thrust away;
and if the force with which the quicksilver descends
be great enough, which is greater or less as it
descends from a place of greater or less height, it
will make the air penetrate the quicksilver in the
422
PHYSICS.
PAI2o IV vesse^ and go up into the cylinder to fill the place
'—'—' which they thought was left empty. But because
menTs fof esta- ^ie quicksilver hath not in every degree of height
Wishing of va- force enough to cause such penetration, therefore
cuum invalid. # p r 7
in descending it must of necessity stay somewhere,
namely, there, where its endeavour downwards,
and the resistance of the same to the penetration
of the air, come to an equilibrium. And by this ex¬
periment it is manifest, that this equilibrium will be
at the height of twenty-six inches, or thereabouts.
tii. The third experiment is, that when a vessel
hath as much air in it as it can naturally contain,
there may nevertheless be forced into it as much
water as will fill three quarters of the same vessel.
And the experiment is made in this manner. Into
the glass bottle, represented (in figure 2) by the
sphere F G, whose centre is A, let the pipe B A C
be so fitted, that it may precisely fill the mouth of
the bottle ; and let the end B be so near the bot¬
tom, that there may be only space enough left for
the free passage of the water which is thrust in
above. Let the upper end of this pipe have a
cover at D, with a spout at E, by which the water,
when it ascends in the pipe, may run out. Also let
H C be a cock, for the opening or shutting of the
passage of the water between B and D, as there
shall be occasion. Let the cover D E be taken off,
and the cock H C being opened, let a syringe full
of water be forced in; and before the syringe be
taken away, let the cock be turned to hinder the
going out of the air. And in this manner let the
injection of water be repeated as often as it shall
be requisite, till the water rise within the bottle;
for example, to GF. Lastly, the cover being
THE WORLD AND THE STARS. 423
fastened on again, and the cock H C opened, the PA^ IV*
water will run swiftly ont at E, and sink by little ^—■—-
and little from G F to the bottom of the pipe B. Joints fKta-
From this phenomenon, they argue for the neces- blishi»g °f
. _ cuura invalid.
sity of vacuum m this manner. The bottle, from
the beginning, was full of air; which air could
neither go out by penetrating so great a length of
water as was injected by the pipe, nor by any other
way. Of necessity, therefore, all the water as high
as F G, as also all the air that was in the bottle
before the water was forced in, must now be in the
same place, which at first was filled by the air
alone; which were impossible, if all the space
within the bottle were formerly filled with air pre¬
cisely, that is, without any vacuum. Besides,
though some man perhaps may think the air, being
a thin body, may pass through the body of the
water contained in the pipe, yet from that other
phenomenon, namely, that all the water which is
in the space B F G is cast out again by the spout at
E, for which it seems impossible that any other
reason can be given besides the force by which the
air frees itself from compression, it follows, that
either there was in the bottle some space empty,
or that many bodies may be together in the same
place. But this last is absurd ; and therefore the
former is true, namely, that there was vacuum.
This argument is infirm in two places. For first,
that is assumed which is not to be granted; and
in the second place, an experiment is brought,
which I think is repugnant to vacuum. That
which is assumed is, that the air can have no pas¬
sage out through the pipe. Nevertheless, we see
daily that air easily ascends from the bottom to the
424
PHYSICS.
PA^e IV* suPer^c^es a river, as is manifest by the bubbles
-——- that rise; nor doth it need any other cause to give
ments fdr "esta-lt this motion, than the natural endeavour down-
biishing of va- wards of the water. Whv, therefore, may not the
cuum invalid. J ' .
endeavour upwards of the same water, acquired by
the injection, which endeavour upwards is greater
than the natural endeavour of the water down¬
wards, cause the air in the bottle to penetrate in
like manner the water that presseth it downwards;
especially, seeing the water, as it riseth in the
bottle, doth so press the air that is above it, as that
it generateth in every part thereof an endeavour
towards the external superficies of the pipe, and
consequently maketh all the parts of the enclosed
air to tend directly towards the passage at B ? I
say, this is no less manifest, than that the air which
riseth up from the bottom of a river should pene¬
trate the water, how deep soever it be. Wherefore
I do not yet see any cause why the force, by which
the water is injected, should not at the same time
eject the air.
And as for their arguing the necessity of vacuum
from the rejection of the water; in the first place,
supposing there is vacuum, I demand by what
principle of motion that ejection is made. Certainly,
seeing this motion is from within outwards, it must
needs be caused by some agent within the bottle ;
that is to say, by the air itself. Now the motion
of that air, being caused by the rising of the water,
begins at the bottom, and tends upwards ; whereas
the motion by which it ejecteth the water ought to
begin above, and tend downwards. From whence
therefore hath the enclosed air this endeavour to¬
wards the bottom? To this question I know not
the world and the stars. 425
what answer can be given, unless it be said, that part iv.
the air descends of its own accord to expel the ^—-
water. Which, because it is absurd, and that the ®^ ta_
air, after the water is forced in, hath as much room wishing of va-
. . cuum invalid.
as its magnitude requires, there will remain no
cause at all why the water should be forced out.
Wherefore the assertion of vacuum is repugnant
to the very experiment which is here brought to
establish it.
Many other phenomena are usually brought for
vacuum, as those of weather-glasses, ceolipyles,
wind-guns, &c. which would all be very hard to be
salved, unless water be penetrable by air, without
the intermixture of empty space. But now, seeing
air may with 110 great endeavour pass through not
only water, but any other fluid body though never
so stubborn, as quicksilver, these phenomena prove
nothing. Nevertheless, it might in reason be
expected, that he that would take awTay vacuum,
should without vacuum show us such causes of
these phenomena, as should be at least of equal, if
not greater probability. This therefore shall be
done in the following discourse, wrhen I come to
speak of these phenomena in their proper places.
But first, the most general hypotheses of natural
philosophy are to be premised.
And seeing that suppositions are put for the true
causes of apparent effects, every supposition, except
such as be absurd, must of necessity consist of
some supposed possible motion ; for rest can never
be the efficient cause of anything ; and motion sup-
poseth bodies moveable ; of which there are three
kinds, fluid, consistent, and mixed of both. Fluid
are those, whose parts may by very weak endeavour
426
PHYSICS.
part iv. be separated from one another; and consistent
-—A- those for the separation of whose parts greater
force is to be applied. There are therefore de¬
grees of consistency; which degrees, by com¬
parison with more or less consistent, have the
names of hardness or softness. Wherefore a fluid
body is always divisible into bodies equally fluid,
as quantity into quantities ; and soft bodies, of
whatsoever degree of softness, into soft bodies of
the same degree. And though many men seem to
conceive no other difference of fluidity, but such
as ariseth from the different magnitudes of the
parts, in which sense dust, though of diamonds,
may be called fluid; yet I understand by fluidity,
that which is made such by nature equally in every
part of the fluid body; not as dust is fluid, for so
a house which is falling in pieces may be called
fluid; but in such manner as water seems fluid,
and to divide itself into parts perpetually fluid.
And this being well understood, I come to my
suppositions.
Six supposi- 5. First, therefore, I suppose that the immense
salving of the space, which we call the world, is the aggregate of
natui°e?ena °f bodies which are either consistent and visible,
as the earth and the stars; or invisible, as the
small atoms which are disseminated through the
whole space between the earth and the stars; and
lastly, that most fluid ether, which so fills all the
rest of the universe, as that it leaves in it no empty
place at all.
Secondly, I suppose with Copernicus, that the
greater bodies of the world, which are both con¬
sistent and permanent, have such order amongst
themselves, as that the sun hath the first place,
THE WORLD AND THE STARS.
42/
Mercury the second, Venus the third, the Earth PAR2gIv*
with the moon going about it the fourth, Mars the v—^
fifth, Jupiter with his attendants the sixth, Saturn
the seventh; and after these, the fixed stars have
their several distances from, the sun.
Thirdly, I suppose that in the sun and the rest
of the planets there is and always has been a
simple circular motion.
Fourthly, I suppose that in the body of the air
there are certain other bodies intermingled, which
are not fluid; but withal that they are so small,
that they are not perceptible by sense; and that
these also have their proper simple motion, and
are some of them more, some less hard or con¬
sistent.
Fifthly, I suppose with Kepler that as the dis¬
tance between the sun and the earth is to the
distance between the moon and the earth, so the
distance between the moon and the earth is to the
semidiameter of the earth.
As for the magnitude of the circles, and the
times in which they are described by the bodies
which are in them, I will suppose them to be such
as shall seem most agreeable to the phenomena in
question.
6. The causes of the different seasons of the Possible causes
t n . -i i ... r> 1 i of the motions
year, and ot the several variations oi days and am)uai and di.
nights in all the parts of the superficies of the apparent°f
earth, have been demonstrated, first by Coper- direction, sta.
i ti i 1 tion, and retro-
nicus, and since by Kepler, Galileus, ana others, gradation of the
from the supposition of the earth's diurnal revolu- planets-
tion about its own axis, together with its annual
motion about the sun in the ecliptic according to
the order of the signs ; and thirdly, by the annual
428
PHYSICS.
PA?JIV" revoluti°n °f the same earth about its own centre,
-—r—- contrary to the order of the signs. I suppose with
r^the^motion^ Copernicus, that the diurnal revolution is from the
diurnal &c m°tion of the earth, by which the equinoctial
circle is described about it. And as for the other
two annual motions, they are the efficient cause of
the earth's being carried about in the ecliptic in
such manner, as that its axis is always kept parallel
to itself. Which parallelism was for this reason
introduced, lest by the earth's annual revolution
its poles should seem to be necessarily carried
about the sun, contrary to experience. I have, in
art. 10, chap, xxi, demonstrated, from the suppo¬
sition of simple circular motion in the sun, that the
earth is so carried about the sun, as that its axis is
thereby kept always parallel to itself. Wherefore,
from these two supposed motions in the sun, the
one simple circular motion, the other circular
motion about its own centre, it may be demon¬
strated that the year hath both the same variations
of days and nights, as have been demonstrated by
Copernicus.
For if the circle a bed (in fig. 3) be the ecliptic,
whose centre is e, and diameter aec; and the
earth be placed in a, and the sun be moved in the
little circle f g h i, namely, according to the order
J\ g, h, and /, it hath been demonstrated, that a
body placed in a will be moved in the same order
through the points of the ecliptic a, b, c, and d,
and will always keep its axis parallel to itself.
But if, as I have supposed, the earth also be
moved with simple circular motion in a plane that
passeth through a, cutting the plane of the ecliptic
so as that the common section of both the planes
THE WORLD AND THE STARS. 429
be in a c, thus also the axis of the earth will be part iv.
26.
kept always parallel to itself. For let the centre -—*—-
of the earth be moved about in the circumference
of the epicycle, whose diameter is I a h, which is a
part of the strait line lac; therefore I ah, the
diameter of the epicycle, passing through the
centre of the earth, will be in the plane of the
ecliptic. Wherefore seeing that by reason of the
earth's simple motion both in the ecliptic and in
its epicycle, the strait line I ah is kept always
parallel to itself, every other strait line also taken
in the body of the earth, and consequently its axis,
will in like manner be kept always parallel to
itself; so that in what part soever of the ecliptic
the centre of the epicycle be found, and in what
part soever of the epicycle the centre of the earth
be found at the same time, the axis of the earth
will be parallel to the place where the same axis
would have been, if the centre of the earth had
never gone out of the ecliptic.
Now as I have demonstrated the simple annual
motion of the earth from the supposition of simple
motion in the sun; so from the supposition of
simple motion in the earth may be demonstrated
the monthly simple motion of the moon. For if
the names be but changed, the demonstration will
be the same, and therefore need not be repeated.
7- That which makes this supposition of the Jhe supposi-
sun's simple motion in the epicycle fghi pro- motion, why
bable, is first, that the periods of all the planets hkely"
are not only described about the sun, but so de¬
scribed, as that they are all contained within the
zodiac, that is to say, within the latitude of about
sixteen degrees; for the cause of this seems to
430
PHYSICS.
PA^JIV' depend upon some power in the sun, especially in
1—that part of the sun which respects the zodiac.
tion of simple Secondly, that in the whole compass of the heavens
likely"' why ^ere appears no other body from which the cause
of this phenomenon can in probability be derived.
Besides, I could not imagine that so many and such
various motions of the planets should have no
dependance at all upon one another. But, by sup¬
posing motive power in the sun, we suppose mo¬
tion also ; for power to move without motion is no
power at all. I have therefore supposed that there
is in the sun for the governing of the primary
planets, and in the earth for the governing of the
moon, such motion, as being received by the pri¬
mary planets and by the moon, makes them neces¬
sarily appear to us in such manner as we see them.
Whereas, that circular motion, which is commonly
attributed to them, about a fixed axis, which is
called conversion, being a motion of their parts
only, and not of their whole bodies, is insufficient
to salve their appearances. For seeing whatsoever
is so moved, hath no endeavour at all towards those
parts which are without the circle, they have no
power to propagate any endeavour to such bodies
as are placed without it. And as for them that
suppose this may be done by magnetical virtue, or
by incorporeal and immaterial species, they sup¬
pose no natural cause ; nay, no cause at all. For
there is no such thing as an incorporeal movent,
and magnetical virtue is a thing altogether un¬
known ; and whensoever it shall be known, it will
be found to be a motion of body. It remains,
therefore, that if the primary planets be carried
about by the sun, and the moon by the earth, they
THE WORLD AND THE STARS. 431
have the simple circular motions of the sun and PA IV'
the earth for the causes of their circulations. -J-'—'
Otherwise, if they be not carried about by the sun Simple
and the earth, but that every planet hath been ^jy"' why
moved, as it is now moved, ever since it was
made, there will be of their motions no cause
natural. For either these motions were concreated
with their bodies, and their cause is supernatural;
or they are coeternal with them, and so they have
no cause at all. For whatsoever is eternal was
never generated.
I may add besides, to confirm tbe probability of
this simple motion, that as almost all learned men
are now of the same opinion with Copernicus con¬
cerning the parallelism of the axis of the earth, it
seemed to me to be more agreeable to truth, or at
least more handsome, that it should be caused by
simple circular motion alone, than by two motions,
one in the ecliptic, and the other about the earth's
own axis the contrary way, neither of them simple,
nor either of them such as might be produced by
any motion of the sun. I thought best therefore
to retain this hypothesis of simple motion, and
from it to derive the causes of as many of the
phenomena as I could, and to let such alone as I
could not deduce from thence.
It will perhaps be objected, that although by
this supposition the reason may be given of the
parallelism of the axis of the earth, and of many
other appearances, nevertheless, seeing it is done
by placing the body of the sun in the centre of that
orb which the earth describes with its annual mo¬
tion, the supposition itself is false; because this
annual orb is eccentric to the sun. In the first
432
PHYSICS.
PA126 IV' P^ace' therefore, let us examine what that eccen-
v—r—- tricity is, and whence it proceeds.
The cause of 8. Let annual circle of the earth abed (in
tlie eccentricity
of the annual fig. 3) be divided into four equal parts by the strait
motion of the }jnes a (, an(j f} ^ cutting one another in the centre
e ; and let a be the beginning of Libra, b of Ca¬
pricorn, c of Aries and d of Cancer ; and let the
whole orb abed be understood, according to Co¬
pernicus, to have every way so great distance from
the zodiac of the fixed stars, that it be in compa¬
rison with it but as a point. Let the earth be now
supposed to be in the beginning of Libra at a.
The sun, therefore, will appear in the beginning
of Aries at c. Wherefore, if the earth be moved
from a to b, the apparent motion of the sun will be
from c to the beginning of Cancer in d; and the
earth being moved forwards from b to c, the sun
also will appear to be moved forwards to the be¬
ginning of Libra in a; wherefore c d a will be the
summer arch, and the winter arch will be a b c.
Now, in the time of the sun's apparent motion in
the summer arch, there are numbered 186f days ;
and, consequently, the earth makes in the same
time the same number of diurnal conversions in
the arch abc ; and, therefore, the earth in its mo¬
tion through the arch c da will make only l78i
diurnal conversions. Wherefore the arch abc
ought to be greater than the arch c da by 8|- days,
that is to say, by almost so many degrees. Let
the arch a r, as also c s, be each of them an arch
of two degrees and Wherefore the arch
r b s will be greater than the semicircle abc
by degrees, and greater than the arch s d r
by 81 degrees. The equinoxes, therefore, will be
the world and the stars.
433
in the points r and s; and therefore also, when part iv.
the earth is in r, the sun will appear in Where- •—^—-
fore the true place of the sun will be in t, that is Eccentricity
to say, without the centre of the earth's annual of the annual
. . n motion of the
motion by the quantity of the sine of the arch a r, earth,
or the sine of two degrees and 16 minutes. Now
this sine, putting 100,000 for the radius, will be
near 3580 parts thereof. And so much is the ec¬
centricity of the earth's annual motion, provided
that that motion be in a perfect circle ; and s and
r are the equinoctial parts. And the strait lines
s r and c a, produced both ways till they reach the
zodiac of the fixed stars, will fall still upon the same
fixed stars ; because the whole orb abed is sup¬
posed to have no magnitude at all in respect of
the great distance of the fixed stars.
Supposing now the sun to be in c, it remains
that I show the cause why the earth is nearer to
the sun, when in its annual motion it is found to
be in d, than when it is in b. And I take the cause
to be this. When the earth is in the beginning of
Capricorn at b, the sun appears in the beginning
of Cancer at d; and then is the midst of summer.
But in the midst of summer, the northern parts of
the earth are towards the sun, which is almost all
dry land, containing all Europe and much the
greatest part of Asia and America. But when the
earth is in the beginning of Cancer at d, it is the
midst of winter, and that part of the earth is towards
the sun, which contains those great seas called the
South Sea and the Indian Sea, which are of far
greater extent than all the dry land in that hemi¬
sphere. Wherefore by the last article of chapter
xxi, when the earth is in d, it will come nearer to
vol. i. f f
434
PHYSICS.
part iv. its first movent, that is, to the sun which is in t;
26.
-—r——- that is to say, the earth is nearer to the sun in the
theeccenu°itity midst of winter when it is in d, than in the midst
&c- of summer when it is in b ; and, therefore, during
the winter the sun is in its Perigceum, and in its
Apogceum during the summer. And thus I have
shown a possible cause of the eccentricity of the
earth ; which was to be done.
I am, therefore, of Kepler's opinion in this, that
he attributes the eccentricity of the earth to the
difference of the parts thereof, and supposes one
part to be affected, and another disaffected to the
sun. And I dissent from him in this, that he thinks
it to be by magnetic virtue, and that this magnetic
virtue or attraction and thrusting back of the earth
is wrought by immateriate species ; which cannot
be, because nothing can give motion but a body
moved and contiguous. For if those bodies be not
moved which are contiguous to a body unmoved,
how this body should begin to be moved is not
imaginable; as has been demonstrated in art. 7,
chap, ix, and often inculcated in other places, to
the end that philosophers might at last abstain from
the use of such unconceivable connexions of words.
I dissent also from him in this, that he says the
similitude of bodies is the cause of their mutual
attraction. For if it were so, I see no reason why
one egg should not be attracted by another. If,
therefore, one part of the earth be more affected
by the sun than another part, it proceeds from
this, that one part hath more water, the other more
dry land. And from hence it is, as I showed above,
that the earth comes nearer to the sun when it
shines upon that part where there is more water,
THE WORLD AND THE STARS. 435
than when it shines upon that where there is more PAI^FIV*
dry land. ^^—-
9. This eccentricity of the earth is the cause The cause why
1 the moon hath
why the way 01 its annual motion is not a perfect always one and
circle, but either an elliptical, or almost an ellip- SmeTtowuds
tical line ; as also why the axis of the earth is notthe earth-
kept exactly parallel to itself in all places, but only
in the equinoctial points.
Now seeing I have said that the moon is carried
about by the earth, in the same manner that the
earth is by the sun; and that the earth goeth about
the sun in such manner as that it shows sometimes
one hemisphere, sometimes the other to the sun ;
it remains to be enquired, why the moon has
always one and the same face turned towards the
earth.
Suppose, therefore, the sun to be moved with
simple motion in the little circle f g h i, (in fig. 4)
whose centre is t; and let be the annual
circle of the earth; and a the beginning of Libra.
About the point a let the little circle I k be de¬
scribed ; and in it let the centre of the earth be
understood to be moved with simple motion ; and
both the sun and the earth to be moved according
to the order of the signs. Upon the centre a let the
way of the moon m n op be described ; and let q r
be the diameter of a circle cutting the globe of the
moon into two hemispheres, whereof one is seen by
us when the moon is at the full, and the other is
turned from us.
The diameter therefore of the moon qor will be
perpendicular to the strait line t a. Wherefore the
moon is carried, by reason of the motion of the
earth, from o towards/j. But by reason of the
F F 2
436
physics.
part iv. motion of the sun, if it were in p it would at tlie
26.
*—r—- same time be carried from p towards o ; and by
dufmoorThath these two contrary movents the strait line q r will
&c- be turned about; and, in a quadrant of the circle
7u n op, it will be turned so much as makes the
fourth part of its whole conversion. Wherefore
when the moon is in p, q r will be parallel to the
strait line m o. Secondly, when the moon is in m,
the strait line q r will, by reason of the motion
of the earth, be in m o. But by the working of the
sun's motion upon it in the quadrantp m, the same
q r will be turned so much as makes another quarter
of its whole conversion. When, therefore, the moon
is in in, q r will be perpendicular to the strait line
o m. By the same reason, when the moon is in n,
q r will be parallel to the strait line in o; and, the
moon returning to o, the same q r will return to
its first place; and the body of the moon will in
one entire period make also one entire conversion
upon her own axis. In the making of which, it is
manifest, that one and the same face of the moon
is always turned towards the earth. And if any
diameter were taken in that little circle, in which
the moon were supposed to be carried about with
simple motion, the same effect would follow;
for if there were no action from the sun, every
diameter of the moon would be carried about
always parallel to itself. Wherefore I have given
a possible cause why one and the same face of the
moon is always turned towards the earth.
But it is to be noted, that when the moon is
without the ecliptic, we do not always see the same
face precisely. For we see only that part which is
illuminated. But when the moon is without the
THE WORLD AND THE STARS.
437
ecliptic, that part which is towards us is not exactly PA^F IV-
the same with that which is illuminated. v—'—'
10. To these three simple motions, one of the
sun, another of the moon, and the third of the of the ocean-
earth, in their own little circles f g h i, I h, and
q r, together with the diurnal conversion of the
earth, by which conversion all things that adhere
to its superficies are necessarily carried about with
it, maybe referred the three phenomena concern¬
ing the tides of the ocean. Whereof the first is
the alternate elevation and depression of the water
at the shores, twice in the space of twenty-four hours
and near upon fifty-two minutes; for so it has
constantly continued in all ages. The second, that
at the new and full moons, the elevations of the
water are greater than at other times between.
And the third, that when the sun is in the equi¬
noctial, they are yet greater than at any other
time. For the salving of which phenomena, we
have already the four above-mentioned motions;
to which I assume also this, that the part of the
earth which is called America, being higher than
the water, and extended almost the space of a
whole semicircle from north to south, gives a stop
to the motion of the water.
This being granted, in the same 4th figure, where
lb he is supposed to be in the plane of the moon's
monthly motion, let the little circle Id he be de¬
scribed about the same centre a in the plane of the
equinoctial. This circle therefore will decline from
the circle / b h c in an angle of almost 28| degrees ;
for the greatest declination of the ecliptic is 23|,
to which adding 5 for the greatest declination of
the moon from the ecliptic, the sum will be 28^
438
PHYSICS.
PA?6T IV* degrees. Seeing now the waters, which are
—•—- under the circle of the moon's course, are by
ofThe'^des reason of the earth's simple motion in the plane of
of the ocean, the same circle moved together with the earth, that
is to say, together with their own bottoms, neither
outgoing nor outgone; if we add the diurnal
motion, by which the other waters which are under
the equinoctial are moved in the same order, and
consider withal that the circles of the moon and
of the equinoctial intersect one another; it will be
manifest, that both those waters, which are under
the circle of the moon, and under the equinoctial,
will run together under the equinoctial; and con¬
sequently, that their motion will not only be swifter
than the ground that carries them; but also that
the waters themselves will have greater elevation
whensoever the earth is in the equinoctial. Where¬
fore, whatsoever the cause of the tides may be,
this may be the cause of their augmentation at
that time.
Again, seeing I have supposed the moon to be
carried about by the simple motion of the earth in
the little circle lb he; and demonstrated, at the
4th article of chapter xxi, that whatsoever is
moved by a movent that hath simple motion, will
be moved always with the same velocity ; it follows,
that the centre of the earth will be carried in the
circumference lb he with the same velocity with
which the moon is carried in the circumference
mnop. Wherefore the time, in which the moon
is carried about in m n op, is to the time, in which
the earth is carried about in I b k c, as one circum¬
ference to the other, that is, as a o to a h. But
a o is observed to be to the semidiameter of the
TIIR WORLD AND THE STARS.
439
earth as 59 to 1 ; and therefore the earth, if a h be pART IV-
26.
put for its semidiameter, will make fifty-nine revo- —•—'
tions in I b k c in the time that the moon makes ofTheMdes
one monthly eireuit in inn op. But the moon of the ocean'
makes her monthly circuit in little more than
twenty-nine days. Wherefore the earth shall make
its circuit in the circumference lb lie in twelve
hours and a little more, namely, about twenty-six
minutes more; that is to say, it shall make two
circuits in twenty-four hours and almost fifty-two
minutes; which is observed to be the time between
the high-water of one day and the high-water of
the day following. Now the course of the waters
being hindered by the southern part of America,
their motion will be interrupted there ; and con¬
sequently, they will be elevated in those places,
and sink down again by their own weight, twice in
the space of twenty-four hours and fifty-two mi¬
nutes. And thus I have given a possible cause of
the diurnal reciprocation of the ocean.
Now from this swelling of the ocean in those
parts of the earth, proceed the flowings and ebbings
in the Atlantic, Spanish, British, and German seas;
which though they have their set times, yet upon
several shores they happen at several hours of the
day. And they receive some augmentation from
the north, by reason that the shores of China and
Tartary, hindering the general course of the waters,
make them swell there, and discharge themselves
in part through the strait of Anian into the
Northern Ocean, and so into the German Sea.
As for the spring tides which happen at the
time of the new and full moons, they are caused
by that simple motion, wrhich at the beginning I
440
PHYSICS.
rA:^JIV- supposed to be always in the moon. For as, when
v I showed the cause of the eccentricity of the earth,
o?thetfdM ^ derived the elevation of the waters from the
of the ocean, simple motion of the sun ; so the same may here be
derived from the simple motion of the moon. For
though from the generation of clouds, there appear
in the sun a more manifest power of elevating the
waters than in the moon ; yet the power of in¬
creasing moisture in vegetables and living creatures
appears more manifestly in the moon than in the
sun; which may perhaps proceed from this, that
the sun raiseth up greater, and the moon lesser
drops of water. Nevertheless, it is more likely,
and more agreeable to common observation, that
rain is raised not only by the sun, but also by the
moon; for almost all men expect change of weather
at the time of the conjunctions of the sun and
moon with one another and with the earth, more
than in the time of their quarters.
In the last place, the cause why the spring tides
are greater at the time of the equinoxes hath been
already sufficiently declared in this article, where I
have demonstrated, that the two motions of the
earth, namely, its simple motion in the little circle
Ibkc, and its diurnal motion in I dice, cause
necessarily a greater elevation of waters when the
sun is about the equinoxes, than when he is in
other places. I have therefore given possible causes
of the phenomenon of the flowing and ebbing of
the ocean.
Cause of the 11. As for the explication of the yearly preces-
precession of - n •• 7 • , , 1
the equinoxes. Sl0n °J t'ie equinoctial points, we must remember
that, as I have already shown, the annual motion
of the earth is not in the circumference of a circle,
THE WORLD AND THE STARS.
441
but of an ellipsis, or a line not considerably dif- rA^ IV-
ferent from tliat of an ellipsis. In the first place, "—■—'
therefore, this elliptical line is to be described. precision ofG
Let the ecliptic ^ vp s (in fig. 5) be divided the e(iuinoxes*
into four equal parts by the two strait lines a h and
vi1 s, cutting one another at right angles in the
centre c. And taking the arch b cl of two degrees
and sixteen minutes, let the strait line cle be
drawn parallel to a b, and cutting vs <3 in f; which
being done, the eccentricity of the earth will be
cf. Seeing therefore the annual motion of the
earth is in the circumference of an ellipsis, of
which v$> is the greater axis, a b cannot be the
lesser axis ; for a b and vf 3 are equal. Where¬
fore the earth passing through a and b, will either
pass above vf, as through g, or passing through vf,
will fall between c and a; it is no matter which.
Let it pass therefore through g; and let gl be
taken equal to the strait line v? ® ; and dividing
g I equally in i, g i w ill be equal to f, and i I
equal to f ; and consequently the point i will
cut the eccentricity cf into two equal parts ; and
taking i h equal to if, h i will be the whole
eccentricity. If now a strait line, namely, the
line ^ i r, be drawn through i parallel to the
strait lines a b and e d, the wTay of the sun in
summer, namely, the arch ^ g will be greater
than his way in wTinter, by 8-} degrees. Where¬
fore the true equinoxes will be in the strait line
^ i v ; and therefore the ellipsis of the earth's
annual motion will not pass through a,g, b, and I;
but through ^ g, <y and /. Wherefore the annual
motion of the earth is in the ellipsis ^ g v I; and
cannot be, the eccentricity being salved, in any
442
PHYSICS.
part iv. otlier line. And this perhaps is the reason, why
r—Kepler, against the opinion of all the astronomers
precissi(mofhe of former time, thought fit to biseet the eccentri-
the equinoxes. city of the earth, or, aeeording to the ancients, of
the sun, not by diminishing the quantity of the
same eccentricity, (because the true measure of that
quantity is the difference by which the summer
arch exceeds the winter arch), but by taking for
the centre of the ecliptic of the great orb the point
c nearer to f, and so placing the whole great orb
as much nearer to the ecliptic of the fixed stars
towards ®, as is the distance between c and i.
For seeing the whole great orb is but as a point in
respect of the immense distance of the fixed stars,
the two strait lines ^ v and a h, being produced
both ways to the beginnings of Aries and Libra,
will fall upon the same points of the sphere of the
fixed stars. Let therefore the diameter of the
earth m n be in the plane of the earth's annual
motion. If now the earth be moved by the sun's
simple motion in the circumference of the ecliptic
about the centre i, this diameter will be kept
always parallel to itself and to the strait line g I.
But seeing the earth is moved in the circumference
of an ellipsis without the ecliptic, the point n,
whilst it passetli through ^ vs v, will go in a lesser
circumference than the point m\ and consequently,
as soon as ever it begins to be moved, it will
lose its parallelism with the strait line vs 25; so
that mn produced will at last cut the strait line
g I produced. And contrarily, as soon as m n is
past op, the earth making its way in the internal
elliptical line v / ===, the same m n produced to¬
wards m, will cut Ig produced. And when the
TIIE WORLD AND THE STARS. 443
PART IV.
earth hath almost finished its whole circumfer- > 26' . -
ence, the same m n shall again make a right angle Cause of the
. ° precession of
with a line drawn from the centre i, a little short the equinoxes,
of the point from which the earth began its motion.
And there the next year shall be one of the equi¬
noctial points, namely, near the end of nji; the
other shall be opposite to it near the end of x.
And thus the points in which the days and nights
are made equal do every year fall back ; but with
so slowr a motion, that, in a whole year, it makes but
51 first minutes. And this relapse being contrary
to the order of the signs, is commonly called the
precession of the equinoxes. Of which I have
from my former suppositions deduced a possible
cause ; which was to be done.
According to what I have said concerning the
cause of the eccentricity of the earth; and according
to Kepler, who for the cause thereof supposeth one
part of the earth to be affected to the sun, the other
part to be disaffected; the apogseum and peri-
gaemn of the sun should be moved every year in
the same order, and with the same velocity, with
which the equinoctial points are moved ; and their
distance from them should always be the quadrant
of a circle; which seems to be otherwise. For
astronomers say, that the equinoxes are nowr, the
one about 28 degrees gone back from the first star
of Aries, the other as much from the beginning of
Libra ; so that the apogseum of the sun or the
aphelium of the earth ought to be about the 28th
degree of Cancer. But it is reckoned to be in the
7th degree. Seeing, therefore, we have not suffi¬
cient evidence of the 6™' (that so it is,) it is in vain
to seek for the Bior'i (why it is so.) Wherefore, as
444
PHYSICS.
rA1v6 lVj l°ng as the motion of the apogseum is not observ ■
v—.—- able by reason of the slowness thereof, and as long
prTccssbifof as remains doubtful whether their distance from
the equinoxes, the equinoctial points be more or less than a
quadrant precisely; so long it may be lawful for
me to think they proceed both of them with equal
velocity.
Also, I do not at all meddle with the causes
of the eccentricities of Saturn, Jupiter, Mars, and
Mercury. Nevertheless, seeing the eccentricity of
the earth may, as I have shewn, be caused by the
unlike constitution of the several parts of the earth
which are alternately turned towards the sun, it
is credible also, that like effects may be produced
in these other planets from their having their su¬
perficies of unlike parts.
And this is all I shall say concerning Sidereal
Philosophy. And, though the causes 1 have here
supposed be not the true causes of these phe¬
nomena, yet I have demonstrated that they are
sufficient to produce them, according to what I at
first propounded.
light, heat, and colours. 445
CHAPTER XXVII.
of light, heat, and of colours.
1. Of the immense magnitude of some bodies, and the unspeak¬
able littleness of others.—2. Of the cause of the light of the
sun.—3. How light heateth.—4. The generation of fire from
the sun.—0. The generation of fire from collision.—6. The
cause of light in glow-worms, rotten wood, and the Bolognan
stone.— 6. The cause of light in the concussion of sea
water.—8. The cause of flame, sparks, and colliquatiori.—9.
The cause why wet hay sometimes burns of its own accord ;
also the cause of lightning.—10. The cause of the force of
gunpowder; and what is to be ascribed to the coals, what to
the brimstone, and what to the nitre.—11. How heat is caused
by attrition.—12. The distinction of light into first, second,
&c.—13. The causes of the colours we see in looking through
a prisma of glass, namely, of red, yellow, blue, and violet colour.
14-. Why the moon and the stars appear redder in the hori¬
zon than in the midst of the heaven. —15. The cause of p^RT jy
whiteness.—16. The cause of blackness. 27.
1. Besides the stars, of which I have spoken in oftheimmense
the last chapter, whatsoever other bodies there be some bodies,
in the world, they may be all comprehended under speakaWeHuie-
the name of intersidereal bodies. And these I haveness of others*
already supposed to be either the most fluid sether,
or such bodies whose parts have some degree of
cohesion. Now, these differ from one another in
their several consistencies, magnitudes, motions,
and figures. In consistency, I suppose some bodies
to be harder, others softer through all the several
degrees of tenacity. In magnitude, some to be
greater, others less, and many unspeakably little.
For we must remember that, by the understanding,
446
physics.
part iv. quantity is divisible into divisibles perpetually.
-—<—' And, therefore, if a man conld do as much with his
magnitude,e&ct hands as he can with his understanding, he would
be able to take from any given magnitude a part
which should be less than any other magnitude
given. But the Omnipotent Creator of the world
can actually from a part of any thing take another
part, as far as we by our understanding can con¬
ceive the same to be divisible. Wherefore there is
no impossible smallness of bodies. And what
hinders but that we may think this likely ? For we
know there are some living creatures so small
that we can scarce see their whole bodies. Yet
even these have their young ones ; their little veins
and other vessels, and their eyes so small as that
no microscope can make them visible. So that we
cannot suppose any magnitude so little, but that
our very supposition is actually exceeded by nature.
Besides, there are now such microscopes com¬
monly made, that the things we see with them ap¬
pear a hundred thousand times bigger than they
would do if we looked upon them with our bare
eyes. Nor is there any doubt but that by aug¬
menting the power of these microscopes (for it
may be augmented as long as neither matter nor
the hands of workmen are wanting) every one of
those hundred thousandth parts might yet appear
a hundred thousand times greater than they did
before. Neither is the smallness of some bodies
to be more admired than the vast greatness of
others. For it belongs to the same Infinite Power,
as well to augment infinitely as infinitely to dimi¬
nish. To make the great orb, namely, that whose
radius reacheth from the earth to the sun, but as a
LTGHT, HEAT, AND COLOURS. 447
point in respect of the distance between the sun par2T iv-
and the fixed stars ; and, on the contrary, to make 7^—-
a body so little, as to be in the same proportion magnTtude,Tc!
less than any other visible body, proceeds equally
from one and the same Author of Nature. But this
of the immense distance of the fixed stars, which
for a long time was accounted an incredible thing,
is now believed by almost all the learned. Why
then should not that other, of the smallness of some
bodies, become credible at some time or other ?
For the Majesty of God appears no less in small
things than in great; and as it exceedeth human
sense in the immense greatness of the universe,
so also it doth in the smallness of the parts thereof.
Nor are the first elements of compositions, nor the
first beginnings of actions, nor the first moments
of times more credible, than that which is now
believed of the vast distance of the fixed stars.
Some things are acknowledged by mortal men
to be very great, though finite, as seeing them to
be such. They acknowledge also that some things,
which they do not see, may be of infinite magni¬
tude. But they are not presently nor without great
study persuaded, that there is any mean between
infinite and the greatest of those things which
either they see or imagine. Nevertheless, when
after meditation and contemplation many things
which we wTondered at before are now grown more
familiar to us, we then believe them, and transfer
our admiration from the creatures to the Creator.
But how little soever some bodies may be, yet I
will not suppose their quantity to be less than is
requisite for the salving of the phenomena. And
in like manner I shall suppose their motion, namely,
448
PHYSICS.
Iv- their velocity and slowness, and the variety of their
*—r— figures, to be only such as the explication of then-
natural causes requires. And lastly, I suppose,
that the parts of the pure aether, as if it were the
first matter, have no motion at all but what they
receive from bodies which float in them, and are
not themselves fluid.
Of the cause Having laid these grounds, let us come to speak
of die°f causes ; and in the first place let us inquire what
may be the cause of the light of the sun. Seeing,
therefore, the body of the sun doth by its simple
circular motion thrust away the ambient ethereal
substance sometimes one way sometimes another,
so that those parts, which are next the sun, being
moved by it, do propagate that motion to the next
remote parts, and these to the next, and so on
continually; it must needs be that, notwithstand¬
ing any distance, the foremost part of the eye
will at last be pressed ; and by the pressure of
that part, the motion will be propagated to the
innermost part of the organ of sight, namely, to
the heart; and from the reaction of the heart, there
will proceed an endeavour back by the same way,
ending in the endeavour outwards of the coat of
the eye, called the retina. But this endeavour
outwards, as has been defined in chapter xxv, is
the thing which is called light, or the phantasm
of a lucid body. For it is by reason of this phan¬
tasm that an object is called lucid. Wherefore
we have a possible cause of the light of the sun;
which I undertook to find.
How light 3. The generation of the light of the sun is ac¬
companied with the generation of heat. Now
every man knows what heat is in himself, by feeling
LIGHT, HEAT, AND COLOURS.
449
it when he stows hot; but what it is in other part iv.
b ... . . 27.
things, he knows only by ratiocination. For it is -—«—'
& ' 11 T - How light
one thing to grow hot, and another thing to heat heateth.
or make hot. And therefore though we perceive
that the fire or the sun heateth, yet we do not
perceive that it is itself hot. That other living
creatures, whilst they make other things hot, are
hot themselves, we infer by reasoning from the
like sense in ourselves. But this is not a necessary
inference. For though it may truly be said of
living creatures, that they heat, therefore they
are themselves hot; yet it cannot from hence be
truly inferred that fire heateth, therefore it is
itself hot; no more than this, fire causeth pain,
therefore it is itself in pain. Wherefore, that is
only and properly called hot, which when we feel
we are necessarily hot.
Now when we grow hot, we find that our spirits
and blood, and whatsoever is fluid within us, is
called out from the internal to the external parts
of our bodies, more or less, according to the degree
of the heat; and that our skin swelleth. He,
therefore, that can give a possible cause of this
evocation and swelling, and such as agrees with
the rest of the phenomena of heat, may be thought
to have given the cause of the heat of the sun.
It hath been shown, in the 5th article of chapter
xxi, that the fluid medium, which we call the air,
is so moved by the simple circular motion of the
sun, as that all its parts, even the least, do per¬
petually change places with one another; which
change of places is that which there I called fer¬
mentation. From this fermentation of the air, I
have, in the 8th article of the last chapter, demon-
VOL. I. G G
450
physics.
part iv. strated that the water may be drawn up into the
s—t—' clouds.
And I shall now show that the fluid parts may,
in like manner, by the same fermentation, be drawn
out from the internal to the external parts of our
bodies. For seeing that wheresoever the fluid
medium is contiguous to the body of any living
creature, there the parts of that medium are, by
perpetual change of place, separated from one
another ; the contiguous parts of the living creature
must, of necessity, endeavour to enter into the spaces
of the separated parts. For otherwise those parts,
supposing there is no vacuum, would have no place
to go into. And therefore that, which is most fluid
and separable in the parts of the living creature
which are contiguous to the medium, will go first
out; and into the place thereof will succeed such
other parts as can most easily transpire through
the pores of the skin. And from hence it is ne¬
cessary that the rest of the parts, which are not
separated, must altogether be moved outwards, for
the keeping of all places full. But this motion
outwards of all parts together must, of necessity,
press those parts of the ambient air which are
ready to leave their places; and therefore all the
parts of the body, endeavouring at once that way,
make the body swell. Wherefore a possible cause
is given of heat from the sun; which was to be
done.
o^fire^om'the 4' have now seen how light and heat are
sun. generated; heat by the simple motion of the me¬
dium, making the parts perpetually change places
with one another; and light by this, that by the
same simple motion action is propagated in a
light, heat, and colours.
451
strait line. But when a body hatli its parts so PAI^IV-
moved, that it sensibly both heats and shines at •—^—-
the same time, then it is that we say fire is
generated. sun-
Now by fire I do not understand a body distinct
from matter combustible or glowing, as wood or
iron, but the matter itself, not simply and always,
but then only when it shineth and heateth. He,
therefore, that renders a cause possible and agree¬
able to the rest of the phenomena, namely, whence,
and from what action, both the shining and heating
proceed, may be thought to have given a possible
cause of the generation of fire.
Let, therefore, ABC (in the first figure) be a
sphere, or the portion of a sphere, whose centre is
D ; and let it be transparent and homogeneous, as
crystal, glass, or water, and objected to the sun.
Wherefore, the foremost part ABC will, by the
simple motion of the sun, by which it thrusts
forwards the medium, be wrought upon by the
sunbeams in the strait lines E A, F B, and G C;
which strait lines may, in respect of the great dis¬
tance of the sun, be taken for parallels. And
seeing the medium within the sphere is thicker
than the medium without it, those beams will be
refracted towards the perpendiculars. Let the
strait lines E A and G C be produced till they cut
the sphere in H and I; and drawing the perpen¬
diculars A D and C D, the refracted beams E A and
GC will of necessity fall, the one between AH
and A D, the other between C I and C D. Let
those refracted beams be A K and CL. And again,
let the lines D K M and D L N be drawn perpen¬
dicular to the sphere; and let A K and C L be
gg 2
452
PHYSICS.
par2{ IV- produced till tliey meet with the strait line B D
v- produced in O. Seeing, therefore, the medium
ofSri within the sphere is thicker than that without it,
sun- the refracted line A K will recede further from the
perpendicular K M than K O will recede from the
same. Wherefore K O will fall between the re¬
fracted line and the perpendicular. Let, therefore,
the refracted line be K P, cutting F O in P; and
for the same reason the strait line L P will be the
refracted line of the strait line C L. Wherefore,
seeing the beams are nothing else but the ways in
which the motion is propagated, the motion about
P will be so much more vehement than the motion
about A B C, by how much the base of the portion
A B C is greater than the base of a like portion in
the sphere, whose centre is P, and whose magnitude
is equal to that of the little circle about P, which
compreliendeth all the beams that are propagated
from ABC; and this sphere being much less than
the sphere ABC, the parts of the medium, that is,
of the air about P, will change places with one
another with much greater celerity than those
about ABC. If, therefore, any matter combustible,
that is to say, such as may be easily dissipated, be
placed in P, the parts of that matter, if the pro¬
portion be great enough between A C and a like
portion of the little circle about P, will be freed
from their mutual cohesion, and being separated
will acquire simple motion. But vehement simple
motion generates in the beholder a phantasm of
lucid and hot, as I have before demonstrated of
the simple motion of the sun; and therefore the
combustible matter which is placed in P will be
made lucid and hot, that is to say, will be fire.
LIGHT, I1EAT, AND COLOURS.
453
Wherefore I have rendered a possible cause of fire ; PA^7T IV*
which was to be done. •—*—-
5. From the manner by which the sun generateth T!le generation
J ° of fire from col-
fire, it is easy to explain the manner by which fire lision.
may be generated by the collision of two flints.
For by that collision some of those particles of
which the stone is compacted, are violently sepa¬
rated and thrown off ; and being withal swiftly
turned round, the eye is moved by them, as it is in
the generation of light by the sun. Wherefore they
shine ; and falling upon matter which is already half
dissipated, such as is tinder, they thoroughly dis¬
sipate the parts thereof, and make them turn round.
From whence, as I have newly shown, light and
heat, that is to say fire, is generated.
6. The shining of glow7-worms, some kinds of P,e cause, of
° # light in glow-
rotten wood, and of a kind of stone made at Bo- worms, rotten
, , , , wood, and the
logna, may have one common cause, namely, the Boiognanstone
exposing of them to the hot sun. We find by expe¬
rience that the Bologna stone shines not, unless it be
so exposed; and after it has been exposed it shines
but for a little time, namely, as long as it retains
a certain degree of heat. And the cause may be
that the parts, of which it is made, may together
with heat have simple motion imprinted in them
by the sun. Which if it be so, it is necessary that
it shine in the dark, as long as there is sufficient heat
in it; but this ceasing, it will shine no longer.
Also we find by experience that in the glow-worm
there is a certain thick humour, like the crystalline
humour of the eye; which if it be taken out and
held long enough in one's fingers, and then be
carried into the dark, it will shine by reason of the
warmth it received from the fingers; but as soon
454
PHYSICS.
part iv. as it is cold it will cease shining. From whence,
1—r—- therefore, can these creatures have tlieir light, but
from lying all day in the sunshine in the hottest
time of summer ? In the same manner, rotten
wood, except it grow rotten in the sunshine, or be
afterwards long enough exposed to the sun, will
not shine. That this doth not happen in every
worm, nor in all kinds of rotten wood, nor in all
calcined stones, the cause may be that the parts,
of which the bodies are made, are different both
in motion and figure from the parts of bodies of
other kinds.
The cause of 7. Also the sea water shineth when it is either
cussion of sea dashed with the strokes of oars, or when a ship in
its course breaks strongly through it; but more or
less, according as the wind blows from different
points. The cause whereof may be this, that the
particles of salt, though they never shine in the
salt-pits, where they are but slowly drawn up by
the sun, being here beaten up into the air in greater
quantities and with more force, are withal made
to turn round, and consequently to shine, though
weakly. I have, therefore, given a possible cause
of this phenomenon.
The cause of jf suc|1 matter as is compounded of hard little
name, spcarks, # # 1
& coiiiquation. bodies be set on fire, it must needs be, that, as they
fly out in greater or less quantities, the flame which
is made by them will be greater or less. And if
the ethereal or fluid part of that matter fly out
together with them, their motion will be the
swifter, as it is in wood and other things which
flame with a manifest mixture of wind. When,
therefore, these hard particles by their flying out
move the eye strongly, they shine bright; and a
LIGHT, HEAT, AND COLOURS.
455
great quantity of them flying- out together, they PART lv-
make a great shining body. For liame being *——'
nothing but an aggregate of shining particles, the e ca"1,°rk°sf
greater the aggregate is, the greater and more& colligation,
manifest will be the flame. I have, therefore,
shown a possible cause of flame. And from hence
the cause appears evidently, why glass is so easily
and quickly melted by the small flame of a candle
blown, which will not be melted without blowing
but by a very strong fire.
Now, if from the same matter there be a part
broken off, namely, such a part as consisteth of
many of the small particles, of this is made a spark.
For from the breaking off it hath a violent turning
round, and from hence it shines. But though
from this matter there fly neither flame nor sparks,
yet some of the smallest parts of it may be carried
out as far as to the superficies, and remain there
as ashes ; the parts whereof are so extremely small,
that it cannot any longer be doubted how far na¬
ture may proceed in dividing.
Lastly, though by the application of fire to this
matter there fly little or nothing from it, yet
there w ill be in the parts an endeavour to simple
motion ; by which the whole body will either be
melted, or, which is a degree of melting, softened.
For all motion has some effect upon all matter
whatsoever, as lias been shown at art. 3, chap. xv.
Now if it be softened to such a degree, as that the
stubbornness of the parts be exceeded by their
gravity, then we say it is melted ; otherwise, soft¬
ened and made pliant and ductile.
Again, the matter having in it some particles
hard, others ethereal or watery; if, by the appli-
456
physics.
part iv. cation of fire, these latter be called out, the former
27. .
v—r—' will thereby come to a more full contact with one
another; and, consequently, will not be so easily
separated ; that is to say, the whole body will be
made harder. And this may be the cause why the
same fire makes some things soft, others hard.
The cause why <)_ is known by experience that if hay be laid
wet hay some- J 1 J
times bums of wet together in a heap, it will after a time begin
also the cause to smoke, and then burn as it were of itself. The
of hghtmng. Cfmse wiiereof seems to be this, that in the air,
which is enclosed within the hay, there are those
little bodies, which, as I have supposed, are moved
freely with simple motion. But this motion being
by degrees hindered more and more by the de¬
scending moisture, which at the last fills and stops
all the passages, the thinner parts of the air ascend
by penetrating the water; and those hard little
bodies, being so thrust together that they touch
and press one another, acquire stronger motion;
till at last by the increased strength of this motion
the watery parts are first driven outwards, from
whence appears vapour; and by the continued
increase of this motion, the smallest particles of
the dried hay are forced out, and recovering their
natural simple motion, they grow hot and shine,
that is to say, they are set 011 fire.
The same also may be the cause of lightning,
which happens in the hottest time of the year,
when the water is raised up in greatest quantity
and carried highest. For after the first clouds are
raised, others after others follow them ; and being
congealed above, they happen, whilst some of them
ascend and others descend, to fall one upon another
in such manner, as that in some places all their parts
LIGHT, HEAT, AND COLOURS.
45 /
are joined together, in others tliey leave hollow part iv-
spaces between them; and into these spaces, the *—-
ethereal parts being forced out by the compressure
of the clouds, many of the harder little bodies are
so pent together, as they have not the liberty of
such motion as is natural to the air. Wherefore
their endeavour grows more vehement, till at last
they force their way through the clouds, sometimes
in one place, sometimes in another; and, breaking
through with great noise, they move the air vio¬
lently, and striking our eyes, generate light, that
is to say, they shine. And this shining is that we
call lightning.
10. The most common phenomenon proceeding ^ieth^srcce
from fire, and yet the most admirable of all others,of gunpowder:
is the force of gunpowder fired ; which being com- be ascribed to
pounded of nitre, brimstone and coals, beaten small, ^thrtrhn!1^
hath from the coals its first taking fire ; from the stone, and what
~ to the nitre.
brimstone its nourishment and flame, that is to say,
light and motion, and from the nitre the vehe¬
mence of both. Now if a piece of nitre, before it
is beaten, be laid upon a burning coal, first it melts,
and, like water, quencheth that part of the coal it
toucheth. Then vapour or air, flying out where the
coal and nitre join, bloweth the coal with great
swiftness and vehemence on all sides. And from
hence it comes to pass, that by two contrary mo¬
tions, the one, of the particles which go out of the
burning coal, the other, of those of the ethereal
and watery substance of the nitre, is generated
that vehement motion and inflammation. And,
lastly, when there is no more action from the nitre,
that is to say, when the volatile parts of the nitre
are flown out, there is found about the sides a cer-
458
PHYSICS.
PAIiT IV- tain white substance, which being thrown again
r—- into the fire, will grow red-hot again, but will not
the6 force6 of dissipated, at least unless the fire be augmented,
gunpowder,&c. If now a possible cause of this be found out, the
same will also be a possible cause why a grain of
gunpowder set on fire doth expand itself with
such vehement motion, and shine. And it may be
caused in this manner.
Let the particles, of which nitre consisteth, be
supposed to be some of them hard, others watery,
and the rest ethereal. Also let the hard particles
be supposed to be spherically hollow, like small
bubbles, so that many of them growing together
may constitute a body, whose little caverns are
filled with a substance which is either watery, or
ethereal, or both. As soon, therefore, as the hard
particles are dissipated, the watery and ethereal
particles will necessarily fly out; and as they fly,
of necessity blow strongly the burning coals and
brimstone which are mingled together ; whereupon
there will follow a great expansion of light, with
vehement flame, and a violent dissipation of the
particles of the nitre, the brimstone and the coals.
Wherefore I have given a possible cause of the
force of fired gunpowder.
It is manifest from hence, that for the rendering
of the cause why a bullet of lead or iron, shot from
a piece of ordnance, flies with so great velocity,
there is no necessity to introduce such rarefaction,
as, by the common definition of it, makes the
same matter to have sometimes more, sometimes
less quantity ; which is inconceivable. For every
thing is said to be greater or less, as it hath more
or less quantity. The violence with which a bullet
LTGHT, IIEAT, AND COLOURS.
459
is thrust out of a gun, proceeds from the swiftness part iy*
of the small particles of the fired powder ; at least v^
it may proceed from that cause without the suppo¬
sition of any empty space.
11. Besides, by the attrition or rubbing of one How heat
• • is Ctiuscd Ibv
body against another, as of wood against wood, we attrition,
find that not only a certain degree of heat, but fire
itself is sometimes generated. For such motion
is the reciprocation of pressure, sometimes one way,
sometimes the other ; and by this reciprocation
whatsoever is fluid in both the pieces of wood is
forced hither and thither ; and consequently, to an
endeavour of getting out; and at last by breaking
out makes fire.
12. Now light is distinguished into, first, second, The distinction
third, and so on infinitely. And we call that first fiLt^econdAc.
light, which is in the first lucid body; as the sun,
fire, &c. : second, that which is in such bodies, as
being not transparent are illuminated by the sun;
as the moon, a wall, &c.: and third, that which is in
bodies not transparent, but illuminated by second
light, &c.
13. Colour is light, but troubled light, namely, The causes of
such as is generated by perturbed motion; as shall see in looking
be made manifest by the red, yellow, blue and pur- m^of giaM,18"
pie, which are generated by the interposition of a e!iolv3 bK*ee&
diaphanous prisma, whose opposite bases are violet colour,
triangular, between the light and that which is
enlightened.
For let there be a prisma of glass, or of any other
transparent matter which is of greater density than
air; and let the triangle ABC be the base of this
prisma. Also let the strait line D E be the dia¬
meter of the sun's body, having oblique position to
460
physics.
part iv. the strait line A B ; and let the sunbeams pass in
*—r-—- the lines D A and EBC. And lastly, let the strait
th'e^ofoimf, &c. lines DA and EC be produced indefinitely to F
and G. Seeing therefore the strait line D A, by
reason of the density of the glass, is refracted to¬
wards the perpendicular ; let the line refracted at
the point A be the strait line A H. And again,
seeing the medium below A C is thinner than that
above it, the other refraction, which will be made
there, will diverge from the perpendicular. Let
therefore this second refracted line be A I. Also
let the same be done at the point C, by making the
first refracted line to be C K, and the second C L.
Seeing therefore the cause of the refraction in the
point A of the strait line of A B is the excess of the
resistance of the medium in A B above the resist¬
ance of the air, there must of necessity be reaction
from the point A towards the point B ; and conse¬
quently the medium at A within the triangle ABC
will have its motion troubled, that is to say, the
strait motion in A F and A H will be mixed with
the transverse motion between the same A F and
A II, represented by the short transverse lines in
the triangle AFH. Again, seeing at the point A
of the strait line A C there is a second refraction
from A H in A I, the motion of the medium will
again be perturbed by reason of the transverse re¬
action from A towards C, represented likewise by
the short transverse lines in the triangle A H I.
And in the same manner there is a double pertur¬
bation represented by the transverse lines in the
triangles CGK and C K L. But as for the light
between A I and C G, it will not be perturbed;
because, if there were in all the points of the strait
LIGHT, HEAT, AND COLOURS.
461
lines A B and A C the same action which is in the PAR^'IY-
points A and C, then the plane of the triangle CGK -—■—'
would be everywhere coincident with the plane of th^oTom",L.
the triangle AFH; by which means all would ap¬
pear alike between A and C. Besides, it is to be
observed, that all the reaction at A tends towards
the illuminated parts which are between A and C,
and consequently perturbeth the first light. And
011 the contrary, that all the reaction at C tends
towards the parts without the triangle or without
the prisma ABC, where there is none but second
light; and that the triangle AFH shows that per¬
turbation of light which is made in the glass itself;
as the triangle A H I shows that perturbation of
light which is made below the glass. In like manner,
that CGK shows the perturbation of light within
the glass ; and C K L that which is belowT the glass.
From whence there are four divers motions, or four
different illuminations or colours, whose differences
appear most manifestly to the sense in a prisma,
whose base is an equilateral triangle, when the
sunbeams that pass through it are received upon a
white paper. For the triangle A F H appears red
to the sense; the triangle A H I yellow; the tri¬
angle CGK green, and approaching to blue; and
lastly, the triangle C K L appears purple. It is
therefore evident that when weak but first light
passeth through a more resisting diaphanous body,
as glass, the beams, which fall upon it transversely,
make redness ; and when the same first light is
stronger, as it is in the thinner medium below the
strait line A C, the transverse beams make yellow¬
ness. Also when second light is strong, as it is in
the triangle C G K, which is nearest to the first
462
physics.
part iv. light, the transverse beams make greenness; and
-—^— when the same second light is weaker, as in the
triangle C K L, they make a purple colour.
Why the moon 14. From hence may be deduced a cause, why
cind tlic stars
appear redder the moon and stars appear bigger and redder near
than1 inTi!eZ°n ^ie horizon than in the mid-heaven. For between
heaven°fthe ^ie eYe an^ the apparent horizon there is more
impure air, such as is mingled with watery and
earthy little bodies, than is between the same eye
and the more elevated part of heaven. But vision
is made by beams which constitute a cone, whose
base, if we look upon the moon, is the moon's face,
and whose vertex is in the eye; and therefore,
many beams from the moon must needs fall upon
little bodies that are without the visual cone, and
be by them reflected to the eye. But these reflected
beams tend all in lines which are transverse to the
visual cone, and make at the eye an angle which is
greater than the angle of the cone. Wherefore,
the moon appears greater in the horizon, than when
she is more elevated. And because those reflected
beams go transversely, there will be generated, by
the last article, redness. A possible cause there¬
fore is shown, why the moon as also the stars ap¬
pear greater and redder in the horizon, than in the
midst of heaven. The same also may be the cause,
why the sun appears in the horizon greater and of
a colour more degenerating to yellow, than when
he is higher elevated. For the reflection from the
little bodies between, and the transverse motion of
the medium, are still the same. But the light of
the sun is much stronger than that of the moon;
and therefore, by the last article, his splendour
LIGHT, HEAT, AND COLOURS. 463
must needs by this perturbation degenerate into PAI^ IY-
yellowness. -—.—-
But for tlie generation of these four colours, it is
not necessary that the figure of the glass be a
prisma; for if it were spherical it would do the
same. For in a sphere the sunbeams are twice
refracted and twice reflected. And this being ob¬
served by Des Cartes, and withal that a rainbow
never appears but when it rains ; as also, that the
drops of rain have their figures almost spherical; he
hath shown from thence the cause of the colours
in the rainbow; which therefore need not be
repeated.
15. Whiteness is light, but light perturbed by The cause of
the reflections of many beams of light coming to
the eye together within a little space. For if glass
or any other diaphanous body be reduced to very
small parts by contusion or concussion, every one
of those parts, if the beams of a lucid body be
from any one point of the same reflected to the eye,
will represent to the beholder an idea or image of
the whole lucid body, that is to say, a phantasm
of white. For the strongest light is the most white;
and therefore many such parts will make many
such images. Wherefore, if those parts lie thick
and close together, those many images will appear
confusedly, and will by reason of the confused light
represent a white colour. So that from hence
may be deduced a possible cause, why glass beaten,
that is, reduced to powder, looks white. Also why
water and snow are white; they being nothing but
a heap of very small diaphanous bodies, namely, of
little bubbles, from whose several convex superficies
there are by reflection made several confused plian-
464
PHYSICS.
part iv. tasms of the whole lucid body, that is to say, white-
'—ness. For the same reason, salt and nitre are white,
as consisting of small bubbles which contain within
them water and air; as is manifest in nitre, from
this, that being thrown into the fire it violently
blows the same ; which salt also doth, but with less
violence. But if a white body be exposed, not to
the light of the day, but to that of the fire or of a
candle, it will not at the first sight be easily judged
whether it be white or yellow ; the cause whereof
may be this, that the light of those things, which
burn and flame, is almost of a middle colour between
whiteness and yellowness.
The cause of 16. As whiteness is light, so blackness is the pri¬
vation of light, or darkness. And, from hence it is,
first, that all holes, from which no light can be re¬
flected to the eye, appear black. Secondly, that
when a body hath little eminent particles erected
straight up from the superficies, so that the beams
of light which fall upon them are reflected not to
the eye but to the body itself, that superficies
appears black ; in the same manner as the sea
appears back when ruffled by the wind. Thirdly,
that any combustible matter is by the fire made to
look black before it shines. For the endeavour of
the fire being to dissipate the smallest parts of such
bodies as are thrown into it, it must first raise and
erect those parts before it can work their dissipa¬
tion. If, therefore, the fire be put out before the
parts are totally dissipated, the coal will appear
black ; for the parts being only erected, the beams
of light falling upon them will not be reflected to
the eye, but to the coal itself. Fourthly, that burn¬
ing glasses do more easily burn black things than
light, heat, and colours.
465
white. For in a white superficies the eminent PA!g Iv*
parts are convex, like little bubbles; and there- v—*—'
fore the beams of light, which fall upon them, are y^kcnaeusss®of
reflected every way from the reflecting body. But
in a black superficies, where the eminent particles
are more erected, the beams of light falling upon
them are all necessarily reflected towards the body
itself; and, therefore, bodies that are black are
more easily set on fire by the sun beams, than
those that are Wxxlte. Fifthly, that all colours
that are made of the mixture of white and black
proceed from the different position of the particles
that rise above the superficies, and their different
forms of asperity. For, according to these differ¬
ences, more or fewer beams of light are reflected
from several bodies to the eye. But in regard
those differences are innumerable, and the bodies
themselves so small that we cannot perceive them;
the explication and precise determination of the
causes of all colours is a thing of so great difficulty,
that I dare not undertake it.
vol. i.
h h
466
PHYSICS.
CHAPTER XXVIII.
OF COLD, WIND, HARD, ICE, RESTITUTION OF
BODIES BENT, DIAPHANOUS, LIGHTNING AND
THUNDER ; AND OF THE HEADS OF RIVERS.
1. Why breath from the same mouth sometimes heats and some¬
times cools.—2. Wind, and the inconstancy of winds, whence.
3. Why there is a constant, though not a great wind, from
east to west, near the equator.—4. What is the effect of air
pent in between the clouds.—5. No change from soft to hard,
but by motion.—6. Wliat is the cause of cold near the poles.
7. The cause of ice ; and why the cold is more remiss in rainy
than in clear weather. Why water doth not freeze in deep
wells as it doth near the superficies of the earth. Why ice is
not so heavy as water; and why wine is not so easily frozen
as water.—8. Another cause of hardness from the fuller con¬
tact of atoms; also, how hard things are broken.—9. A third
cause of hardness from heat.—10. A fourth cause of hardness
from the motion of atoms enclosed in a narrow space.—11.
How hard things are softened. —12. Whence proceed the
spontaneous restitution of things bent.—13. Diaphanous and
opacous, what they are, and whence.—14. The cause of light¬
ning and thunder..—15. Whence it proceeds that clouds can
fall again after they are once elevated and frozen.—16. How
it could be that the moon was eclipsed, when she was not dia¬
metrically opposite to the sun.—17. By what means many
suns may appear at once.—18. Of the heads of rivers.
PAR28R' 1 • A s, when the motion of the ambient ethereal
v ' ' substance makes the spirits and fluid parts of our
AVhy breath ,
from the same bodies tend outwards, we acknowledge heat; so,
umSheT™6 by the endeavour inwards of the same spirits and
and some- humours, we feel cold. So that to cool is to make
times cools. 3
the exterior parts of the body endeavour inwards,
OF COLD, WIND, ETC.
46/
by a motion contrary to that of calefaction, by lv-
which the internal parts are called outwards. He, -—-—"
therefore, that would know the cause of cold, must
find by what motion or motions the exterior parts
of any body endeavour to retire inwards. To
begin with those phenomena which are the most
familiar. There is almost no man but knows, that
breath blown strongly, and which comes from the
mouth with violence, that is to say, the passage
being strait, will cool the hand; and that the
same breath blown gently, that is to say, through
a greater aperture, will warm the same. The
cause of which phenomenon may be this, the breath
going out hath two motions ; the one, of the whole
and direct, by which the foremost parts of the
hand are driven inw ards ; the other, simple motion
of the small particles of the same breath, which,
(as I have shown in the 3rd article of the last
chapter, causeth heat. According, therefore, as
either of these motions is predominant, so there is
the sense sometimes of cold, sometimes of heat.
Wherefore, when the breath is softly breathed out
at a large passage, that simple motion which causeth
heat prevaileth, and consequently heat is felt; and
when, by compressing the lips, the breath is more
strongly blown out, then is the direct motion pre¬
valent, which makes us feel cold. For, the direct
motion of the breath or air is wind; and all wind
cools or diminisheth former heat.
2. And seeing not only great wind, but almost wind, and the
any ventilation and stirring of the air, doth refri- SsfXLe!
gerate ; the reason of many experiments concern¬
ing cold cannot well be given without finding first
what are the causes of wind. Now, wind is
H H 2
4G8
PHYSICS.
PA2J IV* not^ng e^se but ^ie direct motion of tlie air thrust'
■—' forwards ; which, nevertheless, when many winds
hiconstancVofconcur, may be circular or otherwise indirect, as
winds, whence, jj. js *n whirlwinds. Wherefore, in the first place
we are to enquire into the causes of winds. Wind
is air moved in a considerable quantity, and that
either in the manner of waves, which is both for¬
wards and also up and down, or else forwards
only.
Supposing, therefore, the air both clear and
calm for any time how little soever, yet, the
greater bodies of the world being so disposed and
ordered as has been said, it will be necessary that
a wind presently arise somewhere. For, seeing
that motion of the parts of the air, which is made
by the simple motion of the sun in his own epicycle,
causeth an exhalation of the particles of water
from the seas and all other moist bodies, and those
particles make clouds; it must needs follow, that,
whilst the particles of water pass upwards, the
particles of air, for the keeping of all spaces full,
be jostled out on every side, and urge the next par¬
ticles, and these the next; till having made their
circuit, there comes continually so much air to the
hinder parts of the earth as there went water from
before it. Wherefore, the ascending vapours move
the air towards the sides every way ; and all direct
motion of the air being wind, they make a wind.
And if this wind meet often with other vapours
which arise in other places, it is manifest that the
force thereof will be augmented, and the way or
course of it changed. Besides, according as the
earth, by its diurnal motion, turns sometimes the
drier, sometimes the moister part towards the sun,
of cold, wind, etc.
469
so sometimes a greater, sometimes a less, quantity paiit iv.
of vapours will be raised ; that is to say, sometimes -A-*
there will be a less, sometimes a greater wind.
Wherefore, I have rendered a possible cause of
such winds as are generated by vapours ; and also
of their inconstancy.
From hence it follows that these winds cannot
be made in any place, which is higher than that to
which vapours may ascend. Nor is that incredible
which is reported of the highest mountains, as the
Peak of Teneriffe and the Andes of Peru, namely,
that they are not at all troubled with these incon¬
stant winds. And if it were certain that neither
rain nor snow were ever seen in the highest tops
of those mountains, it could not be doubted but
that they are higher than any place to which
vapours use to ascend.
3. Nevertheless, there may be wind there, though ^Vhythere
not that which is made by the ascent of vapours, though not a
yet a less and more constant wind, like the con- e^'to
tinued blast of a pair of bellows, blowing- from the west> near the
A ° equator.
east. And this may have a double cause ; the one,
the diurnal motion of the earth; the other, its
simple motion in its owrn epicycle. For these
mountains being, by reason of their height, more
eminent than all the rest of the parts of the earth,
do by both these motions drive the air from the
west eastwards. To which, though the diurnal
motion contribute but little, yet seeing I have
supposed that the simple motion of the earth, in its
own epicycle, makes two revolutions in the same
time in which the diurnal motion makes but one,
and that the semidiameter of the epicycle is double
to the semidiameter of the diurnal conversion, the
470
PHYSICS.
part iv. motion of every point of the earth in its own
v—•—- epicycle will have its velocity quadruple to that of
the diurnal motion; so that by both these motions
together, the tops of those hills will sensibly be
moved against the air; and consequently a wind
will be felt. For whether the air strike the sen¬
tient, or the sentient the air, the perception of
motion will be the same. But this wind, seeing it
is not caused by the ascent of vapours, must neces¬
sarily be very constant,
what is the ef- 4 When one cloud is already ascended into the
lect of air pent J
in between the air, if another cloud ascend towards it, that part of
the air, which is intercepted between them both,
must of necessity be pressed out every way. Also
when both of them, whilst the one ascends and
the other either stays or descends, come to be
joined in such manner as that the ethereal sub¬
stance be shut within them on every side, it will
by this compression also go out by penetrating the
water. But in the meantime, the hard particles,
which are mingled with the air and are agitated,
as I have supposed, wdth simple motion, will not
pass through the water of the clouds, but be more
straitly compressed within their cavities. And
this I have demonstrated at the 4th and 5th articles
of chapter xxn. Besides, seeing the globe of the
earth floateth in the air which is agitated by the
sun's motion, the parts of the air resisted by the
earth will spread themselves every way upon the
earth's superficies ; as I have shown at the 8th
article of chapter xxi.
fro°mchsaonfteto 5. We perceive a body to be hard, from this,
hard, but by that, when touching it, we would thrust forwards
that part of the same which we touch, we cannot
OF COLD, WIND, ETC.
471
do it otherwise than by thrusting forwards the PAI^IV*
whole body. We may indeed easily and sensibly •—<—'
thrust forwards any particle of the air or water
which we touch, whilst yet the rest of its parts
remain to sense unmoved. But we cannot do so
to any part of a stone. Wherefore I define a hard
body to be that whereof no part can be sensibly
moved, unless the whole be moved. Whatsoever
therefore is soft or fluid, the same can never be
made hard but by such motion as makes many of
the parts together stop the motion of some one
part, by resisting the same.
6. Those things premised, I shall show a possible What is„the,
. cause of cold.
cause why there is greater cold near the poles of near the poles,
the earth, than further from them. The motion
of the sun between the tropics, driving the air
towards that part of the earth's superficies which
is perpendicularly under it, makes it spread itself
every way; and the velocity of this expansion of
the air grows greater and greater, as the superficies
of the earth comes to be more and more straitened,
that is to say, as the circles which are parallel to
the equator come to be less and less. Wherefore
this expansive motion of the air drives before it
the parts of the air, which are in its way, con¬
tinually towards the poles more and more strongly,
as its force comes to be more and more united,
that is to say, as the circles which are parallel to
the equator are less and less; that is, so much the
more, by how much they are nearer to the poles
of the fearth. In those places, therefore, which are
nearer to the poles, there is greater cold than in
those which are more remote from them. Now this
expansion of the air upon the superficies of the
472
PHYSICS.
PA^s IV" eart^' ^rom east to west, doth, by reason of the
--r—- sun's perpetual accession to the places which are
successively under it, make it cold at the time of
the sun's rising and setting ; but as the sun comes
to be continually more and more perpendicular to
those cooled places, so by the heat, which is gene¬
rated by the supervening simple motion of the
sun, that cold is again remitted ; and can never be
great, because the action by which it was generated
is not permanent. Wherefore I have rendered a
possible cause of cold in those places that are near
the poles, or where the obliquity of the sun is great.
tiiq cause of j# How water may be concealed by cold, may
ice ; ana why ... .
the cold is more be explained in this manner. Let A (in figure 1)
remiss in rainy , -i t-» i i **ni
than in clear represent the sun, and i3 the earth. A will there-
wate^cioti^not f°re be much greater than B. Let E F be in the
weU^6 a sit doth plane °f the equinoctial; to which let GH, IK,
near the super- and L C be parallel. Lastly, let C and D be the
earth, why ice poles of the earth. The air, therefore, by its
MTOterfaS action in those parallels, will rake the superficies
why wine is not 0f the earth ; and that with motion so much the
so easily frozen
as water. stronger, by how much the parallel circles towards
the poles grow less and less. From whence must
arise a wind, which will force together the upper¬
most parts of the water, and withal raise them a
little, weakening their endeavour towards the
centre of the earth. And from their endeavour
towards the centre of the earth, joined with the
endeavour of the said wind, the uppermost parts
of the water will be pressed together and coagu¬
lated, that is to say, the top of the water will be
skinned over and hardened. And so again, the
water next the top will be hardened in the same
manner, till at length the ice be thick. And this
OF COLD, WIND, ETC.
473
ice, being now compacted of little hard bodies, PA^T IV-
28.
must also contain many particles of air received
into it. Tfhe car
of ice, &c.
As rivers and seas, so also in the same manner
may the clouds be frozen. For when, by the
ascending and descending of several clouds at the
same time, the air intercepted between them is by
compression forced out, it rakes, and by little and
little hardens them. And though those small drops,
which usually make clouds, be not yet united into
greater bodies, yet the same wind will be made;
and by it, as water is congealed into ice, so will
vapours in the same manner be congealed into
snow. From the same cause it is that ice may be
made by art, and that not far from the fire. For
it is done by the mingling of snow and salt
together, and by burying in it a small vessel full of
water. Now while the snowT and salt, which have
in them a great deal of air, are melting, the air,
which is pressed out every way in wind, rakes the
sides of the vessel; and as the wind by its motion
rakes the vessel, so the vessel by the same motion
and action congeals the water within it.
We find by experience, that cold is always more
remiss in places where it rains, or where the
weather is cloudy, things being alike in all other
respects, than where the air is clear. And this
agreeth very well with what I have said before.
For in clear weather, the course of the wind which,
as I said even now, rakes the superficies of the
earth, as it is free from all interruption, so also it
is very strong. But when small drops of water
are either rising or falling, that wind is repelled,
474
PHYSICS.
rA^s IV broken, and dissipated by them ; and the less the
s—■—' wind is, the less is the cold.
We find also by experience, that in deep wells
the water freezeth not so mnch as it doth upon the
superficies of the earth. For the wind, by which
ice is made, entering into the earth by reason of
the laxity of its parts, more or less, loseth some of
its force, though not much. So that if the well be
not deep, it will freeze ; whereas if it be so deep,
as that the wind which causeth cold cannot reach
it, it will not freeze.
We find moreover by experience, that ice is
lighter than water. The cause whereof is manifest
from that which I have already shown, namely, that
air is received in and mingled with the particles of
the water whilst it is congealing.
Lastly, wine is not so easily congealed as water,
because in wine there are particles, which, being
not fluid, are moved very swiftly, and by their
motion congelation is retarded. But if the cold
prevail against this motion, then the outermost
parts of the wine will be first frozen, and after¬
wards the inner parts ; whereof this is a sign, that
the wine which remains unfrozen in the midst will
be very strong.
ofhardnessUSe have seen one way of making things hard,
from the fuller namely, by congelation. Another way is thus,
atoms! Also Having already supposed that innumerable atoms,
ar*broXings some harder than others and that have several
simple motions of their own, are intermingled with
the ethereal substance ; it follows necessarily from
hence, that by reason of the fermentation of the
whole air, of which I have spoken in chapter xxi,
some of those atoms meeting with others will
OF COLD, WIND, ETC.
4/5
cleave together, by applying themselves to one IV-
another in such manner as is agreeable to their -—^—-
motions and mutual contacts ; and, seeing there is ofhardnesT&c6
no vacuum, cannot be pulled asunder, but by
so much force as is sufficient to overcome their
hardness.
Now there are innumerable degrees of hardness.
As for example, there is a degree of it in water,
as is manifest from this, that upon a plane it may
be drawn any way at pleasure by one's finger.
There is a greater degree of it in clammy liquors,
which, when they are poured out, do in falling
downwards dispose themselves into one continued
thread; which thread, before it be broken, will by
little and little diminish its thickness, till at last it
be so small, as that it seems to break only in a
point; and in their separation the external parts
break first from one another, and then the more
internal parts successively one after another. In
wax there is yet a greater degree of hardness. For
when we would pull one part of it from another,
we first make the whole mass slenderer, before we
can pull it asunder. And how much the harder
anything is which we would break, so much the
more force we must apply to it. Wherefore, if we
go on to harder things, as ropes, wood, metals,
stones, &c., reason prompteth us to believe that the
same, though not always sensibly, will necessarily
happen; and that even the hardest things are
broken asunder in the same manner, namely, by
solution of their continuity begun in the outermost
superficies, and proceeding successively to the
innermost parts. In like manner, when the parts
of bodies are to be separated, not by pulling them
476
PHYSICS.
PAIl28.1V' asunder, but by breaking them, the first separation
v T ' will necessarily be in the convex superficies of the
bowed part of the body, and afterwards in the
concave superficies. For in all bowing there is in
the convex superficies an endeavour in the parts to
go one from another, and in the concave superficies
to penetrate one another.
This being well understood, a reason may be
given how two bodies, which are contiguous in one
common superficies, may by force be separated
without the introduction of vacuum ; though
Lucretius thought otherwise, believing that such
separation was a strong establishment of vacuum.
For a marble pillar being made to hang by one of
its bases, if it be long enough, it will by its own
weight be broken asunder ; and yet it will not
necessarily follow that there should be vacuum,
seeing the solution of its continuity may begin in
the circumference, and proceed successively to the
midst thereof.
a third cause g Another cause of hardness in some things
of hardness, 0
from heat. may be in this manner. If a soft body consist of
many hard particles, which by the intermixture of
many other fluid particles cohere but loosely to¬
gether, those fluid parts, as hath been shown in
the last article of chapter xxi, will be exhaled;
by which means each hard particle will apply itself
to the next to it according to a greater superficies,
and consequently they will cohere more closely to
one another, that is to say, the whole mass will be
made harder.
dr hTrdnessUSe 1 some things hardness may be made
from the mo- to a certain degree in this manner. When any
ion o atoms Sllbstance hath in it certain very small bodies
OF COLD, WIND, ETC.
4 77
intermingled, which, being moved with simple mo- rA^g IV"
tion of their own, contribute like motion to the v—*—
parts of the fluid substance, and this be done in a space,
small enclosed space, as in the hollow of a little
sphere, or a very slender pipe, if the motion be
vehement and there be a great number of these
small enclosed bodies, two things will happen ; the
one, that the fluid substance will have an endeavour
of dilating itself at once every way ; the other,
that if those small bodies can nowhere get out,
then from their reflection it will follow, that the
motion of the parts of the enclosed fluid substance,
which was vehement before, will now be much
more vehement. Wherefore, if any one particle
of that fluid substance should be touched and
pressed by some external movent, it could not yield
but by the application of very sensible force.
Wherefore the fluid substance, which is enclosed
and so moved, hath some degree of hardness.
Now, greater and less degree of hardness depends
upon the quantity and velocity of those small
bodies, and upon the narrowness of the place both
together.
11. Such things as are made hard by sudden How hard
heat, namely such as are hardened by fire, are softened,
commonly reduced to their former soft form by
maceration. For fire hardens by evaporation, and
therefore if the evaporated moisture be restored
again, the former nature and form is restored
together with it. And such things as are frozen
with cold, if the wind by which they were frozen
change into the opposite quarter, they will be un¬
frozen again, unless they have gotten a habit of
new motion or endeavour by long continuance in
478
PHYSICS.
PA2? IV' hardness- Nor is it enough to cause thawing,
v—.—' that there be a cessation of the freezing wind ; for
the taking away of the cause doth not destroy a
produced effect; but the thawing also must have
its proper cause, namely, a contrary wind, or at
least a wind opposite in some degree. And this
we find to be true by experience. For, if ice be
laid in a place so well enclosed that the motion of
the air cannot get to it, that ice will remain un¬
changed, though the place be not sensibly cold.
whence Pr°- 22. Of hard bodies, some may manifestly be
ceeds thespon- J J
taneous resti- bowed; others not, but are broken in the very
tution of things r ci-it
bent. first moment of their bending. And of such
bodies as may manifestly be bended, some being
bent, do, as soon as ever they are set at liberty,
restore themselves to their former posture; others
remain still bent. Now if the cause of this resti¬
tution be asked, I say, it may be in this manner,
namely, that the particles of the bended body,
whilst it is held bent, do nevertheless retain their
motion ; and by this motion they restore it as soon
as the force is removed by which it was bent. For
when any thing is bent, as a plate of steel, and, as
soon as the force is removed, restores itself again, it
is evident that the cause of its restitution cannot be
referred to the ambient air ; nor can it be referred
to the removal of the force by which it was bent;
for in things that are at rest the taking away of
impediments is not a sufficient cause of their future
motion; there being no other cause of motion, but
motion. The cause therefore of such restitution is
in the parts of the steel itself. Wherefore, whilst
it remains bent, there is in the parts, of which it
consisteth, some motion though invisible; that is to
OF COLD, WIND, ETC.
4 79
say, some endeavour at least that way by which PAI^;IV"
the restitution is to be made; and therefore this —'
endeavour of all the parts together is the first ^edTthePsp°on-
beginning of restitution ; so that the impediment tut^no/tWngs
being removed, that is to say, the force by which tent,
it was held bent, it will be restored again. Now
the motion of the parts, by which this done, is
that which I called simple motion, or motion
returning into itself. When therefore in the bend¬
ing of a plate the ends are drawn together, there
is on one side a mutual compression of the parts;
which compression is one endeavour opposite to
another endeavour : and on the other side a divul-
sion of the parts. The endeavour therefore of the
parts on one side tends to the restitution of the
plate from the middle towards the ends; and on
the other side, from the ends towards the middle.
Wherefore the impediment being taken away, this
endeavour, which is the beginning of restitution,
will restore the plate to its former posture. And
thus I have given a possible cause why some bodies,
when they are bent, restore themselves again;
which was to be done.
As for stones, seeing they are made by the
accretion of many very hard particles within the
earth; which particles have no great coherence,
that is to say, touch one another in small latitude,
and consequently admit many particles of air; it
must needs be that, in bending of them, their
internal parts will not easily be compressed, by
reason of their hardness. And because their co¬
herence is not firm, as soon as the external hard
particles are disjoined, the ethereal parts will
480
physics.
PA28TIV" necessarily break out, and so the body will sud-
-—^—' denly be broken.
Fnd^opTcous, 13- Those bodies are called diaphanous, upon
what they are, which, whilst the beams of a lucid body do work,
and whence. . .
the action of every one of those beams is propa-
gated in them in such manner, as that they still
retain the same order amongst themselves, or the
inversion of that order; and therefore bodies,
which are perfectly diaphanous, are also perfectly
homogeneous. On the contrary, an opacous body
is that, which, by reason of its heterogeneous
nature, doth by innumerable reflections and refrac¬
tions in particles of different figures and unequal
hardness, weaken the beams that fall upon it before
they reach the eye. And of diaphanous bodies,
some are made such by nature from the beginning;
as the substance of the air, and of the water, and
perhaps also some parts of stones, unless these
also be water that has been long congealed. Others
are made so by the power of heat, w7hich congre¬
gates homogeneous bodies. But such, as are made
diaphanous in this manner, consist of parts which
were formerly diaphanous.
Hghtning and w^at manner clouds are made by the
thunder. motion of the sun, elevating the particles of water
from the sea and other moist places, hath been
explained in chapter xxvi. Also how clouds come
to be frozen, hath been shown above at the 7th
article. Now from this, that air may be enclosed
as it were in caverns, and pent together more and
more by the meeting of ascending and descending
clouds, may be deduced a possible cause of thunder
and lightning. For seeing the air consists of two
OF COLD, WIND, ETC.
481
parts, tlie one ethereal, which has no proper mo- PA^IV*
tion of its own, as being a thing divisible into the ■ *—
least parts ; the other hard, namely, consisting of
many hard atoms, which have every one of them
a very swift simple motion of its own: whilst the
clouds by their meeting do more and more straiten
such cavities as they intercept, the ethereal parts
will penetrate and pass through their watery sub¬
stance ; but the hard parts will in the meantime
be the more thrust together, and press one another;
and consequently, by reason of their vehement
motions, they will have an endeavour to rebound
from each other. Whensoever, therefore, the com¬
pression is great enough, and the concave parts of
the clouds are, for the cause I have already given,
congealed into ice, the cloud will necessarily be
broken; and this breaking of the cloud produceth
the first clap of thunder. Afterwards the air,
which wras pent in, having now broken through,
makes a concussion of the air without, and from
hence proceeds the roaring and murmur which
follows; and both the first clap and the murmur
that follows it make that noise which is called
thunder. Also, from the same air breaking through
the clouds and with concussion falling upon the
eye, proceeds that action upon our eye, which
causeth in us a perception of that light, which we
call lightning. Wherefore I have given a possible
cause of thunder and lightning.
15. But if the vapours, which are raised into whence it
itt ,i . , i proceeds that
clouds, do run together again into water or be clou(ls can fall
congealed into ice, from whence is it, seeing both
ice and water are heavy, that they are sustained in elevated and
' , . frozen.
the air ? Or rather, what may the cause be, that
VOL. T. 1 I
482
PHYSICS.
PA^8 IV' being once elevated, they fall down again ? For
v—«—' there is no donbt but the same force which could
proceeds that carry up ^iat water, could also sustain it there,
clouds, &c. Why therefore being once carried up, doth it fall
again ? I say it proceeds from the same simple
motion of the sun, both that vapours are forced to
ascend, and that water gathered into clouds is
forced to descend. For in chapter xxi, article
11, I have shown how vapours are elevated;
and in the same chapter, article 5, I have also
shown how by the same motion homogeneous
bodies are congregated, and heterogeneous dissi¬
pated ; that is to say, how such things, as have a
like nature to that of the earth, are driven towards
the earth; that is to say, what is the cause of the
descent of heavy bodies. Now if the action of the
sun be hindered in the raising of vapours, and be
not at all hindered in the casting of them down, the
water will descend. But a cloud cannot hinder
the action of the sun in making things of an
earthly nature descend to the earth, though it may
hinder it in making vapours ascend. For the
lower part of a thick cloud is so covered by its
upper part, as that it cannot receive that action of
the sun by which vapours are carried up ; because
vapours are raised by the perpetual fermentation
of the air, or by the separating of its smallest parts
from one another, which is much weaker when a
thick cloud is interposed, than when the sky is
clear. And therefore, whensoever a cloud is made
thick enough, the water, which would not descend
before, will then descend, unless it be kept up by
the agitation of the wind. Wherefore I have ren¬
dered a possible cause, both why the clouds may
of cold, wind, etc.
483
be sustained in tlie air, and also why they may fall PA^g IV*
down again to the earth ; which w as propounded •———-
to be done.
16. Granting that the clouds maybe frozen, it is How it; could
.. be that the
no wonder it the moon have been seen eclipsed at moon was
such time as she hath been almost two degreesshewasnot16"
above the horizon, the sun at the same time appear- ™toly
ing in the horizon; for such an eclipse was ob- the su»-
served by Msestlin, at Tubingen, in the year 1590.
For it might happen that a frozen cloud was then
interposed between the sun and the eye of the
observer. And if it were so, the sun, which was
then almost two degrees below the horizon, might
appear to be in it, by reason of the passing of his
beams through the ice. And it is to be noted that
those, that attribute such refractions to the atmos¬
phere, cannot attribute to it so great a refraction
as this. Wherefore not the atmosphere, but either
water in a continued body, or else ice, must be the
cause of that refraction.
17- Again, granting that there may be ice in the By what means
many suns may
clouos, it will be no longer a wonder that many appear at once,
suns have sometimes appeared at once. For look¬
ing-glasses may be so placed, as by reflections to
show the same object in many places. And may
not so many frozen clouds serve for so many look¬
ing-glasses ? And may they not be fitly disposed for
that purpose ? Besides, the number of appearances
may be increased by refractions also; and there¬
fore it would be a greater wonder to me, if such
phenomena as these should never happen.
And wrere it not for that one phenomenon of the
new star which was seen in Cassiopea, I should
think comets were made in tin; same manner,
i i 2
484
PHYSICS.
PA:2s IV' namely, by vapours drawn not only from the earth
'—*—' but from the rest of the planets also, and congealed
into one continued body. For I could very well
from hence give a reason both of their hair, and of
their motions. But seeing that star remained
sixteen whole months in the same place amongst
the fixed stars, I cannot believe the matter of it
was ice. Wherefore I leave to others the disquisi¬
tion of the cause of comets; concerning which
nothing that bath hitherto been published, besides
the bare histories of them, is worth considering.
Of the heads ig. The heads of rivers may be deduced from
ot rivers. . J mi
ram-water, or trom melted snows, very easily; but
from other causes, very hardly, or not at all. For
both rain-water and melted snows run down the
descents of mountains; and if they descend only
by the outward superficies, the showers or snows
themselves may be accounted the springs or foun¬
tains ; but if they enter the earth and descend
within it, then, wheresoever they break out, there
are their springs. And as these springs make
small streams, so, many small streams running
together make rivers. Now, there was never any
spring found, but where the wTater which flowed to
it, was either further, or at least as far from the
centre of the earth, as the spring itself. And
whereas it has been objected by a great philoso¬
pher, that in the top of Mount Cenis, which parts
Savoy from Piedmont, there springs a river which
runs down by Susa ; it is not true. For there are
above that river, for two miles length, very high
hills on both sides, which are almost perpetually
covered with snow; from which innumerable little
streams running down do manifestly supply that
river with water sufficient for its magnitude.
sound, odour, etc.
485
CHAP. XXIX.
of sound, odour, savour, and touch.
1. The definition of sound, and the distinctions of sounds.
2. The cause of the degrees of sounds.—3. The difference be¬
tween sounds acute and grave.—4. The difference between
clear and hoarse sounds, whence.— 5. The sound of thunder
and of a gun, whence it proceeds.—6. Whence it is that pipes,
by blowing into them, have a clear sound.—7. Of reflected
sound.—8. From whence it is that sound is uniform and last¬
ing.—9. How sound may be helped and hindered by the wind.
10. Not only air, but other bodies how hard soever they be,
convey sound.—11. The causes of grave and acute sounds,
and of concent.—12. Phenomena for smelling.—13. The first
organ and the generation of smelling.—14. How it is helped
by heat and by wind.—15. Why such bodies are least smelt,
which have least intermixture of air in them.—16. Why odo¬
rous things become more odorous by being bruised.—17. The
first organ of tasting; and why some savours cause nauseous-
ness.—18. The first organ of feeling; and how we come to the
knowledge of such objects as are common to the touch and
other senses.
1. Sound is sense generated by the action of the partiv.
29
medium, when its motion reacheth the ear and the —-
rest of the organs of sense. Now, the motion of T,he defimtlon
«/ o t/ ' of sound, and
the medium is not the sound itself, but the cause the distinction
r • -nil 1--I- i • -i0^ sounds'
ot it. For the phantasm which is made m as, that
is to say, the reaction of the organ, is properly that
which we call sound.
The principal distinctions of sounds are these;
first, that one sound is stronger, another weaker.
Secondly, that one is more grave, another more
acute. Thirdly, that one is clear, another hoarse.
Fourthly, that one is primary, another derivative.
486
physics.
part iv. Fifthly, that one is uniform, another not. Sixthly,
- that one is more durable, another less durable. Of
all which distinctions the members may be sub-
distinguished into parts distinguishable almost in¬
finitely. For the variety of sounds seems to be not
much less than that of colours.
As vision, so hearing is generated by the motion
of the medium, but not in the same manner. F or
sight is from pressure, that is, from an endeavour ;
in which there is 110 perceptible progression of any
of the parts of the medium; but one part urging
or thrusting 011 another propagateth that action
successively to any distance whatsoever; whereas
the motion of the medium, by which sound is made,
is a stroke. For when we hear, the drum of the
ear, which is the first organ of hearing, is stricken;
and the drum being stricken, the pi a mater is also
shaken, and with it the arteries which are inserted
into it; by which the action being propagated to
the heart itself, by the reaction of the heart a phan¬
tasm is made which we call sound; and because
the reaction tendeth outwards, we think it is
without.
The cause of 2. And seeing the effects produced by motion
soundfre6S ° are greater or less, not only when the velocity is
greater or less, but also when the body hath greater
or less magnitude though the velocity be the same;
a sound may be greater or less both these ways.
And because neither the greatest nor the least
magnitude or velocity can be given, it may happen
that either the motion may be of so small velocity,
or the body itself of so small magnitude, as to pro¬
duce no sound at all; or either of them may be so
SOUND, ODOUR, ETC.
487
great, as to take away the faculty of sense by PA^1V*
hurting the organ. -—.—-
From hence may be deduced possible causes of ^decrees0of
the strength and weakness of sounds in the follow- sounds,
ing phenomena.
The first whereof is this, that if a man speak
through a trunk which hath one end applied to the
mouth of the speaker, and the other to the ear of
the hearer, the sound will come stronger than it
would do through the open air. And the cause,
not only the possible, but the certain and manifest
cause is this, that the air which is moved by the
first breath and carried forwards in the trunk, is
not diffused as it would be in the open air, and is
consequently brought to the ear almost with the
same velocity with which it was first breathed out.
Whereas, in the open air, the first motion diffuseth
itself every way into circles, such as are made by
the throwing of a stone into a standing water,
where the velocity grows less and less as the undu¬
lation proceeds further and further from the be¬
ginning of its motion.
The second is this, that if the trunk be short,
and the end which is applied to the mouth be wider
than that which is applied to the ear, thus also the
sound will be stronger than if it were made in the
open air. And the cause is the same, namely, that
by how much the wider end of the trunk is less
distant from the beginning of the sound, by so
much the less is the diffusion.
The third, that it is easier for one, that is within
a chamber, to hear what is spoken without, than
for him, that stands without, to hear what is spoken
within. For the windows and other inlets of the
488 physics.
PA 29^IV" movef^ a*r are as ^ie "wide end of the trunk. And
>—t—- for this reason some creatures seem to hear the
better, because nature has bestowed upon them
wide and capacious ears.
The fourth is this, that though he, which standeth
upon the sea-shore, cannot hear the collision of
the two nearest waves, yet nevertheless he hears
the roaring of the whole sea. And the cause seems
to be this, that though the several collisions move
the organ, yet they are not severally great enough
to cause sense ; whereas nothing hinders but that
all of them together may make sound.
The difference 3. That bodies when they are stricken do yield
between sounds jn . , ,
acute and grave some a more grave, others a more acute sound, the
cause may consist in the difference of the times in
which the parts stricken and forced out of their
places return to the same places again. For in
some bodies, the restitution of the moved parts is
quick, in others slow. And this also may be the
cause, why the parts of the organ, which are moved
by the medium, return to their rest again, some¬
times sooner, sometimes later. Now, by how much
the vibrations or the reciprocal motions of the
parts are more frequent, by so much doth the
whole sound made at the same time by one stroke
consist of more, and consequently of smaller parts.
For what is acute in sound, the same is subtle in
matter; and both of them, namely acute sound
and subtle matter, consist of very small parts, that
of time, and this of the matter itself.
The third distinction of sounds cannot be con¬
ceived clearly enough by the names I have used of
clear and hoarse, nor by any other that I know;
and therefore it is needful to explain them by
SOUND, ODOUR, ETC.
489
examples. When I say hoarse, I understand whis- Iv*
pering and hissing, and whatsoever is like to these, v—-—J
by what appellation soever it be expressed. And JetwcensoumL
sounds of this kind seem to be made by the force acuteandsrave
of some strong wind, raking rather than striking
such hard bodies as it falls upon. On the con¬
trary, when I use the word clear, I do not under¬
stand such a sound as may be easily and distinctly
heard; for so whispers would be clear; but such
as is made by somewhat that is broken, and such
as is clamour, tinkling, the sound of a trumpet, &c.
and to express it significantly in one word, noise.
And seeing no sound is made but by the concourse
of two bodies at the least, by which concourse it is
necessary that there be as well reaction as action,
that is to say, one motion opposite to another; it
follows that according as the proportion between
those two opposite motions is diversified, so the
sounds which are made will be different from one
another. And whensoever the proportion between
them is so great, as that the motion of one of the
bodies be insensible if compared with the motion
of the other, then the sound will not be of the same
kind; as when the wind falls very obliquely upon
a hard body, or when a hard body is carried swiftly
through the air; for then there is made that sound
which I call a hoarse sound, in Greek avpiy/uog.
Therefore the breath blown writh violence from the
mouth makes a hissing, because in going out it
rakes the superficies of the lips, whose reaction
against the force of the breath is not sensible.
And this is the cause why the winds have that
hoarse sound. Also if two bodies, how hard soever,
be rubbed together with no great pressure, they
490
PHYSICS.
PARTiv. make a hoarse sound. And this hoarse sound,
1' when it is made, as I have said, by the air raking
the superficies of a hard body, seemeth to be
nothing but the dividing of the air into innumera¬
ble and very small files. For the asperity of the
superficies doth, by the eminences of its innumera¬
ble parts, divide or cut in pieces the air that slides
upon it.
The difference 4. Noise, or that which I call clear sound, is
between clear 8c
hoarse sounds, made two ways ; one, by two hoarse sounds made
by opposite motions; the other, by collision, or by
the sudden pulling asunder of two bodies, whereby
their small particles are put into commotion, or
being already in commotion suddenly restore
themselves again; which motion, making impres¬
sion upon the medium, is propagated to the organ
of hearing. And seeing there is in this collision
or divulsion an endeavour in the particles of one
body, opposite to the endeavour of the particles of
the other body, there will also be made in the
organ of hearing a like opposition of endeavours,
that is to say, of motions; and consequently the
sound arising from thence will be made by two
opposite motions, that is to say, by two opposite
hoarse sounds in one and the same part of the
organ. For, as I have already said, a hoarse sound
supposeth the sensible motion of but one of the
bodies. And this opposition of motions in the
organ is the cause why two bodies make a noise,
when they are either suddenly stricken against one
another, or suddenly broken asunder.
The sound of 5. This being granted, and seeing withal that
a'-m^whenc^ thunder is made by the vehement eruption of the
it proceeds. air out of the cavities of congealed clouds, the
sound, odour, etc.
491
cause of the great noise or clap may be the sudden part iv.
breaking asunder of the ice. For in this action it —r—'
is necessary that there be not only a great c-oncus- "tdof
sion of the small particles of the broken parts, but a sun' whence
also that this concussion, by being communicated
to the air, be carried to the organ of hearing, and
make impression upon it. And then, from the
first reaction of the organ proceeds that first and
greatest sound, which is made by the collision of
the parts whilst they restore themselves. And
seeing there is in all concussion a reciprocation
of motion forwards and backwards in the parts
stricken; for opposite motions cannot extinguish
one another in an instant, as I have shown in the
11th article of chapter viii ; it follows necessarily
that the sound will both continue, and grow weaker
and weaker, till at last the action of the recipro¬
cating air grow so weak, as to be imperceptible.
Wherefore a possible cause is given both of the
first fierce noise of the thunder, and also of the
murmur that follows it.
The cause of the great sound from a discharged
piece of ordnance is like that of a clap of thunder.
For the gunpowder being fired doth, in its en¬
deavour to go out, attempt every way the sides of
the metal in such manner, as that it enlargeth the
circumference all along, and withal shorteneth the
axis; so that whilst the piece of ordnance is in
discharging, it is made both wider and shorter
than it was before; and therefore also presently
after it is discharged its wideness will be dimi¬
nished, and its length increased again by the resti¬
tution of all the particles of the matter, of which it
consisteth, to their former position. And this is
492
PHYSICS.
part iv. done with such motions of the parts, as are not
s—>—' only very vehement, but also opposite to one
another; which motions, being communicated to
the air, make impression upon the organ, and by
the reaction of the organ create a sound, which
lasteth for some time ; as I have already shown in
this article.
I note by the way, as not belonging to this
place, that the possible cause why a gun recoils
when it is shot off, may be this; that being first
swollen by the force of the fire, and afterwards
restoring itself, from this restitution there pro¬
ceeds an endeavour from all the sides towards the
cavity; and consequently this endeavour is in
those parts which are next the breech; which
being not hollow, but solid, the effect of the resti¬
tution is by it hindered and diverted into the
length; and by this means both the breech and
the W'hole gun is thrust backwards ; and the more
forcibly by how much the force is greater, by
which the part next the breech is restored to its
former posture, that is to say, by how much the
thinner is that part. The cause, therefore, why
guns recoil, some more some less, is the difference
of their thickness towards the breech; and the
greater that thickness is, the less they recoil; and
contrarily.
whence it is 6. Also the cause wrhy the sound of a pipe, which
biown'g^into is made by blowing into it, is nevertheless clear, is
clear'sounda same with that of the sound which is made by
collision. For if the breath, when it is blown into
a pipe, do only rake its concave superficies, or fall
upon it with a very sharp angle of incidence, the
sound will not be clear, but hoarse. But if the
SOUND, ODOUR, ETC.
493
angle be great enough, the percussion, which is rAI^J 1V-
made against one of the hollow sides, will be re- •—^—'
verberated to the opposite side; and so successive
repercussions will be made from side to side, till at
last the whole concave superficies of the pipe be
put into motion; which motion will be recipro¬
cated, as it is in collision; and this reciprocation
being propagated to the organ, from the reaction
of the organ will arise a clear sound, such as is
made by collision, or by breaking asunder of hard
bodies.
In the same manner it is with the sound of a
man's voice. For when the breath passeth out
without interruption, and doth but lightly touch
the cavities through which it is sent, the sound it
maketh is a hoarse sound. But if in going out it
strike strongly upon the larynx, then a clear
sound is made, as in a pipe. And the same
breath, as it comes in divers manners to the palate,
the tongue, the lips, the teeth, and other organs of
speech, so the sounds into which it is articulated
become different from one another.
7. I call that primary sound, which is generated ®^nedflected
by motion from the sounding body to the organ in
a strait line without reflection; and I call that
reflected sound, which is generated by one or more
reflections, being the same with that we call echo,
and is iterated as often as there are reflections
made from the object to the ear. And these re¬
flections are made by hills, walls, and other resist¬
ing bodies, so placed as that they make more or
fewer reflections of the motion, according as they
are themselves more or fewer in number; and
they make them more or less frequently, according
494
PHYSICS.
paiit iv. as they are more or less distant from one another.
-—Now the cause of both these things is to be sought
for in the situation of the reflecting bodies, as is
usually done in sight. For the laws of reflection
are the same in both, namely, that the angles of
incidence and reflection be equal to one another.
If, therefore, in a hollow elliptic body, whose in¬
side is well polished, or in two right parabolical
solids, which are joined together by one common
base, there be placed a sounding body in one of
the burning points, and the ear in the other, there
will be heard a sound by many degrees greater
than in the open air; and both this, and the burn¬
ing of such combustible things, as being put in
the same places are set on fire by the sun-beams,
are effects of one and the same cause. But, as
when the visible object is placed in one of the
burning points, it is not distinctly seen in the other,
because every part of the object being seen in
every line, which is reflected from the concave
superficies to the eye, makes a confusion in the
sight; so neither is sound heard articulately and
distinctly when it comes to the ear in all those
reflected lines. And this may be the reason why
in churches which have arched roofs, though they
be neither elliptical nor parabolical, yet because
their figure is not much different from these, the
voice from the pulpit will not be heard so articu¬
lately as it would be, if there were no vaulting at all.
From whence 8. Concerning the uniformity and duration of
it is that sound -iii i • i i
is uniform and sounds, both which have one common cause, we
may observe, that such bodies as being stricken
yield an unequal or harsh sound, are very hetero¬
geneous, that is to say, they consist of parts which
SOUND, ODOUR, ETC.
495
are very unlike both in figure and hardness, such PAIJ,gIv*
as are wood, stones, and others not a few. When -—*—-
these are stricken, there follows a concussion of ^soimd
their internal particles, and a restitution of them j®s^orm and
again. But they are neither moved alike, nor
have they the same action upon one another;
some of them recoiling from the stroke, whilst
others which have already finished their recoilings
are now returning; by which means they hinder
and stop one another. And from hence it is that
their motions are not only unequal and harsh, but
also that their reciprocations come to be quickly
extinguished. Whensoever, therefore, this motion
is propagated to the ear, the sound it makes is
unequal and of small duration. On the contrary,
if a body that is stricken be not only sufficiently
hard, but have also the particles of which it con-
sistetli like to one another both in hardness and
figure, such as are the particles of glass and metals,
which being first melted do afterwards settle and
harden; the sound it yieldeth will, because the
motions of its parts and their reciprocations are
like and uniform, be uniform and pleasant, and be
more or less lasting, according as the body stricken
hath greater or less magnitude. The possible
cause, therefore, of sounds uniform and harsh, and
of their longer or shorter duration, may be one
and the same likeness and unlikeness of the inter¬
nal parts of the sounding body, in respect both of
their figure and hardness.
Besides, if two plane bodies of the same matter
and of equal thickness, do both yield an uniform
sound, the sound of that body, which hath the
greatest extent of length, will be the longest heard.
496
physics.
tart iv. For the motion, which in both of them hath its
29. . . i
*—r—beginning from the point of percussion, is to be
kfcthlfsoundProPagated in the greater body through a greater
lasting0"" and sPace> and consequently that propagation will re¬
quire more time ; and therefore also the parts
which are moved, will require more time for their
return. Wherefore all the reciprocations cannot
be finished but in longer time ; and being carried
to the ear, will make the sound last the longer.
And from hence it is manifest, that of hard bodies
which yield an uniform sound, the sound lasteth
longer which comes from those that are round and
hollow, than from those that are plane, if they be
like in all other respects. For in circular lines
the action, which begins at any point, hath not
from the figure any end of its propagation, because
the line in which it is propagated returns again to
its beginning ; so that the figure hinders not but
that the motion may have infinite progression.
Whereas in a plane, every line hath its magnitude
finite, beyond which the action cannot proceed.
If, therefore, the matter be the same, the motion
of the parts of that body whose figure is round
and hollow, will last longer than of that which is
plane.
Also, if a string which is stretched be fastened
at both ends to a hollow body, and be stricken, the
sound will last longer than if it were not so fas¬
tened ; because the trembling or reciprocation
which it receives from the stroke, is by reason of
the connection communicated to the hollow body;
and this trembling, if the hollow body be great, will
last the longer by reason of that greatness. Where-
sound, odour, etc.
497
fore also, for the reason above mentioned, the part iv.
29
sound will last the longer. •—^—-
9. In hearing it happens, otherwise than in How sound
seeing, that the action of the medium is made and hindered*
stronger by the wind when it blows the sameby the wmd*
way, and weaker when it blows the contrary wTay.
The cause whereof cannot proceed from anything
but the different generation of sound and light.
For in the generation of light, none of the parts
of the medium between the object and the eye are
moved from their own places to other places sen¬
sibly distant; but the action is propagated in
spaces imperceptible ; so that no contrary wind
can diminish, nor favourable wind encrease the
light, unless it be so strong as to remove the
object further off or bring it nearer to the eye.
For the wind, that is to say the air moved, doth
not by its interposition between the object and the
eye work otherwise than it would do, if it were
still and calm. For, where the pressure is perpetual,
one part of the air is no sooner carried away, but
another, by succeeding it, receives the same impres¬
sion,which the part carried away had received before.
But in the generation of sound, the first collision
or breaking asunder beateth away and driveth out
of its place the nearest part of the air, and that to
a considerable distance, and writh considerable
velocity ; and as the circles grow by their remote¬
ness wider and wider, so the air being more and
more dissipated, hath also its motion more and
more weakened. Whensoever therefore the air is
so stricken as to cause sound, if the wind fall upon
it, it will move it all nearer to the ear, if it blow
VOL. I. K K
498
PHYSICS.
part iv. that way, and further from it if it blow the con-
wtrary way ; so that according as it blows from or
towards the object, so the sound which is heard
will seem to come from a nearer or remoter place;
and the reaction, by reason of the unequal distances,
be strengthened or debilitated.
From hence may be understood the reason why
the voice of such as are said to speak in their bel¬
lies, though it be uttered near hand, is neverthe¬
less heard, by those that suspect nothing, as if it
were a great way off. For having no former thought
of any determined place from which the voice
should proceed, and judging according to the
greatness, if it be weak they think it a great way
off, if strong near. These ventriloqui, therefore,
by forming their voice, not as others by the emis¬
sion of their breath, but by drawing it inwards,
do make the same appear small and weak; which
weakness of the voice deceives those, that neither
suspect the artifice nor observe the endeavour
which they use in speaking; and so, instead of
thinking it weak, they think it far off.
Not only air, i(). As for the medium, which conveys sound,
but oilier bo- . . "1 1
dies, how hard it is not air only. Jbor water, or any other body
convey sound!' h°w bard soever, may be that medium. For the
motion may be propagated perpetually in any hard
continuous body; but by reason of the difficulty,
with which the parts of hard bodies are moved, the
motion in going out of hard matter makes but a
weak impression upon the air. Nevertheless, if one
end of a very long and hard beam be stricken, and
the ear be applied at the same time to the other end,
so that, when the action goeth out of the beam, the
SOUND, ODOUR, ETC.
499
air, which it striketh, may immediately be received PA^ IV*
by the ear, and be carried to the tympanum, the ■——»—'
sound will be considerably strong.
In like manner, if in the night, when all other
noise which may hinder sound ceaseth, a man lay
his ear to the ground, he will hear the sound of
the steps of passengers, though at a great distance;
because the motion, which by their treading they
communicate to the earth, is propagated to the ear
by the uppermost parts of the earth which receiveth
it from their feet.
11. I have shown above, that the difference be- The causes of
grave and acute
tween grave and acute sounds consisteth in this, sounds, and of
that by how much the shorter the time is, in which concent•
the reciprocations of the parts of a body stricken
are made, by so much the more acute will be the
sound. Now by how much a body of the same
bigness is either more heavy or less stretched, by
so much the longer will the reciprocations last;
and therefore heavier and less stretched bodies,
if they be like in all other respects, will yield a
graver sound than such as be lighter and more
stretched.
As for the concent of sounds, it is to be con¬
sidered that the reciprocation or vibration of the
air, by which sound is made, after it hath reached
the drum of the ear, imprinteth a like vibration
upon the air that is inclosed within it; by which
means the sides of the drum within are stricken
alternately. Now the concent of two sounds
consists in this, that the tympanum receives its
sounding stroke from both the sounding bodies in
equal and equally frequent spaces of time ; so that
when two strings make their vibrations in the same
k K 2
500
physics.
part iv. times, the concent they produce is the most exqui-
-—^—- site of all other. For the sides of the tympanum,
JraveaXSiuothat ™ to say of the organ of hearing, will be
sounds, and of stricken by both those vibrations together at once,
concent. J ^
on one side or other. For example, it the two
equal strings A B and C D be stricken together, and
the latitudes of their vibrations E F and G H be
also equal, and the points E, G, F and H be in the
concave superficies of the tympanum, so that it
receive strokes from both the strings together in E
and G, and again together in F and H, the sound,
which is made by the vibrations A B
of each string, will be so like, C D
that it may be taken for the G E
same sound, and is called uni¬
son ; which is the greatest con¬
cord. Again, the string AB
retaining still its former vibra¬
tion E F, let the string C D be
stretched till its vibration have
double the swiftness it had be¬
fore, and let E F be divided equally in I. In what
time therefore the string C D makes one part of
its vibration from G to H, in the same time the
string A B will make one part of its vibration from
E to I; and in what time the string CD hath made
the other part of its vibration back from H to G,
in the same time another part of the vibration of
the string AB will be made from I to F. But the
points F and G are both in the sides of the organ,
and therefore they will strike the organ both to¬
gether, not at every stroke, but at every other
stroke. And this is the nearest concord to unison,
and makes that sound which is called an eighth.
K
I
L
sound, odour, etc.
501
Again, the vibration of the string A B remaining tart iv.
still the same it was, let C D be stretched till its ^—'
vibration be swifter than that of the string A B in
the proportion of 3 to 2, and let E F be divided
into three equal parts in K and L. In what time
therefore the string C D makes one third part of
its vibration, which third part is from G to H, the
string A B will make one third part of its vibra¬
tion, that is to say, two-thirds of E F, namely, EL.
And in what time the string C D makes another
third part of its vibration, namely H G, the string
A B will make another third part of its vibration,
namely from L to F, and back again from F to L.
Lastly, whilst the string C D makes the last third
part of its vibration, that is from G to H, the
string A B will make the last third part of its
vibration from L to E. But the points E and H
are both in the sides of the organ. Wherefore, at
every third time, the organ will be stricken by the
vibration of both the strings together, and make
that concord which is called a fifth.
12. For the finding out the cause of smells, I Phenomena
shall make use of the evidence of these followingof stne mg*
phenomena. First, that smelling is hindered by
cold, and helped by heat. Secondly, that when
the wind bloweth from the object, the smell is the
stronger ; and, contrarily, when it bloweth from
the sentient towards the object, the weaker ; both
which phenomena are, by experience, manifestly
found to be true in dogs, which follow the track
of beasts by the scent. Thirdly, that such bodies,
as are less pervious to the fluid medium, yield less
smell than such as are more pervious; as may be
seen in stones and metals, which, compared with
502
PHYSICS.
PAI29IV' P^ants an<l living creatures, and their parts, fruits
s—r—- and excretions, have very little or no smell at all.
Fourthly, that such bodies, as are of their own
nature odorous, become yet more odorous when
they are bruised. Fifthly, that when the breath
is stopped, at least in men, nothing can be smelt.
Sixthly, that the sense of smelling is also taken
away by the stopping of the nostrils, though the
mouth be left open.
The first organ 13. By the fifth and sixth phenomenon it is
^T10- manifest, that the first and immediate organ of
smelling. smelling is the innermost cuticle of the nostrils, and
that part of it, which is below the passage common
to the nostrils and the palate. For when we draw
breath by the nostrils we draw it into the lungs.
That breath, therefore, which conveys smells is in
the way which passeth to the lungs, that is to say,
in that part of the nostrils which is belowT the pas¬
sage through which the breath goetli. For, nothing
is smelt, neither beyond the passage of the breath
within, nor at all without the nostrils.
And seeing that from different smells there must
necessarily proceed some mutation in the organ,
and all mutation is motion ; it is therefore also ne¬
cessary that, in smelling, the parts of the organ,
that is to say of that internal cuticle and the
nerves that are inserted into it, must be diversely
moved by different smells. And seeing also, that
it hath been demonstrated, that nothing can be
moved but by a body that is already moved and
contiguous ; and that there is no other body con¬
tiguous to the internal membrane of the nostrils
but breath, that is to say attracted air, and such
little solid invisible bodies, if there be any such, as
sound, odour, etc.
503
are intermingled with the air; it follows neces- partiv.
29
sarily, tliat the cause of smelling is either the v^—-
motion of that pure air or ethereal substance, or Jnhde thf ge?a"
the motion of those small bodies. But this motion nera'ion of
smelling.
is an effect proceeding from the object of smell,
and, therefore, either the whole object itself or its
several parts must necessarily be moved. Now,
we know that odorous bodies make odour, though
their whole bulk be not moved. Wherefore the
cause of odour is the motion of the invisible parts
of the odorous body. And these invisible parts do
either go out of the object, or else, retaining their
former situation with the rest of the parts, are
moved together with them, that is to say, they have
simple and invisible motion. They that say, there
goes something out of the odorous body, call it
an effluvium; which effluvium is either of the
ethereal substance, or of the small bodies that are
intermingled with it. But, that all variety of
odours should proceed from the effluvia of those
small bodies that are intermingled with the ethe¬
real substance, is altogether incredible, for these
considerations ; first, that certain unguents, though
very little in quantity, do nevertheless send forth
very strong odours, not only to a great distance of
place, but also for a great continuance of time, and
are to be smelt in every point both of that place
and time ; so that the parts issued out are sufficient
to fill ten thousand times more space, than the
whole odorous body is able to fill; which is impos¬
sible. Secondly, that whether that issuing out be
with strait or with crooked motion, if the same
quantity should flow from any other odorous body
with the same motion, it would follow that all
odorous bodies would yield the same smell. Thirdly,
504
PHYSICS.
pa^t IV- that seeing those effluvia have great velocity of
s—r—' motion (as is manifest from this, that noisome
odours proceeding from caverns are presently
smelt at a great distance) it would follow, that, by
reason there is nothing to hinder the passage of
those effluvia to the organ, such motion alone
were sufficient to cause smelling ; which is not so;
for we cannot smell at all, unless we draw in our
breath through our nostrils. Smelling, therefore,
is not caused by the effluvium of atoms; nor,
for the same reason, is it caused by the effluvium
of ethereal substance ; for so also we should smell
without the drawing in of our breath. Besides,
the ethereal substance being the same in all odo¬
rous bodies, they would always affect the organ in
the same manner; and, consequently, the odours of
all things would be alike.
It remains, therefore, that the cause of smelling
must consist in the simple motion of the parts of
odorous bodies without any efflux or diminution
of their whole substance. And by this motion
there is propagated to the organ, by the interme¬
diate air, the like motion, but not strong enough
to excite sense of itself without the attraction of
air by respiration. And this is a possible cause of
smelling.
Howsmeiiingis 14, The cause why smelling is hindered by cold
helped by heat jo j
and by wind, and helped by heat may be this ; that heat, as hath
been shown in chapter xxi, generateth simple
motion; and therefore also, wheresoever it is
already, there it will increase it; and the cause of
smelling being increased, the smell itself will also
be increased. As for the cause whv the wind
•/
blowing from the object makes the smell the
stronger, it is all one with that for which the at-
SOUND, ODOUR, ETC.
505
traction of air in respiration doth the same. For, part iv.
he that draws in the air next to him, draws with it «—
by succession that air in which is the object.
Now, this motion of the air is wind, and, when
another wind bloweth from the object, will be in¬
creased by it.
15. That bodies which contain the least quan- why suchbo-
tity of air, as stones and metals, yield less smell smelt,'which
than plants and living creatures ; the cause may l^ermixtire51
be, that the motion, which causeth smelling, is aof air in them*
motion of the fluid parts only; which parts, if
they have any motion from the hard parts in which
they are contained, they communicate the same to
the open air, by which it is propagated to the
organ. Where, therefore, there are no fluid parts
as in metals, or where the fluid parts receive 110
motion from the hard parts, as in stones, which
are made hard by accretion, there can be no smell.
And therefore also the water, whose parts have
little or no motion, yieldeth no smell. But, if the
same water, by seeds and the heat of the sun, be
together with particles of earth raised into a plant,
and be afterwards pressed out again, it will be
odorous, as wine from the vine. And as water
passing through plants is by the motion of the
parts of those plants made an odorous liquor; so
also of air, passing through the same plants whilst
they are growing, are made odorous airs. And
thus also it is with the juices and spirits, which are
bred in living creatures.
16. That odorous bodies may be made more why odorous
, , . . -ir- j.1 • j.1 < things become
odorous by contrition proceeds from this, that ni0re odorous,
being broken into many parts, which are all odor- when bruisecL
ous, the air, which by respiration is drawn from
the object towards the organ, doth in its passage
506
PHYSICS.
PA^o IV* touc^ nPon those parts, and receive their motion.
-—r—- Now, the airtoucheth the superficies only ; and a
body having less superficies whilst it is whole, than
all its parts together have after it is reduced to
powder, it follows that the same odorous body
yieldeth less smell whilst it is whole, than it will do
after it is broken into smaller parts. And thus
much of smells.
The first organ 17- The taste follows ; whose generation hath
why "some' sa-d this difference from that of the sight, hearing, and
vours cause smelling, that by these we have sense of remote
nauseousness. .
objects ; whereas, we taste nothing but what is
contiguous, and doth immediately touch either the
tongue or palate, or both. From whence it is evi¬
dent, that the cuticles of the tongue and palate,
and the nerves inserted into them are the first
organ of taste ; and (because from the concussion
of the parts of these, there followeth necessarily a
concussion of the pi a mater) that the action com¬
municated to these is propagated to the brain, and
from thence to the farthest organ, namely, the
heart, in whose reaction consisteth the nature of
sense.
Now, that savours, as well as odours, do not
only move the brain but the stomach also, as is
manifest by the loathings that are caused by them
both; they, that consider the organ of both these
senses, will not wonder at all; seeing the tongue,
the palate and the nostrils, have one and the same
continued cuticle, derived from the dura mater.
And that effluvia have nothing to do in the
sense of tasting, is manifest from this, that there
is no taste where the organ and the object are not
contiguous.
sound, odour, etc.
50 7
By what variety of motions tlie different kinds part iv.
J J 29.
of tastes, which are innumerable, may be distin- -—»—'
gnished, I know not. I might with others derive
them from the divers figures of those atoms, of
which whatsoever may be tasted consisteth; or
from the diverse motions which I might, by way of
supposition, attribute to those atoms; conjecturing,
not without some likelihood of truth, that such things
as taste sweet have their particles moved with slow
circular motion, and their figures spherical; which
makes them smooth and pleasing to the organ;
that bitter things have circular motion, but vehe¬
ment, and their figures full of angles, by which
they trouble the organ ; and that sour things have
strait and reciprocal motion, and their figures long
and small, so that they cut and wound the organ.
And in like manner I might assign for the causes
of other tastes such several motions and figures of
atoms, as might in probability seem to be the true
causes. But this would be to revolt from philoso¬
phy to divination.
18. By the touch, we feel what bodies are cold Tfhe first organ
or hot, though they be distant from us. Others, how we came
as hard, soft, rough, and smooth, wre cannot feel fedge^o^sudi
unless they be contiguous. The ors;an of touch is obJects as fre
Jo ~ common to the
every one of those membranes, which being con-touch and to
- other senses.
tmued from the pia mater are so diffused through¬
out the whole body, as that no part of it can be
pressed, but the pia mater is pressed together with
it. Whatsoever therefore presseth it, is felt as
hard or soft, that is to say, as more or less hard.
And as for the sense of rough, it is nothing else
but innumerable perceptions of hard and hard
succeeding one another by short intervals both of
508
PHYSICS.
part iv. time and place. For we take notice of rough and
-—A- smooth, as also of magnitude and figure, not only
by the touch, but also by memory. For though
some things are touched in one point, yet rough
and smooth, like quantity and figure, are not per¬
ceived but by the flux of a point, that is to say,
we have no sense of them without time ; and we
can have no sense of time without memory.
CHAPTER XXX.
OF GRAVITY.
1. A thick body doth not contain more matter, unless also more
place, than a thin.—2. That the descent of heavy bodies pro¬
ceeds not from their own appetite, but from some power of
the earth.—3. The difference of gravities proceedeth from the
difference of the impetus with which the elements, whereof
heavy bodies are made, do fall upon the earth.—4. The cause
of the descent of heavy bodies.—5. In what proportion the
descent of heavy bodies is accelerated.'—6. Why those that
dive do not, when they are under water, feel the weight of the
water above them.—7. The weight of a body that floateth, is
equal to the weight of so much water as would fill the space,
which the immersed part of the body takes up within the
water.—8. If a body be lighter than water, then how big
soever that body be, it may float upon any quantity of water,
how little soever.—9. How water may be lifted up and forced
out of a vessel by air.—10. Why a bladder is heavier when
blown full of air, than when it is empty.—11. The cause of
the ejection upwards of heavy bodies from a wind-gun.
12. The cause of the ascent of water in a weather-glass.
13. The cause of motion upwards in living creatures.—14. That
there is in nature a kind of body heavier than air, which never¬
theless is not by sense distinguishable from it.—15. Of the
cause of magnetical virtue.
a thick body \, JN chapter xxi I have defined thick and thin,
doth not con- r ... .
tain more mat-as that place required, so, as that by thick was
OF GRAVITY.
509
signified a more resisting body, and by thin, a body part iv.
less resisting; following the custom of those that 1^
have before me discoursed of refraction. Now ifter'unlessalso
more place,
we consider the true and vulgar signification ofthan a thin-
those words, we shall find them to be names col¬
lective, that is to say, names of multitude ; as thick
to be that, which takes up more parts of a space
given, and thin that, which contains fewer parts of
the same magnitude in the same space, or in a
space equal to it. Thick therefore is the same
with frequent, as a thick troop; and thin the same
with unfrequent, as a thin rank, thin of houses;
not that there is more matter in one place than in
another equal place, but a greater quantity of some
named body. For there is not less matter or body,
indefinitely taken, in a desert, than there is in a
city; but fewer houses, or fewer men. Nor is
there in a thick rank a greater quantity of body,
but a greater number of soldiers, than in a thin.
Wherefore the multitude and paucity of the parts
contained within the same space do constitute
density and rarity, whether those parts be sepa¬
rated by vacuum or by air. But the consideration
of this is not of any great moment in philosophy;
and therefore I let it alone, and pass on to the
search of the causes of gravity.
2. Now we call those bodies heavy, which,That th®de-
'' . scent of heavy
unless they be hindered by some force, are carried bodies proceeds
towards the centre of the earth, and that by their appetite?1
own accord, for aught we can by sense perceive to ts°™e
the contrary. Some philosophers therefore have eartll«
been of opinion, that the descent of heavy bodies
proceeded from some internal appetite, by which,
when they were cast upwards, they descended
510
PHYSICS.
tart iv. again, as moved by themselves, to such place as
<—y~—- was agreeable to their nature. Others thought
they were attracted by the earth. To the former
I cannot assent, because I think I have already
clearly enough demonstrated that there can be 110
beginning of motion, but from an external and
moved body; and consequently, that whatsoever
hath motion or endeavour towards any place, will
always move or endeavour towards that same place,
unless it be hindered by the reaction of some
external body. Heavy bodies, therefore, being
once cast upwards, cannot be cast down again
but by external motion. Besides, seeing inanimate
bodies have no appetite at all, it is ridiculous to
think that by their own innate appetite they should,
to preserve themselves, not understanding what
preserves them, forsake the place they are in, and
transfer themselves to another place; whereas
man, who hath both appetite and understanding,
cannot, for the preservation of his own life, raise
himself by leaping above three or four feet from
the ground. Lastly, to attribute to created bodies
the power to move themselves, what is it else than
to say that there be creatures which have no
dependance upon the Creator ? To the latter,, who
attribute the descent of heavy bodies to the attrac¬
tion of the earth, I assent. But by what motion
this is done, hath not as yet been explained by any
man. I shall therefore in this place say some¬
what of the manner and of the way by which the
earth by its action attractetli heavy bodies.
The difference
3. That by the supposition of simple motion in
proceedethS ^ie sun> homogeneous bodies are congregated and
from the differ- heterogeneous dissipated, lias already been demon-
OF GRAVITY.
511
strated in the 5th article of chapter xxi. I have PA^ IVi
also supposed, that there are intermingled with the -—.—'
pure air certain little bodies, or, as others call them, ^cpeet°sf ^uh
atoms; which by reason of their extreme small- which th,e ele;
J _ _ ments, whereot
ness are invisible, and differing from one another heavy bodies
r . , are made, do
m consistence, figure, motion, and magnitude ; faii up0n the
from whence it comes to pass that some of them earth"
are congregated to the earth, others to other
planets, and others are carried up and down in the
spaces between. And seeing those, which are car¬
ried to the earth, differ from one another in figure,
motion, and magnitude, they will fall upon the
earth, some with greater, others with less impetus.
And seeing also that we compute the several
degrees of gravity 110 otherwise than by this their
falling upon the earth with greater or less impetus;
it follows, that we conclude those to be the more
heavy that have the greater impetus, and those to be
less heavy that have the less impetus. Our inquiry
therefore must be, by what means it may come to
pass, that of bodies, which descend from above to
the earth, some are carried with greater, others
with less impetus; that is to say, some are more
heavy than others. We must also inquire, by what
means such bodies, as settle upon the earth, may
by the earth itself be forced to ascend.
4. Let the circle made upon the centre C (in The cause of
fig. 2) be a great circle in the superficies of the heavy bodies,
earth, passing through the points A and B. Also
let any heavy body, as the stone A D, be placed
anywhere in the plane of the equator; and let it
be conceived to be cast up from A D perpendicu¬
larly, or to be carried in any other line to E, and
supposed to rest there. Therefore, how much
512
physics.
PA?JTV' sPace soever the stone took up in A D, so much
-—- space it takes up now in E. And because all place
SeVescent^f supposed to be full, the space A D will be filled
heavy bodies, tiie a^r wj1ic}1 flows into it first from the nearest
places of the earth, and afterwards successively
from more remote places. Upon the centre C let
a circle be understood to be drawn through E; and
let the plane space, which is between the superficies
of the earth and that circle, be divided into plane
orbs equal and concentric ; of which let that be the
first, which is contained by the two perimeters that
pass through A and D. Whilst therefore the air,
which is in the first orb, filleth the place A D, the
orb itself is made so much less, and consequently
its latitude is less than the strait line A D. Where¬
fore there will necessarily descend so much air
from the orb next above. In like manner, for the
same cause, there will also be a descent of air from
the orb next above that; and so by succession
from the orb in which the stone is at rest in E.
Either therefore the stone itself, or so much air,
will descend. And seeing air is by the diurnal
revolution of the earth more easily thrust away
than the stone, the air, which is in the orb that
contains the stone, will be forced further upwards
than the stone. But this, without the admission
of vacuum, cannot be, unless so much air descend
to E from the place next above ; which being done,
the stone will be thrust downwards. By this
means therefore the stone now receives the begin¬
ning of its descent, that is to say, of its gravity.
Furthermore, whatsoever is once moved, will be
moved continually (as hath been shown in the
19th article of chapter viii) in the same way, and
OF GRAVITY.
513
with the same celerity, except it be retarded or Iv-
accelerated by some external movent. Now the v—>—'
air, which is the only body that is interposed be- th/descentof
tween the earth A and the stone above it E, will heavy bodies'
have the same action in every point of the strait
line E A, which it hath in E. But it depressed the
stone in E; and therefore also it will depress it
equally in every point of the strait line E A. Where¬
fore the stone will descend from E to A with acce¬
lerated motion. The possible cause therefore of
the descent of heavy bodies under the equator, is
the diurnal motion of the earth. And the same
demonstration will serve, if the stone be placed in
the plane of any other circle parallel to the equator.
But because this motion hath, by reason of its
greater slowness, less force to thrust off the air in
the parallel circles than in the equator, and no
force at all at the poles, it may well be thought
(for it is a certain consequent) that heavy bodies
descend with less and less velocity, as they are
more and more remote from the equator; and that
at the poles themselves, they will either not descend
at all, or not descend by the axis ; which whether
it be true or false, experience must determine. But
it is hard to make the experiment, both because
the times of their descents cannot be easily mea¬
sured with sufficient exactness, and also because
the places near the poles are inaccessible. Never¬
theless, this we know7, that by how much the
nearer we come to the poles, by so much the
greater are the flakes of the snow that falls; and
by how much the more swiftly such bodies descend
as are fluid and dissipable, by so much the smaller
are the particles into which they are dissipated.
VOL. I. L L
514
physics.
part iv. 5. Supposing, therefore, this to be the cause of
v—r—- the descent of heavy bodies, it will follow that
Jortiondic1'^-their motion will be accelerated in such manner,
scent of heavy as tliat the spaces, which are transmitted by them
bodies is accele-
rated. in the several times, will have to one another the
same proportion which the odd numbers have in
succession from unity. For if the strait line EA
be divided into any number of equal parts, the
heavy body descending will, by reason of the per¬
petual action of the diurnal motion, receive from
the air in every one of those times, in every several
point of the strait line E A, a several new and
equal impulsion ; and therefore also in every one
of those times, it will acquire a several and equal
degree of celerity. And from hence it follows, by
that which Galileus hath in his Dialogues of Mo¬
tion demonstrated, that heavy bodies descend in
the several times with such differences of trans¬
mitted spaces, as are equal to the differences of
the square numbers that succeed one another from
unity; which square numbers being 1,4,9,16,
&c. their differences are 3, 5, 7, &c.; that is to say,
the odd numbers which succeed one another from
unity. Against this cause of gravity which I have
given, it will perhaps be objected, that if a heavy
body be placed in the bottom of some hollow
cylinder of iron or adamant, and the bottom be
turned upwards, the body will descend, though the
air above cannot depress it, much less accelerate
its motion. But it is to be considered that there
can be no cylinder or cavern, but such as is sup¬
ported by the earth, and being so supported is,
together with the earth, carried about by its
diurnal motion. For by this means the bottom of
of gravity.
515
the cylinder will be as the superficies of the earth; pakt iv-
and by thrusting off the next and lowest air, will ^'
make the uppermost air depress the heavy body,
which is at the top of the cylinder, in such manner
as is above explicated.
6. The gravity of water being so great as bv whythose that
~ , . . , tit diye'do not'
experience we find it is, the reason is demanded by when they are
many, why those that dive, how deep soever they feeith<Twe7ght
go under water, do not at all feel the weight of
the water which lies upon them. And the cause
seems to be this, that all bodies by how much the
heavier they are, by so much the greater is the
endeavour by which they tend downwards. But
the body of a man is heavier than so much water
as is equal to it in magnitude, and therefore the
endeavour downwards of a man's body is greater
than that of water. And seeing all endeavour is
motion, the body also of a man will be carried
towards the bottom with greater velocity than so
much water. Wherefore there is greater reaction
from the bottom; and the endeavour upwards is
equal to the endeavour downwards, whether the
water be pressed by water, or by another body
which is heavier than water. And therefore by
these two opposite equal endeavours, the endeavour
both ways in the water is taken away; and con¬
sequently, those that dive are not at all pressed
by it.
Coroll. From hence also it is manifest, that
water in water hath no weight at all, because all
the parts of water, both the parts above, and the
parts that are directly under, tend towards the
bottom with equal endeavour and in the same
strait lines.
l l 2
516
PHYSICS.
PAI3o lV' ^^ a body uPon the water> weight of
— that body is equal to the weight of so much water
bohdyWthatflo°It- as would fill the place which the immersed part of
eth, is equal to the body takes up within the water.
the weight of so .
much water as Let EF (in fig. 3) be a body floating in the water
spa el'' w hk: h't h c A B C D ; and let the part E be above, and the
onhe'bodypart °^lpr Part ^ under the water. I say, the weight
takes up within 0f the whole body E F is equal to the weight of so
the water. J . .
much water as the space F will receive. For
seeing the weight of the body E F forceth the
water out of the space F, and placeth it upon the
superficies A B, where it presseth downwards; it
follows, that from the resistance of the bottom
there will also be an endeavour upwards. And
seeing again, that by this endeavour of the water
upwards, the body E F is lifted up, it follows, that
if the endeavour of the body downwards be not
equal to the endeavour of the water upwards,
either the whole body E F will, by reason of that
inequality of their endeavours or moments, be
raised out of the water, or else it will descend to
the bottom. But it is supposed to stand so, as
neither to ascend nor descend. Wherefore there
is an equilibrium between the tw o endeavours ;
that is to say, the weight of the body E F is equal
to the weight of so much water as the space F will
receive ; which was to be proved,
if a body be g_ From hence it follows, that any body, of how
lighter than . ...
water, then how great magnitude soever, provided it consist of
bodyTeVu may matter less heavy than water, may nevertheless
quantity of"7 fl°at upon any quantity of water, how little soever,
water, how Let A B C D (in fig. 4) be a vessel; and in it let
.little soever. x ° y
EFGH be a body consisting of matter which is
less heavy than water; and let the space AGCF
OF GRAVITY.
517
be filled with water. I say, the body EFGH will PARsT iv-
not sink to the bottom D C. For seeing the matter -—<—-
of the body EFGH is less heavy than water, if the
whole space without A B C D were full of water,
yet some part of the body EFGH, as E FI K,
would be above the water; and the weight of so
much water as would fill the space I G H K would
be equal to the weight of the whole body EFGH;
and consequently G H would not touch the bottom
D C. As for the sides of the vessel, it is no
matter whether they be hard or fluid; for they
serve only to terminate the water; which may be
done as well by water as by any other matter how
hard soever ; and the water without the vessel is
terminated somewhere, so as that it can spread no
farther. The part therefore E FIK will be extant
above the water A G C F which is contained in the
vessel. Wherefore the body EFGH will also
float upon the water A G C F, how little soever
that water be ; which was to be demonstrated.
9. In the 4th article of chapter xxvi, there is Howwatermay
brought for the proving of vacuum the experiment forced out of a
of water enclosed in a vessel; which water, the vessel by air>
orifice above being opened, is ejected upwards by
the impulsion of the air. It is therefore demanded,
seeing water is heavier than air, how that can be
done. Let the second figure of the same, chapter
xxvi be considered, where the water is with great
force injected by a syringe into the space F G B.
In that injection, the air (but pure air) goeth
with the same force out of the vessel through the
injected water. But as for those small bodies,
which formerly I supposed to be intermingled with
air and to be moved with simple motion, they
518
physics.
part iv. cannot, together with the pure air, penetrate the
-■—r— water ; but remaining behind are necessarily thrust
beTftedup^nd together into a narrower place, namely into the
forced out of a space which is above the water F G. The mo-
vesse byair' tions therefore of those small bodies will be less
and less free, by how much the quantity of the
injected water is greater and greater ; so that by
their motions falling upon one another, the same
small bodies will mutually compress each other,
and have a perpetual endeavour of regaining their
liberty, and of depressing the water that hinders
them. Wherefore, as soon as the orifice above is
opened, the water which is next it will have an
endeavour to ascend, and will therefore necessarily
go out. But it cannot go out, unless at the same
time there enter in as much air; and therefore
both the water will go out, and the air enter in,
till those small bodies which were left within the
vessel have recovered their former liberty of mo¬
tion ; that is to say, till the vessel be again filled
with air, and no water be left of sufficient height
to stop the passage at B. Wherefore I have shown
a possible cause of this phenomenon, namely, the
same with that of thunder. For as in the gene¬
ration of thunder, the small bodies enclosed within
the clouds, by being too closely pent together, do
by their motion break the clouds, and restore them¬
selves to their natural liberty; so here also the
small bodies enclosed within the space which is
above the strait line F G, do by their own motion
expel the water as soon as the passage is opened
above. And if the passage be kept stopped, and
these small bodies be more vehemently compressed
OF GRAVITY.
519
by the perpetual forcing in of more water, they PA^ Iv-
will at last break the vessel itself with great noise. —'
10. If air be blown into a hollow cylinder, or%aU*r
into a bladder, it will increase the weight of either blown fail of
of them a little, as many have found by experience, uVemptyf611
who with great accurateness have tried the same.
And it is no wonder, seeing, as I have supposed,
there are intermingled with the common air a great
number of small hard bodies, which are heavier
than the pure air. For, the ethereal substance,
being on all sides equally agitated by the motion of
the sun, hath an equal endeavour towards all the
parts of the universe; and, therefore, it hath no
gravity at all.
11. We find also by experience, that, by the The cause of
force of air enclosed in a hollow cannon, a bullet wardsofHy
of lead may with considerable violence be shot out ^nd-gun!™ a
of a gun of late invention, called the wind-gun. In
the end of this cannon there are two holes, with
their valves on the inside, to shut them close ; one
of them serving for the admission of air, and the
other for the letting of it out. Also, to that end
which serves 'for the receiving in of air, there is
joined another cannon of the same metal and big¬
ness, in which there is fitted a rammer which is
perforated, and hath also a valve opening towards
the former cannon. By the help of this valve the
rammer is easily drawn back, and letteth in air
from without; and being often drawn back and
returned again with violent strokes, it forceth some
part of that air into the former cannon, so long,
till at last the resistance of the enclosed air is
greater than the force of the stroke. And by this
520
PHYSICS.
VAJ3o IV' means men think there is now a greater quantity of
-—^—- air in the cannon than there was formerly, though
SejecTioUc.it; were full before. Also, the air thus forced in,
how much soever it be, is hindered from getting
out again by the aforesaid valves, which the very
endeavour of the air to get out doth necessarily
shut. Lastly, that valve being opened which was
made for the letting out of the air, it presently
breaketh out with violence, and driveth the bullet
before it with great force and velocity.
As for the cause of this, I could easily attribute
it, as most men do, to condensation, and think
that the air, which had at the first but its ordinary
degree of rarity, was afterwards, by the forcing in
of more air, condensed, and last of all, rarified
again by being let out and restored to its natural
liberty. But I cannot imagine how the same place
can be always full, and, nevertheless, contain some
times a greater, sometimes a less quantity of matter;
that is to say, that it can be fuller than full. Nor
can I conceive how fulness can of itself be an effi¬
cient cause of motion. For both these are impos¬
sible. Wherefore we must seek out some other
possible cause of this phenomenon. Whilst, there¬
fore, the valve which serves for the letting in of
air, is opened by the first stroke of the rammer,
the air within doth with equal force resist the enter¬
ing of the air from without; so that the endeavours
between the internal and external air are opposite,
that is, there are two opposite motions whilst the
one goeth in and the other cometh out; but no
augmentation at all of air within the cannon. For
there is driven out by the stroke as much pure air,
which passeth between the rammer and the sides
OF GRAVITY.
521
of the cannon, as there is forced in of air impure by PA^0T IY-
the same stroke. And thus, by many forcible -—^—-
strokes, the quantity of small hard bodies will be
increased within the cannon, and their motions also
will grow stronger and stronger, as long as the
matter of the cannon is able to endure their force ;
by which, if it be not broken, it will at least be
urged every way by their endeavour to free them¬
selves ; and as soon as the valve, which serves to
let them out, is opened, they will fly out with
violent motion, and carry with them the bullet
which is in their way. Wherefore, I have given
a possible cause of this phenomenon.
12. Water, contrary to the custom of heavy The cause
bodies, ascendeth in the weather-glass ; but it doth of water in a
it when the air is cold : for when it is warm it des- weather-slass-
cendeth again. And this organ is called a ther¬
mometer or thermoscope, because the degrees of
heat and cold are measured and marked by it. It is
made in this manner. Let AB CD (in fig. 5) be
a vessel full of water, and E F G a hollow cylinder
of glass, closed at E and open at G. Let it be
heated, and set upright within the water to F ; and
let the open end reach to G. This being done, as the
air by little and little grows colder, the water will
ascend slowly within the cylinder from F towards
E; till at last the external and internal air coming
to be both of the same temper, it will neither as¬
cend higher nor descend lower, till the temper of
the air be changed. Suppose it, therefore, to be
settled anywhere, as at H. If now the heat of the
air be augmented, the water will descend below H ;
and if the heat be diminished, it will ascend above
522
physics.
VAR3o IV' Which, though it be certainly known to be
v—true by experience, the cause, nevertheless, hath
not as yet been discovered.
In the sixth and seventh articles of chapter
xxviii, where I consider the cause of cold, I have
shown, that fluid bodies are made colder by the
pressure of the air, that is to say, by a constant
wind that presseth them. For the same cause it
is, that the superficies of the water is pressed at F;
and having no place, to which it may retire from
this pressure, besides the cavity of the cylinder
between H and E, it is therefore necessarily forced
thither by the cold, and consequently it ascendeth
more or less, according as the cold is more or
less increased. And again, as the heat is more in¬
tense or the cold more remiss, the same water
will be depressed more or less by its own gravity,
that is to say, by the cause of gravity above expli¬
cated.
Cause of mo- 13. Also living creatures, though they be heavv,
tion upwards in . . . in * • i
living creatures can by leaping, swimming and flying, raise them¬
selves to a certain degree of height. But they
cannot do this except they be supported by some
resisting body, as the earth, the water and the air.
For these motions have their beginning from the
contraction, by the help of the muscles, of the body
animate. For to this contraction there succeedeth
a distension of their whole bodies; by which dis¬
tension, the earth, the water, or the air, which sup-
porteth them, is pressed; and from hence, by the
reaction of those pressed bodies, living creatures
acquire an endeavour upwards, but such as by
reason of the gravity of their bodies is presently
of gravity.
523
lost again. By this endeavour, therefore, it is, that pART iv.
living creatures raise themselves up a little way by t—-
leaping, but to no great purpose : but by swimming
and flying they raise themselves to a greater height;
because, before the effect of their endeavour is quite
extinguished by the gravity of their bodies, they
can renew the same endeavour again.
That by the power of the soul, without any ante¬
cedent contraction of the muscles or the help of
something to support him, any man can be able to
raise his body upwards, is a childish conceit. For
if it were true, a man might raise himself to what
height he pleased.
14. The diaphanous medium, which surrounds Thatthefe in„
. ... nature akind of
the eye on all sides, is invisible; nor is air to be body heavier
. , . n . t than air, which
seen m air, nor water m water, nor anything but nevertheless is
that which is more opacous. But in the confines dJtinguiSiabie
of two diaphanous bodies, one of them may be dis-from ir-
tinguished from the other. It is not therefore a
thing so very ridiculous for ordinary people to
think all that space empty, in which we say is air;
it being the work of reason to make us conceive
that the air is anything. For by which of our
senses is it, that we take notice of the air, seeing
we neither see, nor hear, nor taste, nor smell, nor
feel it to be anything r When we feel heat, Ave do
not impute it to the air, but to the fire : nor do we
say the air is cold, but we ourselves are cold ; and
when we feel the wind, we rather think something
is coming, than that any thing is already come.
Also, we do not at all feel the weight of water in
water, much less of air in air. That we come to
know that to be a body, which we call air, it is by
524
PHYSICS.
PA 30 IV' reasoninS ' it is from one reason only, namely,
—-i— because it is impossible for remote bodies to work
nature,1C&c.S m uPon our organs of sense but by the help of
bodies intermediate, without which we could have
no sense of them, till they come to be contiguous.
Wherefore, from the senses alone, without reason¬
ing from effects, we cannot have sufficient evidence
of the nature of bodies.
For there is underground, in some mines of coals,
a certain matter of a middle nature between water
and air, which nevertheless cannot by sense be
distinguished from air ; for it is as diaphanous as
the purest air ; and, as far as sense can judge,
equally penetrable. But if we look upon the effect,
it is like that of water. For when that matter
breaks out of the earth into one of those pits, it
fills the same either totally or to some degree ; and
if a man or fire be then let down in it, it extin¬
guishes them in almost as little time as water would
do. But for the better understanding of this phe¬
nomenon, I shall describe the 6th figure. In which
let A B represent the pit of the mine; and let part
thereof, namely C B, be supposed to be filled with
that matter. If now a lighted candle be let down
into it below C, it will as suddenly be extinguished
as if it were thrust into water. Also, if a grate filled
with coals thoroughly kindled and burning never
so brightly, be let down, as soon as ever it is below
C, the fire will begin to grow pale, and shortly
after, losing its light, be extinguished, no otherwise
than if it were quenched in water. But if the grate
be drawn up again presently, whilst the coals are still
very hot, the fire will, by little and little, be kindled
OF GRAVITY.
525
again, and shine as before. There is, indeed, be- part iv.
tween this matter and water this considerable dif- -—-—-
ference, that it neither wetteth, nor sticketh to such To!* m
things as are put down into it, as water doth;
which, by the moisture it leaveth, hindereth the
kindling again of the matter once extinguished.
In like manner, if a man be let down below C, he
will presently fall into a great difficulty of breath¬
ing, and immediately after into a swoon, and die
unless he be suddenly drawn up again. They,
therefore, that go down into these pits, have this
custom, that as soon as ever they feel themselves
sick, they shake the rope by which they were let
down, to signify they are not well, and to the end
that they may speedily be pulled up again. For
if a man be drawn out too late, void of sense and
motion, they dig up a turf, and put his face and
mouth into the fresh earth ; by which means,
unless he be quite dead, he comes to himself again,
by little and little, and recovers life by breathing
out, as it were, of that suffocating matter, which he
had sucked in whilst he was in the pit; almost in
the same manner as they that are drowned come
to themselves again by vomiting up the water. But
this doth not happen in all mines, but in some only ;
and in those not always, but often. In such pits
as are subject to it, they use this remedy. They
dig another pit, as D E, close by it, of equal depth,
and joining them both together with one common
channel, E B, they make a fire in the bottom E,
which carries out at D the air contained in the pit
D E ; and this draws with it the air contained in
the channel E B; which, in like manner, is fol-
526
physics.
part iv. lowed by the noxious matter contained in C B; and,
30
—A- by this means, the pit is for that time made health¬
ful. Out of this history, which I write only to
such as have had experience of the truth of it,
without any design to support my philosophy with
stories of doubtful credit, may be collected the fol¬
lowing possible cause of this phenomenon ; namely,
that there is a certain matter fluid and most trans¬
parent, and not much lighter than water, which,
breaking out of the earth, fills the pit to C; and
that in this matter, as in water, both fire and
living creatures are extinguished.
°f 15. About the nature of heavy bodies, the greatest
virtue. difficulty ariseth from the contemplation of those
things which make other heavy bodies ascend to
them; such as jet, amber, and the loadstone. But
that which troubles men most is the loadstone,
which is also called Lapis Herculeus ; a stone,
though otherwise despicable, yet of so great power
that it taketh up iron from the earth, and holds it
suspended in the air, as Hercules did Antseus.
Nevertheless, we wonder at it somewhat the less,
because we see jet draw up straws, which are heavy
bodies, though not so heavy as iron. But as for
jet, it must first be excited by rubbing, that is to
say, by motion to and fro ; whereas the loadstone
hath sufficient excitation from its own nature, that
is to say, from some internal principle of motion
peculiar to itself. Now, whatsoever is moved, is
moved by some contiguous and moved body, as
hath been formerly demonstrated. And from hence
it follows evidently, that the first endeavour, which
iron hath towards the loadstone, is caused by the
OF GRAVITY.
52 7
motion of that air which is contiguous to the iron: part iv.
also, that this motion is generated by the motion of
the next air, and so 011 successively, till by this
succession we find that the motion of all the inter- virtue-
mediate air taketh its beginning from some motion
which is in the loadstone itself; which motion,
because the loadstone seems to be at rest, is in¬
visible. It is therefore certain, that the attractive
power of the loadstone is nothing else but some
motion of the smallest particles thereof. Sup¬
posing, therefore, that those small bodies, of which
the loadstone is in the bowels of the earth com¬
posed, have by nature such motion or endeavour
as was above attributed to jet, namely, a reciprocal
motion in a line too short to be seen, both those
stones will have one and the same cause of attrac¬
tion. Now in what manner and in what order of
working this cause produceth the effect of attrac¬
tion, is the thing to be enquired. And first we
know, that when the string of a lute or viol is
stricken, the vibration, that is, the reciprocal mo¬
tion of that string in the same strait line, causeth
like vibration in another string which hath like
tension. We know also, that the dregs or small
sands, which sink to the bottom of a vessel, will
be raised up from the bottom by any strong and
reciprocal agitation of the water, stirred with the
hand or with a staff. Why, therefore, should not
reciprocal motion of the parts of the loadstone con¬
tribute as much towards the moving of iron r For,
if in the loadstone there be supposed such reciprocal
motion, or motion of the parts forwards and back¬
wards, it will follow that the like motion will be
528
physics.
part iv. propagated by the air to the iron, and consequently
that there will be in all the parts of the iron the
of the cause same reciprocations or motions forwards and back-
oi magnetical -l
virtue. wards. And from hence also it will follow, that
the intermediate air between the stone and the
iron will, by little and little, be thrust away; and
the air being thrust away, the bodies of the load¬
stone arid the iron will necessarily come together.
The possible cause therefore why the loadstone
and jet draw to them, the one iron, the other
straws, may be this, that those attracting bodies
have reciprocal motion either in a strait line, or in
an elliptical line, when there is nothing in the na¬
ture of the attracted bodies which is repugnant to
such a motion.
But why the loadstone, if with the help of cork
it float at liberty upon the top of the water, should
from any position whatsoever so place itself in the
plane of the meridian, as that the same points,
which at one time of its being at rest respect the
poles of the earth, should at all other times respect
the same poles, the cause may be this; that the reci¬
procal motion, which I supposed to be in the parts
of the stone, is made in a line parallel to the axis
of the earth, and has been in those parts ever since
the stone was generated. Seeing therefore, the
stone, whilst it remains in the mine, and is carried
about together with the earth by its diurnal mo¬
tion, doth by length of time get a habit of being
moved in a line which is perpendicular to the line
of its reciprocal motion, it will afterwards, though
its axis be removed from the parallel situation it
had with the axis of the earth, retain its endeavour
OP GRAVITY.
529
of returning to that situation again; and all en- R'
deavour being the beginning of motion, and nothing ^
, . , ■, ..iii Of the cause
intervening that may hinder the same, the load- of magnetical
stone will therefore return to its former situation.virtue"
For, any piece of iron that has for a long time rested
in the plane of the meridian, whensoever it is forced
from that situation and afterwards left to its own
liberty again, will of itself return to lie in the
meridian again; which return is caused by the
endeavour it acquired from the diurnal motion of
the earth in the parallel circles which are perpen¬
dicular to the meridians.
If iron be rubbed by the loadstone drawn from
one pole to the other, two things will happen; one,
that the iron will acquire the same direction with
the loadstone, that is to say, that it will lie in the
meridian, and have its axis and poles in the same
position with those of the stone; the other, that
the like poles of the stone and of the iron will
avoid one another, and the unlike poles approach
one another. And the cause of the former may be
this, that iron being touched by motion which is
not reciprocal, bat drawn the same way from pole
to pole, there will be imprinted in the iron also an
endeavour from the same pole to the same pole.
For seeing the loadstone differs from iron no other¬
wise than as ore from metal, there will be no
repugnance at all in the iron to receive the same
motion which is in the stone. From whence it
follows, that seeing they are both affected alike by
the diurnal motion of the earth, they will both
equally return to their situation in the meridian,
whensoever they are put from the same. Also, of
VOL. I. M M
530
PHYSICS.
PAR30 IV ^ie ^atter ^is maY be the cause, that as the load-
—.— stone in touching the iron doth by its action im-
onnagneticS print in the iron an endeavour towards one of the
virtue. poles, suppose towards the North Pole; so reci¬
procally, the iron by its action upon the loadstone
doth imprint in it an endeavour towards the other
pole, namely towards the South Pole. It happens
therefore in these reciprocations or motions for¬
wards and backwards of the particles of the stone
and of the iron betwixt the north and the south,
that whilst in one of them the motion is from north
to south, and the return from south to north, in
the other the motion will be from south to north,
and the return from north to south ; which motions
being opposite to one another, and communicated
to the air, the north pole of the iron, whilst the
attraction is working, will be depressed towards
the south pole of the loadstone; or contrarily, the
north pole of the loadstone will be depressed
towards the south pole of the iron ; and the axis
both of the loadstone and of the iron will be situate
in the same strait line. The truth whereof is
taught us by experience.
As for the propagation of this magnetical virtue,
not only through the air, but through any other
bodies how hard soever, it is not to be wondered
at, seeing no motion can be so weak, but that it
may be propagated infinitely through a space filled
with body of any hardness whatsoever. For in a
full medium, there can be no motion which doth
not make the next part yield, and that the next,
and so successively without end; so that there is
no effect whatsoever, but to the production thereof
OF GRAVITY.
531
something is necessarily contributed by the several part iv.
motions of all the several things that are in the -—^
world.
And thus much concerning the nature of body Conclusion,
in general; with which I conclude this my first
section of the Elements of Philosophy. In the
first, second, and third parts, where the principles
of ratiocination consist in our own understanding,
that is to say, in the legitimate use of such words
as we ourselves constitute, all the theorems, if I be
not deceived, are rightly demonstrated. The fourth
part depends upon hypotheses; which unless we
know them to be true, it is impossible for us to
demonstrate that those causes, which I have there
explicated, are the true causes of the things whose
productions I have derived from them.
Nevertheless, seeing I have assumed no hypo¬
thesis, which is not both possible and easy to be
comprehended; and seeing also that I have rea¬
soned aright from those assumptions, I have withal
sufficiently demonstrated that they may be the true
causes ; which is the end of physical contempla^-
tion. If any other man from other hypotheses
shall demonstrate the same or greater things,
there will be greater praise and thanks due to him
than I demand for myself, provided his hypotheses
be such as are conceivable. For as for those that
say anything may be moved or produced by itself \
by species, by its own power, by substantialforms,
by incorporeal substances, by instinct, by anti-
peristasis, by antipathy, sympathy, occult quality,
and other empty words of schoolmen, their saying
so is to no purpose.
532
PHYSICS.
?A so 1V ^11(^ llow ^ Proceed t° the phenomena of man's
-—v— body ; where I shall speak of the optics, and of
Conclusion, dispositions, affections, and manners of men,
if it shall please God to give me life, and show
their causes.
END OF VOL. I.
london:
c. UK HaRBSjST. martin's lank, charing cross.
YALE UNIVERSITY
A39M2_00ST7?iTTb
i * -»V * «J'
YALE
m
■ ?-|.