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A^l '-'

ENGLISH WORKS

THOMAS HOBBES

OF MALMESBURY ;

NOW FIRST COLLECTED AND EDITED

SIR WILLUM MOLESWORTH, BART.

VOL. I.

LONDON:

JOHN BOHN.
HENRIETTA STREET, COVENT GARDEN.

MOCCCXXXIX.

Lonxnr:
c uoHimM, PUMnm, n. UAxra*t lam.

GEORGE GROTE, ESQ.

MP. FOR THE CITY OF LONDON.

Dear Grote,

I dedicate to you this edition of
the Works of Hobbes j 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 hav^e 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 poUtical life.

Yours, truly,

William Moles worth.

February 25th, 1839*
79, Eaton Square, London,

ELEMENTS OF PHILOSOPHY.

THE FIRST SECTION,

CONCERNING BODY,

WRITTEN IN LATIN
Br

THOMAS HOBBES OF MALMESBURY,

AND

TRANSLATED INTO ENGLISH.

THE

TRANSLATOR TO THE READER.

If, when I had finished my 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,
XVIII and xx, almost wholly changed, you may nevertheless
remain assured, that as now I present it to you, it doth not at
all vary firom 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 be published in a separate Tolume, with other works of a
similar description. W. M.

THE AUTHOR'S EPISTLE DEDICATORY,

TO Tfl£

RIGHT HONORABLE, >nr MOST HONORED LORD,

WILLIAM, EARL OF DEVONSHIIIE,

This first section of the Elements of PMIosophy^ the
monument of my service and yom* Lordship's boxmty^
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 fiill ; 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 oflFensive 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. 1 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, w^ere 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, Arls-
tarehus, and Philolaus. After him, the doctrine of
the motion of the earth being now receivedj and a dif-
ficult question thereupon arising concerning the de-
scent of heavy bodies, Galileus in oui* time, striving
with that difficulty, was the first that opened to us the
gate of natural philosophy universal, which is the
knowledge of the nature of mot htL So that neither can
the age of natural philosophy be reckoned higher than
to him. Lastly, the science of mans boily^ the most
profitable part of uatiu"al 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 Bloody
and of the Generation of Living Creatures ; w ho is
the only man I know, that conquering enx^r, hath
established a new doctrine in his life-time. Before
these, there was nothing certain in natural philosophy

BPISTLE 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 phUosophy in general have,
for so little time, been extraordinarily advanced by
Joannes Keplerus, Petrus Gassendus, and M arinus
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 Cite. 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 fiill 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

EPISTLE DEDICATORY*

them to be tanght, 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, bora
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 divinitij^ walking on one foot
firmly, which is the Holy Scripture, but halted on
the other rotten foot, which the Apostle Paul called
t?am, and might have called pernicioiiH philosophy ;
for it hath raised an infinite number of controversies
in the Cliristian 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 fortime. Against this Empum I think

EPISTLE DEDICATORY.

there cannot be invented a better exorcism, than to
distinguish between the rules of religion, that is, the
mles of honouring God, which we have from the
laws, and the rules of phDosophy, 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 Emj}nsa ; not by skirmish, but by letting in the
light upon her. For 1 am confident, if any con-
fidence of a writing can proceed from the writer's
circumspection, and diffidence, that in the
three former parts of this book all that I have said
is sufficiently demonstrated from definitions ; and all
ill the fourth part from suppositious not absurd.
But if there appear to your Lordship anything less
fiiUy demonstrated tliaii to satisfy every reader, the
cause was this, that I professed to write not all to

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
wTitten 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 on 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 acknow^ledge ; and for which^ with my
prayers to Almighty God for your Lordship's safety,
I shall^ to my pow er, be always thankful.

Your Lordship^s most humble servant,

THOMAS HOBBES.

April 2», 165.'!,

THE

AUTHOR'S EPISTLE TO THE READER.

Think not, Courteous Reader, 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 nighty Xheji/rmament^ the lumi-
naries^ sensible creatures^ man ; and, after the creation, the
commandment. Therefore the order of contemplation will
be, reason, de/inition, space, the stars, sens^ible 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 HEADER.

the Grounds of Philosophy, I distinguish the iii05l common
notions by accurate definition, for the avoiding of confusion
and obscurity. The third part concerns the expansion of
Bpace, that is Georaetr)^ The fourth contains tlie 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 thhrd section, the doctrine of Subject mn 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 tlie
method you will like, I would very fain commend philosophy
to youj that is to say, the study of wisdom, for want of which
we have all suffered moch damage lately* For even they, that
study wealths 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 tliey, that love to be
employed In public business, aim at anything but place
wherein to show their wisdom. Neither do voluptuous men
neglect philosophyj 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 HRST,

OR LOGIC.

CIAT. PAGE.

1. Of I^OBophy 1

2. Of Name8 13

8. Of Ph>po8itioii 29

4. Of Syllogism - . . 44

5. Of Erring, Falsity, and Captions . . . 55

6. Of Method 65

PART SECOND,

OR THE FIRST GROUNDS OF PHILOSOPHY.

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 Ph>portion

144

14. Of Strait and Crooked, Angle and Figure

176

TITLES OF THE CHAPTERS-
PART THIRD,

OF THE PROPORTIOKS OF MOTIONS AND MAGKITUDSS.

15. Of the NaUire, Properties, and divers Considerations of

Motion and Endeavour - 203

16. Of Motion Accelerated and Unifonn, and of Motion by

Concourse 218

17. Of Figures Deficient 246

18. Of tlie Equation of Strait hinm wi^ the Crooked Linen

of Parabolas, and otlier Figures made in imitation of
Parabola.^ 268

1 9. Of Angles of Incidence and Reflection, equal by Suppo-

sition 273

20. Of the Dimenaion of a Circlcj and the Division of Angles

or Arches 287

2L Of Circular Motion ' , . 317

22. Of other Variety of Motions , . , . . 333

gS. Of tlie Centre of Equiponderation of Bodies pressing

downwards in Strait Parallel Lines .... 350

24. Of Refraction and Reflection * . . . • 374

PART FOURTH,

OF PHYSICS, OR THE PHKKOMENA OF KATURE.

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 Thtinder, and of the
Heada of Rivens 466

29. Of Sound, Odour, Savour, and Touch • . . ^85
SO, Of Gravity ... 508

;^ 1-5

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 l
in the same manner as com and wine are said to — i! — -
have been in the world in ancient time. For from i^^^^^^^^^--
the beginning there were vines and ears of com
growing here and there in the fields ; but 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
imknown 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

COMPUTATION OB LOGIC.

PART I. of sowing and planting, that is, of improving their
-^ — reason. And from heuec it comes to pass, that

Introduction, t^jgy y^]^Q 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 voilgar, yet full of uncertainty, and
carelessly received, do nothing but dispute and
wrangle, like men that are not wtII in their wits.
I coiifesSj 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
Httle.

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 oidy not beautiful, but poor,
arid, and, in appearance, deformed. Nevertheless,
there being certainly some men, though but few,
who are deUghted 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,

PART I,
1.

from the very definition of philosophy, which is

this. — ,— ^

2. Philosophy is such knotvled^e of effects or definition of

. . . Philosophy

appearances^ as we acquire by true rattocinuiion expkinea.

from the knowledge we have Jirsi of their causes
or gejieration: And again., of such causes or gene-
rations as may he from knowing first their effects.
For the better nnderstandiug of which definition,
we must considefj first, that although Sense and
Memor)^ of things, which are common to man and
all living creatures, be knowledge, yet because they
are pven 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

I we have already had experience of, Pni deuce 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 w^hen one thing is taken out of another.
Ratiocination J 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 isnothiug but addition of equals one
to another, and di^dsioii 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 oiu" mind, R»t^<'dnati
vse add and substract in our silent thoughts, with-
out the use of words, it will be necessary for me

B 2

PART h to make intelligible by an example or two. If
^~^—r — ' therefore a man see sometliiiiic afar off and ob-
^thrlS^nd^ scurely, although no appellation had yet been f^iven
to anything, he willj notwithstanding, have the
same idea of that thing for which now, by im-
posing a name on it, we call it bod//. 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 animatetL 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 thii'd idea,
though it have yet no appellation, namely, that for
which we now call anything rationaL Lastly,
when, by looking fidly 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,
(immiitedj rational^ are in speech compounded into
this one name, botlfi-ftninuiied-rfiiionat^ or num.
In like manner, of the several conceptions of Jour
,fides, equcdity 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 ;
''f, as he goes away, he follow him with his

OF PHILOSOPHY. 5

eyes only, he will lose the idea of those things part i.
which were signs of his being rational, whilst, ^ — '^-^
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, hody^ motion y 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 cow-
sider, in Greek XoyiUtrQai, in which language also
(mWoylUaQai signifies to compute, reason, or reckon.

4. But effects and the appearances of things to Properties,
sense, are faculties or powers of bodies, which ^*^ *^*^
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

PART I. columiij and other fixed bodies j and so that motion
H ^ — r" — or going is the property thereof, as being proper
^L to living creatures, and a faculty by which they

^F make ns distinguish them from other bodies.

Howpropcrtici 5. How the knowledge of any eflfect may be
Generation/ gotten from the knowledge of the generation
and contraniy. thereof, may easily be understood by the example
of a circle : for if there be set before ns 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 tme 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 immoved, 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 radiL 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

OF PHILOSOPHY,

6, The end or scope of philosophy is, that we part r.
may make use to our benefit of effects formerly ^—J;.^-
aeen; or that, by appiicatioo of bodies to one scope of
another, we may produce the like effects of those ^^^'**°P**y-
we conceive in our mind, as far forth as matter,
streng:th, 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 bidden 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 eud 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 perforui-

Cing of some action, or thing to be done*
I 7* But what the utility of philosophy is, espe- uriiuy t.f
cially of natural philosophy and geometry, will be * ^'^^p y-
hest 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, wth that of others which want the same.
Now, the greatest commodities of mankind are the
; namely, of measuring matter and motion ; of
oving ponderous bodies ; of architecture ; of
vigation ; of makiug iustraments for all uses ;
calculathig the celestial motions, the aspects of
e stars, and the parts of time ; of geography, &e.
By which sciences, how great benefits men receive
more easily understood than expressed. These
snefits are enjoyed by almost aU the people of

COMPUTATION OR LOGIC,

PART T.

■Utility of
Philosophy,

Europe^ by most of those of Asia, and by some of
Africa : but the Americans, and they that live near
the Poles, do totally want them* But why ? Have
they shaqier wits than these ? Have not all men
one kind of soul, and the same faculties of mind r
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 industiy, 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 canse,
therefore, of civil war is, that men know 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 cleai' and exact method ? For
what shall we say ? Could the ancient masters of
Greece, Egypt, Rome, and others, persuade the
unskilfid multitude to their innumerable opinions
concerning the nature of their gods, which they

OP PHILOSOPHY. 9

themselves knew not whether they were true or part i.

false, and which were indeed manifestly false and . V -.

absurd; and could they not persuade the same l^/!}**^®^
multitude to civil duty, if they themselves had ^^ ^'
understood it? Or shall those few writings of
geometricians which are extant, be thought suflS-
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,

Subject of
Philosophy.

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 snch 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.

8, The subject of Philosophy, or the matter it
treats of, is every body of which we can conceive
any generation, and which we may, by any consi-
deration thereof, compare with other bodies, or
w hich is capable of composition and resolution ;
that is to say, ever)^ 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 generatioiij 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, iugenerable, incom-
prehensible, and in whom there is nothing neither
to divide nor compound, nor any generation to be
conceived.

It excludes the doctrine of migelsy and all such
things as are thought to be neither bodies nor
ies of bodies ; there being in them no place ,
composition nor division, nor any capa-
and less, that is to say, no place for

Bfi hi.siory^ as well natural liVA poUHcaly
useful (nay necessai'y) to philosophy ;

OF PHILOSOPHY.

experience, or

PART t

Hall

because such knowledge is but
authority, and not ratiocinatiou.

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

;se, 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 diri-
nations rather than sciences, axe excluded.

Lastly, the doctrine of Gaits war. ship 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-
e.

9. The principal parts of philosophy are two. Pmruof
For two chief kinds of bodies, and very different *"^^ ^
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 hod I/, the other is called a commomvealthy and
is made by the wills and agreement of men. And
from these spring the two parts of philosophy,
called natural and clviL 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 politicsy or
simply civil philo.wphy. In the first place, there-

COMPUTATION OR LOGIC.

PART I. fore (after I have set down such premises as ap*
V — !; — . pertaiu to the nature of philosophy in general), I
will discourse of bodies naiurat ; in the seeond^
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 Uke this my definition of philosophy,
and will say, that, from the liberty which a man
may take of so definiug 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 piinciples.

OF NAMES. 13

CHAPTER II.

OF NAM£S.

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. — ^. 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
Uni vocal and Equivocal. — 13. Absolute and Relative Names. —
H. 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 p^^t ^•
are, and how much the recovery of them depends ^ — ^ — '
upon chance, there is none but knows by infallible JJ^geMiWe
experience in himself. For no man is able to re- Moniments

1 - . . , .11, 0' Marks

member quantities without sensible and present for the help
measures, nor colours without sensible and present ^ ^^^^'
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

PART I.

A Mark
defined.

Necessity of
Marks for the

also registered every one in its own order. These
moniraents I call mahks^ namely, sensible things
taken at pleasure, that, by the sense of thenij such
thoughts may be recalled to our mind as are like
those thoughts for which we took them,

2. Again, though some one man^ of how exeel-
significatioQ of lent a wit soever, should spend all his time partly

the conceptions * , i i * - i j»

ofthcMiod. m reasomng, and partly in inventnig marks for
the help of his memory^ and advancing himself in
learning ; who sees not that the benefit he reaps to
himself wiU 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 antececlents of tkeir
conseqiieuts^ and the consequents of their aniece-
dents, as often as we obserue 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 w ithout 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 hmig iipj signi-
fies that wine is to be sold there ; a stone set in
the ground signifies the bound of a field; and

COMPUTATION OR LOGIC.

PART I
2.

4, A NAME is a word taken at pleasure to serve
% — Jhr a mark, which may raise in our mind a thought
Defimijon /;^.^, /q some tkotij^ht we had before^ and which
being pronouueea to others^ may be to them a
sign of what 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-
trar)% judging it a thing that may be assumed as
unqnestionable. 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 tilings
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 w ere 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 bad 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, hy-
per holes, eissoeides, quadrat ices, &c. or to call
one magnitude A, another B. ^^

OF NAMES. 17

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 ; ^*"Jf ' ,
for that the sound of this word */ow^ should be "<>* of things,

- , - 111- ^"^ of our

the sign of a stone, cannot be understood m any cogiutiuns.
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 "H^^H^^
these, a many a treey 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 u^m^ 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
beeUy &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

COMPUTATION OR LOGIC.

aud Kcgative,

PART L example, we subs tract 2 aud l\ from 5, and so
nothing remaining, we would call that substrac-
tion to miiidj this speech nothing remains, and in
it the word nothing is not unusefuh And for the
same reason we say truly, le.^s 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 memor}\ 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.
NamesPosiHve 7. The first distinction of names is, that some
are positive, or ajfirmative^ 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 nof-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
nposed upon certain things, and afterwards upon
it things, the names of blacky bliu\ trans-

OF NAMES*

parent^ c^'r. the infinite dissimilitudes of these i'art l
with white could not be comprehended in any one — l-
name, save that which had in it the negation of
white, that is to say, the name not -while, or some
other equivalent to it, in which the word white is
repeated, such as unlike to white, %"€. And by
these negative names^ we take notice ourselves,
and signify to others what w^e have not thought of.

8. Positive and negative names are cow/rf/- ^<^^^^^^^'^^«7

- uamea.

mctory 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 he^ and not be, speak obscurely;
but they that say, whatsoecer 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 whatsoerer,
the other not, is the original and foimdation of all
ratiociuatiou, 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 comraou
mauy things, as a man^ a tree ; others proper to ^^^''^'
one thing, as he that writ the Hi ad. Homer, this
man, thai man. And a common name, being the

c 2

COMPUTATION OR LOGIC*

PART r.
2,

name of many things severally taken , but not
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
nniversiil name ; and therefore this w ord 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 w'hen a thing creature^ a sfonCy a
spirit, or any other thing, is said to be universal,
it is not to be nnderstood, that any man, stone,
&c. ever was or can be universal, but only that
these words, living creature^ stone^ S^^c. are nni-
f)€rsal 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. Mare
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
se, or lioUf because it comprehends them all :
therefore a more common name, in respect of
IS common, is called the genus, or a general
e ; and this in respect of that, the species^ or
'cial name.
And from hence proceeds the third distinc-
names, which is, that some are called
f the ^firsi% others of the second intention.

OF NAMES. 21

Of the first intention are the names of things, part i.
a iwflw, stone, 8fc. : 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, Sfc. 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.

1 1 . Fourthly, some names are of certain and universal,
determined, others oi uncertain and undetermi7ied\^^^^^li^
signification. Of determined and certain signifi- '^j!^^^'**^
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

COMPUTATION OR LOGIC.

name ; secondly, a common name set by ita€
without any note either of universality or partici
larity, as man^ stone, and is called an indefini
name ; but both particular and indefinite nami
are of uncertain sig^nificationj because the hear(
knows not w hat thing it is the speaker would hai
him conceive ; and therefore in speech, particuh
and indefinite names are to be esteemed equivalei
to one another. But these words^ a//, every ^ som
4^c, which denote universality and particularit
are not names, but parts only of names ; so thj
ef>erT/ man, and that man which the hearer co\
ceives in his mind^ are all one ; and some ma\
and that man which the speaker thought qf^ signh
the same. From whence it is evident, that tl
use of signs of this kind, is not for a man*s ow
sake, or for his getting of knowledge by his ow
private meditation (for every man has his ow
thoughts sufficiently determined without such hel|
as these) but for the sake of others ; that is, ft
the teaching and signifying of our conceptions!
others ; nor w^erc they invented only to makel
remember, but to make us able to discourse wh
others, ^

12. Fifthly, names are usually distinguishe
And equivocal into univocal and equivocal. Univocal are Xhoi
which in the same train of discourse signil
always the same thing ; but equivocal those whic
mean sometimes one thing and sometimes anothi
Thus, the name triangle is said to be unitocm
because it is always taken in the same sense ; an
parabola to be equivocal^ for the signification
has sometimes of allegory or similitude, and somi
times of a certain geometrical figure. Also ever

Ntmeii
uiilvoeal

OF NAMES. 23

metaphor is by profession equivocal. But this p-a.rt i.
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 ^*^*j!g"^*i^g
relative. Relative are such as are imposed for names,
some comparison, as father, son, cause, effect,

like, unlike, equal, unequal, master, servant, 8fc.
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 ahso--
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 crompoMwrf^rf simple and
names. But here it is to be noted, that a name is namw."" ^
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 accoimted 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

COMPUTATION OR LOGIC.

coiiipou

PART I. i^y the joining of another name to it, is made less
^ — ^ — ' universal, and signifies that more conceptions than
^'"^^^^ ndti ^^^^ ^'^^^ i^ the mind, for which that latter name
Avas added. For 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 hodtf. Body,
therefore, is a simple jtame, being put for that
first single conception ; afterwards, npon the sight
of such and such motion, another conception
arises, for which he is called an an i mated body ;
and tliis I here call a compomtded nami\ as I do
also the name animfdy which is equivalent to an
avi mated body. And, in the same manner, an
animated rational bady^ 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
compoundt d 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,
^hen, by the addition of magnitude, comes
mtity, and by more or less quantity to
or rarity ; and again, by the addition
be figurate, and after this, by the
iight or colour, to become lucid or
ough such has been the philosophy

OF NAMES. 25

15. The writers of logic have endeavoured to part i.
digest the names of all the kinds of things into ^ — r — -
certain scales or degrees, by the continual subor- ^6^^^^"®''*
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 inanimatedy and so on till they
come to individitals. In like manner, in the
scale of quantities, 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

i Not ani-
/ mated.

/Not living
i Animated Creature.

I Living f Not Man
Creature

Man

Not Peter.
Peter.

I Quantity, or so much.

Absolutely, as «"»"J"y'°'««'
Both Accident and Body J ^/ ^ ^'"^'^y' «' «"<='»

Comparatively, which is called
V their Relation.

COMPUTATION OR LOGIC.

PART I.
2.

A predicament
described.

The Form of the Predicament of Quantity.

Quantity

^Not continual,
as Number.

V Continual

I 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,
and solidy 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.

'Perception
by Sense

Quality

Sensible
Quality

'Primary

(Secondary

Seeing.

Hearing.

Smelling.

Tasting.

^Touching.

Imagination.

AflTection

( 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

PART I.
The Form of the Predicament of Relation. 2,

RelatioD of '

Magnitudes, as Equality and Inequality.
Qualities, as Likeness and Unlikeness.

Order

ny .t (In Place.

Together {i„ Time.

^Not t<^;eUier

InPlacejg— •
UnTin.e{JX-

16. Concerning which predicaments it is to be some things
noted, in the first place, that as the division is concerning
made in the first predicament into contradictory p"^*^'"*®"**-
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 line and
not'line, and again, not4ine 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 ; but 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 ; but, 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 maybe searched out and determined
by such distinctions as these ; or that arginnents

COMPUTATION OR LOGIC.

PART I.
2.

Predicaments,

Hiay be taken from heuce (as some have done
ridiculously) to prove that the kinds of things are
not 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) li^ht in the predicament
of quaUiiea^ while another places the same iu 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

Lastly, I confess I have not yet seen any great
use of the predicaments in phUosophy< 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.

OP PROPOSITION. 29

CHAPTER III.

OF PROPOSITION.

]. Divers kinds of speech. — 2. Proposition defined.— S. Subject,
predicate, and copula, what they are ; and abstract and con-
crete what The use and abuse of names abstract.— 5. Pro-
podtion, universal and particular. — 6. Affirmative and negative.
—7. True and false. — 8. True and false belongs to speech,
and Dot 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 ^ — r — -
the desires and aflfections of men ; such are, first, J^Jpgech"^^
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
aflfections. Speech may also be absurd and in-
significant ; as when there is a succession of

COMPUTATION OR LOGIC.

PART L words, to which there can be no succession of
^^1 — ' 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 w ant the end of speech (which
is signification) yet it is speech ; and is used by
WTiters of 7ueiapky^ie,s almost as fiequently as
speech significative. In philosophy, there is but
one kind of speech useful, which some call in Latin
dictum^ others enunfiditim el proni(ncmimu ; but
most men call it proposition^ and is the speech of
those that affirm or deny, and expresseth truth or
falsity,
Propoiition 2. A PROPOSITION IS (t Speech eonsistinsc of

defined, / ^ ''

two twmes copulated^ hy which he that speaketh
signijies he conceives the latter name to he 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^ inan is a living creature^ in which
two names are copulated by the verb is^ is a pro-
positiotiy 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
namCy 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, a^ in this

OF PROPOSITION. 31

proposition, vian walketh (which is equivalent to tart i.
this, man is walking) ; the termination by which it ^ — r —
is said he walketh^ 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 t *,
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 S€dd nuin
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 Jnd io*puia.
are the subject , and the predicate, and their ^^j^j^^lJJ^*^'*'
copulation ; both which names raise in oiu* mind «nd 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,
\pgX searches farther what it is to be a body, or to
be moveable, that is, wherein consists the diflfer-
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, 8fc. 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

COMPUTATION OR LOGIC.

RART i.

Subject.

which we suppose to have a being, and is there-
fore called tlie .subject y in Latin sttpposiluw^ and
in Greek vwoKu^n'ov ; as bodij, moveahlej moved,
fignratey a cubit high, hot, cold, like, equals
Appiiis^ Lcntulus^ and the like ; and, ahsiract
is that w^hich in any subject denotes the cause
of the concrete name, as to he a hody^ to be
moveable, to be moved, to be ^figuratCj to he of
such quantittj, to be hot, to be cold, to be like,
to he eqnal, to he Appius^ to he Lenlulus, S^^c.
Or names equivalent to these, which are oiost
commonly called ahstraci names, as corporiettf,
mohiUty, motion, figure, qnantit}/^ heat, cold,
Ukcness, equaliti/j 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 mot ion ^ or to be Hviug 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 onr mind any visible thing, that thing appears
to us, or is conceived by us, not io 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
extended, or the extension or corporieiy of it.
So when we see a thing appear sometimes here,
sometimes tliere, 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 with
the causes of onr conceptions, namely, some
power of action, or affection of the thing eon-

OF PROPOSITION. 33

ceived, which some call the manner by which any ^art 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 csui never be abstracted.
4. There is also this diflference betwixt concrete ^f 'J** ,

. and abose of

and abstract names, that those were invented names aimtnct
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

COMPUTATION OR LOGIC,

PAKT h
3.

from lieuce 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 tlie 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 esty as essence,
essentiality, entity ^ entitative ; besides reality ^
aliquiddity, quiddity^ ^c, which could never
have been heard of tamong 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^ entity^ and other the like barbarous
terms.

b. There are many distinctions of propositions,
first iSj that some are nniversul^
others particular^ others indejijiite, and others
singular I and this is commonly called the dis-
tinction of quantity. An tmiversal 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 vrith
the sign of a particular name, as sotne fuan is
learned. An indefinite proposition has for its
subject a common name, and put without any
sign, as ?/ian is a living creature, man is learned.

Proposition,

^^"i^o^?"'^ whereof the

OF PROPOSITION. 35

And a singular proposition is that whose subject part i.
is a singular name, as Socrates is a philosopher^ ^ — ^ — -
this man is black.

6. The second distinction is into affirmative ^^^^^"^y
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 ^rw^. True & false.
anotheryb^^. 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 ox 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 y one

D 2

COMPUTATION OR LOGIC.

PART I.
8.

True Be false
belongs to
sptfech, and
not to things.

tkhg, and a vrrtj things are equivalent to one
another, it is but trifling and childish ; for who
does not know, that €i mun^ one mauy and a verij
mmiy signify the same.

8. And from hence it is evident^ that truth and
falsity ha%'e no place but amongst such living
creatures 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
tine 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 fi*ora man, so
from man also is derived the ugly absurdity of
false opinions. For speech has something in it
like to a spider's w^eb, (as it was said of old of
Solaris 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 trutlis 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 iivhig creaiure^

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

'^^'^ 7jrm«/7/ and noi primary. Primary is that

»^tio!i, wherein the subject is explicated by a predicate of

many names^ as man is a hody^ animated^

OF PROPOSITION. 37

rational; for that which is comprehended in the part i.
name man, is more largely expressed in the names - — r — -
bodi/y animated, and rational, joined together;
and it is called primary, because it is first in ratio-
cination ; for nothing can be proved, without
nnderstanding 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 pri^nary 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 he 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 "ontfnge^ *
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

PART 1. the same thing ; as man is a living creatnre is a
- — r — ' necessary proposition, because at what time
rJclT^'J^T soever we suppose the name man agrees with any
contingtot thing, at that time the name living^reatnre 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 black ; which
may perhaps be true now, but false here^ter.
Again, in every necessarij 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-
turCy for the name rational'liiing-creaturey or
man^ is compounded of these two, rational and
Uving'-creature. 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
wmK, that proposition is not to be called necessary,
but contingenty though it should happen to be true
always. And therefore those propositions only
are necessary y 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, [f many then living-crea-
ture; but that any man^ or living-creatiirey shoidd
exist eternally, is not necessary.
St^oncfti & 1 1 . A sixth distinction of propositions is into
lypot Ltica , f*(if^gQfi^Qi 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 contlitionally, as, if any thing he a

OF PROPOSITION. 39

fium, the same is also a living-creature, if any-
thing he a man, the same is also not-a-stone.

A categorical proposition, and an hypothe-^^l^^fj^
tieal answering it, do both signify the 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 livings
creature ; but in contingent propositions, though
this be true, every crow is black, yet this, if any
thing he 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 tnily
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 seeing every proposition may be, and '''''« ««."?«
uses to be, pronounced and written in many forms, diversely
and we are obliged to speak in the same manner 1*^°"*^""*'* •
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

COMPUTATION OR LOGIC.

PART L
3.

predicate, both which must be separated aud
clearly distinguished one from another* For
example, if this proposition, man can not shy be
compared with this, man cannot siuj their dif-
ference will easily appear if they be reduced ta
these^ man is able not to sin^ and, man h not able
to sin, where the predicates are manifestly dif-
ferent. But they ought to do this sileiitly by
themselves, or betwixt them and their masters
only ; for it will be thought both ridiculous and
absurdj for a man to use such language publicly.
Being therefore to speak of eguipollent proposi-
tions^ I put in the first place all those for equipol-
hnty that may be reduced purely to one and the
same categorical proposition,
PfopofliUons 13, Secondly^ that which is categorical and

that may be * • i j * • i i • i

reduced 10 necessary, is equipollent to its hypothetical pro-
te^ori^Ipm- Position ; as this categorical, a right-Uned tri'
poBition, are Qpglf^ J^q^ {(^ three angles equal to two rii^ht

Chqiupollent. ^ ^ . r »

angles, to this hypothetical, if any figure be a
right'lhied triangle^ the three angles of it are
equal to two right angles,
Univers*! 14. Also^ any two universal propositions, of

ISnvetteTby which the terms of the one (that is, the subject
^^^**^^^^^^ and predicate) are contradictory to the terms of
equiswUent the Other, and their order inverted, as these, erertf
man is a living creature^ and everff thing that is
not a liring-creature is not a man, are equipollent.
For seeing every man is a liring 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 llving-creatttre ;

after

OF PROPOSITION. 41

wherefore every thing that is not a living-crea- part i.
ture, is not a many is a true proposition. likewise — r — -
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
fio 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 as SJ^TJ wme,
some nations do, or before it, as it is in Latin and nemtion'be
Greek, if the terms be the same, are equipollent : ^^ore or ai

r 1 si A • tl»e copula,

as, for example, man ts not a tree, and, man ts
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. Simply c^n-
some blind thing is a man, are equipollent ; for ^^u*^
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 aub-
and are placed in the same order, but varied either ^l^au^^n-
by quantity or quality, some are called subaltern, g^^J^f^
others contrary, others subcontrary, and others propositions.
contradictory.

Subaltern, are universal and particular propo-
sitions of the same quality ; as, every man is a

VMvr I. Ihing creature^ sotne man is a Ihing crectture ;

-^ — r — ' or, no man is ivise^ some nmn is not wise. Of
propositioni. t^egg^ jf xh^ miiversal be true, the particular mil
be true also.

Contrary y are universal propositions of different
quality j as^ every man is happy ^ no man is
happy. And of these, Lf one be true, the other
is false : also, they may both be false, as in the
example given.

Subcontraryy are particular propositions of
different quality ; as, some man is learned^ some
man is not learned ; which cannot be both false^
but they may be both true.

Contradictory are those that differ both in
quantity and quality ; as, every man is a living
creature J some man is not a living-creature ;
which can neither be both true, nor both false.
Consequence, 18. A propositiou is said to Jhllow from two
other propositions, when these being granted to
be true, it canuot be denied but the other is true
also. For example, let these two propositions,
every man is a living creature^ and, every living
creature is a body, be supposed true, that is, that
body is the name of every living creature^ and
living creature the name of every man. Seeing
therefore, if these be understood to be true, it
cannot be understood that body is not the name of
every many that is, that every man is a body is
false, this proposition will be said to follow from
those two, or to be necessarily inferred ivom them.

^awty 19, That a true jiroposition may follow from

tmot follow rt I ... , .. , ^

m truth, ndse propositions, may happen sometimes ; but
false from time, never. For if these, every man
is a stone, and every stone is a living creature^

OF PROPOSITION. 43

(which are both false) be granted to be true, it is part i.
granted also that living creature is the name of — ^ — -
eterf/ stone, and stone of everi/ man, that is, that
living creature is the name of every man ; that
is to say, this proposition every man is a living
creature, is true, as it is indeed true. Wherefore
a true proposition may sometimes follow from
fialse. But if any two propositions be true, a
false one can never follow from them. For if
true follow from false, for this reason only, that
the false are granted to be true, then truth from
two truths granted will follow in the same manner.

20. Now, seeing none but a true proposition How one
will follow from true, and that the understanding fJ^tbriauL
of two propositions to be true, is the cause of °^ another.
understanding that also to be true which is
deduced from them ; the two antecedent propo-
sitions are commonly called the causes of the
inferred proposition, or conclusion. And from
hence it is that logicians say, thejrremises are
causes of the conclusion ; which may pass, though
it be not properly spoken ; for though understand-
ing be the cause of understanding, yet speech is
not the cause of speech. But when they say, the
cause of the properties of any thing, is the thing
itself, they speak absurdly. For example, if a
figure be propounded which is triangular ; seeing
every triangle has all its angles together equal
to two right angles, from whence it follows that
all the angles of that figure are equal to two right
angles, they say, for this reason, that that figure
is the cause of that equality. But seeing the
figure does not itself make its angles, and there-
fore cannot be said to be the efficient-cause, they

COMPUTATION OR LOGIC,

PART I

IJuw one
propasition

call it thtformal^mfse ; wliereas indeed it is no

cause at all ; nor does the property of any figure

follow tlie figure, but has its being at the same

18 the cause time with it ; only the knowledere of the fissure

of another, i , , t -

goes before the knowledge ot the properties ;
and one knowledge is truly the cause of another
knowledge, namely the efficient cause.

And thus much concerning proposilimi ; which
in the progress of philosophy is the first step,
like the mo\ing 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.

Dcfiuition

1, The definition of gyllogi^m. — *2, In a syllogism there are but
three terms, — !L Major, minor, and middle terra ; also major
and minor proposition^ what they are, — \, Tiie middle ttrm in
every syllogism ought to h\* detennitied in both the propositions
to one and the same thing. — 5. From two partieukr propo-
sitions nothing can be concluded. — 6. A fiyllogisiu is the col-
lection of two propositions into one sum. — 7. The figure of a
syllogism, what it is. — 8, What b in the mind answering to a
syllogism. — 9* The first indirect figure, how it is made, —
10. The second indirect figure, how made.^ — 1 1. How the third
indirect figure is made. ^-12. There are many moods in every
figure, but most of them useless in philosophy .^^ 13. An
hypothetical i»ylIogisni when cciuipollent to a categorical.

1. A SPEECH, consisting of three propositions,
from two of which the third follows, is called a
SYLLOGISM : and that which follows is called the
conclusion ; the other two premises. For example,

OF SYLLOGISM. 45

this speech, every man is a living creature,
every living creature is a body, therefore, every
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«»yiioR»*«n

I . ^ ,, J there are but

term conmion, no conclusion can follow ; 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
flant 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 Major, minor
in the conclusion, is commonly called the major ; term; 'also
that which is the subject in the conclusion, ^^Jji^n^'
the minor, and the other is the middle term; ^"^^^^^^y^^-
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

PART I. living creature the middle term. Also of the

^ — r — premises, that in which the major term is found,

is called the major propositimi^ and that which

has the minor term ^ the minor proposition.

The middle 4, If the middle term be not in both the pre-
term m avevy • i ^ - 1 , , - ,

syllogism to mises detennined to one and the same singular
irCh'^p'^o things no conclusion will follow, nor syllogism be
^f anTthe ^^^^' ^or let thc minor term be man, the middle
same thing, term Uviug creature^ and the major term lion;
and let the premises be^ man is a living creature^
some living creature is a lion^ yet it w ill 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 j
ought to be either universal or singtdarj but not
particular nor indefinite. For example, this syl-
logism, every man is a living creature^ some living
creature is Jom\footedy therefore some man is
fonr-footedy is therefore faulty, because the middle
term, living creature, is in the first of the premises
determined only to many for there the name of
living creature is given to man only, but in the
latter premise it may be xmderstood of some other
living creature besides man. But if the latter
premise had been universal^ as here, every man is
a living creature y every living creature is a body ^
therefore every man is a body, the syllogism had
been true ; for it would have foUowed that body
had been the name of every living creaturej that
is of Jna7i ,* that is to say, the conclusion every man
is a body had been true. Like^vise, when the
middle term is a singular name, a syllogism may
be made, I say a tnie syllogism, though useless in
philosophy, as ilmysome ma7i is Socrates ^ Socrates

OF SYLLOGISM. 47

is a philosopher, therefore, some man is a philo- part i.
sopher ; for the premises being granted, the con- — r — -
elusion cannot be denied.

5. And therefore of two premises, in both From two
which the middle term is particular, a syllogism propptions
cannot be made ; for whether the middle term be S^ ronduded.
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

Same man is learned, f term 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 limng^reature,
Every horse is a living-creature,

In both which the middle
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- \ j^ ^„^ ^f ^^^^^ ^^^ ^y^,^
creature, Uerm is the subject, and in

Some hvtng creature ts \ ti,e other the predicate,
four-footed, ) ^

the conclusion will not follow, because the name

Bitione into
one sum

The ig;urc of
a syllagiaui
wh&t It is*

living creaiifre being not determined^ it may in
one of tliem be understood of mmij in the other of

A syllogism is g^ ^Qw it IS manifest from what has been said,

the collection n .

of twopropo- that a syllogism is nothing but a collection of the
sum of two propositions, joined together by a
common term, which is called the middie term.
And as proposition is the addition of two names,
so syllogism is the adding together of three,

7- Syllogisms are usually distinguished according
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 diflferences of propositions in
qumitity and quaUtyn 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 maUy the middle term, living creature^
and the major term, iorfy, 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 hoily, 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 terra unirersal^ 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 imiversal name,

OP SYLLOGISM,

PART
4.

the Jigurc will be in the second mood., as, every
man h a Ihing creature^ every Ihing ereaiure is
not a tree^ in which the niajor 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
moody in which all the propo.sitions 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 Jiint^ this syllogism,
no man is a stone^ whatsoever is not a stone is
not a Jiint^ therefore, 7io man is a flinty is true,
though it consist of three negatives. But in phi-
bsophy, the profession whereof is to estabUsh
imiversal 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 Jigurey besides that, in which all
the propositions are both universal and aflfirm'
ative.

8. The thoughts in the mind answering to awhaiii
rect syllogism, proceed in this manner ; first, auawenng to
there is conceived a phantasm of the thing named, * ^y"**^"'"'-
with that accident or quality thereof, for which it
i|s 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
fsame proposition is the predicate ; thirdly, the
f thought returns of the same thing as having that

VOL. 1. E

COMPUTATION OR LOGIC,

PART I. accident in it^ for which it is called by the name,
^ — ^ — ' 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 trae.
For example^ when this syllogism is made, rnan is
a livhg creature^ a living creatnre is a hody^
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
wan; 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 hody ; 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, many living creature^ and
hoihjy are names of the same thing, and that there-
fore man is a litring creatnre 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 ill- 9. The rcst of the figures arise either from the
how^ made!* 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,

hrom whence follow three other figures; of part l
which, two are iufiected^ and the third inverted, ^~- — '
The first of these three is made by the eonversiou Jj^ecffi^re ,
of the major proposition. For let the minor, ^'^^ "*»**«• '
middle, and major terms stand in direct order,
thus, man is a lirinff ereaturey 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 crertture, a stone is not
a living creature \ and this is the second figure,
or the first of the indirect figures ; in which the
inclusion w^ill 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,
hi like manner this direct syllogism, man is not a
tree, is not a pear-treey 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

irect figure ought to be negative. For though

s direct, man is a living creature, is a body, be
ade indirect, by converting the major i>ropo-
sition, thus,

Mem is a living creature^

Not a body is not a Umng creature y

Therefore, Et^ery man is a body ;

et this conversion appears so obscure, that

E 2

COMPUTATION OR LOGIC.

Second indi-
rect figure
how made.

PART L this mood is of no use at all. By the oonversio^
* — r — ' of the major proposition, it is manifest, that iu this
figure, the middle term is always the predicate in
both the premises.

10. The second indirect figure is made by con-
verting the minor proposition, so a.s 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

Evert/ man ?,v a living creature^
Evertf livlmj creature h a body^

by conversion of the minor proposition, will stand

thus,

Some livhftj creature i/t a man^
Every tmng creature is a hody^
Therefore, Smne man is a body.

For every man Is a living creature cannot be
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 ereaiure^ and consequently
the conclusion w ill be some man is a hody^ seeing
the minor term ///r/n, which is the subject in the
conclusion, is a particular name.
?;7*';V*''^ IL The third indirect or inverted fimure, is

indirect figure ^ " '

iimadc. made by the conversion of both the premises.
For example, this direct syllogism.

Every man is a Ihiny creature^
Every Hviny creature is not a stmte^
Therefore, Every man is not a Hone^

being inverted, wnll stand thus,

OF SYLLOGISM. 53

Every stone is not a living creature, PART I.

Whatsoever is not a living creature, is not a man, - ^* --
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
nambered 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 Xoyo/iaxta, 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.

12. In every one of these figures there are ny moods"!^
many moods, which are made by varying the pre- but'^mMt'^'of
mises according to all the differences they are ?*®™ "*®^^**
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

conversiou

^ — '^ — * the premises in the direct figure.

Aiihypotheti- 13, As it WRS sliowed before, that in necessary

whcnequipoi- propositious a catcgoncal and hypothetical propo-

go'JJj^if "**" sitioii 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^
Evert/ Uvmg creature is a bodi/j
Tlierefore, Everif man is a hody^

is of equal force with this hypothetical syllogism :

If any thing he a mmu fheftame is also a limngerealure.
If any thing be a Vmng creature^ the same is a body^
Therefore, If any thing be a mauy the same is a body.

In like manner, this categorical syllogism in an

indirect figure,

No stone is a livirig creature^
Every man is a Umng 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 stoney the same is not a living creature^
Therefore, If any thing he a stone^ the same is not a man.
Or, If any thing be a mavy 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 i.
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 ERRING, FALSITY, -AND CAPTIONS.

1. Eniiig and falsity how they differ. Error of the mind by
itself without the use of words, how it happens. — 2. A seven-
fold incohereney of names, every one of which makes always
a false proposition. — 3. Examples of the first manner of inco-
hereney.— 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. — II. Of the
fault of a syllc^sm 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 Ernng & fai-
denying, but also in perception, and in silent differ.^'Errw
cogitation. In affirming and denying, when they ^^eiif! wuhou^t
call any thing by a name, which is not the name ^<*^ ^f^ ?^

i '!• fi •! n ^ n words, how it

thereof ; as if from seemg the sun first by reflec- happens,
tion in water, and afterwards again directly in the

Erring^ and
lalsity how
they diflcr.

firmament, we should to both those appearances
give the name of sun, and say there are two suns ;
w^hich none but men can do, for no other living
creatures have the use of names. This kind of
error only deserves the name of Jalsiiy^ 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 Jalsehjy by
their own negligence, in departing from such
appellations of things as are agreed upon, and are
not deceived neither by the tilings, 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 w^ater, 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 w^e
feign the mind of the promiser to be such and such ;
or lastly, when from any sign we vainly imagine
somethhig 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 \\q feign such things as
are but mere images to be something more than
images. But neither things^ nor imaginations of

O* BRRING^ FALSITY, ETC. 57

things, can be said to be false, seeing they are parti.
truly what they are ; nor do they, as signs, pro- -— t^ — ^
inise 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
Msity 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
m syllogizing, consist either in the falsity of the n"^e"*^ai7of
premises, or of the inference. In the first of these '^I'^ly^ J?^^®
cases, a syllogism is said to be faulty in the p'°p°"^°°-
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. Ill, art. 7) in which are copulated two
names of one and the same thing; and that always
false, in which names of diflFerent things are copu-
lated, look how many ways names of diflFerent
things may be copulated, and so many ways a
false proposition may be made.

Now> all things to which we give names, may be

COMPUTATION OF LOGIC.

PAKT 1.
5,

reduced to these four kinds , namely^ bodies, acci-
ilents^ phantasms^ miAuames themselves; and there-
fore, in every true proposition, it is necessary that
the names copulated, be both of them names of
hodiesyOV both names of €tc€idenis jOrhoth. names of
phantasmsy or both names of mimes. For names
otherwise copulated are incoherent, and constitute
a false proposition. It may happen, also, that the
name of a bodi/, of an accident, or of ^ phantasm^
may be copulated with the name of a speech. So
that copulated names may be incoherent seven
manner of ways.

1 . If the name of a Body

2. If the Dame of a Body

3. If the iianie of a Body

4. If the iiaDie of an Aceident

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 AecideDt
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*

Examples
of the first

Of all v^hich I will give some examples.

3. After the first of these ways propositions are
manner of false, whcH abstract names are copulated with
inuo cTuncy. ^Qjj^,^gj.^ names ; as (in Latin and Greek) esse est
ensy essentia est e?is, to tI ^v hvuI (i.) ; quiddiias
est ens, and many the like, which are fonnd in
Aristotle's MefapIiT/sics, Also^ the umlersiamUug
tvorkethy the understanding nmlerstandeik^ the
sight seeth ; a body is magnitude^ a body is
f/uaniity, a body is extension ; to be a man is a
many whiteness is a white things &c. ; which is
as if one should say^ the runner is the running,
or the walk walheth. Moreover, essence is sepa-
rated ^ substance is abstracted : and others like
these, or derived from these^ (witli which common

OF ERRING, FALSITY, ETC.

philosophy abounds,) For s^eeing no subject of i'art l
an accident (that is, no hody) is an accident : no * — ^ — '
name of an accident ought to be given to a hodtf^
Hor of a hody to an accident.
4, False, in the second manner, are such propo- The secomh
itions as these ; a ghost is a hody^ or a spirit ^
that is, a thin body ; sensible species fly up and
daum in the air^ or are moved hither and thither^
whicb is proper to bodies ; also, a shadow is
L jior^rf, or is a hody ; light is moved, or is a
^Hotify; colour is the object of sights sound of
^m hearing ; space or place is extended; and inmi-
^ 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.

K 5. False propositions of the third kind, are such ^^'^ ^**'''**
^bs these ; genus est ens^ miicersale est ens^ ens
Bile ente pr€edicatm\ For genus ^ and universale^
and predicarey are names of names, and not of
things. Also, number is inflftite, 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- '^^^ *"^"'^'''
sitions as these, an object is of such magnitude or
^gure as appears to the beholders ; colour, light,
undj are in the object ; and the like. For the
le object appears sometimes greater, sometimes

PABTL
5.

Tlie fiah.

Th«dxth.

The seven tb.

lesser, sometimes square, sometimes round, accor-
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,

7. Propositions are false in the fifth manner,
when it is said that ike dvjinition is the essence of
u thing ; whiienessy or some other accident, is
the genus J or iinirersaL For detiuitiou is not the
essence of any thing, but a speech signifying
what we conceive of the essence thereof ; and so
also not w^hiteness itself, but the w ord whiteness,
is a genus, or an universal name.

8. 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,
w^iich 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 idect thereof.

9. They err in the seventh manner, that make
this distinction between things that have being,
that some of them ejcist by themselves^ others by
accideut ; namely, because Socrates is a ?nan 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 ; w^hereby,
seeing wt^rf^y.v^/ry, contingent ^by itselj\ by accident^
are not names of things, liut uf propositions, they

OF ERRING^ FALSITY, KTC.

(il

PARTL

any ffihf^ fJmt has being, exists hy acci-
drnt^ copulate the name of a proposition with the
ame of a thing. In the same manner also, they
err, which place some ideas in the nnderstanding,
others in the fancy ; as if from the understanding
of tills 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 tilings one after another, winch we observe
in the same idea of man ; so that this proposition,
nan h 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 ^^^^^^J ^jf*
With names of bodies, names ot accidents with detected by
names of accidents, names of names with names of terms with
names, and names of phantasms with names of '^^*'"'^***"**
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

th those names, and again the definitions of

ch names as are in the former definition, and so
proceed by a continual resolution till we come to

simple name, that is, to the most general or
lOst universal name of that kind ; and if after all

COMPUTATION OR LOGIC

PART T* this, the truth or falsity thereof be not evident,
* — r^— ' we must search it out by pliilosophy, and ratioci-
Bation, 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,
or this ftoU 1 1 , That fault of a syllogism which lies hid in
coii*i?tj^**in the form thereof, will always be found either in

with one of the

Z^'r^iwnf the implication of the copula

the copuJa.

Of the

fault which
conaiats m
equivocation.

one

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,

77te luifid foucheth the pen^
Tlie pen ioucheth the papcr^
Therefore, Tfie hand foucheth the paper ;

the fallacy will easily appear by reducing it, thus :

The handf isj iotiching the pen^
Thepeny t>, touching tite paper ^
Therefore, 77ie handy i#> touching the paper ;

where there are manifestly these four terms, tfie
handy touching the peuy 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,

t2. And though there may be fallacy in equi-
voccol terms, yet in those that be manifestly such,
there is none at all ; nor in metaphors, for they
profess the transferring of names from one thing

OF ERRING, FALSITY, ETC.

Vie

to another. Nevertheless, sorii(*times equivocals i*art i.
(and those not very obscure) may deceive ; as in — ^ — -
this argiunentation : — // belongs to metaphysics
(q treat of principles ; hut the first princijde of
all, iSy that the same thing cannot both ewist and
not exist at the same time ; and therefore it
hlongs to metaphysics to treat whether the same
ling 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 tlie
eating of principles belongs to primary science,
le understands by principles, causes of things,
and certain existences which he calls primary ;
but where he says a primary 2>f'oposition is a
principle^ by principle, there, he means the
l)eginning and cause of knowledge, that is, the
iinderstanding of words, which, if any man want,
is incapable of learning*
13. But the captions of sophists and sceptics. Sophistical

ii-ii i»ii 1*1 1 captions

by which they were wont, ot old, to dende and are oftener
oppfjse truth, were faidty for the most part, not Jjlttttl/lhan
in the form, but in the matter of syllogism; ^ii^ j^'J^^^nogu^^^^^^^^
they deceived not others oftener than they were
themselves deceivedi For the force of that famous
argument of Zeno against motion, consisted in
this proposition, whatsoever may be diimled 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 notliing 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

COMPUTATION OR LOGIC.

PART 1

number, or be infinite, becanse I can divide and
' — r — ' subdivide it as often as I please ; for how many
parts soever I make, yet tbeir 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
conciuaion. 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
pliilosophy; 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.

I. Method and science defined. — 2. It is more easily known
concerning singular, than universal things, that they are ; and
coQtrarily, it b more easily known concerning universal, than
siognlar things, why they are, or what are their causes. —
5. 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
sjmtheticaL — 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. — 1 1 . 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
necessary for me to repeat the definition of philo-
sophy, delivered above (Chap, i, art. 2.) in this ^i^n^^des^cd.
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

COMPUTATION OR LOGIC-

PART I
6.

of such production^ as has been or may be, frot
the knowledge we Juwe of the effects. Method^
dTfiDfid therefore, in the study of phOosophy, f^ il
shortest way of Jinding out effects by their knot
causes^ or of causes by their known effects. Bt
we are then said to know any effect^ when wc
know that there be causes of the same, and
what subject those causes are, and in what su\
jeet they produce that effect, and in what manne
they work the same. And this is the science ol
causes, oFj as they call it, of the Stou. All othc
science, which is called the on, is either percej
tion by sense, or the imagination, or memor
remaining after snch perception. '

The first beginnings, therefore, of knowledge,
are the phantasms of sense and imagination ; and
that there be snch phantasms we know well enough
by nature ; but to know why they be, or froi
what causes they proceed^ is the work of ratioc
nation ; which consists (as is said above, in
1st Chapter, Art, 2) in composition, and divisit
or resolntion. There is therefore no method, by
which we find out the causes of things, but is
either eompositire or resolutive, or partly cofA
positive, and partly resolutive. And the resolutive
is commonly called analytical method, as thfi
compositive is called syntheiicaL
It is easier *^* '* ^^ commou to all sorts of method, to pi
known con- ^ecd from known things to unknown ; and this
lar ihnti uni- manifest from the cited definition of philosoph]
tYIuhey afe; But iu kuowlcdgc by sense, the whole object is
?t***u''"e^ilr^*^^^ known, than any part thereof ; as when viJ
known coii^ see a man, the conception or whole idea of thaf

cerning uni- • r- i i i •

venaithaoBin man IS nrst or more knov^n, than the partici

OF METHOD.

of his being Jigurati% animal e, and rational; ^^^ ^•
that is^ we first see the whole man, and take ^^---'^ — -
notice of his being, before we observe in him those why "tbly ^i
other particulars. And therefore in any know-°^.*^*^ *™
ledge of the Sn, or that any thing is^ the beginning
of our search is from the whole idea \ and con-
trarily, in our knowledge of the Ston, or of the
causes of any thing, that is, in the sciences, we
hav« 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 shoiilderSj his arms, &c. but his figure,
quantity, motion, sense, reasoTi, 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 knoiATi 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
miirerjfai) are more known to us than the parts^
that is, such things as have names less universal,

F 2

COMPUTATION OR LOGIC,

PARTI.

(which I therefore call Ahigfifar) ; and the causes
of the parts are more known to nature than the
cause of the whole ; that is, universals than
singulars.
What it h 3. In the study of philosophy, men search after

philosopher* . . i , / ^ . ,

«eek to know, science Cither simply or indetinitely ; that is, to
know as much as they can, without propounding
to themselves any hmited question ; or they
enquire into the cause of some determined appear-
ancCj or endeavour to find out the certainty of
something: in question, as what is the cause of
lighiy of keaty of grariii/, of a Jigiire 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

Thefirrtptrt, 4, But to thosc that scarch after science inde-
cM*^^^^ finitely, which consists in the knowledge of the
cohered, is causcs of all thiiigs, as far forth as it may be
lyucai. 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
universal things, or of such accidents as are
oaon to all bodies, that is, to all matter, before
r can know the causes of singular things, that
those accidents by which one thing is distin-
?d from another. And, again, they must
'hat those universal things are, before they
V their causes. Moreover, seeing universal

OP METHOD.

things are contained in the nature of singular part l
things, the knowledge of them is to be acquired - I ^
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 matte r^ namely,
line, plain, (which contains superficies) termi-
nated, angle ^ straighiness, 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 gokh he may, by resolving,
come to the ideas o( salid, visible, heavy, (that is,
tending to the centre of the earth, or do wti wards)
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 w^hat those things are, whose causes being
first know n 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, Tbo highest

1 1-1 \ 'p ^causes, and

at least, that nave any cause) are manifest of moat imkersai
themselves, or (as they say commonly) know n to TreTnTwn by
nature ; so that they need no method at all ; for '^^^»^^^«^«-
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 w hich they are

COMPUTATION OE LOGfC.

fiT r* made ; and motion cannot be understood to have
^ any other caui^e besides motion ; nor has the
variety of those tilings we perceive by sense, as of
colours^ sounds y j^avonrsj &c- any other canse than
motion, residing partly in the objects that work
upon our senses, and partly in ourselves, in suck
manner, as that it is manifestly some kind of
motion, though we cannot, without ratiocination,
come to know what ki!id. 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.
lethijd from g^ gy the loiowledere therefore of universals,

principles ■' ^

found out, and of their causes (which are the first principles
sdence simply, hy which wc kuow the Stfjn of thiugs) we have in
wiiat ii IS. ^j^^ j^^,g^ place their definitions, (which are nothing
but the explication of our simple conceptions.)
For example^ he that has a true conception of
plac€y cannot be ignorant of this definition, place
is that space which is possessed or filled ade-
quately hy 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 ^notion of a pointy superficies
by the motion of a line^ and one motion hy another

OF METHOD.

PART T.
6.

motion, &c. It remains, that we enquire what
motion begets such and such effects ; as, what
motion makes a straight line, and what a circular ; JJ^J^^p^i^^^
what motion thrusts, what draws, and by what ^o'^"^ out,
way; w^hat makes a tmng which is seen or heard, science simply,
to be seen or heard sometimes in one manner,
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 w^hat effects, what figures, and what properties,
they produce ; from which kind of contemplation
sprung that part of philosophy which is called
geometrt/.

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 w hich 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

PAET I. drawn that part of philosophy which treats of
motion.
Mcthoti from Ju the third place we must proceed to the

pnociplea * «

found out, enquiry of snch effects as are made by the motion
scfeacf simply, of thc parts of any body, as, how it comes to
wbantis. pj^gg^ ^^^^^ things when they are the same, yet
seem not to be the same, bnt changed* And here
the things we search after are sensible qualities,
snch as lights colon r^ transfMtrenct/^ opacity^
soundy odour y .savour ^ heat^ eold^ and the like;
which because tliey cannot be known till we
know the causes of sense itself^ therefore the
consideration of the causes of seeing ^ hearings
smeUing^ tiLsthig^ and iouchmgy 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 i)hilosapljy 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 lie drawn from the parts of the
sciences above mentioned, or no reason at all will
be given, but all left to nncertain conjecture.

After physics we must come to moral philo-
sophy ; in which w^e are to consider the motions
of the mind, namely, uppelifc^ aversion, love^
benevole7ice, hope^ J^^^^ ^^g^^'i emnlation^ ent^y^
^c. ; what causes they have, and of what they
be causes. And the reason why these are to
be considered atler physics is, that they have

OP MBTHOD. 73

their causes in sense and imagination, which are part i,
the subject of physical contemplation. Also the — r — -
reason, why aD 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 eflFect. And because all
appearance of things to sense is determined, and
made to be of such and such quality and quantity
by compoimded 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
ignorant of geometry, do but make their readers
and hearers lose their time.

7. Civil and moral philosophy do not so adhere That method
to one another, but that they may be severed, tu^ sdence,
For the causes of the motions of the mind are fr^^ selsf to
known, not only by ratiocination, but also by the principles, is
experience of every man that takes the pains to andagain,that
observe those motions within himself. And, Tt ' pnndpies
therefore, not only they that have attained the ^' "y''^'^*^*^-
knowledge of the passions and perturbations of
the mind^ by the synthetical method^ and from
the very first principles of philosophy, may by

PART L proceediug in the same way, come to the causes

— ^^— ' and necessity of constituting commonwealths, and

ofciviiTndn'L to get the knowledge of what is natural right, and

iTrocle'tUur*^' ^^^^ ^^^ ^^^'^^ duties ; and, in every kind of

from sense govemment, what are the rights of the common-

to pnnciplcSi , , i n i i i i - *

m analytical ; wcalth, and all othcr knowledge appertaining to
rhlirbegirta ^1^"^^ philosophy ; for this reason, that the princi^
fs^yQ^^etic^ ^^ ^^^ politics cousist iu 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,
geofuetry and pkt/sics^ may, notwithstanding,
attain the principles of civil philosophy, by the
analytical metJiod. For if a question be pro-
pounded, aSj whether such an aciiou he just or
unjitat; if that unjust be resolved hitojiici against
lau\ and that notion law into the command of him
or them that have coercive power; and that
power be derived from tlie wills of men that con-
stitute such power, to the end they may live in
peace, they may at last come to tliis, 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.

"anmytical, atid partly synthetical ; namely, that part i.
i\hich proceeds from sense to the inventiou of — ^ —
principles, analytical ; and the rest synthetical.

8. To those that seek the cause of some certain The method
aud propounded appearance or effect, it happens, °at* whether
sometimes, that they know not whether the thing, p^JiJl^'^^',^
whose cause is sought after, be matter or body, or'^'***^*"^"^-'
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 liave 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 maimer, the phan-
tasm of the same thing appears sometimes round,
by being seen afar oflF, 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 w ith the idea
we have before us ; and if it agree with the pro-
perties of matter or body, then it is a body ; other-

COMPUTATION OB LOGIC,

PART I.
6.

wise it is an accident. Seeing, therefore, matter
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 syntketical,

^^ seeking ^' ^^^ **' ^^^^'^ ^^ ^ doubt made concerning the
mhethcT any subiect of aov kuowai accidcut (for this mav be
in thiiorin aouDted sometimes, as in the precedent example,
subject. (JQ^|J^ j^^^j ]j^ ijiade in what 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 true ratiocination the sun be
found to be greater than its apparent magnitude,
then that magnitude is not in the sun ; if the sim
be in one determined straight line, and one deter-
mined distance J 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.

'nitude but the sentient itself. And this method, part i.
in regard the subject is divided into parts, is — "t — -
analytical ; and in regard the properties, both of
the snbject 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 th^c^lsf Tf
mto our mind an exact notion or idea of that p°J ^^^S
which we call cause, namely, that a cause is the
mm 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 he understood hut
that the effect existeth with them ; or thai
it can possibly exist if any one of them be
absent. This being kno^Ti, in the next place we
must examine singly every accident that accom-
panies or precedes the effect, as far forth as it
j^eems to conduce in any manner to the production
of the same, and see wlietlier the propounded
effect may be conceived to exist, without the

stence of any of those accidents ; and by this
separate such accidents, as do not concur,
from such as conctir to produce the said effect ;
which being done, we arc 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,
Jthen that aggregate of accidents is the entire
luse, 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

ui tut
^«ffect
^nistei

COMPUTATION OR LOGIC,

PART h out . first, we examine things without us, and find
* — * — that whensoever light appears, there is some prin-
i^^Mcbing for cip^ object, as it were the fomitaio of light,
It^ '"''^^5,?/ without which we cannot have any perception 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
obsen e 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 efl^ecting of light, but only action (or a certain
motion), w^hich cannot be conceived to be w^anting,
whensoever the efi^ect 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 liath
in "f ^r winch yet is but a trifling exception,
84 1 '^Mt^ h meant thereby but the cause of

li^ man should say that the cause of

' sun which prodnceth it) ; it

hat the action, by which light
tion only in the parts of the

OF METHOD.

object. Which being: understood, we may easily part r,
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 wliich we
demonstrate.

U. In the method of invention, the use of^^'^i^^^-r^

' to ID vent] on

words consists in this, that they may serve for a^'^^f'^s; to

, - 1*1 1 1 /» 1 demon strati on

marks, by which, whatsoever we have round out ns sigQa.
may be recalled to memory ; for without this all
our inventions perish, nor will it be possible for
m 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

COMPUTATION OR LOGIC.

PART I.
6.

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,
imlike the former, or the same in diflferent situa-
tion, should be offered to his consideration, he
would not know readOy 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 no 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
faTy^t^S 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 sened 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

I
I

OP METHOD. 81

need no demonstration, though they need expli- part r.
cation. The whole method, therefore, of demon- — r —
stration, is synthetical^ consisting of that order of
speech which begins from primary or most
universal propositions, which are manifest of
themselyes, 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- ^^efinitiong
nitions, whereof there are two sorts; one of primary,
names, that signify such things as have some con- pro^^ritl^g.
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 J 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 well 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

COMPUTATION OR LOGIC.

PART L the circumduction of a straight line in a plane, &c,

^ — ^ — Besides definitions, there is no other proposition

fn'iy"a«*'' ^^at ought to be called primar)% or (according

primary, ^o scvcre tmth) Ije received into the number of

«c universal ^ ^ ^

propoaitjona. 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
petifiom^ or po*^tnfatff, (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 theoremh^ which are speculations, but of
problems^ which belong to practice, or the doing
of something. But as for those common received
opinions. Nature fihhors vaadtif, Nature doth
tiotft!/ig hi rahi, and the like, which are neitber
evident in themselves, nor at all to be demon-
strated, and wliich 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 ; wliich 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.

ce ; which is against the scope and intention part r.
of demonstration, ^ — ^ — '

14. Now, seeinff definitions (as I have said) are The na*^*^

. . , . * . . , / & definition

pnnciples, or primary propositions, they are there- of a dcfimtion.
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 conceptions^ the
definition is nothing but a resolution of tliat name
into its most universal parts. As when we define
man, sa3^g man is a hody animated, seniient^
rational^ those names, body animated^ Sfc, are
parts of that whole name man ; so that definitions
of this kind always consist of genus and difference;
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
differetice are put together, and yet make no
definition ; as these words, a straight tine, 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 diflFerent 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

P^^T I. proposition^ tvhose predicate resolves the ^ffhjeet.

-^ wheti it may ,
the mme.

and when

Properties of 15^ The propcrtics of a definition are :

rirst, 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 uied about the name defined.

Secondly, that it gives an universal notion of
the thing defined, representing a certain universal
picture thereof, not to the eyCj 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
wlietlier 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 mil 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-
nothiug, then certainly the definition is
for the natm-e of a definition consists in
it exhibit a clear idea of the tiling defined;
les are either known by themselves, or
I e not principles,
ily, that, in philo§ophyi definitions are

OF METHOD.

before defined names. For in teaching philosophy, part t.
the first beginning is from definitions ; and all pro- - — r — -
gression in the same, till we come to the knowledge f^Xn'^fo*!!?^
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 ;
Day, when the names of the parts of any speech
be explicated, it is not necessary that the definition
1 ghould be a name compounded of them. For
I example, when these names, equilateralj quadri'
I kteraly right-angled^ are sufiiciently understood,
litis not necessary in geometry that there should
[be at all such a name as .square; for defined
aes are received in philosophy for brevity's
ake 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
efined ; as a parabola and an hifperhole have
bne definition in geometry, and another in rhetoric ;
/or definitions are instituted and serve for the
aderstanding of the doctrine which is treated of,
id, therefore, as in one part of philosophy, a
lefinition may have in it some one tit name for
be more brief explanation of some proposition in
>metry ; so it may have the same Uberty in
other parts of philosophy ; for the use of names is
particidar (even where many agree to the settling
them) and arbitrary.

Sixthly, that no name can be defined by any
ae word ; because no one word is sufficient for
resolving of one or more words.

Seventhly, that a defined name ought not to be
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 1^. Auy two definitions, that may be com^
dcmoDatrntion. poundcd ittto a syllogism, produce a conclusion ;
whichj 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-
fion h a sifllogimu, or series of sijliogisms,
derived mid continued ^ Jrom the defimimns of
names, to the fast eoficfmion. 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 aTroS€i£(c? and the Latins
demonstration was understood by them for that
sort only of ratiocination, in which, by the de«
e: of certain lines and figures, they placed
g they were to prove, as it were before
yes, which is properly awo^uKvifuv^ or to
the figure ; yet they seem to have done it
reason, that unless it were in geometry,
li only there is place for such figures)
no ratiocination certain, and ending in

OF MKTHOD-

PART L
6,

anence, their doctrines coiicemiiig all other things
being nothing but controversy and t laniour ;
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

b would be true,
17* It is proper to methodical demonstration, Properties o
First, that there be a true succession of one ItXordl^of
reason to another, according to the rules of syllo- ^?'"^* ^^ ^^
gizing delivered above-
Secondly, that the premises of aU 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 pkce those things be demonstrated,
which immediately succeed to universal definitions
(in which is contained that part of philosophy
which is called philoiiophki prima). Next, those
things which may be demonstrated by simple
motion (in which geometry consists). After
geometry, such things as may be taught or shewed
by manifest action, that is, by thrusting from, or
pulling towards. And fifter these, the motion or
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 ke])t in all sorts of
hilosophy, is evident from hence, that such things

COMPUTATION OE LOGIC.

TAET I.
6.

of a
demo list rat) an

as I liave said are to be taught last, caimot be de-
monstrated, till such as are propouuded 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 w^ork.

18* Besides those paralogisms, whose fault lies
either in the falsity of the premises, or the want
of tnie composition, of which I have spoken in
the precedent chapter, there are two more, wliich
are frequent in demonstration ; one whereof is
commonly called petit h principii ; the other is
the supposing of a JaJjie eauJie ; and these do not
only deceive unskilful learners, but sometimes
masters themselves, by making them take that for
well demoustrated, 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 demoustrated; and thus, by putting for the
cause of the thing sought, either the thing itself or
some eflFect 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 owai cause.

Of a J'a/tsc came 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, ^^^ ^'
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 roimd, 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
munoved, 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 ^y ,*^« *?*:
m this place of that art of the geometricians, of geometn-
which they call logistica, that is, the art, by^tJeated°of
which, from supposing the thing in question to be "^ ^" ^^^'
trae, they proceed by ratiocination, till either they
come to something known, by which they may

demonstrate the truth of the thing sought for ; or

to something which is impossible, from whence

i^c^im^eu^o^ ^^^y c^oWect that to be false, which they supposed

of g«3metri. true. But this art cannot be explicated here, for

ciaus cannot ^ • n i

be treated of this reason, that the method of it can neither be
IS p ce. pj-g^^^i^jgpjj^ jjpi- untlerstood, unless by such as are
well versed in geometry ; and among geometri-
cians themselves, they., that have most theorems in
readiness^ are the mo*st ready in the use of this
iogisiica ; 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 geometTician. It happens
besides, that for the finding out of equations, there
is no certain metliod, but he is best able to do it^
that has the best natural wit.

PART II.

TH£

FIRST GROUNDS OF PHILOSOPHY.

CHAPTER VII.

OF PLACE AND TIME.

1. 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. —
IS. 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 be ™e?o^ex-
annihilated. But, if such annihilation of all things »^°ce, may
be supposed, it may perhaps be asked, what would be understood
remain for any man (whom only I except from ^^ ^°°*p^^^*
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.^

PART IL

Thingi that
hmvt no ex-
istence, may
jievcTtheleM
be yiiderstood
■nd computed.

I say, therefore, there would remain to that man
ideas of the worlds and of all such bodies as he
had, before their annihilation, seen with his eyes^
or perceived by any other sense i that is to say,
the memory and imagination of magnitudes,
motions^ sounds^, colours, &c, 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 wdll
appear as if they were external, and not at all
depending upon any power of the mind^ And
these are the things to w^hich 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 tldnk 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 iSj be brought into account, either as
ntemal accidents of om- mind, in which manner
e consider them when the question is about
•e faculty of the mind ; or as species of external
^, not as really existing, but appearing only
ist, or to have a being without us* iVnd in
auner we are now to consider them.

OF PLACK AND TIME.

is Space.

2: If tlierefore we remember, or liave a phantasm part ir
of any thing that was iti the world before the
supposed annihilatiou of the same ; and consider,
not that the thing was such or such, but only that
it had a being without the mind, we liave 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
tliink 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 w hich is
once in it. And this is of itself so manifest, that
I should not think it needed any explaitiing at all,
but that 1 find space to be falsely defined by
certain philosophers, who infer from thence, one,
that the world is infinite (for taking Apace to be
the extension of bodieSj and thinking extension
may enerease 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 w orld were to be
ereatedj he says, that seeing there is nothing
Tvithout this world, and therefore (according to liis
definition) no space, that new world must be
placed in nothing ; but in nothing nothing can be
placed ; w hich he aflSrms 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*

PHILOSOPHY,

PART IT.

Time,

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 k the pffanifLHm of ft ffiinj; exkfing without
the mind simphf ; that is to say, that phantasm,
in which we consider no other accident, but only
that it appears without us.

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 AriHtotles 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 he gun ^ 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 he^ 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,

mutations made in our mind ? Time therefore ^^^J ^^•
is a phantasm, but a phautasm of motion, for if —-7^ — '
we would know by what moments time passes
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 jihan-
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 snccession
in the motion of a body, in as much as it is first
^bere then there. Wherefore a complete definition
time is such as this, time is the phantasm of
^fare and after in motion ; which agree^i wath
is definition of Arisfotle, time is the number of
QtioH aceording to former and latter : for that
^numbering is an act of the mind ; and therefore
. is all one to say, time is the number of ^notion
ecor fling to former and latter ; aiul 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,
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 fart, but that which is com-

ared with something that contains it.

5, And therefore to make parts^ or to part or
DIVIDE space or time, is nothing else but to con-
sider one and another within the same ; so that

Part.

DiviiioiL

Division.

Onp

* if any man dhhic space or time, the diverse
conceptions he has are more, by one, than the
parts he makes ; for his first 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
fon^ ards as loiijsj as he goes on in di\iding.

Bnt 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,

6. 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, a!id 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 coukl not be conceived
that there were another. The common definition
of one, namely, that one is that whieh is nmliridedy
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.

7. Number is one and one^ or one one and one^
and so forwards ; namely, one and one make the
number /iro, and one one and one the number
three ; so are all other numbers made ; w hich is
all one as if we should say, miml}er is unities.

pCoraposition. ^^ To COMPOUND spacc of spaces, or time of
times^ is first to consider them one after another.

Number.

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 ^ — r — '
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
divman 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 no
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 no soul can be a whole
sonl. 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 ybwr,
as two, is a part of the number eight ; ior four is
VOL. I. n

Spai!«t and
times con-
tiguaua and

Beg:iiii)}ng,
end, way

made of tarn and two ; but eight is oompnunded
of tivOy tivo^ m\AJmu\ and therefore two^ which
is a part of the part four^ is also a part of the
whole eight.

10, Two spaces are said to be contiguous,
when there is uo other space betwixt them. But
two times, betwixt which there is no other time, are
called immediatCj as A B, B C. .

And any two spaces, as well as

times, are said to be continual^ when they have
one common part, as A C, B D, ^ x^ p ^i

where the part B C is common; —

and more spaces and times are continual, when
every two which are next one another are
continual

i 1 , That part which is between tw o other parts,
finite> infinite, is Called a MEAN ; aud 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
tos:ether taken are the way. Also, extreme parts
and limits are the same thing. And from hence
it is manifest, that tie ginning and eml depend
upon the order in which we number them ; and
that to terminate or limit space and time, is the
same thin^ with imagining their heginning and
end : as also that every thing is finite or infi-
nite, according as w^e imagine or not imagine it
limited or terminated every way ; and that the
limits of any number are unities, and of these,
that which is the first in our numbering is the
fjeginning, and that which we number last, is the
end. When we say number is injinite^ we mean
only that no number is expressed ; for when we

OF PLACE AND TIME. 99

speak of the numbers two^ three, a thousand^ &e. part n.
they are always ^wi7^. But when no more is said ^ — r — -
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 he finite in power , what ii infi.
or terminable, when there may be assigned a Nothb|°i^fi^-
number of finite spaces or times, as of paces or ^y ^^ ^
hours, than which there can be no greater number ^« either whole

' ^ , or one; nor in-

of the same measure in that space or time ; and finite spaces^
u^nite in power is that space or time, in which '"^"'"^y-
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
Humbered more paces or hours than any number
that 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 be infinite, then there may be
taken in it some part which is distant from us an
iifinite number of paces : but no such part can
he 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 firom infinite, he determines the same, that
is, he makes it finite.
Of infinite space or time, it cannot be said that

H 2

PAFT n. it is a ivhole or one : not a tvhole. because not
" — i — ' 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 coueeived that there are two
spacesj 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 ima^ne,
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, Aud, therefore, that which is commonly

ToThc leMt! ^*^^t^i *l*^t space aud time may be 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 inlo 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
r* 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 ti.
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.

]. 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. — 1 1 . 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. — IS. 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,
m which we supposed nothing remaining without
us, but aU those things to be destroyed, that, by

PART IK existing heretofore, left images of themselves m
^-'-— ' 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 w ith 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 ^^ufMisting 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 bo€ly is thatj which having no dependance
tipon our thought, is coincident or coextended
with some part of space.

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 renio\aag 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
e same manner imderstand him that says any

Accident
lie fined.

OF BODY AND ACCIDENT,

103

Wng Is extended, or moved, or not moved ? But part il

most men will have it be said that {in accident h - — ^ — '

mmHhing, namely, some part of a natural thing, ^Xld!'

when, indeed, it is no pait of the same. To satisfy

these men, as well as may be, they answer best

that define an accident to be t/te manner by which

untj body is conceived ; which is all one tis if they

should say, an accident is that f acuity of any

^Kior/^, by which if works in us a conception of

^mkself. Which definition ^ though it be not an

^■inswer to the question propounded, yet it is an

Hmswer 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 w ell answered thus : it happens

from the extension of that body. Or, how comes

it to pass that the whole body^ by snccessionj is

seen now here^ now there ? and the aiisw er w ill be,

by reason of its motion. Or, lastly, whence is it

that any body possesseth the same space for

tometime ? and the answer w ill be, because it is

not mored. For if eonceniing the name of a

body, that is, concerning a concrete name, it be

! asked, what is it ? the answ er must be made by

definition ; for the question is concerning the

^lignification of the name. But if it be asked

Hisonceming an abstract name, what is it ? the

Hteuse is demanded why a thing appears so or so.

H^ if it be iisked, what is hard? The answer

1 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

How an aici-
df ut inty be
unflerstoocl
to bu in its
subject.

TAKT iL whole give place. Wherefore, I define an accident
" — • — to be the mifuner of our conception of body,

3. When an accident is said to be in a hody, it
is not so to be understood, fu* if any thing were
contained in that body ; a.s if, for example, redness
were in blood, in the same manner, as l>lood is in
a bloody clothj that is, as a part in the whole ;
for so, an accident would be a I)ody also. But, as
magnitnide, or rest, or motion, is in that which is
great, or w^hieh resteth, or which is moved^ (whieli,
how^ it is to be understood, every man understands)
so also, it is to be understood, that every other
accident /*y in its subject. And this, also, is
explicated by Aristotle no otlierwise than nega-
tively, namely, that an accident is in its suhjeci,
not fM" any part thereof^ but so as that it may be
aicay^ the subject still remaining ; which is riglit,
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, £is to /)e at rest, to he mored^ 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 eJLtension, 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

la iiniiiov

liable*

bv ratiocination, whether these verv accidents are part h.
ant also certain motions either of the mind of the ^ — r — -
pereeiver, or of the bodies themselves which are
perceived ; for in the search of this, a great part
of natnral phihisophy consists.

4< The extenalon of a body, is the same thing Mag^iiiude,
with the magnitude of it, or that which some call
real space. But tliis magnitude does not depend
u|ion our cogitation, as imaginary space doth ; for
this h* 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 w ord I here u!iderstand ^^^^^^*> ^^^^^^^ ii
imaginary space, which is coincident with thei
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, tliat 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 sucli 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, i]esides,^i/f/6'£' is immovahle ;
for, seeing that whi(*h is moved, is understood to
be carried fiom ]ilm*e to place, if place were
moved, it would also be carried from i)lace to

PART IL place, so that one place must have another place,
and that place another place, and so on infinitely,
f ii, md that it which is ridiculous. And as for those^ that^ by
h immovable, making pkice to be of the same nature mth 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 immovaf>le, because space in general is
considered there ; if he had remembered that
nothing is general or miiversal besides names or
signs, he would easLly have seen that that space,
which he says is considered in general, is nothing
hut a phautfusm, in the mind or the memory, of a
body of such magnitude and such figure. And
whilst another says: real space is made immovable
I)y 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 peri)lexed words? Lastly, the
nature of pltiee does not consist in the superjicies
of the ambient y but in solid npace ; for the whole
placed body is coextended with its whole place,
and every part of it with evei7 answering part of
the same place ; but seeing every placed body is a
solid thing, it cannot be understood to be coex-
tended with supeiiicies. Besides, how c<an 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 jiarts of a body cannot leave the

OF BODY AND ACCIDENT.

107

superficies of an external part eoiitiguons to it ; PAnT n
and, therefore, it follows, that if place be the — -^.^^ —
mperficies 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, ^Ij'^^*^^'*'
is Ciilled full, and that which is not so possessed,
Is called empty,

7* Here, there^ in the country^ hi the city^ and Ji^r^i ^^^^''^
other the Mke names, by w hich answer is made to wiiat they *
the question where h it ? are not properly names *^*^"' ^'
of place, nor do they of themselves bring into the
mind the place that is sought ; for here and there
signify nothing, nnless the thing be shewn at the
same time with the finger or something else ; but
[ivhen the eye of him that seeks, is, by pointing or
>me other sign, directed to the thing sought, the
place of it is not hereby defined by him that
{inswers, but found out by him that asks the ttues-
tion. Now such shewings as are made by words
only, as when wt say, in the country^ or rVi the
itity, are some of greater latitude than others, as
iirhen we say, /// the country^ in the cify^ in such a
itreet^ in a house^ in the chamher, in heil, &c-
For these do, by little and little, direct the seeker
nearer to the proper place ; and yet they do not
detennine the same, but only restrain it to a lesser
ispace, and signify no more, than that tbe place of
■the tiling is within a certain space designed by
those words, as a part is in the whole, ilnd all
such names, by which answer is made to the ques-
tion where ? have, for their highest gennn^ the
name somewhere. From whence it may be under-
I stood, that whatsoever is somewhere, is in some
place properly so called^ wliich place is part of

108

PHILOSOPHY.

muny p J aces.

PART I J. that greater space that is signified by some of these
^^ — T — ' names, iu the country^ in the eitif, or the like.
Many bodies g, A bodv, and the raaffnitude, and the place

ciDnot be 111 i* * i i i i i

one place, nor thereof, are divided by one and the same act of
* ^ *" 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.

9. Two bodies are said to be contiguous to one
another, and contimial, in the same maimer as
spaces are ; namely, thoHe are contiguousy between
which there is no space. Now, by space I under-
stand, here as formerly, an idea or phantasm of a
body. Wierefore, 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

ConLiguous
and cotitifiual,
wfant they are*

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 ^ — ^
tliat affect metaphysical subtleties wander from
truth, as if they were led out of their way by an
\^is 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 Thcdeflnitioii

7 _7 •• /» -f j^i. of motion. No

me place, and acquiring of another ; and that motion inteUi-
plaee which is relinquished is commonly called the ^"e.**"' '''^^
terminus a quo, as that which is acquired is called
the terminus ad quern ; 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 \ n xi J v^ ^

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,

whose part G H B D 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-

PART 11.

What it is
to be at rest,
to bave been
moved, iLnd
to be mo?€tL
No motion to
be conceived
without the
coniceptiou of
paat and future.

ceived that any thmg can be moved Tvitbout time;
for time is, by the definition of it, a phantasm, that
is, a conception of motion ; and, therefore, to con-
ceive that any tiling may be moved without time,
were to conceive motion without motion, which is
impossible,

1 1 . T/ffit M' said to be at rest, which, during
any time^ is in one p/ace ; and thai to tie niorrd^
or to hare hern moved^ which^ whether it in* now
at rest or mo red, iras Jhrmrrlif in another place
than that which it is now in. From which defini-
tions it may be inferred, first, that whatsoever i»
mored, has fjeen moved ; for if it be still in the
Sfirae place in which it was formerly, it is at rest,
that iSj it is not moved, by the definition of rest ;
but if it be in another place, it has been moved,
by tlie definition of moved. Secondly, that what
is moved y 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. Tliirdly, that whatsoever is
moved, is not in one place during any timCy 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 he moved ^ it is moved either in the place
where it is^ or in the place where it is not ; both
whicfi 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 ^^^^'^ ^^

not. Indeed it caunot 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 sand 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, w^hen any body is moved, ^'^ solid!'
tlie magnitude of it be not at all considered, the «'^at they are.
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 w^hich sense the earth is called a pointy
and the way of its yearly revolution, the ecliptic
line. But if a body, which is moved, be considered
a8 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 KHperfieies^ consisting of two
dimensions, erne whereof to every several part of
Ae 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 soUd^
consisting of three dimensions, any two whereof
are applied whole to every several part of the
third.

112

PHILOSOPHY,

PART IL
8,

^ Id bodies
and ma^tii-
todes, wbal
they are.

^^VOae and th&
^r lame body
h&a Always one
*ud the iaiue
tnagmtude.

But if a body be considered as soltd^ then it is
not possible that 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
it is a body, than the three which 1 have now-
described ; though, as it shall be shewed hereafter,
reheity, which is motion according to lengthy
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.

13* BodicH^ how many soever they be, that
can fill every one the place of every one, are said
to be equal every one to every other. Now, one
body may fill the same place which another body
fiUeth, 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 tlie parts. And one
body iH greaier than another body, ichen a part
of thai is equal to all thi.s ; and iejfjs, when all
that /*¥ equal to a part of this. Also, magnituden
are equals or greater^ or lesser, than one another,
for the same consideration, namely, when the
bodies, of which they are the magnitudes, are
either equals or greater, or less, &c.

14. One and the same body is always of one
and the same magnitude* For seeing a body
and the magnitude and place thereof cannot be

OF BODY ANI> ACCIDENT.

113

'e coinoident, if any body he understood to be at part il
that iSj to remain in the same place during - — r — ^

le time, aud the magnitude thereof be in one
KTi of that time greater, and in another part less,
bat body s place, which is one and the same, will

coincident sometimes with greater, sometimes
with le^ 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 magnitnde, and have greater or less mag-
nitude bestowed upon it, making use of this
principle for the explication of the nature of rarum
and densum,

15. Motion, in as much as a certain length may ^f^"^\^>

^IB a certain time be transmitted by it, is called

^P^KLociTY or swiftneHs : &c. For though swift

Hbe very often understood with relation to jslmver

Qt kss swifts as great is in respect of less, yet

^UPvertheless, as magnitude is by philosophers taken

olutely for extension, so also velocity or smift-
^** may be put absolutely for naotion according to
ength.

15. Many motions are said to be made in equal ^^i"*^^
times, when every one of them begins and ends le
together wnth sonu* other motion, or if it had
W^u together, would also have ended together
^'ith 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 son or of
the hand ; and if two or more motions begin and
end Tvath this motion, they are said to be made in

VOL. I. I

reatLT, and
e&Sf in timet,
what they are*

PART n. equal times ; from whence also it is easy to under-
^' — -^ — 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, be^an
sooner.
EqiiAi greater. J 7. Motions 3X6 SEid to be eonaUv swift, when

and ksa, m ve- . , . , . ,

locitj, whau eqnal lengths are transmitted in eqnal 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 nnj/orm
swiftness or motion ; and of motions fioi 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 iiHiJormly.
Erjiiai, grtatcr, 18. But motiou is said to be greater, less, and
motionrwiiaf- ^^^^^s ^^^ ^^^J ^'^ Tcgurd of the length which is
transmitted in a certain time, that is, in regard of
swiftness only, but of swiftness appUed 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 t
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 eqnal to one another, and another thing for
them to be equally swift. And this is manifest m
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 tlie
motion of both is greater than the motion of one

OF BODY AND ACCIDENT.

115

of them, namely, double. Wherefore motions are
mid to he simply equal to one another^ when the
9wjfiness qf one^ computed in every part of* its
magmtude^ is equal to the swiftness qf the other
wmputed also in every part of its magnitude:
imd greater than one another^ when the swiftness
of one comptited a^ above, is greater than tfw
miftness qf the other so computed ; and less,
when less. Besides, the maguitude of motion
computed in this manner is that which is commonly
called FORCE,

19. Whatsoever is at rest^ will always he at
rent^ unless there he some other fmdy besides it^
whichy by endeavouring to get into its place by
motion^ steers it no longer to remain at rest.
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 auy 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 reai^on of its motion one way would be the
same with the reason of its motion every other
way, wherefore it would be moved dike ail ways
at once ; which is impossible.

In like manner, whatsoever is viovefl, will

ulways he moved ^ except there be some other body

Ihesides it, which causeth it to rest. For if we

suppose nothing to be without it, there will be no

easoQ why it should rest now, rather than at

1 2

PART n.

That which
i» &t rest will
atwuys b« «td
rest, except it '
be moved bf
sotneextemu
thing.

That which is
moved will al-
ways be mot ed,
ualesa it I>C! hin-
dered by Mome
external tkiiig.

another time ; wherefore its raotion would cease
in every particle of time alike i which is not
inteUigilile.

Accidenu ire 20, When wc sav a living creature, a tree, or any
d^^dvy«d.b"at other specified body is generated or destroyed^
"^* ^ it is not to be so understood as if there were made
a body of that which is not-body, or not a body of
a body, but of a liiing creature not a living crea-
ture, of a tree not a tree, &c. that is, that those
accidents for which w^e call one thing a liHng
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 oiur mind
that a point may swell to a huge bulk, and that
this may agaui 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 uot-man ; but that which is once
called body, can never be called not-body. But it
is manifest, that all other accidents besides magm-
tude or extension may be generated and destroyed ;

BODY AND ACCIDENT.

as when a white thinsr is made black, the whiteness i*art ii,
that W&5 in it periisheth, and the blackness that ^- /— *
was not in it is now generated ; and therefore
bodies, and the accidents under whieh they appear
diversely, have this difference, that bodies are
things, antl not generated ; accidents are generated,
and not things.

21- And therefore, when any thing appears ^^"^ '^'^^,^*^'''^^
otherwise than it did by reason of other and other fromiumbjfct
accidents, it is not to be thonght 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 ma<ster of a family in his house,)
but that one accident perisheth, and another is
g:enerated. 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
accident is moved ; as when, instead of saying,
figure is an accident of a hoily curried ftwmj^ we
say, a body carries away its figure.
I 23. Now that accident for which we give a
Lxertain name to any body, or the accident which
Htoenominates its subject, is commonly called the
f ESSENCE thereof; as rationality is the essence of
^u man ; whiteness, of any white thmg, and exten-
^hiou the essence of a body. And the same essence,
^■n as much as it is generated, is called the form.
^P^gain, a body, in respect of any accident, is called
^^the SUBJECT, and in respect of the form it is
called the matter.

Nor be moved.

E»e)itcef fonnp
and matt eft
what thoj are.

•k

118

PHILOSOPHY.

Pirst mat'
lcr» whaL

PART II. Alsoj the production or perishinsf of any accident
^— >A-^ 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 MetaphynicH^ says,
that whatsoever is made of any thing ought not to
be called Unvi)^ but fViivivov; as that which is made
of wood, not £uXov, but SiXivov, that is, not wood,
but wooden. ^

24. And as for that matter which is common to
all things, and which philosophers, following Aris-
totle, usually call materia primal that is, first
matter^ it is not any body distinct from all other
bodies, nor is it one of them. \\Tiat 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 geaeraJ^
if we use that of materia prima^ we do well. For
as when a man not knowing which was firsts
water or ice, would find out which of the t\^ o 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. Wlierefore 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
^ir forms and accidents certain.

OF BODY AND ACCIDENT. 119

Materia prinuiy therefore, is body in general, ^^^^ ii.
that is, body considered universally, not as having ' — ^
neither form nor any accident, but in which no
form nor any other accident but quantity are at all
considered, that is, they are not drawn into argu-
mentation.

25. Prom what has been said, those axioms may ,?^ili:£
be demonstrated, which are assumed by Euclid in »;y p*rt there-

of, why demon-

the beginning of 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
leam 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.

PART IF.

OF CAUSE AND EFFECT.

1< Action and passion, what they are. — 2, Action and passion
mediate and immediate. — 3, Cause simply taken* Cause
without w iiich no effect folloivs, or cause necessary by sup-
position. ^ — A. Cause efficient and inatedaL — 5, An entire
cause, h always sufficient to produce its effect. At the same
instant that the cause is entire, the effect is produced. Every ■
eiFeet has a necessary cause.^6- The generation of effects is I
continual. What is the beginning in causation. — 7* No cause
of motion but in a body contiguous an«i nioved.^ — 8. The same _
agents and patients, if alike disposed, produce like effects ■
though at different times. ^ — 9, All mutation is motion*
lO. Contingent accidents, wlmt they are.

1. A BODY is said to work upon or aef^ that is to
say, do something to another liody, when it either
generates or destroys some accident in it : and the

Actio u
and paacian,

vhat they ire* body ill whirh an accident is generated or destroyed
is said to stiff er^ 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 J
called the agent ; and the body in wliich motion
is so generated, is called the patient ; so fire that
wai'ms 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.

2. When an agent and patient are contiguous to

one another^ their action and passion are then said

to be immediate, otherwise, mediate ; and w hen I

another Ijody, lying betwixt the agent and patient,

Js contiguous to them both, it is then itself both an

OF CAUSE AND EFFECT.

121

agent and a patient; an agent in respect ot tlie
body next after it, upon which it wt)rks, and a
patient in respect of the body next before it, from
whieh it suffers* Also, if many bodies be so
ordered that evtTy two which arc next to one
another be contiguous, then all those tliat 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 prmhfce its deter-
mined or certain eifect in the patient, according to
Jome 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
ahke, would produce like effects in all patients.
And therefore the fire, for example, docs not warm,
because it is a body, but because it is hot ; nor
does one body put forward anotlicr body because it
is a body, but because it is moved into the place
of that other body. The cause, therefore, of all
pfieets consists in certain accidents both in the
agenti? and in the patients ; which when they are
all present, the effect is produced ; but if any one
of them be wanting, it is not produced; and that
accident either of the agent or patient, without
which the effect cannot be produced, is called
muaa mne qua mm^ or eause neces^Hury by mp-
position^ as also the e&ihse req/iLsife Jor the pro-
diiction of (he effect. But a cause simply, or an
mtire cauaej m the aggregate of all the aeekientH
ih of the agent H how many soever they f^e, and
af ttte j/atient^ put together ; whiek when they

of the PART TL

Cause simply
tiiken.

Ciiusc? without
Hhich 110 effect
folio w»| or
eau5ieneceji!»ary
by Mippujiiiiofl.

are all supposed to he preHent^ if cmnioi he imder-
" '— ^ stood but that the effect Ls' produced at the same
instant ; and if any one of them he wanting^ it
cannot be understood hut that the effect is not
produced.

^d materia *" ^' ^^^ aggregate of accidents in the agent or
agents, requisite for the production of the eflfeet,
the eflfeet 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 &ay the effect being
produced ; for where there is no effect, there can
be nu 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, wbieh
in the next precedent article I called an entire
cause. And from hence it is manifest, that the
eflFect we expect, though the agents be not defective
on their part, may nevertheless be finistrated by a
defect in the patient ; and when the patient is
sufficient, by a defect in the agents.

AnEniirecause 5^ ^u entire causc is always suflScient for the

18 alwitya tuf- *i* \ rr* i_

flcieiit to pro- productiou of its effcct, if the effect be at all
'''^^' possible. For let any effect whatsoever be pro-
pounded to be produced ; if the saine 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.

It follows also from heuce, that in whatsoever
instant the cause is entire, in tht* same instant the
effect is produced. For if it be not produced,
ethinp: h still wanting^, which is requisite for
tie production of it ; and therefore the cause was
not entire, as was supposed.

And seeing a necessary cause is defined to be
that, which being supposed, the effect cannot but
lUow ; this also may be collected, that whatsoever
■pct is produced at any tiuie^ the same is produced
a necessary cause. For whatsoever is produced^
in as much ns it is prodxiced, had an entire cause,
that is, had all those thin^, which l)ping supposed,
it cannot be understood but that the effect fol-
l«ms ; that is, it had a n<*cessary cause. And in the
same manner it may be shewn, that whatsoever
■ects are hereafter to be produced, shall have a
ecessarv' cause ; so that all tht^ effects that have
\ beni, or shall be produced, have their necessity in
[ thiups antecedent.

^H 6. And from this, that whensoever the cause is

^entire, the effect is produced in the same inst^int,

^jt is manifest that causation and the production

^K^ effects consist in a certain continual progress ;

80 that as there is a continual mutation in the

ap^nt or agents, by the w^orking of other agents

npon them, so also the patient, upon whicli 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

^l^at of such bodies as are next to it, and again, of

^Bieh other bodies as are next to them, increjise

more and more accordingly ; which is already no

little arerument that all mutation consists in motion

PART IL
9,

Attlie same in-
Btant that the
cause is entire,
the effect is pro-
duced*

Erery effect
has a nccct*
sary cauae.

The izenera-
lion of effects
18 contiyuaL
What 15 the
beginning in.
cauAation.

oaly ; the tnitli whereof slmll be further demon-
strated in the ninth article. But in this prog-ress
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.

7* There can be no cause of motion, except in a
body contiguous and moved. For let there be
any two bodies 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- i'art ir.
p^ate of all such accidents, which being; supposed — ^-^
to be present, it cannot be conceived but that the
effect wiU 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, which is at rest,
there can be no cause of motion. Wherefore there
k no cause of motion in any body, except it be
contiguous and moved.

The same reason may serve to prove that what*
Merer is moved, will always be moved on in the
same way and with the same velocity, except it
be hmdered 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
lias written that things moved are more resisted
by things at rest, than by things coutrarily moved ;
for this reason, that he conceived motion not to be
so contrary to motion as rest. That w hich deceived
him was, that the words re.^t and mollon are but
coiitradictory names; whereas motion, indeed, is
not resisted by rest, but by contrary motion.

8. But if a body work upon cinother body at one Tin? same
time, and afterwards the same body work upon the paSrsf
name body at another time, so that both the asrent ^'^J^"^ ^'sp^-
and patient, and all their parts, be in all things as ^*ke efn^cts,
they were ; and there be no difference, except only hxtnx times. ' ,
in time, that is, that one action be former, the

126

FHILOSOPHY,

PART IL
9.

All mutation
U motion.

igcnt

otlier later in time; it is manifest of itself, that the
efferts will be equal and like, as not diflfering in
anything besides time. And as effects themselves
proceed from their causes, so the diversity of them
depends upon the diversity of then* causes also.

9. This being true, it is necessary that mutation
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 w liich appears to our
senses otherwise than it appeared formerly. Se-
condly, both those appearances are eflFects pro-
duced in the sentient; and, therefore, if they be
(hfferent, it is necessary, by the preceding article,
that either some part of tlie agent, which was for-
merly at rest, is now moved, and so the mutation
consists in this motion ; or some part, which w^as
formerly moved, is now otherwise moved, and so
also the mutation consists iu 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 figain, mutation is
motion ; or lastly, it happens iu 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, eitlier 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 jjroceed from it ; seeing
neither motion nor mutation can be caused by it.

10. Accidents, in respect of other accidents
which precede them, or are betcjre them iu time.

OP POWER AND ACT. 1 27

and upon which they do not depend as upon their part ii.
causes, are called contingent accidents ; I say, in ' — A-'
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.

OP POWER AND ACT.

1. Power and cause are the same thing. — 2. An act is prodaced
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 effecty are Power and
POWER and act ; nay, those and these are the same thing. *
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 eflFect in the
patient, then we say that agent has power to pro-
duce that eflFect, if it be applied to a patient. But,
as I have shewn in the precedent chapter, those
accidents constitute the efl&cient 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 diflFerence, that cause is

»AaT II. so called in respect of the effect already produced,
— -r^ and power in respect of the isame effect to be pro-
duced hereafter ; so that eanse 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 itj 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 patieftty commoidy called passive power^
and materhil eaww^ are the same thing ; but with
this different consideration, that in cause the past
timCj 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 accideuti?, 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 au effect, so in respect of the
power, it is called an act.
itrtiipro- 2. As therefore the effect is produced in the
same instant in which the cause is entire, so also
eiy act that may be produced, is produced in the
e instant in which the power is plenar\ . And
sre can be no effect but from a sufficient and
<ary cause, so also no act can be produced but
lent powder, or that power by which it
>t but be produced*

OP POWER AND ACT. 129

3. And as it is manifest, as I have shewn, that partii.

10,

the efficient and material causes are severally and — ^ — '
by themselves parts only of an entire cause, and ^^{y/^^et
cannot produce any effect but by being joined ^® p"^ °°^y
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 , \f it he 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 produc- 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 thmgs 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 ncces-
whereof it is impossible to hinder ; and therefore ungent, whTC

VOL. I. K

PART II. every act, that shall be produced, shall necessarily

^ be produced ; for, that it shall not be produced ^ is

M^y^nrcon* inipossible ; because, as h already demonstrated,

dngent, what, evefy possiblc act shall at some time be produced ;

uay^ this proposition, zvkat j^/iaii he^ .shall be^ is as

necessfiry a proposition as this, a man is a man.

But here^ perhaps, some man may ask whether
those future things, which are commoidy called
eonfhff^epif.^^ 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 eaj^iml or eon tiitgent^ \yhereof
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 no,
they say it is possible it never was done.

Wherefore, all propositions concerning future
things, contingent or not contingent, as this, it
will rain tomorrow^ or this, to marrow the funt
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 clauses. But
there are some, w^ho though they confess this %vhole
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 rainy or, tomorrow

OF POWER AND ACT,

131

PART

10.

it will not rain, to be either of tliem tnie by it.^elf ;
because they say neither this> nor that is true defer-
miftafeftf. But what is this detenniimiely frue^h\it
true upon our knowledge, or evidently true t And
therefore they say no 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 w ords, with wliich they endea-
vour to hide their own ignorance.
C, In the 9th article of the precedinsc chapter. I A^^^^^-e power

i_ 1 1 1 rt- * i 11 * consists iu

nave shewn that the emcrent cause or 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-
stoodj that all active power consists in motion also ;
and that power is not a certain accident^ which
thffers from all acts, but is, indeed, an act, namely,
motion, which is therefore called power, because
another act shall be produced by it af^terwards.
For example, if of three bodies the first put
fon^ard 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
«mie second body.

7. The writers of metaphysics reckon up two cauae, fommi
other causes besides the efficient and material^ what they are.
namely, the essence, which some call \\w formal
eame, and the end, or Jinal eat(*He ; both which
are nevertheless efficient causes. For when it is
Haid the essence of a thing is the cause thereof, as
to be rational is the cause of man, it is not intel-
ligible ; for it is aU one, as if it were said, to be a

K 2

PART 11.

man is the cause of man ; which is not well said.
And yet the knowledge of the emence 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. Kjinal came 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 XI.

OF IDENTITY AND DIFFERENCE,

1. What it is for one thing to difier from another. — % To difFor
in number, magnitude, speci*^4^, and genus, what.- — i\. What is
relation, proportiou, and relatives. — ^Is Proportionals, what* —
5* The proportion of magnitudes to one another, wherein it
consists.^^. Relation is no new accident, hut one of those
that were in the relative before the relation or eomparison wa»
made. Also the causes of accidents in the correlatives, are the
cause of relation,^?. Of the beginning of individuation.

whaiitis 1, Hitherto I have spoken of body smiply, and
lo dider frwm accidents common to all bodies^ as 7nagnitn(h\
motion^ rest, fief ion j passion ^ power ^ possible, S/'C. ;
and I should now descend to those accidents by
which one body is distinguished from another, bat
that it is first to be declared what it is to be dis-
tinct and not distinct, namely^ what are t-he same
and DIFFERENT; for this also is common to all
bodies, that they may be distingnished and differ-
enced from one another. Now, two bodies are
said to differ from one another, w^hen something
may be said of one of them, which cannot be said
of the other at the same time.

another.

1, it is manifest that no two part il
bodies are the same ; for seeing tliey are two, they ^ — ^ — -
are in two places at the sanit* time ; as that, which is i^j^ntmber,
the same, is at the same time in one and the same n^»«^it^ti«^.

tpecies, and

place. All bodies therefore differ from one another genus, what.
in number^ namely, as one and another ; so that
the ^same and il\fferent in number^ are names
opposed to one another by contradiction.

In tnagnitude bodies differ when one is greater
than another, as a cubit hii*j;^ and two aihits hmfi^,
of two pound weight, and of t/iree pound iveight>
And to these, equals are opposed.

Bodies, which differ more than in magnitude, are
cidled 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 y when their difference is perceived by one
and the same sense, as wfiife and hiack ; and in the
fftnus, when their difference is not perceived but
by divers senses, as wkite and hot.

3, And the likeness^ or unfikeness, equality ^ or AVhfttis
inequalitij of one body to anotlier, is called their po^prrtion,
RELATION ; and the bodies themselves relatives or *"'^ '*®**^'^®**
correlatives ; Aristotle calls them ra irpo^ rl ; 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 niagui-
tude of the consequent by their difference only,
or compared also with their difference , For ex-

Proporlion.'
als, what*

PART II. ample, the jwoportion of three to two consists
- — r^ only in this, that three eiceed^ tw o by unity ; and
the proportion of two to five in tliis, that two,
compared with five, is dejicieut 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,
4. Besides, of unequals, some are more, some
lessj and some equally unequal ; so that there is
proportion of proper tioiu\ as well as of magm~
tudes ; namely, where two imequals have relation
to two other unequals ; as, when the inequality
which is between 2 and 3, is compared with the
inequality wliich 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 proportionah ; otherwise they are not pro-
portionals.
Thepropoi- 5, The proportion of the antecedent to the con-
nitudwTo*^' sequent consists in their difference, not only
whei^Q k*''^* ^™P^y ^^^^1 t>^t also as com]>ared 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 thi'ee, yet there is not the same inequality ;

OF IDENTITY AND DTFFEEENCE.

and therefore the proportion of two to five is not
in all relation the same with. that of nine to twelve,
but only in that which is called arithmetical.

6. But we must not so think of relation, as if it
were an accident differing from all the other acci-
dents of the relative ; but one of them, namely,
that by which the comparison is made. For ex-
ample, the likeness of one white to another whiie^
or its unlikeness to blacky is the same accident
with its whifeneHH ; and equalily and inequalifffy
the same accident w ith tlie magnitude of the tiling
compared, though under another name : for that
which is called white or ^reat^ w hen it is not com-
pared with sometliing else, the same when it is
compared, is called like or unlike^ etjual or uu-
equuL And from this it follow s that the causes
of the accidents, which are in relatives, are the
causes also of likeness^ unUke)u\sH^ equalitf/ 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 ride, their conginiity ;
if incongruous^ their incongruity. And thus much
concerning coinparisoH of one body with another.

7- But the same body may at d liferent times be
compared with itself. And from hence springs a
great controversy among philosophers abtmt the
heginuing of indiinduationy namely, in what sense
it may be conceived that a body is at one time the
same, at another time not the same it was formerly.
For example, whetlier a man grown old be the
same man he was whilst he was young, or another
nun ; or whether a city he in different ages the
8amej or another city. Some place individuity m

PART n.

H«]a(ion it no
new ftccid«rnt,
but one of those
Cba^wereinllie
relative, before
the reUtJoD or
com parison wm
made. Also the
causea of acci-
dents in corre-
latives xrt the
cauae of relft-
tion*

Of the begin.
liing of iradi-
vidtiattoii*

PART I L
11.

Of the beg^n*
ning of indi-
viduation.

the unity of 7natfer ; others, in the xmity Qi/ormX
and one says it consists in the unity of the aggre-
gate of all the accidents together. For matter^
it 18 pleaded that a lump of wax> whether it be
spherical or cubical, is the same wax, because the
same matter. For Jbrniy that when a man is epro\^^
from an infant to be an old man, though his matter
be changed, yet he is still the same numerical
man : for that hfentftf/ywhich cannot be attributed
to the matter, ought probably to be ascribed to the
form. For the aggregate of aeeidentSj 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
hiditfdtilti/y 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 chiference 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, wTre, 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 it.

11.

and so there would have been- two ships numerically

the same^ which is absurd. But, according to the ninp^of^j^-'

third opinion, nothing would be the same it vitiuation.

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 diflFerence 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,
^ — r^— whether the same water, or other water, or some*
Sa^onSdi"' thiiig else than water, flow from thence ; and that
TidiMOon. the same city, whose acts proceed continually from
the same institution, whether the men be the same
or no. 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 ¥rill
be partly the same^ and partly not the same.

Of^JMIlCity.

CH.VPTER XIL

OF ^rAXTlTY.

1. The dtiliiiidi>Q of ijuaiicicy. — ^i. T^e exposidoa of qoanthj,
what it isk — 5» How line* superticies* and solid, are exposed.
♦. How time i$e\poi$ed. — 5» How number is e3q>o6ed. — 6. How
vebettT is expoeed.— T. How weight ise^iposed — S. How d^
proportioa of ma^niitudes b e^Lputwd.— 9- How the proportioii
^ timeti and Yekxruiftis is e:Lpu>ed.

MuhM 1. What and how manifold dimtrmsi4m i&, has

beea said in the ti^ chapter, namely, that thare are

iwiisioasy line or Ien^:th> superficies, and

Vf cMie (tf whkh,. if it be determined, that

JiitH of it be QMide known, is eommoiily

tmtifg: f» hf ^/mmmtitf allmeii under-

OF QUANTITY.

139

PART in

stand that which is signified by tliat word, by
which answer i^ made to the question, IloiP mnch
is it? Whensoever, therefore, it is asked, for
example, How long is the Joume;/ ? it is not
answered indefinitely, length ; nor, when it is
ajsked, How big is the field? is it answered inde-
finitely, .siiperficirs ; nor, if a man ask. How great
is the bulk ? indefinitely, solid : but it is answered
determinately, the journey is a hundred miles ; the
fieUl is a hundred aeres ; the bulk is a hundred
cubical feet ; or at least in some such manner, tlmt
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-
swn, whose limits are set aut, either by their
pheey or by some comparison,

2. And quantity is determined two ways; one, Tite expoii-
by the sense, when some sensible object is set tity! whru is.
before it \ n^ 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
m 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 luaiiy thousand paces ;
■tbat is, by comparing the way with a pace, or some
other measure, determined and known by exposi-
tian ; or the quantity of it is to some other quan-

vrlmt U Ja.

PARTiL tity known by exposition, as the diameter of a
^ — ^-^ square is to the side of the same, or by some
JmXtyr ^"^ ^^'^^^ ^^^ ^^^ means. But it is to be understood,
that the quantity exposed must be some standing
or permaTient thins:, sueh bs is marked out in
consistent or durable matter ; or at lea.st something
which is revoeaWe 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 with 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 tlie equality or inequality thereof
w ith au exposed quantity.

3. Lines, sitperficicfi^ and soIt(hy are exposed,
first, by vmfion^ in sucli manner as in the 8th
' chapter I ha^T said they are generated ; but so as
that the marks of such motion be permanent ; as
w hen 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 thit*kness ; 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 arc to be understood very short
lines ; and, by superficies, very thin solids.
Thirdly, lines and snpeiiicies 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-

How line,
«uperficie]S^
and aoHcls,
are exposed.

OF QUANTITY,

4- Time is exposed, not only by tlie exposition twkt
of a line, but also of some moveable thiiisr, which ^ — ^

II.

is moved uniformly upon that line, or at least is i> ^jTMSed*
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
ex^iosition of time that a line be described ; but
we must also liave 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
^aid. Let the conception of unijonu motion upon
that line^ be time. For though the circles in dials
he 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 Huw number
of points, or of the names of number, ofH\ two, '^ ^^^"^^^ '
three ^ ^'c. ; and those points must not be conti-
jE^uous, so as that they cannot be distinguished by
notes, but they must be so placed that they may
be (tiscerned one from another ; for, from this it
in, that number is called di.screet quautiti/ ,
whereas all quantity, which is designed by motion,
is called contimml qmtntitif. 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
munber we are at beyond two or three ; which
also appear to us in tliis manner, not as number,
but as figure.

PART II,

How velocity
u exposed.

How wdgbt
IS exposed.

How the pro-
portion of
magnitudes
15 exposed.

6. For the exposition of reheihj^ wliich, by the
definition thereof, is a motion which, in a certain
tinie^ passeth over a certain space, it is requisite,
not only that time be exposed^ but that there he
also exposed that space which is transmitted by
the body, whose velocity we 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 i^

we would expose the velocity of the C D

body A, we draw^ two lines A B

and C Dj 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, ot
what matter soever^ so it be always alike heavy.

8. The proportwu of two masrnitudes is then
exposed, when the magnitudes themselves are ex-
posed, namely, the proportion of equality, when
the magnitudes are equal ; and of inequality, w^hen
they are unequaL For seeing, by the 5th article
of the preceding chapter, the proportion of two
unequal magnitudes consists in their difi'erence,
comimred 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 eqnals,

G F

OF QUANTITY. 143

which consists in this, that there is no diflFerence part ir.
of magnitude betwixt them, is exposed at the — ^r^ — '
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
be unequal, the proportion which
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 no
cMFerence ; 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- How the pro-
form velocities, is then exposed, when two lines umw and
are exposed by which two bodies are understood Jj^g^^pojed.
to be moved uniformly ; and therefore the same
two Unes 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 J^
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.

PART ir.
12.

of the times. To conclude, if the same lines, A
and 1\ be considered as drawn iu the same time,
then, seeing their velocities are greater, or equal,
or less, according as they pass over iu 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 ANALOCflSM, OR THE SAME PROPORTION.

J) 2, 3, 4', The nature and definition oF proportion, arithmetical
and geometricnl. — 5, Tlie definition , and sonu* properties of
the same arithmetical proportion. — 6, 7. The definition and
transmu tilt ions of analogisni, or the pame i^eo metrical propor^
tion. — 8, 9. The definitions of hyperiogism and hypologbro,
tliat hf of greater and less proportion, and their transinutd'^
tioni*-— 10, 11, V2. Comparison of analogical quantities, ac^
eordingto magnitude^- — 13, H, 15, Composition of proportions.
16, 17, IS, 19» 20. 21, '2% 2% 24-, 25. The detinition and
|)roperties of continual proportion.— 26, 27, 28, 29. Corapa-
rison of arithaictical and geometrical proportions.

[Note, that in this chapter Oie sign + signifies thot tlic quitntitiet betwixtl
which it ia put, are added together; and this sign — the remainder after J
the latter i^uaiitity h taken out of tlie former. So that A + B is equal t»T
botli A and B together: and where you see A—B, there A is the whole,
B the part taken out of it, iind A — 0 the remainder. Also* two letters, set
together without any sign, signily, unless they belong to a figure, that tmfl I
of the quiititilies is multiplied by the otlier ; as A B signifies the prodact of |
A multiplied by 15.]

Tiie nature ] , Great aiid little ave not intelligible, but by com-

and dt'fiuitian , t*t » i • i i i

orpruporiit>ii, parison. Now tlaat, to which they are compared,
I'^'glometricai. '^ somcthiiig exposed ; that is, some magnitude

either perceived by sense^ or so defined by words, ■
that it may be coniprehended by the mind. Also
that, to which any magnitude is compared, is either

OF ANAL06ISM.

145

13.

■greater or less, or equal to it. And therefore pro- part il
portion (which, as 1 have shewn, ii^ 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, whicli is the pro-
portion of the less to the greater.

Again, every one of these proportions is two-
fold ; for if it be asked eoDceniinja; any magnitude
la^ven, how great it is, the answer may be made
bjr comparing it two ways ; first, by saying it is
|2:reater 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-
miaate proportion, and of a different kind from
die former, and called geometrical proportion^
and most commonly proportion .s-implt/.

2. Proportion* whether it be arithmetical or The natiire
geometrical, cannot be exposed but in two magiii- fio„ of p«^-
tudes, (of which the former is commonly called the p^'^*^"^ ^"^^
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

twice numbered ; so that there ^ill be always four
terms.

3- Of two proportions, whether they be arith-
metical or geometrical, when the magnitudes com-
pared in both (which Euclid, in the fifth definition
of his sixth book, calls the qtiantitieH of prapor-
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 heuce it may
be understood, that the proportions of excess and
defect are quantity, being capable of more and
less ; but the proportitin of equality is not quan-
tity, because not capable neither of more, nor of
lens^ 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 ^ame witli that of the second
antecedent to the second consequent. And when
four magnitudes are thus to one another in geo-
metrical proportion, they are csiled proportionals ;
and by some, more hvie^y ^anaiogi^m. And greater

OF ANAL06ISM. 147

proportion is the proportion of a greater ante- partii,

IS.

cedent to the same consequent^ or of the same
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 cidled
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-
qnent is less than that of the second to the second,
Uie four magnitudes may be called hypologism.

5. One arithmetical proportion is the ^a^Tt^ with The definiUon
another arithmetical proportion, when one of the pertiwTf th^
antecedents exceeds its consequent, or is exceeded J^®cd*^
by it, as much as the other antecedent exceeds its potion-
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 diflFerence on both sides be the
8ame 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
win be equal to the product of half the number of
the t^ms multiplied by the sum of the extremes.

L 2

For if A. B : : C, D ; : E. F be arithmetically pro^
portional, the couples x\-fF, B + E, C + D will be
L^dwmrpX *^q^^l ^^ o»^ another ; and their sum will be equal
ptrticiof, ace, to A-f F, multiplied by the number of their combi-
natiooS) that is, by half the number of the terms.
If, of four unequal magnitudes^ any two, together
taken, be equal to the other tw^o together taken,
then the greatest and the least of them w^ill be in
the same combination. Let the unequal magni-
tudes be A, B, C, I) ; 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 lea.st, the sum of the means, the
diiference of the two greatest, and the difference
of the two leajst, 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; 1 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,
X — 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 they be not, let A. C : : D. E (supposing E to be part ii.
greater or less than B) be arithmetically propor- — ^ — '
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.

6. One geometrical proportion is the same with The deanition
another geometrical proportion; when the sametationsofana-
cause, producing equal eflFects in equal times, de- iaS'Jgijorai'
termines both the proportions. *"2^ p™*

* * portion.

If a point uniformly moved describe two lines,
either with the same, or diflPerent 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, A D,
AG, be described with uniform motion; and let
there be taken in them two parts A B, A E, 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, A C in the line A D, and the con-
qnents AE, AF in the line A G) that AB. AC::
AE. AF are proportionals. For seeing (by the
8th chap, and the 1 5th 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 AB 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 AC, whe-

150

PHILOSOPHY,

PART IL

ther it be proportion of equality, or of excess
defect, is determined by the velocities and tim^

The definiti.B j^y ^i^;^,^ ^ g ^ fj ^^^ described; but seeing the

■ &c.

Rud trail smuta«

^^ * *^*^°' ^^^i*^^i ^^ the point A upon A B and A C is uni^
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 AF 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
tliey liave 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
yrm motion, the differenc*e of the velocities in
A B, A E are described, will be the same

OF ANAL06I8M. 151

with the diflFerence of the velocities, in which A C, part ii.

13.

AF are described. Wherefore the cause which ^ ■ » ' ■ ^
determines the proportion of A B to A E, is the J^^dVlnimu^
same viith that which determines the proportion of ^^/^^^^Jj^ »°*i<>"
AC to AF; and therefore AB. AE::AC. AF,^^' ""
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-
iion, that is, by tummg the antecedents into con-
sequents. For if in the last mialogism^ I had for
A B, A C, put by inversion AC, 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.

CoroQ. III. 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
sabtraction 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-
*«»»• or quantity, the products or quotients will be
rtionals. For the multiplication and division

of proportionals, is the same \vith the addition aud
subtraction of them.
"L^rlm"ja^ Co^oU. V. If there be four proportionals, they
tioiii of atido' ^Y^l ^]^Q 1^^ proportionals by eompomtioH^ that isj,
by eompoundins an antecedent of the antecedeot
and c*onseqnent pnt 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.

CoroU. VL 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 proporfton hif eompofiition^

Corolh V!K If there be four proportionals, they
will also be proportionals by division, that is, by
taking the remainder after the cousequent 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 i : D — C D. For in all these divisions,
proportionals are, by the very supposition of the
dogism A. B r : C. D, taken from A and B, and
C and D.

roll. VI n. If there be four proportionals, they

ilso be proportionals by the conversion of

fion^ that is, by inverting the di\"ided pro-

or by taking the whole for antecedent,

difference or remainder for consequent.

OF ANALOGISM. 163

As, if A. B : : C. D be proportionals, then A. A— B part ii.
;;a C— D, as also B.A— B::D.C— D will be — ^

- - -n • ^1- • _x. J "L The defiDition

proportionals. For seeing these inverted be pro- andtransmma-
portionals, they are also themselves proportionals, ^^m, &JI°'^^"

C!oroll. 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 diflFerence of the first quantities will be to
the second, as the sum or diflFerence of the third
quantities is to the fourth. Let A. B : : C. D and
E.B::F.Dbe analogisms; IsayA+E.B: :C+F.D
are proportionals. For the said analogisms will
by permutation be A. C : : B. D, and E. F : : B. D ;
aud 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 coroU.) 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 : : C. D and
E. F : : G. H be two analogisms, and A. E : : 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 Ah- E. B+F : :
C+G. D+H will be proportionals. For seeing

154

PHILOSOPHY,

PART IT. A, E : : C. G are proportionals, A + E, E : : C +G.
''^ — r-^ will also by composition l)e proportionals, and I
permutation A+E, C + G: : E. G mli be propo;
tionals ; wherefore, also A +E. C+G : : F. H wi
be proportionals. Again, seeing, as is shown ahovi
B. F : : 1), H are proportionals, B + R F : : D + H. 1
will also by composition be proportionals ; and b
permutation B + F. D + H : : F. II will also be pre
portionals; wherefore A + E. C + G : : B + F. D + 1
are proportionals ; wliich remained to be proved.
CorolL By the same reason, if there be never a
many analogisms, and the antecedents be propoi
tional to the antecedents, it may be demonstrate
also that the consecpients will be proportional t
the consequents, as also the sum of the antece
dents to the sum of the consequents, ^M

Tiie definition 8. Ill an hyperlogism, that is, where the pro
LdTjypob-'"^ portion of the first antecedent to its consequen
orffrrour^nd ^^ Si^^^^^tcr than the proportion of the second ante
jcMpropf>riioii, cedent to its consequent, the permutation of th

and their traiiF- . ^ it»j^

mutations, proportioiials, and the addition or proportionals t
pro])or tionals, and substraction of them ft"om ori'
another, as also their composition and division, nm
their multiplication and division by the same mim
ber, produce always an hjqierlogism. For suppose
A. B : : CD and A.C : : E. F be analogisms, A-f E. I
: : C'H-F. D will also be an analogism ; but A + E
B : : C* D will be an hyperlogism ; wherefore b]
permutation, A + E. C : : B. 1) is an hyperlogism
because A. B : : C. D is an analogism. Secondly, t
to the h}T>erlogism A+E. B::C.D the proper
tionals G and H be added, A + E + G. B : : C + H. E
will be an hyperlogism, by reason A+E+G
B : : C + F + H, D is an analogism, iVlso, if G and

OF ANALOGI8M. 155

H be taken away, A+E— G. B : : C— -H. D will be part il
an hyperlogism ; for A+E— G.B : : C+F— H. D - '*

13.

18 an analogism. Thirdly, by composition A+E^jfy^^°^!^"
+B. B : : C+D. D will be an hyperlogism, because hypoiogiam&c!
A+E+B. B : : C+F+D. D is an analogism, and so
it will be in all the varieties of composition.
Fomthly, 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 hyperiogism ; 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 \ A+^E. B:: :^C.D is an hyperlogism,
because |- A. B : : \- C. D is an analogism.

9. But if A+E. B:: CD 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,
byconversion of proportion, A+E. A+E—B : : C.
C-D 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).

PART IL 10, If there be four proportionals, the difFereuce
' — ^ — of the two first 3 to the difFereuce of the two last,
of'nmlogkal ^'iH ^^^ ^s the first anteeedent is to the second
arcTrdhfe' anteced(*nt, 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. D ; that is, the differences are proportional to
the consequents^ and therefore they are so also to
the antecedeuts.

i 1 . Of four proportionals, if the first be greater
than the second, the third also shall be greater
tlian 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 foiu'th. In the same manner 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 : ; CD, 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

of the second 3 B ; the product <ilso

M C, will be greater than the product

3 D. But if the produ(^t of the first

the product of the sec^ond, then the

jc third will be less than that of the

id lastly, if the products of the first

OF ANAL0GI8M. 157

and second be equal, the products of the third and part ii.
fourth shall also be equal. Now this theorem ^ — ^-^
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 D will be proportionals ; and again,
by permutation, 2 A. 3 B : : 2 C. 3 D 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 compoaition
three things whatsoever that have any proportion ^ p~p°'^*"**
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,
AB, 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,
A T> n rfc as of the second A C to the

third A D, 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 diflFerence of the times in which
they are described ; that is, A B has to A C such
proportion as is determined by the diflferent times
of their description ; and A C to AD such propor-

PART IL
13.

Composirion
of proportion B,

tion as is determined l)y their times. But the
proportion of A B to A D is such as is determiued
by the diflfereiice of the times in which A B and
A D are described ; and the diflference of the times
in which AB and AC 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 A B to
A D. Wherefore, by the definition of the same
proportion, delivered above in the 6th article, the
proportion of A B to A C together with the pro-
portion of A C to A D, 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 exeesSj 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 A D and
A B. Wher€*fore 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 1) 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

^
^

are described, is excess of time ; for there goes ^*^^^ El-
more time to the description of A D than of A B ; — -^
and tlie diflFerence of the times in which A B and of^pro|)or^^^^^
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 eqnal to D C, by which the
first A D exceeds the third A C ; and therefore the
proportions of the first A 1) 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 1) to the tliird A C.
Wherefore, if any three magnitudes, &c.

Corolh 1. If there be never so many magnitudes
having proportion to one another^ tlie 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 yon come to the last ;
or, the proportion of the first to the last is the
^^ianie with the sum of all the intermediate propor-
Htions. For any number of magnitudes having pro-
"portion to one another, as A, B, t\ D, E being
p propounded, the proportion of A to E, as is newly
Hribown, is compounded t)f the proportions of A to D
I 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-
fcfKirtions of A to B, and of B to C.
' CorolL lu From hence it may be understood

^-liow any tw o proportions may be compounded. For
^Pr the proportions of A to B, and of C to D^ be
propounded to be added together, let B have to
mething else, as to E, the same proportion which
has to D, and let them be set in this order.

PART 1 1.

Coraposilion
of propc>rtioni<

A^ B, E ; for so the proportion of A to E will evi-
dently be the sum of the two proportions of A to B,
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
(jf 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 ont of another. For if the pn)portion 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 Dj 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, whit^h is cviW^A pertitrbed
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 figgregates 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-
)rtionals.

OF ANALOGISM,

IGI

14. If any two qnantities be made of the mutual tart if.
mdtiplication of many quantities, which have pro- — .— -
portion to one another, and the efficient quantities ^^ p^^*^^^^^^
on both sides be equal in number, the proportion
*the products will be compounded of the several
^portions, 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 A B to C D is
compo\mded 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
i?et also in order. Then {by „* '

eorolL iv* of the 6th art,) it will * '

k as A B the first quantity to CB
the second quantity in the first order, so A to C in
the second order ; and asrain, as CB to C D in the
first order, so B to D, that is, Itj^ construction,
80 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 propoition 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 A B F, and
3, each of them made of three efficients, the
of A, B and F, and the second of C, D and
say, the proportion of A B F to C D G is
lounded 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.

PART IL
13.

Compotition

A B F,
CBR
C D F.
CDG.

before ; and as B is to D, so let C be to another
quantity E ; and again, as F is to G, so let E be to
of proponioiw another, H ; and let the first order stand thus,
ABF.CBF, CDFandCDG;
and the second order thus,
A, C, E, H, Then the propor-
tion of A B F to C B F in the
first order, will be as A to C in
the second ; and the proportion of C B F to C D F
in the first order, as B to D, that isj as C to E (by
construction) in the second order ; and the pro-
portion of C D F 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 A B F, C D G : : A. H will be
propo»-tionals. 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 CD 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
*^re made by the duction of two straight Hues into
e another, and all solids which are equal to
res so made, have their proportions compounded
le proportions of the efficients ; and in the
1 place, why the multiplication of two or
fractions into one another is the same thing

with the composition of the proportions of their part h.
several numerators to their several denoeiinators. -— r-^
For example, if these fractions I, f , f be to be ^;;j^^^^^
multiplied into one another^ the numerators I, 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 ^. In like manner, if the proportions of 1
to 2, of 2 to 3, and of 3 to 4, be to be corn pounded,
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 ;
aud 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
b, the proportion of equality. And from hence

tthe cause is evident why two equal products have
their efficients reciprocally proportional- For, for
the making of two products equal, the jnoportions
of their efficients must be such, as being com-
pouuded may make the proportion of equality,
which canuot be except one be the inverse of the
other ; for if betw ixt A and A any other quantity,
as C, be interposed, their order will be A,C, A, and
e later proportion of C to A w ill be the inverse
the former proportion of A to C.

M 2

proportion.

PART II. 1(5. A proportion is said to he multiplied by a
' — r-^ — ' number, when it is so often taken as there be
L*!i^tSrtieB unities in that number ; and if the proportion be
nfconibnai gf thc greater to the less, then shall also the
quantity of the proportion be increased by the
multiplication ; but when the proportion is of tlie
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 I 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 l>e divided, when between two quantities
are intei-posed 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.

Tliis 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-
tioa, sometimes substraction ; which soundeth
absurdly to suc^li 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 ([uautities, the proportion of the first to

the last is compounded of the proportions of the part \h
first to the second, and of tlie second to the third, — ^r^-^
and so ou to the last, without regarding their ^|^J^^^j|;^^*^^^
equality, excess, or defect ; so that if two propor- of contuiuai

Kons, one of inequality, the other of equality, be ^^''^''
ided 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 eompouiuled, the pro-
portion of the first to the second is as niucli as the
sum of both, because proportion of equality, being
not quantity, neither augmeiiteth quantity nor
tesseneth 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
proix>rtion 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 otber quantities, hath a greater proportion to
tlie leaser of them than to tlie 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-
paimded, 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, <j, 8, 4,
thcD shall the proportion of the first to the third

FART II.
13.

be erreater than that of the first to the second^ as
6 hath a greater proportion to 4 than to 8 ; the
The ciefinihnii re^sou whefeof is manifestly this, that the less any

and propertiifs j 7 j

of continual nuantity is deficient of another, or the more one

proportion. 1 , 1 1 . j* . 1

exceedeth another, the proportion of it to that
other is the greater.

Suppose now three quantities in continual pro-
portion, A B 4, AC G, AD 9. Because therefore
AD is greater than AC, but not greater than A D,
the proportion of A D to A C will be (by Euclid,
V. 8) greater than that of AD to A D ; and like-
wise, because the proportions of AD to AC, 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 A D 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

PART ir,
of A B to A C than that of A B to A D. There is — ^
therefore some point (which suppose be E) in such wd%/ope'iiw
distance from C, as that the proportion of A B to ^*'*^*^»^^""'»*

A C will be twice as great as that of A II to A E.
That considered, let him determine the length of
the line AE, and demonstrate, if he can, that A E
is greater or less than A D.

By the same method, if there be more quantities
than three, as A, 13, 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 midtitude, to the proportion of
Ato 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 on 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, 1), 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 1) to A, and
^M 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 I) to B is subdupUcale
ufthe proportion of E to C. And in the same
' manner, if there were three terms on eitlier side,
it might be demonstrated that the proportion of

proportiOQ.

The de tin ill on
And pro|)eriie»

proportion.

D to B would be subtriplicate of that of the ex-
tremes, &c,

18. If there be never so many continual propor-
tionals, as the first, i^econd, third, &c. their diflFer-
enees will be proportiooal to them. For the second,
third, &c. are severally coni?equents of the preceding,
and antecedents of the foUowing proportion. But
(by art, x.) the diflFerence 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 terra 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 uf
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 proportionjUs ; and again,
by composition A + 2 B+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 qu«antity
than the other two put together. Let A. B : : C\ D
be continual proportionals ; whereof let the great-
est be A, and the leiii^t 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.
1 1 y greater than C— 1>. Add B on both sides, and
A will be greater than C + B—D. And again, add
1> on both sides, and A+D will be greater than
B + C ; which was to be proved.

2L If there be four proportionals, the extremes part n.
miikiplied into one another, and the means multi- ^ — ^-^
plied into one another, will make equal products, ^j ''^^^^^^^^
Let A, B : : C. D be proportionals ; I say A D is of co.itinuai
equal tx) BC. For the proportion of AD to EC
k compounded, by art. 1 3, of the proportions of
A to Bj 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 1) to B C is the pro-
portion of equality. Whereibre they are eqnal.
b 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 proportitin 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 raeans^ as C to D.
For seeing the proportion of A to B is duplicate of
the proportion of C to 1), 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 tliat 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, (>, A 1> is to AE as D to E, that is,
^ C to D ; and therefore A 1) is to B C, which as
I have shown is equal to A E, as C to D ; which
i^as to be proved.

Moreover, if the proportion of the first A to
the second B be trijjlicati^ of the proportion of

The definition
and properties
of eontinual
proportJwu.

the third C to the fourth D, the product of the
extremes to the product of the means will be
duplicate of the proportion of the third to the
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 fonr 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 subdupUcate of the proportion of A to B,
or of C to D. iVlso, the proportion of C to F is
subdupUcate 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.

i;i

A C. A B, B C are in continual proportion. And if
tiie same line A C be again clivided * B C

in D, so as that AC. CD. AD be - — i — i

continual proportionals; then also ^

A C. A B- A D will be continual proportionals ;

and in like manner, though in eoutrary order,

CA* CD, CB will be continual proportionals;

which cannot happen in any line otherwise

divided.

25, If there l>e 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. B. 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 R. Wherefore, by
equality, A, D : : E, C are proportionals.

26. If any two unequal quantities be made ex-
tremes, and there be interposed betwixt them any
number of means in geometrical proportion, and
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 inteqiosed three
means, B, C, D, in geometrical proportion, and as
many more, F, G, H, in arithmetical proportion ;
I say B w ill be less than F, C than G, and D than
[ H* For first, the difference betw een A and F is the
i same with that between F and G, and with that
between G and H, by the definition of arithme-

PART IT.
13.

Comparison of
aritlimetical
and geometric
caJ proportioii.

PART IT.

tical proportion ; and therefore, the difference of
the proportionals which stand next to one another,
at^i'^etftJi ^^ ^^ ^^ difference of the extremes, is, when there is
atici geometri- but One mean, half their difference ; when two, a
third part oi it ; w hen three, a quarter, &c. ; so that
in this example it is a quarter. But the difference
between D and E, by art. 1 7, is more than a
quarter of the difference be-
tween the extremes, because
the proportion is geometrical,
and therefore the difference
between A and D is less than
three quarters of the same
difference of the extremes. In
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 lai^tly, if the difference between A and C be
divided into two equal parts, that the difference
betw een 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 togetlier 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 foiu^h part. Also, that C, that is A with
something else which is less than tw o fourth parts
of the said difference, is less than G, that is, than
A together w ith the said two-fourths. And lastly,
that D, which exceeds A by less than three-fourths

of the said lUft'erence, is less than H^ which ex-

jcls the same A l)y three entire fourths of the
id difference. And in the same manner it would

PART IL
13,

be if there were four means, saving that instead ^rhml^ic^^^ ""^
of fourths of the difference of the extremes we are ^^^ geonietri-

calproportMjnt,

to take nftli parts ; and so on.

2". 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 tlie 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 hj'perlogism 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 pai't of
each there be interposed two means^ one in geome-
tricaJ, 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
bet\%ixt A B and A E let A G be a mean in geo-
metrical proportion^ and A H a mean in arithme-

Coin pari son of
ttrithmetical
and geometric
CAlpropDrlioiia.

betwixt A B and A F let

A I be a mean in geo- 'T^ —

metrical proportion, and

A K a mean in arithmetical

H G is greater than K L

For in the first place we have
this analogism * *

-I-

-J -

F I K B
proportion ; I say

A B. A G : : B G.

article 18,

G E, by

Then by composition we have
this

And by taking the halves of
the antecedent;* this third .

And by conversion a fourth .
And by diviiiion this tiFth

A B+AG. A B::BG-hGE
that b, B E. B G.

iAB+i AG.A B::1BG +
iGE. that is, BH. BG.

AB.iAB + 4AG::BG.BH.

iAB^i AG. i AB+i A G
::HG. BH.

And by doubling the first an-
tecedent and the first con-
sequent ....... AB-

Abo by the mme niethotl may
be found out thij» analogism

AG.AB + AG;;HG-BH.

AB— ALAB+AI::KI.BK.

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 AG^ 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 A I 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 B K ; for B H is the
half of BE, as B K is the half of B F, which, by

OF ANALOGI8M. 175

supposition, is less than B E. Wherefore H G is part ii.
greater than K I ; which was to be proved. ^ — r^

CoroU. It is manifest from hence, that if any aii^m'^ti«d
quantity be supposed to be divided into equal ^^pJ^J^^JiL
parts infinite in number, the diflference between
the arithmetical and geometrical means will be
infinitely littie, 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 one 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
qoantities.

1 70

PniLOSOPHY-

CHAP. XIV.

OF STRAIT AND CROOKED, ANGLE AND
FIGURE-

L The definition and properties of a strait line, — 2, The deHiH-

tion and properties oCa plane superficies.— 3. Several sorts of
crooked Fines* — 4, Tlie definition and properties of a circular
line. — 5. The properties of a strait line taken in a plants
6. The definition of tangent lines. — 7. The definition of an
angle, and the kinds thereof- — 8. In concentric circles, arehtt*
of the same angle arc 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 Ihe sanie.
12. The general definition of parallels, and the properties *if
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 ofl^ 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 ditTer-
ent kind from that of an angle simply so called ; and tiiat 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 k determined.—
2L What is like situation ; what is figure; and what are like
figures.

K Between two points given^ the shortest line is
that, whose extreme points cannot be drawn far--
ther asnnder without altering the quantity, that is,
of a Wait line, ^qthout 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

The definition
mid properties

OF STRAIT AND CROOKED.

17:

^ and propcrtiea
of A strait Ime.

hftween its extreme points ; so that any one line, ^^^^t' u-
wliether it lie extended or bowed, has always one * — ^^-^
aodthe same length, because it can have but ont '^^'"''^''"''^°
greatest distance between its extreme points.

And seeing the action, by which a strait line is
made crooked, or contrarily a crooked Une is made
strait, is nothing but the bringing of its extreme
points nearer to one another, or the setting of
tliem f\irther asunder, a crooked Une may rightly
be defined to be ihai^ ivhose extreme points may
he understood to be drawn ^farther asunder ;
and a strait line to be thai^ whose extreme
points cannot he draivn further asunder ; and
mmparathebj, a more crooked^ to he that line
whose extreme points are nearer to one another
than those of the other, supposiuff both the lines
to be of equal length. Now, howsoever a line
be bow ed, it makes always a sinus or cavity, some-
times on one side, sometimes on another ; so that
tile same crooked line may either have its whole
cavity on one side only, or it may have it part on
<^>iie side and part on the other side. Which
being well understood, it will be easy to under*
stand the following comparisons of strait and
rrooked 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 Hne 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
s shorter than the crooked line, which had tin*
e extreme points. And for the same reason,
VOL. 1. N

^*^^J ^^ if ^^^ crooked lines have their extreme points
^^ — ■— common, and both of them have all their cavity on

and propertks OHC and the Same side^ the outermost of the two

of « .trait ihie. ^^.ju ^^ ^^^ bogest line.

Secondly, a strait line and a perpetually crook-
ed line caimot be coincident, no, not in the least
part. For if they should, then not only some
strait line would have its extreme points common
wth 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 t\\o 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
supei-ficies, 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 ail the parts should not constitute a strait

OF STRAIT AND CROOKED.

179

line, they would altogether be longer than the f*AaT il
whole line, ^ — ^-^

2. A plmie or a plane superficies^ u thai which The definition
\u described by a strait line so moiwd^ that all ©? a pbn7iu-
Ihe several points thereof describe sereral strait p®""*^^*^**
titles. 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 8ame comparisons, which were made of
strait and crooked lines, may also be made of plane
and crooked supeiiicies.

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 hues, of which the crooked superficies con-
siv^ts, be extended, they will be found to be longer
than those of which the plane superficies consists,
which carmot be extended, because they are strait,
L Secondly, two superficies, whereof the one is
"plane, and the other continually crooked, cannot
lje coincident, no, not in the least part. For if they
Were coincident, they would be equal ; nay, the
sanie superficies would be both plane and crooked,
which is impossible.

Thirdly, within the same terminating lines
there can be no more than one plane supei*ficies ;
ause there can be but one least superficies
within the same.

S^cvfral sorts of
crooked lme£.

Dell 1 11 1 ion and
proporliea of a
circular line.

Fourthly, no number of plane superficies can
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 supeilicies ; 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
platie 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 Hues; if every part
should not constitute a plane superficies, all the
parts put together would not be equal to the
whole,

3. Of straitness, w-hether it be in lines or in
superficies, there is but one kind ; but of crooked*
ness there are many kinds ; for of crooked mtigni-
tudes, some are congruous, that is, are coincident
w hen they are applied to one other ; othei^ are
incongruous. Again, some are oftomftspuQ or uni-
form^ that is, have their parts, howsoever taken,
congruous to one another ; others are avoftoto^afHtq
or of several forms. Moreover, of such as are
crooked, some are continually crooked, others have
parts which are not crooked.

4. If a strait line be moved in a plane, in such
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 w^as first
moved, it will describe a plane superficies, wiiich
will be terminated every way by that crooked line,
which is made by that end of the strait line which

carried round. Now this snperficies is called
a CIRCLE ; and of this eirclej the unmoved point is
the centre; the crooked line which terminates it, definition and

' properties ol a

the perimeter ; and every part of that crooked circtii&t line.
liwe, a circumference or arch ; the strait line, wliich
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
circiimferencej 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
aud all those have one common centre.

Wherefore in eveiy circle, all strait lines from
the centre to the curcumference are equal. lu)r
they are all coincident with the radius which
generates the circle.

Also the cHameter divides both the perimeter
aiid 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
iJiameter, will be cut by them in two points.
^^Tierefore, seeing all the strait lines from the
peatre to the circumference are equal, a part of
the same strait hue will be equal to the whole;
^hich is impossible.
For the same reason the perimeter of a circle

182

PHILOSOPHY.

PART ih will be uniform, that is, auy one part of it will be

^ — -r^ coincident with any other equal part of the same.
Tiie proper- ^ From heucc may be collected this property

Ilea, ol a strait ^ J ^ ^ . ,

I line taken in gf a stfait line> namely, that it is all contained iu
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.
Wlierefore 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 hue between auy 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 bt* either coincident with that
line, that is, it will be the same line ; or it will not

PART II.

he coincident, and then it will be in neither, or
but in one of those planes.

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 r/.ri> of that
motiou. 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,
e understood to be in the same plane in which
Jhe 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 point^s will be all in that plane ;
and consequently the tliree strait lines which con-
nect those points, will also be in the same plane.

6. Two lines are said to fouch one another, DeBnition of
vhich being both drawn to one and the same
point, will not cut one anotlier, though they be
(iroduced, 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 congniity ; 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 maidfest
from this, that there can be no congruity between
a strait line and a line that is continnally crooked ;
tor otherwise the same line might be both strait

tangent lines.

tlicruor.

PART u. and crooked. Besides, when a strait line touches
- — r^ — ' a crooked hne, if the strait line be never so little
lb moved about upon the point of contact, it will cut

^^^H the crooked line ; for seeing it touches it but in
^^^H one point, if it incline any way, it will do more
^^^P 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.

Jratfaric*'" 7- An angle, according to the most general
mui ihe kimii acceptation of the word, may be thus defined ;
when two lines^ or mam/ .super^eies^ concur in one
sole pointy ami diverge everi/ 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-
Jicial angle ; or by the concurrence of superficies,
and then it is called a solid angle.

Again, from the two ways by w hich 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 w hole 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 Hexion or curvation in every
imaginable point ; and the quantity of this sepa-
ratiou is tluit, which is railed an angle of con-
tingrnee,

Hesid( s, of superficial angles sim[>ly so cidled^

OF STRAIT AND CROOKED.

185

le^ which are in a plaiit* supprficies, are plane ;
and thosej which are nut plane, are deiioiriinated
from the superficies in which they are.

Lastly, those are 8trait4inecl 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 of ^» tcmocmric
concentric circles, have the same proportion to one of ihc same
another, which their wliole perimeters have to one olle Luiiicrl**
another^ For let the point A (in the first figure) :[,,:.^;:,-:,[:l
be the centre of the two circles B C D and E F G, ^re
in which the radii AEB and AFC intercept the
arches B C and E F ; 1 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 AFC 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 descril)e the
pmmeter 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 BCD 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 perinu'ters are described
ill one and the same time, an(i 1 joth the archt\s iu
otJe and the same time ; and therefore the propor-
tions of the perimeter B C D to the arch B C, and
fjf the perimeter E F G to the arch E l\ are
Mil determined bv the same (*ause. Wherefore

TAET IL
1 4.

B C D. B C : : E F G. E F are proportionals (by the
6th art. of the last chapter), and by permutation
B C D, EFG : : B C. E F will also be proportionals ;
which was to be demonstrated.
«f^I.?'mS ^' ^^thing is contributed towards the quantity
ia what it of ail angle, 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 R 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 f/uautity of an angle is an arch
or circtfmference of a circle^ lietermined by its
proportioit io the whole perimeter. So that when
an arch is intercepted between two strait line^
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
wheii 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 theu* concurrence ; for
the least distance is to be considered as a strait
line, seeing no crooked line can l>e imagined so
httlcj but that there may be a less strait line. And

OF STRAIT AND CROOKED

187

Ithough the least strait line cannot be pveii, i'aht
?caus>e the least given line may still he divided, - — r^
we may come to a part so small, aii 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, ils 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 rertica! angleJi^ ThediMinciioii
such as are ABC, DBF in the second figure, are piyso^lioT'
equal to one another. For if, from tlie two semi-
perimeters DAC, 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
ohliqite. A right angle is thidy whose (juavtitij is
the Jmirih part of the perimeter. And the lines,
which make a right angle, are said to be perpefi-
dicnlar 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 ricute
angle. From whence it foUow.s, 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 riglit angles ; for if that point be

PART IL

made the eentre, that strait hne will be the dia-2
meter of a circle, by whose circumference the
quantity of an angle is determined ; and that dia-
meter will divide the perimeter into two eqnal
parts.

1 L If a tangent be made the diameter of a

Of iitrait lines

frojn tlie ceu- . , i - , . ^

tn? of a circle circlc, whose Centre is the point of contact, a
i^f the aS. 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 perpentliciilar 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 per|iendicnlar ; 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 thcu
same. ^1

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. .

, pa rail cl 8-

12- There is in Euclid a definition of strait-
lined parallels ; but I do not find that parallds in
g^eneral are anywhere defined ; and therefore for J^.lfnftl^rof
an universal definition of them, I say that atw two i»*faiipi»;

^ J *^ the proper-

//V/£\v whatsoever, .strait or crooked, a^ aha r//^w licaofsirait
tito superficies y are parallel ; when two cf/ual
»trait linejfy wheresoever they Jail upon them,
make always espial angles with each of them.

From which definition it follows ; first, that any
two strait lines^ not inclined opposite way8» 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 w ill also be equal and
parallel. For the perpendiculars B E and D F
being drawn, the right angles E B D and F 1) 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 fail upon G. Wherefore AG will be either
p-eater or less than B D ; and therefore the angles
E B A and F D C are not equal, as was supposed*
WTierefore A B and C D are equal ; which is the
first.

Again, because they make equal angles with the
perpendiculars B E and L) F ; therefore the angle
CDH will be equal to the angle ABD, and^ by
the definition of paraUels, A B and C D will be
parallel ; which is the second.

That plane, which is included both ways
wiihin jmrallel lines Js called a pakallelogram.

PART }L
14,

General defi-
nition of pa-
TftJlcla, &uc»

CorolL I. From this last it follows, that the
angles A B I) and C D H are equal, that is, that
a strait line, as B H, falling upon two parallels, as
A B and C D, makes the internal angle A B D
equal to the external and opposite angle C D H,

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 i ertical
angle A G F, which is alternate to G F D.

CorolL m. 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 A G F. Wherefore
both the angles G F C and A G F, which are in-
ternal on the same side of the line F G, are equal
to tw^o 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
triatigle 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 t^ o 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 lOth article. Wherefore taking

remain tlie anj2:les A aud V, part
|ual to the angle D ; which is the second. — -^

CoroU. V, If the angles A and B be equal, the ^j^j^^^fj^
sides A C and C B will also be equal, because A B ''^ii*^^''* ^•
and E F are parallel ; and, on the contrary, if the
sides A C and C B be equals 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 angle^5 G and B
equal, the sides G € 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
be, by the 1 Ith article.

CorolL VK From hence it is manifest, that if
two radii of a circle be connected by a strait line,
the angles they make with that conneeting line
will be equal to one another ; and if there be
added that segment of the circle, which is sub-
tended by the same hne which connects the radii,
then the angles, which those radii make with the
cmimference, will also be equal to one another.
For a strait line, which subtends any arch, makes
equal angles with the same ; because, if tlie arch
and the subtense be divided in the middle, the two
halves of tlie segment will be congruous to one
another, by reason of the uniformity both of the
drcamference of the circle, and of the strait line.
i 1 3. Perimeters of circles are to one another^ as The eircumfe-
their semidiameters are. For let there be any two ^citslre to*^©nc
circles, as, in the first figure, B C 1) the greater, Sl'^^^e^^trl!
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
perimeter BCD has to the perimeter E F G.

lURTIL
14.

For the magnitude of the semi diameters A C and
A E Is determined by the distance of the points
C and E fi-om 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 B C 1)
and E F G, as welt 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 1 3th chapter and
6th article, the perimeters of circles and their
semidiameters are proportionals ; which was to be
proved.
L™TtTifcrr>o. ^"^^ ^^ ^^^^ strait lines, which constitute an angle,
miiei lo the \y^ eut bv strait~Uned parallels, the hitercepted pa-
another, as the rallels wul be to ouc auother, as the parts w nich
they cut oflF from the vertex. Let the strait lines
A B and A C, in the (Hh 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 oflF from the vertex
in either of those lines, as in A B, may be A B
and A D. I say, the parallels B C and D E are to
one another, as the parts A B and A D* For let
A B be chvided into any number of equal parts, as
into A F, F Dj 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 L If now the pomt
A be understood to be moved uniformly over A B,
same time B be moved to C, and all 1

parts of the
sidfS wliirh
they cut ofT
from the vertex.

equal swiftness over F G, 1) 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
yjjiother, as their velorities are ; a'ld when A is in
Fp, D will be iu K ; when A is in 1), 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 veloeity in B C, so
is AD to D E ; but as the veloeity 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
m off trom 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*
tioiials.

The subtenses of equal angles in different cireles,
as the strait lines B C and FE (in %. I), are to
nue 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
bp preiportional to the semidiameters, that is, to

»thp perimeters, that is, to the arches which they
«^ibteud.
15. If in a circle any number of equal subtenses ^> "^''^^ '''^**^-
■X* placed immediately after one another, and strait iitietiietircum-
^nieg be drawn from tlie extreme pomt of the first ck bmade.

subtense to the extreme points of all the rest, the
first subtense being produced will make with the
^«Mof''nJiTaTt®^^*^^d subtense an external angle double to that,
lifieihecirtum- v^r]iic}j jg made bv the same first subtense, and a

feience of a cir- ^

de is made, taugeut 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 A B (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 G C D, and H 1) E- Lastly, let
tilt* tangent K B be drawn, making with the first
sul)tense the angle K B C, I say the angle G C D
is doul)le 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
draw n 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 GOD is double to
either of the angles C B D or C D B ; and there-
fore also the angle G C D is double to the angle
LC D, that is. to the angle K B C, Again, C D is
parallel to B E, by reason of the equality of the
angles C B E and DEB, and of the strait lines

C B and D E ; and therefore the anecles G C D and part il
G B E are equal ; and consequently G B E, as also — A-'
D E B is double to the angle K B C. But the ex- tionira mndt
teraal angle HDE is equal to the two internal J-^^^^^^^'j^^^^^^^
DEB and D B E ; and therefore the angle H D E <^i*^ '' ™'^^«-
is triple to the angle K B C, &c. ; which was to be
proved.

CorolK 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 II. If the tangent B K be moved in the
circumference with uniform motion about the
centre B, it w ill in equal times cut off ec^ual archer ;
and wiU pass over the w hole perimeter in the same
time in w hich itself describes a semiperimeter about
the centre B.

Coroll. 111. From hence also we may under-
stMid, what it Ls 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 ad flit ion 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 %vliich is triple,
ami 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
TTia^nitude ; but if they be understood to be the
'mt that can be, that is, as so many points, then

o 2

TART IL
14.

the figure described will not he rectiliueal, 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 he 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 linCj 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
w'hole perimeter of that which is from the centre,
the arches, which it cuts oflF in the perimeter whose
centre is A, wilt be double to those, which 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 hist 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 C A D insisting upon the
same arch C D is also double to the same angle
C B D or K B C^ ^
'That an ingle 16- Au angle of contingence, if it be compared
Ti q^imt'i"y!but ^'ith an angle simply so called, how little soever,
kfn/ from that ^^ ^^^^^ proportion to it as a point has to a line;
of auangicfiim that is, uo proportion at all, nor any cmantity. For

plj *o called ; i .- * • i , . ,

and that it can urst, au angle oT contingence is made by coutnmal
*'"' flexion ; so thfit in the generation of it there is no

OF STRAIT AND CROOKED.

PARTIL

enrcnlar motion at all^ in which roii«iiits the nature
of an angcle simply so called ; and therefore it can-

1 ' 1 • 1 * T • ^*^^ away

not be compared with it aceonhiig to quantity, any Uiiug

Secondly, seeing the extenial angle made by ^ ^™°^ '^^ ""**"*

subtense produced and the next subtense is equal

to an angle from the centre insisting upon the

uime arch, a.s in the last figure the angle G C D is

equal to the angle C A D, the angle of contuigence

will be equal to that angle fi'om the centre, which

is made by A B and the same A B ; for no 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 eontingence 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 t^\o lines, which con-
cur in one sole point ; and seeing one opening is
L^ greater than another, it cannot be denied, but that
Bb} the ver)' generation of it, an angle of contin-
^M gence is quantity ; for w heresoever there is greater
^ and less, there is also quantity ; t>ut this quantity
^ consists in greater and less flexion ; for how much
B the greater a circle is, so much the nearer comes
the circumference of it to the nature of a strait
liDe ; for the circumference of a circle being made
hy the curvation of a strait line, the less that strait
. liDe is, the greater is the curv^ation ; and therefore,
B when one strait line is a tangent to many circles,
^ the angle of eontingence, which it makes with a
^i^s circle, is greater than that w'hieh it makes
i^ith a greater circle.

Nothing therefore is added to or taken from an
'"^Qgle simply so called, by the addition to it or

PART n.

II.

That th*" mcli-
□alinn of planes
is anc^le aimplj

A Kolld angle
wIiAi iL is.

taking from it of never ^o many angles of coiitin'
geiiee. And as an angle of one sort can never be
equal to an angle of the other sort, so they canuot
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 wiiich is
made between G B and B K is equal to that w hich
is made between G B and the arch B C.

17- An angle, which is made by two planes, is
commonly called the inclination of those planes ;
and because 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.

1 8, 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

14.

Miat vertex to the whole superficies of the sphere,
Xu which sense, solid angles are to one another i\s
It lie spherical bases of solids j which have their ver-
tex in the centre of the same sphere.

19. All the ways, by which two lines respect one ^^^^ ^^i^"-'

,' ' •* *■ nature of

another, or all the variety of their position, may aayaiptoiei.

fee comprehended under four kinds ; for any two

lines whatsoever are either parallels, or being pro-

cluced, if need be, or moved one of theai 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 usymptoieH. The

nature of parallels, angles, and tangents, has been

already declared. It remains that I speak briefly

of the nature of asymptotes,

Asymptosy depends upon this, that quantity is
infinitely divisible. And from hence it follows, that
auy 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
liae 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 Hth
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 eterniilly,
and there be drawn from the point F, the strait
line F F at any angle A F F ; 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
iaiBe A F, the crooked line A B C I) E, and the
strait line F F, will be fi.Hympioie.s^ that is, they

PAriT 11, ^iu always come nearer and nearer together, but
never touch one another. Now, because any Hue
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-
Iht, 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
asyuiptosy of hyperbolic lines, it is understood
they should apj>roach to a distance less than any
given quantity.

Firuatbn, iiy 20. SITUATION IS the relation of one place to

what It la t.f I

deiBrminird. unotker ; aud where there are many places, their
situation is determined by four things ; by their
diHtaneeiifrom one another ; by sever al distaneen
Jrom a place a.^.'figned ; by the order of strait
lines drawn Jrom a place assigned to the places
of them idt ; and by the angles which etre made
hi/ the lines so drawn. For if their distanceSj
order, and angles, be given, that is, be certainly
known, their several places will also be so certainly
kiH>wn, as that they can be no other.

Whatk 2h Points, how many soever they be, have like

like situation : , . ^ ^ ,

vtUaiifingmv: situation with an equal number of other pohits,
like figures, w hen all the strait lines, that are drawn trora some
one point to all these, have severally the same
proiKjrtion to those, that are drawn in the same
order and at equal angles from some o!ie 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 I> let the strait lines D A, D B, and
D C l^e drawn ; and let there be an equal number
jjf other points, as E, F, and G, and from so

iilM

drawn, so that the angles A D B and B 1) C be -

severally and iu the same order equal to the angles ^tu-'IfioV'tThat

E H F and F H G, and the strait lines D A, D B, » ^g"^'^^' > *^*=-

and 1) 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 equiility of the angles A D B and E H F ; and

the point G will be in the strait line D C\ by reason

of the ejuality of the angles B 1) C and F H G ;

aiid the strait lines A B and E F, as also B C and

FG, wiU be parallel, because A D. E H : : B D.

F H : : C D. 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-

tiond to the distances between the points E and F,

and the points F and G. Wherefore, in the situa-

tiou 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

Bothing but the inequality of their distances from

one another, and of their distances from the points

Dand H. Now, in both the orders of points, those

inequalities are equal ; for A B. B C : : E F. F G,

T^hich are their distances from one another, as

ttW) D A. D B. D C : : H E, H F, H G, which are

their distances from the assumed points U and

H, are proportionals. Their difference, therefore,

consists solely in the magnitude of their distances.

, by the definition of iiki\ (chapter i, article 2)

le thinp>, which differ only in magnitude, are

Wherefore the points A, B, and C, have to

H eoti8

PART
14.

'^- one another like situation with the points E, F,
— ' and Gv or are placed alike ; which was to be proved,
whai Figure h quantity^ determined by the sitimtion
*^^' or placing of all its extreme points. Now I call
those points extreme, which are contiguous to the
place which is without the figure. In lines there-
fore and superficies, all points may be called ex-
treme ; hut 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 hut magnitude.

And like figures are alike placed^ w hen 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 evei*y other plane,
when nothing but planeness is considered; if the
lines, which include planes, or the superficies,
w^hich include solids, have their proportions known,
it will not be hard to know whether any figure
be like or uidike to another propounded figure.

And thus much concerning the first grounds of
philosophy. The next place beh>ngs to geometry ;
in which the quantities of figures are sought out
from the proportions of lines and angles. Where-
fore it is necessary tor him, that would study geo-
metry, to know first what is the nature of quantity,
proportion, angle and figure. Ha\ing 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.

!• 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 tilings in order to be treated of are part iil

MOTION and magnitude, which are the most Ifl^

common accidents of all bodies. This place there- Repetition
fore most properly belongs to the elements ofcipieeofthe
geometry. But because this part of philosophy, moriorfor-
baving been improved by the best wits of all ages, ^er^y'^t^own.
tes afforded greater plenty of matter than can well

PART III. be thinst together witliiii the narrow limits of this
^ — - discourse, I thought fit to admonish the reader,
ofTme''' that before he proceed farther, he take into Ms
pri»dpi«, itc, bauds 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 follow ing
chapters, shall Ije such oidy as is new, and con-
ducing to luitural 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, tliat the
reader in going on may have their light nearer at
hand.

First, therefore, in chap. vin. art. ID, motion is
defined to be the contintfa! prhationof one place^
and (icqutsifion of another.

Secondly, it is there shown, that whatsoever /a*
mo red is moved in time.

Thirdly, in the same chapter, art. 11, I have
defined rest to be when a body remain^s Jar some
time in one plaee.

Fourthly, it is there shown, that whatsoever is
moved is not in any determined place ; as also
that the same has /}een mored^ is still moved^ and
will tjet 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
t\\i\fHtnre time.

Fifthly, in art. 1 5 of the same chapter, I have
defined velocity or swiftness to he motion con-
sidered as power, namely^ that power by whieh a
body moved may in u certain time transmit a

MOTION AND ENDEAVOUR,

2o:>

cert ft in length ; which also may more hriefly be part iil

eonDciated thus, rehcift/ i/t the quantity of moiion "-— — ^

determined by time mid fine. ' f^^me'"

Sixthly, in the same chapter, art, 15, I h;iveP'''"*^^P^**»^^

shown that motion is the measure of time.
Seventhly, in the same clnipter^ art. 17, I have

defined motions to be equally swift, when in equal

times equal len^^rths are transmitted by them,
Eiehthly, in art. 1 8 ot the same cliapter, motions

are defined to be equals when the .swiftness of one
moved body, computed in every part of its mag-
nitude^ is equal to the swiftness of another^ eom-
pftted also in every part of its inagnitnde. From
whence it is to be noteri, that motions equal to
one another^ and ^notions equally swifts do not
signify the same thing ; for when two horses draw
abreast, the motion of both \s greater than the
motion of either of them singly ; but the swiftness
of both together is but equal to that of cither.

Ninthly, in art. 1J> of the same chapter, I have
^hown, that whatsoever is at rest will always he
(itrest^ unless there be some other body besides
it, which by getting into its place supers it no
longer to remain at rest. And that whatsoever is
mved^ will always he 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
ntrestj the immediate efficient cause of that motion
w in some other moved and contiguous body.

Eleventhly, I have shown in the same place, that
ttliatsoever is moved^ will always be moved in the
^nme way^ and with the same swiftness, if it be

not hindered by some other moved find contignom
body.
Other prin- 2. To whicli principles I shall here add those

ciplPB added / J^

to them. that follow. FiTst, I defiiie endeavour to be
motion made in less space mid time than can be
given ; that is, less than can be determined or
assigned by e.vposition or number ; that is, motion
made throngh the length of a potnt^ and in an
instant or point of time. For the explainiog of
which definition it most 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 indivisilile 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
he compared with another endeavour, and one j
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-^

tion which the perimeters have to one another, part in,
And in the same mimiier, if two motions begin —-^ — '
and end both together, their endeavours will be ^^es^dXd
equal or unequal, according tu the propoition of tothem.
their velocities ; as w^e see a bidlet of lead descend
with greater endeavour than a ball of w^ool.

Secondly, I define impetus, or quickuess of
motion^ to he the swiftneM or rehcity of the body
mrcd^ but considered hi the several points of
that time in which it is moreeL In which sen^e
■ impetus is nothing else hut 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 timCj and equal
to the product of a line representing the time,
mult i plied into a line representing the arith-
mtically mean impetus or quickness. Which
arithmetical mean, w hat it is, is defined in the 29th
article of chapter xni.

And because in equal times the ways that are
passed are as the veh>cities, and the impetus is the
velocity they go withal, reckoned in all the several
points of the times, it foUoweth that during any
time whatsoever, howsoever the impetus be in-

■ creased or decreased, the length of the way passed
over shall be increased or decretised in the same
proportion ; and the same line shall represent
both the way of the body moved, and the several
impetus or degrees of sw iftness wherewith the way

^is passed over.
And if the body moved be not a point, but a
Mfait line moved so as that every point thereof
liiake a several strait line, the plane described by
Its motioUj whether uniform, accelerated, or re-

tarded, shall be greater or less, the time being the

same, in the same proportion with that of the

eip^'el added f^f^p^^f^^ reckoned in one motion to the impetus

to them. reckoned in the other. For the reason is the same

ill 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 impcfufi or
quicknesses reckoned through one line, to the
several impefus reckoned through another.

This understood, let A BCD, (in figure 1 ^ chap.
XVII.) be a parallelogram; in which suppose the
side AB 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 B EFC; and suppose the time of
this motion designed by the line V 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 O
taken at adventure in the line C I), draw O R pa-
rallel to BD, cutting the line BEFC in E, and
the side A B in R. And again, from the point d
taken also at adventure in the line C D, draw CI 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 C D, cutting A C in G and H. Lastly, suppose
the same construction done in all the points possi-
ble of the line H E F C. I say, that as the propor-
tions of the swiftness wherewith Q F, 0 E, 1> B,
and all the rest supposed to be thrawn 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

supposed to he drawn parallel to the line of rARTiiL
ae C D and terminated in the line B E F C, the —r^
rate to tlie aggregate, so is the area or plane ^^H ^^^^;^
D B E F C to the area or plane A C F E B. For ^ '^lem.

■as A B decreasing continually by the line B E F C
Taiiisheth 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 deseribeth 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 ;
aud their swiftnesses are therefore equaL Again,
because in the time G E the point O deseribeth the
^lineOE, and in the same time the point S de-
Bi»cribeth the hne S E, the line O E shall be to the
Hue S E, as the swiftness wherei^ith O E is de-
scribed to the swiftness wherewith SE is described.
In tike manner, because in the same time H F the
|>omt Q deseribeth the line QF, and the point R
the line R F, it shall be as the swiftness by which
QF is described to the swiftness by which R F is
described, so the line itself Q F to the Hue itself
IRF; and so in all the lines that can possibly be
Idrawn parallel to B I) in the joints w here they
Icut the line B E F C. But all the parallels to B 1),
asSE, RF, A C, and tlie rest that can possibly be
imm firom the line A B to the line B E F C, make
I the area of the plane A B E F C ; and all the paral-
lels to the same B D, as QF, OE, DB ami the
rest dra^^Ti to the points where they cut the same
htie B E F C, make the area of the plane B E F C 1),
As therefore the aggregate of the swiftnesses
wherewith the plane B E F C D is described, is
t<J the aggi'egate of the swiftnesses wherewith
VOL. I. p

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

cipiti iSd^ the aggregate of the times represented by the pa-
toihem. ^,^,^,1^ ^g^ QE j^p ^^^^ ^l^g j,^g^^ makethalso

the area A C FE B. And therefore^ as the aggre-
gate of all the lines Q F, O 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 SAviftness 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 Aj and so of all the rest taken toge-
ther, are the proportions of the plane DBEFC
to the plane A B E F C. Bnt the lines Q F, O E,
I> 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 ;
antl therefore the proportion of the plane DBEFC
to the plane A B E*F C is the proportion of all the
veloeities taken together to all the times taken
together* Wherefore, as the proportions of the
swiftnesses, &c. ; w'hich 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, w hich are here parallelograms, will be there

MOTION AND ENDFAVOUK.

211

1;>.

Offier pritT-

eyliiiders, the diameters of wliose liases are the
lint's H t\ GE, A B, &c, and the altitude a point,
that is to say, a qnantity less than any quantity
that can possibly be named ; and the lines Q F, O Ej ^"^ ^^^^'^
D B, &c* small solids %\ hose len^^hs and breadths
arc less than any quantity that can be named.

But this is to lie noted, that unless the propor-
tion of the sum of the swiflrnesses to the proportion
of the snm of the times be determined, the pn)]>(>r-
tioii of the figtire DBEFC to the figure ABEFC
cannot be determined.

Thirdly, I defiue resistance to be the endea-

rmr of one vmred body either wholhf or in part

emir art/ to the endeavour of another moved hodtf^

which touch eth the uime, I say, wholly contrar)^

when the endeavour of two bodies proceeds in the

»ame strait line from the opposite extremes, and

contrary in part, when two bodies have their en-

\ deavour in two lines, which, proceedinj? from the

■tttreme poiuts of a strait line, meet without the

HKime<

^ Fourthly, that I may define what it is to press,
; I my, that of two moved bodies one premcs the
otker, when with i7.v endearonr it makes either all
or part of the other body to go out of* its place.

Fiftlily, a body, which is pressed and not
icliolly removed, is said to restore itselj\ when^
the pressing body being taken away^ the parts
HlPAirA were moved rfo, fry reason of the internal
^institution of the pressed body, return every one
^kto its own place. And this we may observe in
Ht)mijB:s, in blown bladders, and in many other
^Wies, whose parts yield more or less to the en-
vour which the pressing body makes at the

Tir.

PART 111. first arrival; but afterwards, when the pressing
- — ^ — body is removed, they do, by some force withhi
them, restore themselves, and give their whole
body the same figure it had before.

Sixthly, I define force to be ike impetns or
quickness of juotmn midiipUed either into itself y
or into the magmtude of the movent^ by means
whereof the said movent works more or less upon
the body that resist\s it.
Certain thecK 3^ Havius: premised thus much, I shall now

rema concern- ,

ing the nature demonstrate^ first, that if a pomt 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 poiut. 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
impetns. For nothing doubled is still nothing;
and for the same reason it shall never move it vdth
that impetus, how many times soever it be midti^
plied, because nothhig, 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.

Secontlly, that when a point moved, how^ little
soever the impetus thereof be, falls upon a point of
any body at rest, how hard soever that body be, it
w ill 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

PAUT
15.

of them have do effect, tlie nggrpgate of them all
to^jether shall have no effeet as many times told as
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 ; tliat 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
uothiiifT at all, nor is of any efficacy ; and that
nothing but motion je:ives motion to such things

she 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 1 1 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.

CorulL 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 ^'""^.^ ..

~ COD 9 1 ut' rations

respects; first, as in a body uudhidedy that is^ of"»wtioii«.
considered as a point ; or, as in a dimded body.
Ill an undivided body, when we suppose the way,
V 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
^f motion, it is in body, considered as undivided.

PART III. sometimes nnijorm aud sometimes multiform.

' — ^ — form is tliat by which equal lines are always

conalderaijoDH transmitted in equal times ; and \multiform^ when

or motion. [jj Qjjg ^^jjjjg more, in another time le^ss 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 otlier line ;
and from hence one motion is calhd perpefidieular^
another oblique^ another paraUeL

Fifthly, from the position of the movent in re-
spect of the moved body ; from whence one motion
Impulsion or driving, another tract ion or drawing.
Pul^ioHy when the movent makes the moved body
go before it ; and tract ion ^ w^hen it makes it follow.
Again, there are two sorts of puixion ; one, when
the motions of the movent and moved body begin
both together, which may be called trumon or
thrufiting and rection ; the other, when the movent
is first moved, and afterwards the moved body,
which motion is called perctMsion 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 m the excess of motion which the morefit
has afwve the motion or endeavour of the resist in^^
hodtj.

Seventhly, it may be eoiisidered from the diver- part iir.
sity of the medium ; as one motion may be made — -—^
in vacuity or empty place ; another in a ^fluid ;
another in a consistent niediumy that is, a medium
\^hose 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,

b. AD endeavour tends towards that part, that is tko way by
to say, in that way which is determined by the ^^j^^^^^^^^^^^
motion of the movent, if the movent be but one ; b^^;^/'^"^*^^i
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,

t J 1 -. . * r- ii 1* i whick IS made

body has its motion rrora the concourse oi two by codcouw,
movents, as soon as in any point thereof the force *'''V*^^^^.***'*'

' J r vents ceasing',

of one of the movents ceases, there immediately }^^ eudcavour
the former endeavour of that body will be changed way by wiiich
into an endeavour in the line of the other movent.
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
Wried by that wind alone which blows still. And
in the describing of a circle, where that which is
^oved has its motion determined by a movent in a

216

MOTIONS AND MAGNITUDES^

PART 111.
15.

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 ; w liicli 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.
All imdcRvoiir 7 ^[] endeavour, whether strong^ or weak, is

IS profMigated ^ ,

in inflnUiim. propagatcd to infiiiite distance ; tor 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 filledj yet, seeing endeavour is motion,
that w hich 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 ftiil
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
})art of the full medium removes that w hich is next
iit, the MiMMid ;n I [upves that part which is next

MOTION AND ENDEAVOUR.

217

I
I

and therefore all endeavour, whether it be in
empty or in full spaee, proceeds not only to any
dititance, how great soever, but also in any time,
Ijow little soever, that is, in an instant. Nor makes
it any matter, that endeavour, by proceeding, grows
weaker and w eaker, till at last it can no longer be
perceived by sense ; for motion may be insensible ;
mid I do not here examine things by sense and ex-
perience, but by reason,

8- When two movents are of equal magnitude,

tlie swifter of tliem works with greater force than

the slower^ uprju a body that resists their motion.

Also, if two movents have equal velocity, the

p-eater of them works with more force than the

less. For where the magnitude is equal, the movent

of greater velocity makes the greater impression

upon that body upon which it falls ; and w here the

Telocity is equal, the movent of greater magnitude

felling upon the same point, or an equal part of

another body, loses less of its velocity, because the

resisting body works only upon tliat 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, til! they also come to

lie toui*hed ; and their force has some effect,

1i\Tierefore, 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

t-ffeet upon the walL

PART iir.

15,

Uow mucU
g^rcater the
ifelocUy or
ma^nirwde is
of a movent,
so inueh the
i^reater is th»'
efficacy lliei
of upon nnj
other Itody
in its way.

w*^^

218

MOTIONS AND MAGNITUDES,

CHAPTER XVL

PART III,
16.

Thovdocityof
cuiy h*idy, in
what time so-
wer ii be com-
puted^ is that
\4bich is made
of the multi-
plication of
the impL'ius
or quickness

OF MOTION ACCELERATED AND UNIFORM, AND
OF MOTION BY CONCOURSE.

L The velocity of any body, in what time soever it be computed,
ia tliat whicli is made of the multiplication of the impetus, or
quickneas of its motion into the time, — 2-5* In all rnotioii,
the lengths which are passed through are to one anotlier, as the
products made by the impetus multiphed into the time, — 6. If
two bodies be moved with uniform motion through two lengths^
the proportion of tho.Hc lengtfis to one another will be com-
|jounded 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 pmpor lions of length to
length, and iinpetnn to impetus reeipnjcally taken ; also the
pioportion of tiieir impetus to one another will be eompounded
of tlie proportions of length to length, and time to time reci-
prneally taken, — 8. If a body be carried on with uniforni motion
by two moventH together^ which meet in an angle, the line by
which it passes will he 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.

L The velocity of any body, in whatsoever time it
be moved, has its quantity determined by the sum
of all the several quicknesses or impetus, which it
bath in the several points of the time of the body's
motion. For seeing velocity, (by the definition of
it, chap. VI II, art. 15) is that power by which a
body can in a certain time pjiss through a t*ertain
length ; and quickness of motion or impetus, (by

ACCELERATED AND UNIFORM MOTION. 219

cljap* XV, art- 2, num. 2) is velocity taken in one iaixt hi

[loint of time only, all the impetus, together tiiken

into the tiu»i?.

in all the points of time, will be the same thing "'"^"^^t^""

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
^■sy in the same proportion continually with the
^Kimes which are passed, the whole velocity of the
^Tnotion shall be represented by a triangle, one side
thereof is the whole time, and the other the
■^reBtest impetus acquired in that time ; or else by
» parallelogram, one of whose sides is the whole
time of motion, and the other, half the greatest
impetus ; or histly, by a parallelogram having for
one side a mean proportional between the whole
time and the half of that time, and for the other
5side the half of the greatest impetus. For both
these parallelogrmns 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-
^juired impetus ; as is demonstrated in the ele-
ments of geometry.

2- In all uniform motions the lengths which are lu all moiion*
transmitted are to one another, tus the product of wl^j '.["fr^'J^^
the mean impetus multiplied into its time, to the ''^^^^''"^^"*

^'^'il ^"^ product of the mean impetus multiplied also into
— ^ — ' its time.
iwX^tro'ducu For let AB (in fig. 1) be the time, and A C the
Zi^Lnmukh ™P^^^is by which any body passes with uniform
plied into time, motion througli the length D E ; and in any part of
the time A B, fis in the time A F, let another body
be moved with miifonn 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
lenf^th A F. Seeing, therefore, when boches 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 foUoweth, 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 A 1 is to the parallelogram A H,
that is, as the product of the time A B into the
mean impetus A C is to tlie product of tlie time
A F into the same impetus A C. Again, let it be
su])posed 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
A C, and the other in hke 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 MOTION.

fJuct of the menn impetus into the time h to tlir partiii.
product of the mean imjietus into the timcj so the - — r^— ^
iength transmitted in the time A B with the iiope- ^"ekijuiS
trus AC, to the length transmitted in the time A F
v^th the impetus AL; which was to be demon-
strated,

CcrolL Seeing, therefore, in uniform motion, as
1:1.11s been shovra, the lengths transmitted are to
c:>iie another as the parallelograms which are made
l:>y the multiplication of the mean impetus into the
times, that is, by reason of the equality of the im-
^J:*etiis all the way, as the times themselves, it will
^^Lso be, by permutation, as time to length, so time
t:^ length ; and in general, to this place are appli-
^liable all the properties and transmutations of ana-
Xogisms, which I have set down and demonstrated
in chapter xin,

3> In motion begun from rest and uniformly
accelerated, that is, where the impetus increaseth
I 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 : hut 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 L 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-

PART TIL petus increase iinifomily, till in the last point F of
the time A F the impetns acquired be F K ; and

16.

Tn ^11 motion, \^i D E be the length passed throush in the time

the lengtjis,&c, ^ n i n

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 impetns increasing'dlH
from rest in A till there be acquired the impetns
FK, will be as the triangle ABI 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
B I to F K. \\Tierefore, as ABI is to A F K, so
let D E be to D P ; for so, the length transmitted
in the time AB 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
ABI 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
imprfns increasing to F K ; and therefore the
length D E which is transmitted in the time A B

with impetus increasing to B I, to the len^h D P parthi
which is trausmitted in the time A F witli impetus --

16,

increaising to F K, is as the product which i.s made Jj"^i\InJfiJ*^4'^;
of the time A B multipUed into its mean impetus,
to the product of the time A F multiplied also into
its mean impetus ; which w as to be proved,

CoroU, 1. 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- 11. In motion uniformly accelerated, the
lengths transmitted in equal tiroes^ taken iu conti-
nual succession from the beginning of motion, are
as the differences of square numbers beginning
from unity, namely, as 3, 5, 7j &c. For if in the
first time the length transmitted be as 1, in the
first and second times the length transmitted w ill

» 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,
Ind 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
Brest, the length transmitted is to another length
transmitted imiformly in the same time, but with
such impetus as was acquired by the accelerated
motion io the last point of that time, as a triangle
to a parallelogram, which have their altitude and
base common. For seeing the length D K (in %• 1 )
■ 13 passed throngh with velocity as the triangle
W Al B 1, it is necessary that for the passing through
of a length which is double to I) E, the velocity be
as the parallelogram A I ; for the parallelogram A I
is double to the triangle A B L
4. In motion, which beginning from rest is so ac-

PART in. lemtedj that the impetus thereof increases conti-
nually in proportion duplicate to the proportion of

L^ouigXAc! 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 ordiiuitely applied to
that point F. Seeing therefore the proportion of
F K to B I is supposed to be duplicate to that of
A F to A B, the proportion of A F to A B w ill be
subduplicate to that of F K to B I ; and that of
AB to AF will be (by chap. xiii. ait Ul) 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 a|>-
plied, the strait line designing that impetus w ill be
in the same parabolical line A K I. Wherefore the
mean impetus multiplied into the whole time A B
will l>e the parabola A K I B, equal to the paralle-
logram A M, w hich parallelogram has for one side
A B and for the other the line of
h is two-thirds of the im-
parabola is equal to tw^o-

thirds of that parallelogram with which it has its paetiil
altitude and base common. Wherefore the whole * — ' — ^
locity in A B will be the parallelogram A M, as JhVilS^^^
being made by the multiplication of the impetus
AL into the time AB, And in like manner, if
FN be taken, which is two-thirds of the impetus
FK, and the parallelogram F O be completed, F O
will be the whole velocity in the time A F, as being
made by the uniform impetus A O or F N multi-
plied hito tlie time A F, Let now^ the length
Imnj^niitted in the time A B and with the velocity
AM be the strait line DE; 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 A N, or as
the parabola A K I B to the parabola A K F, so is
DE to D R 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. Now 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 F Lj so by construction is D E to D G ;
and as F L is to A N (seeing the time in both is the
same» namely, A F), so is the length DG 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
AN, that is, as the mean impetus AL multiplied
into ite time A B, is to the mean impetus A O
multiplied into A F, so is D E to 1) P ; which was
to be proved,

CorolL !• Lengths transmitted w^ith motion so
accelerated, tliat 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

p\RT iir.

10.
In&llmoiion

point of their times. For as thp length D E is to
the length DP, so is the parallelofrram AM to thi
parallelogram A N, and so the parabola A K I B
to the parabola A K F. But the proportion of the
parabola A K I B to the parabola A K F is triplicatt
to the proportion which the base B I has to the
base FK. Wlierefore also the proportion of DI
to D P is triplicate to that of B I to F K. A

CoroU, H. Lengths transmitted in equal limeA
succeeding one another from the I)eginning, b^
motion so accelerated, that the proportion of th-
impetus be duplicate to the proportion of th^
times, arc to one another as the differences of cubi^
numbers beginning at unity, that is as 7, 1^? 37, &c
For if in the first time the length transmitted be a^
ly the length at the end of the second tune will b<
a.s 8, at the end of the third time as 27? and at th(
end of the fourth time as 64, &c. ; which are cubi(
numbers, whose differences are 7^ 19, 3/, &e.

CoroU. III. In motion so accelerated, as that tin
length transmitted be always to the length trans
mitted in duplicate proportion to their times, thi
length uniforTidy transmitted in the whole tim^
and with impetus all the way equal to that whici
is last acquired, is as a parabola to a parallelograB
of the same altitude and base, that is, as 2 to 3
For the parabola AKIB is the impetus increxisin^
in the time A B ; and the parallelogram A I is th
greatest uniform impetus multiplied into the sam
time AB- Wherefore the lengths transmitted wi]
be as a parabola to a parallelogram, &c., that is
as 2 to 3. fl

a. If 1 should proceed to the explication of swl
motions as are made by impetus increasing in pro

ACCRLKRATED AND UNIFORM MOTION.

22/

ion triplicate, quadniplicato, qiiiiituplicate, Ike, I'art hi
''that of their times, it would l)e

ifi.

labour infinite
and unnecessary. For by tlie same methfid by
which I have computed such lengths, as are trans-
mitted with impetus increasing; in single and dupli-
c^ate proportion, any man may compute such as are
transmitted with impetus increasing in triplicate,
quadruplicate, or what other proportion he pleases.
In making w^hich computation he shall find, that
where the im|i€tus increase in proportion triplicate
to that of the times, tliere 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 time*;. And in like manner, when* 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
leagthi* transmitted will be in quintuplictite j>ro-
portion to that of the times ; and so on continually.

6. If two bodies with uniform motion transmit iitwo badie*
two lengths, each with its own iiopetus and time, uniJlVrnir^'*
the proportion of the lenarths transmitted wiU 1m^ titm iimiuirh

* * ^ ^ I wo lunfjLhs,

eompomided of the proportions of time to time, t^efoporriun

and nnpetus to impetus, directly taken. to on« anotiier,

Lt^t two bodies be moved uniformly (as in fig. 3), pllIJerrTVh.

one ui the time A B with the impetus A C, the F"P«rtioo« of

* tIDR* to tJITR",

otlier in the time A 1) with the impetus A E. I •^ad unpeius
say the lengths transmitted have their proportion direcUy ukL-n.
to one another compounded of the prtjportious of
A U to A 1), and of A C to \ E. For let any
Ipiigth whatsoever, as Z, be transmitted by one of
the bodies in the time A B vvitli the i:

22H

MOTIONS AND MAGNITUDES.

Tf ti^o boiHes
lie iiioveH, jtc,

PART iTL other body in the time A D with the impetus A E ;
and let the parallelograms A F and A G be com-
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 A G ; the proportion of Z to X w ill be
compounded of the same proportions^ of w hich tlie
proportion of A F to A G is compounded ; but the
proportion of A F 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 impetus
A C to the impetus A E. Wlierefore 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 w ere as A B to AD (in the
same fig, 3) so reciprocally A E to AC, the pro-
portion of A F to AG would be compounded of
the proportions of A B to A D, and of A C to AE,
that is, of the proportions of A B to A D, and of
A D to A B. \\Tierefore, A F would be to A G as
A B to AB, that is, equal ; and so the two products
made by the multiplication of impetus into time
^ oi^ifl bf> <>qiiol : fiTul by consequent, Z would be

e moved in the same
j ictus, the lengths trans-
to impetus. For if the

ACCELERATED AND UNIFOBM MOTION. 229

time of both of them be AD, and their different PAiri in
impetus be A E and A C, the proportion of AG to — ^—
DC i^ill be coinponiided of the proportions of A E
to AC and of A 1) to A D, that is, of the propor-
tioDs of A E to A C and of A C to A C ; and so
the proportion of A G to D C, that is, the propor-
tiuti of lenf]fth to lenii:tli, will be as A E to A C, that
is, as that of impetus to impetus. In like manner,
if two bodies be moved nniformly, 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 A B and AD, the proportion of A F
to DC will be compounded of the proportions of
AB to AD and of AC to AC; that is, of the
proportions of A B to A D and of A D to AD;
and therefore the proportion of A F to DC, that is,
of leugth to length, will be the same with that of
AB to A D, whieh is the proportion of time to time.

7. If two bodies pass through two lengths with iftwo bodin
uniform motion, the proportion of the times iuiwoJeugth^
which they are moved will be eompounded of the ^i.'fiJJ^fu/o"
proportions of length to length and impetus to pf^porti«»' «t'
impetus reciprocally taken. one iinoih..r,

For let any two lengths be given^ as (in the same pouiKied^nfthe
%. :i) Z and X, and let one of them be transmitted I^^IX^'" ^*
mth the impetus A C, the other with the impetus ^^"^^^^^ »"<' »**
AE, I say the proportion of the tmies m which tui rccjpro-
they are transmitted, will be compounded of the aiViht pV«jMir-
proportions of Z to X, and of AE, which is the ^.^[.i'^j;,.^
impetus with which X is transmitted, to AC, the'*"**^^^^^^^'^ *^*^
impetus With which Z is transmitted, tor seeing the propurtioua

of length to

A F is the product of the impetus A C multiplied Wugth,

znd

230

MOTIONS AND MAGNITUDES.

PART 11 L
16.

Uma to Lime
rcdprocally

into the time A B, the time of motion through Z
will l)e a line, which is made by the application of
the paraUelogram AF to the strait line AC, which
Hne is 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 multiphed 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 compoinided 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, A G, D C, and it will be manifest that
tlie proportion of AF to DC is compounded of the
proportions of A 1^^ to A (! and of AG to 1) C ; but
A F is to D C as A B to A D ; wherefore also tlie
proi)ortion of A B to A D is compounded of the
l>roportious 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 D C^ 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 ; w hich w as to be demon-
strated.

In the same manner it may be proved, that in
two uniforiu motions the proportion of the impetus
is compounded of the proi>{>rtiuns of length to
time reciprocally taken.

the same fig. 3) to be

oetus with %vhicli the

and A E to be the

ACCELERATED AND UNIFORM MOTION, 231

time, and A D the impetus with which the length X
is passed through, the deinoostration will proceed
as in the la^t article,

8. If a body be carried by two movents toge-
ther, w%ieh move with strait and uniform motion^
and concur in any given angle, the Hne by wMch
that body passes will be a strait line-
Let the movent A B (in fig. 4) have strait and
uniform motion, and be moved till it come hito the
place C D ; and let another movent A C, having
likewise strait and uniform motion, and making
with the movent A B any given angle C A B> be
understood to he moved in the same time to D B ;
and let the body be placed in the point of their
enuconrse, A. I say the line w hich that body de-
sLTihes with its motion is a strait line. For let the
piirallelo£:ram 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^
patting A D in G ; and through tlie point G let H I
be drawTi parallel to the strait Hnes 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
i^iU be in C D ; and in like manner, when A C is
inBD, the body will be in B D, But AB is in
CD at the same time when AC is in B I) ; and
tJierefore the body w ill be in C D and B D at the
^me 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
ill which they are transmitted, when A C is in E F,

PART HI,
16.

If a. Wdy bf?

carried ou with
uuifomi motion
by two uio-
veDti together,
which meet in
Ml angle, tlic
line by which it
pasj^es will be a
strait VwiL', aiib*
tending llie
complement of
that aii^te to 2
right angles.

233

MOTIONS ANl

riTDES.

VXKT III.

W a b(K]y be
carried by two
itioTentfl toge-
ther, one of
tijcm being
moved widi
umfoTmf ibe
otJii?! with ac-
celerated mo-
lioo, and tbe
ljra|H)rtioo of
tbeir lengths to
their times be-
iiig explicable
ill unmbeni,
liow to find ont
what line that
body dfltchbefl.

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 J
time A H, Wlierefore A B will be in H I in the i
same time in which A C is in E F, so that the body
wM at the same time be in E F and H I, and there- I
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 hne A D,
though the motion be not uniform, provided it i
have like acceleration ; for the proportion of A B
to A E will always be the same with that of A C -
to A H. 1

9. If a body be carried by two raoveuti^ toge-
ther^ which meet in any g^iven angle, and are
moved, the one uniformly, the other with motion
uniformly accelerated from rest, that is, that the
proportion of their impetus be as that of their
times, that is, that the proportion of their lengths
be duplicate to that of the lines of their times, till
the line of greatest impetus acquired by accelera-
tion be equal to that of the line of time of the xini- I
form motion ; the line in which the body is carried
will be the crooked hne 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 wirh nnifonn motion to C D;
nd let anothiT in ii the strait line A C be

supposed h time to BD,

but f <\> that is.

ACCELERATED AND UNIFORM MOTION. 233

mih such motion, that the proportion of the part iil
spaces which are transmitted be always duplicate ^ — '^ —
to that of the times, till the impetus acquired be ^^i^^^^^
B D equal to the strait line A C ; and let the
^^emiparabola A G D B be described. I say that by
'i::he concourse of those two movents^ the body will
"be carried through the semiparabolical crookerl
line A G D. For let the parallelogram A B D C be
^c^ompleted ; 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,
tihrough the point G let H I be draw^i parallel to
t:he 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 couimon point
G* And so it will always be, in wliat 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 w as to be demonstrated.
p 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
bereasing 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
ore 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 ini-

urquired. j

rait line AB (in the 0th figure) be moved i

PART III.
16,

imiformly to C D ; and let another movent A C be
>- — , — ' moved at the same time to B D with motiou so
cl^i^it*" at!celerated, that the proportion of the leng^ths
transmitted be everywhere triplicate to the pro^
portion of their times ; and let the impetus acquired
in the end of that motion be B D, 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
ifay, that by the concourse of the two movent*
together, the body will be always in that crooked
line A G D, For let the parallelogram A B D C he
completed ; and from the point E, taken any w here
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 w hen
A C is in E F, A B will be in HI; and therefore
the moved body will be in the common point G.
And so it w ill always be, in w hat part soever of
A B the point E be taken ; and by consequent, the
body wiU always be in the crooked line AGD;
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,
w hich 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-
cale, triplicaiey &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

and time be added together ; and let their sum be

the deijouiiuator of a fraction, whose uuinerator

must be the number of the length. Seek this frac- eLri^d^'^flt^

tion in the table of the third article of the xviith

chapter ; and the line sought will be that, which

denominates the three-sided figure noted ou the

left hand ; and the kind of it will be that, w hich is

nmnbered above over the fraction* For example,

let there be a concourse of two movents, w^hereof

one is moved uniformlv, the other with motion so

accelerated, that the spaces are to the times as 5

to 3. Let a fraction be made w hose denominator

kthe sum of 5 and 3, and the numerator 5, namely

the fraction |. Seek in the table, and yon 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 couctmrse 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
uthtT 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 jmrallcl spaces, less
thau if both the movents had uniform motion ; and
^till less and less, as the moticm of the other
movent is more and more accelerated.

Let the body be placed in A, (in the /th figure)
and be moved by two movents, by one with uni-
form motion from the strait line A B to the strait
line CD iiarallel to it; ami by the other witli any
atcderation, from the strait line A C to the strait

PART III.

If 4 hody be
earned, ^c.

line BD parallel to it; and in the parallelo2:rani
A B 1) C let a space be taken between any two pa-
rallels E F and G H. F say, that whilst the movent
A C passes through the latitude w hich is between
E F and G H, the body is less moved forwards from
A B towards C 1), 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 AC toward.^ BD, 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
A B towards C D ; and let the strait line A F be
drawn and produced indeterminately, cutting G H
in H. Seeing therefore, it is as A E to A G, so E F
to G H ; if A C should descend towards B D with
uniform motion, the body in the time G H, (for I
make AC and its parallels the measure of time,)
would be tbund in the point H. But because AC
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 I K,
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 Fand G H, tJie body is
less moved forwards from A B towards C D, than

ACCELERATED AND UNIFORM MOTION.

PART

I IT,

it would have been, if Ix^tli the motions had been
uniform ; whieh was to be demonstrated.

13. Any length being given, whieh is imssed 'a^,^f^^
through in a given time with unifonii motion^ to
find out what length shall be passed through in the
same time with motion iniiformly aeeelerated, that
is, with snch motion that the proportion of the
knphs pa.ssed 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 whole time
of the motion.

Let A B (in the 8th figure) be a length, trans-
riiitted w ith 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 A B D C 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 jirodueed till it
meet with 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-
tiou, so let A H be to A I, that is, let A I be the
balf 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,
tWt is, equal to B E ; and I K equal to B F ; for
BD, that is, G H, B F, and B E, that is, I L, being
continual proportionals, A H, A B and A I will
al^o he continual proportionals. But as A B is to

PART III. A I, that is, as A M is to A B, so is B D to I K, and
^ — , — ' so also k G H^ that is, B D to B F; and therefore
ILJi^lc*' ^ ^ ^^^^ ^ ^ ^^^ equal. Now the proportion of
A H to A I is duphcate to the proportion of A B
to A h that is, to that of B I) 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. Tlie body therefore proceeding
from rest in A, with motio!i uniformly accelerated
in the time A C, when it has passed throui^h the
length A H, will acquire the impetus G H equal to
the time A C, that is, such impetus, as that witli it
the body will pass throus^h the length A C in the
time A C. Wlierefore any length being given, &c.,
which was propounded to be done.

14. Any length being given, which in a given
time is transmitted witli 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 triphcate 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 miiforra 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
impetus last acquired be equal to the time

he parallelogram ABDC be completed;

} D be so (Uvided in E, that B E be a third

le whole B D ; and let B F be a mean pro-

jH>rtionaI between B D and B E; and let A F bo part in.
drawn and produced till it meet the strait line C D ^— A^
in G; and lajstly, let the parallelogram A C G H be H^^^fX^
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 A I be to another,
A N ; and from the point N let N Q be draw n pa-
rallel to A C, cutting A G, A D, and F K prodxiced
in P, M, and O ; and last of all, let F O and L M
lie drawn, which will be equal and parallel to the
strait lines B N and IN. 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 a,
N O3 N M and N P, will be continual proportionals,
and in the same proportion with A H, A B, A I
and A N. Wlierefore the proportion of A H to
A N is the same with that of B D, that is, of N ti
to N P J 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
19 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 w hose 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 tlie lengths
acquired in any of the times triplicate to the pro-

If B body bt;
carried, fitc.

portions of the times themselves. Wlierefore A H
is the length required to be found out.

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 U be made to a third, and
again so that third to a fourth, and that fourth to
a fifthj A Nj so that the proportion of A H to AN
be quadruplicate to that of A H to A B, and the
parallelogram N B F 0 he 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 serv^es for the find-
ing ont 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
AC 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 drav^ii and pro- ^—r^ — -
duced till it meet with C D produced in G ; and ^i^'^jj'**
making A M a mean proportional between A H
and A B, let it be m A M to A B, so A B to A I ;
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
wliich 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 draw n 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 AC 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
beAH.

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
AB might have three of the same proportions,
of i\hich A H to A I 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 leugth be trmuimiited
^^ that time with uniform motion ; as for exmnple^
if the time be AC, and the length A B, the 6 f rait
AG, which determines the length C G or A Hj

VOL. 1. R

PART 11 L transmitted in ike same time AC with any acce-
^ — ^ — • lerated motion^ shall so cut B D in Fj that B F
JfrriedfL^'' vyA«// be a mean proportional between B D and
B E, B E being so taken in B D, that the propor-
tion of fengih to length be everywhere to the pro-
port ion of time to tinier as the whole BD is to itJt
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 A I 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 unifornnly accelerated ; I say, that as
A H is to A B, so will A B be to A L

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 shaU be a mean proportional
between A H and A I. Ag^n, because A I and A B
are described by the same motions, if I K be

ACCELERATED AND rxiFORM MOTION, 243

irawn parallel and equal to B F or AM, and part hi.

vided in the midst at N, and betwee!i I K and ^ — ^^
I N be taken the mean proportional I L, the strait J^^^^-^^f
line A F wili^ by the demonstration of the same
art, 13, pass through L. And dividing A B in the
midst at O, the line A I will be a mean proportional
between A B and A O. Where A B is di\ided in
I and O, in like manner as A H is divided in B and
I ; and as A H to A B, so is A B to A I. Which

as 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 nni-
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

k 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
Tvhole length transmitted, as it would do if the
other motion were also uniform, and tlie 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,
(it) the I Ith figure) be understood to be moved with
uniform motion till it be coincident vdxh 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

PART III. be moved with motion uniformly accelerated, till
it be coincident with B D ; then dividing: A B in

If A body be
carried, &c.

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 nnderstood to
be moved in the same time AC 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 FG. ]
For seeing A F is a mean proportional betw een
the whole A B and its half A E, B D will (by the
1 3th 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 lengthy trans-
mitted by A C with motion uniformly accelerated,
will exceed the length transmitted by the same
AC 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 FG, the part F B, seeing it concurs not at all
with the motion of the side A C w hich 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 put 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.

wards, as is sho\\^i in the 12th article. When part in,
therefore AC is in FG with accelerated motion, — ^^ — -
the body will not be in tlie side C D at the point G, J^^^J.^^
bat at the point D ; so that G D will be the excess^
by which the length transmitted with accelerated
motion to B D exceeds the length transmitted with
uniform motion to F G ; so that the body by its
acceleration avoids the action of the part A F, and
comes to the side C D in the time A C, and makes
the length C D, w hich is equal to the length A B.
lYherefore uniform motion from A B to C D in the
time A C, w orks no more in the w hole length A B
upon the body miiformly accelerated from A C to
BD, than if AC were moved in the same time
with uniform motion to FG; the diflFerence con-
sisting only in this, that when A B works upon the
body uniformly moved from AC to FG, that, by
i^hich 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 w hole length
AB or CD, 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, &c. ; which w^as to be demoostrated.

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
toAE 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
^ the same method, that the side, which is moved

imifonn motion, works as much with its c
course throug:h 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 coneemiug motion by concourse.

ofn I

CHAP. XVIL

OF FIGURES DEFICIENT-

. Definitions of a deficient figure; of a complete figure; oftbe
complement of a cleficieot figure; and of proportiotjsi which
are proportional and commensurable to one anotUer. — 2. The
proportion of a deficient figure to ita complement- — 3. The
proportions of deficient figures to the parallelograms m 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 propoition the same figures exceed a
slrait-Iined triangle of the same altitude and base.— 7* A table
of solid deficient figures described in a cyUnder. — ^8. Id 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 tlie 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 propoi^ition demonstrated in art, 2 confirmed
from the elements of philosophy.^13. An uimsuaJ way of
reasoning concerning the equality between the superficies of a
portion of a sphere and a circle, — IK 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.

. I CALL those deficient ^gures which may be

' understood to bt* generated by the uniform motion

of some quantity, which decreases continually, till
at last it have no magnitude at alL

And I call that a eomplete Jigurey answering to
a deficient figure, w4iich is generated wath the
same motion and in the same time, by a quantity
which retains always its whole mtignitude*

The complement of a deficient figure is that which
being added to the deficient figure makes it com-
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 dnplicate to the second, and again, the
third be duplicate to the fourth^ those proportions
are Baid to be proportionaL

And commemurable proportions are those^ which
are to one another as immber to nimiber. 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 I ;
and therefore I call those three proportions com^
mnmrable,

2. A deficient figure, which is made by a quantity
eoutiuually decreasing to nothing by proportions
everywhere proportional and commensurable, is to
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-
liished in the same time.

Let the quantity A B (in fig. 1), by its motion
through the altitude AC, describe the complete
figure A D ; and again, let the same quantity, by

PART in*

17.

Dofiivitiooaof a
completc^figuie;
of tlie com pie*
meut of II duti-
dent figure^
and of propor-
tioaa which are
proportional k.
cominensuTaUe
to one aaother.

The proportioi.
of a dffiinent
ifif uru (0 its
ooQiplemenL

PART nr.

TfiT pioponioo riT^ll
r»fu deficient ^^^^
figure to its
complement.

decreasing continually to nothing in C, describe
the deficient figure A B E F C, whose complement
be the figure B D C F E. Now let A B be
supposed to be moved till it lie in GK, so that the
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
G E, And in like manner, let H I 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 H I to H F ; and let the same be done in as
many parts of the strait line A C as is possible ;
and a line be drawn throiigh 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. fl

For (by art, 2, chapter xv,) the proportion o"
the complement B E F C D to the deficient figure
A B E F C is aU the proportions of D B to B A,
OE to EG, GFto FH, 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 FC. And seeing the pro-
portions of D B to O E, and of D B to Q 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 triplicate of the proportions of
aF to D B, andof OE to DB kc. and therefore
the deficient figure A B E F C, which is the aggre-

Q F, 0 E, D B, &c. ; which was to be proved.
It follows from hence* that the same complement f"^^"'

• r>rvnii'l

gate of all the lines H F, GE, A B, &c. is triple part in.
to the complement B E F C D made of all the lines ' — ^^—^

Tbe ppopuriioo
of a deficient
ure to ita
^ _____ ^ __., ^ _ , , _, , t 1 coniplemenU

B E F C D IS I of the whole parallelogram- And
fay the same method may be calcidated in all other
deficient figures, generated as above declared, the
proportion of the parallelogram to either of its
parts ; as that when the paridlels increase from a
point in the same proportion, the parallelogram
wriU be divided into two equal triangles ; when
one increase is double to the other, it wiU 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 drawTi cutting G K in
L, and through L let M N be drawn parallel to the
strait line A C ; w herefore the parallelograms G M
andLD will be equal. Then let LK 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 betw een G K and G L is to the arith-
ttietical proportion between G K and the same GK
wanting the third part of L K ; and K E w ill be
somewhat greater than a third of L K. Seeing
How the altitude A G or M L iSj by reason of the
continual decrease, to be supposed less than any
tioantity that can be given ; L K, which is inter-
cepted between the diagonal BC and the side BD,

The pro|,K)ition
of a dt-ticiejiC
figure to ki

iny quantity
given ; and consequently, if G be pat so near to A
in g^ as that the difference between C g and C A
be less than any quantity that can be assigned,
the difference also between C / (removing L to /)
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 A to G / %vill still be
triplicate to the proportion of G k to G e^ and the
difference between k and e^ the third part of k /,
will be less than any quantity that can be given ;
and therefore the parallelDgram 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
eqnal to G E, cutting C B in P, the crooked line in
F, and O E in I, and the proportion of Cg to C H
will be triplicate to the proportion of H F to H P,
and 1 F wQl be greater than the third part of P L
But again, setting H in /i so near to g^ as that the
difference betwx^en Ck and Cg may be but as a
point, the point P will also in p be so near to /,
as that the difference between Cp and CI will be
but as a point; and drawmg kp till it meet with
B D in /, cutting the crooked line in /, juid having
drawn eo parallel to B D, cutting DC in o, the pa-
rallelogramy'o w ill differ less from the third part afl
the parallelogram g /\ than by any quantity that
be given. And so it w ill be in all other spaces
rated in the same maimer. Wlierefore the
ences of the arithmetical and geometrical
g, which are but as so many points B,^,y, &c^

OF FIGURBS DEFICIENT.

251

(seeing the whole figure is made up of .so many part hi.
iudivisible spaees) will constitute a certain line, ' — ^ — '
such as is the line BE FC, which will divide the
complete figure A D into two parts, w hereof one,
namely, A B E F C, which I call a deficient figui"e,
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 thu
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-
«>t: rated that the deficient figure had been quadruple
^o its complement ; and so in any other proportion*
^^Hierefore, a deficient figure, which is made, &€•
^^tich was to be demonstrated.

The same rule holdeth also in the diminution of
^He bases of cylinders, as is demonstrated in the
^^cond article of chapter xv.

3, By this proposition, the magnitudes of all The pmportion
^f?ficient figures, when the proportions by which gur^IrruTc pa-
tljeir bases decrease continually are proportional J^^ilieh^^^^
^o tliose by w hich their altitudes decrease, mav be de«cribed, nn
Compared with the magnitudes of their comple-
^:*ients ; and consequently, with the magnitudes of
tilieir complete figures. And they will be found to
\ie, as I have set them down in the following
tables ; in which 1 compare a parallelogram w ith
tihree-sided figures; and first, with a strait-lined
tiriangle, made by the base of the parallelogram
continually decreasing in such manner, that the
altitudes be always in proportion to one another

PART III, aj3 the bases are, and so the triangle will be equal
■— r— - to its complement ; or the proportions of the alti-
Jf^'dXr^"" tudes and bases wiU be as 1 to I, and then the
figures, Btc. 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
midtiplied, 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 ajs 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 tshows that the
deficient figure is to the parallelogram as f to U
or as 5 to 7-

OF FIGURES DEFICIENT.

253

1 1

3 3 4 fi 6 7

: : : : :

*♦[:::::

♦ *:f

til

: : :

i i

tVA

i tV

iVA

VVr

-Ar

PARTIIL

J^amilelogram ,,....
Stmit-sideH triangle , . , •
n»ree-sided figure of I mean
Hiree-sided figure of 2 means .
TTiree-sided figure af 3 means .
1 rhree-sided figure of 4 means .
tree-sided figure of 5 means .
r*hree-sided figure of 6 means ,
T*liree-flided figure of 7 means .

' 4. Now for the better understanding of theDescriptu

niature of these three-sided figures, 1 will show f^^^^^T "^^

ti.Qw they may be described by points ; and first, **^'^''"'

Pilose which are in the first column of the table.

Any parallelogram being described, as A B C D

(in fip^e 2) let the diagonal BD 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 B H H D be drawn ;

and the figure B H H D C will be that which I call

a three-sided figure of one mean, because in three

proportionals, as E F, EG and EH, there is but

one mean, namely, E G ; and this three-sided

figure will be f 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 B 1 1 D 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

PART JIJ,

DetcripiioD Sc
producLiun of
the same
Hgures.

MOTIONS AND MAGNITUDES,

proportions be further continued in E F, there will
be made the rest of the three-sided figures of the
first column ; which I thus demonstrate. Let there
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 D C or K M to
K Nj 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 K M be placed. Wliere-
fore the figure B H H D € is double to its comple-
ment B H H D A, and consequently f of the whole
parallelogram. In the same manner, if through I
be drawn O P I Q parallel and equal to C D, it may
be demonstrated that the proportion of O (i to
O Pj that is, of B C to B O, is triplicate that of
O Q to O I, and therefore that the figure B II D C
is triple to its complement B I I 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 BC D (in fig. 3) be a parallelogram, who^e
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 D C, I draw
E F as often as is necessary, cutting B D in G ; and
between E F and E G, I take three proportionals
E H, E I 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

make the prnelMction i»f

points

s the second ; an(
drawn through all the points K, ,.i*. «i**.v^ ^.^^ p,,
figure BKDC the third of those three-sided ;J;;^"»<=
fipires. The first of these, seeing the proportion
of E F to E G is qiiadmplicate of that E F to E H,
win 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 4 or
f of the parallelogram. The third, seeing the pro-
proportion of E F to EG is that of E F to E K ns
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 Fbe 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
FE and FG ; the crooked line BKD produced,
will pass through the point H, For if the strait
line B H be drawn, cutting CD in I, and H L be
dfawii 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
% in quadruplicate proportioTi to that of F E to
FH, or of C D to C I. Wherefore if the line BKD
be produced according to its generation, it will
upon the point H,

PART rn.

17-

The drawiDg
of tangents
to ihem»

In what pro por-
tion the same
i^gures exceed

5. A strait line may be drawn so as to touch
the crooked line of the said figure in any point, in
this manner. Let it be required to draw a tangent
to the line B K 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 Hne 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 G and RP. Wherefore {by
corolL of art* 28, chapter xiir.) R K will be less
than R O ; and therefore M D wiU fall without the
figure. Now if M D be produced to N, F N will
be the greatest of three means in arithmetical pro-
portion between F E and F G ; and F H will be the
greatest of three means in geometrical proportion
between the same FE and FG. Wherefore (by
the same corolL of art. 28, chapter xiii.) F H 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.

6. The proportion of a deficient figure to its
complement being known, it may also be known

OF FIGURES DEFICIENT.

257

what proportion a strait-lined triangle has to the part iir.

excess of the deficient figure above the same tri- — r^—

angle ; and these proportions I have set down in frflngu oToie

the following table ; where if you seek, for ex- J^^d ^Me?"^*

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 A ; by which is signified, that that three-

aded figure exceeds the triangle by two-tenths or

by one-fifth part of the same triangle.

The triangle ....
A three-sided fig. of 1 mean .
A three-sided fig. of 2 means
A three-sided fig. of 3 means !

^\k three-sided fig. of 4 means

6 7

1 : : : : :

f • • *

f
*

: : :

: :

A =

A A

A three-sided fig. of 5 means
A three-sided fig. of 6 means
.A three-sided fig. of 7 means

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 des^M m'*a
figures, namely, the proportions between them and <^y^*°^«'-
a cylinder. As for example, in the concourse of
the second column with the three-sided figures of
four means, you have i ; which gives you to un-
derstand, that the solid of the second three-sided
figure of four means is to the cylinder as * to 1, or
as 5 to 9.

VOL. I.

Inwhatpi'opor*
tloa the same
figures exceed
m cone of the
■ame alutude
and baset

A cylinder .•♦.*.

A cone .•.,».,
- A three-sided fig. of 1 mean

A three-sided fig-, of 2 means

A tin'ee-sided fig. of 8 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

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 :

The Cone * . , .
_Of the solid of a three-sided)

♦

1 : : ; :

* " •

AjA

J •

tVA-VW

•
•

rt-

tVIt^

•flr

•frliViV

I'd

figure of 1 mean
Ditto ditto, 2 means

Ditto ditto, S means

Ditto ditto, 4 means

Ditto ditto, 5 means

Ditto ditto, 6 means

Ditto ditto, 7 means

S 6

. . .

1: : : : : : t

-sided)

: : : :

* ■ «

aI : :^

How i plain
deficient figure
may be describ-
ed in aparalle-
lograiLit to that
it be to a tri-
angle of the
tame base and
aUitude» as an*
other deficient
figuFCf plain or
■olid, twice ta-
ken, h to the
»• deficit tit

9. If any of these deficient figures, of w hieh I
have now spoken, as A B C D (in the 5th figure) be
inscribed within the complete figure B E» having
A D C E for its conaplemeut ; and there be made
upon C B produced the triangle A B I ; and the
parallelogram A B 1 K be completed ; and there be
drawn parallel to the strait line C I, any number
of linesj as M F, cutting every one of them the

OF FIGURES DEFICIENT. 259

crooked line of the deficient figure in D, and the pakt hi,
strait lines A C, A B and A I in H, G, and L ; and ^ — A—
as G F is to G D, so G L be made to another, G N ; &Km!
and through all the points N there be drawn the piete figure, in

" * which It u de-

line A N I : there will be a deficient figure A N I B, icribed.

virliose complement will be AN IK. I say, the
figure A N I B is to the triangle A B I, as the de-
ficient %ure A B C D twice taken is to the same
deficient figure together with the complete figure
B E.

For as the proportion of A B to A G, that is, of
G ]\1 to G L, is to the proportion of G M to G N,
»o is the magnitude of the figure A N I B to that
of* its complement A N I K, by the second article
c^f* this chapter.

But, by the same article, as the proportion of
A. B to AG, that is, of GM to GL, is to the pro-
portion of G F to G D, that is, by construction, of
G L to G N, so is the figure A B C D to its comple-
uxent ADCE.

And by composition, a^ 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 A B C D.

And by conversion, as the proportion of G M to
GL 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
GN, which is the proportion compounded of both,
*o is the deficient figure A B C D to the complete
fipireBE.

But it was, as the proportion of G M to G L to
that of GMto GN, so the figure ANIB to its
complement A N I K. And therefore, A B C D. B E
• • ANIB. A NIK are proportionals. And by com-

PART IlL
17.

The tntnsfer-
ring of ceriain
pioperties
t»r deficient
figures des-
cribed ill & pa*
rnlklog7ani to
llic propor-
imnB of spAccs
Lrat I emitted
Viiih several
degrees of
velocity.

position, ABCD + BE. ABCD:: BK. AN IB
are proportionals.

Aiid by doubling the consequents, A B C' D +
B E, 2 A B C D : : B K. 2 A N I B are proportionals.

And by taking the halves of the third and the
fourth, A B C D + B E. 2 A B C D : : A B I. A N I B
are also proportionals ; which was to be proved,

10- From what has been said of deficient fi^^ures
deseribed in a parallelograin, may be found out
what proportions spaces, transmitted with accele-
rated motion in determined times, have to the
times themselves, according as the moved body is
accelerated in the several times Avith one or more
degrees of velocity.

For let the parallelogram A BCD, in the 6th
figure, and in it the three-sided figure D E B C be
described ; and let F G be draw^i aiiyw here 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
w ill be triple to its complement B E D A ; and in
Uke manner, I E being draw^n parallel to B C, the
three-sided figure E K B F w ill 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 E K B F,
will be to one another as the parallelograms AC
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 E K B F will be quadruplicate to the proper- part hi.
tion of the same DC to F E. And by the same — A—
ixiethod, may be found out what proportion any of ^® ^*"^J|^"a
the said three- sided figures has to any part of the Pi°P^/^i*»

^ "^ . ,. of deficient

B^tme, cut off from the vertex by a strait line pa- figures, &c.
x«Uel 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
CD. Seeing therefore the proportion of BC to
BF 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 E K B F 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 DEBC 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 E K B F to the figure DEBC will be
quadruplicate to that of B F to B C ; that is, the
pi'oportion of the sum of the velocities in the time
B F, to the sum of the velocities in the time B C,
^ni be quadruplicate to the proportion of B F to

17.

The traii*f«r
ring of certain
properlic*
of deficient

B C. Wherefore if a body be moved from B w
velocity so iiicreasingj tliat the velocity acquired
in the time B F be to the velocity acquired in the
time B C in triplicate proportion to that of the
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 tlierefore the proportion of C D, that is, of
F G to FE is snbtripHcate to that of B C to B F ;
the proportion of F G to F E will also be subtripli-
cate to that of FG to FH. Wherefore the propor-
tion of F G to F H is triplicate to that of F G, that
is, of CD to FE. But in four continual propor-
tionals, of which the least is the first, the propor-
tion of the first to the fourth, (by the IGth article
of chapter xin), is subtriplicate to the proportion
of the third to the same fourth. Wherefore the
proportion of F H 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,
B F to B C ; w hich was to be proved.

It may from hence be collected, that when the
velocity of a body is hu^reased in the same propor-
tion with that of the times, the degrees of velocity
above one another proceed as numbers do in ini-
tiate succession from unity, namely, as 1, 2, 3, 4,
And when the velocity is increased in pro-

OF FIGURES DBFIGIBNT. 268

portion duplicate to that of the times, the degrees part iit.
proceed as numbers from unity, skipping one, as * — ^

1. 3, 6, 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.

11. Moreover, it is to be noted that as in quan- or deficient fi-
tities, which are made by any magnitudes decreas- S7^h^!^r***
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
semldiameters 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

tJ'VA W Vf^a M.\fM.

The propoti'
tion ciemon-
Htrott d ill art
2 cohfinncd
frorti Uie ele^
menu of |thi'

spaces as may be described by a line or siipei
decreasing either in magnitude or power ; Si
if the proportions, in which they decreas
commensurable to the proportions of the till
which they decrease, the magnitudes of the fi
they describe will be known,

12. The truth of that proposition, which
. monstrated in art. 2, which is the foundation
. that has been said concerning deficient fi{
may be derived from the elements of philos
as having its original in this ; timf all eqt
ami ineqiudity between two effect\s\ thai i
proportion y proceeds from y and is deter mint
the equal and unequtd eauh*es of those effe\
Jrom the proportion which the causes^ con
to one effect J hate to the causes which coi
the producing of the other effect ; and that
fore the proportions of quantities are the
with the proportions of their causes. Si
therefore, two deficient figures, of which ■
the complement of the other, are made, ot
motion decreasing in a certain time and propo:

OF FIGURES DEFICIENT. 265

is, the proportions of the remainders of all the partiil
times and altitudes, may be other proportions than - — A-'
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 areAnunuioii
determinable from the knowledge of their causes, ki'g <^nccrn?ng
namely, from the comparison of the motions by i!ltwTn*^£J
which they are made ; and that more easily than "J^JJ^rtSn
from the common elements of geometry. For of*"P^.«'«
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 B A C (in fig. 7) be a portion
of a sphere, whose axis is A E, and whose base is
BC; and let AB be the strait line drawn from
the pole A to the base in B ; and let AD, equal to
AB, touch the great circle B AC in the pole A.
It is to be proved that the circle made by the
i^us AD is equal to the superficies of the portion
BAC. 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
ABthe superficies of a right cone. Now seeing
^th the strait line A B and the arch A B make

An uniisuiil
way of feasors

one and the same revolution^ and both of them
have; the same extreme points A and B, the eause
why the splierical superficies, which is made by
the arch, is greater than the conical snperficies,
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 DAB (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 1) A B is the cause why the
supeiiicies of the portion, described by the arch
A B, 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 lioth 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 supeiiicies 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 tbe
superficies of the portion and that of the circle
do equally exceed the superficies of the cone.
Wherefore the cin^lc made bv A D or A B, and

OF FI0URE8 DEFICIENT. 267

the spherical superficies made by the arch A B, are ^'^^t hi.

17.

equal to one another ; which was to be proved.

14. If these deficient figures, which I have de- How from the
scribed in a paraUelogram, were capable of exact defid'e^nlfiju °m
description, then any number of mean propor- ^^ » paraiicio-

* ■' * ■■• gniin»any num-

tionals might be found out between two strait lines ber of mean
^TCD. For example, in the parallelogram A B C D, mTy^folTnd
(in figure 8) let the three-sided figure of two means two^l^natrtit
l>e described (which many caU a cubical parabola); ^'°**"-
and let R and S be two given strait lines ; between
^vvhich, 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
BA, and cut the crooked line in E ; then through
B let G H be drawn parallel and equal to the strait
Kne A D, and cut the diagonal B D in I ; for thus
^e have G I the greatest of two means between
GH and G E, as appears by the description of the
figure in article 4. Wherefore, if it be as G H to
GI, 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
inay be found out, by the description of a three-
sided figure of four means ; and so any other num-
ber of means, &c.

rARTIH,
18.

OF THE EQUATION OF STRAIT LINES WITH THE
CROOKED LINES OF PARABOLAS AND OTHER
FIGURES iMADE IX IMITATION OF PARABOLAS.

1 ♦ To find the strait line equal to the crooked line of a semi pa-
rabola*— 'i. To find a strait line equal to the crooked line of
the first seniip^rabolaster, or to tiie crooked line of any other
of the deficient figures of the table of the 3d article of the
precedent chapter.

1, A PARABOLA being given, to find a strait line
equal to the crooked line of the semiparabola.
lineeqiiat'oihe Let the paraboHcal line given be ABC (in
1^5.11^^^^^^^^ 1), and the diameter found be AD, and the

base drawn DC; and the parallel o^gram ADCE
being completed^ draw the strait line AC. Then
dividing A D into two equal parts in F, draw F H
equal and parallel to 1) 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 O,
AP and P C\ 1 say that the two lines A P and
P C, taken together as one line, are equal to the
parabolical line A B O C\

For the line A B O 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 STRAIT LINES^ ETC. 269

lately applied in the semiparabola will design the part hi.
jarts of time wherein the body, that describe th - — ^ — '
;he line A B O C, is in every point of the same ; so Hne, &c** "**
iat as at the end of the time AE or DC it is in C,
$0 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
vrill design also the continual augmentation of the
impetus, till it be at the greatest, designed by the
base DC. 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 FK; and the line
AO will be uniformly described by the concourse
of the two uniform motions in A F and F O in the
time FO. From O draw O L parallel to E C,
cuttmg A C in L ; and draw L N parallel to D C,
catting 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 H, F P and F O ; and a strait
fine parallel to E C passing through M will fall on
P; and therefore O P will be the increase of im-
petus gained in the time F O or I L. Lastly, pro-
duce PM to CD in Q; and QC or MN or PH will
he the increase of impetus proportional to the time
FP or I M or D Q. Suppose now uniform motion
from H to C in the time P H. Seeing therefore in
4e time F P with uniform motion and the impetus
Jucreaaed in proportion to the times, is described

270

MOTIONS AND MAGNITUDES,

PART IIF.
18.

To find a aUftll
liue equal lii
LhiT crooked line
of llie first se-
mi parabolasier
or lo the crook-
ed liiu* of any
other of the de-
fidciit figures
ik( the J able oi
art.3 of I he pre-
ceding chapter.

the straight hue A P ; and in the rest of the time
and impetus, namely, PH, is described the line
C P uniformly ; it followeth that the whole line I
A PC is described with the whole impetus, and in
the same time wherewith is described the parabo- _
lical line A B C ; and therefore the line A P C, 1
made of the two strait Imes A P and PC, is equal
to the parabolical line ABC; which was to be
proved.

2. To find a strait line equal to the crooked line
of the first semiparabolaster. ■

Let A B C be the crooked line of the first semi-
parabolaster ; A I) the diameter ; D C the base ;
and let the parallelogram completed be A D C E,
whose diagonal is A C. Divide the diameter into
two equal parts in F, and draw F H equal and
parallel to DC, cutting AC in K, the crooked Une
in 0, 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, J
and the strait line E C in N ; and produce it on "
the other side to A D in 1. Lastly, through the
point M draw P M (i parallel and equal to H C^ f
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 Kne A B O C*

For the line ABO C, being the crooked Une 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
Test in A to D, so as that the impetus inereaseth
in proportion perpetually triplicate to that of the M
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

tbut of the times of their traiisinissbn ; for as the part iir.
impetus or quicknesses increase, so the lengths ' — r^
transmitted increase also. And because the mo- i^*Jt.^'"a[^,.* ' ''^
tion from A to E is uniform, the line AE may
^ene to represent the time, and consequently the
nines, ordinately drawn in the semiparabolaster,
^iU design the parts of time wherein the body,
l)eginning from rest in A, describeth by its
oiaotion the crooked line ABO C. And because
DC, which represents the greatest acquired im-
j)etns, is equal to A E, the same ordinate lines will
Tepresent the several augmentations of the impetus
increasing from rest in A. Therefore, supposing
liuiform motion from A to F, in the time F K there
IV ill be described, by the concourse of the two
liniform motions A F and F K, the line A K uni-
formly, and K O will be the increase of impetus h
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 1) hi 1, the crooked line ABC in
M, and the strait line E C in N ; and through the
point M the strait line P M Gl pai^allel 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 tlie time F P will be uniformly
described the strait line A P ; and L M or O P
ml] be the increase of impetus to be added for the
time F O. And because the proportion of I N to
1 L is triplicate to the proportion of I N to I M,
the proportion of F H to F O 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.

PART III. So that FH being equal to DC, which designed
'^ — ^ — ' the whole impetus acquired by the acceleration,

To find a strait ti« • ■ i? * ^ i. i.

Vme, «tc. there is no more increase of impetns 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 ABO 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-
ius, and in our time Bonaventura) thinking the
hry of the ancients had reached to all that
J be done in geometiy, thought also, that
it 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- ^ — ^^
pated by some of those ancients themselves, whe- jfng^'J^.* "^^
ther there might be any equality at all between
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 des{>ised, as he had reason. And there is a
late writer that granteth that between a strait
and a crooked line there is equality; but now,
says he, since the fall of Adam, without the spe-
cial assistance of Divine Grace it is not to be
found.

CHAPTER XIX.

OP 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-

litelyy 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

inddeDce. — 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 wiU make an angle equal to twice that angle, which is

made by two strait lines drawn from the centre to the points of

inddence, together with the angle which the incident lines

themselwes 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* ami the i
they make be less than twice the angle at the centre, the line«
reflected from them and meeting within the circle will make an
angle, which being added to the angle of the incitlent lines will
be equal to twice the angle at the centre. — 6* If through any
one point two une(|ual chorda 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 tlte two former
lines reflected, — 7* In equal chorda the same h 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 maj
contain any angle given, — 9. If a strait line falling upon the
circumference of a circle be produced till it reach the semU
diameter, and that part of it, which is intercepted between
the circumference and the semidiameter, be equal to that part
of the gemidiameter 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 !ine»
be drawn to the circumference, and their reflected lines meet
in the circumference of the same circle, the angle made by tfie
reflected lines will be a third part of the angle made by the in-
cident lines.

PART III, Whether a body falling upon the superficies of

'^^ /-^ another body and being reflected from it, do make

kci^denc^^ equal angles at that superficies, it belongs not to

and reflection* this pkce 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-
phedt not to the finding out of the causes, but
some consequences of the same.

I call an tuigle of incidence, that which is made

OF ANGLES OF INCIDENCE, ETC.

275

PAKT Xtt

10*

en a strait line and another line^ strait
looked, upon which it falls, and which I call t
line reflecting ; and an angle of rejiection equal
to it, that which is made at the same point between
the j?trait line which is reflected and the line
reflecting,

1. If two strait lines, which fall upon another i/twoitrwt
strait line, be parallel, their reflected lines shall beupoa»*JSfcr
^1^0 parallel. ^fiK

Let the two strait lines AB and CD (in fi^r. nii net reflected

^ 0/3 ffom them

wliicli fall upon the strait line EF, at the points »haiuiso be
B and D, be parallel ; and let the lines reflected ^*
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 D F 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
U H are pm^lleL

2, If two strait lines drawn from the same '/**»<» Bteait
point fall upon another strait line, the lines re- from one point
fl<^cted from them, if they be drawn out the other Slcr"£i°t(l^
Way, will meet m an angle equal to the angle of the ^|^^^
JJicident lines. them, if they

From the point A (in fig. 2) let the tv^o strait the other w»/,
Knes A B and AD be drawn ; and let them faUSrequl" ^
upon the strait line E K at the points B and D ; b*"'j;°^I*„^/^
and let the lines B I and D G be reflected from iiwidence.
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 BED
shall be equal to the angle BAD.

For the angle of reflection I B K is equal to the

T 2

PART HI. angle of incidence ABE; and to the aii^le I
' — T^ — ^ 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
proved-

Coroll. I. If the strait line A F be draw n, it w ill
be perpendicular to the strait line E K. For both
the angles at E will be equal, by reason of the
equality of the tw o 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
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.
^^*'!?f^^ 3. If from tw'O points taken without a circle,

paraJlel lines, ^ * ^ '

drawn not op- tw^o strait parallel lineSj draw^n not oppositely, but
from the same from the saoic pafts, fall upon the circumference ;
ih*e^irl*l!LiX" t^^^ \in^s reflected from them, if produced they
ren-e of a cir- meet w ithiu the circle, w ill make an angle double

cIp» the Imea ^ ^ ^ f-^

reflected from to that w hich is Qiadc by two Strait lines drawn
dicTd tije7' frf>^ ^h^ centre to the points of incidence.
fhrci^kjlm Let the two strait parallels A B and D C (m

make an angle fipr^ 3) fall upou thc circumfeTence BC at the

double to that ^. '' '

which i* made points B and C ; and let the centre of the circle be
liaeTdrair E ; aud Ict A B reflected be B F, and D C reflected
from the centre ^^^QQ^ aud Ict the liucs FB and GC produced

LUU

to the poioLs
incidence.

meet within the circle in H ; and let E B and E C

ftw:

he connected. I say the angle F H G is double to ^art hi.
the angle B E C. ^ — .^—

For seeing A B and D C are parallels, and E B Jl^Veui
mts A B in B, the same E B produced will cut ^^**"* ^*^
DC somewhere; let it cut it in D; and let DC
be produced howsoever to I, and let the intersec-
tion of D C and B F be at K, The angle therefore
I C H, being external to the triangle C K H, will
be equal to the two opposite angles C K H and
CHK. Again, ICE being external to the triangle
CDE, 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 C K H
and CHK are equal to the two angles at D and E

ice 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 altem 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 I C falling
pon 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 froL th* Tame
the lines reflected from them being produced meet ^JJ-deTIiUpln
within the circle, they will make an angle equal to ^^ ^^^^^^

278

MOTIONS AND MAGNITUDES.

PART

19.

HL twice tliat angle, which is made by two strait lines
drawn from the centre to the points of incidence,

Hi!r*rXcted^ together with the aiigU* which the incident lines
frora them themselvcs make.

bung pioduced * r^ i

meet within Let the two strait lines A B and AC (in fig. 4)
win inak^jlil^^ be drawn fi'om the point A to the eircimiference
?wfce Z^.T- <^f the circle, whose centre is D ; and let the lines
gie, which b reflected from them be BE and CG, and, beinjs:

made hy two i

strait lines produccd^ make w ithm the circle the angle H ;
cemre t'o'^the ^ also let the two Strait lines D B and 1) C be drawn
dStoytle^fr^^'^ the centre D to the points of incidence B
^^J^ *^^ *"g^e and C. I say, the angle H is equal to twice the
jpcident angle at D together with the angle at A.
irifJjmSc, For let AC be produced howsoever to L There*
fore the angle ICH, which is external to the
triangle C K H, will be equal to the two angles
C K H and 0 H K. Again, the angle I C D, 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 ICD, 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
ALB, is equal to the two angles LAB and LB A;
and consequently CLD twice taken is equal to
L A B and L B A twice taken. Wherefore CKH
and C H K are equal to the angle C D L together
with LAB and L B A twice taken. Also the
angle C K H is equal to the angle LAB once and
A B K^ that is, L B A twice taken, Wlierefore
the angle C H K is equal to the remaining angle
C D L, that iSj to the angle at D, twice taken, and

OF ANGLES OF INCIDENCE, ETC,

279

the angle L A B, that is^ the angle at A, once
taken ; which was to be proved.

CorolL If two strait converging lines, as IC and
MB, fall upon the concave circumference of a
circle, their reflected linet^, as C H and H H, will
meet in the auja^le H, equal to twice the angle D,
together with the angle at A made by the ineithnt
lines produced. Or^ if the incident lines be H B
and I C, 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
niade 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
N B H, that is, K B A. But the angles K 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
upon tlie 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, w^ill be equal to twice the angle at the centre.

Let the two lines AB and AC (in fig. 5), drawn
from the point A, fall upon the concave circum-
ference of the circle whose centre is D ; and let
their reflected lines B E and C E meet in the point
E ; also let the angle A be less than tmce the
angle D, I say, the angles A and E together
taken are equal to twice the angle D.

For let the strait lines A B and E C cut the

PART itu

19.

If two fitrait
lines drawn
from one point
fall upon Ine
concave cir-
cumferuace of
a circlet aq^
the angle they
make be less
ihtm twice the
angle at the
centre, the
lines reflected
fionv tjjem and
meeting within
the circle will
make an angle^
wbicli heiug
Added to tbe
anf^le of the m*
cidetultnei will
be «qiLAl to
twieo the angle
ut the centre*

If two strait
linea drawn
from one, ^c,

Strait lines D C and D B in the points G and H ;
and the angle B H C will be equal to the two
angles E B H and E ; also the same augle B H C
will be equal to the two angles D and DCH ; and
in like maimer 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,
ACD and A, are equal to the four angles D, DCH,
DBG and D. If, therefore^ equals be taken away
on both sidesj namely^ on one side ACD and
EBH, and on the other side DCH and DBG,
(for the angle E B H is equal to the angle D B G,
and the angle ACD equal to the angle D C H),
the remainders on 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 tw ice the angle D, the reflected Unes
B E and C E will be parallel ; and if it be less,
they will eoncurj 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,
f 6. If through any one point two unequal chords
be drawn cutting one another, either within the
circle, or, if they be produced, w ithout it, and the
centre of the circle be not placed betw een them,
and the lines reflected from them concur where-

OF ANGLES OF INCIDENCE, ETC,

281

fleeted.

Miner; there cannot, through the point through pabtiu,
which the former lines were drawn, be drawn ^ — r^
another strait line, whose reflected line shall pass f^*""* betwe^Q

^ * them, and the

through the point where the two former reflected iii<^* reflected

J. from them con-

lines concur. curwbereio-

Let any two unequal chords, as B K and C H cl^noftlwugh
(in fig. 6), be drawn through the point A in the [^rorlT^ i *
circle B C ; and let their reflected lines B D and tiie two former
CE meet in F; and let the centre not be betwxa^n drawn^^i^
AB and AC ; and from the point- A let any other ^[w.trdtiine
strait line, as AG, be drawn to the circumference w>io»e reflected
oetween B and C. I say, GN, which passes through tiie
through the point F, where the reflected lines B I) of rhTiwo^fo"!
and C E meet, will not be the reflected line of A G. '^''' ^^^^» '*'■

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 M B L equal to the angle

^ B G, A G and M L will also be equal, and, pro-

<3ucing G A to the circumference in I, the whole

Xines LO and G I will in like manner be equal.

iJut L O is greater than G F N, as shall presently

>5e demonstrated ; and therefore also G I is greater

than GN. Wherefore the angles NGC and 1GB

sre not equal. Wherefore the line G F N is not

xeflected from the line of incidence A G, and con-

sjequently no other strait line, besides A B and

AC, 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
GN; which I shall do in this manner. LO and

I PART m. G N cut one another iu P ; and P Lis greater thaa
— ^— P G. Seeing now L P, P G : : P N, P O are propor-
tionals, therefore the two extremes L P and P O
together taken, that is L O, are greater than P G
and PN together taken, that is, GN; which re*
raained to be proved.
Saiquakiiords 7, B^t if two eoual chords be drawn throuerh one

1110 tame la not , ■* . ^

||rQ0, point within a drc^le, 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 iu 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 K 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; a,s also
the angle B C H to the angle EDI; and the angle
A M C to its verticle angle F M D ; 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 Ihie F D ; and
the chords C G and G D being drawn, will also be
e€[ual ; and also the angles F D G and ACQ, in the
equal segments G D I and G C B* Wherefore the
strait lines F G and A G are equal ; and, therefore,
the ajigle 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 € G ; which was to be proved.

OF ANGLES OF INCIDENCE, ETC.

283

Coroll, By the very sight of the figiire it is maui- ^^^^ i^^*
fest, that if G be not the middle point between C " *- ^
and D, the reflected Hue G h will not pass through
the point F.

R. Two points in the circumference of a circle J^^ p^^^!* ^«-
being given to draw two strait lines to them, so as circumference
tbftt their reflected lines may be parallel > or con-arawtwo'^strau

fr*:*^ ^»«* »»is.l^ ^»«»^n lines to them I

tarn any angle given. ^ thattheir«

In the circumference of the circle, whose centre ^^^^^^ ^^"^

. may coutam

is A, (In the 8th figure) let the two poiuti<i B andanyangiegiveii^
Che 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 hues may, first,
be parallel.

Let A B and A C be drawn ; as also any incident
line D C, with its reflected Une 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 DC produced in I. Lastly, pro-
(kcing A B indefinitely to K, let G B be dra\^n 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 hue H B. I say, DC and H B are two
incident hnes, whose reflected lines C F and B G
re parallel.

For seeing the angle E C D is double to the angle

AC, the angle H IC is also, by reason of the
parallels E C and H I, double to the same BAG;
therefore also F C and G B, namely, the lines re-
Hected from the incident lines D C and H B, are
Iiarallel. Wherefore the first thing required is
ine*

Secondly, let it be required to draw to the points
B and C two strait lines of incidence, so that the

PART iiT, lines reflected from them may contain the given

^—r- — ' angle Z.
TnggLtTnlh; To the angle ECD made at the point C, let there
ciruumfereiice b^v added OH onc sidp the ansrle D C L equal to half

of a circle, &c. ^ , i i

Z, and ou the other side the angle EC M equal to
the angle DCL; and let the strait lineBN be
drawn parallel to the strait line C M; and let the
angle K B O be made equal to the angle N B K ;
which being done, B 0 w ill be the line of reflection
from the line of incidence N B. Lastly, from the
incident line LC, let the reflected line CO be
drawn, cutting B O at O, and making the angle
C OB, 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 LC M is, by construction, equal to twice the
angle B A C, together w^ith 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 B A C, together w ith the angle Z.
And seeing the two strait lines O C and O 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 N P L will be equal to twice the angle
BAG together with the angle C O B. But I have
already proved the angle NPL to be equal to twice
the angle B A C together with the angle Z* There-
fore the angle COB is equal to the angle Z; w^here-

re, twH> points in the circumference of a circle

given, Ihave drawn,&c,; which was to be done,

: if it be required to draw^ the incident lines

. point within the circle, so that the Unes re-

from them may contain an angle equal to

tlie an^le Z, the same method is to be usedj saving
tliat in this ease the angle Z is not to l)e added to
twice the angle B AC, but to be taken from it.

9, If a strait line, faUing upon the circumference
Vi a circle^ be produced till it reach the seniidia
teeter, and that part of it which is intercepted be-
tween the circumference and the semidiameter be
equal to that part of the semidiameter which is
between the point of concourse and the centre^ the
Inflected line will be parallel to the semidiameter.

Let any hne AB (in the 9th figure) be the semi-
diameter of the circle whose centre is A : and upon
tlie circumference B D let the strait line C D tall,
and be produced till it cut A B in E, so that E I)
and E A may be equal ; and from the incident line
CD let the iine DF be reflected- 1 say, A B and
I) F will be parallel.

Let A G be drawn through the point D, Seeing,
therefore, E D and E A are equal, the angles EDA
and E AD will also be equaL But the augles FDG
and EDA are equal ; for each of them is half the
angle EDH or FDC. Wierefore the angles FDG
and EAD are equal; and consequently DF and
A B are parallel ; w hich w^as 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
He less than E D^ then B A and DH being produced
will concur.

10, If from a point within a circle two strait
lines be drawu to the circumference, and their re-
flected Unes meet in the circuinference of the same
circle, the angle made by the lines of reflection will
be a third part of the angle made by the Unes of
iiiL'ideuce*

If a strait line
frilling upon the
c i re umfe fence
of a circle be
produced till it
reach the semj-
dmmeter, nnd
that pan of it,
which is inter-
cepted between
the circurafer-
enceand thelle-
^]idiamctf?r, be
equal to that
part of the te-
rn idinmeter
which is he-
fween the point
of concoitrie
and the central
the reflected
line will be pa-
ralkl to the &e*
mldkmeter.

If from a point
within a circle
two jitrait linei
be drawn to tlie
circuniference,
and their re-
l!ected lines
meet ID thecii-

PART III,
19.

From the point B (in the 10th figure) taken
within the circle whose centre is A, let the two
cwmfereucfl of gtrait lincs B C and B D be drawn to the circum-

thc same circle,

ihe angle made ferencc ; and let their reflected lines C E and D E
liuea wJi he& meet in the circumference of the same circle at the
SCdt'Iy point E. I say, the angle C E D will be a third
Hnet'"'^'^'"* part of the angle C B D/

Let A C and A D be dra\^Ti. Seeing, thereforei
the angles CED and CBD together taken are
equal to twice the angle CAD {as has been de-
monstrated in the 5th article) ; and the angle
CAD twice taken is quadruple to the angle CED;
the angles CED and CBD together taken will
also be equal to the angle CED four times taken;
and therefore if the angle C E D be taken away on
both sides, there wiU remain the angle C B D on
one side, equal to the angle CED thrice taken ott
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 folloT?^dng chapter.

DIMENSION OF A CIRCLB^ ICTC. 28/

CHAPTER XX.

OF THE DIMENSION OF A CIRCLE, AND THE
DIVISION OF ANGLES OR ARCHES.

1. The diineliflioii of a circle never determined in numbers by
Architnedet and oth6r8«--S. 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. — i. The third
attempt ; and some things propounded to be further searched
iato^ — 5* The equation of the spiral of Archimedes with a
stiait line.— 6. Of the analysis of geometricians by the powers
of lines.

1. In the comparing of on arch of a circle with a ^^^^/"*
strait line, many and great geometricians, even ^"7"^^.
from the most ancient times, have exercised their of a*cireUMver
wits ; and more had done the same, if they had ^^^'J^fy*''
not seen their pains, though undertaken for the Archimedes
common good, if not brought to perfection, vilified
by those that envy the praises of other men.
Wongst 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 diflFering
fi'om 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,
Ludovictts Van Cullen and Willebrordus Snellius,

PART in* ^vith joint endeavour, have come yet nearer tzrJi

20.

the truth ; and pronounced from true prineiplesr^

^ttrr^^l^^^ the arch of a quackant, putting, as befor^

determined io 10,000,000 for radius, differs not one whole nnitryl

niunbera by i-i*f-iijl

Archimedeii from the numbcr 15,707,963 ; which, if they ha.ci
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 len^gth of the circumference were
exactly set out, either by nimibers, or mechanically,
or only by chance, yet this would contribute no
help at all towards the section of angles, unless
happily these two problems, fo diride a giren
angle according to any proporiion unsigned, and
tojind a strail line equal to the arch 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, 1

DIMENSION OF A CIRCLE, ETC.

289

thought fit to attempt the same by geometry, pabtitt
and in this ch^ipter to make trial whether it might ^ 1 1^^
not be performed by the drawing of strait and
circular lines.

2. Let the square ABC D (in the first figure) The nm
be described ; and with the radii A B, B C, aud thl^ finding
D C, the three arches B D, C A, and AC ; of which ^4,;^^^ j
let the two B D and CA cut one another in E, and^^^^cie byline*.
the tw o 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 B H D 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 O P be draw^
parallel and equal to the side BC, cutting the
arch B F D in F, and the sides A B and D C in O
and P. Therefore OF will be the sine of tbe 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 midcUe 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 B F ; and consequently that B V, the triple of
BT^ is equal to the arch of the quadrant B FED.
Let T F be produced till it meet the side B A
oduced in X ; and dividing O F in the middle

VOL. 1. u

PART in. in z, let Q Z be dvRwa and produeed till it meet

^— - — ' with the side B A produced. Seeing therefore the

ftttem^Vor Strait lincs R S and O F are parallel, and divided

?«lSfth!f ^^ ^^^ ^^^®^ ^^ ^ ^^^ Z, QZ produced will tall

diineniion of a upoii X, and X Z Q produced to the side B C wUl

circle by lines. m^ m . t - i -

cut B T m the nudst in «•

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 r/, with the
radius a Z, let the arch Z F be dravvTi ; wliich arch
Z F will therefore be equal to the arch Q F, the
half of the arch B F, Again, let the strait line
Z 0 be cut in the midst in i, and the strait line
b O in the midst in c ; and let the bisection be
continued in this manner till the last part O r 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 O c. If, therefore, upon
the centre r/, with the radius d O, be di-awn the
arch O c, and upon the rest of the equal parts
of the strait line 0 F be drawn in like manner so
many equal arches, idl those arches together taken
will be equal to the whole arch B F, and the lialf
of them, namely^ those that are comprehended
between O 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 O c be of the
strait line 0 F, the same part will the arch O e be
of the arch B F, though both the arch and the
chord be infinitely bisected. Now seeing the
arch O e is more crooked than that part of the
arch BF which is equal to itT and seeing also

DIMENSION OF A CIRCLE, ETC.

291

'20.

ofl

circle hy line*.

that the more the strait line X ^ is produced, tlie
more it diverges from the strait line X O, if the
points O and e be understood to be moved for- ^ttcmpJVor
wards with strait motion in X O and X r, the ^^'"^ '^"fi"^

' om of the

archOc* will thereby be extended by little and Jmitnsion*
little^ till at the la.st it come somewhere to have
the same crookedness with that part of the arch B F
which is equal to it. In like manner, if the strait
Uiie X A be drai^ii, and the point h be understood
to be moved forwards at the same time, the arch
c b will also by little and Utile 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 OF. It is also
manifest, that by strait motion in X O and X Z
all those small arches will lie in the arch B F»
in the points B, Q and F* And though the same
small equal arches shoidd not be coincident with
the equal parts of the arch B F in all the other
jioints thereof, yet certainly they \v]l\ constitute
Xv»o crooked lines^ not only equal to the two
arches B Q and Q ¥, and equally crooked, but
^so having their cavity towards the same pjirts ;
^^hich how it should be, unless all those small
arches should be coincident with the arch B F in
all its pomts, is not imaginable. They are there-
fore coincident, and all the strait hues drawn
from X, and passing through the points of division
of the strait line O F, w ill also divide the arch
B F into the same proportions into which O F is
divided*

Now seeing X b cuts off from the point B the

u 2

292

MOTIONS AND MAGNITUDES,

^^^ "^* fourth part of the arch B F, let that fourth part

— ' — be B e ; and let the sine thereof, J^ e^ be produced

^^L^pTVor to FT in g, for so fe will be the fourth part of

the findiag the strait line / e^, because as O A is to O F, so is

out of the , €/ O 3 »

djmenaion of i/ €? to J g. But B T is greater than^ «-; and
y ines. therefore the same B T is greater than four sines
of the fourth part of the arch B F, And in hke
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
sme of one of those small arches, so many times
taken as there be parts made of the whole arch
B F. Wherefore the strait hue B T is not less
than the arch B F. But neither can it be greater,
because if any strait Hne whatsoever, less than
B Tj be dra^Ti 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 everyw here 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 BV, 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*

the Riidolpliine numbers. Nevertheless, if in the pa^t '^I-
place of BT, another strait line, though never so — ^ — -
little less, be substituted, the division of angles is attcm^ for
immediately lost, as may by any man be demon- Jj;^^ l^^^
strated by this very scheme. dimenjiioii or a

"^ .i* 1*11* -I- ^"^^^ byline*.

Howsoever, if any man thmk this my strmt hue
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 B V 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
I of the arch B F, should cut off in the tangent B V
^■60 many strait lines equal to them, seeing the con-
p nected straight line X V passes not through the
point D, but cuts the strait line A D produced in /;
' and consequently require some determination of
this problem. Concerning which, I will say what
I think to he 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 ; otherwse
not. For A V being connected, cutting the arch
BHD in I, if XC 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
tnie that the arch B F is equal to the strait line BT ;
aiid drawing X K it would cut the arch B 1 in the
midst in i ; also drawing A i and producing it to the
tangent B C in A, the strait line B k will be the

PART 111. tangent of the arch B i, (which arch is eqnal to
half the radius) and the same strait line B k will
be equal to the strait line i 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 A shall not cat 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 oflF in the tangent by the strait
lint XE produced to the tangent, may also be
known ; and it will be found to be less than B A ;
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 fuUy in
the next article concerning the magnitude of the
arch B I.

And let this be the fii^t attempt for the findings
out of the dimension of a circle by the section of
the arch B F.

The second g ] j^j^g^p qq^ attempt the same by argunient*^

iindiiig QUI cf drawn from the nature of the crookedness of the^
ofacirciufrojii circle itsclf; but I shall first set down some pre —
liliVoTthr'*^ mises necessary for this speculation ; and

First, if a strait line be bowed into an arch of
a circle equal to it^ as when a stretched thready
w^hich touch eth 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 j
and consequently the crookedness of the arch of a

11)1 lure of
trtHtkedness,

rirele k everym here uniform ; which needs no other i*art kt.

* 20.

demonstration than this, that the perimeter of a ^—^^^^
circle is an uniform line. «t'n:;""rU,.

Secondly^ and consequently : if two unequal *^"tiing, &c.
arehes of the same circle be made by the bow ing 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
to the crookedness of the lesser arch, as the
greater fireh is to the lesser arch.

Thirdly : if two miequal circles and a strait line
touch one another in the same point, the crooked-
ness of any arch taken in the lesser circle, will be
greater than the crookedness of an arch equal to it
t:aken in the greater circle, in reciprocal proportion
lio that of the radii with which the circles are
described; or, which is all one, any strait line
l>eing drawn from the point of contact till it cut
fcoth the circumferences, as the part of that strait
Xine 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

\wo circles, touching one another^ and the strait

line A D in the point A ; and let their centres be

^UE 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 crookethiess

I of the arch A H, as A E is to A F. For let the

^■itrait line A D be supposed to be equal to the arch

^■k£, and the strait line A G to the arch A C ; and

VAKTiiL let AD, for example, be double to AG. There-
* — ^ — ' fore, by reason of the likeness of the arches A B
!itnpub?tbe «iiid A C, the strait line A B will be double to the
finding, &c. gtrait line AC, and the radius A E double to the ra-
dius A F, and the arch A B double to the arch A H.
And because the strait line A D 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 w ill be equal to
the flexion of the strait line A D into the crooked
line A B, But the flexion of the strait hue 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
vas to be demonstrated.
Let the square A B C D be again described (in
e third figure), and in it the quadiants A B D,
2 A and D A C ; and dividing each side of the
are A B C D in the midst in E, F, G and H, let

DIMENSION OF A CIRCLE, ETC.

297

EG and F H be cotinected, which will cut one an- part iil
other in the centre of the square

at I, and divide

20.

rhe arch of the quadrant AB D into three equaPJ^^^^^^J ji,^

parts in K and L. Also the diagonals A C and fi«dmg, ^.

B D being drawn will cut one another in I, and

divide the arches B K D 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

ill O ; 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 pohit E in the arch E F, and let the

arch a h be taken equal to the arch O n, and let

B a and B h be drawn and produced to the arch

AN in c and d; and lastly, let the strait line kd

be drawTi. I say the strait Une A d is equal to the

arch A N or B M.

1 have proved in the preceding article, that the
arch E O is twice as crooked as the arch B P, that
b to say, that the arch E O is so much more
crooked than the arch B P, as the arch B P is more
crooked than the strait hue E a. The crookedness
therefore of the chord E a, of the arch B P, and of
the arch EO, are as 0, 1,2, Also the difference
l>eti\ een 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 A, are as 0, 1, 2. So
aUo the difference between the arches AN and
I AN, the difference between the arches A N and
Ac, and the difference between the arches AN
and A r/, are as 0, 1 , 2 ; and the strait line A e is
double to the chord B P or E a^ and the strait line
A d double to the chord E 6.

Again, let the strait Hne B F be divided in the
toidst in Q, and the arcli B P in the midst in II ;

he second at-
tempt for the

and describing the quadrant BQS (whose arch
U S is a fourth part of the arch of the quadrant
B M Dy as the arch B R is a fourth part of the arch
B M, which is the arch of the semiquadrant A B M)
let tlie chord S e equal to the chord B R be set off
from the point S in the arch S Q ; and let B e be
drawn and produced to the arch AN in J*; which
being done, the strait line A^/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 AJ^ above the crook-
edness of the arch Ac 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 mil be double to the arch cj*. WTierefore
the arch c d is divided in the midst in J] and the
arch Ny is | of the arch N d. And in like manner
if the arch B R be bisected in V, and the strait
line B Q in X, and the quadrant B X Y be de-
scribed, and the strait line Y^ equal to the chord
B V be set off from the point Y in the arch Y X,
it may be demonstrated that the strait line B^
being drawn and produced to the arch A N, will
cut the arch J^d into t^o 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
^' =■ A d is equal to so many equal chords of
the arch B M, as may be made by
IS. Wherefore the strait line A d
reh B M or A N, that is, to half
uadrant A B D or B C A.
.»h being given not greater than

the arch of a quadrant (for being made greater^ it ^^i}T nr.
comei? again towards the radius BA produced, -- ^ '-^
from which it receded before) if a strait line double J^^^Tr'^^hl
to the chord of half the given arch be adapted ^ladiog, ate,
from the beginning of the arcli, and by how much
the arch that is subtended by it is greater than the
gi?en archj by so much a greater arch be sub-
tended by another strait line, tMs 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 B H D, and A V be

oounected cutting the arch B H D in I, it may be

asked what proportion the arch B I has to the

arch 1 D. Let therefore the arch A Y be divided

in the midst in o, and in the strait line A I) let

kp be taken equal, and A q double to the drawn

chord Ao. Then upon the centre A, with the

radius Ay, let an arch of a circle be drawn cutting

the arch A Y in r, and let the arch Y r be doubled

atf; which being done, the drawn strait line A^

(by what has been last demonstrated) will be

^ual to the arch AY. Again, upon the centre A

with the radius A t let the arch / u be draw n

Hitting A D in ?^ ; and the strait line A u will be

eqoal to the arch A Y, From the point u let the

strait line us ht drawn equal and parallel to the

strait Une A B, cutting M N in a\ and bisected by

MN in the same point .r. Therefore the strait

line Ax being drawn and produced till it meet

with B C produced in V, it will cut off B V double

to B*, that is, equal to the arch B H D- Now let

the point, w here the strait Une A V cuts the arch

BHD, be I ; and let the arcli DI be divided in

the midst in */; and in the strait line D C, let U z

PART ]iL be taken equal, and D S double to the drawn chord
D y ; and upon the centre D with the radius D S

^VrrAhe ^^^ ^^ ^^^^ ^^ ^ circle be drawn cutting the arch
finding, kc. B H D 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 7fi will (by the last foregoing
corollary) be equal to the arch D I. If now the
stniit lines D m and C V be equal, the arch B I
will be equal to the radius A B or B C ; and con-
sequently X C being dramn, will pass through the
point I. Moreover, if the semicircle B H D € being
completed, the strait lines €1 and BI 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 / being connected w ill be parallel to
the strait line 6 1, and being produced to B C in *,
will cut off the strait line B k equal to the strait
line kl, and equal also to the strait line A y cut
off in A D by the strait line E L All which is
manifest, supposing the arch B I and the radius
B C to be equal.

But that the arch B I 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, I, namely, that the strait lines

drawn from X through the equal parts of O 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

rtp far as B C in the tangent, and BI in the

insomuch that no inequality between

^ I and the radius B C can be discovered

the hand or by ratiocination. It is

to be further enquired, Avliether the

DIMENSION OF A CIRCLE, ETC.

301

ine A V cut the arch of the quadrant in I part hi,
in tiie same proportion as the point C dividei^ tlie — ^U— '
stmit 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 V is eqxial to the
rchBHD,
4, I shall now attempt the same dimension of a ^^ third »u

, ^ ^ ^ lempt ; And

Circle another way, assuming the two following some tMng»

1 propounded

lemmas. ta be further

Lemma i. If to the arch of a quadrant, and the *""'"^'^^ ^^^'^
radius, there be taken in continual proportion a
third line Z ; then the arch of the seraiquadrant,
half the chord of the quadrant, and Z, w411 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
^ch 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
rnh 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
lo the sine of 45 degrees), so is half the chord of
the quadrant to Z ; wiiich was to be proved*

Lemma ii. The radius, the arch of the semi-
qaadrant, the sine of 43 degrees, and the semi-

iius, are proportional.

8(ffi MOTIONS AND MAGNITUDES.

For seeing the sine of 45 degrees is a mean
proportional between the radius and the semi-
Jttei^t! &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
tlie 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 ;
w hich was to be demonstrated.

Let now A B C D (in fig, 4) be a square ; and
with the radii A B, B C and D A, let the three
quadrants A B D^ 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^ di\idethe square A BCD
into four equal squares. Tliey will therefore cut
the arch of the quadrant A B D 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 draw^, 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 drawna 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 Une 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 Ime B P ; I
say then the strait line B P w ill be equal to the
arch BMD.

For seeing P B A and A D R are like triangles,

DIMENSION OP A CIRCLE, ETC.

303

it ffill be as P B to the radius B A or A D, so A D part iil

20.

to D R ; and therefore as well P B, A D and D R, - — ^ — -
asPB, AD (or AQ) and QH are in continual ^^;:^JJ^^^
proportion ; and producing HO to D C in T, DT
wiJ] be equal to the sine of 45 degrrees, 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. D F are
proportionals. And thus it will be, whether 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 Q to H Q, or so B C to C P. And then will
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
drawTi (by the construction of fig. 1) a strait line
equal to the arch B M D, then D R to H Q, and
also the half of the strait line B P to D S, w ill
always be so equal, that no inequality can be dis-
covered between them, I w ill therefore leave this
to be further searched into. For though it be
ahnost out of doubt^ that the strait line B P and
tlie areh B M I) 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.

The third
attempt, Stc

In B A produced let A V be taken equal to the
sine of 45 degrees ; and drawing and producing

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 DC in T, so that DT will be equal
to the sine of 45 degrees, or to the strait line A V ;
also the strait line V 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

V H is equal to three squares of the semiradius.
But H I is a mean proportional bet^^een 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 HV, But
because A D is cut in the midst in H, therefore V 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 43 de-
grees ; for so V X will be equal to the radius ; and
it will be as V A to A H the semiradius, so V X the
radius to X N the sine of 45 degrees. Wherefore

V H produced passes through N. Lastly, upon the
centre V with the radius V A let the arch of a circle
be dravvn cutting V H in Y ; which being done,

V 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, AH which was to be demonstrated.

I will here add certain problems, of which if

any analyst can make the construction, he wiU

thereby be able to judge clearly of what I have now

ar the dimension of a circle. Now

DIMENSION OF A CIRCLE, KTC,

305

these prablems are nothing else (at least to sense) part iil
but certain symptoms accompanying the construe- — ~^ — .
tion of the first and tliird fio;ure of this chapter. ^^^ ^*"/**.

Describing, therefore, again, the square A BCD
(la fig. 5) and the three quadrants A B D, BC A
and D AC, let the diagonals AC and BD be drawn,
cutting the arches B H D and C I 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 eqmd squares,
and trisecting the arches B H D and CIA, namely,
B H D in K and M, and C I A in M and O. 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 G R be double to ft 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 V
be made triple to B S, and consequently, (by the
second article of this chapter) equal to the arch
BD. 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 draw^n,
catting 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 woidd determine their
difference.

Secondly, if in the side D A, the strait line D a
he taken equal to D Z, and V a be drawn ; why
V a and V B should be equal ; or if they be not
equal, what is the difference.

VOL. I. X

PART IIL
20.

The third
attempt^ &c.

Thirdly, drawing Z h parallpl and equal to the
side C B^ eutting the arch B H D in c, and draw-
ing the strait line A r, and producing it to B V 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 takiTig (in e A produced) ef equal to the dia-
gonal AC, and conncctingy'C ; whyy*C should
pass through a (which point being given, the length
of the arch B H D is also given) and c ; and why
fe and ^/> should be equal; or if not^ why un-
equal.

Fifthly, drawingy'Z, I desire he would show,
w hy it is equal to B V, or to the arch B D ; or if
they be not equal, what is their difFerence.

Sixthly, .^ranting y'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 B D «- being completed,
why g 1 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 //, why A r/, which
is equal to the arch B I), should pass through that
point h.

Eighthly, upon the centre of the square A BCD,
wliich let be k^ the arch of the quadrant E / L being
draw n, cutting e K produced in /, 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 1> V, or to the arch
B H. and drawing w/? parallel and equal to the

OIMRNSTON OF A CIRCLE, ETC.

307

side B A, cutting the arch B I) in o, why the strait part hi.
line which connects V / should pass through the — ^.^— -

point O. Tl.e third

Tenthly, I would know^ of him why the strait
line which connects r/ H shouhl be equal to B ;// :
or if not, how much it differs from it.

The analyst that can solve these problems w ith-
out knowing: tirst 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 ordinarj^ geometry i^
able to perform. But if the dimension of a circle
cannot be tbund 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 squfiring 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 beinsc'^^^*^^!!^^'^?",*'^

f '**♦ , . Iht' spiral of Ar-

iouiidj that strait line is also found, which touches chiincdc-» with
a spiral at the end of it^ first conversion. For upon
the centre A (in fig. 6) let the circle B C D E be de-
scribed ; and in it let Archimedes' spiral A FG H B
be drawTi, Ijeginning 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 1 I>e equal to the perimeter
B C D E B. Tlierefore I B being draw n will touch

X 2

PART in. the spiral A F G H B in B ; which is demonstrated
' — -r-^ by Archimedes in liis book De Spiralihus.

JAir^ptlllaf \iid for a strait line equal to the given spiral

Archimerks A F G H B, it niav be found thus.

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 rectaugle 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 Dioved by the concourse of t^ o
movents, the one from I M to K L w ith velocity
encreasing continually in the same proportion with
the timeSj the other from M L to I K uniformly,
that both those motions begin together in M and
end in K; Galiteus 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 li ; 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 FG H B, are like. But the unifonn motion
in A B concurring with circular motion iu the peri-
meters of all the concentric circle^^ describes that

DIMENSION OF A CIRCLE, ETC.

309

the '^'^'^''i™'^*^^^

withastraitUnif

circle, whose centre is A^ and perimeter B C 1) E ; i'^^i^t in.
mid, therefore, that circle is (by the coroU. of art. ^- — 1~^
1, chap. XVI) the aggregate of aU the velocities to^ ^^Jj^^^^^^^
gether taken of the point A whilst it describes
spiral A FG 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 Hne M K is described, is to the
whole velocity with which the spiral line AFGH B
is described in the same time, as the rectangle
I K L M is to the circle B C D E, that is to the
triangle A I B, But because A I 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 v elocity 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 MethocL^^J^^^^^^^y^^*
that which I should there have spoken of the ana- cians by the
lytics of geometricians I thought fit to defer, tjg. p^*^"^*"* ^^^
cause I could not there have been understood, as
not ha\ang then so much as named ilneji, snperfi^
cieSj solids^ equal and unequal^ %'e. Wherefore I
will in this place set down my thoughts concern-
ing it.

Analynh 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

PART 111. thost* definitions, and so on. till we come to some

20,

* — ^ — ' things known, the compt>sition whereof is the
n7?eor.trr'*t^^*»c>iistration of the tnith or falsity trf the first
cinns by rho siipDosition ; and this composition or demonstration

powers of luies. ^ ^ ^ /• •

is that we call Si/nfhesh. Amdiftica^ therefore, is
that art, by which our rcitsoii proceeds from some-
thing supposed, to principles^ thai is^ to prime
propositions, or to such as are known by these, till
we have so many known propositions as ar^ snffi*
cient for the demonstration of the truth or falsity
of the thing supposed. Sijnthetiea is the art itself
of demonstration. Synthesis, therefore, and ana-
lysis, differ in nothing, but in proceeding forwards
or backwards ; and LogiHiiea 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
enmiciated hyiiothetically, 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 othenvise, 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 resolvhig, we say, these two
rectangles are equal, and therefore their sides are
reciprot ally proportional, we must necessarily in
componndhig say, the sides of these rectangles are
reciprocally proportional, and therefore the rect-
angles themselves are equal ; w hich we could not
say, unless rectangles hare their aides reeipro-

DIMENSION OF A CIRCLE, ETC. 311

^aUy proportional^ and rectangles are equal, partiit.
Tvere terms convertible. - — ^ — '

Now in every analysis, that which is sought is ^/g^ometri^"'
tie proportion of two quantities ; by which pro- cian« ^y the
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,
«r the construction of the problem.

And seeing analysis is reasoning from something
supposed, till we come to principles, that is, to
de&iitions, 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
Dttake construction. But he must of necessity stay,
when he comes to prime propositions ; and these
«re definitions. These definitions therefore must
contwn the efficient cause of his construction ; I
s^y of his construction, not of the conclusion which
he demonstrates ; for the cause of the conclusion
18 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.

cians by ihe
powers of lines

PAJOiT III. But the cause of his construction is iu the thuigs
^ — r^ — ' themselves J and consists in motion, or in the eon-
Ko'm^^'rlr'' course of motions. Wherefore those propositions,
in which analysis ends, are definitions, but such as
signii^^ in what manner the construction or gene-
ration of the thing proceeds. For otherwise, w hen
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
scientifieal ; and no demonstration is scientifical^
but that which proceeds from the knowledge of the
causes from which the construction of the problem
is driiwn. To collect therefore what has been said
into few words; analysis h rafiocinatioujrom
the supposed constniction or generation of a thing
to the efficient eause or coefficient causes of that
which is constructed or generated. And SYN-
THESIS is ratiocination Jrom the first causes i^
the constntetion^ continued through all the middle
causes till we come to the thing itself* which is
constrneted or generated.

But because there are many means by w hich 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

other strait^ may be made congruous or coincident p^»^J in.
And this method Arc^himedes made use of in his -- V—
book De Spfraiibm. Also the equality or inequa- ^;*^*'*
lity of two quantities may be found out and *='a°« i^y f^^

♦ « power* of Itnea.

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
Arcliimedes often. Lastly, the same is performed
hy 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
contiuued from the point retain everjrvvhere the
same proportion, they shall all terminate in a strait
line. Aud the same happens if the point be taken
between two circles. So that the places of all their
poiuts of termination make either strait lines, or
L'ircuniferences of eireles, and are called plane
pluceit. So also when strait parallel lines are
applit^d to one strait line, if the parts of the strait
'ine t(j which they are applied be to one another in
proportion duplicate to that of the contiguous
applied hues, they will all terminate in a conical
f^tion ; which section, being the place of their
termination, is called a solid place, because it
^rves for the finding out of the quantity of any

314

MOTIONS

MAGNITUDES^

PART irt. equation vvhieli consists of three dimensions. There
^-'r -- are therefore three ways of tiiiding out the cause of

ofglomlirl^^'^ equality or inequality between two given quantities;

cUm by the namely, tirst, by the eomputation of motions ; for

powers uf lines. ^ ^ j i ^

by equal motion, and equal tmie, equal spaces are
described; and ponderation is motion. Secondly,
by indivkihies : because all the parts together
taken are equal to the whole. And thirdly, by the
powers : for when they are equals theh" roots also
are equal ; and contrarilyj the powers are equal,
when their roots are equal. But if the question
be mucli complicated, there caniuit by any of these
ways be constituted a certain rule, from the sup^
position of which of the nnknown 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 geometritnan, as synthesis dothj
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 Vieta, though Vieta
was a most admirable geometrician ; but he that
has Vieta for his master, not so, without Euclid,

And as for that part of analysis which w orks 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
exten iitaiued in the doctrine of

DIMENSION OF A CIRCLE, ETC. 315

rectangles, and rectangled solids. So that although ^^^^ ^^^•
they come to an equation which determines the * — r^
quantity sought, yet they cannot sometimes hy ^/^omrtri^""
art exhibit that quantity in a plane, but in some ^^^^^J^f^n^j^
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
T)isection, 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 soUdy
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 solidy 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

ciauit by the
|ii)werii of Hties

PART III, little after he says, *' of this kind are the spiral
- — -^ lines J the quadratrieefs^ tlie conchoeides^ and the
orgeLnri?"' chsoeides. And geometricians think it no small
fault, when for the finding out of a plane problem
any man makes use of eonics, or new lines." Now
he raidcs the trisection of an angle among solid
problems, and the quinqueseetion among hneary.
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, w hen
he found out the trisection of an angle by the
help of an hyperbole? Or am I in the wTong,
who think 1 have found out the construction of
both these problems by the ride and compass only?
Neither they, nor L 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 cnssiek ; which are, as 1 may say, the
braehygraphy 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 CIRCULAK MOTION*

317

>y RjTnbols iif very remote propositions; yet
whether such discourse deserve to be thought very
p»rofitable, when it is made without any ideas of
36 things themselves, I know not*

PART TIL

CHAPTER XXI.

OF CIRCULAR MOTION.

In simple motion, every strait line talien in the btnly moved
ik 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
itimes it make equal angles, every strait line taken in that
epicycle will be so carried, tiiat it will always be parallel to the
jjlaees id which it formerly wa&.™3. The properties of simple
motion.— 4, If a fluid be moved with simple circular motion,
^1 the points taken in it will describe their circleii in times
jjroportional to the distances from the centre. — 5. Simple
fnotiou difeipatea heterogeneous and congregatos homogeneous
iDodies.— 6. If a circle made by a movent moved with simple
:imotion be commensurable to another circle made by a point
'^^'hich 13 carried about by the same movent, all the points of
^oth the circles will at some time return to the same situation,
"J. If a sphere have simple motion, its motion will more
dissipate heterogeneous bodies by how much it is more remote
^Voui the poles.— H, 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 superflcies of that body. — 9. Cir-
cular motion about a fixed centre casteth off" by the tangent
such things as lie upon the circumference and »«tick not to it*
I0» Such things, as are moved with simple circular motion,
beget simple circular motion.^ — II, 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 beif^^'nipi;

.1 - ,.11 , - 1 ' 1 mouoti, &c,

that, in which the several points taken m a moved

PART ITI.

In simple
motioni every
«tmit line
taken in the
body moved
IB so carried,
that it is always
parallel to
tile places
in which it
fonnerly wm*

body do in several equal times describe several
equal arnhes. And therefore in simple circular
motion it is necessary that every strait line taken
in the moved body be always earned parallel to
itself; which I thus demonstrate.

First, let A B (in the first figure) be any strait
line taken in any sohd body ; and let A D be any
arch drawn upon any centre C and radius CA,
Let the point B be understood to describe towards
the same pai'ts the arch B E, hke and equal to the
arch A D. Now in the same time in which the
point A transmits the arch A D, 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 AB will be in DE ; and
therefore A B and 1) E are equal And seeing the
arches AD and BE are like and equal, their subtend-
ing strait hues A3 and BE will also be equal ; and
therefore the four-sided figure A B D E will be a
paraUelogram. Wherefore A B is carried parallel
to itself. And the same may be proved by tlie
same method, if any other strait line be taken in
the same moved body in which the strait line AB
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 aud B E were not arches of circles, but
of parabolas, ellipses^ or of any other figures,

OP CIRCULAR MOTION.

319

both the\% and their subtenses, find the strait
Lines wliich join them, woiUd be equal and paniUeL

Coroil. II, It is also mauifestj that the radii
3f the equal cireles A D and B E, or the axis of a
ssphere, will be so carried, as to be always parallel
:o 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
^nal to the strait line F E or C D ; and the aufcle
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 € G E (seeing B E and A D are
parallel) w ill be equal to the angle D A C. But
the angle E B F is equal to the same angle D A C ;
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
figure) whose centre is A, and radius A B ; and
upon the centre B and any radius B C let the
epicycle CDE be described. Let the centre B
be understood to be carried about the centre A,
and the whole epicycle with it till it lie coincident
with the circle FG H, whose centre is 1 ; and let
B A I be ^ny angle given. But in the time that
the centre B is moved to I, let the epicycle CDE
have a contrary revolution upon its own centre,
namely from E by D to C, according to the same
proportions ; that is, in such manner, that in both
the cireles, equal angles be made in equal times.
I say E C, the axis of the epicycle, will be always
■fcrried parallel to itself. Let the angle F I G be
made equal to the angle B A I ; IF and A B w ill

PART III.
2L

I f circalar
motioD he
made about a
resting centre^
and in that
circle there be
an epicycle
whose revolu-
tion is mode
the contrary
way, in such
manner that lu
equal times it
make equal
angle »» every
Ktralt line
Utken in that
epicycle will
be &u earned,
that it will
£Llwaya be
parallel to
the placea
in which it
formerlv waa.

PART

If eircutar
motion, &.c<

r ni. therefore be parallel ; and how much the axis
A G has departed from its former place A C (the
measure of which progression is the angle C A G,
or C B D, which I suppose equal to it) so much iu
the same time has the axis I G, the same with B C,
departed from its own former situation. Where-
fore, iu 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 iu 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 h manifest, that those
two animal motions which Copernicus ascribes
to the earthj are reducible to this one circular
simple motion, by which all the points of the
moved body are carried always mth equal velocity,
that is^ in equal times they make equal revolutioi
uniformly.

This, as it is the most simple, so it is tlie most
frequent of all circular motions ; being the same
which is used by all men when they tnni anything
round with their arms, as they do in grinding o:
sifting. For all the points of the thing mov<
describe lines which are like and equal to on
another* So that if a man had a ruler^ in whict*-
many pens' points of equal length were fastened
he might with this one motion write many lines
at once.

PART 1 U.
2L

3. Having shown wliat simple motion is, I will
here also set down ^ome properties of the same.

First, when a body is moved with simple motion Pfop'^i^^it^s of
m a flmd medmm which hath no vacmty^ 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 c^ome into the same.

For (in the same second figure) let any body,
as K L M N, be understood to be moved with
simple circular motion ; and let the circle, which
every point tliereof 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 thcretbre it is carried to K, the
parts of the fluid medium on that side will go
back ; and, supposing all space to be fnll, 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 evei*y 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. WTierefore 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- J, Y

PART lU.

21*

If & fluid be
moved with
limple circulm-
modotif all Ihe
points taken in
it will dcAcribe
thtiT circles iti
times propor-
tional to the
distances from
the centre.

make perpetual returns to their former places,
hath 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.

4. Secondly, supposing the same simple motion
in the air, water, or other fluid medium ; the
parts of the medium^ which adhere to the moved
body, will be carried about with the same motion
and vekicity, so that in what time soever any point
of the movent finishes its circle, in the same time
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 di:stances
from the movent. For every ptunt of the fluid
ambient, as long as it toueheth 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 J'art irr.
mo%'ent be carried <about. Now that part of the - — .- — -
fluid ambient, which is not the nearest but almost
the nearest, receiving its degree of velocity from
the nearej^t, 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 nut
describe its circle ; yet it shall describe snch 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 simpk motioa
placed m a fluid medimn, congregates or gathers rog^neous md
into one place such things as naturally float in that homJ^ulmis
medium, if they be homogeneous ; and if they be ^^^'^^'^^
iieterogeneous, it separates and dissipates them.
But if such things as Ik* heterogeneous do not
float, but settle, then the same motion stirs and
mingles them disorderly together. For seeing
bodies, which are unhke to one another, that is,
heterogeneous bodies, are not unlike in that they
are bodies ; for bodies, tis bodies, have no diflfer-
ence ; but only from some special cause, that is,
from some internal motion, or motions of their
smallest parts (for 1 have shown in chap, ix, art. 9,
that all mutation is such motion), it remains that

Y 2

»PAKTiiL heterogeneous bodies have their imlikeness or
' — * — ' diflfereiice from one another from their internal or
iTpateThet?- specificol motions. Now bodies which have such
rogeneoua, &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
ahke and together with tliem; and afterwards
meeting with more bodies like themselves, they
will nnite and become greater bodies. Wherefore
homogeneous bodies are congregated, and hetero-
geneons dissipated by simple motion in a medium
where they natxirally 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 imite, 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 ferment at ion ^

from the Ijxinifervere ; as the Greeks have their

ZifiJj, which signifies the same, from Ztw ferveo.

For seething makes all the parts of the water

^auge their places ; and the parts of any thing,

s thrown into it, will go several ways ac-

to their several natures. And yet all

or seething is not caused by fire ; for new

ikI many other things have also their fi^

f mentation cand fervour, to which fire eontrihiites
ttle, and sometimes nothing. But when in fer-
[ mentation we find heat, it is made by the fer-
mentation,

6. Fourthly, in what time soever the movent,
whose centre is A (in fig. 2) moved in K L N^ shall,
by any number of revolutions, that is, when the
perimeters BI aiid KLN be commensurable, have
described a line equal to the circle which passes
through the points B and I ; in the same time all
the points of the floating body, whose centre is B,
shall return to have the same situation in respect
of the movent, from which they departed. For
seeing it is as the distance B A, that is, as the
radius of the circle which passes through B I is to
the perimeter itself B I, so the radius of the circle
KLN is to the perimeter KLN; 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 penmeter
B I 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

If a circle made
by a movptit
moved witli
simple motian,
be cGDinieosu*
nble to ano-
ther circle
made by a point
which id car-
ried about by
the some mo-
vent, all the
poiQta of bolh
the circlei will
at iome time
return to the
same ftkuation.

PART IIL points will never return to have the same situation

21.

or confipn^iration in respect of one another.

have *?ropTe ^' '^ Simple motion, if the body moved be of a
motion its mo- spherical fiimre, it hatli less force towards its poles

tion will more , i - .in t - i

disaipate bete- thaii tow^ards its middle to dissipate heteroge-

rogeneous bo- ^ .1 l j-

dJea by bow neoiis, or to congregate homogeneous bodies.
wmo^te'f^r'' Let there l}e a sphere (as in the third figure)
the poiea . whose centre is A and diameter B C ; and let it be
conceived to be moved with simple circular motion;
of which motion let tlie axis be the strait line D E,
cutting tlie 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 H F
and A B are equal, a circle described upon the
centre H with the radius HF or HG, will be equal
to the circle whose centre is A and radius AB.
And if the same motion be continued, the point B
w ill at the end of another half revolution return to
the place from wlience it began to be moved ; and^
therefore at the end of half a revolution, the point &-
will be carried to F^ and the whole hemisphere DBC«
into that hemisphere in which are the points L,
and F. \Mierefore that part of the fluid medium
which is contiguous to the point F, will in the same^
time go bat^k the length of the strait line B F ; ancf
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

OP CIRCULAR MOTION. 327

the pole E, and through it let the strait line I K be part iii.
drawn parallel to the strait line B F, cutting the ^ — r-^
arch F L in K, and the axis H L in M ; then con-
necting H K, upon H F let the perpendicular K N
l>e drawn. In the same time therefore that B
comes to F the point I will come to K, B F and
I K being equal and described with the same velo-
cit:y. Now the motion in I K to the fluid medium
Ti-IK)n which it works, namely, to that part of the
na^dium which is contiguous to the point K, is
<^l>lique, whereas if it proceeded in the strait line
ti K it would be perpendicular ; and therefore
tl^e motion which proceeds in I K has less power
tli^m that which proceeds in H K with the same
^^locity. But the motions in H K and H F do
^cjually thrust back the medium; and therefore
t\ie part of the sphere at K moves the medium
l^ss than the part at F, namely, so much less as
K N is less than H F. Wherefore also the same
motion hath less power to disperse heterogeneous,
and to congr^ate homogeneous bodies, when it is
nearer, than when it is more remote from the
poles ; which was to be proved.

CoroU. 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 eflFect. 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 no endeavour at all.

8. If in a fluid medium moved about, as hath cfreu^^Tmotion
been said, with simple motion, there be conceived of a fluid body
to float some other spherical body which is not fluid, a body which u
the parts of the medium, which are stopped by that ^^^^°^^ ^*

328

MOTIONS AND MAGNITUDES.

PART III.
21.

Rmd body will
spread i 1st If
upon the £ti^
pcrficies of
that body.

Circular mo-
tion about a
fixed centre
caatetli ©n" by
the tttiigfiit
nucb thingA as
]ic upon the cir-
cumference it
stick not to it.

body, will eiideav our to spread themselves everj
way upon the superficies of it. And this is manifed
enough by experience, namely, by the spreading
of water poured out upon a pavement* But thi
reason of it may be this. Seeinec 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 nol
liquid, as G, so that it cannot go on ; and seeing
the small parts of the medium cannot go forvvardsi
nor can they go directly backwards against thi
force of the movent; it remains, therefore, tha
they diffiise themselves upon the superficies of thai
body, as towards O and P ; which was to bi
proved, i

9. Compounded circular motion, in which all thi
parts of the moved body do at once describe cir-
cumferences, some jjreater, others less, accordinj
to the proportion of their several distances fron
the common centre, carries about with it suci
bodies, as being not fluid, adhere to the body sc
moved; and such as do not adhere, it casteth foP
wards in a strait line which is a tangent to
point from which they are cast oflF.

For let there be a circle whose radius is A
fig. 4) ; and let a body be placed in the circumfi
rence in B, which if it be fixed there, will neees^
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 bodj
will continue its motion in the tangent B C. Foi
let both the radius A B and the sphere B be coa
ceived to consist of hard matter ; and let us supi
pose the radius A B to be stricken in the point i

h top
to thi

!umf©^

OF CIRCULAR MOTION. 329

by some other body which falls upon it in the tan- part hi.
g:ent D B. Now, therefore, there will be a motion -—^ — '
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
pixts further fi*om the centre of its motion such
tilings as touch, but do not stick fast to its super-
ficies ; and the more, by how much the distance is
^eater 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 ^"^** things a«

* •' are moved with

nioved with simple circular motion, and in the same simple circular
i^acdium there float another sphere whose matter is ^miTcircXr
not fluid, this sphere also shall be moved with sim- "*^^^°"-
pie circular motion.

Let B C D (in fig. 6) be a circle, whose centre is
^y and in whose circumference there is a sphere
^ moved, that it describes with simple motion the
*^e perimeter BCD. Let also E F G be another
^liere of consistent matter, whose semidiameter is
** H, and centre H ; and with the radius A H let
^e circle HI be described. I say, the sphere

FART HI. E FG will, by the motioo of the body in B C I
'^ — <-^ he moved in the circumfereivce H I with simpt

For seeiiie: the motion in B C D (by art. 4 of thi
chapter) makes all the points of the fluid mediui
describe in the same time circular lines equal t
one another, the points E, H and G of the strai
hne EHG will in the same time describe with equi
radii equal circles. Let E B be drawn equal an
parallel to the strait line A H ; and let A B be cor
nected, which will therefore be equal and paralle
to E H ; and therefore also, if upon the centre I
and radius B E the arch E K be drawn equal to th
arch H I, and the strait hues A I, B K and 1Kb
draT?vu, B K and A I will be etjuaJ ; and they wi!
also be parallel, because the two arches E K ani
H I, that is, the two andes K BE and I A H ar
equal ; and, consequently, the strait lines A I
and K I, which connect them, wiU also be equa
and parallel. Wherefore KI and E H are parallel
Seeing, therefore, E and H are carried in the sam
tinie to K and I, the whole strait hue I K will b
parallel to E H, from whence it departed. And
therefore, seeing the sphere E F G is supposed t
be of consistent matter, so as all its points keej
always the same situation, it is necessary that ever
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 move*
with simple circular motion ; which was
demonstrated.
iftiiM which is 11. If in a fluid medium, whose parts are i
ZnZt iirrd by a body moved with simple motion, there floa'
atid ihe oiiitr another body, which hath its superficies eithei

toJ|

OF CIRCULAR MOTION,

331

wholly bard, or wholly fluid, the part^ of this body r*AET xil
shall approach the centre equally on all sides; t!iat ^^r^
is to say, the motion of the body shall be circular, J^l^i,,I]'^f|*i J**
and concentric with the motion of the movent, ^'f perfectly
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 LM N (iniig 2*) whose 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.

Wlierefore let the superficies of the body at I be
fluid on one side, and hard on the other. And
first, let the fluid side be towards tlie 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 w ill 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.

MOTIONS AND MAGNITUDES

PART III. which is contigiioiis to the fluid part, has no cans

" "■'-' at all of elevation, that is to say, no eudeavou jc-mr

n that which is 4.^„,^-j^^ A
«o moved, acc.^*^^^*^^^ ^*

Secondly, let the hard superfcies of the body a-^^^t
I be towards A. By reason, therefore, of the saic^ Mii
change of place of the parts which are contiguous ^p^^JIS
to it, the hard superficies must, of necessity^ seeiii^^ «g
by supposition there is no empty space, either com^ ^Mit
nearer to A, or else its smallest parts must suppl^r^Jy
the contiguous places of the medium, which other-^:*B
wise would be empty. But this cannot be, by rea— -*=i-
son of the supposed hardness ; and, therefore, th^ ^e
other must needs be, namely, that the body coni^ ^me
nearer to A. Wherefore the body at I has greateit^^M
endeavour towards the centre A, when its harr::»^
side is next it, than wiien it is averted from it ^z^rt.
But the body in I, while it is moving in the circum .^n-
ference of the circle I B, has sometimes one side^^e,
sometimes another, turned towards the centre; auttlaJ,
therefore, it is sometimes nearer, sometimes fiirar-
ther off from the centre A. Wherefore the bod-^Qy
at I is not carried hi the circumference of a perfea^ ct
circle ; which was to be demonstrated*

OF OTHER VARIETY OF MOTION. 333

CHAPTER XXIL

OF OTHER VARIETY OF MOTION.

•1* Endeavour and pressure how they differ. — 2. Two kinds of
mediums in which bodies are moved. — 3. Propagation of mo-
tioUf 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 doth not penetrate it, the action of the pressing body is
perpendicular to the superficies of the body pressed. — 7. When
Si hard body, pressing another body, penetrates the same, it
cloth 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 ^^^2.^^^'
endeavour to be motion through some length, ' — ■ — '
though not considered as length, but as a point, pressure how
WTiether, therefore, there be resistance or no re- ^^^ ^*^*''
sistance, the endeavom- will still be the same. For
simply to endeavour is to go. But when two bodies,
'^ving opposite endeavours, press one another, then
*^e endeavour of either of them is that which we
^^ pressure, and is mutual when their pressures

Two kinils of
t]it:diuiii«» ill
which budies
arc aioved»

Propagation
of motioti,
vrbal it is.

AVhat motion
bodies have
wkeaLheypress
one HQOtlicn

2. Bodies moved, and also the mediums in which
they are moved, are of two kinds. For either they
have their parts coherent in such manner^ as no
part of the moved body will easily yield to the
movent, except the whole body yield also, and such
are the tilings we call hard : or else their parts,
while the whole remains unmoved, will easily y^eld
to the movent, and these w*e call fluid or sqff
bodies. For the wonhfluid^ ^^if^, iougk^ and hard,
in the same manner as great and liftie^ are used
only comparatively ; and are not different kinds,
but different degrees of qujdity,

3. To doy and to m{ffer^ is to move and to be
moved ; and nothing is moved but by that which
toneheth 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
Ijody, but mediately ; namely so, as that the first
moveth the second, the second the third, and so
on, till the last of aU be touched, WTien there-
fore one body, having opposite endeavour to an-
other body, moveth the same, and that moveth a
tliird, and so on, I call that Retion propagation of
motion.

4. When two fluid bodies, which are in a free
and open space, press oiu^ another, their parts will
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 tlie
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

I
I

OF OTHER VAEIETY OF MOTION* 335

in a line peq>enfiicular to the bodies pressing. But t'Airr in
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
xesistance of the foremost parts» but proceeds for
3ome time ; and therefore^ seeing they must have
some place in whieh they may be moved, and that
"there is no 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 eifect 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 disceni any
swelling: outwards towards the sides, as we per-
ceive in two bodies of wax ; yet we know well
enough by reason, tliat some tumour must needs be
there, thougli it be but httle.

S. But when the space is enclosed, and both the riuta bodies,
"bodies be fluid, they will, if they be pressed toge- prl^Vspd'Tog^
ther, penetrate one another, though differeiitly, ^^^^^^^^^^^
according to their different endeavours. For sup-
pose a hollow cylinder of hard matter, well
stopped at both ends, but filled first, below mth
some heai^^ 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
die it must open a passage through itself, by

■

aaS IMWHWS A3n> MMaMTmiBSw

?.ueFnT: wUrb die iehor boi^ mov^ aaceniL For

two hnifies. disc, wbaee puts are mast emaSb

^^^ntfttLwittbetiieiiratcfiy^ wfaick bring
Jf""^^^ kbiiflCnecesarTduitciieputsaf dieodicr
'Ni«jM]»& aof iepMataoa at alL And dmefoce whe
fiqnors. wyrii are encioftiefi in the same
efaaoee their places, there b no need tihsl
MiaDest part» shoofai be mmsied widi one aa
fer a way bem^ opened thronsh one of die
parti of the odier need not be xparated.

Now if a fluid body, wiiuji is not enclosed
a hard bodj, its endeaTonr will indeed be ti
the internal ports of tiiat hard body ; but
exrinded fay the resstance of it^ the parts
fluid body will be moTed erery way accorc
the superficies of the hard bodr^ and that e
if the pfeaBUfe be perpendicnlar ; for when
parts of the cause are equals the effects i
equal also. But if the pressure be not pei
cular^ then the an^es of the incidence bei:
equal, the expansion also will be uneqoaly n
greater on that side where the ans:ie is g
because that motion is most direct which pr
by the directest line.
^^^^^ 6. If a body, pressing another body, c
other and doth penetrate it, it wiU nevertheless firive to the

not penetrate * , _ ' -, -, -, -■

i^ the action of presseth an endeavour to yield, and reced
lodjfu^^n- str^ut line perpendicular to its superficies i

OF OTHER VARIETY OF MOTION. 33/

endeavour to yield or recede in a strait line per- part iir

pendicular to the line A D.

For let AB be perpendicular to AD, and let J^^^/^^^"^^^^
B A be produced to F. If therefore A F be coin-
cident with A E, 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 AD 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 FA. Wherefore the body
E, after it is in A, will have only that perpendicu-
lar endeavour which proceeds from the motion in
^ A, that is, in A B ; which was to be proved.

7. If a hard body falling upon or pressing an- when a hard
^t:her body penetrate the same, its endeavour anotiier^b^d"?
^^Pter its first penetration will be neither in the J^^®J'^*^^'j^^J
^^clined line produced, nor in the perpendicular, not penetrate it
*^Xit sometimes betwixt both, sometimes without f>^^n"e88hn!ji

Let E A G (in the same fig. 1) be the inclined
*itie produced ; and first, let the passage through
'•^e medium, in which E A is, be easier than the
passage through the medium in which AG is. As

VOL. I. Z

When n haril
body, &e.

soon therefore as the body is within the medkira
in which is A G, it will find greater resistance to
its motion in D A and its parallels, than it did
whilst it was above A 1) ; and therefore below A D
it will proceed with slower motion in the parallels
of DA, 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 1) 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 betw^een A B and A G, as in the
Hue 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 endf^avour will
not proceed from A in E A produced, but beyond
it, as in A L 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 rejraction^ 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 iu the

OP OTHER VARIETY OF ^iOTION. 339

first mediuia, by departing farther from tlie per- I'^^^'r iit.
pendicular. -- T-^

8- By the 6th theorem it is manitestj that the Motion some-
force of the movent may be so placed, as that to that of the
the body moved by it may proceed in a way almost *"°'"^"**
directly coutraiy to that of the movent, as we bee
in the motion of ships.

For let A B (in %, 2) represent a ship, whose
length from the prow to the poop is A B, and let
the wind he 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 B I be
drawTi; and let the sail be conceived to be spread
in the same line B 1, and the wind to fall upon it
in the points L, M and B; from which points, per-
pendicular to B 1, let B K, M Q and L P l)e drawn.
Lastly, let E N and G O be drawTi perpencMcuIar to
B G, and cutting B K in H and K ; and let H N
and K O be made equal to one another, and seve-
rally equal to B A. 1 say, the ship BA^ by the
w ind falling upon it in C B, D E, F G, and other
lines parallel to them, will be carried forwards
almost iippositc tu 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 perpenclicular 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 hties M G and L F, which are
parallel to B K. Let now the measure of the time

z 2

340

MOTIONS AND MAGNITUDES*

PART in, be B G, which is divided in the middle in E ; and

^ — ^ let the point B be carried to H in the time B E.

fimw ^4po»^i« ^^^ ^^ ^^^^ time, therefore^ by the wind blowing

mmlTx. "^ **"" ^^ ^ ^ ^^^ ^ ^' ^^^ ^ ^^^y ^"^^^ ^^^^^^ ^ ^^y

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 eqnal 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 O, as it may
be moved forwards iu 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 A B C, 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 no bosom at all ; for otherw ise
the strait lines L P, M Q and B K will not be per-
pendicular to the plane of the sail, but falling below^
P, (i 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 experieuce found

OF OTHER VARIETY OF MOTION. 341

this to be so true, that I could scarce oppose the part hi.
board to the wind in any obliquity, though never ^ — A-
so small, but the waggon was carried forwards J^ci^'^oppSrite
\^Y it. ^ ^^^ ®^ ^*

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 C A (in fig. 3.) Let
C E be drawn parallel to A B, and D A perpendi-
ctUar 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 C A 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 C E and B A are parallels ;
and, therefore, the stroke in C A is made by the
faction which is in E A only. But the velocity or
foxce of the perpendicular stroke in E A, to the
Velocity or force of the stroke in D A, is as E A to
t> A. Wherefore the oblique stroke in C A is weaker
*^an the perpendicular stroke in D A, in the pro-
Portion of E A to D A or C A ; which was to be
P'Toved.

9. In a full medium, all endeavour proceeds as i? • fou me-
^^ir as the medium itself reacheth ; that is to say, if ii'^i^tS
^lie medium be infinite, the endeavour will proceed ^"^**"^"***

PARTHL

22,
V ^ _^

In ft full me-
d I unit motion
is pro pa If; Med
to oDj distADce.

For whatsoever endeavoureth is moved, and
therefore whatsoever standeth in its way it maketh
it yield, at least a little, namely, so far as the movent
itself is moved forwards. But that which yieldeth
is also movedjandeonseqnently maketh that to yield
which is in its way, and so on sneeessively as long
as the medium is fidl ; that is to say, infinitely^ if
the full medium be infinite ; which w^as to be
proved.

Now although endeavour thus perjietually pro-
pagated do not alw^ays 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 may be 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 which was
last proved, that all endeavour proceeds infinitely.
Let it be conceived therefore to be removed from
our eyes to any distariee 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.

?^J!!!Il!!". 1<^ Between the deirrees of hard and soft are

bat ihcy arc. thosc thiugs which wc Call toifgk, tough being that
which may be bent without being altered from

OF OTHBR VARIETY OP MOTION. 343

what it was; and the bending of a line is either pamiii.
the adduction or diduction of the extreme parts^ * — . — '
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.

1 L Dilatation and contraction ^bs elm all Jlexian, ^^If^^^^^
supposes necessarily that the internal parts of the suppose mu-

, V 1 11.1 . ^1 ^1 Nation of the

body bowed do either come nearer to the external smallest paru
parts, or go further from them. For though flexion Se^'lltuaUon.
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, imless something be
taken from it and added to the convex line, be the
more crooked, that is, the greater of the two.
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 suficr such transposition
of their parts, they are called brittle ; and the
peat force they require to make them yield, makes
them also with sudden motion to leap asunder, and
^^^^ in pieces.

12. Also motion is distinguished into pulsion ah traction
^d traction. And pulsion, as I have already de- " ^
fcied it, is when that which is moved goes before
that which moveth it. But contrarily, in traction

MOTIONS AND MAGNITUDES.

PART in.
22*

the movent s:oes before tliat which israoved. Never-
theless, consicku'iitg it with greater attention, it
seemeth to be the same with pulsion. For of twc
parts of a hard body, when that which is foremost*^.
drives before it the medium in which the motion
made, at the same time that which is thrust for — ^•:a*-
wards thrust eth the next, and this again the next ^i^^ t,
and so on successively. In which action, if we sup— ^^rz)-
pose that there is no place void, it must needs be-^^^»e,
that by continual pulision, namely, when that actior^r^ri
lias gone round, the movent will be behind tha-^sat
part, which at the first seemed not to be thnis -^^t
fonvards, but to be drawn ; so that now the bodj'^-^p'?
which was drawn, goes before the body whicM^ -^
gives it motion ; and its motion is no longer trac-
tion, but pulsion.
Snrii tilings as ] 3, Such tliiiigs as are removed from their
orS'^re^torii places by forcible eompression or extension, and,a^
wTmorloJi in ^^^^^ ^^ ^^^c force is taken away, do presently retui
their iuu-niai ^^^f\ rcstorc themselvcs to their former sitnatio:

have the beginning of their restitution within them -
selves^ namely, a certain motion m their iuten)^k-J
parts, which was there, when, before the taking^?
away of the force, they were compressed, or ex^ —
tended. For that restitution is motion, and tha^^^
which is at rest cainiot be moved, but by a inove«t3-
and a contiguous movent. Nor doth the cause c^^'f
their restitution proceed from the taking away c^^
the force by which they were compressed or es
tended ; for the removing of impediments hath nc
the efficacy of a cause, as has been shown at
end of the 3rd article of chap, xv* The caoa
therefore of their restitution is some motion eithc
of the parts of the ambient^ or of the parts of tli

OF OTHER VARIETY OF MOTION,

34:

body corapressed or extended. But the parts of pm^t hl
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.

1 4. In the carriage of bodies, if that body, which n^oui^ that

, . ,^ , 1 iL 1^ which carrielh

carries another, hit upon any obstacle, or be by another be
any means suddenly stopped, and that which is Jl7y clrried
carried be not stopped, it will go on, till its motion *'^'^ p"<^ecd.
be by some external impediment taken away.

For I have demonstrated (chap, viii, 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 Uttle 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

*rfec effects of

periruBiioi] not
to bt' compjircd
with ijiusc of
weiglit.

that carries them ; namely, the more remote tlm^
parts are, the less will be their degrees of veloeit^^.
It is necessary, therefore, that the body, which -^s
carried^ be left accordingly more or less behiii^^^
And this also is manifest by experience, when ^^sit
the starting forward of the horse the rider falle^^tlx-
back wards, f

15, In peramifouy therefore^ when one hazard
body is in some small part of it stricken by auotli. ^?r
with great force J it is not necessary that the whc^^le j
body should j^cld to the stroke W'ith the sat^M^^e f
celerity with which the stricken part yields. F*€3r
the rest of the parts receive their motion from ttie
motion of the part stricken and yielding, whic^h
motion is hss propagated every way towards tli^
sides, than it is directly fon\ards. And hence i^
isj that sometimes very hard bodies, which being
erected can hardly be made to stand, are mcpT^
e^isily broken than thrown down by a \ioleut
stroke ; when^ nevertheless^ if all tlieir parts tog:^-
ther were hy any weak motion thrust forw^ards^?
they would easily be east down,

16. Though the diflFerence between ^r;/Wow axid
percussion consist only in this, that in tmsioii tb^
motion both of tlie movent and moved body begiw
both together in their very contact ; and in percuB-
sion the striking body is first moved, and aftc?!*-*
wards the body stricken ; yet their effects are bo
different, that it seems scarce possible to compai*^
their forces with one another. I say, any effect ^
percussion being propounded, as for example, tli*
stroke of a beetle of any weight assigned, l^J
which a pile of any given length is to be drivexJ

into earth of any tenacity given, it seems to me
very hard, if not impossible, to define with what
weighty or with what stroke, and in what time, the
same pile may he driven to a depth assigned into
the same earth. The canse 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 w^eight is as a solid thing, being
measured by the dimension of the whole body.
And there is no comparisou to be made of a solid
body with a length, that is, with a line.

J 7. If the internal parts of a body be at rest, or
retain the same situation with one another for any
time how little soever, there cannot in those parts
be generated any new motion or endeavourj w hereof
the efficient cause is not w ithout 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 he moved, that part must of necessity receive
its motion from some moved and contiguous body.
But by supposition, there is no such moved and
contiguous part witViin the body. Wlierefore, 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, In hard bodies, therefore, which are com-
pressed or extended, if, that which compresseth or
extendeth them being taken away, they restore

PART UL

Motion cannot
be^in £rst in
the internal
parti of a body*

PART II L
2%

themselves to their former place or sitxiation, \13^ aV
must needs be that that endeavour or motion ot ^
their internal parts, by which they were able tcz^^o
recover their former places or situations, was no0" ^^
extinguished when the force by which they were^^r?
compressed or extended was taken away. There- ^!^-
fore, when the lath of a cross-bow bent doth, asL
soon as it is at liberty, restore itself, though to him
that judges by sense, both it and all its parts see
to be at rest ; yet he, that judging by reason dotr^h'
not account the taking away of impediment for amn^-an
efficient cause, nor conceives that without an effi^ — i
cient cause any thing can pass from rest to motioi]^~:3],
will conclude that the parts were already in motioc^ ^n
^ before they began to restore themselves. f

■ctilTpTOcc^d '^- ^4^^^^^* ^^^^ react ion proceed in the sam* ^«e

1! lime line, but from opposite terms. For seeing reaction i J^^
nothing but endeavour in the patient to restore itsel^W
to that situation from whicli it was forced by th^ ^^
agent ; the endeavour or motion both of the agen t
and patient or reagent will be propagated betwee^cn
the same terms; yet so, as that in action the teni»^ i
Jrom which^ is in reaction the term to which, Anc^
seeing all action proceeds in this manner, not onl^S^'
between the opposite terms of the whole line iwra
which it is propagated, but also in all the parts c^^
that line, the term^ from ivhieh and to ivhich^ho\9^
of the action and reaction, will be in the same lin^*^ -
Wherefore action and reaction proceed in the sam -^^
line, &c.

20. To what has been said of motion, I will ad
what I have to say concerning habit* Hahii
therefore, is a generation of motion, not of motio:

Habif,
what ii is.

OP OTHER VARIETY OP MOTION. 349

simply, but an easy conducting of the moved body i'art iil
in a certain and designed way. And seeing it is — r-^
attained by the weakening of such endeavours as ^^^l\ j^
divert its motion^ therefore such endeavours are
to be weakened by little and little. But this cannot
l>e done but by the long continuance of action, or
l>y actions often repeated; and therefore custom
begets that facility, which is commonly and rightly
oalled habit ; and it may be defined thus: habit
^ motion made more easy and ready by custom ;
that is to say J hy perpetual endeavour ^ or by
iterated endeavours in a way differing from that
»J« which the motion proceeded from the beginnings
^Mnd opposing such endeavours as resist. And to
Xioake 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
^virliere he would make the second stroke, without
-taking it back by a new endeavour, and, as it were
Ijeginning again, pass from the first to the second.
!Nor will he be able to go on to the third place
-wthout another new endeavour ; but he will be
^rced to draw back his hand again, and so suc-
oessively, by renewing his endeavour at every
stroke ; till at the last, by doing this often, and by
€?cmpounding many interrupted motions or endea-
vours into one equal endeavour, he be able to make
liis hand go readily on from stroke to stroke in
tlxHX order and way which was at the first designed.
N'oT are habits to be observed in living creatures
^my, but also in bodies inanimate. For we find
^Hat when the lath of a cross-bow is strongly bent.

350

MOTIONS AND MAGNITUDES,

HtthiT,

whit k 18*

i^» and would if the impediment were removed retur^»^
— again with great force ; if it remain a long tin^_e
bent, it will get such a habit, that when it is loose-^ d
and left to its own freedom, it will not only ncrril:
restore itself, but will require as much force fcr^r
the bringing of it back to its first posture, as it d^_cl
for the bending of it at the first.

CHAP. XXIIL

OF THE CENTRE OF EQUIFONDERATION ; C^F
BODIES PRESSING DOWNWARDS IN STRA M'f
PARALLEL LINES,

, Definitions and suppositions.— 2. TNvo planes of eqnipondp »^'
tion are not pamlleL — 3. The centre of equiponderatiun is ^^
every plane of eqiiiponderation. — 4, The moments of eq«-*^
ponderants are to one another as their distances from C ^^
centre of the scale. — 5, 6, The inonients of unequal pc^ ^'
demote have their proportion to one another compound ^^^
of the proportions of their weights and dis^tances from t.-^'*^
centre of the scale, — 7* If two ponderants have tlieir weigh '•-t^
and distances from the centre of the scale in reciprocal pr""^^
portion, they are equally poised; and contrarily* — ^8, Ift^'^
parts of any ponderant press the beams of the scale ev^^
where equally, all the part-s cut off> reckoned from the cend^^**
t if the scale, will have their moments in the same proportl^c^u
M itli that of the parts of a triangle cut off from the vertex ^-^J
strait lines parallel to tiie baise.— 9. The diaoieter of equipc^ ^*
deration of figures, which are deficient according to cooime'^^'
surable proportions of their altitudes and bases, divide? t^^M^
axis, 8o that the part taken next the vertex is to the other pfc^^*'
of the eotuplete figure to the deficient figure.^ — 10. The d'
meter of equiponderation of the complement of tlie half of a
of the said deficient figures, divides that line which is dra'
through the vertex parallel to the base, so that the part n€
the vertex is to the other part as the complete figure to c^ J^"^

CENTRE OF EQUIPONDERATION.

351

23.

complement,— 11- The centre of equipoiideration ''if the half PART f 11.

of any of the deficient figures in the iirst row of the table of
art. 3, chap, xvu, may be found out by the numbers of the
second row. — 12, The centre of equip on deration of the half
of any of the figures of the second row of the same table, may
l>e 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, tin? centre of the excess of the
saine figure above a triangle of the same altitude and base is
also known. — 14, The centre of equipoii deration of a soUd
sector h in the axis so divided, that the part next the vertex

be to the whole axiij> wanting half the axis of the portion of Uie

sphere^ as 3 to i.

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. Equipomlerai'ion is when the endeavour of
one body, which presses one of the beams, resists
the endeavour of auotlier body pressinjaf the other
beam, so that neither of them is moved ; and the
bodies, when neither of them is moved, are said to
be equally polwd.

III. Weight is the aggregate of all the endea-
vours, by which all the points of that body, which
presses tlie beam, tend downwards in lines parallel
to one another ; and the body which presses is
called the ponderant,

IV. Moment is the power which the ponderant
has to move the beam, by reason of a determmed
situation,

V. The plane of equiponderafion is that by
w^hich the ponderant is so divided, that the mo-
ments on both sides remain equal.

Definitiooa.

VI. The diameter of equiporideration is tbz:Ae
common section of the two planes of equiponder=r
tion, and is in the strait line by which the weigr— 1
is hanged.

VII. The eentre of equiponderation is the corzann?
moil point of the two diameters of equiponderatio*

Suppoaitioits*

Two pknes
of equipoude
ration are
nut paroJkl,

SUPPOSITIONS.

I. When two bodies are equally poised, if weisr^i^lif
be added to one of them and not to the oth"
their eqniponderation ceases,

II. Wien two ponderants of equal magnitnc 3C^
and of the same species or matter, press the be^^m
on both sides at equal distances from the centre of ^
the scale, their moments are equal. .\lso w^h -^n \
two bodies endeavour at equal distances from t ^^
centre of the scale, if they be of equal magnitu^^de
and of the same species, their moments are equ^^*

2* No two planes of equiponderation are parallel*
Let A B C D (in fig, I) be any ponderant wha»^t-
soever; and in it let EF be a plane of eqmpotmi-
deration; parallel to which, let any other plague
be drawn, as G H* I say, G H is not a plane <5f
equiponderation. For seeing the parts AEF P
and E B C F of the ponderant A B € 1) are equaB^ly
poised ; and the weight E G 11 F is added to t^^^

part AEFD, and nothing is added to the p»-'r^
E B C F, but the weight E G H F is taken frt^ J^
it J therefore, by the first supposition, the par^^
A G H D and G B C H will not be equally poisec::3»
and consequently G H is not a plane of equipoud ^•■
ration. Whereto re^ no two planes of equipoud^"
ration are parallel ; which was to be proved.

CENTRE OP EQUIPONDERATION. 353

3. The centre of equiponderation is in every partiii.

23.

plane of equiponderation.

For if another plane of equiponderation be 2J,yrd«a.''^
taken, it will not, by the last article, be parallel to ^°^"^*|," 7*J^?[
tiie former plane ; and therefore both those planes pondcration.
y^^nil cut one another. Now that section (by the
6th definition) is the diameter of equiponderation.
-/Vgain, if another diameter of equiponderation be
't^en, it will cut that former diameter; and in
t:liat section (by the 7th definition) is the centre of
equiponderation. Wherefore the centre of equi-
ponderation is in that diameter which lies in the
Baid plane of equiponderation.

4. The moment of any ponderant applied to one The momente
point of the beam, to the moment of the same or ^Yre^°one
^n equal ponderant applied to any other point of 5j^a°t!^*ci"^m
the beam, is as the distance of the former point Jj« J^^^ ®^
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
AB in the points B and C ; also let the strait lines
BD and CE be diameters of equiponderation;
and the points D and E in the ponderants D and E
be their centres of equiponderation. Let A GF be
drawn howsoever, cutting D B produced in F, and
E C in 6 ; and lastly, upon the common centre A,
Itt the two arches B H and C I be described, cut-

VOL, I. A A

354

MOTIONS AND MAGNITUDES,

PART TTL
22.

ting A G F in H and L I say, the moment of the
ponderant D to the moment of the pouderaot E
is as A B 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, tliese 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 A C ; for A B. A C : :
B F. CG : : B H. C I. 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 C I ; which was to
be demonstrated.
*The moments 5^ Uneoual pouderants, when they are applied
deraots have to scveral poiuts of the beam, and hang at liberty,
lioTto^Xr that is, so as the line by which they hang be the
pou^deVrn^^ of eqniponderation, whatsoever be the

proportion* orfiori^u'e of the ponderant, have their moments to

their weights ^ , . . 1 1 i* 1

and distances onc auothcr ui proportiou compouuded of the
"he icide"*^* proportions of the ir distances firom the centre of
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. 1 say,
the proportion of the moment of the ponderant C

CENTRE OP EQUIPONDERATION.

355

to the momeut of the pouderant D, is compounded of part in
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 J^'^^'^^^J^^.
species, of the magnitude C to the magnitude D. deruiu, &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
equfd to C, the moment of C to the moment of
F+D will be (by the last article) as BG is to EIL
Now as F + D is to D, so let E H be to another
E I ; 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 E I 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, WTierefore unequal ponderants, when they
are applied, &c. Which was to be proved.

6. The same figure remaining, if I K be drawn
parallel to the beam A B, and cutting AG in K ;
and K L be drawn parallel to B G, cutting A B in
L, the distances A B and A L 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 L to which K L is equal. But as the distance
k B from the centre is to the distance A L fix)m
the centre, so is B G, the moment of the 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-

,» ^ 1 ^ . . 1 ^. rant* hare their

distances irom the centre ui reciprocal proportion, wei^hu and
and the centre of the scale be between the points Sr"«trrTf
to which the ponderants are applied, they will be ^^'^ •<=**", *"

* , ♦ ,, reciprocal pro-

equally poised* And contrarily, if they be equally portion, they
poised, their weights and cUstances from the centre polied^and

ipf the scale will be in reciprocal proportion.

A A 2

caalranl^.

■Mniua& ASS HAfianruDBS.

HK * pwup of die sisie (in the same third
be A^ nir beamAB : and let any ponderant
3G !br :t5 mniiBHit,, be applied to th^
3: iJB» ler JOT ocfaor pondarant D, whos^
If 5I^':w^aiipuHLtotfaepointE. Throvagl^
z3£r^pmBL I .« I BL *}e iiiawn parallel to the beaiC-
^3. -max: VG in Bl; also let KL be drawtB-
to EEG» KL^wiil tfaim.be the moment of th^
D : ind br the last article^ it will be a9
3 (jL :iie Timwit of the pondennt C in the poinC::^
3 -o LX ate •nuimsnt oi the ponderant D in ths
jamr 3. ?« .V3 ro A L. On die other aide of thcr
.*escrr'yf oxe^cme* lee AN be taken eqnal to AL;^
jsmd. :o rbe imm N let there be applied the ponde-
Twaa: 0, jutubc :o die pundnrant C the proportion
}f A3 TO A ^. I >aT« the ponderants in B and K
^vijI ~]^ eoLoaifr pii»KL For the proportion of the
niomear if die punderanc O* ui the point N^ to the
Timntqir yi die pumlerant C in the point B, is hy
die idi jTOirse. jompuunded of the proportions of
die -Tpjshr L^ n> die weiaic C» and of the distance
anm die jenne jr die >cale A N or A L to the
•tistance iom die o«icre of the scale A B. But
-seeinfl: ^«^ Jave :^pus^ed« that the distance AB
Ji die di:scuii*e VX is in reciprocal propor-
rlnu of die ^«^a:tir O :u the weight C, the propor-
den of die moment of die ponderant O, in the point
X. ro die moment of die ponderant C, in the point
B, win be compoanded of the proportions of A B
to A X. and of A X to A B- Wherefore, setting in
order A B. A X. A R the moment of O to the mo-
ment of C win be as the tir^ to the last, that is, as
A B to A B- Their moments therefore are eqnal ;
and conaeqnentlT the plane which passes through

CENTRE OF EQUTFONDERATION,

357

A will (by the fifth defiiiitioii) be a plane of equi- paet iil

ponderation, Wherefore they will be equally ^ — r— '

poised ; as wavS to be proved.

Now the converse of this is manifest. For if

there be equiponderation and the proportion of the
iveights and distances be not reciprocal, then both
tte 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
cli^stances 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 ^f ^^ v^ ^f

any pondermt

stp plied a parallelogram, or a parallelopipedum, or prMathcbeam*
a- prisma, or a cylinder, or the superficies of a everywhere
cyhnder, or of a sphere, or of any portion of a !^'j^ *^y**^®
sphere or prisma ; the parts of any of them cut rMtoned from

^~, - 1 1 n 1 11 <n 1 ^ . centre of the

Off With planes parallel to the base will have their 8caJe,wiiihavo

moments in the same proportion with the parts of i^thel^pro.

a triangle, which has its vertex in the centre of the fh^^7tii^^

Soale, and for one of its sides the beam itself, which ofa triangle cut

pcirts are cut off by planes parallel to the base. vertex by strait

First, let the rectangled paraOelogram A B C D ^TbLT**'**"

QxTMx figure 4) be applied to the whole length of

tlxe 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 Gbe

produced to AE in the point H. I say, the mo-

^nentof the whole A BCD to the moment of its

PART HI. part A F, is as the triangle ABE to the triang:!?
^ — .-^ A G H, that is, in proportion duplicate to that of
i!i/pomremi"t[ ^^^ distances from the centre of the scale*
^^' 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 C B to be B E, 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 w^ill be so many strait lines in
the triangle A B E drawn parallel to the bavse B E;
all which parallels together taken are the moment
of the whole parallelogram A B C D ; and the same
parallels do also constitute the superficies of the
triangle ABE. TMierefore 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 A G H ; and therefore the
moment of the w hole parallelogram to the moment
of a parallelogram which is part of the same, is 8&
the triangle ABE 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 ^ ' ^iMh^ame parallels, by which the

%^^^^m f^r^ tnt off, as Archunedes
fore when the parts
are equal and at equal

CENTRE OP EQUIPONDERATION.

distances from the centre of tte j^cale, their mo-
ments also are equal, in the same manner as they
are in paraUelograms.

Secondly, let the parallelogram A K I B not be
rectangled ; the strait line I B mil 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 I B
w ill do the like. Whatsoever therefore the moment
be which is assigned to the strait line I B, as here,
for example, it is supposed to be B E, if A E be
drawn y the moment of the whole parallelogram A I
will be the triangle ABE; and the moment of the
part A L will be the triangle A G H. Wherefore
the moment of any ponderant, which has its sides
equally applied to the beanij 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 oflF by a plane which is parallel to the base,
9. Tlie centre of equiponderation of any figure,
which is deficient according to commensurable
proportions of the altitude and base diminished,
and whose complete figure is either a parallelogram
or a cylinder, or a parallel op ipedum, divides the
axis, so, that the part next the vertex, to the other
part, is as the complete figure to the deficient
figure.

For let C I A P E (in fig. 5) be a deficient figure,
whose axis is A B^ and whose complete figure is
C D F E ; and let the axis A B be so divided in Z,
that AZbe to ZB as CDFE is to ClAPE. I

PART ni,

23,

The diameter of
equipondera-
tion of figrirea
which are defi-
cient af cording
to com mensu-
rable propor-
tions of thetr
altitudes and
baaea, dividei
the axis, BO that
tlic part taken
next tlie vertex
is to the other
part as the com-
plete figure to
the defecient fi-
gure«

MOTIONS AND MAGNITUDES.

pARTiiT. say, the centre of eqnipoiideration of the figure

^-^ C I A P E will be in the point Z.

^^'equiponde' Fifst, that the Centre of equiponderatiou of the

mion, Ace. fitrure C I A 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 fig:ure
C D F E* Let the axis A B be equally di\ided in
Lj and let G L H be drawn parallel and equal to
the strait line CE, cutting the crooked Une
C I A PE in I and P, and the strait lines A C and
AEin K and M, Moreover, let ZO 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 hnes 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 fiennre
CIAPE* Now the triangle ABE is (by the
9th article of chapter xvn) to the three-sided
figure ANEB, as ABCD + AICB is to AICB
twice taken, that is, as C D F E + CI A P E is to
CI APE twice taken. But as CI APE is to
CDF Ej that is, as the weight of the deficient
figure is to the weight of the complete figure, so is
CI APE twice taken to CDFE twice taken.
Wherefore, setting in order CDFE+CIAPE.
2 C I A P E. 2 C D F E ; the proportion of C D FE +
CIAPE to CDFE twice taken will be com-
pounded of the proportion of CDFE + CI APE
to CIAPE twice taken, that is, of the proportion

of the triangle ABE to the tbree-sided fip:ure partiil
A. K E B, that is, of the moment of the complete ^ — -^ — '
figure to the moment of the deficient figure, and of ^^^^^)p^JJ*]^[
ttxe proportion of C I A PE twice taken to C D FE ^^^'on^ *'^'
t^^ce taken, that is, to the proportion reciprocally
t^i^lten of the weight of the deficient figure to the
w^^ight of the complete figure.

Again, seeing by supposition A Z. Z B : : C D F E.
C I A P E are proportionals ; A B. A Z : : C D F E +
C TAP E, C 1) F E will also, by compounding, be
px^cportionals* And seeing A L is the half of A B,
.V LAZ::CI>FE + CIAPE. 2CDFEwillalso
b^ proportionals. But the proportion of CDFE -h
C 3 A P E to 2 C D F E is compounded, as was but
txow shown, of the proportions of moment to mo-
na^nt, &e,, and therefore the proportion of A L to
^ Z is compounded of the proportion of the mo-
tticnt of the complete figure CDFE to the moment
of the deficient figure C I A PE, and of the pro-
portion of the weight of the deficient figure CIAPE
to the weight of the complete figure C L> F E ; but
the proportion of AL to AZ 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 reeiprocally taken, that is to say, of the
^veight C I A P E to the weight CDFE. There-
fore the remaining proportion of A L to B Z, that
^s> of L B to B Z, is the proportion of the moment
^^ the weight CDFE to the moment of the weight
^' I A P E. But the proportion of A L to B Z is
^oitipouiided of the proportions of A L to A Z and
*^f A Z to Z B ; of which proportions that of AZ to
2 B is the proportion of the w eight C D F E to the
"^ eight CIAPE. WTierefore (by art. 5 of this

362

MOTIONS AND MAGNITUDES*

PART irr. chapter) the remaiiiiTis: proportion of AL to AZi is

*— C^ the proportion of the distances of the points Z ^tsA

The difttnfter j^ from the centre of the scale, ivhich is A. A^ nd,

of equip{inde-

Tation, &c, therefore, (by art. 6) the weight CI APE shall h ^ing
from O in the strait line O Z. So that O Z is one
diameter of equiponderation of the weight CIA^Tfi,
But the strait line AB is the other diameter of ec|n/.
ponderatiou of the same weight CI APE. Where-
fore {by the /th definition) the point Z is the cenfre
of the same equiponderation ; w hich 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, ns the complete fignre C D F E is to the
deficient figure CI APE ; which is that which wa.^
to be demonstrated.

Co roll I. The centre of equiponderation of any of
those plane three-sided figures, which are compared
w ith their complete figures in the table of art. 3,
chap. XVII, 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 cohimn
with the row of three-sided figures of four iDeaiis
this fraction 4, which signifies that that tigurt i^
to its parallelogram or complete figure as f to
unity, that is, as f to |, or as 5 to 7 ; and, there-
fore the centre of equiponderation of that fip:urt*
divides the axis, so that the part next the vertt^>
is to the other part as 7 to 5.

CorolL II, The centre of equiponderation of any
of the solids of those figures^ winch are eontainf*^

CENTRE OF BQUIPONDBRATION. 963

in the table of art. 7 of the same chap, xvii, is ^^^J ^^^
exhibited in the same table. For example, if the ^ — '^— '
centre at 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
^vide 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 f , that is i 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 ratwn of°Sir
are contained in the table of art. 3, chap, xvii, Se^hlafTlny
divides that line which is drawn through the ver- f}^^^ "'** ^^

o ficient figures,

tex parallel and equal to the base, so that the part divide that
next the vertex will be to the other part, as the drawn throigh
complete figure to the complement. JaUerto^the^*^

For let A I C B (in the same fiff. 5) be the half \"«» «i ^^^\

^ o / 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, asthe complete
whose axis is AB, and base BC, having A D ^^^^1^^^"^^
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 A I 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

MOTIONS AND MAGNITUDES.

PART m. A Q in its complement ; aiiJ so in all the rest of
^ — r^ — the ordinate lines and altitudes the mutation is
^^q^lp;";^^^ such, that that line, which is ordinately applied in
ration, itc the fijTiirej 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, iTae
proportion of AD to AQ in the complement AI CD,
which is the same with that of B C to L I, will be
subduplicate to that of C D 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 A IC D are its moment ; and all
the ordinate lines in AlCB are its moment. AMiere-
fore the moments of the complements of the halves
of deficient figures in the table of art. 3 of chap.
XVII, 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 A I C D.

Coroll, The diameter of equiponderation of these
halves may be found by the table of art. 3 of chap*
XV n, 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^

CBNTRE OF BQUIPONDERATION. 365

plete figure as * to i; that is 3 to 5. Wherefore part hi.
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 strdt
line drawn firom the vertex parallel to the base, so
tliat the part next the vertex will be to the other
part as 5 to 2. And, in like manner, any other of
tJie said three-sided figures being propounded, if
tlie numerator of its fraction found out in the table
l>e taken firom the denominator, the strait line
drawn from the vertex is to be divided, so that the
pMurt next the vertex be to the other part, as the
denominator is to the remainder which that sub-
^:raction leaves.

11. The centre of equiponderation of the half of '"^•«*°^ ®^

*• *- equiponderA-

^Lnyof those crooked-lined figures, which are intionofthehiaf
^lie first row of the table of art. 3 of chap, xvii, is Sefiden^t fi-*
xnthat strait line which, being parallel to the axis, SS^rowJf^the
divides the base according to the numbers of the taWeof art 3,

-. . 1 1 . - 1 -I 1 chapter xvii,

xraction next below it m the second row, so that may be found
tie numerator be answerable to that part which is num^w of the
towards the axis. .econdrow.

For example, let the first figure of three means
\>e 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
ABED is to the figure A B C D (by the table) as
5 to 4, the same ABED will be to the comple-
ment BCDE as 5 to 1. Wherefore, if FG be
Arawn 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 BC D will, by the pre-
cedent article, be somewhere in the same FG.
Again, seeing, by the same article, the complete

MOTIONS AND MAGNITUDES,

PART iir. fig;ure ABED, is to the complement B C D E as
^ — r^—- 5 to 1, therefore if BE and A D be divided in I
5p^«dera-^ ^iid H ES 3 to 1 , the centre of equiponderatioii of
tion, ^c. ^i^p complement B C D E will be somewhere id the
strait line which connects H and L Let now the
strait line L K be drawn through M the centre of
the complete figure, paraUel to the base ; and tbe
strait line N O through the same centre M, perpeu*
dicularto it; and let the strait lines LK and FG cut
the strait line H I in P and Q. Let P R be taken
quadmple to PU ; and let RM be drawn and pro-
duced to FG in S. R M therefore will be to MS
as 4 to 1, that is, as the figure A B C D to its com-
plement B C D E. Wierefore, 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 wOl either be the centres of equiponderation
of their own figures, or those centres will be in some
other points of the diameters of equiponderation
H I 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 H I and FG, and retain-
ing the proportion of M R to M S, that is, of the
figure A B C D to its complement BCDE. The
centre, therefore, of equiponderation of the figurt
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 whicli IN is four. But I N or PM Is 3
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 EQUIPONDERATION.

367

will be 2^. Wherefore Q S is 21 of tliose parts

of which F G is 6. But FQ is 1 ; and, tlierefore,

[FS is 31 . Wierefore the remaininj^ part GS is 2h

I So that FG is so di\ided in S, that the part to-

['Wards the axis h in proportion to the other part,

^as 2^ to 3|, that is as 5 to 7 ; which answereth to

the fraction f in the second row, next under the

I 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
liparabola will be divided into 3 and 5 ; and the
^liiiBe 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 eqinponderation of the half of
any of the figures in the second row of the same
table of art, 3, chap, xvii, is in a strait line parallel
to the axis, and dividing the base according to the
numbers of the fraction in the fi>urth row, two
places lower, so as that the numerator be answer-
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 BCD {in
fig. 7) ; whose complement is BCD E, 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 riglit
angles ; and because A B E D is to A B C D as 5 to

PART HL
23.

The cEctre of
e qui pondc ra-
tion of the half
of any of tlie
figuitrs of the
second row of
tliP same table
may he found,
out hj the
numbejsof the
fourth row»

368

MOTIONS AND MAGNITUDES.

PART iir. 3 let AB be divided in G, so that AG to BG b

^ — p^^ 3 to 5 ; and let F G be dra\^ii parallel to the \> ase,
Jq^pTderi^^Also because ABED is (by art 9) to B& Dfi
tion, &c, as 5 to 2^ let B E be divided in the point I, so ^«iiat
B I be to I E as 5 to 2 ; and let I H be dr^3iro
parallel to the axis, cutting LK and F G in P anci Q,
Let now PR be so taken, that it be to P Q as 3 to
2j and let RM be drav^n and produced to FG in S*
Seeing, therefore, RP k to PQ, that is, RM to
MS, as A BCD is to its complement BCD6^
and the centres of equiponderation of A B C D sm^^^i
BODE are in the strait lines FG and H I, atiC^
the centre of equiponderation of them both togi
ther in the point M ; R will be the centre of the
complement BODE, and S the centre of the
figure A B C D, And seeing P M, that is I N, is
to Q S, as R P is to R Q ; and I N or PM 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
FG, 4 ; and FS, 9 ; and G S, 5. Wherefore the
strait line ST being drawn parallel to the axis,
will divide the base A D into 5 and 9. But the
fraction I is found in the fourth row of the table,
two places below the fraction ^ 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 A, two places below in
the fourth row. And the same way sen es for all
the rest of the figures in the second row. In like
manner, the centre of equiponderation of the thirds
three-sided figure of three means w ill be found to^

CENTRE OF EQUIPONDERATION. 369

be in a strait line parallel to the axis, dividing the I'art iir.
base, so that the part next the axis be to the other ' — r^
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
FG, which are both known.

13. The centre of equiponderation of the half of ^h®. «*'»*'* «f

pquiponuen-

any of the figures, which (in the table of art. 3, tionofthehtif
chap, xvii) are compared with their parallelo- Sgurcs in the"
grams, being known; the centre of equiponderation J^J^g knoJm,
of the excess of the same figure above its triangle ^^ centre of

, ° ° the excess of

IS also known. the same figure

For example, let the semiparabola A B C D (in gie of ie^i
% 8) be taken, whose axis is A B ; whose com-^^"f,X^
plete figure is ABED; and whose excess above ^°°^"-
its triangle is B C D B. Its centre of equiponde-
radon 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 H I 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 A BD will be I. Again,
let K L be drawn parallel to the base, so that
AK be to A B as 2 to 5 ; and M N parallel to the
axis, BO 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
ABCD is N; and therefore we have the centres
of equiponderation of the semiparabola A B C D,
wid of its part the triangle A B D. Tliat we may
^iow find the centre of equiponderation of the
I'emaining part B C D B, let IN be drawn and
produced to O, so that N 0 be triple to I N ; and

VOL. I. BE

IT HI-
SS.

>-»^ — — '

centre of
Hiiitlcra-
of ihe

MOTIONS AND MAGNITUDES,

0 will be the centre sought for. For seeing tlje
weight of A B D to the weight of B C D B is j„
proportion reeiproeal to that of the strait line ^q
to the strait line IN; and N is the centre of tbv
whole, and I the centre of the triangle A B D ; ()
will be the centre of the remaining part, name/v,
of the figure B D C B ; wkich was to be found*

CoroU, The centre of eqniponderation of the
figure B D C B is in the concourse of two strait
lines, whereof one is parallel to the bai^e^ and
divides the axis, so that the part next the bage be
f or V*i* of the whole axis; the other is parallel to
the axis, and so divides the base, that the part
towards the axis be |, or l^ of the whole baso.
For draw ing O P parallel to the base, it will be as

1 N to N O, so F K to K P, that is, so I to 3, or

5 to 1 5. But A F is A, or ^ of the whole A B ;
and A K is ttj or i ; and F K A ; and K P A ;
and therefore A P is A of the axis A B. Also AH
is ij or t/i : and A M |, or A of the whole bast' ;
and therefore O Q being drawn parallel to the
axis, M Q, which is triple to H M, w ill be iV.
Wherefore A fel is H, or i 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.
XV M, have their centres of equipon deration in t^vo
strait lines, which divide the axis and base accord-
ing to those fractions, T\'hich add 4 to the nuuie-
rators of the fractions of a parabola ^*, and H ; and

6 to the denominators, in this manner : —

In a parabola, the axis i!, the base B.

In the first three-sided figure, the axis H, the base i|.
In the second three-«ided figure, the axis i?, the bade J|, Ac.

And by the same method, any man, if it k

worth the pains, may find out the centres of eqni- paut iir.
poiideratiou of the excesses above their triangles - — .-^
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 t?m^fj^^soM
right cone, whose vertex is the centre of the IJ'J^'^^'JidS
sphere, and the portion of the sphere whose base ^'^^^ ^\^ p^^

/ ' ^ '^ ^ . . . n«^t the Tertex

is the same with that of the cone, divides the strait be to the whole
Hue which is made of the axis of the cone and half SniirLx^fof
the axis of the portion together taken, so that the ^^l ^^^^^ ^^
part next the vertex be triple to the other part, or ^ t" *•
to the whole strait line as 3 to 4.

For let A B C (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 wliich
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-
deratiun of the sector A BC,

Let the strait line F H be drawn of any length,
making right angles with A F at F ; and drawing
the strait line A H, let the triangle AFH be made.
Then upon the same centre A let any arch I K be
drawn, cutting AD in L ; and its chord, cutting
AD in M ; and dividing ML equally iu N, let NO
be drawn parallel to the strait line FH, and meet-
ing with the strait line A H in 0.

Seeing now B D C is the spherical supei-ficies of
tlie portion cut off with a plane passing through
BC, and cutting the axis at right angles; and
seeing F H divides E D, the axis of the portion,

B B 2

FART II L into two equal parts in F ; the eentre of equipon-

^ — '-^ deration of the superficies B D C will be in F (by

Jj'nTralTof^^t. 8); and for the same reason the centre ol

a solid, ate. equiponderatioD of the superficies I L K, K beinfi

in the strait line A C, will be in N, And in like

manner J if there were drawn, between the centn

of the sphere A and the outermost spherical super

fieies of the sector, arches infinite in number, thi

centres of equiponderation of the spherical super

fieies, in which those arches are, w ould be founc

to be in that part of the axis, which is intereeptec

between the superficies itself and a plane passing

along by the chord of the arch, and cutting thi

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
fieies ILK in proportion dupHcate to that of thi
arch BDC to the arch I L K, that is, of B E t(
I M, that is, of F H to NO; let it be as F H U
N 0, so N O to another N P ; and again, as N O tc
N P, so N P to another N Q ; and let this be done
in all the strait Hues parallel to the base F H tha
that can possibly be drawn between the base ant
the vertex of the triangle A F H, If then througl
all the points Q there be drawn the crooked lint
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 sectoi
A B C D is compounded ; and by consequent, th«
moment of the sector it?elf. Let now F H be un^
iderstood to be the semidiameter of the base of i
[tight cone, whose side is AH, and axis

CENTRE OF EQUIPONDERATION. 373

Wherefore, seeing the bases of the cones, which p^i^t hi.
pass through F and N and the re^t of the points ^ — ^
of the axis, are in proportion duplicate to that of p^ndwa^Jon^^Jf
the strait lines .FH and N O, &c., the moment of »«>"d»^<^
all the bases together, that is, of the whole cone,
will be the figure itself A F H Gl A ; and therefore
the centre of equiponderation of the cone A F H is
the same with that of the solid sector. Wherefore,
seeing A G is |^ 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
addeth nothing to half the axis of the portion.
Wherefore, if in the axis of the hemisphere there
be taken from the centre i of half the axis, that is,
% of the semidiameter of the sphere, there will be
the centre of equiponderation of the hemisphere.

374

MOTIONS AND MAGNITUDES,

CHAPTER XXrv.

OF REFRACTION AND REFLECTION.

PART III.

* i -

Definitions.

1 . DefinitioTi&i,— 2. In perpendicular motion there is no refrae-
t!on.— 3. Tilings thrown out of a thinner into a thicker me-
dium are so refracted that tlie angle refracted is greater thao
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, sis the density of the first medium is to the rlensity 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 tite 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 ill
a thinner, the otlier in a thicker medium, the sine of the angle
of inclination will be o mean proportiotial 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, find the proportion of their densities be the same witii
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 npon
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 s^^ame happens in the generation of motioD
in the line of incidence.

DEFINITIONS.

I. Refraction is the breaking of that strait
line, ill 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 Une of
incidence ; the latter the refracted line.

OF RBVRACTION AND REFLBCTION. 375

III. The point of refraction is the common partiil
point of the line of incidence, and of the refracted - — r^

line. Definition..

IV. The refracting s^uperjiciesy which also is
tlie 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 diflfe-
reiit medium.

VI. 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
ABF. The refracted line is BF. 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 CBF. The angle refracted is HBF. 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.

PART UT.
2*,

In perpenJi-
eulaT mutioii
there is no
jrcrnctiGQ*

X. And that medium in which there is equal re-
sistance everywhtn'e^ is a homogeneous medium.
All other mediums are heterogeneous.

2, If a body pass, or there be generation of mo-
tion fi'om one medium to another of diflFerent
density, in a line perpendicular to the separating
superficies, there will be no refraction.

For seeing on every side of the pei-pendieular
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
no cause from which refraction may be inferred to
be on one side of the peqiendicular, which will
not conclude the same refraction to be on the
other side. Which being so, refraction on one
side w ill destroy refraction on the other side ; and
consequently either the refracted line will be
everj'wbere, wliich is absurd, or there will be no
refracted line at all ; w hich 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, ivhether 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
^^^^'^a'^liiiXfi^^ internal parts, as a stone, be moved obliquely
~"* *'th«l ^^^^ of the thinner medium, and proceed penetrating
re- the thicker medium, and the thicker medium be
r tiian such, as that its internal parts being moved restore
iiSnf themselves to their former situation ; the angle
refracted will be e:reater than the angle of incli-

nation.

OF REFRACTION AND REFLECTION. 3/7

For let D BE (in the same first figure) be the part in.
separating superficies of two mediums ; and let a — ^ — -
body, as a stone thrown, be understood to be ™X j*^;;;,^^';
moved as is supposed in the strait line ABC; and »"^^ a timker

, . ^ , . , , . ,. . , . medium, &c.

let A B be in 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-
ciilar 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
AG, 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
BF, and make the angle H B F greater than the
angle H B C.

And from hence may be derived the cause, why
bodies which are throwTi in a Very oblique line, if
either they be any thing flat, or be thrown w ith
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

PART III,

24.

Endeavour,
which from ojie
point tpnclcth
every way, will
bcMj refracted,
ft! that the sine
of the angle re*
fracted will he
to the tine of
the angle of in-
ch^natian^asthe
density *>f tlie
first medium is
to the dtnisity

second
reti-

Ukeii,

MOTIONS AND MAGNITUDES-

water ; into which, from the point C^ let a stone be
thrown in the strait Mne C A, making with the line
B A produced a very little ane^le CAD; and pro-
dncine: B A indefinitely to D, let C D be drawn per-
pendicular to it, and A E parallel to C D. The ^toue
therefore %vill 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 frina
the motion in C D and all its parallels downwarcb,
as soon rs the stone falls upon A^ there will be
reaction upwards, because the water restorer itself
to its former situation. If now the stone be thrown
with sufficient obliquity, that is, if the strait linf
C D be short enough, that is, if the endeavour of
the stone downwards be le.ss 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 dovvTiwards, 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.

4. If from a point, whatsoever the medium bej en-
deavour be propagated every way into all the part>
of that medium ; and to the same endeavour there \
be obliquely opposed another medium of a different
nature, that is^ either thinner or thicker; that
endeavour will be so refracted, that the sine of the
angle refracted^ to the sine of the angle of incline- i
tion^ will be as the density of the first medium to "
the density of the second medium, reciprocally
tak(*n.

First, let a body be in the thimier medium in

OF REFRACTION AND REFLECTION. 379

(fig. 3), and let it be understood to have endeavour pam hi,

24.

every way, and consequently, that its endeavour
proceed in the lines A B and A h ; to which let wWchr&IJ^
B& the superficies of the thicker medium be
obliquely opposed in B and A, so that A B and A h
be equal ; and let the strait line B A be produced
both ways. From the points B and A, let the per-
pendiculars B C and he he drawn ; and upon the
centres B and A, and at the equal distances B A and
b A, let the circles A C and A c be described, cutting
B C and A r in C and c, and the same C B and c A
produced in D and rf, as also A B and A A produced
in E and e. Then from the point A to the strait
lines B C and A c let the perpendiculars A F and A /"
be drawn. A F therefore wU be the sine of the
angle of inclination of the strait line A B, and kf
the sine of the angle of inclination of the strait
line A A, which two inclinations are by construc-
tion made equal. I say, as the density of the
medium in which are B C and A er is to the density
of the medium in which are B D and A rf, so is the
Bine of the angle refracted, to the sine of the angle
of inclination.

Let the strait line F G be drawn parallel to the
Wrait line A B, meeting with the strait line A B
produced in G.

Seeing therefore A F and B G are also parallels,
they will be equal ; and consequently, the endea-
vour in A F 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

PART in.

propafi:atefl less in B G than in A F, accordiiijE: to
the proportion which the density of the mediimiju
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 BG is, be to the density of the
medium, in which A F is, as B G is to B H ; 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 B D ; whicli
being done, B H and IK mil be equal ; and I K
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 I K. Seehig therefore in the time A B,
which is the radius of the circle, the endeavour is
propa£:ated in A F in the thinner medium^ it uill
be propagated in the same time, that is, in the
time B I in the thicker medium from K to L
Therefore, B I is the refi-acted 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 I K is to A F, as the
density of the medium in which is A F to the
density of the medium in which is I K ; it will be
as the density of the medium in which is A F or
BC to the density of the medium in which is
I K or B D, so the sine of the angle refracted to
the sine of the angle of inelination. And by the
same reason it may be shown, that a.s the density
of the thinner medium is to the density of the
thicker medium, so will K I 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 iil
both the mediums were of the same density, the - — ^^-^
tndeavour 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 BH. But because the
density of the medium, in which is IK, to the
density of the medium, in which is L M, is as B H
to B G, that is, to A F^ the endeiivour w^ill be pro-
pagated fiirther in the mediiim 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 ML 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 antrle refracted to the sine of
the angle of inclination ; which was to be demon-
strated.

In this demonstration, I have made the sepa-
rating supeiiicies B h plane by construction , But
though it were concave or convex, the theorem
would nevertheless be tnie. 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- Timsmeofthe
nation is to the sine or the angle reiractert n\ m mie inciiiia-
another inclination, as the sine of the angle of that ^^^^^ Jr* ^i^''^ f^!
inclination to the sine of the ande of this inch- ^''^''^V^'^""*^^^"

° another ini^li-

nation.

382

MOTIONS AND MAGNITUDES.

PART III.

2i.

nation, aa the
sine of ihe an-
gle ul that in-
clinalion is to
the sine of the
wtiglc of ihlB
iaclinatioi}.

If Uo lines
of incidence,
having e[][Ual
inclinutiou, he
one in a thinner
the other in a
thicker nie-
diutiif the sine
of the flijglc of
indinaljon nill
be a mean pro-
porlioual he*
tweeu the two
iines of the re-
fracted angles.

For seeing the sine of the refracted angle is to
the sine of the angle of inclination, whatsoever
that inclination be, as the density of one medium
to the density of the other medium ; the propor-
tion of the sine of the refracted angle, to the sine of
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. Wlierefore
refracted angles in different inclinations are as the
sines of the angles of those inelinatious ; whiclt
was to be demonstrated.

a. If two lines of incidence, having equal inch-
nation, be the one in a thinner, the other in a.
thicker medium, the sine of the angle of their in-
clination will be a mean proportional between the
two sines of their angles refracted. ■

For let the strait line A B (in fig. 3) have its in-
clination in the thinner medium, and be refracted
in the thicker medium in B I ; and let E B have as
much incHnation 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
' *^ker medium to the density of the thinner me-
o R S to A F. But it is also as the den-
he same thicker medium to that of the
iner medium, so A F to I K. Wherefore
: A F. I K are proportionals ; that is, RS^ fl
IK are in continual proportion, -^^-^ ^ ^
au proportional; which was to be

and A F

7. If the ane^le of inctioatiou be semirect, and
the line of inclination be lu the thicker medium,
and the proportion of the densities be as that of a
diagonal to the side of its square, and the sepa-
rating superficies be plain, the refracted hue will
be in that separating superficies.

For in the circle A C (fig. 4) let the angle of in-
clination A B C be an angle of 45 degrees. Let
C B be produced to the (circumference in D ; and
let C E^ the sine of the angle E B C, be drawn, to
which let B F be taken equal in the separating
line B G. B C E F will therefore be a parallelo-
gram, and F E and B C\ that is F E and B G equaL
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 mc^dium, in which C is, to the den-
sity of the medium in which I) is ; and so also the
sine of the angle refracted to the sine of the angle
of inclination. Drawing therefore FD^ and from
D the line D II perpendicular to A B produced,
J) 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,
I as A G is to F E, that is as B G is to D H ; and
■feeing D H is the sine of the angle of inclination,
BG will therefore be the sine of the angle re-
fracted. Wlierefore 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 %vhen the
inclination is greater than 45 degrees, as also

PART III.
21.

If llie uiiglo of
inclinalion ho
acimrcQl, and
the tine of in-
clinaliDti he ill
the ihicker me-
diuin^ and ihe
pro port ion of
their densities
be till? same
with that of the
diagt^nal lo the*
side of a square,
and the scpu-
radng 8Uper>
fictes be plain,
tlic refrat'ted
line will be In
the separating
RLiperncies.

r JiL

•ody be
iu tt
tie upon
bodvi
not pe-
itjbutbc
il from
sngle of
m will
I to tbe
»f inci-

when it is less, provided the density be greater, it
may happen that the refraction will not enter the
thinner medium at all.

8* If a body fall in a strait line upon anotkr
body, and do not penetrate it, but be reflected
from it, the angle of reflection will be equal to
the angle of incidence*

Let there be a body at A (in fig. 5), which fail-
ing with strait motion in the line A C upon another
body at C\ passeth no farther, but is reflected ; aud
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 liie
D F let C G be taken equal to CD; and let the
perpendicular G E be raised, cutting C E in E,
This lieing 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 L 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-
eflicient or concurrent motions, the one in A H
parallel to D G, the other in A D perpendicular to
the same DO; of which two motions that in AH
works nothing upon the body A after it has been
moved as far as Cj because, by supposition, it do^
not pass the strait line DG ; whereas the enc
vour in A D, that is in H C, worketh further to-
wards L But seeing it doth only press and not

OF REFRACTION AND REFLECTION-

385

PART m,

24.

penetrate, there will be reaction in H, which
rauseth 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 eqnal to the two motions it
had formerly in A H and H C, Wlierefore it will
be carried on in C E. The angle therefore of re-
flection will be ECG, equal, by construction, to
the angle A C D ; 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, Th^^ame hap-

X r J ^ pens lu the

but some endeavour only be propagated from A to generation of
C, the demonstration will nevertheless be the \\m of inci.
same. For all endeavour is motion ; and when it *^^°*^^'
hath reached the solid body in C^ it presseth itj
and endeavoureth further in C L 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.
6) to the circumference in B, and the centre of
the sphere be C, and the line CB be drawn, as
VOL. I. c c

noted, that if CB be prodoced howsoevCT
the endeavour iu the liue G B C will procei
from the perpendicular reaction in G B ; ai
therefore there will be no other endeavour
point B towards the parts which are with
sphere, besides that which tends towi
centre.

And here I put an end to the third pat
discourse ; in which I have considered mot
magnitude by themseh es in the abstract
fourth and last part, concerning the pketiom
nature, that is to say, conceniing the motjl
magnitudes of the bodies which are part?
world, real and existent, is that which folloi

1 * The connexitin of what hath been said with that which fol-
loweth.' — 2. The investigation of the nature of sense, anil the
definition of s^Dse.— 3. Tlte subject and object of senm,
4. Tiie organa of sense, — 5. All bodies are not indued with
sense.— 6, But one phantasm at one and the aaine time*
7» Imagination the remains of past een»e» which also is memory.
Of sleep,— 8. How pliantaams succeed one anothcr.^ — 9.
Dreams, w!ience they proceed. — ^10, Of the senses, their kinds,
their organs, and phantasms proper and common- — ^U. The
magnitude of images, liow and by what it is determined.

12. Pleasure, pain, appetite and aversion^ what they are.

13. Deliberation and will, what.

L I HAVE, ill the first chapter, defined philosophy part rv.
to be knowledge of effects acquired by true ratio- -
cinat ion, from kuowledgefir^t had of their causes ^\^^^l^^
and generation; and of such causes or genera- ^'^''^^''''^'^^^

J r^ r , , , ^ , , , that which

tions as may fje^ Jromjormer fmoivledge of their foUoweth.
effects or appearances. There are, therefore,
two methods of philosophy ; one^ from the geiie-
ration of things to their possible effects ; and the

cc2

other, from their eflFects or appearances to some
possible generation of the same. In the former

^""whrrhi^tS ^^ ^^^^^ ^^^ ^^^^ ^^ ^^^ ^^^^ principles of our
been said with ratiocination j namely definitions, is made and con-
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 eflFects of nature, which we know by sense, some
w ays 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 wdthout such general propositions as
have been already demoiistrated, 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

The connexion

OF SENSE AND ANIMAI. MOTION.

such things as appear, or are shown to us by na-
ture, we call phenomena or appearances.

Of all the phenomena or appearances which are ^/^' ^j^^j ^^^
near us, the most admirable is apparition itself, f^irw^dch^ou
Tu faiv((F9at ; namely, that some natural bodies have lowcth.
in themselves the patterns almost of all things, and
Dthers of uoue 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 J 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-

n or endeavour (and endeavour also is motion)

The

in the internal parts of the thing that is altered,
hath been proved (in art, 9, chap, viii) from this,
tion Pf' ihf n^ ^^^^ whilst even the least parts of any body remain
ttiw of jente, in the Same situation in respect of one another, it

and the defim- - i , , .

tiou Qf senge. cannot be said that any alteration, unless perhaps
that the vthole 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, w^e derive tlie phantusm or idea
that is made in us by our sen?**^
soever it be, is that

Sense, therefore, is some interual motion in the partiv,
sentient, generated by some internal motion of the * — ^ — '
parts of the object, and propagated throngh 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 1 shall give you the whole definition
^f sense, as it is drawn from the expUcation of the
Bauses thereof and the order of its generation, thus:
SENSE t* a phanta^m^ made by the reaction (tml
etidearoiir oMtward.H in the organ of sense , caused
by an endeaiour inuardsjrom the object, remain-
itigfor some time more or less. _

■ 3. The subject of sense is the sentient itself, The object and
namely, some living creature ; and we speak more object of n—
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,
^or light and colour, and heat and sound, and

i*ART iv. other qualities which are commonly called sensible,

are not objects, but phantasms in the sentients.

»od objS ^^^ ^ phantasm is the act of sense, and differs no
•eiue. Otherwise from sense than^yf^^r?, that is, being: a

doing, differs from /actum ejtse, that is, being
done ; which cHfference, in things that are done in
an instant, is none at all ; ^nd 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 on
to the last, and that to any di.stance, 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.

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 li\ing
creatures are found to be certain spirits and mem-
branes, which, proceeding from the pia 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

Tlie organi of
•ease.

la hnrt or obstructed, that the motion can be partiv

2a.

propagated no further, no i^ense 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 ^^^ ^^^^
by reaction, nevertheless it is not necessary that *ith 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 phantiism 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
[ijense 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 j 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

«hich is commonly so called, hath necessarily
ime memory adhering to it, by which former and
later phantasms may be compared together, and
distinguished from one another.

PART IV.

^ r — '

All Ibodies &re
not endaed
wiiU icnse.

But ODG pfaaii«
Usm at one aud
Ihe same time.

Sense, therefore, properly so called, must ne-
cessarily have ill it a perpetual variety of phan-
tasms, that they may be discerned one from
another. For if we should suppose a man to be
made with clear eyes, and all the rest of his org^ans
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
fig^ure^ 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 ray 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.

6. And yet such is the nature of sense, that it
does 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 diride
its place ; and when we reckon many bodies, we
must oecessarOy 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

PART IV.

25,

understand the same number of motion*?
also ; and ss oft as we comit many motions, so oft
we reckon many times. For though the object we ^^^^^t^c^/e^^'^d
look upon be of divers colours, yet with those ^^e wme Ume,
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 tit to receive
any other impression from whatsoever other ob-
jects, to what sense soever those objects belong.
lind hence it is, that an earnest studying of one
object, takes away the sense of all other objects for
the present. For fftmly is nothing else but a pos-
■ision 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 Htupidm in Junam-
htilo animum occuparat'' For what is stupor but
that which the Greeks call aimwBnam^ 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 pjige be
presented to our eye ; and though every several
■ptter be distinctly wTitten there, yet when we look
^pon the whole page at once, we read nothing.

396

PHYSICS.

PART IV.
25,

Imagtnationy
the remain B of
past sense i
which alio
ii memozy*
Of i]««p.

From hence it is manifest, that every endeavour
of the organ outwardSj is not to be called sense,
but that onlyy which at several times is by vehe-
mence made strong-er 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 ^dth his
excess of brightness.

7* But the motion of the organ, by which a
phantasm is made^ is not commonly called sense,
except the object be present. And the phantasm J
remaining after the object is removed or past by, ^
is called ^/« wry, and in Latin imagmatio ; which_a
word, because all phantasms are not images, dath_^
not fully answer the signification of the wordyV/wcjr^
in its general acceptation. Nevertheless I majr*
use it safely enough, by understanding it for the^

Greek *aiTacria.

Imagination therefore is nothing else hutsens^
decnyingy or weakened^ by the absence of th^
object. But what may be the cause of this decaj^
or w eakening ? Is the motion the weaker, because^
the object is taken aw ay ? If it were, then phan-
tasms would always and necessarily be less clear
in the imagination, tbfin 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 tlie less
predominant. Whereas in sleep, the passages

OF SENSK AND ANIMAL MOTION.

397

PART l\\
25.

being shut up, external action doth not at all
disturb or hinder internal motion*

If this be true, the next thing to be considered, ^f"^«*'P-
^ili 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 nerv^es is necessarily
iatercepted. 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 awake th. 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 aHowpMn-
thing as many perhaps think it, that phantasms in one"^^o'^w!
this their great variety proceed from one another;
and that the same phantasms sometimes bring into
the mind other phantasms like themselves, and at

other times extremely unlike. For in the motion
of any continued body, one part follows another by
cohesion ; and therefore, whilst we turn our eyes
and other 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 w aking
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 ^ai^atitidm and mem hiisse^ Janet/ and
memory y differ only in this, that memory supposetli
the time past, which fancy doth not. In memory,
the phantasms we consider are as if they were w oni
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 m 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 part iv.
are not diseerned, by reason of their reraoteness ; so ^ — A
in memory, many accidents and places and parts
of things, which were formerly perceived by sense,
are by leni^th of time decayed and lost.

The perpetual arising of phantasms, both in
sense and imagination, is that w^hich we commonly
caU discourse of the mind, and is common to men
with other living creiitures. For he that thinketh,
compareth the phantasms that pass, that is, taketh
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 ha\e 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
lucidj is nothing else but the memory both of a
heating, and of an enlightening object,

9. The phantasms of men that sleep, are dreams. I>reamfl,

. 11, - whence they

Concermng 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
Itered through sleepiness; and sometimes also

OF SENSE AND ANIMAL MOTION,

401

some of those pbniitasms that are still in motion ^'^^ ^^•
in the brain ; and whim any internal motion of the ^-^^' — '
heart reacheth that membrane, then the predomi- whence \hty
nant motion in the brain makes the phantasm, p^"*^^*^^
Now the motions of the heart are appetites and
aversions, of which I ^hall presently speak further.
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 lieat in the heart, whatsoever
be the cause of it, is generated anger and the
image of an enemy » in sleep. And as lov^e and
beauty stir up heat in certain organs ; so heat in
the same organs, from whatsoever it proceeds,
often canseth 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-
ta.sm, 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 lis, because admiration requires that the
things appearing be new and unusual, which can

VOL. I- DO

402

PHYSICS.

PART IV. happen to none but those that remember former
* — T-^ appearances ; whereas in sleep, all things appear
as present.

But it is here to be observ ed, that certain dreams,
especially such as some men have when they are
between sleeping and waking, and such as happen
to those that have no 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
A^oices 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
liallowed 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 gho.sts and incorporeal sub-
siances.

thek kioSr^*' 10. In most living creatures there are observed
iiieir organa five kiutls of scuses, which are distinguished by their
proper and ' orgaus, aud by their different kinds of phantni^ms ;
commou. namely, ^tigkt^ hearing:, mueif, fa^fe, and touch;
(lud 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 waterj^ 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 on the other side* by the

OF SENSE AND ANIMAL MOTION.

403

wliary processes, and the coat of the crystalline partiv
humour ; the crystalline, which, hanging in the ^

midst between the ciliary processes, and being ^^ir WndT^a*
almost of spherical figure, and of a thick con-
sistence, is enclosed on all sides with its own traus-
pareut coat ; and the vitreous or glassy humour,
which tilleth 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 ehoroeides^
which is a part of the pia maler^ saving that it is
covered with a coat derived from the marrow of
the optic nerve, which is called the rethm ; and
this choroeiiies^ seeing it is part of the pia wafer, is
continued to the beginning of the medntla ."ipinulh
within the scull, in which all the nerv^es w hich are
witbin 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 retiformu^ or
whether they be other arteries which are inserted
into the substance of the bram. 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 reacheth from tlic eye to the

D D 2

TART IV, roots of the nerves is proper only to sight. Tlie
* — r— proper organ of hearing is the tympanum of the
thlir^i^nar^c' ^^*" ^^^ *^^ ^^^ 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 neitlier 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 in 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 €?2oc, and ■
ilStoXov^ and ISia ; and by the Latins, species and

names signii

pearance

Tlie phantasm, which is made by hearing;, is ,^eii k1 Jr&c!
sound ; by smell, odour ; by taste, savour ; and by
touch, hardness and softness, heat and cold, wet-
ness, oiliness, and many more, w hich 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 mnmal
tnotions. For seeing in all sense of external things
there is mutual action and reaction, that is, two
endeavours opposing one anotlier, 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
intenial organ, the endeavour outwards will pro-
ceed in a soUd angle, which will be greater, and
consequently the idea greater, than it would have

Jgen if the impression had been weaker.

^pi 1 . From hence the natural cause is manifest, ThemigmudA
first, why those things seem to be greater, which, LdT/ whaVlr
cteteris paribus^ are seen in a greater angle : ^* <*ctemiitieci

PART IV*

25.

secondly, why in a serene cold night, when the
moon doth not shincj 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 morf
pressing, helpeth or strengtheiieth the action of tk
stars upon our eyes ; in so much as stars may then
Ix* seen which are seen at no other time. And thi;^
may suffice to be said in general conceniing 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 w hen I come to speak of man,
Fli»iiire»p«in, 12. But there is another kind of sense, of which
aveman' what I Will say Something in this place, namely, tlie
they arc. geusc of plcasurc and pain, proceeding not from
the reaction of the heart outw^ards, but from con*
tinual action from the outennost part of the oi^n
towards the heart. For the original of life being
in tlie heart, that motion in the sentient, which h
propjigated to the heart, must necessarily make
some alteration or divei*sion of vital motion, namely,
by quickening or slackening, helping or hinderinf^
the same. Now when it Iielpeth, it is pleasure;
and when it hindereth, it is pain, trouble, grief,
&c. And as phantasms seem to be without, by
reason of the endeavcmr outwards, so pleasure and
pain, by reason of tlu^ endeavour of the organ iu-
wards, seem to be within ; namely, there where the
first cause of the pleasme or pain is ; as when the

pain proceeds from a wound, we think the pain tart iv,
and the wound are both in the same place. - *.'--

Now vital motion is the motion of the blood, ^!'^1'''' ^'"^

' appetite, aijcl

perpetually circulatmg (as hath been shown from **^<^'8ion, 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 ; wliich 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. But 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 preserv ation and
augmentation of that motion, by the help of the
ner\^es. 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 wimt
pleaseth it. And this first endeavour, when it
tends towards such things as are known by expe-
rience to be })leixsant, is called appetite,^ that is, an
approaching ; and when it shuns w hat is trouble-
some, aversiouy or flying from it. And little m-
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 memoi7 ; 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-

PART rv,

25.

Drlib<:ffttion
jixid ^lUf v^'liAt

oat such 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 of harm, yet some-
times they approach and sometimes retire from
the same thing, as their doubt prompts them. But
iifterwards, by acciistomiDg 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 nerv es 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 foDowed by contraction and extension of the
limbs, which is animal motion.

13. The considerations of appetites and aver-
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 pleaseth, or to avoid that
which displeaseth them. Appetite, therefore, and
aversion are sim]>ly so called as long as they follow

OF SENSE AND ANIMAL MOTION.

409

Sr\ tlifif Dclibcmtion

not deliberation. But if deliberation have gone patitiv,
before, then the lai>t act of it, if it be appetite, is ^
called will; if aversion, miwillmgness
the same thing is called both will and appetite ;
but the consideration of them, iiamely^ before and
after deliberation, is divers. Nor is that which is
done within a man whilst he willeth any thing,
diflFerent from that which is done in other living
creatures, whilst, deliberation having preceded,
they have appetite.

I Neither is the freedom of wilUng or not w illing,
greater in man, than in other living creatures. For
where there is appetite, the entire caiLsc of appetite
hath preceded \ and, consequently, the act of
appetite could not choose but follow, that is, hath
of necessity followed (as is shov^n 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 w illing, but of doing
what they w ill, 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 it^i 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 hopt% and from
aversion, fear. For where there is no hope, it is
not to be called fear, but hate ; and where no 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.

PART IV. a certain fniitioii of good or e\il ; as anger isave
sionfrom some iramiueiit evil, but such as is joined

and wil* wbaL with appetite of avoiding that e\il by force, Bm
because the passions and perturbations of the mina
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 whicfl
is couceruiug mau. As for those objects, if there
be any such, which do not at all stir the mind, w^
are said to contemn them, f

And thus much of sense in general. In the next
place I sliall speak of sensible objects.

CHAPTER XXVL

OF THE WORLD AND OF THE STARS.

1 . Tlie magnitude linil duration of the world, ijiscrij table. — 2. Ni
place in the world empty, — IL The arguments of Lucretius foi
vacuum^ invalid. — 4-. Othor arguments for the eetablishing i
vacuum, invalid*— 5. Six suppositions for the salving of tb
phenomena of nature.^6. Possible causes of the niotiona
annual and diurnal ; and of the apparent direction, station, and
retrogradation of the planets, — 7. The sappositioii of sirapk*
motion, why likely. — 8. The cause of the eccentricity of tlie
annual motion of the earth. — 9, The eause why the moon bath
always one and tlie same face turned towards tlie eartli.
10. The cause of the tides of the ocean. — 1 L The cause of t
precession of the equinoxes.

The magnitude 1. CONSEQUENT to the Contemplation of sense
X^ ioriir'iii- *^^ cootemplatioii of bodies, which are the efficie
Bcrutabie. causes OF objects of sense. Now every object

either a part of the whole world, or an agg^regate"
of parts^ The greatest of all bodies, or sensibl
objects^ is the world itself; wliich we behold wl

THE WORLJJ AND THE STARS* 411

we look round about us from tliis point of the same ^art iv.
which we call the earth. Conceroiut; the world, ^ — r — -
as it is one aggregate of many parts, the thhigs ^^^^.j^^'^^f
that fall under inquiry are but few ; and those we *^^ *•;'/'*» *""
can determine, none. Of the whole w orld we may
inquire w hat is its magnitude, what its duration^
and how many there be, l)ut 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 takeUj as I have shown before in chapter vii-
All other phantasms are of liodie^s or objects, as
they are distinguished from one another ; as colour,
the phantasm of coloured bodies ; somid, of bodies
that move the sense of hearing, &c. The questions
concerning the mjiguitude of the w orld are whether
it be finite or infinite, full or not full ; concerning
its duration, wiiether it had a beginnings 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 no controversy at all. Also if it had a begin-
ning, then by what cause and of w hat matter it was
made ; and again, from whence that cause and
that matter liad 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 tht^ uiu-
vei^ial 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,
wv learn it from onr phantasms ; and of infinite,
whether magnitude or tunc, there is no phantasm

PART IV.
26.

MfUtlbltli

at all ; so that it iis impossible either for a man oi^
P ' — ^^-^ any other creatiire to have any conception of inl
^d drrationlf ^**^- And though a man may from some effecF
tb« world, in- proceed to the immediate cause thereof, and from
that to a more remote cause, and so ascend conti-
nually by right ratiocination from cause to caus^H
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 au^
end or not. But whether we suppose the w orld
be finite or infinite, no absurdity will follow. For"
the same thinf^s 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 fii'st 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 w hich is already
moved. The questions therefore about the mag-
nitude and beginning of the w orld, are not to be
determined by philosophers, but by those that are
laT;\fully 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 w hen he had delivered
up the world to the disputations of men, it was
pleasure t tiMMMieming the nature i

infinit 'y to himself, should,

as Uir ' ' ed by thos

whos» ordering

d those that

boast they have demonstrated, by reason?^ drawn ^^^J ^^•
from natural things, that the work! had a beginning, ■ *» "
They are contemned l>y idiots, because they under- J^'^^^^^^ttntf
stand them not ; and by tlie learned, because they '^^« ^";^^* i°-
understand them ; by both deservedly. For who
can commend him that demon^^trates 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

B 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 aw ay eternity from the Creator of the world ?
From this absurdity therefi^re they run into another,
being forced to call eternity nunc sians^ 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 theti 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,

H that reason thus absurdly, are not idiots, but,
which makes the absurdity unpardonable, geome-
tricians, and such as take upon them to be judges.

PART IV.
2(i.

No place In Ihe
worlJ empty.

impertinent, hut ^exere judges of other men's
demonstrations. The reason is this, that as soon
as they are entangled in the words injimte and
eternal^ of which we have in our mind no idea, but
that of our own insufficiency to comprehend tliem,
they are forced either to speak something absurd,
or, which they love w orse, to hold their pe^ce. For
geometry hath in it somewhat like wine, which,
when new, is windy ; but afterwards thougli 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
begiiining 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 law s. And so I pass on to such things as it is
not unlawful to dispute of,

2. Concerning the world it is further questioned,
whether the parts thereof be contiguous to one
another, in such manner as not to admit of the lexist
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

water will not flow out at any of the holes m the part iv>

^ _ .^ , ^ i . . . 2(i,

bottoDi B.

anrt im No plactMii the
^•^^ ^ world empty.

But if the finger be removed to k*t in
the air above, it will run out at them all ;

^coon 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 dov^n 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

prays^ 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 w^e see it does in a vessel whose mouth
is wide enough, when we turn suddenly the bottom
upwards to pour out the w^ater ; for then the air
being forced by the w eight of the w^ater, 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 natund motion of the w ater,
w hich is a heavy body, downwards, w ould not be
liindered*

3. On the contrary, for the establishing of va- tJ»«"^>?^»i«

^ i * of Lucretius

cxium, many and specious arguments and experi- for vacuum
ments have been brought. Nevertheless there
seems to be something WTmtiug in all of them to

FART IV. conclude it finnly- Tliese arguments for vacuum

^ — ' — ' are partly made by the followers of the doctrine

Ir^LSc^otTr'' of Epicurus ; who taught that the world consists

^^^"^rT"^^ of very 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 caUs

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 no beginning anywhere ; and consequeiidy
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 J 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 begiiniing 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

md aften\ ards moved, than that they were first part iv,
moved, and rested, if ever they rested at all, after- ^ — ^ — -
wards. Neither doth there appear any cause, why iJ?L«Sr''
the matter of the world should, for the admission f°^ v,^cu«m

invaliu.

of motion, be interraiugled v^ith 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 officitLm est quoniam premere omnia deorsum ;
Contra autem natura manet sine pondpre inanis ;
ErgOi quod magnum est iiecjuej leviusque videtur,
Nirairnni plus esse aibi dedarat iuauL*,^ — L 363-66.

That is to say, seemg the office mid propertij of

VOL. I. E E

. And thus much of the arguments of Lucretius,
us now consider the arguments which are
uilais ^for"tho ^rawu tVom the experiments of later wTiters.

eatabiisiung of j^ fhc first experiment is this : that if a hollow

vouuum|in valid i ^ i • • i i i

vessel be thrust nito 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 m the air. Also, that when the air is com-
pressed to a certain degree, it can receive no further
compression J its small particles not suflFering 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 vahd. But
it is sufficiently known, that air will penetrate
water by the application of a force equal to tlie
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 penetratt*dj 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.
Tliere is therefore such an equilibration between
them, as that the naturjd endeavour of the water
downwards is equal to the endeavour by which
the same water is to be penetrated to the increased
depth.

n. The second experiment i

THE WORLD AND THB STARS.

419

PART IV.

2d.

That is, if two flat bodies he suddenif/ pulled
a^Hunder^ of ueceA^sify the air wwa7 come hetween
them toflJl tip ike space they left empty. ^«/ J/T.^cTbT'*
with what celerity soever the air floiv in^ yet it ?^'" *'ac«"m
cantiot in one instant of time fill the whole space,
hut first one part of ity then successively all.
Which nevertheless is more repugnant to the opi-
nion of Epicunis, than of those that deny vacnum.
For tliou^li it be tnie, that if two bodies were of
infinite hardness, and were joined together by their
superficies which w ere most exactly plane, it would
be impossible to pull them asunder, in regard it
could not be done !)ut 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 ru)t true. But if we sup*
pose, as Epicunis 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
pidled asunder by any force whatsoever. But this
15 repugnant to daily experience.

E E 2

PART IV.

OibenffH-
BiCMti for the

[Of

are

4. And thus much of the arguments of Lucretius!
Let us now consider the arguments which are.
drawn from the experiments of later writers

I. TTie first experiment is this : that if a hoUo
vrasel be thrust into water with the bottom ui
wards, the water will ascend into it ; which th
say it could not do, unless the air within were thrust
together into a narrower place ; and that this wer^S
also impossible, except there were little empt^^
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 dowTiwards, 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 tlie 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 conca^

cylinder of sufficient length, made of glass, that ^^^ ^•
the experiment may be the better seen, having — - — '
one end open and the other closfe shut, be filled ^^®J ^f Veti
with quicksilver, and the open end being stopped ^uo^" ui^Lld
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 w ay for the descent of the quicksilver, see it
descend into the vessel nnder it, till there be only
so much remaining withhi the cyMnder 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
empt)' of all body. But in this experiment I find
no necessity at all of vacuum. For w^hen the
quicksilver which is in the cylinder descends, the
vessel under it must needs be filled to a greater
heiglit, 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-
sw^ered but this, that it thnisteth away the next
air, and that the next, and so successively, till
there be a return to the place w here 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, wliich 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

^*^^6 ^^' ^'^^^^^y ^^d S^* ^P ^^^^ ^^^ cylinder to fill the place
^^ — ^ which they thought was left erapty. But because
2i^^^ foT^ta. the quicksilver hath not in every degree of height
biishing ofy^^fQYce enouerh to cause such peuetration, therefore
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
w ater 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,
w hen 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 betw een 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 aw a/, let the cock be turned to liinder 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 G F. Lastly, the cover being

&steued on again, and the cock H C opened, the part iv
water will run swiftly out at E, and sink by httle - — -- — -
and little from G F to the bottom of the pipe B. ^f^ fofeit-

From this phenomenon, they arenie for the neces- fe^is*»i"g of y*-
sity of vacuimi m this manner- The bottle, from
the beginning, w as full of air ; w hich air could
neither go out by penetrating so great a length of
water as was injected by the pipe, nor by any other
w ay. 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 fi*om 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 foUow^s, that
either there w^as 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

PART IV. Buperficies of a river, as is manifest by the bubbles
^— -r^-^ that rise ; nor doth it need any other cause to ^ve
m^n^ foreau- ^^ ^^*^ motion, thau the natural endeavour down-
biishiDg of ya- ^^xai*(jg of the water* >\Ti\\ therefore, may not the
endeavour up%vards 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 dov^Tiwards ;
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 pait 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. WTierefore
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 w^ater ; 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 w ater,
begins at the bottom, and tends upwards ; whereas
the motion bv which it eiecteth the water ouffht to

begin above, and tend downwards. From whenc
therefore hath the enclosed air this endeavour to-
wards the bottom? To this question I know* not

what answer can he mven, 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 ^e'Jt^'^rofrsta-

air, after the water is forced in, hath as much room b^i»h*ug of \^-

, cuum niTaliiL

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-gfffsses^ {eollpyles^
tvhid'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% seehig
air may with no 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 away 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, when 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
causesof 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, fliikly conHistent^ and mLved of hoik. Fluid
are those, whose parts may by very weak endeavour

PHYSICS.

I IV.

Six suppou^
tiorki for the
JvJDg of the
bcnomena i>f
ftture.

be separated from one another ; and consistent
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, haTe the
names of hardness or softness. Wherefore a flnid
body is always divisible into bodies equally flnid,
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 diflFerence of Jiukiitijy 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 Jluidity^
that w hich 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 weO understood, I come to my
suppositious.

b. First, therefore, I suppose that the immense
space, which we call the world, is the aggregate of
all bodies which are either consistent and visible,
as the earth and the stars ; or in\'isible, 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 miiverse, as that it leaves in it no empty
place at all.

Secondly, I suppose with Copernicus, that the
greater bodies of tlie world, which are both con-
sistent imtl permanent, have such order amongst
themselves, as that the sun hath the first place,

THE WORLD AND THE STARS.

427

Mercury the second, Venus the third, the Earth
with the moon goiui^ about it the fourth, Mars the
fifth, Jupiter with his attendants the sixth, Saturn
the seventh ; and after these, the fixed stars have
their several distances irom 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
year, and of the several variations of days and
nights in all the i>arts of the superficies of the
earth;, have been demonstrated, first by Coper-
nicus, and since by Kepler, Galileus, and others,
from the supposition of the earth s diurnal revolu-
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

PART rv.

26.

Possible caiuai
of die in olio OS
Antiu&i and di-
urtiaJ ; and of
tlie apparent
direction, sti-
tion^ and retro-
gradation of the
planets.

PART IV. revolution of the same earth about its own centre,
^ — contrary to the order of the signs. I suppose with
fnhf m^DHs! Copernicus, that the diurnal revolution is from the
Turaai &c ™<^^^^>i of tii^ earth, by which the equinoctial
circle is described about it. x\nd 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. WTierefore,
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 abed (in fig. 3) be the ecliptic,
whose centre is e^ and diameter aec; and the
earth be placed in at, and the sun be moved in the
little circle J'g h /, namely, according to the order
y, gj hy and i, it hath l)een demonstrated, that a
body placed in a will be moved in the same order
through the points of the ecliptic rsr, A, r, and f/,
and will always keep its axis parallel to itself.

But if, as 1 have supposed, the earth also be
moved with simple circular motion in a plane that
passeth through r/, 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 ^art ry.

. 20*

kept always parallel to itself* For let the centre ' — .- — '
of tlie earth be moved about in the circumference
of the epicycle, whose diameter is Ink, which is a
part of the strait line lac ; therefore / a k^ 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 lak 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
win 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 1 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 'i'ii« ^uppoii-
* , , .^ . ^ • , yi , . tioii of simple

sun s simple motion m the epicycle fg h t pro- modon, why
bable, is first, that the periods of all the planets ^'^'*^'
are not only described about the sun, but so de-
scribed, as that they are all contained within the
zodiac, that is to say, v^ithin the latitude of about
sixteen degrees; for the cause of this seems to

PART IT* depend upon some power in the siin, especially in
tliat part of the smi which respects the zodiac.
Secondly, that in the whole compass of the heavens
*% there appears no other body from which the cause
of this phenomenon can in probability be derived.
Besides, I ooold not imagine that so many and such
varioas motions of the planets should hare no
dependance at all npon one another. But, by sup-
posing motive power in the sun, we suppose mo-
tion also ; for power to move mthout 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, w hich 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, bath no endeavour at all towards those
parts which are without the circle, they ha%^e 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,
'' **refore, that if the primary planets be carried
ut hy the sun, and the moon by the earth, they

have the simple circular motions of the sun and ^^^ '^*
the earth for the causes of their circulations. — ^ — '
Otherw ise, if they be not carried about by the sun ^^^ ^f afmpie
and the earth, but that every planet hath been J^^^^p* '"^y
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 coeteroal with them, and so they have
no cause at €ilL 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 w^ay, neither of them simple,
nor either of them such as miglit be produced by
any motion of the sun. I thaught 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 tbe 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

PART IV, place, therefore, let us examine what that eccen-

' — -- — tricity is, and whence it proceeds.
The cauM of g. Let the amiual circle of the earth abed (in
of the aoTjuai fig. 3) be divided into four equal parts by the strait
mo^on of ihe y^^^^ ^ ^ ^^^^ ^ ^^ cutting ODC anothcf in the centre

e ; and let a be the beghniing of Libra, h 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 by the apparent motion of the sun wiU be
from e to the beginning of Cancer in d ; and the
earth being moved forwards from b to r, the sun
also will appear to be moved forwards to the be-
ginning of libra in a ; wherefore eda will be the
summer arch, and the winter arch will be a be.
Now, in the time of the sun's apparent motion in
the summer arch, there are niunbered 1 86i days ;
and, consequently, the earth makes in the same
time the same number of diurnal conversions in
the arch u h e ; and, therefore, the earth in its mo-
tion through the arch eda will make only 178^
diurnal conversions* Wherefore the arch abc
ought to be greater than the arch c da hy 8i days,
that is to say, by almost so many degrees. Let
tlie arch a r, as also c s, be each of them an arch
of two degrees and A- WTierefore the arch
r b» will be greater than the semicircle abc
by 4i degrees, and greater than the arch sdr
by 8i degrees. The equinoxes, tlierefore, will be

THE WORLD AND THE STARS.

433

in the points r and s ; and therefore also, when part iv,
the earth is iu r, the suu will appear in s, WTiere- — ^^—
fore the true place of the sun will be in t, that is ^eecc^atridj
to say, without the centre of the earth's annual ^^ ^« «»i"»*^

.*, , ^* , ^, motjon of uie

motion by the quantity of the sme of the arch a /% earth,
or the sine of two degrees and 16 minutes. Now
this sine, putting JOOjOOO 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 Uy produced both ways till they reach the
zodiac of the fixed stars, will fall still upon the same
fixed stars ; because the whole orb a h c d 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 r, 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 cl, than when it is in i. And I take the cause
to be this. When the earth is in the beginning of
Capricorn at A, 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 rf, 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 f/, it will come nearer to
VOL. 1, jr F

PART IV. its first movent, that h, to the sun which is in f ;

^ — r — ' that is to say, the earth is nearer to the sun in the

rhe^cccenu^ki^^ °^ ^^ Winter when it is in rf, than in the midst

^^' of summer when it is in b ; and, therefore, dnring

the winter the sun is in its Perig^umy and in its

Apog€eum 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 aflFected, and another disaflfected to the
sun. And I dissent from him in this, that he thinks
it to be by magnetic virtue, and that this magnetic
viitue 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
simihtude 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,
''Herefore, one part of the earth be more affected
the sun than another part, it proceeds from
that one part hath more water, the other more
and. And from hence it is^ as I showed above,
the earth comes nearer to the sun when it
8 upon that part where there is more water,

THE WORLD AND THE STARS.

435

PART IV.

t]iaB when it shines upon that where there is more
dry land.
9. This eccentricity of the earth is the cause The catnc why

, . the moon hftth

why the way of it» annual motion m not a perfect aiwaya one and
circle, but either an elliptical, or almost an ellip- turned "toward!
tical line ; as also why the axis of the earth is not *^^ ^^^^
kept exactly paraUel 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 sfime 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 circley^'* h ?, (in fig. 4)
whose centre is /; and let r^-^'/f be the annual
circle of the earth ; and ri the beginning of Libra.
About the point a let the little circle / 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 o^ 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 fiom us.

The diameter therefore of the moon q or will be
perpendicular to the strait line / a. Wherefore the
moon is carried, by reason of the motion of the
earth, from o towards p. But by reason of the

FF 2

PART IV.
26.

motion of the sun, if it were in p it would at the
same time be carried from p towards o ; aiid by
Sl^mooriTaih these two contrary movents the strait line q r will
**• be turned abont ; and, in a quadrant of the circle

rnnopj 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. Seeondly, w^heii the moon is in w,
the strait line q r w ill, by reason of the motion
of the earth, be in m o. But by the w orking of the
sun's motion upon it in the quadrant jl> m, the same
qr will be turned so much as makes another quarter
of its whole conversion. ^Vhen, therefore^ the moon
is in ?/i, 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 mo ; 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 w^hieh, it is
manifest^ that one aiul the same face of the moon
is always turned towards the earth. And if any
diameter were taken in that Ottle circle, in which
the moon were supposed to be carried about with
simple motion, the same eflFect would follow;
for if there w^ere 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 w^hich 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 ^-^^ ^^'
the same with that which is illuminated. ^— A-/

10. To these three simple motions, one of the 2?5jrtidM
sun, another of the moon, and the third of the ®^^« <>««*»•
earth, in their own little circles f g h i, Ik, 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, may be 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 sim 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
lhkc\& supposed to be in the plane of the moon's
monthly motion, let the little circle Idke be de-
scribed about the same centre a in the plane of the
equinoctial. This circle therefore will decline from
the circle IhkcmKn 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.

PABT IV.
•Id,

The cause
of thtf tides
of tbe o«c«n*

degrees. Seeing now the waters, which are
under the circle of the moon's course, are by
reason of the earth's simple motion in the plane of
the same circle moved together with the earth, that
is to say^ together with their own bottoms, neither
ontgoing 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 ont^ another; it will be
manifest, that both those waters, which are under
the circle of the moon, and under the equinoctial,
will nm 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 Ihkc ; and demonstrated, at the
4 th article of chapter xxt, that whatsoever ii
moved by a movent that hath simple motion, wUl
be moved always with tlie same velocity ; it follows
that the centre of the earth will be carried in the
circumference tbkc with the same velocity with
which the moon is carried in the circumference
vtnop. Wherefore the time, in which the moon
is carried about in m n op, is to the time, in whidi
the earth is carried about in / A ^ ^, as one circuiii-
ference to the other, that is, a^ no to a k* W
a o is observed to be to the semidiameter of At

THE WORLD AND THE STARS.

439

earth as 59 to 1 ; and therefore the earth, if a k be ^ AaT iv.

put for its seniidiameter, will make fifty-iiiiie revo- ^

tions in IbJcc in the time that the moon makes (jfJ^hrSSet
one monthly circuit in mnop. But the moon**^*****""
makes her monthly circuit in little more than
twenty-nine days. Wherefore the earth shall makt*
its circuit in the circumference lb kc 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 fitty-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
Tartar y, 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, which at the beginning I

suppos^ed to be always in the mooti. For as, when
I showed the cause of the eccentricity of the earth,
Jnhrtides I derived the elevation of the waters from the
of Uie 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 clonds, 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
snn ; which may perhaps proceed from this, that
the snn 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
Ibkcy and its diurnal motion in hike, 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.

IK As for the explication of the yearly precex-
mon of the equhwetial pointJi, we must remember
that, as I have already shown^ the annual motion
of the earth is not in the cuTumference of a circle.

but of ail ellipsis, or a line not considerably dif- pa^t i^'*
fereiit from that of an ellipsis. In the first place^
therefore, this elliptical hne is to be described. pA^rwionlf*'
Let the ecliptic ^ yf r ^ (in fig. 5) be divided t^** *=i"^»**"*'
into four equal parts by the tw o strait lines a h and
'^ e, cutting one another at right angles in the
centre c. And taking the arch h d of two degrees
and sixteen minutes, let the strait hue de be
drawn parallel to a h^ and cutting v$ ^ in^*; which
being done^ the eccentricity of the earth will be
cj\ Seeing therefore the annual motion of the
earth is in the circumference of an ellipsis, of
which <f qa is the greater axis, a b cannot be the
lesser axis ; for a h and v: -3 are equaL Where-
fore the earth passing through a and A, 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 ^/ be
taken equal to the strait line yf © ; and dividing
■1^ / equally in i^ g i will be equal to yf^, and i I
equal toj^^ ; and consequently the point / will
cut the eccentricity c/into tw-o equal parts; and
taking i// equal to ij] hi will be the whole
centricity. If now a strait line, namely, the
line -Q: i t, be drawn through 1 parallel to the
strait lines a h and e rf, the way of the sim in
summer, namely, the arch ^ ^ t, will be greater
than his way in winter, by 8} degrees, Where-
re the true equinoxes will be in the strait line
-^ i r ; and therefore the ellipsis of the earth's
annual motion will not pass through a,g, 6, and i ;
but through ^ g^y and /. Wherefore the annual
motion of the earth is in the ellipsis ^ g *r I ; and
cannot be, the eccentricity being salved, in any

other line. And this perhaps is the reason, why
Kepler, against the opinion of aU the astronomers
p^dLiJif«>?"^^ former time, thought fit to bisect the eccentri-
ihe equiDoxet. city of the eaith, or, according to the ancients* of
the sun, not by diminishing the quantity of the
same eccentricity, (because the true measure of that
quantity is the diflFerence 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 J\ 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 wliole great orb is but as a point ill
respect of the immense distance of the fixed stars,
the two strait lines f^ t and a 6, being produced
both ways to the beginnings of Aries and Libra,
will fall upon the fcame points of the sphere of the
fixed stars. Let therefore the diameter of the
earth mn 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 echptic
about the centre /, this diameter will be kept
always parallel to itself and to the strait line gU
But seeing the earth is moved in the circumference
of an ellipsis without the ecliptic, the point «,
whilst it passetb through ^ \% t, will go in a lesser
circumference than the point m\ and consequently,
as soon a« ever it begins to be moved, it will
lose its parallelism with the strait line V5 ^ ; so
that mn produced will at last cut the strait line
g I produced. And contrarily, as soon ajs m n is
past cp, the earth making its way in the internal
elliptical line r / ^, the same m n produced to-
wards m^ will cut Ig produced. And when the

THE WORLD AND THE STARS.

443

PART IV.

26.

earth hath almost finished its whole eircumfer-

euce, the same m n shall a^aiii make a rig-ht anerle ^^""^^^ ^^ ^^*

• IT jn 1 -1-1 precession of

With a une drawn from the centre t, a httle short the equinoxci.
of the point from wliich the earth began its Diotion.
And there the next year shall be one of the equi-
noctial points, namely, near the end of nt; the
other shall be opposite to it near the end of x*
And thus the points in which the days and nights
are made eqnal do every year fall back ; but with
80 slow 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 1 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 apogfeum and peri-
g^um of the sim should be moved every year in
the same order, and with the .same velocity, with
which the equinoctial paints are moved ; and their
distance from them should always be the quadrant
of a circle ; which seems to be otherv^'ise. For
astronomers say, that the equinoxes are now, 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 apoggeuni of the sun or the
aphelium of the earth ought to be about the 28tli
degree of Cancer. But it is reckoned to be in the
7th degree. Seeing, therefore, we have not suifi-
cient evidence of the ort (that so it is,) it is in vain
to seek for the SIotI (why it is so.) Wherefore, as

Cause of tlie

p recess! on of

the eqiJiaoxes. f^^Q

long as the motion of the apogseum is not observ
able by reason of the slowness thereof, and as long
as it remains doubtful whether their distance from
equinoctial points be more or less than a
quadrant precisely ; so long it may be law^il 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 eaith 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 I 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 XXVIL

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.— 5. 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 coUiquation. — 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 dbtinction 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 . ««, jy
whiteness. — 16. The cause of blackness. 27.

1. Besides the stars, of which I have spoken in oftheimmeMe
the last chapter, whatsoever other bodies there be some bodies,
in the world, they may be all comprehended under I^uwe uuie.
the name of intersidereal bodies. And these I have "•" ^^ o^«"-
already supposed to be either the most fluid sether,
or such bodies whose parts have some degree of
cohesion. Now, these diflfer from one another in
their several consistencies, magnitudes, motions,
WiA 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,

PART IV. quantity is divisible into di% isibles perpetually.

^ — T^ — Andj therefore, if a man could do as much mth his
mSt!idr&c!liands 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 \dsible. So that we
caimot 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

LIGHT, HEAT, AND COLOUKS.

point in respect of the distance between the sun part rv.

and the fixed stars ; and, on the contrary, to make ^ — -r^— '

a body m httle, as to be in the same proportion magnitude, &c

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 lon,s: time was accounted ao 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 r

For the Majesty of God appears oo 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 wondered 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,

Of the caiise
ctf the light
of llie sim«

PART 17, their velocity and slowness, and the variety of their
^ — r^— ' figures, to be only such as the explication of their
natural causes requires. And lastly, I suppose,
that the parts of the pure sether, 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.

2. Having laid these grounds, let us come to speak
of 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 tlie endeavour outwards of the coat of
the eye, called the retbm. 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 caUed lucid. Wherefore
we have a possible cause of the light of the sun ;
which I undertook to find.

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

How light

LIGHT J HEAT, AND COLOURS,

449

it when he s^rows hot; but what it is in other partiv.
thinffs, he knows only bv riitioeination. For it is „^-"^f^
one thing to g:row 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 h 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 J that fftet/ fieaf, i here fore thetj
are them^ehes hot ; yet it cannot from hence be
truly inferred that fire heateth^ therefore it h
itaelf hot ; no more than this, fire cauHeth pahij
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 snn.

It hath been shown, in the 5th article of chapter
txij 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.

27.

PART IV, strated that tlie water may be drawn up into the
clotids.

And I sliall now show that the fluid parts may,
in like manner, by the same fermentation, be drawn
out from the internal to the extenial 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 clmii^e of place, separated from one
another ; the contiguous ])arts of the li\ing creature
must, of necessity, endeavour to enter intothespaces
of the separated parts. For other^vise those parts,
supposing there is no vacuum, would have no place
to go into. And therefore that, which is most fluid
and separable in tlie 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 trom 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 paits together must, of necessity,
press those parts of the ambient air which are
ready to leave their places ; and therefore all the
jiarts of the body, endeavouring at once that w^ay,
make the body swelL Wherefore a possible cause
is given of heat from the sun ; which was to be
done,

2?fiif^om1he ^* ^^^ h^^^^ ^^^^^^ ^^^^* 1^^^' 1^^^*^ ^^^ ^^^^ ^^^

fun. 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

gtrait line. But when a body hath its parts so partiv.
moved, that it sensibly both heats and shines at ^- — A--
the same time, then it is that we say fire isJ^lrf^^X
generated. *""•

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 oifire.

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
G C will of necessity fall, the one between A H
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 AK and CL be

6G2

PART n\ produced till they meet with tlie stmit line B D
' — "^^^ produced in O. Seeing, therefore, the medium
ofa«^om*thBwi*^liiii the sphere is thicker than that without it,
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 pei-peiidicular. Let, therefore,
the refracted line be K P, cutting FO in P; and
for the same reason the strait line LP 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
comprehendeth all the beams that are propagated
from A B C ; and this sphere being much less than
tlie sphere A B C, the parts of the medium, that is,
of the air al>out P, will change places with one
another with much greater celerity than those
about A BC. If, therefore, any matter combustible,
that is to say, sucli 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 iti the beholder a phantasm of
lucid ami hot, as I have before demonstrated of
the simple motion of the sun ; and therefore the
combustible matter which is placed in P w ill be
made lucid and liot, that is to say, will be tire.

LIGHT, HEAT, AND COLOURS. 453

Wherefore I have rendered a possible cause of fire ; ^-^^^ ^^•
which was to be done. -- /-^

5. From the manner by which the snn generateth ^firrfro„jJ.o"
fire, it is easy to explain the manner by which fire ii«on.
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, Ught and
heat, that is to say fire, is generated.

6. The shinine: of fflow-worms, some kinds of J^* ?*"" °^

° ° ' light in glow-

rotten wood, and of a kind of stone made at Bo- worms, rotten

11 1. ^1 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 crystalUne
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

"PART IV. as it is cold it will cease shining. From whence^

ti-^ therefore, can these creatures have their 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, AIso the sea water shineth when it is either
cJwiou of^KK dashed with the strokes of oars, or when a ship in
^^^' its course breaks strongly through it ; but more or

^^|b less, according as the wind blows from different

^^H points* The cause whereof may be this, that the

^^^B particles of salt, though they never shine in the

^^^H salt-pits, where they are but slowly drawn up by

^^^B the sun, being here beaten up into the air in greater

^^H qtiantities and with more force, are withal made

^^^1 to turn round, and consequently to shine, though

^^^^ weakly* I have, therefore, given a possible cause

^■^ of this phenomenon.

IHlVlBirk^^ 8. If such matter as is compounded of hard little
^ac cuiiiiiuatiou. bodies be set on fire, it must needs be, that, as they
fly out in greater or less quantities, the flame which
is made bv them will be in'eater or less. jVnd 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. WTien,
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 iv.
make a great shining body. For flame being ^ — A- ^
nothing but an aggregate of shining particles, the Jame,*^rarkf
greater the aggregate is, the greater and more ^ coiiiquaiioiu
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 ofl^, 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 will 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 has 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-

PART IV. cation of fire, these latter be called out, the foroier

27. * . ,

^ — . — ' 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 cj^ It ig koowu bv experience that if hay be laid

wpt hay soiTK?- . .

tmics burni of \\'et together in a heap, it will after a time begin
ai»f> die cause to suioke, and tlien burn as it w^ere of itself. The
onigiitning. ^.^^^^ whereof seems to be this, that in the air,
which is enclosed within the hay, there are those
little bodies, w hich, 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
Ijodies, 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 w atery parts are first driven outwards^ from
w hence 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 o!i fire.

The same also may be the cause of lightning,

which happens in the hottest time of the year,

>vhen the water is raised up in greatest quantity

firried hie:hest. For after the first clouds are

others after others follow them ; and being

ed above, they happen, whilst some of them

nd others descend, to fall one upon another

nanner, as that in some places all their part5

LIGHT, HEAT, AND COLOURS. 457

are joined together, hi others they 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 ^*j^^*^^'^®^^
from fire, and yet the most admirable of all others, of gunpowder;
is the force of gunpowder fired ; which being com- be Mcnbed to
pounded of nitre, brimstone and coals, beaten small, lo*thrbriin?*'
hath from the coals its first taking fire ; from the JoTc'n°frr.^*'
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-

tain white substance, which being thrown again
into the fire, will grow red-hot again, bnt will not

o^r fowG of '^^ dissipated, at least unless the fire be augmented.

gunpowderAc. 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 eonsisteth, be
supposed to be some of them hard, otlu?rs 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
tilled 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 tly,
of necessity blow strongly the burning coals and
brimstone which are mingled together ; w hereupon
there will follow a great expansion of light, with
vehement flame, and a \ iolent dissipation of the
particles of th(* 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

LIGHT, HEAT, AND COLOURS. 459

is thrust out of a gun, proceeds from the swiftness part iv.
of the small particles of the fired powder ; at least — r^
it may proceed from that cause without the suppo-
sition of any empty space.

1 1 . Besides, by the attrition or rubbing of one How heat
body against another, as of wood against wood, we "ttritl^.
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, iTie distinction
third, and so on infinitely. And we call that first fiwtjfecondl&c
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

- . , - t "I . 1 11 ^^6 colours we

such as IS generated by perturbed motion ; as shall see in looking
be made manifest by the red, yellow, blue and pur- m^of gfaw?''
pie, which are generated by the interposition of a "eiTow^wue'*&
diaphanous prisma, whose opposite bases are ^Jo^ct 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
tiir ; and let the triangle A B C 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

PART IV. tiie strait line A B ; and let the sunbeams pass in
- ^^'-' the lines D A and E B C. And lastly, let the strait
ii!ico7o"u^'^J"i^s DA and E C 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 peq>endjcular ; let the line refracted at
the point A be the strait line A H. And again,
seeing the mediiitn below A C is thinner than that
above it, the other refraction, which %\ill be made
there, will diverge from the perpendicular. Let
therefore this second refracted line be A L 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 tlie triangle ABC
will have its niotion 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
AH, 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 C G K 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 i

LIGHT, HEAT, AND COLOURS. 461

lines A B and A C the same action which is in the part iv.
points A and C, then the plane of the triangle C G K ' — ^^
would be everywhere coincident with the plane of thecoiouw'&c.
the triangle A F H ; 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
on 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 A F H 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 C G K shows the perturbation of light within
the glass ; and C K L that which is below the glass.
From whence there are four divers motions, or four
diflferent illuminations or colours, whose diflFerences
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 C G K green, and approaching to blue ; and
lastly, the triangle CKL 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.

YAWt IV.

ST.

Wig Uktmooa
IkMi in tlio

light, the transverse beams make greenness ; and
who!) the same second light is weaker, as in the
triangle C; K L, they make a purple colour.

14. From hence may he deduced a cause, why
the moon and stars appear bigger and redder near
the horizon than in the mid-heaven. For between
the eye and the apparent horizon tliere is more
impure air, such as is mingled with watery aod
eaithy little bodies, tlian is between the same eye
and the more elevated part of heaven. But vision
18 made bv beams which constitute a cone, whose
l>ase* if we look upon the moon, is the moon's faee,
and whose vertex is in the eye; and therefore,
many beams from the moon must needs fall upon
little iHidies that are i^ithout the visual cone, and
be by them reflected to the eye. But these reflected
h^invT teod all in lines which are transverse to the
nsMJ cmie, and make at the eye an angle which is
gltiUi^r lltau the angle of the cone. WTierefore,
tht^ UiHvn appears greater in the horizon, tban when
ditt iai mDre ^Tated. And because those reflected
Imum gi> ttanaiw^iy, there will be generated, by
%h» la$l arude^ reduesss A }x)ssible cause there-
fiw b «Im>wii« wby the moon as also the stars ap-
jffmt fpmtn and rakkr in tlie horizon, than in the
ttM^ fif ImiTtiu IW same also may be the cause,
\^Uy the j^un appMi^ m die horizon greater and of
a c^ihMir aione dcgmewtiiig to yellow, than when
^ i^ h^;iNr ele!f«i«L F^ tiie refiectioo from tht*
W>iH^ Wtufii^ aid Ike tnnsrorae moiion of
"diuai^ are itffl die same. Bm the light of
m i» aiwk itro^pfr than that of the moon;
thtHrrAxts. hr the kfeSI aitkle, his splendour

LIGHT, HEAT, AND COLOURS. 463

must needs by this perturbation degenerate into part iv.
yellowness. ^ — r^

But for the 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 came 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 phan-

PART IV.

27.

The c*iiie of
blacknesi.

tasms of the whole lucid body, that is to say, white-
Dess, 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 bein^ 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.

16. As whiteness is light, so blacJcnesfi is the pri-
vation of lightj or flarkness. 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 rufiledbythe wind. Thirdly,
that any combustible matter is by the tire made to
look 1 J lack 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 tire be put out before the
parts are totally dissipated, the coal will appear
black ; for the parts Iieiiig 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 part iv.
parts are convex, like little bubbles; and there- ^ — ^
fore the beams of light, which fall upon them, are bu^j^n*^ ^^
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 white. Fifthly, that all colours
that are made of the mixture of white and black
proceed from the diflFerent position of the particles
that rise above the superficies, and their diflFerent
forms of asperity. For, according to these diflFer-
ences, more or fewer beams of light are reflected
from several bodies to the eye. But in regard
those diflFerences 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 difliculty,
that I dare not undertake it.

VOL. I. HE

AND OF THE HEADS OF ElVERS.

1 • Why breath from the same mouth sometimes heats and some-
times cools. — 2* Wifid, and the inconstancy of wmdij, whence.
3. Why there is a coDstant, though not a great wind, from
east to west, near the eqnator. — 4. What is the effect of air
pent in between the clouds, — 5. No change from soft to hard)
but by motion.^O* Wfiat 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. W^hy water doth not freeze in deep
wells as it doth near the superficies of the earth. Wliy ice is
not so heavy as water ; and why wine is not so easily frozen
as water. — S* Another cause of hardness from the fuller con-
tact of atoms; abo, 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.*-! L
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 tliat clouds can
fall again after they are once elevated and frozen.— -16. How
it coutd be that the moon was eclipsed> when she was not dia-
metrically opposite to the sua. — ^17. By what means many
suns may appear at once. — 18. Of the heads of rivers.

1. As, when the motion of the ambient ethereal
substance makes the spirits and fluid parts of our
fro J th^* a'mc bodies tcnd outwards, we acknowledge heat ; so,
tTmeL^helT* ^y the endcavour inwards of the same spirits and
aod tome- humouTs, we feel cold. So that to cool is to make

times cooli, '

the exterior parts of the body endeavour inwards.

PART IV.
28.

OF COLD, WIND, ETC-

467

by a motion contrary to that of calefactioii, by partiv,
which the internal parts are called outwards. He, — . — '
therefore, that w ould know the cause of cold, mnst
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 maybe this, the breath
going out hath tw^o 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 showii in the 3rd article of the last
chapter, caxLseth heat. According, therefore, as
either of these motions is predominant, so there is
the sense sometimes of cold, sometimes of heat*
Wherefore, w hen the breath is softly breathed out
at a large passage, that simple motion which causeth
heat prevaileth, and consequently beat is felt ; and
when, by compressing the lips, the breath is more
strongly blown out, then is the direct motion pre-
valent, which makes ns feel cold* For, the direct
motion of the breath or mr is wind ; and all wind
cools or diminisheth former heat.

2. And seeing not only great wind, but almost ^Vind, and the
any ventilation and stirring of the air, doth refri-* wimTa' th7mM.
gerate ; the reason of many experiments concern-
ing cold cannot well be given writhont finding first
W'hat are the causes of wind. Now% wind is

M u 2

468

PHYSICS.

PART IV<
28.

iiotbiug eke but the direct motion of the air thrust
forwards ; which, oevertheless, when many winds
i!lconiia''o°cVo^*^*^^^^^^ may be circiilar or otherwise indirect, as
wiuds, whence. i|. jg j^j whirlwduds. Wherefore, hi the first place
W'e are to enquire into the causes of wintls. Wind
is air moved in a considerable quantity, and that
either in the manner of w aves, 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 iis 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 aD otlier 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. ^^Hierefore, the ascending vapours move
the air towards the sides everyway ; and all direct
motion of the air being wind, they make a wind.
And if this wind meet often with other vapours
** in other places, it is manifest that the
w^ill be augmented, and the way or
hanged. Besides, according as the
urnal motion, turns sometimes the
js the moister part towards the sun,

OF COLD, WINDj ETC.

469

SO sometimes a greater, sometimes a less, quantity part iv.
of vapours will be raised ; that is to say, sometimes ^"/-^
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 camiot
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 Teneriife and the Andes of Peru, namely,
that they are not at aU 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 y^'h tliere
not that w hich is made by the ascent of vapours, though not &
yet a less and more constant wind, like the con- ^^^m lahwo
tinned blast of a pair of bellows, blowine; from the^^*^*'^*'^''^^*

■^ JO equator.

east. And this may have a double cause j the one,
the diurnal motion of the earth ; the other, its
simple motion in its own 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 w^hich, though the diurnal
motion contribute but little, yet seehig I have
supposed that the simple motion of the earth, in its
owTi epicycle, makes two revolutions in the same
time in which the diurnal motion makes but one,
and that the semitliamcter of the epicycle is double
the semidiameter of the diurnal conversion, the

470

PHYSICS.

«loud«.

PART lY, motion of every point of the earth in its own
^ / - 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 w ind
will be felt. For whether the air strike the sen-
tient, or the sentient the air, the perception of
motion wiU be the same* But this wind, seeing it
is not caused by the ascent of vapours, must neces-
sarily be very constant.
What is (lie cf^ 4 When one cloud is already ascended into the

feet 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 w ay. Also
when both of them, whUst 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, with simple motion, will not
pass through the w ater of the clouds, but be more
straitly compressed within their cavities. And
this I have demonstrated at the 4th and 5th articles
of chapter xxii. 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 w ay npon the
earth's superficies; as I have shown at the 8tti

bill bs

notion.

^ to be hard, from this,
would thrust forwards
we touch, we cannot

OF COLD, WIND, ETC.

471

do it otherwise than by thrusting forwards the pahtiv.
whole body. We may indeed easily and sensibly ^ — r^
thrust forwards any particle of the air or water
which we touchy 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 ^^»J ^"^ *^«^
cause why there is greater cold near the poles of nearthcpoiei.
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 earth. In those places, therefore, which are
nearer to the poles, there is greater cold than in
those which are more remote fr€>m them. Now this
ijBxpansion of the air upon the superficies of the

472

PHYSICS.

TART IV.
28.

Thfr cftiise of
ice ; and why
tbecohi 19 more
Temiss in rainy
than in dear
weather. Why
wafer doth Dot
free I e in deep
wells, as it doth
near the Buper-
fleies of the
earth. \^'hy ke
. ii not so heavy
[m water; and
why wine is not
so easily frozen
«s water.

earth, from east to west, doth, by reason of the
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
s\in, that cold is agaiu remitted ; and can never be
great, because the action by which it was generated
is not permanent. Wlierefore 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.
7. How water may be congealed by cold, may
be explained iu this manner. Let A (in figure 1 )
represent the sun, and B the earth. A will there-
fore be much greater than B, Let E F be in the
plane of the equinoctial ; to which let G H, I K,
and L C be parallel. Lastly, let C and D be the
poles of the earth. The air, therefore, by its
action in those parallels, will rake the superficies
of the earth ; and that with motion so much the
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, thllNifttHiflHlBiifiP of the water will be

so again, the

1 in the saiue

^ilui-k. And thi9

I
I

(
I

ice^ being now compacted of little bard bodies,
must also contain many particles of air received
into it.

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 wiU
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 fidl of
water. Now while the snow and salt, which have
in them a great deal of air, are melting, the air,
which is pressed out every way m 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 ahke 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.

PART IV,

The cttute
of ice, &C.

474

PHYSICS,

^^^l ^^' broken, and dissipated by them ; and the less the
"^^ — wind is, the less is the cold.

We find also by experience, that in deep wells
the water fi^eezeth not so much as it doth upon the
superficies of the earth. For the mnd, 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 fi-eeze ; whereas if it be so deep,
as that the wind w hich causeth cold cannot reach
itj it will not freeze.

We find moreover by experience, that ice is
lighter than water* The cause whereof is manifest
firom that which I have already shown, namely, that
air is received in and mingled mth 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 smftly, 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 wiU
be very^ strong.
Auoiiicrcflusc g^ ^^ h^ve secu one way of making things hard,

of hardness i i - i

froni the fuller namely, by congelation. Another way is thus.
ItonT *Aiao Haviug already supposed that innumerable atoms,
*ruokcn^"^ ^^™^ harder than others and that have several
simple motions of then: own, are intermingled with
mibstance; it follows necessarily from
m of the fermentation of the
I have spoken in chapter xxi,
meeting with others will

OF CO LB J WIND, ETC.

473

cleave together, by applying themselves to one part iv.
another in such manner as is agreeable to their ^^ — r^—
motions and mutual contacts ; and, seeing there is ^^^^^^^^^^^
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 dra\^Ti 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, wUl 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

47ft

PHYSICS-

PART IV.

28.

asunder, but by breaking theEi^ the first separation
will necessarily be in the convex superficies of the
bowed part of the body, and aften\ards in the
concave superficies. For in all bowiiig 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
eoramon superficies, may by force be separated
without the introduction of vacuum ; though
Lucretius thought otlienvise, believing that such
separation Wfis 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.

9. Another cause of hardness in some things
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 eacli 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*

A fourth cQuao | Q^ Aoraiu, iu somc thine:s hardness may be made

from the ma- to a ccrtaiu degree m this manner. When any

fluid substance hath in it certain verv small bodies

A third cause
of hartlnesa,
from heaL

OF COLD, WIND, ETC.

477

intermingled J wliich, beins: moved witli simple mo- ''ART it.
tion of their own, contribute like motion to the ' — -r-—
parts of the fluid substance, and this be done lE a^^^^^p^^^
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 numtier 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,
w^hich was vehement before, will now be much
more vehement. WTierefore, 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 naiTowness of the place both
together.

1 1 . Such things as are made hard by sudden np^ hard
heat, namely such as are liardened by fire, aresofteaiX^
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 J 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.

PART IV,
28,

that hardness. Nor is it enough to cause thawing,
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 contraiy 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 remfiin un-
changed, though the place be not sensibly cold.
Whence pro- |2, Qf hard bodics, some may manifestly be
lancoiis reHij. bowcd ; othcrs not, but are broken in the very
beuu ^ '"^' 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 maimer,
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

I
I

say, some endeavour at least that way by wliich ^^^^ i^-
the restitution is to be made ; and therefore this — -^—^
endeavour of all the parts together is the first ceedTth^^ijTonp
beginning of restitution ; so that the impediment J^^^^y^ln^
being removed, that is to say, the foree by which bent,
it was held bentj 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 di\ail-
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,
wOl 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 w as 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

•ADil whence.

necc^ssar

deuly be broken.
Kp^a^s. 13. Those bodies are called diuphauoHH, upon
»hat thej are, which, whilst the beains of a lucid body do work,
the action of every one of those beams is propa-
gated in thein 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 opacouJi body
is thatj 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 fi*om the beginning ;
as the substance of the air, and of the w ater, 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, which congre-
gates homogeneous bodies. But such, as are made
diaphanous in this manner, consist of parts which
were formerly diaphanous.

14. In what manner clouds are made by the
motion of the suu^ 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 ihunder
ftud Ughtning, For seeing the air consists of two

The cause of
ligbtmng and
thunder*

OF COLD, WIND, ETC. 481

parts, the one ethereal, which has no proper mo- pa^t iv.
tion of its own, as being a thing divisible into the ' — A-'
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 wiU in the meantime
be the more thrust together, and press one another;
and consequently, by reason of their vehement
motions, they wUl 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 was 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 npon 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
clouds, do run together again into water or bCcfouds canflu
congealed into ice, from whence is it, seeing both Jf ""^.c^once
ice and water are heavy, that they are sustained in ^^*^^ ^^
the air ? Or rather, what may the cause be, that

VOL. T. II

482

PHYSICS,

^^^8 ^^* being once elevated, they fall down again ? For
' — ' — ' there is no doubt but the same force which could
pIo^c?e^dVthai ^^^''H^ ^P ^^^^ water, could also sustain it there<
clouds. &c Why therefore being once carried up, doth it fall
again ? I say it prneeeds 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
hke nature to that of the earth, are driven towards
the earth ; that is to say, what is the cause of the
descent of hea>7 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 snn 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
' '^k cloud is interposed, than when the sky is
^nd therefore, whensoever a cloud is made
ugh, the water, which w ould not descend
1 then descend, unless it be kept up by
n of the wind- Wherefore I have ren-
rssible cause, both why the clouds may

OF COLD, WIND, ETC.

483

5? snstained in the air, and also why thpy may fall part iv.
down again to the eartli ; which was proponnded *—---—
to be done.

16. Grantins: that the clouds may be frozen, it in ^^«7 it f^^i^
tio wonder if the moon na?e been seen eclipsed at moon *»»
6uch time as slie hath been almost two degrees shewL'uor^
febove the horizon, the snn at the same time appear- fpp'i','^^^
ing in the horizon ; for such an eelipse Mm ob- ^^e sun.

.terred by Maestlin, at Tubingen, in the year 1590.
For it might happen that a frozen eloiid was then
interposed between the sun and the eye of the
obserrer. And if it were so, the sun, which was
Ithen almost two degrees below" the horizon, might
jhppear 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
an this. Wherefore not the atmosphere, but either
water in a continued body, or else icCj must be the
cause of that refraction.

17. Again, gi-anting that there may be ice in the By whatmeatu
rlouds, it Win l)e no longer a wonder that many appear at once.
inns have sometimes appeared at once. For look-
ing-glasses may be so placed, as by reflections t6
show the same object in many places. And may
Bot so many frozen clouds serve for so many look-
ing-glasses ? And may they not be fitly disposed for
that purpose r 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 were it not for that one phenomenon of the
new star which was seen in Cassiopea, I should
tliink comets were made in the same manner,

t I 2

484

PHYSICS.

TART IV.

28.

or the licadA
of riyers.

namely, by vapours drawn not only from tlie 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
w^as ice, ^Vherefore I leave to others the disquisi-
tion of the cause of comets; concerning which
nothing that hath hitherto been published, besides
the bare histories of them^ is worth considering.

18, The heads of rivers may be deduced from
rain-water, or from 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 j 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 sti'eams running
together make rivers. Now, there was never any
spring found, but where the water which flow^ed 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
'^nvoy from Piedmont, there springs a river which
down by Susa ; it is not true. For there are
^ that ri> er, for two niUcs length, very high
n both sides, which are almost perpetually
d with snow^ ; from which innumerable little
as running do\Mi do manifestly supply that
with water sufficient for its magnitude.

SOUND, ODOUR, ETC. 485

CHAP. XXIX.

OF SOUND, ODOUR, SAVOUR, AND TOUCH.

. 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 oil^l^^^^^H

the medium is not the sound itself, but the cause thedistincUon

of sounoB.

of it. For the phantasm which is made in us, 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 derivatives

486

PHYSICS.

PAET IV,
29.

The ctuae of
the degrcci of

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. For
sight is from pressure, that is, from an endeavour ;
in wiiicli there is no perce|)tible progression of any
of the parts of the medium ; Imt one part urging
or thrusting qn another propagateth that actipn
successively to any distance whatsoever ; whereas
the motion of the metlinm, 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 strickep;
and the dmm being stricken, the piu 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 >Yhich we call sound ; and because
the reaction tendeth outwards, we think it is
without.

2. And seeing the effects produced by motion

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.

^ise neither the greatest nor the least

velocity can be given, it may happen

notion may be of so small velocity,

F of so smaU magnitude, as to pro-

.all 1 or either of tl

may

great, as to take away the faculty of senge by
hurting the organ.

Prom hence may be deduced possible causes of J%e^^J^[f
the strength and weakness of sounds in the follow- *^'^^^'
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 diffiiseth
itself evei7 way into circles, such a^ 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 w ill be stronger than if it w ere 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

PART IV.
29.

moved air are as the wide end of the trank. And
for this reason some creatures seem to hear the
better, because nature has bestowed upon them
wide and capacious ears.

The fourth is tliis^ that though he, which standeth
upon the sea-shore, cannot hear the colHsion 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 bodics whcu they are stricken do yield
acuTeiDd grave some a morc 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, w^hich are moved
by the mechum, return to their rest again, some-
times sooner, sometimes later. Now% by how^ much
the ^ ibrations 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 w4iat 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 kiiow ;
and therefore it is needful to explain them by

PABT IV,
29.

Xnd The difference
HUU tctweeri sounds

examples. WTien I say hoarse, I understand whis-
pering and hissing, and whatsoever is like to these,
by what appellation soever it be expressed,
sounds of this kind seem to be made by the force »'^»»**'"'^i"^«
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 mider-
stand such a sound as may be easily and distinctly
heard ; for so whispers w^ould be clear ; but such
as is made by somewhat that is broken, and such
as is clamour, tinkling, the sound of a tnimpet, &c.
and to express it significantly in one w^ord, noise.
And seeing no sound is made but by the concourse
of two bodies at the least, by w^hich 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 w iU he difi^erent from one
another. And whensoever the proportion bet\\^een
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 or^pi-y/ioc.
Therefore the breath blown with violence from the
mouth makes a hissingj 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 nibbed together with no great pressure, they

490

PHYSICS.

^^^V^* make a hoarse sound. And this hoarse sound
^— ^^ when it is made, as I have said, by the air rakir
the superficies (rf 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 difTfrencu 4. JVo/><% OF that which I Call clear sound,

Be; twi?en clear ^ i ^ 11 ^ 1

hoarse sounds, made two ways ; one, by two hoarse sounds made
whence, j^y QppQgjte 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 ; v^hieh motion, making imprej^
sion upon the medium, is propagated to the organ
of hearing. And seeing there is in this collision
or divulsion an endeavour in the partiele^s 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 wiE 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.
of 5. This being granted, and seeing withal that
^^°J thunder is made by the vehement eruption of the
air out of the cavities of congealed clouds, the

1- '

cause of the great noise or clap may be the sudden
breaking asunder of the ice. For in this action it

PAET IV.

29.

the
fee

■jfi necessary that there be not only a great concus- ^^u^jj^crald ^of
HlipQ of the small particles of the broken parts, but ? g""* whence

^F^ 1 1 • -11. . 1 ^^ procecda.

n|l$o that this concnssion, by being communicated

ffto 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

reatest sound, which is nmde by the collision of

||he parts whilst they restore themselves. And

leeipg there is iu all concussion a reciprocation

of motion forwards and backwards in the parts

|K fitricjien ; for opposite motions cannot extinguish

H pne another in an instant, as I have shown in the

■ ) 1th article of chapter viii ; it follows necessarily

Htbat the sound will botli continue, and grow weaker

^nd weaker, till at last the action of the recipro-

■ pating air grow so wealij as to be imperceptible.
"Wherefore a possible cause is given both of the
first fierce noise of th^ tliupder, and also of the
innrmur that follows it*

The cause of the great sound from a discharged
piece of ordnance is like that of a clap of thunder,
■ror the gunpowder being fired doth^ in its en-
deavour to go ontj attempt every way the sides of
»ihe metal in such manner, as that it enlargeth the
ciLrcoimference all along, and withal shorteneth the
mas; 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,
29,

done with such motions of the parts, as are not
only very vehement, but also opposite to one
another ; which motions, being communicated to
the air J make impression upon the organ, and by
the reaction of the organ create a sound, which
lasteth for some time ; as I have already sho^n in
this article.

I note by the way, as not belonging to this
l)lace, 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 whole 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 pait. 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.

6. Also the cause w hy the sound of a pipe, which
bi"f»5:"iiito is made by blowing into it, is nevertheless clear, is
dw'aownd.* *^^ ^^^^ ^*^^ ^^^^ 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 nghJllMlnri but hoarse. But if the

Whence it is

SOUND, ODOUR, ETC. 493

angle be great enough, the percussion, which is pa^ iv.
made against one of the hollow sides, will be re- ' — r^
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 thdX primary sound, which is generated of "fl«cted
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

PART IV, as they are more or less distant frora one another,
^ — r^ 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 eqnal 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-beam5,
are effects of one and the same cause. Butj as
when the visible object is placed in one of the
Inirning 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 concate
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
he neither elliptical luir parabolical, yet becanse
their figure is not much difl^erent 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 wiience 8. Couceniine: the uniformifH and ihtration of

ii is that sotiiid 111 tT • i 1

is uniform and sounds, both wluch havc one common cause, we
'**^^^* may observe, that such bodies as being stricken

yield an unequal or harsh sound

geneous, that is to say, tl

SOUND, ODOUR, ETC. 496

are very unlike both in figure and hardness, such part nr.
as are Wood, stones, and others not a few. When — ^-^r^ — '
these are stricken, there follows a concussion of .^^j{*^^'^
their internal particles, and a restitution of them }][,^''°"° ■"**
again. But they are neither moved alike, not
have they the satne action upon one another;
some of them recoiling from the stroke, whilst
others which have already finished their recoililigs
ate now returning ; by which means they hinder
lOid stop one another. And frotn hence it is that
their motions are not only Unequal and harsh, btit
also that their reciprocations cohie to be quickly
extinguished. Whensoever, therefore, this motion
is propagated to the ear, the sound it makes id
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-
sisteth like to One another both in hardness and
figure, such as are the particles of glass and metals,
ifrhich being first melted do afterwards settle and
harden; the sound it yieldeth will, because the
motions of its parts and their reciprocations art
like and uniform, be uniform and pleasant, and be
Inore 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 unifomi
sound, the sound of that body, which hath the
greatest extent of length, will be the longest heard.

496

PHYSICS.

lasting.

PART IV, For the motion, wbicli in both of them hath its

20. .

- — r^ — ' begiuning from the point of percussion, is to be
k7s"ihrt''L"uod P^c>pagated in the greater body through a greater
it uniform and spacc, and Consequently that propagation mil re-
quire more time ; and therefore also the parts
which are moved, wilt reqnire more time for their
return. ^Tierefore 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-

fore also, for the reason above mentioned,
sound will last the longer.

9, In hearing it happens, otherwise than in How sonnd
seeing, that the aetion of the medium is made "iY bloderEd
stronger by the wind when it blows the same ^^ ^^* ''^'**
way, and weaker when it blows the contrary way.
The cause whereof eannot proceed from anything
but the different generation of sound and light.
For in tlie 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 others ise than it would do, if it were
still and calni. 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 partcarriedaway 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 consideral*le distance, and with 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* 1. K K

498

PHYSICS.

PABT IV, that way, aud further from it if it blow the con-
trary w ay ; so that according as it blows from or
towards the object, so the sound i?vhich 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
w ere a great way off. For having no former thought
of any determined place from which the voice
shotxld proceed, and judging according to the
greatness, if it be m eak 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, l>ut 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, JO, \s for the mcdium, which conveys sound,

but nihcr lio- . . •it-* i i t

dies, how hnrd it IS uot air 0!uy. For water, or any other body
7 sound!' ^^^ liard soever, may be that medium. For the
motion may be propagated peii^etually in any hard
continuous body ; but by retison of the difficulty,
w ith 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 ven,^ lojig 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

«oeve
convey

SOUND, ODOUR, ETC.

499

air, which it striketh, may immediately be received p^i^t iy.
by the ear, and be carried to the tympamim, 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 somid 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.

1 1 . I have shown above, that the difference be- "^^ cau«t ^

^ ^ • 1 • 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 "^^^"^
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, irfter 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 vibraticfus in the same

K K 2

PART IV, times, the concent they produce is the most exqui-

- — '^ — - site of all other. For the sides of the tympanum,

S/ar/"j'tl^at is to say of the organ of hearing, will be

lounds, and of strickeu bv both those vibrations toe:ether at once,

on one side or other. For example, if the two

equal strings A B and C D be stricken together, and

the latitudes of their \dbrations 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,

w^hich is made by the \ibrations 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 i//?/-
son ; w hich is the greatest con-
cord. Again, the string A B
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 L 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 fi'ora
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 somid which is called an els:hf/i.

SOUND, OBOURj ETC,

501

Again, the vibration of the string A B remaining partiv.
still the same it was, let C D be stretehed till its "^ — ^ — '
vibration be swifter than that of the string A B in
the proportion of 3 to 2, and let EF be divided
into three eqnal 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 w hat 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, w hilst 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. WTieretbre, at
every third time, the organ will be stricken by the
vibration of both the strings together, and make
that concord which is called B^Jifth.

12. For the finding ont the canse of sme/h\ I Phcnomen*
shall make use of the evidence of these followin^^
phenomena. First, that smelling is hindered by
cold, and helped by heat. Secondly, that when
the w ind bloweth from the object, the smell is the
stronger ; and, contrarily, w^hen it bloweth from
the sentient towards the object, the w eaker ; both
which phenomena are, by experience^ manifestly
found to l>e 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 Avith

of smellmg.

SOUND, ODOUR, ETC. 503

are intenninfirled with the mr; it follows neces- partiy.
sarily, that the cause of smelling is either the ^ — ^ — -
motion of that pure air or ethereal substance, or ^a ST 2?"
the motion of those small bodies. But this motion «>^'»^.«nof
IS an eflfect 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, tiiough
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 Ls 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,

PART IV.
29.

Howimellingis

that seeing those effluvia have j2:reat velocity of
motion ( as is manifest froni this, that noisome
odours proceeding from caverns are presently
smelt at a great distance) it would follow, that, by
reason there is nothing to binder the passage of
those effluvia to the organ, such motion alone
w^ere sufficient to cause smelling ; w hich is not so ;
for we cannot smell at all, unless we draw in our
breath througli 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 orgau, 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.

14. The cause why smelling is hindered by cold!

helped by heat may be this ; that heat, as hath

iho^\^l in chapter xxi, generateth simple

; and therefore also, wheresoever it is

there it will increase it ; and the cause of

being increased, the smell itself will also

ased. As for the cause why the wind

from the object makes the smell the

it is all one with that for which the at-

same,
he that draws in the air next to him, draws v
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- whyiuchbo-
tity of air, as stones and metals, yield less smell smeirwh^di
than plants and living creatures ; the cause may tTenitxtire'^
be, that the motion, which causeth smelling, is a *^'^^^''^"^^'^"^'
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 w hich it is propagated to the
organ. Where, therefore, there are no fluid parts
as in metals, or where the fluid parts receive no
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 w ith the juices and spirits, w hich are
bred in living creatures*

16. That odorous bodies may be made more why odoroui
odorous by contrition proceeds from this, that mlrf odomiw^
being broken into many parts, which are all odor- '*^^'' bmsiMi.
ous, the air, which by respiration is drawn from
the object towards the organ, doth in its passage

PART IV.
29.

«rhy §oine sa-
vours cause
nauseousaess.

toucli upon all those parts, and receive their motic
Now, the air toucheth the superficies oaly ; and
bodjr havine: less superficies whilst it is whole^ thfl
all its parts together have after it is reduced
powder, it follows that the same odorous
yieldeth less smell whilst it is whole, than it will
after it is broken into smaller parts. And
much of smells.
The first organ 17* The tastc follows ; whose generation hn
Ihy'^ome^f this difference from that of the sight, hearing,

smelling, that by these we have sense of remo^
objects ; whereas, we taste nothing but what
contiguous, and doth immediately touch either
tongue or palate, or both. From whence it is
dent, that the cuticles of the tongue and
and the nerves inserted into them are th
organ of taste ; and (because from the cone
of the parts of these, there foUoweth necessaril
concussion of the pra mater) that the action
mnnicated to these is propagated to the b
from thence to the farthest organ, name!
heart, in whose reaction consisteth the nature
sense.

Now, that savoui'S, as well as odours^ do i
only move the lirain liut the stomach also, m
manifest by the loathings that are caused by
both ; they, that consider the organ of both iksf
senses, will not wonder at all ; seeing the too^
the palate and the nostrils, have one and the
continued cuticle, derived from the dura maifr.
And that effluvia have nothing to do ili
sense of tajsting, is manifest from this, thai til
is no taste where the organ and the object are!
contiguous.

SOUND, ODOURj ETC.

507

Bv what variety of uiotions the different kinds p^^RT* iv»

of tastes, which are innumerable, may be distin- ^ — r^— '
gnishert, I know not. I might with other?^ derive
them from the divers figures of those atoms, of
whieh whatsoever may be tasted eonsistetli; or
from the diverse motions which I oiiglit, by way of
supposition, attribute to those atoms; conjecturing,
not without some likehhoodoftruth,thatsuch things
as taste sweet have their particles moved with shnv
circular motion, and their figures spherical: which
makes them smooth and pleasing to the organ ;
that bitter things have circular motion, but vehe^
inent, and their figures full of angles, by whieh
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 s\ich several motions and figures of
atoms, as might in probability seem to be the true
causes. But this would he to revolt from philoso-
phy to divination,

18. By the toneh^ we feel what bocUes are cold ^^te first organ
or hot, though they be distant from us. Others, bow we came
as hard, soft, rough, and smooth, we cannot feel jodgV °of TZh

The organ of touch is '^^J'*-^^' *» "^

unless they be contiguous. .^. .,^.» .. ....» ,. ^^-^^^^ ^^^^^

every one of those membranes, which being con- to"ch and lo
tinued from t\w pia mfiternve so diffused through-
out the whole body, as that no part of it can be
pressed, but the pia mater is pressed together w ith
it. Whatsoever therefore prcsseth 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,

AETir. time and place. For we take notice of rough and
— '-^ smooth, as also of magnitude and figure, not only
by the touch, but also by memory. For though
some things are touched m 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 w ithout 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 proceedef h from the
diflTerence of the impetus with which the elements, whereof
heavy bodies are made, do fall upon the earth.^^. The cause
of the descent of heavy bodies. — 5* In what proportion Uie
descent of heavy bodies is accelerated,- — 6* Why tho,se that
dive do not, wlien 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, tlian when it is empty • — lU The cause of
the ejection upwards of heavy bodies from a wind-gun.

12, Tlie 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 tJie
cause of magnetical virtue.

wniy 1, In chapter xxi I have defined thick and thm,
nai^ as that place required^ so, as that by thick was

OF GRAVITY,

509

TART IV,
30.

Nnxv ;f ten tinlcss ai^o
^^"^^^^ mare place,

signified a more resiKtins ^Kid>% and by thin, a body
less resisting; following the custom of those that
have before me discoursed of refraction.
we consider the true and vulgar signitication of 'han a thin.
those wordsj we shall find them to be names col-
lective, that is to say, names of multitude ; as ikick
to be thatj which takes up more parts of a space
given, and ikin 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 graiiiy.

2. Now we call those bodies hearu. which, '"'^**^''^^*-

^^ , J scent or lieavy

unless they be hindered by some force, are carried bodies procccdt
I towards the centre of the earth, and that by their own Ipp" tul?^
own accord, for aught we can by sense perceive to ^^J^'^'l\ "^^^
the contrary. Some philosophers therefore have^^'^^-
been of opinion, that the descent of heavy bodies
proceeded from some internal appetite, by which,

f" ™

PAirr iv%

30.

again, as moved by themselves, to such place as
was agreeable to their nature. Others thought

thev were attracted bv the earth. To the former

* If

I cannot assent, because I think I have already
clearly enotigh demonstrated that there can be no
beginning of motion, but fiom 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
extem<il body. HeRyy bodies, therefore, being
once cast upwards, cannot be cast down again
but by external motion. Besides, seehig inanimate
bodies have no appetite at all, it is ridiculous to
think that by their own innate appetite they should,
to preserve themselves, not understandhig 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 o\%'B hfe, raise
himself by leaping above three or four feet from
the ground. Lastly, to attribute to created bodies
the power to move themselves, w hat is it else than
to say that there be creatures which have no
dependance upon the Oeator ? To the latten 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-
hat of the manner and of the way by which the
:h by its action attracteth hea%7 bodies.
That by the supposition of simple motion in
un, homogeneous bodies are congregated and
rogeneous dissipated, has already been demon-

strated in the 5tli article of chapter xxi. I have ^"^^J^^-
also supposed, that there are intermingled with the ^—* — -
pure air certain little bodies, or, as others call them, f^^p^^ras Ihh
atoms ; which by reason of their extreme small- ^^"*^^^ ^^f *'*%

J men Is, whereof

ness are invisible, antl dififerin^ from one another iicav> bodit»»

, * .* - " 1 * 1 ^^*^ made, do

in consistence, iigure, motion, and magnitude ; f^ii upon lUe
from whence it comes to pass that some of them ^'^^^ *
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 wdl fall upon the
eaithj some with greater, others with less impetus.
And seeing also that we compute the several
degrees of gravity no otherwise than by tins 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
w ith le^s 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 cmtse of
fig. 2) ' be a great circle in the supeiiicies of the heavy bodi^,
earth, passing through the points A and B. Also
let any heavy body, as the stone A 1), be placed
anywhere in the plane of the equator; and let it
be conceived to be cast up from A D perpendicu-
larly, (U' to be carried in any other line to E, and
supposed to rest there. Tlieretbrc, how much

soever the stone
space it takes up now in E.

And because all place

^Ve'cent'^If 1® supposed to be full, the space A D will be filled
heavy bodies, by the aiF wliich flows into it first from the nearest
places of the earthy and afterwards successively
from more remote places. Upon the centre C let
a circle be understood to be dra^vn througli 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, w hich is contained by the two perimeters that
pass througrh 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 vacuujn, cannot be, unless so much air descend
to E from the place next above ; which being done,
•Tie will be thrust downwards. By this
^fore the stone now receives the begin-
descent, that is to say, of its gra%'ity.
, whatsoever is once moved, will be
nually (as hath been shoi;\Ti in the
)f chapter viii) in the same way, and

OF GRAVITY.

613

with the same celerit), except it be retarded or ^"^^J^^-
accelerated by some external movent* Now the ^^-^ — '
air, which is the only body that is interposed be- the^aTscmi'of
tween the earth A and the stone above it E, will ^"^y ^^'"•
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 hea^^ 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 know, 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 1.

PART
30.

5. Supposing, therefore, this to be the cause of
the descent of heavy bodies, it will follow that
portilnYh/de- *^^^^*^ motioii Will be accelerated in such manner,
scent of he^vy ^ ^^^^^ i\^^ spaces, which are transmitted by them

bodies IS rtcrcle- » , i * -11 i

rated, ui the sevcral time^^ will have to one another the

same proportion which the odd numbers have hi
succession from unity. For if the strait line EA
be divided into any number of equal parts, the
heai^r 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 impiilsion ; 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 Dkdognes of Mo-
tion demonstrated, that heavy bodies descend in
the several times \vith 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 diflferences are 3, 5, 7, &e. ; 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 w ill 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

above them.

the cylinder will be as the superficies of the earth ; part iv.
and by thrusting off the next and lowest air, will — -^ — -
make the uppermost air depress the hea\7 body,
whieh is at the top of the cylinder, in such manner
as is above explicated.

6, Tlie fi^ravitv of water beiner so ereat as by y^y those that
experience we find it is, the reason is demanded by when they are
many, why those that dive, how^ deep soever they ^eioirwdght
^o under water, do not at all feel the w^eight 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 w ater
as is equal to it m 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 dow nwards, whether the
water be pressed by water, or by anoth(*r body
which is heavier than water. And therefore by
these tw 0 opposite equal endeavours, the endeavour
both ways in the water is taken aw ay ; 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.

LL 2

Thowfight of II
body that float-
ethf ii eqim] to
thewdghtof so
much water as
would till the
space which the
immerKed part
of I he body
Ukea up witbia
tlift water.

i f a boily be
lighter thou
wntcr, then how
hip so^evcr »hat
body be, it may
llo'ti upon any
i^quaiitity of
ItHter. how
I fQ«ver.

that body is equal to the weij^lit of so much water
as would fill the place which the immersed part of
the body takes up within the water.

Let EF (ill fig. 3) be a body floating in the water
A B C D ; and let the part E be above, and the
other part F under the water. I say, the weight
of the whole body E F is equal to the weight of so
much water as the space F will receive. For
seeing the weight of the body EF forceth the
water out of the space F, and plaeeth it upon the
superficies A B, where it presseth doTAaiwards ; it
followSj 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 upwarcLs^
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 two endeaA^ours ;
that is to say, the weight of the body E F is equal
to the w eight of so much water as the space F will
receive ; which was to be pro% ed.

8. From hence it follow s, that any body, of how
great magnitude soever, provided it consist of
matter less hea\7 than water, may nevertheless
float upon any quantity of water, how little soever.

Let A B C D (in fig. 4) be a vessel ; and in it let
E F G H be a body consisting of matter which is
less heavy than water; and let the space AGCF

OF GRAVITY.

5i;

PART
3U.

rv^

be filled with water. I say, the body EFGH will
not siuk 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 fall of water,
yet some part of the body EFGH, as EFIK,
w onld be above the water ; and the weig:ht of so
mueh \% ater as would fill the space I G H K would
be equal to the weight of the w hole body EFGH;
and consequently G H would not touch the bottom
I) C* As for the sides of the vessel, it is no
matter whether they be hard or fluid; for they
«erve only to terminate the w ater ; 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. Tlie part therefore E F 1 K will be extant
above the water A G C F which is contained in the
vessel. Wlierefore 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

, , « , * J* 1 • be lifted op and

brought tor the proving of vacuum the experiment forced out of *

of water enclosed in a vessel ; which water, the ''®*'*^^ ^^ **'*'

orifice above being opened, is ejected upwards by

the impulsion of the air. It is therefore demanded,

seeing water is heavier tlian 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 1 supposed to be intermingled with

air and to be moved with simple motion, they

PART rv.

30.

cannot, together ^^th the pure air, penetrate the
water ; but remaining: behind are necessarily thrust
^^j?'^*^' ^^5 together into a narrower place-, namely into the
fbrced oui of a space wMch is above the water FG. The mo-
tesae y air, ^.j^^^g therefore of those smaD 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
endeavour to ascend, and \\ill therefore nece^saril;
go out. But it cannot go out, unless at the sami
time there enter in as much air; and therefore
both the water wiU 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 breidt the clouds, and restore them-
selves to their natural hberty ; so here also the
Hmall bodies enclosed within the space which is
ve the strait line F G, do by their own motion
il the water as soon as

tndV

passage

>pened

e* And if the passage be kept stopped, ami
small bodies be more vehemently compressed

by the perjietual forcing in of more water, they
will at last break the vessel itself with gp:-eat noise.

10. If air be blown into a hollow cylinder, or
into a bladder, it will increase the weight of either
of them a little, as many have found by experience,
wiio with great accerateness 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 smidl 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
force of air enclosed in a hollow cannon, a bullet
of lead may with considerable violence be shot out
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 sen ing for the admission of air, and the
other for the letting of it out. Also, to that end
which serves tor 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
peribrated, and hath also a valve opening towards
the former cannon. By the lielp 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

Why a bladder
IS heavier when
blown full of
air, thau when
it ia empty*

Tbe cause of
the ejection op
wttrds of heavy
bodies fn>ni a
wtrirl-gtiti.

PART IV,

30.

The cause of
theejecdoD^&c^

means men think there is now a greater quantity of
air in the cannon than there was formerly, thongh
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 w ithcmt ; so that the endeavours
'*^tween the internal and external air are opposite,
it is, there are tw o opposite motions whilst the
; goeth in and the other cometh out ; but no
mentation at all of air within the cannon. For
re is driven out by the stroke as much pure air,
di passetli between the rammer and the sides

OF GRAVITY.

521

of the cannon, as there is forced in of air impure by part iv.
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 w hichj 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 senses to
let them out, is opened, they will fly out with
violent motion, and carry with them the bullet
which is in their way, Wlierefore, I have given
a possible cause of this phenomenon,

12. Water, contraiT to the custom of heavy The cause
bodies, ascendeth in the w eather-glass ; but it doth of water in «
it when the air is cold : for when it is warm it dcs- *'=^*'^"-b1""-
cendeth again. And this organ is called a ther-
mometer or thermoseope, because the degrees of
heat and cold are measured and marked by it. It is
made in this manner. Let A B C D (in fig. 5) be
a vessel full of w ater, 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 w ill
ascend slowiy 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 w ill descend below^ H ;
and if the heat be diminished, it will ascend above

522

PHYSICS-

PART IV,
30.

it. Which, thoug^h it be certainly known to be
true by experience, the canse, 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
show^n, that fluid bodies are made colder by the
pressnre of the air, that is to say, by a constant
whid 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
mil l)e 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 creaturcs, thoudi they be heavy,

Uoa upwards in , , - - • i n > * i

liiringcrcarurea cau by Icapmg, swimmiDg and liying, raise them-
selves to a certain degree of height. But they
cannot do this except tht^y 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 sncceedeth
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
actpiire an endeavoiu' upwards, but such as by
reason of the gravity of their bodies is presently

OF GRAVITY. 523

ost again. By this endeavour, therefore, it is-, that i'art iv.
living ereatnres raise themselves up a little way by — ^- —
leaping, bnt to no great purposf* : bnt by s\\ imming
and flying they raise themselves to a greater height;
beeaus<% 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 soulj 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 diaphanons medium, which surrounds '^^^'^^^^'^f^V"

, . . . • , * . , nature ilkjnd of

the eye on all sides, is invisible ; nor is air to be body heavier

... ^1*1. than air, which

seen m air, nor water m water, nor anything but neverthtUsi ia

that which is more opacous. But in the confines JJ^^tinKutsLabl*

of two diaphanous bodies, one of them may be dis- ^i^^"' i^-

tinguished tVoni 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 Wlieu we feel heat, we 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 soraething

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.

PART IV, reasoning ; but it is from one reason only, namely,
^— ^ becanse it is impossible for remote bodies to work

nauire Tc? "* ^P*^^ 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, w ithout 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 ciinnot 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
iuto 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 dov^n, 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 ^I'mt e^^^!' '"^
things as are put down into it, as water doth ;
which, by the moisture it leaveth, hindereth the
kindhng again of the matter once extinguished-
In like manner, if a man be let down below C, he
w ill 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 w ere let
dowUj to signify they are not w ell, 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 w hich means,
uidess 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 w ater. 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
tlig another pit, as DE, 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.

R.TIV.
SO.

the cause
rnaguetical
Uttf.

lowed by the noxious matter contained in C B; anc
by thifci 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 tinith 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
li\^ng creatures are extinguished.

1 5. About the nature of hea\"y bodies, the greatest
difficulty ariseth from the contemplation of those
things which make other hea\y 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 Heradeus ; a stone,
thougli 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 Antaeus.
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 henoe
it follows evidently, that the first endeavour, which
iron hath towards the loadstone, is caused by the

I
I

OF GRAVITY.

527

motion of that air which is contiguous to the iron ; p^^J ^v.
also, that this motion is generated by the motion of — r—
the next air, and so on successively, till by this ^/^4n^ti^
succession w6 find that the motion of all the inter- ^»^^^-
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 w as 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 v\o\ 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". Wliy, therefore, should not
reciprocal motion of the parts of the loadstone con-
tribute as much towards the mo\ing 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.
30.

Of the caufte
of magnelicaJ
virtue.

propagated by the air to the iron, and consequently
that there will be in all the parts of the iron the
same reciprocations or motions forwards and back-
wards. And from hence also it will follow, that
the intermediate air between the stone and the
iron w ill, by little and little, be thrust away ; and
the air being thrust away, the bodies of the load-
stone and 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
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
r together with the earth by its diunial mo-
oth by length of time get a habit of being J
u a line which is perpendicular to the line
procal motion, it will afters^ ards, though
e removed fi-om the parallel situation it
the axis of the earth, retain its endeavour

OF GRAVITY.

529

of returning to that situation again ; and all en- ^^^J ^^ •
deavour being the beginning of motion , and nothing '^^—
intervening that may hinder the same, the load- oimigneucli
stone will therefore retuni to its former situation. ^^"**^'
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 draT;\Ti 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.

PART IV. the latter this may be the cause, that as the load-
— — stone in touching the iron doth by its action im-
^/mS^eSP''^^*^ ^^ ^-'^^ ^^^^ ^^ endeavour tovvards one of the
virtoe, poles, suppose towards the North Pole; so red-
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 w hilst 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 tow^ards
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 wiU be situate
in the same strait hne< 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

^ody of any hardness whatsoever. For in a

um, there can be no motion which doth

the next part yield, and that the next,

eessively without end ; so that there is

hatsoever, 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 tliiings^
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 it self ^
by species^ by its own power , hy suhstimtial forms ^
by incorporeal suhstauees^ by rnsfmct^ by a7iti-
perislasis^ by aniipathy, sympatkij^ occult qmdity^
and other empty words of schoolmen, their saying
so is to no purpose.

532 PHYSICS.

And now I proceed to the phenomena of man's

body ; where I shall speak of the optics , and of

ConciuMon. ^^ dispositiouSy c^ectioTis, and manners of men,

if it shall please God to give me life, and show

their causes.

END OF VOL. I.

LOVDOX:

c. micHAmM, rr. kaktix's laxf, cH*RiMi ckuss.

CECIL H, GREEN LIBRARY

STANFORD UNIVERSITY LIBRARIES

STANFORD, CALtFORNIA 94305*6004

(650) 723-1493

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