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--=. f :v 



(V^////. 



/ 



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. 






TO 



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. 



Xi 



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 



8 



COMPUTATION OR LOGIC, 



PART T. 

1, 

■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. 



11 



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- 




12 



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. 

2. 



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 



18 



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* 



19 



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 



20 



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 



22 



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 



24 



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. 



26 



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 



28 



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 



30 



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 



32 



COMPUTATION OR LOGIC. 



RART i. 

5. 

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 



34 



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 



36 



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 



40 



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 

3. 



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, 



49 



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 



50 



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, 



&t 






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 



52 



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 



58 



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. 



59 






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 

5. 



^ 






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 




62 



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. 



I 



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 



I 



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 





64 



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 



66 



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. 



67 



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 




68 



COMPUTATION OR LOGIC, 



PARTI. 

6. 



(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 



70 



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. 



71 



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 



I 

I 



OF METHOD. 



75 



"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- 



76 



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. 



n 



'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. 



79 



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 



80 



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 
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 



82 



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. 



83 



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 

g2 




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. 



85 



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- 



87 



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 



88 



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* 




94 



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 



I 



Spai!«t and 
times con- 
tiguaua and 



B 



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 ^^^^'^ ^^ 



8. 



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. 



P 

^^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. 



u; 



h: 



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 



I 



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. 



n. 



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. 

la 



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 

tm tifg : 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, 



HI 



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. 



B 



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, 



A 


B 


C 


D 


E 


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 — 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 



i 



^ 
^ 



^ 



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. 



M 



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 



N 



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 — 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. 



A 


A 


B 


F 


C 


G 


D 


H 


E 


E 



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- 



A 



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 * * 



E 

-I- 



G 

-J - 



H 



B 



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. 

n2 



H. 



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 



H. 



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 



p 



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 



^ 



N 



^ 



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. 



i 



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- 



I 




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, 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 

p2 



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. 



nu 



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. 




I 



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^"" 



1^ 



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, 



I 



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- 






I 



^ 



^ 



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 



2 



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- 



I 



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 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, 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, &ۥ 
^^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 1 


3 3 4 fi 6 7 

: : : : : 


i 




i 




*♦[::::: 


♦ *:f 




* 


til 


A 


: : : 


*i 


i i 


tVA 


: 


*l 


i tV 


tV 


A 


tV 




f 


iVA 


A 


VVr 


-Ar 


A 



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 



I 



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, 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 



I 


2 


3 


4 


5 


6 7 


1 : : : : : 








i| 








* 


f 




f • • * 




f 
* 


* 


+ 




: : : 


i 


A 


A 


: : 




* 


AA 


A 


A = 




A 


A A 


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) 









♦ 


















f 


1 : : ; : 




i 


a! 


* " • 




t 


AjA 




J • 


i 


A 


tVA-VW 




• 
• 


i 


rt- 


tVIt^ 


TV 


tV 




•flr 


A 


•frliViV 


I'd 


A 



I 



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 





1 


2 


s 


4 


S 6 


7 


. . . 


1: : : : : : t 


-sided) 


A 






U 


A 






H 


H 


A 


: : : : 




H 


if 


H 


A 


* ■ « 




H 


H 


H 


a 


aI : :^ 




M 


M 




H 


HA 


I 




U 


31 


M 


HH 


A 



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- 

s2 



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- 



t 



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 ; 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. 



I 



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 



i 



LUU 

i 



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 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 



I 



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 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. 



I 




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 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 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 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 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 



b 



b: 



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 

2L 




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 

2 



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 



I 



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. 



I 



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©^ 



i 



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 fii rar - 
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 
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> M MaMTmiBSw 



?.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 iep M at a oa 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 p fe aBUfe 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 



i 




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 



I 



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: 



I 



I 



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^"^ 



3 




CENTRE OF EQUIPONDERATION. 



351 



23. 



h 



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* 



I 



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. 



I 




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 * pw u p 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 be triple to I N ; and 

VOL. I. BE 



IT HI- 
SS. 

>-»^ — — ' 

centre of 
Hiiitlcra- 
of ihe 



r 



MOTIONS AND MAGNITUDES, 

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 



1 



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}. 



I 



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- 



I 



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- 



TO 



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 




F 




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 : 



I 



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 

I 



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 dcgmewt iiig 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. 



N 



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 
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* 



I 

I 



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 



to 

i 



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 



so 




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- 



I 




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. 



K 

I 

L 



H 



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- 



p 



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 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 



IS 

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 




I 



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 
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. 




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