Skip to main content

Full text of "The English Works Of Thomas Hobbes Vol VII"

See other formats

> DO ^ 

B]<OU 166642 >m 


be returned on or Before tflfdatc last marked below. 










Kl( HARDS, PKIMEU, 100, M >I\HlIN's I ANI 





A QUADRANT . . . . . . . 178 


MATHEMATICS . . . . . . . 181 


DR. WALLIS ....... 357 



AGAINST DR. WALLIS ...... 429 










VV] 1 II 





THAT which I do here most humbly present to 
your sacred Majesty, is the best part of my medi- 
tations upon the natural causes of events, both of 
such as are commonly known, and of such as have 
been of late artificially exhibited by the curious. 

They are ranged under seven heads. 1. Problems 
of gravity. 2. Problems of tides. 3. Problems of 
vacuum. 4. Problems of heat. 5. Problems of 
hard and soft. 6. Problems of wind and weather. 
7- Problems of motion perpendicular and oblique, 
&c. To which I have added two propositions of 
Geometry : one is, the duplication of the cube, 
hitherto sought in vain ; the other, a detection of 
the absurd use of arithmetic, as it is now applied 
to geometry. 

The doctrine of natural causes hath not infal- 
lible and evident principles. For there is no effect 
which the power of God cannot produce by many 
several ways. 

But seeing all effects are produced by motion, 
he that supposing some one or more motions, can 
derive from them the necessity of that effect whose 

B 2 


. cause is required, has done all that is to be ex- 
pected from natural reason. And though he prove 
not that the thing was thus produced, yet he 
proves that thus it may be produced when the ma- 
terials and the power of moving are in our hands : 
which is as useful as if the causes themselves 
were known. And notwithstanding the absence of 
rigorous demonstration, this contemplation of na- 
ture (if not rendered obscure by empty terms) is 
the most noble employment of the mind that can 
be, to such as are at leisure from their necessary 

This that I have done I know is an unworthy 
present to be offered to a king : though considered, 
as God considers offerings, together with the mind 
and fortune of the offerer, I hope will not be to 
your Majesty unacceptable. 

But that w r hich I chiefly consider in it is, that my 
writing should be tried by your Majesty's excellent 
reason, untainted with the language that has been 
invented or made use of by men when they were 
puzzled ; and who is acquainted with all the ex- 
periments of the time ; and whose approbation, if I 
have the good fortune to obtain it, will protect my 
reasoning from the contempt of my adversaries. 

I will not break the custom of joining to my 
offering a prayer ; and it is, that your Majesty will 
be pleased to pardon this following short apology 
for my Leviathan. Not that I rely upon apolo- 
gies, but upon your Majesty's most gracious gene- 
ral pardon. 


That which is in it of theology, contrary to the 
general current of divines, is not put there as my 
opinion, but propounded with submission to those 
that have the power ecclesiastical. 

I did never after, either in writing or discourse, 
maintain it. 

There is nothing in it against episcopacy ; I 
cannot therefore imagine what reason any episco- 
pal man can have to speak of me, as I hear some 
of them do, as of an atheist, or man of no religion, 
unless it be for making the authority of the Church 
wholly upon the regal power ; which I hope your 
Majesty will think is neither atheism nor heresy. 

But what had I to do to meddle with matters of 
that nature, seeing religion is not philosophy, but 
law r 

It was written in a time when the pretence of 
Christ's kingdom w r as made use of for the most 
horrid actions that can be imagined ; and it was 
in just indignation of that, that I desired to see 
the bottom of that doctrine of the kingdom of 
Christ, which divers ministers then preached for a 
pretence to their rebellion : which may reasonably 
extenuate, though not excuse the writing of it. 

There is therefore no ground for so great a 
calumny in my writing. There is no sign of it 
in my life ; and for my religion, when I was at 
the point of death at St. Germain's, the Bishop of 
Durham can bear witness of it, if he be asked. 
Therefore I most humbly beseech your sacred Ma- 4 
jesty not to believe so ill qf me upon reports, that 


proceed often, and may do so now, from the dis- 
pleasure which commonly ariseth from difference in 
opinion ; nor to think the worse of me, if snatch- 
ing np all the weapons to fight against your ene- 
mies, I lighted upon one that had a double edge. 

Your Majesty's poor and 
most loyal subject, 





A. WHAT may be the cause, think you, that stones CHAP. i. 
and other bodies thrown upward, or carried up s ' ' 

x ^ * Plohlrms 

and left to their liberty, fall down again, for aught ouumty. 
a man can see, of their own accord ? I do not 
think with the old philosophers, that they have 
any love to the earth ; or are sullen, that they will 
neither go nor stay. And yet I cannot imagine, 
what body there is above that should drive them 

IJ. For rny part, I believe the cause of their 
descending is not in any natural appetite of the 
bodies that descend ; but rather that the globe of 
the earth hath some special motion, by which, it 
more easily casteth off the air than it doth other 
bodies. And then this descent of those we call 
heavy bodies must of necessity follow, unless there 
be some empty spaces in the world to receive them. 
For when the air is thrown off from the earth, 
somewhat must come into the place of it, in case 
the world be full : and it must be those things 
which are hardliest cast off, that is, those things 
which we say are heavy. 

//. But suppose there b no place empty, (for 


CHAP. i. . I will defer the question till anon), how can the 
probLBmT" earth cast off either the air or anything else ? 
oi gravity. j shall show you how, and that by a familiar 
example. If you lay both your hands upon a basin 
with water in it, how little soever, and move it 
circularly, and continue that motion for a while ; 
and you shall see the water rise upon the sides, 
and fly over. By which you may be assured that 
there is a kind of circulating motion, which would 
cast off such bodies as are contiguous to the body 
so moved. 

A. I know very well there is ; and it is the same 
motion which country people use to pui'ge their 
corn ; for the chaff and straws, by casting the 
grain to the sides of the sieve, will come towards 
the middle. But I would see the figure. 

13. Here it is. There is a circle pricked out, 
w T hose centre is A, and three less circles, whose 
centres are B, C 5 D. Let every one of them repre- 
sent the earth, as it goeth from B to C, and from 
C to D, always touching the uttermost circle and 
throwing off the air, as is marked at E and F. 
And if the world were not full, there would follow 
by this scattering of the air, a great deal of space 
left empty. But supposing the world full, there 
must be a perpetual shifting of the air, one part 
into the place of another. 

A. But what makes a stone coine down, sup- 
pose from G ? 

B. If the air be thrown up beyond G, it will 
follow that at the last, if the motion be continued, 
all the air will be above G, that is, above the stone ; 
which cannot be, till the stone be at the earth. 

A. But why comes it down still with increasing 
wiftness ? 


B. Because as it descends and is already in mo- CHAP. i. 
tion, it receiveth a new impression from the same ^^^ 
cause, which is the air, whereof as part mounteth, of gravity 
part also must descend, supposing as we have done 
the plentitude of the world. For, as you may ob- 
serve by the figure, the motion of the earth, ac- 
cording to the diameter of the uttermost circle, 
is progressive ; and so the whole motion is com- 
pounded of two motions, one circular and the 
other progressive ; and consequently the air as- 
cends and circulates at once. And because the 
stone descending receiveth a new pressure in every 
point of its w r ay, the motion thereof must needs be 

A. It is true ; for it will be accelerated equally 
in equal times ; and the way it makes will increase 
in a double proportion to the times, as hath here- 
tofore been demonstrated by Galileo. I see the 
solution now of an experiment, which before did 
not a little puzzle me. You know T that if two 
plummets hang by two strings of equal length, and 
you remove them from the perpendicular equally, 
I mean in equal angles, and then let them go, they 
will make their turns and returns together and in 
equal times ; and though the arches they describe 
grow continually less and less, yet the times they 
spend in the greater arches will still be equal to 
the time they spend in the lesser. 

B. It is true. Do you find any experiment to 
the contrary ? 

A. Yes ; for if you remove one of the plummets 
from the perpendicular, so as, for example, to make 
an angle with the perpendicular of eighty degrees, 
and the other so as to maUe an angle of sixty 


CHAP. r. degrees ; they will not make their turns and re- 
" ; turns in equal times. 

rioblorns * 

oi gi.mty B. And what say you is the cause of this ? 

A . Because the arches are the spaces which these 
two motions describe, they must be in double pro- 
portion to their own times : which cannot be, un- 
less they be let go from equal altitudes, that is, 
from equal angles. 

B. It is right ; and the experiment does not 
cross, but confirm the equality of the times in all 
the arches they describe, even from ninety degrees 
to the least part of one degree. 

A. But is it not too bold, if not extravagant an 
assertion, to say the earth is moved as a man 
shakes a basin or a sieve ? Does not the earth 
move from west to east every day once, upon its 
own centre ; and in the ecliptic circle once a-year ? 
And now you give it another odd motion. How 
can all these consist in one and the same body ? 

B. Well enough. If you be a shipboard under 
sail, do not you go with the ship ? Cannot you 
also walk upon the deck ? Cannot every drop of 
blood move at the same time in your veins ? How 
siany motions now do you assign to one and the 
same drop of blood ? Nor is it so extravagant a 
thing to attribute to the earth this kind of motion ; 
but that I believe, if we certainly knew what mo- 
tion it is that causeth the descent of bodies, we 
sh ould find it either the same, or more extrava- 
gant. But seeing it can be nothing above that 
w orketh this effect, it must be the earth itself that 
does it ; and if the earth, then you can imagine 
no other motion to do it withal but this. And you 
will wonder more, when by the same motion I 


shall give you a probable account of the causes of CHAP. i. 
very many other works of nature. ' ' 

A ^ i r Problems 

A. But what part of the heaven do you suppose oi gravity. 
the poles of your pricked circle point to ? 

B. I suppose them to be the same with the poles 
of the ecliptic. For, seeing the axis of the earth 
in this motion and in the annual motion, keeps 
parallel to itself, the axis must in both motions be 
parallel as to sense. For the circle which the 
earth describes, is not of visible magnitude at the 
distance it is from the sun. 

A. Though I understand well enough how the 
earth may make a stone descend very swiftly un- 
der the ecliptic, or not far from it, where it throws 
off the air perpendicularly ; yet about the poles of 
the circle methinks, it should cast off the air very 
weakly. I hope you will not say, that bodies de- 
scend faster in places remote from the poles, than 
nearer to them. 

B. No ; but I ascribe it to the like motion in 
the sun and moon. For such motions meeting, 
must needs cast the stream of the air towards the 
poles ; and then there will be the same necessity 
for the descent there, that there is in other places, 
though perhaps a little more slowly. For you may 
have observed, that when it snows in the south 
parts, the flakes of snow are not so great as in the 
north : which is a probable sign they fall in the 
south from a greater height, and consequently dis- 
perse themselves more, as water does that falls 
down from a high and steep rock. 

A. It is not improbable. 

B. In natural causes all you are to expect, is 
but probability ; which is better yet, than making 


THAP. T. gravity the cause, when the cause of gravity is 
t7 M r that which you desire to know ; and better than 

Tioblems ^ J ' 

of gravity, saying the earth draws it, when the question is, 
how it draws. 

A. Why does the earth cast off air more easily 
than it does water, or any other heavy bodies ? 

J3. It is indeed the earth that casteth off that 
air which is next unto it ; but it is that air which 
casteth off the next air ; and so continually, air 
moveth air ; w r hich it can more easily do than any 
other thing, because like bodies are more suscepti- 
ble of one another's motions : as you may see in 
two lute-strings equally strained, what motion one 
string being stricken communicates to the air, the 
same will the other receive from the air ; but 
strained to a differing note, will be less or not at 
all moved. For there is no body but air, that hath 
not some internal, though invisible, motion of its 
parts : and it is that internal motion which distin- 
guisheth all natural bodies one from another. 

A. What is the cause why certain squibs, though 
their substance be either wood or other heavy mat- 
ter, made hollow and tilled with gunpowder, which 
is also heavy ; do nevertheless, when the gunpow- 
der is kindled, fly upwards ? 

B. The same that keeps a man that swims from 
sinking, though he be heavier than so much water. 
He keeps himself up, and goes forward, by beating 
back the water with his feet ; and so does a squib, 
by beating down the air with the stream of fired 
gunpowder, that proceeding from its tail makes it 

A. Why does any brass or iron vessel, if it be 
hollow, float upon the twater, being so very heavy : 


B. Because the vessel arid the air in it, taken CHAP. i. 
as one body, is more easily cast off than a body of !~7; 

J J J Problems 

water equal to it. of Cavity. 

A. How comes it to pass, that a fish, (especially 
such a broad fish as a turbot or a plaice, which are 
broad arid thin), in the bottom of the sea, perhaps 
a mile deep, is not pressed to death with the 
weight of water that lies upon the back of it ? 

B. Because all heavy bodies descend towards 
one point, which is the centre of the earth : and con- 
sequently the whole sea, descending at once, does 
arch itself so, as that the upper parts cannot press 
the parts next below them. 

A. It is evident ; nor can there possibly be any 
weight, as some suppose there is, of a cylinder of 
air or water or any other liquid thing, while it 
remains in its own element, or is sustained and in- 
closed in a vessel by which one part cannot press 
the other. 



A. WHAT makes the flux and reflux of the sea, 
twice in a natural day ? 

B. We must come again to our basin of water ; 
wherein you have seen, whilst it was moved, how 
the water mounteth up by the sides, and withal 
goes circling round about. Now if you should 
fasten to the inside of the basin some bar from the 
bottom to the top, you would see the water, in- 
stead of going on, go back again from that bar 
ebbing, and the water on tjie other side of the bar 


:HAP. JT. to do the same, but in counter- time ; and conse- 
" *~~" quently to be highest where the contrary streams 

Problems 1 J O J 

oi tuies. meet together ; and then return again, marking out 
four quarters of the vessel ; two by their meeting, 
which are the high waters ; and two by their re- 
tiring, which are the low waters. 

A. What bar is that you find in the ocean that 
stops the current of the water, like that you make 
in the basin ? 

B. You know that the main ocean lies east and 
west, between India and the coast of America ; 
and again on the other side, between America 
and India. If therefore the earth have such a 
motion as I have supposed, it must needs carry 
the current of the sea east and west : in which 
course, the bar that stoppeth it, is the south part 
of America, which leaves no passage for the water 
but the narrow strait of Magellan. The tide rises 
therefore upon the coast of America; and the rising 
of the same in this part of the world, proceedeth 
from the swelling chiefly of the water there, and 
partly also from the North Sea ; which licth also 
east and west, and has a passage out of the South 
Sea by the strait of Anian, between America and 

A. Does not the Mediterranean Sea lie also east 
and west ? Why are there not the like tides there ? 

J5, So there are, proportionable to their lengths 
and quantity of water. 

A. At Genoa, at Aucona, there are none at all, 
or not sensible. 

B. At Venice there are, and in the bottom of 
the straits, and a current all along both the Me- 
diterranean Sea and th Gulf of Venice : and it is 


the current that makes the tides insensible at the CHAP. u. 
sides ; but the check makes them visible at the ^^7" 
bottom. ofti ^ s 

A. How comes it about that the moon hath 
such a stroke in the business, as so sensibly to in- 
crease the tides at full and change ? 

B. The motion I have hitherto supposed but in 
the earth, I suppose also in the moon, and in all 
those great bodies that hang in the air constantly, 
I mean the stars, both fixed and errant. And for 
the sun and moon, I suppose the poles of their 
motion to be the poles of the equinoxial. Which 
supposed, it will follow (because the sun, the earth, 
and the moon, at every full and change are almost 
in one straight line) that this motion of the earth 
will then be made swifter than in the quarters. For 
this motion of the sun and moon being commu- 
nicated to the earth, that hath already the like 
motion, maketh the same greater ; and much 
greater when they are all three in one straight 
line, which is only at the full and change, whose 
tides are therefore called spring tides. 

A. But what then is the cause that the spring 
tides themselves are twice a-year, namely, when 
the sun is in the equinoxial, greater than at any 
other times ? 

B. At other times of the year, the earth being 
out of the equinoxial, the motion thereof, by which 
the tides are made, will be less augmented, by 
so much as a motion in the obliquity of twenty- 
three degrees, or thereabout, which is the distance 
between the equinoxial and ecliptic circles, is 
weaker than the motion which is without obliquity. 

A. All this is reasonable enough, if it be pos- 


CHAP. u. sible that such motions as you suppose in these 
bodies, be really there. But that is a thing I have 

s y J o 

of tides some reason to doubt of. For the throwing off of 
air, consequent to these motions, is the cause, you 
say, that other things come to the earth ; and 
therefore the like motions in the sun and moon 
and stars, casting off the air, should also cause all 
other things to come to everyone of them. From 
whence it will follow, that the sun, moon, and 
earth, and all other bodies but air, should pre- 
sently come together into one heap. 

B. That does not follow. For if two bodies cast 
off the aii% the motion of that air will be repressed 
both ways, and diverted into a course towards the 
poles on both sides ; and then the two bodies can- 
not possibly come together. 

A. It is true. And besides, this driving of the 
air on both sides, north and south, makes the like 
motion of air there also. And this may answer 
the question, how a stone could fall to the earth 
under the poles of the ecliptic, by the only casting 
off of air ? 

B. It follows from hence, that there is a certain 
and determinate distance of one of these bodies, 
the stars, from another, without any very sensi- 
ble variation. 

A. All this is probable enough, if it be true that 
there is no vacuum, no place empty in all the 
world. And supposing this motion of the sun and 
moon to be in the plain of the equinoxial, methinks 
that this should be the cause of the diurnal motion 
of the earth ; and because this motion of the earth 
is, you say, in the plain of the equinoxial, the 
same should cause ^also a motion in the moon on 


her own centre, answerable to the diurnal motion CHAP. n. 
of the earth. ]"":; ~~ 


B. Why not ? What else can you think makes <> tui^ 
the diurnal motion of the earth but the sun ? And 
for the moon, if it did not turn upon its own centre, 
we should see sometimes one, sometimes another 
face of the moon, which we do not. 



A. WE/AT convincing argument is there to prove, 
that in all the world there is no empty place ? 

B. Many ; but I will name but one ; and that 
is, the difficulty of separating two bodies hard and 
flat laid one upon another. I say the difficulty, 
not the impossibility. It is possible, without in- 
troducing vacuum, to pull asunder any two bodies, 
how hard and flat soever they be, if the force used 
be greater than the resistance of the hardness. 
And in case there be any greater difficulty to part 
them, besides what proceeds from their hardness, 
than there is to pull them further asunder when 
they are parted, that difficulty is argument enough 
to prove there is no vacuum. 

A. These assertions need demonstration. And 
first, how does the difficulty of separation argue 
the plenitude of all the rest of the w r orld ? 

B. If two flat polished marbles lie one upon 
another, you see they are hardly separated in all 
points at one and the same instant ; and yet 
the weight of either of them is enough to make 
them slide off one from th<^ other. Is not the 


;HAP. in. cause of this, that the air succeeds the marble that 
so slides, and fills up the place it leaves ? 

A. Yes, certainly. What then ? 

B. But when you pull the whole superficies 
asunder, not without great difficulty, what is the 
cause of that difficulty ? 

A. I think, as most men do, that the air cannot 
fill up the space between in an instant ; for the 
parting is in an instant. 

B. Suppose there be vacuum in that air into 
which the marble you pull off is to succeed, shall 
there be no vacuum in the air that was round 
about the two marbles when they touched ? Why 
cannot that vacuum come into the place between ? 
Air cannot succeed in an instant, because a body, 
and consequently cannot be moved through the 
least space in an instant. But emptiness is not a 
body, nor is moved, but is made by the act itself 
of separation. There is therefore, if you admit 
vacuum, no necessity at all for the air to fill the 
space left in an instant. And therefore, with what 
ease the marble coming off presseth out the va- 
cuum of the air behind it, with the same ease will 
the marbles be pulled asunder. Seeing then, if 
there were vacuum, there would be no difficulty 
of separation, it follows, because there is difficulty 
of separation, that there is no vacuum. 

A. Well, now, supposing the world full, l\ow 
do you prove it possible to pull those marbles 
asunder ? 

JS. Take a piece of soft wax ; do not you think 
the one half touches the other half as close as 
the smoothest marbles ? Yet you can pull them 
asunder. But how ? Still as you pull, the w r ax 


grows continually more and more slender; there CHAP. nr. 
being a perpetual parting or discession of the out- ^ Ill | lin 7' 
ermost part of the wax one from another, which the * ^' 
air presently fills ; and so there is a continual less- 
ening of the wax, till it be no bigger than a hair, 
and at last separation. If you can do the same to 
a pillar of marble, till the outside give way, the 
effect will be the same, but much quicker, after it 
once begins to break in the superficies ; because 
the force that can master the first resistance of 
the hardness, will quickly dispatch the rest. 

A. It seems so by the brittlcness of some hard 
bodies. But I shall afterward put some questions 
to you, touching the nature of hardness. But now 
to return to our subject. What reason can you 
render (without supposing vacuum) of the effects 
produced in the engine they use atGresham college? 

B. That engine produceth the same effects that 
a strong wind would produce in a narrow room. 

A. How comes the wind in ? You know the 
engine is a hollow round pipe of brass, into which 
is thrust a cylinder of wood covered with leather, 
and fitted to the cylinder so exactly as no air can 
possibly pass between the leather and the brass ? 

B. I know T it ; and that they may thrust it up, 
there is a hole left in the cylinder to let the air out 
before it, which they can stop when they please. 
There is also in the bottom of the cylinder a pas- 
sage into a hollow globe of glass, which passage 
they can also open and shut at pleasure. And at 
the top of that globe there is a wide mouth to put 
in what they please to try conclusions on, and that 
also to be opened and shut as shall be needful. It 
is of the nature of a pop-gun which children use, 


CHAP. in. but great, costly, and more ingenious. They thrust 
forward and pull back the wooden cylinder (be- 
cause it requires much strength) with an iron 
screw. What is there in all this to prove the pos- 
sibility of vacuum. 

A. When this wooden cylinder covered with 
leather, fit and close, is thrust home to the bottom, 
and the holes in the hollow cylinder of brass close 
stopped, how can it be drawn back, as with the 
screw they draw it, but that the space it leaves 
must needs be empty : for it is impossible that 
any air can pass into the place to fill it ? 

B. Truly I think it close enough to keep out 
straw and feathers, but not to keep out air, nor yet 
matter. For suppose they were not so exactly close 
but that there were round about a difference for a 
small hair to lie between ; then will the pulling 
back of the cylinder of wood force so much air in, 
as in retiring it forces back, and that without any 
sensible difficulty. And the air will so much more 
swiftly enter as the passage is left more narrow. 
Or if they touch, and the contact be in some 
points and not in all, the air will enter as before, in 
case the force be augmented accordingly. Lastly, 
though they touch exactly, if either the leather 
yield, or the brass, which it may do, to the force of 
a strong screw, the air will again enter. Do you 
think it possible to make two superficies so ex- 
quisitely touch in all points as you suppose, or 
leather so hard as not to yield to the force of a 
screw ? The body of leather will give passage 
both to air and water, as you will confess when 
you ride in rainy and windy weather. You may 
therefore be assured that in drawing out their 


wooden leather cylinder, they force in as much air CHAP. in. 
as will fill the place it leaves, and that with as 
much swiftness as is answerable to the strength 
that drives it in. The effect therefore of their 
pumping is nothing else but a vehement wind, a 
very vehement wind, coming in on all sides of the 
cylinder at once into the hollow of the brass pipe, 
and into the hollow of the glass globe joined to it. 

A. I see the reason already of one of their won- 
ders, which is, that the cylinder they pump with, 
if it be left to itself, after it is pulled back, will 
swiftly go up again. You will say the air comes 
out again with the same violence by reflection, and 
I believe it. 

B. This is argument enough that the place was 
not empty. For what can fetch or drive up the 
sucker, as they call it, if the place within were 
empty ? For that there is any weight in the air to 
do it, I have already demonstrated to be impossible. 
Besides, you know, when they have sucked out, as 
they think, all the air from the glass globe, they 
can nevertheless both see through it what is done, 
and hear a sound from within when there is any 
made ; which, if there were no other, but th^re 
are many other, is argument enough that the place 
is still full of air. 

A. What say you to the swelling of a bladder 
even to bursting, if it be a little blown when it is 
put into the receiver, for so they call the globe of 
glass ? 

B. The streams of air that from every side meet- 
ing together, and turning in an infinite number of 
small points, do pierce the bladder in innumerable 
places with great violence ait once, like so many 


CHAP. in. invisible small wimbles, especially if the bladder 
r^i>iomT" ' 3e a l lt:t l e Wwn before it be put in, that it may 
oi vacuum, make a little resistance. And when the air has 
once pierced it, it is easy to conceive, that it must 
afterward by the same violent motion be extended 
till it break. If before it break you let in fresh 
air upon it, the violence of the motion will thereby 
be tempered, and the bladder be less extended ; 
for that also they have observed. Can you imagine 
how a bladder should be extended and broken by 
being too full of emptiness ? 

A. How come living creatures to be killed in 
this receiver, in so little a time as three or four 
minutes of an hour ? 

B. If they suck into their lungs so violent a 
wind thus made, you must needs think it will pre- 
sently stop the passage of their blood ; and that is 
death ; though they may recover if taken out be- 
fore they be too cold. And so likewise will it put 
out fire ; but the coals taken out whilst they are 
hot will revive again. It is an ordinary thing in 
many coal-pits, whereof I have seen the experi- 
ence, that a wind proceeding from the sides of the 
pit every way, will extinguish any fire let down 
into it, and kill the workmen, unless they be 
quickly taken out. 

A. If you put a vessel of water into the receiver, 
and then suck out the air, the water will boil; 
what say you to that ? 

B. It is like enough it will dance in so great a 
bustling of the air ; but I never heard it would be 
hot. Nor can I imagine how vacuum should make 
anything dance. I hope you are by this time satis- 
fied, that no experiment made with the engine at 


Gresham College, is sufficient to prove that there CHAP. in. 
is. or that there may be vacuum. " 7 ' 

y J Fiobh-ms 

A. The world you know is finite, and conse- of \acuum. 
quently, all that infinite space without it is empty. 

Why may not some of that vacuum be brought in, 
and mingled with the air here ? 

B. I know nothing in matters without the 

A. What say you to Torricellio's experiment in 
quicksilver, which is this : there is a basin at A 
filled with quicksilver, suppose to B, and C D a 
hollow glass pipe filled with the same, which if you 
stop with your finger at B, and so set it upright, 
and then if you take away your finger, the quick- 
silver will fall from C downwards but not to the 
bottom, for it will stop by the way, suppose at I). 
Is it not therefore necessary that that space be- 
tween C and D be left empty r Or will you say 
the quicksilver does not exactly touch the sides of 
the glass pipe ? 

B. I will say neither. If a man thrust down 
into a vessel of quicksilver a blown bladder, will 
not that bladder come up to the top ? 

A. Yes, certainly, or a bladder of iron, or any- 
thing else but gold. 

B. You see then that air can pierce quicksilver. 

A. Yes, with so much force as the weight of 
quicksilver comes to. 

B. When the quicksilver is fallen to D, there is 
so much the more in the basin, and that takes 
up the place which so much air took up before. 
Whither can this air go if all the world without 
that glass pipe B C were full ? There must needs 
be the same or as much air qpme into that space, 


CHAP. HI. which only is empty, between C and D : by what 
force ? By the weight of the quicksilver between 
D and B. Which quicksilver weigheth now up- 
ward, or else it could never have raised that part 
higher, which was at first in the basin. So you 
see the weight of quicksilver can press the air 
through quicksilver up into the pipe, till it come 
to an equality of force as in D, where the weight 
of the quicksilver is equal to the force which is 
required in air to go through it. 

A. If a man suck a phial that has nothing in it 
but air, and presently dip the mouth of it into 
water, the water will ascend into the phial. Is 
not that an argument that part of the air had been 
sucked out, and part of the room within the phial 
left empty ? 

B. No. If there were empty space in the world, 
why should not there be also some empty space in 
the phial before it was sucked ? And then why 
does not the water rise to fill that ? When a man 
sucks the phial he draws nothing out, neither into 
his belly, nor into his lungs, nor into his mouth ; 
only he sets the air within the glass into a circular 
motion, giving it at once an endeavour to go forth 
by the sucking, and an endeavour to go back by 
not receiving it into his mouth; and so with a 
great deal of labour glues his lips to the neck of 
the phial. Then taking it off, and dipping the 
neck of the phial into the water before the circu- 
lation ceases, the air, with the endeavour it hath 
now gotten, pierces the water and goes out : and 
so much air as goes out, so much matter comes 
up into the room of it. 




A. WHAT is the cause of heat ? CHAP. iv. 

B. How know you. that any thing is hot but "7 T 

J J jo Problems of 

yOUrSelf ? heat and light 

A. Because I perceive by sense it heats me. 

B. It is no good argument, the thing heats me ; 
therefore it is hot. But what alteration do you 
find in your body at any time by being hot ? 

A. I find my skin more extended in summer 
than in winter; and am sometimes fainter and 
weaker then ordinary, as if my spirits were ex- 
haled ; and I sweat. 

B. Then that is it you would know the cause 
of. I have told you before that by the motion I 
suppose both in the sun, and in the earth, the air 
is dissipated, and consequently that there would 
be an infinite number of small empty places, but 
that the world being full, there comes from the 
next parts other air into the spaces they would 
else make empty. When therefore this motion of 
the sun is exercised upon the superficies of the 
earth, if there do not come out of the earth itself 
some corporal substance to supply that tearing of 
the air, we must return again to the admission of 
vacuum. If there do, then you see how by this 
motion fluid bodies are made to exhale out of the 
earth. The like happens to a man's body or hand, 
which when he perceives, he says he is hot. And 
so of the earth when it seiuleth forth water and 


CHAP. iv. earth together in plants, we say it does it by heat 

rl^T from the sun. 

beat imd light. A. It is very probable, arid no less probable, 
that the same action of the sun is that which from 
the sea and moist places of the earth, but especially 
from the sea, fetcheth up the water into the clouds. 
But there be many ways of heating besides the 
action of the sun or of fire. Two pieces of wood 
will take fire if in turning they be pressed together. 
B. Here again you have a manifest laceration 
of the air by the reciprocal and contrary motions 
of the two pieces of wood, which necessarily 
causeth a coming forth of whatsoever is aereal or 
fluid within them, and (the motion pursued) a dis- 
sipation also of the other more solid parts into 

A. How comes it to pass that a man is warmed 
even to sweating, almost with every extraordinary 
labour of his body ? 

B. It is easy to understand, how by that labour 
all that is liquid in his body is tossed up and down, 
and thereby part of it also cast forth. 

A. There be some things that make a man hot 
without sweat or other evaporation, as caustics, 
nettles, and other things. 

B. No doubt. But they touch the part they so 
heat, and cannot work that effect at any distance. 

A. How does heat cause light, and that partially, 
in some bodies more, in some less, though the heat 
be equal ? 

B. Heat does not cause light at all. But in 
many bodies, the same cause, that is to say, the 
same motion, causeth both together ; so that they 
are not to one another as cause and effect, but are 


concomitant effects sometimes of one and the CHAP. iv. 
same motion. ^^T^T" 

Problems of 
A. HOW? heat and light. 

B. You know the rubbing or hard pressing of 
the eye, or a stroke upon it, makes an apparition 
of light without and before it, which w r ay soever 
you look. This can proceed from nothing else 
but from the restitution of the organ pressed or 
stricken, unto its former ordinary situation of 
parts. Does not the sun by his thrusting back 
the air upon your eyes press them ? Or do not 
those bodies whereon the sun shines, though by 
reflection, do the same, though not so strongly ? 
And do not the organs of sight, the eye, the heart, 
and brains, resist that pressure by an endeavour of 
restitution outwards ? Why then should there not 
be without and before the eye, an apparition of 
light in this case as well as in the other ? 

A. I grant there must. But what is that which 
appears after the pressing of the eye f For there 
is nothing without that was not there before ; 
or if there were, methinks another should see it 
better, or as w r ell as he ; or if in the dark, me- 
thinks it should enlighten the place. 

B. It is a fancy, such as is the appearance of 
your face in a looking-glass ; such as is a dream ; 
such as is a ghost ; such as is a spot before the 
eye that hath stared upon the sun or fire. For all 
these are of the regiment of fancy, without any 
body concealed under them, or behind them, by 
which they are produced. 

A. And when you look towards the sun or 
moon, why is not that also which appears before 
your eyes at that time a fan&y r 


CHAP. IY. B* So it is. Though the sun itself be a real 

Problems o r body, yet that bright circle of about a foot dia- 

fceat and light, meter cannot be the sun, unless there be two suns,, 

a greater and a lesser. And because you may see 

that which you call the sun, both above you in the 

sky, and before you in the water, and two suns, 

by distorting your eye, in two places in the sky, 

one of them must needs be fancy. And if one, 

both. All sense is fancy, though the cause be 

always in a real body. 

A. I see by this that those things which the 
learned call the accidents of bodies, are indeed 
nothing else but diversity of fancy, and are inhe- 
rent in the sentient, and not in the objects, except 
motion and quantity. And I perceive by your doc- 
trine you have been tampering with Leviathan. 
But how comes wood with a certain degree of 
heat to shine, and iron also with a greater degree ; 
but no heat at all to be able to make water shine ? 

B. That which shineth hath the same motion in 
its parts that I have all this while supposed in the 
sun arid earth. In which motion there must needs 
be a competent degree of swiftness to move the 
sense, that is, to make it visible. All bodies that 
are not fluid will shine with heat, if the heat be 
very great. Iron will shine and gold will shine ; 
but water will not, because the parts are carried 
away before they attain to that degree of swiftness 
which is requisite. 

A. There are many fluid bodies whose parts 
evaporate, and yet they make a flame, as oil, and 
wine, and other strong drinks. 

B. As for oil I never saw any inflamed by itself, 
how much soever heated, therefore I do not think 


they are the parts of the oil, but of the combusti- CHAP. i 
ble body oiled that shine ; but the parts of wine "TT^T 

J . Problems ot 

and strong drinks have partly a strong motion of heat a 
themselves, and may be made to shine, but not with 
boiling, but by adding to them as they rise the 
flame of some other body. 

A. How can it be known that the particles of 
wine have such a motion as you suppose ? 

B. Have you ever been so much distempered 
with drinking wine, as to think the windows and 
table move ? 

A. I confess, though you be not my confessor, 
I have ; but very seldom ; and I remember the win- 
dow seemed to go and come in a kind of circling 
motion, such as you have described. But what of 
that ? 

B. Nothing, but that it was the wine that caused 
it ; which having a good ^degree of that motion 
before,, did, when it was heated in the veins, give 
that concussion, which you thought was in the 
window, to the veins themselves, and, by the con- 
tinuation of the parts of man's body, to the brain ; 
and that was it which made the window seem to 

A. What is flame ? For I have often thought 
the flame that comes out of a small heap of straw 
to be more, before it hath done flaming, than a 
hundred times the straw itself. 

B. It was but your fancy. If you take a stick 
in your hand by one end, the other end burning, 
and move it swiftly, the burning end, if the mo- 
tion be circular, shall seem a circle ; if straight, a 
straight line of fire, longer or shorter, according 
to the swiftness of the mctfion, or the space it 
moves in. You know the cause of that. 


CHAP. iv. A. I think it is, because the impression of that 
visible object, which was made at the first instant 

J ' 

ofthe motion, did last till it was ended. For then 
it will follow that it must be visible all the way, 
the impressions in all points of the time being 

B. The cause can be no other. The smallest 
spark of fire flying up seems a line drawn upward ; 
and again by that sw r ift circular motion which we 
have supposed for the cause of light, seems also 
broader than it is. And consequently the flame of 
every thing must needs seem much greater than 
it is. 

A. What are those sparks that fly out of the 

B. They are small pieces of the wood or coals, 
or other fuel loosened and carried away with the 
air that cometh up with them. And being extin- 
guished before their parts be quite dissipated into 
others, are so much soot, and black, and may be 
fired again. 

A. A spark of fire may be stricken out of a cold 
stone. It is not therefore heat that makes this 

B. No it is the motion that makes both the heat 
and shining; arid the stroke makes the motion. 
For every of those sparks, is a little parcel of the 
stone, which swiftly moved, imprinteth the same 
motion into the matter prepared, or fit to re- 
ceive it. 

A. How comes the light of the sun to burn al- 
most any combustible matter by refraction through 
a convex glass, and by reflection from a concave ? 

B. The air moved, by the sun presseth the con- 


vex glass in such manner as the action continued CHAP. iv. 
through it, proceedeth not in the same straight Pr ^^7" 
line by which it proceeded from the sun, but heat ttnd Il 8 llt - 
tendeth more toward the centre of the body it 
enters. Also when the action is continued through 
the convex body, it bendeth again the same way. 
By which means the whole action of the sun -beams 
are enclosed within a very small compass : in 
which place therefore there must be a very vehe- 
ment motion ; and consequently, if there be in that 
place combustible matter, such as is not very hard 
to kindle, the parts of it will be dissipated, and re- 
ceive that motion which worketh on the eye as 
other fire does. 

The same reason is to be given for burning by 
reflection. For there also the beams are collected 
into almost a point. 

A. Why may not the sun-beams be such a body 
as we call fire, and pass through the pores of the 
glass so disposed as to carry them to a point, or 
very near ? 

J3. Can there be a glass that is all pores ? if 
there cannot, then cannot this effect be produced 
by the passing of fire through the pores. You have 
seen men light their tobacco at the sun with a 
burning glass, or with a ball of crystal, held 
which way they will indifferently. Which must be 
impossible, unless the ball were all pores. Again, 
neither you nor I can conceive any other fire than 
we have seen, nor than such as water will put out. 
But not only a solid globe of glass or crystal will 
serve for a burning-glass, but also a hollow one 
filled with water. How then does the fire from the 
sun pass through the glass of water without being 


CHAP. IY. put out before it come to the matter they would 

pr^T have it burn? 

heat and hght A I know not. There comes nothing from the 
sun. If there did, there is come so much from it 
already, that at this day we had had no sun. 



A. WHAT call you hard, and what soft ? 

B. That body whereof no one part is easily put 
out of its place, without removing the whole, is 
that which I and all men call hard ; and the con- 
trary soft. So that they are but degrees one of 

A. What is the cause that makes one body 
harder than another, or, seeing you say they arc 
but degrees of one another, what makes one body 
softer than another, and the same body sometimes 
harder, sometimes softer ? 

B. The same motion which we have supposed 
from the beginning for the cause of so many other 
effects. Which motion not being upon the centre 
of the part moved, but the part itself going in 
another circle to and again, it is not necessary that 
the motion be perfectly circular. For it is not cir 
culation, but the reciprocation, I mean the to and 
again, that does cast off, and lacerate the air, and 
consequently produce the fore-mentioned effects. 

For the cause therefore of hardness, I suppose 
the reciprocation of motion in those things which 
are hard, to be very swift, and in very small 


A. This is somewhat hard to believe. I would CIIAP.V. 
you could supply it with some visible experience, p^^T" 

B. When you see, for example, a cross-bow hai(l <i * ft - 
bentj do you think the parts of it stir ? 

A. No. I am sure they do not. 

B. How are you sure ? You have no argument 
for i^ but that you do not see the motion. When 
I see you sitting still, must I believe there is no 
motion in your parts within, when there are so 
many arguments to convince me there is. 

A. What argument have you to convince me 
that there is motion in a cross-bow when it stands 
bent ? 

B. If you cut the string, or any way set the bow 
at liberty, it will have then a very visible motion. 
What can be the cause of that f 

A. Why the setting of the bow at liberty. 

B. If the bow had been crooked before it was 
bent, and the string tied to both ends, and then 
cut asunder, the bow would not have stirred. 
Where lies the difference ? 

A. The bow bent has a spring; unbent it has 
none, how crooked soever. 

13. What mean you by spring ? 

A. An endeavour of restitution to its former 

B. I understand spring as well as I do endeavour. 

A. I mean a principle or beginning of motion in 
a contrary way to that of the force which bent it. 

B. But the beginning of motion is also motion, 
how insensible soever it be. And you know that 
nothing can give a beginning of motion to itself. 
What is it therefore that gives the bow (which you 
say you are sure was at rest % when it stood bent) 



CHAP. v. its first endeavour to return to its former pos- 

p^T r ~T ture? 

J lohleins ot 

hard ami suit A. It was he that bent it. 

B. That cannot be. For he gave it an endea- 
vour to come forward, and the bow endeavours to 
go backward. 

A. Well, grant that endeavour be motion, and 
motion in the bow unbent, how do you derive from 
thence, that being set at liberty it must return to 
its former posture ? 

B. Thus there being within the bow a swift 
(though invisible) motion of all the parts, and con- 
sequently of the whole ; the bending causeth that 
motion, which was along the bow (that was beaten 
out when it was hot into that length) to operate 
across the length in every part of it, and the more 
by how much it is more bent ; and consequently 
endeavours to unbend it all the while it stands 
bent. And therefore when the force which kept 
it bent is removed, it must of necessity return to 
the posture it had before. 

A. But has that endeavour no effect at all before 
the impediment be removed ? For if endeavour 
be motion, and every motion have some effect more 
or less, methinks this endeavour should in time 
produce something. 

B. So it does. For in time (in a long time) the 
course of this internal motion will lie along the 
bow, not according to the former, but to the new 
acquired posture. And then it well be as uneasy 
to return it to its former posture, as it was before 
to bend it. 

A. That is true. For bows long bent lose their 
appetite to restitution, long custom becoming 


nature. But from this internal reciprocation of the CHAP. v. 
parts, how do you infer the hardness of the whole 1) S T7~ T ~T 

17 J Pinhlfins of 

body ? l'ir<l ami ^oft 

B. If you apply force to any single part of such 
a body, you must needs disorder the motion of the 
next parts to it before it yield, and there disordered, 
the motion of the next again must also be disor- 
dered ; and consequently no one part can yield 
without force sufficient to disorder all : but then 
'the whole body must also yield. Now when a body 
is of such a nature as no single part can be removed 
without removing the whole, men say that body 
is hard. 

A. Why does the fire melt divers hard bodies, 
and yet not all ? 

B. The hardest bodies are those wherein the 
motion of the parts are the most swift, and yet in 
the least circles. Wherefore if the fire, the motion 
of whose parts are swift, and in greater circles, be 
made so swift, as to be strong enough to master the 
motion of the parts of the hard body, it will make 
those parts to move in a greater compass, and 
thereby weaken their resistance, that is to say, soften 
them, which is a degree of liquefaction. And 
when the motion is so weakened, as that the parts 
lose their coherence by the force of their own 
weight, then we count the body melted. 

A. Why are the hardest things the most brittle, 
insomuch that what force soever is enough to bend 
them, is enough also to break them ? 

B. In bending a hard body, as (for example) 
a rod of iron, you do not enlarge the space of the 
internal motion of the parts of iron, as the fire 

* D 3 



CITAP.V. does; but you master and interrupt the motion, 
and that chiefly in one place. In which place the 
motion that makes the iron hard being once over- 
come, the prosecution of that bending must needs 
suddenly master the motions of the parts next unto 
it, being almost mastered before. 

A. I have seen a small piece of glass, the figure 
whereof is this, A A B C. Which piece of glass if 
you bend toward the top, as in C, the whole body 
will shatter asunder into a million of pieces, and 
be like to so much dust. I would fain see you give 
a probable reason of that. 

B. I have seen the experiment. The making of 
the glass is thus : they dip an iron rod into the 
molten glass that stands in a vessel within the 
furnace. Upon which iron rod taken out, there 
will hang a drop of molten but tough metal of the 
figure you have described, which they let fall into 
the water. So that the main drop cornes first to 
the water, and after it the tail, which though 
straight whilst it hung on the end of the rod, yet 
by falling into the water becomes crooked. Now 
you know the making of it, you may consider what 
must be the consequence of it. Because the main 
drop A comes first to the water, it is therefore first 
quenched, and consequently the motion of the 
parts of that drop, which by the fire \vere made to 
be moved in a larger compass, is by the water 
made to shrink into lesser circles towards the other 
end B, but with the same or not much less swiftness. 

A. Why so ? 

B. If you take any long piece of iron, glass, or 
other uniform and continued body; and having 
heated one end thereof, you hold the other end in 


your hand, and so quench it suddenly, though be- OHAP. v. 
fore you held it easily enough, yet now it will * ' ' 

J J ' J Problems ol 

burn your fingers. hard and sort 

A. It will so. 

B. You see then how the motion of the parts 
from A toward C is made more violent and in less 
compass by quenching the other parts first. Be- 
sides, the whole motion that was in all the parts of 
the main drop A, is now united in the small end 
B C. And this I take to be the cause why that 
small part B C is so exceeding stiff. Seeing also 
this motion in every small part of the glass, is not 
only circular, but proceeds also all along the glass 
from A to B, the whole motion compounded will 
be such as the motion of spinning any soft matter 
into thread, and will dispose the whole body of 
the glass in threads, which in other hard bodies 
are called the grain. Therefore if you bend this 
body (for example) in C (which to do will require 
more force than a man would think that has not 
tried) those threads of glass must needs be all bent 
at the same time, and stand so, till by the breaking 
of the glass at C, they be all at once set at liberty ; 
and then all at once being suddenly unbent, like 
ho many brittle and overbent bows, their strings 
breaking, be shivered in pieces. 

A. It is like enough to be so. And if nature 
have betrayed herself in any thing, I think it is in 
this, and in that other experience of the cross- 
bow' ; which strongly and evidently demonstrates 
the internal reciprocation of the motion, which you 
suppose to be in the internal parts of every hard 
body. And I have observed somewhat in looking- 
glasses which much confirm* that there is some 


CHAP.V. such motion in the internal parts of glass, as you 
" " have supposed for the cause of hardness. For let 

Piohlcins ot x A 

hard ami sort the glass be A B, and let the object at C be a 
candle, and the eye at D. Now by divers reflec- 
tions and refractions in the two superficies of the 
glass, if the lines of vision be very oblique, you 
shall see many images of the candle, as E, F, G, 
in such order and position as is here described. 
But if you remove your eye to C, and the candle 
to D, they will appear in a situation manifestly 
different from this. Which you will yet more 
plainly perceive if the looking-glass be coloured, 
as I have observed in red and blue glasses ; and 
could never conceive any probable cause of it, till 
now you tell me of this secret motion of the parts 
across the grain of the glass, acquired by cooling 
it this or that way. 

B. There be very many kinds of hard bodies, 
metals, stones, and other kinds, in the bowels of the 
earth, that have been there ever since the begin- 
ning of the world ; and I believe also many diffe- 
rent sorts of juices that may be made hard. But 
for one general cause of hardness it can be no 
other than such an internal motion of parts as I 
have already described, whatsoever may be the 
cause of the several concomitant qualities of their 
hardness in particular. 

A. We see water hardened every frosty day. 
It is likely therefore you may give a probable cause 
of ice. What is the cause of freezing of the ocean 
towards the poles of the earth ? 

It. You know the sun being always between the 
tropics, and (as we have supposed) always casting 
off the air ; and the earth likewise casting it off 


from itself, there must needs on both sides be a CHAP.V. 
great stream of air towards the poles, shaving the Pl ^j T s () 7" 
superficies of the earth and sea, in the northern h < ml d *>it. 
and southern climates. This shaving of the earth 
and sea by the stream of air must needs contract 
and make to shrink those little circles of the inter- 
nal parts of earth and water, and consequently 
harden them, first at the superficies, into a thin 
skin, which is the first ice ; and afterwards the 
same motion continuing, and the first ice co-ope- 
rating, the ice becomes thicker. And this I con- 
ceive to be the cause of the freezing of the ocean. 

A. If that be the cause, I need not ask how a 
bottle of water is made to freeze in warm weather 
with snow, or ice mingled with salt. For when 
the bottle is in the midst of it, the wind that goeth 
out both of the salt and of the ice as they dissolve, 
must needs shave the superficies of the bottle, and 
the bottle work accordingly on the water without 
it, and so give it first a thin skin, and at last 
thicken it into a solid piece of ice. But how comes 
it to pass that water does not use to freeze in a 
deep pit? 

13. A deep pit is a very thick bottle, and such 
as the air cannot come at but only at the top, or 
where the earth is very loose and spungy. 

A. Why will not wine freeze as well as water? 

7J. So it will when the frost is great enough. 
But the internal motion of the parts of wine and 
other heating liquors is in greater circles and 
stronger than the motion of the parts of water ; 
and therefore less easily to be frozen, especially 
quite through, because those parts that have the 
strongest motion retire to the centre of the vessel. 




CHAP. vi. A. WHAT is the original cause of rain ? And 

' ' how is it generated ? 
lam, wimi, and B. The motion of the air (such as I have de- 

other weather g^J}^ fo y QU already) tending to the dlSUmOn of 

the parts of the air, must needs cause a continual 
endeavour (there being no possibility of vacuum) of 
whatsoever fluid parts there are upon the face of 
the earth and sea, to supply the place which would 
else be empty. This makes the water, and also 
very small and -loose parts of the earth and sea to 
rise, and mingle themselves with the air, and to 
become mist and clouds. Of which the greatest 
quantity arise there, where there is most water, 
namely, from the large parts of the ocean ; which 
are the South Sea, the Indian Sea, and the sea that 
divided Europe and Africa from America ; over 
which the sun for the greatest part of the year is 
perpendicular, and consequently raiseth a greater 
quantity of water ; which afterwards gathered into 
clouds, falls down in rain. 

A. If the sun can thus draw up the water, 
though but in small drops, why can it not as easily 
hold it up ? 

B. It is likely it would also hold them up, if 
they did not grow greater by meeting together, 
nor were carried away by the air towards the poles. 

A. What makes them gather together ? 

B. It is not improbable that they are carried 


against hills, and there stopt till more overtake CHAP. vj. 
them. And when they are carried towards the "~"~ "~7" 

J Problems ot 

North or South where the force of the sun is more win, wind, and 
oblique and thereby weaker, they descend gently ierwea 
by their own weight. And because they tend all 
to the centre of the earth, they must needs be 
united in their way for want of room, and so grow 
bigger. And then it rains. 

A. What is the reason it rains so seldom, but 
snows so often upon very high mountains ? 

B. Because, perhaps, when the water is drawn 
up higher than the highest mountains, where the 
course of the air between the equator and the 
poles is free from stoppage, the stream of the air 
freezeth it into snow. And it is in those places 
only where the hills shelter it from that stream, 
that it falls in rain. 

A. Why is there so little rain in Egypt, and yet 
so much in other parts nearer the equinoxial, as 
to make the Nile overflow 7 the country ? 

B. The cause of the falling of rain I told you 
was the stopping, and consequently the collection 
of clouds about great mountains, especially when 
the sun is near the equinoxial, and thereby draws 
up the water more potently, and from greater 
seas. If you consider therefore that the moun- 
tains in which are the springs of Nile,, lie near the 
equinoxial and are exceedingly great, and near the 
Indian Sea, you will not think it strange there 
should be great store of snow. This as it melts 
makes the rain of Nile to rise, which in April and 
May going on toward Egypt arrive there about 
the time of the solstice, and overflow the country. 

A. Why should not the NMe then overflow that 


CHAP. vi. country twice a year, for it comes twice a-year to 
ij^T^T^ the equinoxial. 

Problems of * 

ram, wmd, and B. From the autumnal equinox, the sun goeth on 
toward the southern tropic, and therefore cannot 
dissolve the snow on that side of the hills that 
looks towards Egypt. 

A. But then there ought to be such another 
inundation southward. 

B. No doubt but there is a greater descent of 
water there in their summer than at other times, 
as there must be wheresoever there is much snow 
melted. But what should that inundate, unless it 
should overflow the sea that comes close to the 
foot of those mountains ? And for the cause why 
it seldom rains in Egypt, it may be this, that there 
are no very high hills near it to collect the clouds. 
The mountains whence Nile riseth being near two 
thousand miles off. The nearest on one side are 
the mountains of Nubia, and on the other side 
Sina and the mountains of Arabia. 

A. Whence think you proceed the winds ? 

B. From the motion, I think, especially of the 
clouds, partly also from whatsoever is moved in 
the air. 

A. It is manifest that the clouds are moved by 
the winds ; so that there were winds before any 
clouds could be moved. Therefore I think you 
make the effect before the cause. 

B. If nothing could move a cloud but wind, 
your objection were good. But you allow a cloud 
to descend by its own w r eight. But when it so 
descends, it must needs move the air before it, 
even to the earth, and the earth again repel it, 
and so make lateral winds every way, which will 


carry forward other clouds if there be any in their CHAP. vi. 
way, but not the cloud that made them. The va- Pr ^^^7" 
pour of the water rising into clouds, must needs r - n wmd ' and 

x other weather. 

also, as they rise, raise a wind. 

A. I grant it. But how can the slow motion of 
a cloud make so swift a wind as it does ? 

B. It is not one or two little clouds, but many 
and great ones that do it. Besides, when the air 
is driven into places already covered, it cannot but 
be much the swifter for the narrowness of the 

A. Why does the south wind more often than 
any other bring rain with it ? 

/>. Where the sun hath most power, and where 
the seas are greatest, that is in the south, there is 
most water in the air ; which a south wind can 
only bring to us. But I have seen great showers 
of rain sometimes also when the wind hath been 
north, but it was in summer, and came first, I 
think, from the south or west, and w r as brought 
back from the north. 

A. I have seen at sea very great waves when 
there was no wind at all. What was it then that 
troubled the water ? 

It. But had you not wind enough presently 
after ? 

A. We had a storm within a little more than a 
quarter of an hour after. 

13. That storm was then coming and had moved 
the w r ater before it. But the wind you could not 
perceive, for it came downwards with the descend- 
ing of the clouds, and pressing the water bounded 
above your sail till it came very near. And that 
was it that made you think th^re was no wind at all. 


CHAP. vi. A. How comes it to pass that a ship should go 
s_-^___ against the wind which moves it, even almost 

Problems of ' 

ram, wma, and point blank^ as if it were not driven but drawn ? 

o ierwea ier. j^ y ou are know first, that what body soever 
is carried against another body,, whether perpendi- 
cularly or obliquely, it drives it in a perpendicular 
to the superficies it lighteth on. As for example, 
a bullet shot against a flat wall, maketh the stone, 
or other matter it hits, to retire in a perpendicular 
to that flat ; or, if the wall be round, towards the 
centre, that is to say, perpendicularly. For if the 
way of the motion be oblique to the wall, the mo- 
tion is compounded of two motions, one parallel 
to the wall, and the other perpendicular. By the 
former whereof the bullet is curried along the wall 
side, by the other it approacheth to it. Now the 
former of these motions can have no effect upon 
it ; all the battery is from the motion perpendicu- 
lar, in which it approacheth, and therefore the 
part it hits must also retire perpendicularly. If it 
were not so, a bullet with the same swiftness would 
execute as much obliquely shot, as perpendicu- 
larly, which you know it does not. 

A. How do you apply this to a ship ? 

B. Let A B be the ship, the head of it A. If the 
\\ind blow just from A towards B, it is true the 
ship cannot go forward howsoever the sail be set. 
Let C D be perpendicular to the ship, and let the 
sail E C be never so little oblique to it, and F C 
perpendicular to E C, and then you see the ship 
will gain the space 1) F to the headward. 

A. It will so ; but when it is very near to the 
wind it will go forward very slowly, and make 
more way with her bide to the leeward. 


J3. It will indeed go slower in the proportion of CHAP. vi. 
the line A E to the line C E. But the ship will not f ' 

. L Problems of 

go so fast as you think sideward : one cause is the iam,wmri,and 

force of that wind which lights on the side of the tierwedl er 

ship itself ; the other is the bellying of the sail ; 

for the former, it is not much, because the ship 

does not easily put from her the water with her 

side ; and bellying of the sail gives some little hold 

for the wind to drive the ship astern. 

A. For the motion sideward I agree with you; 
but I had thought the bellying of the sail had 
made the ship go faster. 

B. But it does not ; only in a fore wind it hin- 
ders least. 

A. By this reason a broad thin board should 
make the best sail. 

13. You may easily foresee the great iucommo- 
dities of such a sail. But I have seen tried in 
little wiiat such a wind can do in such a case. For 
I have seen a board set upon four truckles, with a 
staff set up in the midst of it for a mast, and 
another very thin and broad board fastened to 
that staff in the stead of a sail, and so set as to 
receive the wind very obliquely, I mean so as 
to be within a point of the compass directly oppo- 
site to it, and so placed upon a reasonable smooth 
pavement where the wind blew somewhat strongly. 
The event was first, that it stood doubting whether 
it should stir at all or no, but that was not long, 
and then it ran a-head extreme swiftly, till it was 
overthrown by a rub. 

A. Before you leave the ship, tell me how it 
comes about that so small a thing as a rudder can 
so easily turn the greatest ship. 


CHAP. vj. B. It is not the rudder only, there must also be 
, "7~ r ~~ ' a stream to do it ; you shall never turn a ship 

Problems of J . 

ram, wmd, and with a rudder in a standing pool, nor in a natural 
current, You must make a stream from head to 
stern, either with oars or with sails ; when you 
have made such a stream, the turning of the rud- 
der obliquely holds the water from passing freely, 
and the ship or boat cannot go on directly, but as 
the rudder inclines to the stern, so will the ship 
turn ; but this is too well known to insist upon. 
You have observed that the rudders of the greatest 
ships are not very broad, but go deep into the 
water, whereas western barges, though but small 
vessels, have their rudders much broader, which 
argues that the holding of water from passing is 
the true office of a rudder ; and therefore to a ship 
that draws much water the rudder is made deep 
accordingly ; and in barges that draw little water, 
the rudders being less deep, must so much the more 
be extended in breadth. 

A. What makes snow ? 

B. The same cause which, speaking of hardness, 
I supposed for the cause of ice. For the stream 
of air proceeding from that both the earth and 
the sun cast off the air, consequently maketh a 
stream of air from the equinoxial towards the poles, 
passing amongst the clouds, shaving those small 
drops of water whereof the clouds consist, and 
congeals them as they do the water of the sea, or 
of a river. And these small frozen drops are that 
which we call snow. 

A. But then how are great drops frozen into 
hailstones, and that especially (as we see they are) 
in summer ? 


B. It is especially in summer, and hot weather, CHAP. vi. 
that the drops of water which make the clouds. 1J v 7r" v ~T' 

I ' Problems ol 

are great enough ; but it is then also that clouds ri "> wmd > aud 

i 1 sc 11 J A j other weather 

are sooner and more plentifully carried up. And 
therefore the current of the air strengthened be- 
tween the earth and the clouds, becomes more 
swift ; and thereby freezeth the drops of water, 
not in the cloud itself, but as they are falling. Nor 
does it freeze them thoroughly, the time of their 
falling not permitting it, but gives them only a thin 
coat of ice ; as is manifest by their sudden dis- 

A. Why are not sometimes also whole clouds 
when pregnant and ready to drop, frozen into one 
piece of ice ? 

B. I believe they are so whensoever it thunders. 

A. But upon what ground do you believe it ? 

B. From the manner or kind of noise they make, 
namely a crack ; which I see not how it can possi- 
bly be made by water or any other soft bodies 

A. Yes, the powder they call aurum fulminans, 
when thoroughly warm, gives just such another 
crack as thunder. 

B. But why may not every small grain of that 
aurum fulminans by itself be heard, though a heap 
of them together be soft, as is any heap of sand. 
Salts of all sorts are of the nature of ice. But 
gold is dissolved into aurum fulminans by nitre 
and other salts. And the least grain of it gives a 
little crack in the fire by itself. And therefore 
when they are so warmed by degrees, the crack 
cannot choose but be very great. 


CHAP. vi. A. But before it be aurum fulmmans they use to 
wash away the salt (which they call dulcifying it), 

ram, wind, and and then they dry it gently by degrees. 

other weather T1 __. . , 111 i 

B. That is, they exhale the pure water that is 
left in the powder, and leave the salt behind to 
harden with drying. Other powder made of salts 
without any gold in them will give a crack as great 
as aurum fulmmans. A very great chemist of our 
times hath written, that salt of tartar, saltpetre, 
and a little brimstone ground together into a pow- 
der, and dried, a few grains of that powder will be 
made by the fire to give as great a clap as a 

A. Methinks it were worth your trial to see 
what effect a quart or a pint of aurum fulmmans 
would produce, being put into a great gun made 
strong enough on purpose, and the breech of the 
gun set in hot cinders, so as to heat by degrees, 
till the powder fly. 

B. I pray you try it yourself; I cannot spare so 
much money. 

A. What is it that breaketh the clouds when 
they are frozen ? 

B. In very hot weather the sun raiscth from the 
sea and all moist places abundance of water, and 
to a great height. And whilst this water hangs over 
us in clouds, or is again descending, it raiseth other 
clouds, and it happens very often that they press 
the air between them, and squeeze it through the 
clouds themselves very violently ; which as it 
passes shaves and hardens them in the manner 

A . That has already been granted ; my question 
is what breaks them : 


B. I must here take in one supposition more. CHAP. vi. 
A. Then your basin, it seems, holds not all you Pl ^^7" 
have need of. ram wmd.ana 

r*r * n i p i Til ^ ier weather. 

Jo. It may for all this, tor the supposition 1 add 
is no more but this ; that what internal motion I 
ascribe to the earth, and the other concrete parts 
of the world, is to be supposed also in every, of 
their parts how small soever ; for what reason is 
there to think, in case the whole earth have in 
truth the motion I have ascribed to it, that one 
part of it taken away, the remaining part should 
lose that motion. If you break a loadstone, both 
parts will retain their virtue, though weakened 
according to the diminution of their quantity ; I 
suppose therefore in every small part of the earth 
the same kind of motion, which I have supposed 
in the whole : and so I recede not yet from my 

A. Let it be supposed, and withal, that abun- 
dance of earth, (which I see you aim at), be drawn 
up together with the water. What then ? 

li. Then if many pregnant clouds, some ascend- 
ing and some descending meet together, and make 
concavities between, and by the pressing out of the 
air, as I have said before, become ice; those atoms, 
as I may call them, of earth will, by the straining 
of the air through the water of the clouds, be left 
behind, and remain in the cavities of the clouds, 
and be more in number than for the proportion of 
the air therein. Therefore for want of liberty they 
must needs justle one another, and become, as they 
are more and more straightened of room, more and 
more swift, and consequently at last break the ice 
suddenly and violently, now in one place, and by 
VOL. vit. E 


CHAP. vi. and by in another ; and make thereby so many 
c ^ a P s f thunder, and so many flashes of lightning. 

, wind, and For the air recoiling upon our eyes, is that which 

o icrwea ler. f-} lose fl as } ie s to OUr 

A. But I have seen lightning in a very clear 
evening, when there has been neither thunder nor 

B. Yes, in a clear evening ; because the clouds 
and the rain were below the horizon, perhaps 
forty or fifty miles off ; so that you could not see 
the clouds nor hear the thunder. 

A. If the clouds be indeed frozen into ice, I 
shall not wonder if they be sometimes also so 
situated, as, like looking-glasses, to make us see 
sometimes three or more suns by refraction and 





A. IF a bullet from a certain point given, be shot 
Against a wall perpendicularly, and again from the 
same point obliquely, what will be the proportion of 
the forces wherewith they urge the wall ? For 
example, let the wall be A B, a point given E, a 
gun C E, that carries the bullet perpendicularly to 
F, and another gun D E, that carries the like bul- 
let with the same swiftness obliquely to G ; in what 
proportion will their forces be upon the wall ? 

B. The force of the stroke perpendicular from E 
to F will be greater then the oblique force from E 


to G, in the proportion of the line E G to the line CHAP vr 
E F. ^~ f ^ 

Problems of nm- 

A. How can the difference be so much ? Can tum porppiuiicn 
the bullet lose so much of its force in the way from o"p" Uon" ami 

F to f^ 2 perm sicm , ic- 

ilettu i and ro- 

B. No; we will suppose it loseth nothing of its lractl(11 ' attrw " 

11 ( ^ tion and irpul- 

swiftness. But the cause is, that their swiftness sum 
being equal, the one is longer in coming to the 
wall than the other, in proportion of time, as E G 
'to E F. For though their swiftness be the same, 
considered in themselves, yet the swiftness of their 
approach to the wall is greater in E F than in E G, 
in proportion of the lines themselves. 

A. When a bullet enters not, but rebounds from 
the wall, does it make the same angle going off, 
which it did falling on, as the sun-beams do ? 

B. If you measure the angles close by the wall 
their difference will not be sensible ; otherwise it 
will be great enough, for the motion of the bullet 
grows continually weaker. But it is not so with 
the sun-beams which press continually and equally. 

A. What is the cause of reflection? When a 
body can go no further on, it has lost its motion. 
Whence then comes the motion by which it re- 
boundeth ? 

B. This motion of rebounding or reflecting pro- 
ceedeth from the resistance. There is a difference 
to be considered between the reflection of light, 
and of a bullet, answerable to their different mo- 
tions, pressing and striking. For the action which 
makes reflection of light, is the pressure of the air 
upon the reflecting body, caused by the sun, or 
other shining body, and is but a contrary endea- 
vour ; as if two men should press with their breasts 

E 2 


CHAP.VU. upon the two ends of a staff, though they did not 
i^MbbiiiT^io remove one another, yet they would find in them- 
selves a e:reat disposition to press backward upon 


lar, oblique, &c . . . , . , , , , , 

whatsoever is behind them, though not a total 
going out of their places. Such is the way of 
reflecting light. Now, when the falling on of the 
sun-beams is oblique, the action of them is never- 
theless perpendicular to the superficies it falls on. 
And therefore the reflecting body, by resisting, 
turneth back that motion perpendicularly, as from" 
F to E; but taketh nothing from the force that goes 
on parallel in the line of E H, because the motion 
never presses. And thus of the two motions from 
F to E, and from E to H, is a compounded motion 
in the line F H, which makcth an angle in 15 G, 
equal to the angle F G E. 

But in percussion (which is the motion of the 
bullet against a wall,) the bullet no sooner gocth 
off than it loseth of its swiftness, and inclineth to 
the earth by its weight. So that the angles made 
in falling on and going off, cannot be equal, unless 
they be measured close to the point \\hcrc the 
stroke is made. 

A. If a man set a board upright upon its edge, 
though it may very easily be cast down with a little 
pressure of one's finger, yet a bullet from a mus- 
ket shall not throw it down, but go through it. 
What is the cause of that ? 

B. In pressing with your finger you ypend time 
to throw it down. For the motion you give to the 
part you touch is communicated to every other 
part before it fall. For the whole cannot fall till 
every part be moved. But the stroke of a bullet 
is so swift, as it breaks through, before the motion 


of the part it hits can be communicated to all the CHAP.VII. 
other parts that must fall with it. '7. ' :' 

i 1 roblcms ot 1110- 

A. The stroke of a hammer will drive a nail a tum P ei r endicu 

1 lar, oblique, &c 

great way into a piece of wood on a sudden. What 
weight laid upon the head of a nail, and in how 
much time will do the same ? It is a question I 
have heard propounded amongst naturalists. 

13. The different manner of the operation of 
weight from the operation of a stroke, makes it 
incalculable. The suddenness of the stroke upon 
one point of the wood takes away the time of re- 
sistance from the rest. Therefore the nail enters 
so far as it does. But the weight not only gives 
them time, but also augments the resistance ; but 
how much, and in how 7 much time, is, I think, 
impossible to determine. 

A. What is the difference between reflection and 
recoiling ? 

li. Any reflection may, and not improperly, be 
called recoiling; but not contrariwise every recoil- 
ing reflection. Reflection is always made by the 
reaction of a body pressed or stricken ; but recoiling 
not always. The recoiling of a gun is not caused 
by its own pressing upon the gunpowder, but by 
the force of the powder itself, inflamed and moved 
every way alike. 

A. I had thought it had been by the sudden re- 
entering of the air after the flame and bullet were 
gone out. For it is impossible that so much room 
as is left empty by the discharging of the gun, 
should be so suddenly filled with the air that en- 
tereth at the touchhole. 

13. The flame is nothing but the powder itself, 
which scattered into its smMlest parts, seems of 

54 PHILOSOPHICAL PROBLEMS. greater bulk by much, than in truth it is, because 
^TT^T" they shine. And as the parts scatter more and 

rrobloms of mo- J 1 

tion pcrpciiaicn- more, so still more air gets between them, entering: 

lar, oblique, &c. i i i i i i f 

not only at the touchhole, but also at the mouth 
of the gun, which two ways being opposite, it will 
be much too weak to make the gun recoil. 

A. I have heard that a great gun charged too 
much or tool ittle, will shoot, not above, nor below, 
but beside the mark ; and charged with one cer- 
tain charge between both, will hit it. 

13. How that should be I cannot imagine. For 
when all things in the cause are equal, the effects 
cannot be unequal. As soon as fire is given, and 
before the bullet be out, the gun begins to recoil. 
If then there be any unevenness or rub in the 
ground more on one side than on the other, it shall 
shoot beside the mark, whether too much, or too 
little, or justly charged ; because if the line wherein 
the gun recoileth decline, the way of the bullet 
will also decline to the contrary side of the mark. 
Therefore I can imagine no cause of this event, 
but either in the ground it recoils on, or in the 
unequal weight of the parts of the breech. 
. A. How comes refraction ? 

Ji. When the action is in a line perpendicular 
to the superficies of the body wrought upon, there 
will be no refraction at all. The action w 7 ill pro- 
ceed still in the same straight line, whether it 
be pression as in light, or percussion as in the 
shooting of a bullet. But when the pression is 
oblique, then will the refraction be that way which 
the nature of the bodies through which the action 
proceeds shall determine. 

A. How is light refracted r 


B. If it pass through a body of less., into a body CHAP. VH. 

of greater resistance, and to the point of the \ T ;' 

3 x > Pioidoms of mo- 

superficies it falleth on, you draw a line perpen- tiou pi-rpenduu 

dicular to the same superficies, the action will 
proceed not in the same line by which it fell on, 
but in another line bending toward that perpen- 

A. What is the reason of that ? 

It. I told you before, that the falling on worketh 
only in the perpendicular ; but as soon as the 
action proceedeth farther inward than a mere 
touch, it worketh partly in the perpendicular, and 
partly forward, arid would proceed in the same 
line in which it fell on, but for the greater resist- 
ance which now weakeneth the motion forward, 
and makes it to incline towards the perpendicular. 

A. In transparent bodies it may be so ; but there 
be bodies through which the light cannot pass 
at all. 

B. But the action by which light is made, passeth 
through all bodies. For this action is pressiou ; 
and whatsoever is pressed, presseth that which is 
next behind, and so continually. But the cause 
why there is no light seen through it, is the uri- 
evenness of the parts within, whereby the action is 
by an infinite number of reflections so diverted 
and weakened, that before it hath proceeded 
through, it hath not strength left to work upon 
the eye strongly enough to produce sight. 

A. If the body being transparent, the action 
proceed quite through, into a body again of less 
resistance, as out of glass into the air, which way 
shall it then proceed in the air ? 

B. From the point where ft goeth forth, draw a 
perpendicular to the superficies of the glass, the 


CHAP. vii. action now freed from the resistance it suffered, 
i> "TT^ w iU 8 from that perpendicular, as much as it did 

Problems oi mo- L \ y 

t.on pnpcmiicu- before come towards it. 

iue, c. ^ When a bullet from out of the air entereth 
into a wall of earth, will that also be refracted 
towards the perpendicular, 

B. If the earth be all of one kind, it will. For 
the parallel motion, will there also at the first 
entrance be resisted, which it was not before it 

A. How then comes a bullet, when shot very 
obliquely into any broad water, and having entered, 
yet to rise again into the air ? 

H. When a bullet is shot very obliquely, though 
the motion be never so swift, yet the approach down- 
wards to the water is very slow, and when it com- 
eth to it, it casteth up much water before it, which 
with its weight presseth downwards again, and 
maketh the water to rise under the bullet with 
force enough to master the weak motion of the 
bullet downwards, and to make it rise in such man- 
ner as bodies use to rise by reflection. 

A. By what motion (seeing you ascribe all effects 
to motion) can a loadstone draw iron to it ? 

B. By the same motion hitherto supposed. But 
though all the smallest parts of the earth have 
this motion, yet it is not supposed that their motions 
are in equal circles; nor that they keep just time 
with one another ; nor that they have all the 
same poles. If they had,, all bodies would draw 
one another alike. For such an agreement of 
motion, of way, of swiftness, and of poles, cannot be 
maintained, without the conjunction of the bodies 
themselves in the cetitre of their common motion, 
but by violence. If therefore the iron have but so 


much of the nature of the loadstone as readily to CHAP.VII. 
receive from it the like motion, as one string of a "~~~ """""7" 

^ Problems ot mo- 

lute doth from another string strained to the same in perpradu-u. 
note, (as it is like enough it hath, the loadstone ar ' Ml " L ' 
being but one kind of iron ore),, it must needs after 
that motion received from it, unless the greatness 
of the weight hinder, come nearer to it, because 
at distance their motions will differ in time, and 
oppose each other, whereby they will be forced to 
a common centre. If the iron be lifted up from 
the earth, the motion of the loadstone must be 
stronger, or the body of it nearer, to overcome the 
weight ; and then the iron will leap up to the load- 
stone as swiftly, as from the same distance it would 
fall down to the earth ; but if both the stone and 
the iron be set floating upon the water, the attrac- 
tion will begin to be manifest at a greater distance, 
because the hindrance of the weight i in part 

A. But why does the loadstone, if it float on a 
calm water, never fail to place itself at last in the 
meridiaii just north and south. 

Ji. Not so, just in the meridian, but almost in 
all places with some variations. But the cause. I 
think is, that the axis of this magnetical motion is 
parallel to the axis of the ecliptic, which is the 
axis of the like motion in the earth, and conse- 
quently that it cannot freely exercise its natural 
motion in any other situation. 

A. Whence may this consent of motion in the 
loadstone and the earth proceed ? Do you think, as 
some have written, that the earth is a great loadstone? 

If. Dr. Gilbert, that was the first that wrote any- 
thing of this subject ratioAally, inclines to that 
opinion. Descartes thought the earth, excepting: 


CHAP. vii. this upper crust of a few miles depth, to be of the 
]>r^L^7^ . same nature with all other stars, and bright. For 
tum perpcndicu- my part, I am content to be ignorant ; but I believe 

LIT, oblique, &c J i V IT . 

the loadstone hath been given its virtue by a long 
habitude in the mine, the vein of it lying in the plane 
of some of the meridians, or rather of some of the 
great circles that pass through the poles of the 
ecliptic, which are the same with the poles of the 
like motion supposed in the earth. 

A. If that be true, I need not ask why the filings 
of iron laid on a loadstone equally distant from its 
poles will lie parallel to the axis, but on each side 
will incline to the pole that is next. Nor why by 
drawing a loadstone all along a needle of iron, the 
needle will receive the same poles. Nor why when 
the loadstone and iron, or two loadstones, are put 
together floating upon water, will fall one of them 
astern of the other, that their like parts may look 
the same way, and their unlike touch, in which 
action they are commonly said to repel one another. 
For all this may be derived from the union of their 
motions. One thing more I desire to know, and 
that is ; what are those things they call spirits ? 
I mean ghosts, fairies, hobgoblins, and the like 

tt. They are no part of the subject of natural 

A . That which in all ages, and all places is com- 
monly seen (as those have been, unless a great part 
of mankind be liars) cannot, I think, be super- 

B. All this that I have hitherto said, though 
upon better ground than can be had for a discourse 
of ghosts, you ought fco take but for a dream. 

A. I do so. But there be some dreams more 


like sense then others. And that which is like CHAP.VII. 
sense pleases me as well in natural philosophy, as ri ^ n 7rfmo- 
if it were the very truth. tu>u ppcndicu. 

. _ . lai, oblique, &c 

B. I was dreaming also once of these things ; 
but was wakened by their noise. And they never 
came into any dream of mine since, unless appari- 
tions in dreams and ghost be all one. 



A. HAVE you seen a printed paper sent from Paris, 
containing the duplication of the cube, written in 
French ? 

II. Yes. It was I that writ it, and sent it thither 
to be printed, on purpose to see what objections 
would be made to it by our professors of algebra 

A. Then you have also seen the confutations of 
it by algebra. 

B. I have seen some of them ; and have one by 
me. For there was but one that was rightly cal- 
culated, and that is it which I have kept. 

A . Your demonstration then is confuted though 
but by one. 

B. That does not follow. For though an arith- 
metical calculation be true in numbers, yet the same 
may be, or rather must be false, if the units be not 
constantly the same. 

A. Is their calculation so inconstant, or rather 
so foolish as you make it ? 

li. Yes. For the same number is sometimes so 


CHAP. viii. times so many solids ; as you shall plainly see, if 
r^TT^T" YOU will take the pains to examine first a deinon- 

Tlio Delphi c J JL 

piobinn, in stration I have to prove the said duplication, and 
oMVe'VuL after that, the algebraic calculation which is pre- 
tended to confute it. And not only that this one 
is false, but also any other arithmetical account 
used in geometry, unless the numbers be always so 
many lines., or always so many superficies, or al- 
ways so many solids. 

A. Let me see the geometrical demonstration. 

B. There it is. Read it. 


Let the side of the cube given be V 1). Produce 
V D to A, till A 1) be double to D V. Then make 
the square of A I), namely A B C I). Divide A B 
and C D in the middle at E and F. Draw E F. 
Draw also A C cutting E F in I. Then in the sides 
1) C and A D take B 11 and A S, each of them equal 
to A I or I C. 

Lastly, dnide S D in the middle at T, and upon 
the centre T, with the distance T V, describe a 
semi-circle cutting A D in Y, and 1) C in X. 

I say the cube of D X is double to the cube of 
D V. For the three lines D Y, D X 5 D V are in 
continual proportion. And continuing the semi- 
cirele V X Y till it cut the line R S, drawn and 
produced in Z 5 the line S Z will be equal to D X. 
And drawing X Z it will pass through T. And the 
four lines T V, T X, T Y and T Z will be equal. 
And therefore joining Y X arid Y Z, the figure 
V X Y Z will be a rectangle. 

Produce C D to P so as 1) P be equal to A D. 
Now if Y Z produced* fall on P, there will be three 
rectangle equiangled triangles, D P Y, D Y X, and 


D X V ; and consequently four continual proportio- CHAP. vm. 
nals, D P, D Y, D X, and D V, whereof 1) X is the T 
least of the means. And therefore the cube of p 


D X will be double to the cube of D V. ot the cube. 

A . That is true ; and the cube of D Y will be 
double to the cube of D X ; and the cube of D P 
double to the cube of D Y. But that Y Z produced, 
falls upon P, is the thing they deny, and which you 
ought to demonstrate. 

13. If Y Z produced fall not on P, then draw 
P Y, and from V let fall a perpendicular upon P Y, 
suppose at u. Divide P V in the midst at a, and 
join a u ; which done au will be equal to a V or 
a P. For because V u P is a right angle, the point 
u will be in the semi-circle whereof P V is the dia- 

Therefore drawing V 11, the angle u V R will be 
a right angle. 

A. Why so ? 

Ji. Because T V and T Y are equal ; and T D, 
T S equal ; S Y will also be equal to D V. And 
because D P and li S are equal and parallel, R Y 
will be equal and parallel to P V. And therefore 
V R and P Y that join them will be equal and pa- 
rallel. And the angles P u V, R V u will be alter- 
nate, and consequently equal. But P u V is a right 
angle ; therefore also R V u will be a right angle. 

A. Hitherto all is evident. Proceed. 

B. From the point Y raise a perpendicular cut- 
ting V R wheresoever in t, and then (because P Y 
and V R are parallel) the angle Y t V will be a 
right angle. And the figure u Y t V a rectangle, 
and u t equal to Y V. But Y V is equal to Z X ; 
and therefore Z X is equal Jto u t. And u t must 
pass through the point T (for the diameters of any 


CHAP. vin. rectangle divide each other in the middle), there- 
^TTTiT" fre Z and u are the same point, and X and / the 

The Delphic 

problem, or same point. Therefore Y Z produced falls upon P. 
And D X is the lesser of the two means between 
A D and D V. And the cube of D X double to the 
cube of D V, which was to be demonstated. 

A. I cannot imagine w T hat fault there can be in 
this demonstration, and yet there is one thing 
which seems a little strange to me. And it is this. 
You take B R, which is half the diagonal, and 
which is the sine of forty-five degrees, and which 
is also the mean proportional between the two ex- 
tremes ; and yet you bring none of these proprie- 
ties into your demonstration. So that though you 
argue from the construction, yet you do not argue 
from the cause. And this perhaps your adversaries 
will object, at least, against the art of your demon- 
stration,, or enquire by what luck you pitched upon 
half the diagonal for your foundation. 

B. I see you let nothing pass. But for answer 
you must know, that if a man argue from the nega- 
tive of the truth, though he know not that it is the 
truth which is denied, yet he will fall at last, after 
many consequences, into one absurdity or another. 
F6r though false do often produce truth, yet it pro- 
duces also absurdity, as it hath done here. But 
truth produceth nothing but truth. Therefore in 
demonstrations that tend to absurdity, it is no good 
logic to require all along the operation of the 

A. Have you drawn from hence no corollaries : 

B. No. I leave that for others that will ; unless 
you take this for a corollary, that, what arithmeti- 
cal calculation soever ^contradicts it, is false. 


A. Let me see now the algebraical demonstration CHAP. VITI. 
against it. ^rT?77" 

The Delphic 

B. Here it is : proi>iem, <>r 

T , A T- A T-\ 1 T , ^ duplication 

Let A B or A D be equal to ... 2 O r the cube 

Then 1) F or D V is equal to ... 1 
And B R or A S is equal to the square root of 2 

And D Y equal to 3 

want the square root of . . . . 2 

The cube of A B is equal to ... 8 
The cube of D Y is equal to . . .45 
want the square root of 1682 that is 
almost equal to ..... 4 

For 45 want the square root of 1681 is 

equal to . . . . . . . 4 

Therefore D Y is a little less then the greater of 
the two means between A D arid D V. 

A. There is I see some little difference between 
this arithmetical and your geometrical demonstra- 
tion. And though it be insensible, yet if his cal- 
culation be true, yours must needs be false, which 
I am sure cannot be. 

B. His calculation is so true, that there is never 
a proposition in it false, till he come to the conclu- 
sion, that the cube of D Y is equal to 45, want the 
square root of 1682. But that, and the rest, 'is 

A. I shall easily see that AD is certainly 2, 
whereof D V is 1 , and A V is certainly 3, whereof 
DVis 1. 

B. Right. 

A. And B R is without doubt the square root 
of 2. 

B. Why, what is 2 ? 

A. 2 is the line AD as being double to D V 
which is 1. 


CHAP. vin. B. And so, the line B R is the square root of the 
^TTTT l ine A D. 

The Delphic 

problem, or A. Out upon it, it is absurd. Why do you grant 

duplication . . i . * , > 

of the cube, it to be true in arithmetic ? 

B. In arithmetic the numbers consist of so many 
units, and are never considered there as nothings. 
And therefore every one line has some latitude, 
and if you allow to B I, the semi-diagonal,, the same 
latitude you do to A B, or to B R, you will quickly 
see the square of half the diagonal to be equal to 
twice the square of half A B. 

A. Well, but then your demonstration is not 
confuted ; for the point Y will have latitude enough 
to take in that little difference which is between the 
root of 1681 and the root of 1682. This putting 
off an unit sometimes for one line, sometimes for 
one square, must needs mar the reckoning. Again 
he says, the cube of A B is equal to 8 ; but seeing 
A B is 2, the cube of A B must be just equal to 
four of its own sides ; so that the unit which was 
before sometimes a line, sometimes a square, is 
now a cube. 

B. It can be no otherwise when you so apply 
arithmetic to geometry, as to number the lines of 
a plane, or the planes of a cube. 

A. In the next place, I find that the cube of 
D Y is equal to 45, want the square root of 1682. 
What is that 45 ? Lines, or squares, or cubes ? 

B. Cubes ; cubes of D V. 

A. Then if you add to 45 cubes of D Vthe square 
root of 1682, the sum will be 45 cubes of D V ; and 
if you add to the cube of D Y the same root of ] 682, 
the sum will be the cube of D Y, plus the square 
root of 1682, and thgse two sums must be equal. 


B. They must so. CHAP. vm. 

A. But the square root of 1682, being a line. .^TTT" 

1 y y Iho Delphic 

adds nothing to a cube ; therefore the cube alone problem, or 
of D Y, which he says is equal almost to 4 cubes of Sw cubo. 
of D V, is equal to 45 cubes of the same D V. 

B. All these impossibilities do necessarily follow 
the confounding of arithmetic and geometry. 

A. I pray you let me see the operation by which 
the cube of D Y (that is, the cube of 3, want the 
root of 2) is found equal to 45, want the square 
root of 1682. 

B. Here it is. 


3 V 2 

3 V2 

^18 +2 
9 Vl8 

9 ^72 +2 
3 V2 

V 162 + 12 >/ 8 

27 N/ 648 + 6 

27 V 658 V 1 62 + 1 8 V 8 

.4 . Why for two roots of 1 8 do you put the root 
of 72. 

B. Because 2 roots of 18 are equal to one root 
of four times 18, which is 72 

A. Next we have, that the root of 2 multiplied 
into 2 makes the root of 8. How is that true ? 

B. Does it not make 2 roots of two ? And is 
not B R the root of 2, and 2 1$ R equal to the dia- 



CHAP. viii. goiial ? And is not the square of the diagonal equal 
'rPn77" to 8 squares of 1) V ? 

The Del pine L 

problem, or A. It is true. But here the root of 8 is put for 
of the cube the cube of the root of 2. Can a line be equal to 
a cube ? 

/?. No. But here we arc. in arithmetic again, 
and 8 is a cubic number. 

A. How does the root of 2 multiplied into the 
root of 72 make 12 ? 

13. Because it makes the root of 2 times 72, that 
is to say the root of 144 which is 12. 

A. How does 9 roots of 2 make the root of 162 ? 

B. Because it makes the root of 2 squares of 9, 
that is the root of 162. 

A. How does 3 roots of 72 make the root of 

/?. Because it makes the root of 9 times 72, that 
is of 648. 

A. For the total sum I see 27 and 18, which 
make 45. Therefore the root of 648 together with 
the root of 162 and of 8, which are to be deducted 
from 45, ought to be equal to the root of 1682. 

B. So they are. For 648 multiplied by 162 
makes 1 04976, of which the double root is 648 
and 648 and 162 added together make . . 810 
Therefore the root of 648, added to the root 

of 162, makes the root of ... 1458 
Again 1458 into 8 is 11664. The double 

root whereof is . . . . .216 
The sum of 1458 and 8 added together . 1466 
The sum of 1466 and 216 is 1682, and the 

root, the root of 1682 

A. I see the calculation in numbers is right, 
though false in lines* The reason whereof can be 
no other than some difference between multiplying 


numbers into lines or planes, and multipying lines CHAP. vm. 
into the same lines or planes. " " ' 

x The Del phi i; 

B. The difference is manifest, ror when you proiin,or 
multiply a number into lines, the product is lines ; oiTho cube. 
as the number 2 multiplied into 3 lines is no more 
than 3 lines 2 times told. But if you multiply 
lines into lines you make planes, and if you multi- 
ply lines into planes you make solid bodies. In 
geometry there are but three dimensions, lengths, 
superficies, and body. In arithmetic there is but 
one, and that is number or length which you will. 
And though there be some numbers called plane, 
other solids, others piano-solid, others square, 
others cubic, others square-square, others quadrato- 
cubic, others cubi-cubic, &c., yet are all these but 
one dimension, namely number, or a file of things 

A. But seeing this way of calculation by num- 
bers is so apparently false, what is the reason this 
calculation came so near the truth ? 

/}. It is because in arithmetic units are not 
nothings, and therefore have breadth. And there- 
fore many lines set together make a superficies 
though their breadth be insensible. And the 
greater the number is into which you divide your 
line, the less sensible will be your error. 

A. Archimedes, to find a straight line equal to 
the circumference of a circle, used this way of 
extracting roots. And it is the way also by which 
the table of sines, secants, and tangents have been 
calculated. Are they all out ? 

Ii. As for Archimedes, there is no man that does 
more admire him than I do : but there is no man 
that cannot err. His reasoning is good. But he, 

F 2 


CHAP. viu. as all other geometricians before and after him, 
TJ^Dd^ have had two principles that cross one another 
problem, or when they are applied to one and the same science. 

duplication ... 

of the cube. One is, that a point is no part of a line., which is 
true in geometry, where a part of a line when it is 
called a point, is not reckoned ; another is, that a 
unit is part of a number ; which is also true ; but 
when they reckon by arithmetic in geometry, there 
a unit is sometimes part of a line, sometimes a part 
of a square, and sometimes part of a cube. As for 
the table of sines, secants, and tangents, I am not 
the first that find fault with them. Yet I deny not 
but they are true enough for the reckoning of 
acres in a map of land. 

A. What a deal of labour has been lost by them 
that being professors of geometry have read nothing 
else to their auditors but such stuif as this you 
have here seen. And some of them have written 
great books of it in strange characters, such as in 
troublesome times, a man would suspect to be a 

B. I think you have seen enough to satisfy you, 
that what I have written heretofore concerning the 
quadrature of the circle, and of other figures made 
in imitation of the parabola, has not been yet 

A. I see you have wrested out of the hands of 
our antagonists this weapon of algebra, so as they 
can never make use of it again. Which I consider 
as a thing of much more consequence to the science 
of geometry, than either of the duplication of the 
cube, or the finding of two mean proportionals, or 
the quadrature of a circle, or all these problems 
put together. 













A. I HAVE heard exceeding highly commended CHAP. r. 
a kind of thing which I do not well understand, ouiT^I 
though it be much talked of, by such as have not t o1 nd [ ural P ]U 

J losoph) 

otherwise much to do, by the name of philosophy ; 
and the same again by others as much despised 
and derided : so that I cannot tell whether it be 
good or ill, nor what to make of it, though I see 
many other men that thrive by it. 

Ii. I doubt not, but what so many do so highly 
praise must be very admirable, and what is derided 
and scorned by many, foolish and ridiculous. The 
honour and scorn falleth finally not upon philo- 
sophy, but upon the professors. Philosophy is the 
knowledge of natural causes. And there is no 
knowledge but of truth. And to know the true 
causes of things, was never in contempt, but in ad- 
miration. Scorn can never fasten upon truth. But 
the difference is all in the writers and teachers. 
Whereof some have neither studied, nor care for 
it, otherwise than as a trade to maintain themselves 
or gain preferment ; and some for fashion, and to 
make themselves fit for ingenious company : and 
their study hath not been meditation, but acquies- 
cence in the authority of those* authors whom they 


CHAP. i. have heard commended. And some,, but few, there 
oTthT^Mi ^ e ' ^ iat ^ iave studied it for curiosity, and the de- 
nt natural pin- light which coiiniioiilv men have in the acquisition 

losophy t * ^ 

of science, and in the mastery of difficult and sub- 
til docrines. Of this last sort I count Aristotle, 
arid a few others of the ancients, and some few mo- 
derns : and to these it is that properly belong the 
praises which are given to philosophy. 

A. If I have a mind to study, for example natu- 
ral philosophy, must I then needs read Aristotle, 
or some of those that now are in request ? 

B. There is no necessity of it. But if in your 
own meditation you light upon a difficulty, I think 
it is no loss of time, to enquire what other men 
say of it, but to rely only upon reason. For though 
there be some few effects of nature, especially con- 
cerning the heavens, whereof the philosophers of 
old time have assigned very rational causes, such 
as any man may acquiesce in, as of eclipses of the 
sun and moon by long observation, and by the cal- 
culation of their visible motions ; yet what is that 
to the numberless and quotidian phenomena of na- 
ture ? Who is there amongst them or their suc- 
cessors, that has satisfied you with the causes of 
gravity, heat, cold, light, sense, colour, noise, rain, 
snow, frost, winds, tides of the sea, and a thousand 
other things which a few men's lives are too short 
to go through, and which you and other curious 
spirits admire (as quotidian as they are), and fain 
would know the causes of them, but shall not find 
them in the books of naturalists ; and when you ask 
what are the causes of any of them, of a philo- 
sopher now, he will put you off with mere words ; 
which words, examihed to the bottom, signify not 


a jot more than I cannot tell, or because it is : CHAP. i. 
such as are intriusical quality, occult quality. "~ ' ' 

1 J y m x t J 7 Ot the original 

sympathy,, antipathy, antiperistasis, and the like, of natural phi. 
Wliich pass well enough with those that care not 0&op y * 
much for such wisdom, though wise enough in their 
own ways ; but will not pass with you that ask not 
simply what is the cause, but in what manner it 
comes about that such effects are produced. 

A. That is cozening. What need had they of 
that? When began they thus to play the char- 
latans ? 

J5. Need had they none. Bat know you not 
that men from their very birth, and naturally, 
scramble for every thing they covet, and would 
have all the world, if they could, to fear and obey 
them ? If by fortune or industry one light upon a 
secret in nature, and thereby obtain the credit of 
an extraordinary knowing man, should he not 
make use of it to his own benefit ? There is scarce 
one of a thousand but would live upon the charges 
of the people as far as he dares. What poor geo- 
metrician is there, but takes pride to be thought a 
conjurer? What mountebank would not make a 
living out of a false opinion that he were a great 
physician ? And when many of them are once en- 
gaged in the maintenance of an error, they will join 
together for the saving of their authority to decry 
the truth. 

A. I pray, tell me, if you can, how and where 
the study of philosophy first began. 

B. If we may give credit to old histories, the 
first that studied any of the natural sciences were 
the astronomers of Ethiopia. My author is Dio- 
dorus Siculus, accounted & very faithful writer, 


CHAP. i. who begins his history as high as is possible, and 
* ' ' tells us that in Ethiopia were the first astronomers ; 

Of the 01 iginal 

of natural phi- and that for their predictions of eclipses, and other 
sopiy ' conjunctions and aspects of the planets, they ob- 
tained of their king not only towns and fields to a 
third part of the whole land, but were also in such 
veneration with the people, that they were thought 
to have discourse with their gods, which were the 
stars ; and made their kings thereby to stand in 
awe of them, that they durst not either eat or drink 
but what and when they prescribed ; no nor live, 
if they said the gods commanded them to die. 
And thus they continued in subjection to their 
false prophets, till by one of their kings, called 
Ergamenes, (about the time of the Ptolemies), they 
were put to the sword. But long before the time 
of Ergamenes, the race of these astrologers (for 
they had no disciples but their own children) was 
so numerous, that abundance of them (whether 
sent for or no I cannot tell) transplanted them- 
selves into Egypt, and there also had their cities 
and lands allowed them, and were in request not 
only for astronomy and astrology, but also for geo- 
metry. And Egypt was then as it were an univer- 
sity to all the world, and thither went the curious 
Greeks, as Pythagoras, Plato, Thales, and others, 
to fetch philosophy into Greece. But long before 
that time, abundance of them went into Assyria, 
and had their towns and lands assigned them also 
there ; and were by the Hebrews called Chaldees. 

A. Why so ? 

B. I cannot tell ; but I find in Martinius's Lexicon 
they are called Chasdim, and Chesdim, and (as he 
saith) from one Chesetl the son of Nachor ; but I 


find no such man as Chesed amongst the issue of CHAP. i. 
Noah in the scripture. Nor do I find that there M( T7^ ' 

* Of the ongiiml 

was any certain country called Chaldsea ; though a natural piu- 
town where any of them inhabited were called a OM>piy * 
town of the Chaldees. Martinius saith farther, 
that the same word Chasdim did signify also 

A. By this reckoning I should conjecture they 
were called Chusdim, as being a race of Ethiopians. 
For the land of Chus is Ethiopia ; and so the name 
degenerated first into Chuldim, and then into 
Chaldim ; so that they were such another kind of 
people as we call gipsies ; saving that they were 
admired and feared for their knavery, and the 
gipsies counted rogues. 

13. Nay pray, except Claudius Ptolomseus, author 
of that great work of astronomy, the Almegest. 

A. I grant he was excellent both in astronomy 
and geometry, and to be commended for his Alme- 
gest; but then for his Judiciar Astrologie annexed 
to it, he is again a gipsy. But the Greeks that 
travelled, you say, into Egypt, what philosophy 
did they carry home ? 

B. The mathematics and astronomy. But for 
that sublunary physics, which is commonly called 
natural philosophy, I have not read of any nation 
that studied it earlier than the Greeks, from whom 
it proceeded to the Romans. Yet both Greeks and 
Romans were more addicted to moral than to 
natural philosophy ; in which kind we have their 
writings, but loosely and incoherently, written upon 
no other principles than their own passions and 
presumptions, without any respect to the laws of 
commonwealth, which are the 'ground and measure 


CHAP. i. of all true morality. So that their books tend rather 

or theon nai men * censure than to obey the laws ; 

of natural phi. which has been a great hindrance to the peace of 
osop iy. ^ e western world ever since. But they that seri- 
ously applied themselves to natural philosophy 
were but few, as Plato and Aristotle,, whose works 
we have ; and Epicurus whose doctrine we have 
in Lucretius. The writings of Philolaus and many 
other curious students being by fire or negligence 
now lost : though the doctrines of Philolaus con- 
cerning the motion of the earth have been revived 
by Copernicus, and explained and confirmed by 
Galileo now of late. 

A. But inetliinks the natural philosophy of Plato, 
and Aristotle, and the rest, should have been culti- 
vated and made to flourish by their disciples. 

H. Whom do you mean, the successors of Plato, 
Epicurus, Aristotle, and the other first philoso- 
phers ? It may be some of them may have been 
learned and worthy men. But not long after, and 
down to the time of our Saviour and his Apostles, 
they were for the most part a sort of needy, igno- 
rant, impudent, cheating fellows, who by the pro- 
fession of the doctrine of those first philosophers 
got their living. For at that time, the name of 
philosophy was so much in fashion and honour 
amongst great persons, that every rich man had a 
philosopher of one sect or another to be a school- 
master to his children. And these were they that 
feigning Christianity, with their disputing and readi- 
ness of talking got themselves into Christian com- 
mons, and brought so many heresies into the 
primitive Church, every one retaining still a tang 
of what they had been used to teach. 


A. But those heresies were all condemned in the CHAP. i. 
first Council of Nice. ^^77 'i 

Ot the original 

B. Yes. But the Arian heresy for a long time t natural pin- 
flourished no less than the Roman, and was upheld s 

by divers Emperors, and never fully extinguished 
as long as there were Vandals in Christendom. 
Besides, there arose daily other sects, opposing 
their philosophy to the doctrine of the Councils 
concerning the divinity of our Saviour ; as how 
many persons he was, how many natures he had. 
And thus it continued till the time of Charlemagne, 
when he and Pope Leo the third divided the power 
of the empire into temporal arid spiritual. 

A. A very unequal division. 

B. Why ? Which of them think you had the 
greater share ? 

A. No doubt, the Emperor : for he only had the 

B. When the swords are in the hands of men, 
whether had you rather command the men or the 
swords ? 

A. I understand you. For he that hath the 
hands of the men, has also the use both of their 
swords and strength. 

B. The empire thus divided into spiritual and 
temporal, the freedom of philosophy was to the 
power spiritual very dangerous. And for that 
cause it behoved the Pope to get schools set up 
not only for divinity, but also for other sciences, 
especially for natural philosophy. Which when by 
the power of the Emperor he had effected, out of 
the mixture of Aristotle's metaphysics with the 
Scripture, there arose a new science called School- 
divinity ; which has been th6 principal learning of 


CHAP. i. these western parts from the time of Charlemagne 
7T r till of very late. 

Of the original J 

of natural pin- A. But I find not in any of the writings of the 
Schoolmen in what manner, from the causes they 
assign, the effect is naturally and necessarily pro- 

It. You must not wonder at that. For you en- 
quire riot so much, when you see a change of 
anything, what may be said to be the cause of it, 
as how the same is generated ; which generation 
is the entire progress of nature from the efficient 
cause to the effect produced. Which is always a 
hard question, and for the most part impossible for 
a man to answer to. For the alterations of the 
things we perceive by our five senses are made by 
the motion of bodies, for the most part, either for 
distance, srnallness, or transparence, invisible. 

A. But what need had they then to assign any 
cause at all, seeing that they could not show the 
effect was to follow from it ? 

B. The Schools, as 1 said, were erected by the 
Pope and Emperor, but directed by the Pope only, 
to answer and confute the heresies of the philoso- 
phers. Would you ha\e them then betray their 
profession and authority, that is to say, their liveli- 
hood, by confessing their ignorance ? Or rather 
uphold the same, by putting for causes, strange 
and unintelligible w r ords ; which might serve well 
enough not only to satisfy the people whom they 
relied on, but also to trouble the philosophers 
themselves to find a fault in. 

A . Seeing you say that alteration is wrought by 
the motion of bodies, pray tell me first what I am 
to understand by the foord body. 


B. It is a hard question, though most men think CHAP. i. 
they can easily answer it, as that it is whatsoever of ^^Tai 
they can see, feel, or take notice of by their senses. ' atiir < i1 ? hl - 
But if you will know indeed what is body, we must 
enquire first what there is that is not body. You 
have seen, I suppose, the effects of glasses, how 
they multiply and magnify the object of our sight ; 
as when a glass of a certain figure will make a 
counter or a shilling seem twenty, though you be 
well assured there is but one. And if you set a 
mark upon it, you will find the mark upon them 
all. The counter is certainly one of those things 
we call bodies : are not the others so too ? 

A. No, without doubt. For looking through a 
glass cannot make them really more than they are. 

It. What then be they but fancies, so many 
fancies of one and the same thing in several places ? 

A. It is manifest they are so many idols, mere 

B. When you have looked upon a star or candle 
with both your eyes, but one of them a little turned 
awry with your finger, has not there appeared tw r o 
stars, or two candles ? And though you call it a 
deception of the sight, you cannot deny but there 
were two images of the object. 

A. It is true, and observed by all men. And the 
same I say of our faces seen in looking-glasses, 
and of all dreams, and of all apparitions of dead 
men's ghosts ; and wonder, since it is so manifest, 
I never thought upon it before, for it is a very happy 
encounter, and such as being by everybody well 
understood, would utterly destroy both idolatry 
and superstition, and defeat abundance of knaves 
that cheat and trouble the woHd with their devices. 


CHAP. i. -^- But you must not hence conclude that who- 
* ' soever tells his dream, or sometimes takes his 

Of the original i r 

of natural pin. direction from it, is therefore an idolater, or super- 
losophy stitious, or a cheater. For God doth often ad- 
monish men by dreams of what they ought to do ; 
yet men must be wary in this case that they trust 
not dreams with the conduct of their lives farther 
than by the laws of their country is allowed : for 
you know what God says, Deut. xiii : Jf a prophet 
or a dreamer of dreams give tliee a sign or a won- 
der, and the sign come to pass, yet if he bid thee 
serve other Gods let him be put to death. Here by 
serving other Gods (since they have chosen God 
for their King) we are to understand revolting from 
their King, or disobeying of his laws. Otherwise I 
see no idolatry nor superstition in following a 
dream, as many of the Patriarchs in the Old Testa- 
ment, and of the Saints in the New Testament did. 

A. Yes : their own dreams. But when another 
man shall dream, or say that he has dreamed, and 
require me to follow that, he must pardon me if I 
ask him by what authority, especially if he look I 
should pay him for it. 

B. But if commanded by the laws you live 
under, you ought to follow it. But when there 
proceed from one sound divers echoes., what are 
those echoes ? And when with fingers crossed you 
touch a small bullet, and think it two; and w r hen the 
same herb or flower smells well to one and ill to 
another, and the same at several times, well and ill to 
yourself, and the like of tastes, what are those 
echoes, feelings, odours, and tastes ? 

A. It is manifest they are all but fancies. But 
certainly when the siln seems to my eye no bigger 


than a dish, there is behind it somewhere some- CHAP. i. 
what else, I suppose a real sun, which creates those ^ ' ' 

* . Ol th original 

fancies,, by working, one way or other, upon my of natural pin- 
eyes, and other organs of my senses, to cause that osoply 
diversity of fancy. 

B. You say right ; and that is it I mean by the 
word body, which briefly I define to be any thing 
that hath a being in itself, without the help of sense. 

A. Aristotle, I think, meaneth by body, sub- 
'stance, or subjectum, wherein colour, sound, arid 
other fancies are, as he says, inherent. For the 
word essence has no affinity with substance. And 
Seneca says, he understands it not. And no won- 
der : for essence is no part of the language of man- 
kind,, but a word devised by philosophers out of 
the copulation of two names, as if a man having 
two hounds could make a third, if it w r ere need, of 
their couples. 

B. It is just so. For having said in themselves, 
(for example) : a tree is a plant, and conceiving 
well enough what is the signification of those 
names, knew not what to make of the word is, that 
couples those names ; nor daring to call it a body, 
they called it by a new name (derived from the 
word est\ essentia, and substantia, deceived by the 
idiom of their own language. For in many other 
tongues, and namely in the Hebrew, there is no 
such copulative. They thought the names of things 
sufficiently connected, when they are placed in their 
natural consequence ; and were therefore never 
troubled with essences, nor other fallacy from the 
copulative est. 

VOL. vn. 




CHAP. ii. A. THTS history of the old philosophers has not 
me ou ^ f l ve ? but oll t f tope of philosophy 

,u.<i in, HUM! ot from any of their writings. I would therefore try 

natural plnlo- J J 

sophy if I could attain any knowledge therein by my own 

meditation : but I know neither where to begin., 
nor which way to proceed. 

B. Your desire, you say, is to know the causes 
of the effects or phenomena of nature ; and you 
confess they are fancies, and, consequently, that 
they are in yourself ; so that the causes you seek 
for only are without you, and now you would know 
how those external bodies work upon you to pro- 
duce those phenomena. The beginning therefore 
of your enquiry ought to be at ; What it is you call 
a cause ? I mean an efficient cause : for the philo- 
sophers make four kinds of causes, whereof the 
efficient is one. Another they call the formal 
cause, or simply the form or essence of the thing 
caused ; as when they say, four equal angles and 
four equal sides are the cause of a square figure ; 
or that heaviness is the cause that makes heavy 
bodies to descend ; but that is not the cause you 
seek for, nor any thing but this : It descends be- 
cause it descends. The third is the material cause, 
as when they say, the walls and roof, &c. of a house 
are the cause of a house. The fourth is the final 
cause, and hath place only in moral philosophy. 


A. We will think of final causes upon some other CHAP. n. 
occasion ; of formal and material not at all : 1 seek v ' 

Of tliopimciple 

only the efficient, and how it acteth from the be- an.i method oi 
ginning to the production of the effect. *" 1 plulrt " 

B. I say then, that in the first place you are to 
enquire diligently into the nature of motion. For 
the variations of fancies, or (which is the same 
thing) of the phenomena of nature, have all of 
them one universal efficient cause, namely the va- 
riety of motion. For if all things in the world 
were absolutely at rest, there could be no variety 
of fancy ; but living creatures would be without 
sense of all objects, which is little less than to be 

A. What if a child new taken from the womb 
should with open eyes be exposed to the azure sky, 
do not you think it would have some sense of the 
light, but that all would seem unto him darkness ? 

B. Truly, if he had no memory of any thing for- 
merly seen, or by any other sense perceived, (which 
is my supposition), I think he would be in the 
dark. For darkness is darkness, whether it be 
black or blue, to him that cannot distinguish. 

A. Howsoever that be, it is evident enough that 
whatsoever worketh is moved : for action is motion. 

B. Having well considered the nature of motion, 
you must thence take your principles for the foun- 
dation and beginning of your enquiry. 

A . As how r ? 

B. Explain as fully and as briefly as you can 
what you constantly mean by motion ; which will 
save yourself as well as others from being seduced 
by equivocation. 

A. Then I say, motion is nothing but change of 


CHAP. ii. place for all the effect of a body upon the organs 
n ,"r~^ ". of our senses is nothing but fancy. Therefore we 

Of the principles . . 

and method of can fancy nothing from seeing it moved, but change 

natural plulo- ~ , 

sophy. of place. 

B. It is right. But you must then tell me also 
what you understand by place : for all men are not 
yet agreed on that. 

A. Well then ; seeing we fancy a body, we can- 
not but fancy it somewhere. And therefore I think 
place is the fancy of here or there. 

B. That is not enough. Here and there are not 
understood by any but yourself, except you point 
towards it. But pointing is no part of a definition. 
Besides, though it help him to find the place, it 
will never bring him to it. 

A. But seeing sense is fancy, when we fancy a 
body, we fancy also the figure of it, arid the space 
it fills up. And then I may define place to be the 
precise space within which the body is contained. 
For space is also part of the image we have of the 
object seen. 

B. And how define you time ? 

A. As place is to a body, so, I think, is time to 
the motion of it ; and consequently I take time 
to be our fancy or image of the motion. But is 
there any necessity of so much niceness ? 

B. Yes. The want of it is the greatest, if not 
the only, cause of all the discord amongst philo- 
sophers, as may easily be perceived by their abusing 
and confounding the names of things that differ in 
their nature ; as you shall see when there is occa- 
sion to recite some of the tenets of divers philo- 

A. I will avoid equivocation as much as I can. 


And for the nature of motion, I suppose I under- CHAP. u. 
stand it by the definition. What is next to be v ' ' 

Ol the principles 
done ? and method of 

B. You are to draw from these definitions, and ^1! p u " 
from whatsoever truth else you know by the light 
of nature, such general consequences as may serve 
for axioms, or principles of your ratiocination. 

A. That is hard to do. 

J3. I will draw them myself, as many as for our 
present discourse of natural causes we shall have 
need of; so that your part will be no more than to 
take heed I do not deceive you. 

A. I will look to that. 

B. My first axiom then shall be this : Two 
bodies, at the same time, cannot be in one place. 

A. That is true : for we number bodies as we 
fancy them distinct, and distinguish them by their 
places. You may therefore add : nor one body 
at the same time in two places. And philosophers 
mean the same, when they say : there is no pene- 
tration of bodies. 

B. But they understand not their own words : 
for penetration signifies it not. My second axiom 
is, that nothing can begin, change, or put an end 
to, its own motion. For supposing it begin just 
now, or being now in motion, change its way or 
stop ; I require the cause why now rather than be- 
fore or after, having all that is necessary to such 
motion, change, or rest, alike at all times ? 

A. I do not doubt but the argument is good in 
bodies inanimate ; but perhaps in voluntary agents 
it does not hold. 

B. How it holds in voluntary agents we will 
then consider when our method hath brought us to 


CHAP. n. the powers and passions of the mind. A third 
n "~i axiom shall be this : whatsoever body being at 

Ol (lie principles J ^ 

,mi iiK'tiKMi ot restns afterwards moved, hath for its immediate 
sopin l p " movement some other body which is in motion and 
toucheth it. For, since nothing can move itself, 
the movent must be external. And because motion 
is change of place, the movent must put it from 
its place, which it cannot do till it touch it. 

A. That is manifest, and that it must more than 
touch it ; it must also follow it. And if more parts 
of the body are moved than are by the movent 
touched, the movent is not immediate. And by 
this reason, a continued body, though never so 
great, if the first superficies be pressed never so 
little back, the motion will proceed through it. 

B. Do you think that to be impossible ? I will 
prove it from your own words : for you say that 
the movent does then touch the body which it 
moveth. Therefore it puts it back ; but that which 
is put back, puts back the next behind, and that 
again the next ; and so onward to any distance, 
the body being continued. The same is also mani- 
fest by experience, seeing one that walks with a 
staff can distinguish, though blind, between stone 
and glass ; which were impossible, if the parts of 
his staff between the ground and his hand made 
no resistance. So also he that in the silence of the 
night lays his ear to the ground, shall hear the 
treading of men's feet farther than if he stood 

A. This is certainly true of a staff or other hard 
body, because it keeps the motion in a straight 
line from diffusion. t But in such a fluid body as 
the air, which being put back must fill an orb, and 


the farther it is put back, the greater orb, the mo- CHAP. ir. 
tion will decrease, and in time, by the resistance " ' ' 

5 * J Ofthopimcipies 

of air to air, come to an end. aiitl "^tuoa ot 

jB. That any body in the world is absolutely at "" p " 
rest, I think not true : but I grant, that in a space 
filled everywhere with body, though never so fluid, 
if you give motion to any part thereof, that motion 
will by resistance of the parts moved, grow less 
and less, and at last cease ; but if you suppose the 
space utterly void, and nothing in it, then whatso- 
ever is once moved shall go on eternally : or else 
that which you have granted is not true, viz., that 
nothing can put an end to its own motion. 
A. But what mean you by resistance ? 
It. Resistance is the motion of a body in a way 
w T holly or partly contrary to the way of its movent, 
and thereby repelling or retarding it. As when a 
man runs swiftly, he shall feel the motion of the 
air in his face. But when two hard bodies meet, 
much more may you see how they abate each other's 
motion, and rebound from one another. For in a 
space already full, the motion cannot, in an instant, 
be communicated through the whole depth of the 
body that is to be moved. 

A. What other definitions have I need of? 

B. In all motion, as in all quantity, you must 
take the beginning of your reckoning from the 
least supposed motion. And this I call the first 
endeavour of the movent ; which endeavour, how 
weak soever, is also motion. For if it have no effect 
at all, neither will it do anything though doubled, 
trebled, or by what number soever multiplied : for 
nothing, though multiplied, is still nothing. Other 


CHAP. n. axioms and definitions we will take in, as we need 
r ~~" them, by the way. 

Of the principles ' ^ . J l . 

and method of A. Is this all the preparation I am to make ? 

wph 1 ?hll Ji* No, you are to consider also the several kinds 
and properties of motion, viz., when a body being 
moved by one or more movents at once, in what 
way it is carried, straight, circular, or otherwise 
crooked ; and what degree of swiftness ; as also 
the action of the movent, whether trusion, vection, 
percussion, reflection, or refraction ; and farther 
you must furnish yourself with as many experiments 
(which they call phenomenon) as you can. And 
supposing some motion for the cause of your 
phenomenon, try, if by evident consequence, with- 
out contradiction to any other manifest truth or 
experiment, you can derive the cause you seek for 
from your supposition. If you can, it is all that is 
expected, as to that one question, from philosophy. 
For there is no effect in nature which the Author 
of nature cannot bring to pass by more ways than 

A. What I want of experiments you may supply 
out of your own store, or such natural history as 
you know to be true ; though I can be well con- 
tent with the knowledge of the causes of those 
things which everybody sees commonly produced. 
Let us therefore now enquire the cause of some 
eifect particular. 

B. We will begin with that which is the most 
universal, the universe ; and enquire in the first 
place, if any place be absolutely empty, that is to 
say in the language of philosophers, whether there 
be any vacuum in nature ? 




A. IT is hard to suppose, and harder to believe, CHAP m 
that the infinite and omnipotent Creator of all * - 
things should make a work so vast as is the world 
we see, and not leave a few little spaces with 
nothing at all in them ; which put altogether in 
respect of the whole creation, would be insensible. 

J3. Why say you that? Do you think any 
argument can be drawn from it to prove there is 
vacuum ? 

A. Why not ? For in so great an agitation of 
natural bodies, may not some small parts of them 
be cast out, and leave the places empty from 
whence they were thrown ? 

B. Because He that created them is not a fancy, 
but the most real substance that is ; who being 
infinite, there can be no place empty where He is, 
nor full where He is not. 

A. It is hard to answer this argument, becausb 
I do not remember that there is any argument for 
the maintenance of vacuum in the writings of di- 
vines : therefore I will quit that argument, and 
come to another. If you take a glass vial with a 
narrow neck, and having sucked it, dip it presently 
at the neck into a basin of water, you shall mani- 
festly see the water rise into the vial. Is not this 
a certain sign that you had sucked out some of the 
air, and consequently that some part of the vial 
was left empty ? 


CHAP. in. B. No ; for when I am about to suck, and have 
or vacuum a ^ r * n m y mou th> contracting my checks I drive 
the same against the air in the glass, and thereby 
against every part of the sides of the hard glass. 
And this gives to the air within an endeavour out- 
ward, by which, if it be presently dipped into the 
water, it will penetrate and enter into it. For air 
if it be pressed will enter into any fluid, much 
more into water. Therefore there shall rise into 
the vial so much water as there was air forced into 
the basin. 

A. This I confess is possible, and not improbable. 
It. If sucking would make vacuum, what would 
become of those women that are nurses ? Should 
they not be in a very few days exhausted, were it 
not that either the air which is in the child's mouth 
penetrateth the milk as it descends, and passeth 
through it, or the breast is contracted ? 

A. From what experiment can you evidently 
infer that there is no vacuum ? 

B. From many, and such as to almost all men 
are known and familiar. If two hard bodies, flat 
and smooth, be joined together in a common super- 
iicies parallel to the horizontal plane, you cannot 
without great force pull them asunder, if you apply 
your force perpendicularly to the common super- 
ficies : but if you place that common superficies 
erect to the horizon, they will fall asunder with 
their own weight. From whence I argue thus : 
since their contiguity, in what posture soever, is 
the same, and that they cannot be pulled asunder 
by a perpendicular force without letting in the 
ambient air in an instant, w r hich is impossible ; or 
almost in an instant, which is difficult : and on the 


other side, when the common superficies is erect, CHAP. in. 
the weight of the same hard bodies is able to "~~~" ' 

~ ^ Oi vacuum 

break the contiguity, and let in the air successively ; 
it is manifest that the difficulty of separation pro- 
ceeds from this, that neither air nor any other body 
can be moved to any, how small soever, distance 
in an instant ; but may easily be moved (the hard- 
ness at the sides once mastered) successively. So 
that the cause of this difficulty of separation is this, 
that they cannot be parted except the air or other 
matter can enter and fill the space made by their 
diremption. And if they were infinitely hard, not 
at all. And hence also you may understand the 
cause why any hard body, when it is suddenly 
broken, is heard to crack; which is the swift 
motion of the air to fill the space beween. Ano- 
ther experiment, and commonly known, is of a 
barrel of liquor, whose taphole is very little, and 
the bung so stopped as to admit no air ; for then 
the liquor will not run : but if the tap-hole be large 
it will, because the air pressed by a heavier body 
will pierce through it into the barrel. The like 
reason holds of a gardener's watering-pot, when the 
holes in the bottom are not too great. A third 
experiment is this : turn a thin brass kettle the bot- 
tom upwards, and lay it flat upon the water. It 
will sink till the water rise; within to a certain 
height, but no higher : yet let the bottom be per- 
forated, and the kettle will be full and sink, and 
the air rise again through the water without. But 
if a bell were so laid on, it would be filled and sink, 
though it were not perforated, because the weight 
is greater than the weight of the same bulk of 


CHAP. in. A. By these experiments, without any more, I 
^T^^ am convinced, that there is not actually in nature 
any vacuum ; but I am not sure but that there may 
be made some little place empty, and this from two 
experiments, one whereof is Toricellius' experi- 
ment, which is this : take a cylinder of glass, hollow 
throughout, but close at the end, in form of a sack. 
B. How long ? 

A. As long as you will, so it be more than 
twenty-nine inches. 

B. And how broad r 

A. As broad as you will, so it be broad enough 
to pour into it quicksilver. And fill it with quick- 
silver, and stop up the entrance with your finger, 
so as to unstop it again at your pleasure. Then 
set down a basin, or, if you will, a sea of quick- 
silver,, and inverting the cylinder full as it is, dip 
the end into the quicksilver, and remove your 
finger, that the cylinder may ernpt itself. Do you 
conceive me r For there is so many passing by, 
that I cannot paint it. 

B. Yes, I conceive you well enough. What follows? 

A. The quicksilver will descend in the cylinder, 
not till it be level with that in the basin, according 
to the nature of heavy fluids, but stay and stand 
above it, at the height of twenty-nine inches or 
very near it, the bottom being now uppermost, that 
no air can get in. 

B. What do you infer from this ? 

A. That all the cavity above twenty-nine inches 
is filled with vacuum. 

B. It is very strange that I, from this same ex- 
periment, should infer, and I think evidently, that 
it is filled with air. I pray, tell me, when you had 


inverted the cylinder, full as it was, and stopped CHAP. nr. 
with your finger, dipped into the basin, if you had ^ vacuum. 
then removed your finger, whether you think the 
quicksilver would not all have fallen out ? 

A. No sure. The air would have been pressed 
upward through the quicksilver itself : for a man 
with his hand can easily thrust a bladder of air to 
the bottom of a basin of quicksilver. 

B. It is therefore manifest that quicksilver can 
press the air through the same quicksilver. 

A. It is manifest ; and also itself rise into the air. 

B. What cause then can there be, why it should 
stand still at twenty nine inches above the level of 
the basin, rather than any place else ? 

A. It is not hard to assign the cause of that. 
For so much quicksilver as was above the twenty- 
nine inches, will rise the first level of that in the 
basin, as much as if you had poured it on ; and 
thereby bring it to an equilibrium. So that I see 
plainly now, that there is no necessity of vacuum 
from this experiment. For I considered only that 
naturally quicksilver cannot ascend in air, nor air 
descend in quicksilver, though by force it may. 

B. Nor do I think that Torricellius or any other 
vacuist thought of it more than you. But what is 
the second experiment ? 

A. There is a sphere of glass> which they call a 
recipient, of the capacity of three or four gallons. 
And there is inserted into it the end of a hollow 
cylinder of brass above a foot long ; so that the 
whole is one vessel, and the bore of the cylinder 
three inches diameter. Into which is thrust by 
force a solid cylinder of wood, covered with lea- 
ther so just, as it may in ever^ point exactly touch 


CHAP. in. the concave superficies of the brass. There is 
^T^^ also, to let out the air which the wooden cylinder 
as it enters (called the sucker) drives before it, a 
flap to keep out the external air while they are pull- 
ing the sucker. Besides,, at the top of the recipi- 
ent there is a hole to put into it anything for 
experiment. The sucker being now forced up 
into the cylinder, what do you think must follow ? 

B. I think it will require as much strength to 
pull it back, as it did to force it in. 

A. That is riot it I ask, but what would happen 
to the recipient ? 

7J. T think so much air as would fill the place 
the sucker leaves, would descend into it out of the 
recipient ; and also that just so much from the 
external air would enter into the recipient, between 
the brass and the wood., at first very swiftly, but, 
as the pktce increased, more leisurely. 

A. Why may not so much air rather descend 
into the place forsaken, and leave as much vacuum 
as that comes to in the recipient ? For otherwise 
no air will be pumped out, nor can that wooden 
pestle be called a sucker. 

B. That is it I say. There is no air either 
pumped or sucked out. 

A. How can the air pass between the leather 
and the brass, or between the leather and the 
wood, being so exactly contiguous, or through the 
leather itself? 

B. I conceive no such exact contiguity, nor such 
fastness of the leather : for I never yet had any 
that in a storm would keep out either air or water. 

A. But how then could there be made in the 


recipient such strange alteration both on animate CHAP. in. 
and inanimate bodies ? "77"' ' 

Of \ ucuura. 

B. 1 will tell you how. The air descends out of 
the recipient, because the air which the sucker 
removeth from behind itself,, as it is pulling out, 
has no place to retire into without, and therefore 
is driven into the engine between the wood of the 
sucker and the brass of the cylinder, and causes 
as much air to come into the place forsaken by 
the retiring sucker ; which causeth, by oft repeti- 
tion of the force, a violent circulation of the air 
within the recipient, which is able quickly to kill 
anything that lives by respiration, and make all 
the alterations that have appeared in the engine. 



B. You are come in good time ; let us therefore 
sit down. There is ink, paper, ruler, and compass. 
Draw a little circle to represent the body of the sun. 

A. It is done. The centre is A, the circum- 
ference is L M. 

B. Upon the same centre A, draw a larger cir- 
cle to stand for the ecliptic : for you know the 
sun is always in the plane of the ecliptic. 

A. There it is. The diameters of it at right 
angles are B Z. 

B. Draw the diameter of the equator. 

A. How ? 

B. Through the centre A (for the earth is also 
always in the plane of the equator or of some of its 


CHAP. iv. parallels) so as to be distant from B twenty-three 
"~ T 7^ decrees and a half. 

Of the system O 

of the world. A. Let it be H I : and let C G be equal to B H ; 
and so C will be one of the poles of the ecliptic, 
suppose the north-pole ; and than H will be east, 
and I west. And C A produced to the circum- 
ference in E, makes E the south-pole. 

B. Take C K equal to C G, and the chord G K 
will be the diameter of the arctic circle, and parallel 
to H I, the diameter of the equator. Lastly, upoil 
the point B, draw a little circle wherein I suppose 
to be the globe of the earth. 

A. It is drawn, and marked with / m. And B D 
and K G joined will be parallel ; and as H and I 
are east and w r est, and so are B and D, and G 
and K. 

B. True ; but producing Z B to the circum- 
ference / m in b, the line B b will be in the diame- 
ter of the ecliptic of the earth, and B m in the dia- 
meter of the equator of the earth. In like manner, 
if you produce K G cutting the circle, whose centre 
is G 5 in d and <?, and make an angle n G d equal to 
b B m, the line n G will be in the ecliptic of the 
earth, because G d is in the equator of the earth. 
So that in the annual motion of the earth through 
the ecliptic, every straight line drawn in the earth, 
is perpetually kept parallel to the place from 
whence it is removed. 

A. It is true ; and it is the doctrine of Coper- 
nicus. But I cannot yet conceive by what one 
motion this circle can be described otherwise than 
we are taught by Euclid. And then I am sure that 
all the diameters shall cross one another in the 
centre, which in this figure is A. 


B. I do riot say that the diameters of a sphere CHAP. iv. 
or circle can be parallel ; but that if a circle of a """^ ' 

1 m Ot the system 

lesser sphere be moved upon the circumference of a f tlie worla - 
great circle of a greater sphere, that the straight 
lines that are in the lesser sphere may be kept pa- 
rallel perpetually to the places they proceed from. 

A. How ? And by what motion ? 

B. Take into your hand any straight line (as in 
this figure), the line LAM, which we suppose to be 
the diameter of the sun's body ; and moving it 
parallclly with the ends in the circumference,, so as 
that the end M may withal describe a small circle, 
as M a. It is manifest that all the other points of 
the same line L M will, by the same motion, at the 
same time, describe equal circles to it. Likewise if 
you take in your hand any two diameters fastened 
together, the same parallel motion of the line LM, 
shall cause all the points of the other diameter to 
make equal circles to the same M a. 

A. It is evident; as also that every point of the 
sun's body shall do the like. And not only so, but 
also if one end describe any other figure, all the 
other points of the body shall describe like and 
equal figures to it. 

B. You see by this, that this parallel motion is 
compounded of two motions, one circular upon the 
superficies of a sphere, the other a straight motion 
from the centre to every point of the same super- 
ficies, and beyond it. 

A. I see it. 

B. It follows hence, that the sun by this motion 
must every way repel the air ; and since there is no 
empty place for retiring, the air must turn about in 
a circular stream ; but slower or swifter according 



CHAP. iv. as it is more or less remote from the sun ; and that 

oithe's stem according to the nature of fluids, the particles of 

of the world, the air must continually change place with one 

another ; and also that the stream of the air shall 

be the contrary way to that of the motion, for else 

the air cannot be repelled. 

A. All this is certain. 

B. Well ; then if you suppose the globe of the 
earth to be in this stream which is made by the 
motion of the sun's body from cast to west, the 
stream of air wherein is the eai'th's annual motion 
will be from west to east. 

A. It is certain. 

B. Well. Then if you suppose the globe of the 
earth, whose circle is moved annually, to be I m, 
the stream of the air without the ecliptic falling 
upon the superficies of the earth I m without the 
ecliptic, being slower, and the stream that falleth 
within swifter, the earth shall be turned upon its 
own centre proportionally to the greatness of the 
circles ; and consequently their diameters shall be 
parallel ; as also are other straight lines correspon- 

A. I deny not but the streams are as you say; 
and confess that the proportion of the swiftness 
without, is to the swiftness within, as the sun's 
ecliptic to the ecliptic of the earth ; that is to say, 
as the angle H A B to the angle m B b. And I like 
your argument the better, because it is drawn from 
Copernicus his foundation. I mean the compounded 
motion of straight and circular. 

B. I think I shall not oifer you many demon- 
strations of physical conclusions that are not 
derived from the motions supposed or proved by 


Copernicus. For those conclusions in natural phi- CHAP. TV. 
losophy I most suspect of falshood, which require ^77^ ' 

* J r m * Ol tho system 

most variety of suppositions for their demonstra- of the world. 

A. The next thing I would know, is how great 
or little you suppose that circle a M r 

B. I suppose it less than you can make it : for 
there appears in the sun no such motion sensible. 
Jt is the first endeavour of the sun's motion. But 
for all that, as small as the circle is, the motion 
may be as swift, and of as great strength as it is 
possible to be named. It is but a kind of trembling 
that necessarily happeneth in those bodies, which 
with great resistance press upon one another. 

A. I understand now from what cause pro- 
ceedeth the annual motion. Is the sun the cause 
also of the diurnal motion ? 

B. Not the immediate cause. For the diurnal 
motion of the earth is upon its own centre, and 
therefore the sun's motion cannot describe it. But 
it proceedeth as a necessary consequence from the 
annual motion. For which I have both experience 
and demonstration. The experiment is this : into 
a large hemisphere of w r ood, spherically concave, 
put in a globe of lead, and with your hands hold it 
fast by the brim, moving your hand circularly, but 
in a very small compass ; you shall see the globe 
circulate about the concave vessel, just in the same 
manner as the earth doth every year in the air ; 
and you shall see withal, that as it goes, it turns 
perpetually upon its own centre, and very swiftly. 

A. I have seen it : and it is used in some great 
kitchens to grind mustard. 

B. Is it so ? Therefore take a hemisphere of 

H 2 


CHAP. iv. gold, if you have it, the greater the better, and a 
or the s s7em bullet of gold, and, without mustard, you shall see 
of the world, the same effect. 

A. I doubt it not. But the cause of it is evident. 
For any spherical body being in motion upon the 
sides of a concave and hard sphere, is all the way 
turned upon its own centre by the resistance of the 
hard wood or metal. But the earth is a bullet 
without weight, and meeteth only with air, without 
any harder body in the way to resist it. 

B. Do you think the air makes no resistance, 
especially to so swift a motion as is the annual 
motion of the earth ? If it do make any resistance, 
you cannot doubt but that it shall turn the earth 
circularly, and in a contrary way to its annual 
motion ; that is to say, from east to west, because 
the annual motion is from west to east. 

A. I confess it. But what deduce you from these 
motions of the sun ? 

B. I deduce, first, that the air must of necessity 
be moved both circularly about the body of the 
sun according to the ecliptic, and also, every way 
directly from it. For the motion of the sun's body 
is compounded of this circular motion upon the 
sphere L M, and of the straight motion of its semi- 
diameters from the centre A to the superficies of 
the sun's body, which is L M. And therefore the 
air must needs be repelled every way, and also 
continually change place to fill up the places for- 
saken by other parts of the air, which else would 
be empty, there being no vacuum to retire unto. 
So that there would be a perpetual stream of air, 
and in a contrary way to the motion of the sun's 


body, such as is the motion of water by the sides CHAP. iv. 
of a ship under sail. 

A. But this motion of the earth from west to 
east is only circular, such as is described by a 
compass about a centre ; and cannot therefore 
repel the air as the sun does. And the disciples of 
Copernicus will have it to be the cause of the 
moon's monthly motion about the earth. 

B. And I think Copernicus himself would have 
said the same,, if his purpose had been to have 
shown the natural causes of the motions of the 
stars. But that was no part of his design ; which 
was only from his own observations, and those of 
former astronomers, to compute the times of their 
motions ; partly to foretel the conjunctions, oppo- 
sitions, and other aspects of the planets ; and partly 
to regulate the times of the Church's festivals. But 
his followers, Kepler and Galileo, make the earth's 
motion to be the efficient cause of the monthly 
motion of the moon about the earth ; which with- 
out the like motion to that of the sun in L M, is 
impossible. Let us therefore for the present take 
it in as a necessary hypothesis ; which from some 
experiment that I shall produce in our following 
discourses, may prove to be a certain truth. 

A. But seeing A is the centre both of the sun's 
body and of the annual motion of the earth, how 
can it be (as all astronomers say it is) that the orb 
of the annual motion of the earth should be eccen- 
tric to the sun's body ? For you know that from 
the vernal equinox to the autumnal, there be one 
hundred and eighty-seven days ; but from the 
autumnal equinox to the vernal, there be but one 


CHAP. iv. hundred and seventy-eight days. What natural 
"~^ cause can you assign for this eccentricity ? 

Of the system 7 

of the worki. B. Kepler ascribes it to a magnetic virtue., 
viz. that one part of the earth's superficies has a 
greater kindness for the sun than the other part. 

A. I am not satisfied with that. It is magical 
rather than natural, and unworthy of Kepler. Tell 
me your own opinion of it. 

B. I think that the magnetical virtue he speaks 
of, consisteth in this : that the southern hemisphere 
of the earth is for the greatest part sea., and that 
the greatest part of the northern hemisphere is 
dry land. But how it is possible that from thence 
should proceed the eccentricity (the sun being 
nearest to the earth, when he is in the winter sol- 
stice), I shall show you when we come to speak of 
the motions of air and water. 

A. That is time enough : for I intend it for our 
next meeting. In the mean time I pray you tell 
me what you think to be the cause why the equi- 
noctial, and consequently the solstitial, points are 
not always in one and the same point of the ecliptic 
of the fixed stars. I know they are not, because 
the sun does not rise and set in points diametrically 
opposite : for if it did, there would be no difference 
of the seasons of the year. 

B. The cause of that can be no other, than that 
the earth, which is / m, hath the like motion to 
that which T suppose the sun to have in L M, com- 
pounded of straight and circular from west to east 
in a day, as the annual motion hath in a year ; so 
that, not reckoning the eccentricity, it will be 
moved through the ecliptics in one revolution, as 
Copernicus proveth, "about one degree. Suppose 


then the whole earth moved from H to I, (which is CHAP. iv. 
half the year) circularly, but falling from I to i in *~ Y "" 

J ' J ' Of the system 

the same time about thirty minutes, and as much of the world. 
in the other hemisphere from H to k ; then draw 
the line i k, which will be equal and parallel to H I, 
and be the diameter of the equator for the next 
year. But it shall not cut the diameter of the 
ecliptic B Z in A, which was the equinoctial of the 
former year, but in o thirty-six seconds from the 
'first degree of Aries. Suppose the same done in 
the hemisphere under the plane of the paper, and 
so you have the double of thirty-six seconds, that 
is seventy-two seconds, or very near, for the pro- 
gress of the vernal equinox in a year. The cause 
why I suppose the arch I / to be half a degree in 
the ecliptic of the earth, is, that Copernicus and 
other astronomers, and experience, agree in this, 
that the equinoctial points proceed according to the 
order of the signs, Aries, Taurus, Gemini, &c.from 
west to east every hundredth year one degree or 
very near. 

A. In what time do they make the whole revo- 
lution through the ecliptic of the sky ? 

B. That you may reckon. For we know by 
experience that it hath proceeded about one degree, 
that is sixty minutes, constantly a long time in a 
hundred years. But as one hundred years to one 
degree, so is thirty-six thousand years to three 
hundred and sixty degrees. Also as one hundred 
years to one degree, so is one year to the hundredth 
part of one degree, or sixty minutes ; which is -j^, 
or thirty-six seconds for the progress of one year ; 
which must be somewhat more than a degree 
according to Copernicus, whe, (lib. iii. cap. 2) saith, 


CHAP. iv. that for four hundred years before Ptolomy it was 
"rr^"" one degree almost constantly. Which is well 

Of the system J 

of the World, enough as to the natural cause of the precession of 
the equinoctial points, which is the often-said com- 
pounded motion, though not an exact astronomical 

A. And it is a great sign that his supposition is 
true. But what is the cause that the obliquity of 
the ecliptic, that is, the distance between the 
equinoctial and the solstice, is not always the 
same ? 

B. The necessity of the obliquity of the ecliptic 
is but a consequence to the precession of the equi- 
noctial points. And therefore, if from C, the 
north pole, you make a little circle, C u, equal to 
fifteen minutes of a degree upon the earth, and 
another, u s, equal to the same, which will appear 
like this figure 8, that is, (as Copernicus calls it), 
a circle twined, the pole C will be moved half the 
time of the equinoctial points, in the arc C u, and 
as much in the alternate arc it s descending to s. 
But in the arc s u, and its alternate rising to C, 
the cause of the twining is the earth's annual 
motion the same way in the ecliptic, and makes 
the four quarters of it ; and makes also their revo- 
lution twice as slow as that of the equinoctial 
points. And, therefore, the motion of it is the 
same compounded motion which Copernicus takes 
for his supposition, and is the cause of the preces- 
sion of the equinoctial points, and consequently 
of the variation of the obliquity, adding to it or 
taking from it somewhere more, somewhere less ; 
so as that one with another the addition is not 
much more, nor the "subtraction much less than 


thirty minutes. But as for the natural efficient CHAP.IV. 
cause of this compounded motion, either in the "~ ' 

1 Of the system 

sun, or the earth, or any other natural body, it of the world. 
can be none but the immediate hand of the Creator. 

A. By this it seems that the poles of the earth 
are always the same, but make this 8 in the sphere 
of the fixed stars near that which is called Cyno- 

B. No : it is described on the earth, but the 
annual motion describes a circle in the sphere of 
the fixed stars. Though I think it improper to 
say a sphere of the fixed stars, when it is so un- 
likely that all the fixed stars should be in the 
superficies of one and the same globe. 

A. I do not believe they are. 

B. Nor I, since they may seem less one than 
another, as well by their different distances, as by 
their different magnitudes. Nor is it likely that 
the sun (which is a fixed star) is the efficient cause 
of the motion of those remoter planets, Mars, 
Jupiter, and Saturn ; seeing the whole sphere, 
whose diameter is the distance between the sun 
and the earth, is but a point in respect of the dis- 
tance between the sun and any other fixed star. 
Which I say only to excite those that value the 
knowledge of the cause of comets, to look for it 
in the dominion of some other sun than that 
which moveth the earth. For why may not there 
be some other fixed star, nearer to some planet 
than is the sun, and cause such a light in it as we 
call a cornet r 

A. As how 7 ? 

B. You have seen how in high and thin clouds 
above the earth, the sun-beam^ piercing them have 


CHAP. iv. appeared like a beard ; and why might not such 
or the system a beard have appeared to you like a comet, if you 
of the world. had looked upon it from as high as some of the 
fixed stars ? 

A. But because it is a thing impossible for me 
to know, I will proceed in my own way of inquiry. 
And seeing you ascribe this compounded motion 
to the sun and earth, I would grant you that the 
earth (whose annual motion is from west to east) 
shall give the moon her monthly motion from east 
to west. But then I ask you whether the moon 
have also that compounded motion of the earth, 
and with it a motion upon its own centre, as hath 
the earth ? For seeing the moon has no other planet 
to carry about her, she needs it not. 

B. I see reason enough, and some necessity, 
that the moon should have both those motions. 
For you cannot think that the Creator of the stars, 
when he gave them their circular motion, did first 
take a centre, and then describe a circle with a 
chain or compass, as men do ? No ; he moved all 
the parts of a star together and equally in the 
creation : and that is the reason I give you. The 
necessity of it comes from this phenomenon, that 
the moon doth turn one and the same face towards 
the earth ; which cannot be by being moved about 
the earth parallelly, unless also it turn about its 
own centre. Besides, we know by experience, 
that the motion of the moon doth add not a little 
to the motion of the sea : which were impossible 
if it did not add to the stream of the air, and by 
consequence to that of the water. 

A. If you could get a piece of the true and 
intimate substance 01 the earth, of the bigness of 


a musket-bullet, do you believe that the bullet CHAP. iv. 
would have the like compounded motion to that """T" 11 

r Of the system 

which you attribute to the sun, earth, and moon r 
B. Yes, truly ; but with less strength, according 
to its magnitude ; saving that by its gravity falling 
to the earth, the activity of it would be unper- 

A. I will trouble you no more with the nature 
of celestial appearances ; but I pray you tell me by 
what art a man may find what part of a circle 
the diameter of the sun's body doth subtend in 
the ecliptic circle ? 

jB. Kepler says it subtends thirty minutes, which 
is half a degree. His way to find it is by letting 
in the sun-beams into a close room through a 
small hole, and receiving the image of it upon a 
plane perpendicularly. For by this means he hath 
a triangle, whose sides and angles he can know by 
measure ; and the vertical angle he seeks for, and 
the substance of the arc of the sun's body. 

A. But I think it impossible to distinguish 
where the part illuminate toucheth the part not 

B. Another way is. this ; upon the equinoctial 
day, with a watch that shows the minutes standing 
by you, observe when the lower brim of the sun's 
setting first comes to the horizon, and set the index 
to some minute of the watch ; and observe again 
the upper brim when it comes to the horizon : 
then count the minutes, and you have what you 
look for. Other way I know none. 




CHAP. v. A. I HAVE considered, as you bad me, this 
of ihe motions compounded motion with great admiration. First, 
oi water and air. jt i s that which makes the difference between 
continuum and contiguum, which till now I never 
could distinguish. For bodies that are but conti- 
guous, with any little force arc parted ; but by this 
compounded motion (because every point of the 
body makes an equal line in equal time, and every 
line crosses all the rest) one part cannot be sepa- 
rated from another, without disturbing the motion 
of all the other parts at once. And is not that the 
cause, think you, that some bodies when they are 
pressed or bent, as soon as the force is removed, 
return again of themselves to their former figure ? 
B. Yes, sure ; saving that it is not of themselves 
that they return, (for we were agreed that nothing 
can move itself), but it is the motion of the parts 
which are not pressed, that delivers those that are. 
And this restitution the learned now call the spring 
of a body. The Greeks called it antitypia. 

A. When I considered this motion in the sun 
and the earth and planets, I fancied them as so 
many bodies of the army of the Almighty in an 
immense field of air, marching swiftly, and com- 
manded (under God) by his glorious officer the 
sun, or rather forced so to keep their order in 
every part of every of those bodies, as never to 
go out from the distance in which he had set them. 

B. But the parts, of the air and other fluids 
keep not their places so. 


A . No : you told me that this motion is not CHAP. v. 
natural in the air. but received from the sun. ' r ~" 

\ Oi the motions 

B. True: but since we seek the natural causes oi water and aw 
of sublunary effects, where shall we begin ? 

A. I would fain know what makes the sea to 
ebb and flow at certain periods, and what causeth 
such variety in the tides. 

B. Remember that the earth turneth every day 
upon its own axis from west to east ; and all the 
while it so turneth, every point thereof by its com- 
pounded motion makes other circlings, but not on 
the same centre, which is (you know) a rising in 
one part of the day, and a falling in the other 
part. What think you must happen to the sea, 
which resteth on it, and is a fluid body ? 

A. I think it must make the sea rise and fall. 
And the same happeneth also to the air, from the 
motion of the sun. 

B. Remember, also, in what manner the sea is 
situated in respect of the dry land. 

A. Is not there a great sea that reacheth from 
the straits of Magellan eastward to the Indies, and 
thence to the same straits again ? And is not 
there a great sea called the Atlantic sea that run- 
neth northward to us ? And does not the great 
south sea run also up into the northern seas ? But 
I think the Indian and the South sea of themselves 
to be greater than all the rest of the surface of the 

B. How lieth the water in those two seas ? 

A. East and west, and rises and falls a little, as 
it is forced to do by this compounded motion, 
which is a kind of succussion of the earth, and 
fills both the Atlantic and Northern seas. 


CHAP. v. B. All this would not make a visible difference 
s , between high and low water, because this motion 

Of the motions m 

of water and air. being so regular, the unevenness would not be 
great enough to be seen. For though in a basin 
the water would be thrown into the air, yet the 
earth cannot throw the sea into the air. 

A. Yes ; the basin, if gently moved, will make 
the water so move, that you shall hardly see it rise. 

B. It may be so. But you should never see it 
rise as it doth, if it were not checked. For at the 
straits of Magellan, the great South sea is checked 
by the shore of the continent of Peru and Chili, 
and forced to rise to a great height, and made to 
run up into the northern seas on that side by the 
coast of China ; and at the return is checked again 
and forced through the Atlantic into the British 
and German seas. And this is done every day. 
For we have supposed that the earth's motion in 
the ecliptic caused by the sun is annual ; and that 
its motion in the equinoctial is diurnal. It follow- 
eth therefore from this compounded motion of the 
earth, the sea must ebb and flow twice in the space 
of twenty-four hours, or thereabout. 

A. Has the moon nothing to do in this busi- 
ness ? 

B. Yes. For she hath also the like motion. And 
is, though less swift, yet much nearer to the earth. 
And therefore when the sun and moon are in con- 
junction or opposition, the earth, as from two 
agents at once, must needs have a greater succus- 
sion. And if it chance at the same time the moon 
also be in the ecliptic, it will be yet greater, be- 
cause the moon then worketh on the earth less 


A. But when the full or new moon happen to be CHAP. v. 
then when the earth is in the equinoctial points, ' ' 

^ , r ' Oi the motions 

the tides are greater than ordinary. Why is that ? of water and air. 

B. Because then the force by which they move 
the sea, is at that time, to the force by which they 
move the same at other times, as the equinoctial 
circle to one of its parallels, which is a lesser circle. 

A. It is evident. And it is pleasant to see the 
concord of so many and various motions, when they 
proceed from one and the same hypothesis. But 
what say you to the stupendous tides which happen 
on the coasts of Lincolnshire on the east, and in the 
river of Severn on the west ? 

B. The cause of that, is their proper situation. 
For the current of the ocean through the Atlantic 
sea, and the current of the south sea through the 
northern seas, meeting together, rise the water in 
the Irish and British seas a great deal higher than 
ordinary. Therefore the mouth of the Severn 
being directly opposite to the current from the 
Atlantic sea, and those sands on the coast of Lin- 
colnshire directly opposite to the current of the 
German sea, those tides must needs fall furiously 
into them, by this succussion of the water. 

A. Does, when the tide runs up into a river, the 
water all rise together, and fall together when it 
goes out ? 

B. No : one part riseth and another falleth at 
the same time ; because the motion of the earth 
rising and falling, is that which makes the tide. 

A. Have you any experiment that shows it ? 

B. Yes. You know that in the Thames, it is 
high water at Greenwich before it is high w r ater at 
London-bridge. The water therefore falls at 
Greenwich whilst it riseth all the way to London. 


CHAP. v. But except the top of the water went up, and the 

l wer P art downward, it were impossible. 
ir. ^4. it is certain. It is strange that this one 
motion should salve so many appearances, and so 
easily. But I will produce one experiment of water, 
not in the sea, but in a glass. If you can show me 
that the cause of it is this compounded motion, I 
shall go near to think it the cause of all other 
effects of nature hitherto disputed of. The experi- 
ment is common, and described by the Lord Chan- 
cellor Bacon, in the third page of his natural history. 
Take, saith he, a glass of water, and draw your 
finger round about the lip of the glass, pressing it 
somewhat hard ; after you have done so a few 
times, it will make the water frisk up into a fine 
dew. After I had read this, I tried the same with 
all diligence myself, and found true not only the 
frisking of the water to above an inch high, but 
also the whole superficies to circulate, and withal 
to make a pleasant sound. The cause of the frisk- 
ing he attributes to a tumult of the inward parts 
of the substance of the glass striving to free itself 
from the pressure. 

B. I have tried and found both the sound and 
motion ; and do not doubt but the pressure of the 
parts of the glass was part of the cause. But the 
motion of my finger about the glass was always 
parallel ; and when it chanced to be otherwise, 
both sound and motion ceased. 

A. I found the same. And being satisfied, I 
proceed to other questions. How is the water, 
being a heavy body, made to ascend in small par- 
ticles into the air, and be there for a time sustained 
in form of a cloud, and then fall down again in 
rain ? 


B. I have shown already, that this compounded CHAP. v. 
motion of the sun, in one part of its circumlation, ft - v 7~~" r T"' 

3 * y Of the motions 

drives the air one way, and in the other part, the f water ana air. 
contrary way ; and that it cannot draw it back 
again, no more than he that sets a stone a flying 
can pull it back. The air therefore, which is con- 
tiguous to the water, being thus distracted, must 
either leave a vacuum, or else some part of the 
water must rise and fill the spaces continually for- 
saken by the air. But, that there is no vacuum, 
you have granted. Therefore the water riseth into 
the air, and maketh the clouds ; and seeing they 
are very small and invisible parts of the water, they 
are, though naturally heavy, easily carried up and 
down with the wind, till, meeting with some moun- 
tain or other clouds, they be pressed together into 
greater drops, and fall by their weight. So also it 
is forced up in moist ground, and with it many 
small atoms of the earth, which are either twisted 
with the rising water into plants, or are carried up 
and down in the air incertainly. But the greatest 
quantity of water is forced up from the great South 
and Indian Seas, that lie under the tropic of Capri- 
corn. And this climate is that which makes the 
sun's perigseum to be always on the winter-sol- 
stice. And that is the part of the terrestrial globe 
which Kepler says is kind to the sun ; whereas 
the other part of the globe, which is almost all dry 
land, has an antipathy to the sun. And so you see 
where this magnetical virtue of the earth lies. For 
the globe of the earth having no natural appetite 
to any place, may be drawn by this motion of the 
sun a little nearer to it, together with the water 
which it raiseth. 



CHAP. v. A. Can you guess what may be the cause of 

OfX^Is Win(i? 

of naterand air. B. I think it manifest that the unconstant winds 
proceed from the uncertain motion of the clouds 
ascending and descending, or meeting with one 
another. For the winds after they are generated 
in any place by the descent of a cloud, they drive 
other clouds this way and that way before them, 
the air seeking to free itself from being pent up in 
a strait. For when a cloud descendeth, it makes 
no wind sensible directly under itself. But the air 
between it and the earth is pressed and forced to 
move violently outward. For it is a certain expe- 
riment of mariners, that if the sea go high when 
they are becalmed, they say they shall have more 
wind than they would ; and take in their sails all 
but what is necessary for steering. They know, it 
seems, that the sea is moved by the descent of 
clouds at some distance off: which presseth the 
water, and makes it corne to them in great waves. 
For a horizontal wind docs but curl the water. 

A. From whence come the rivers ? 

B. From the rain, or from the falling of snow on 
the higher ground. But w^hen it dcscendeth under 
ground, the place where it again ariseth is called 
the spring. 

A. How then can there be a spring upon the top 
of a hill? 

B. There is no spring upon the very top of a 
hill, unless some natural pipe bring it thither from 
a higher hill. 

A. Julius Scaliger says, there is a river, and in 
it a lake, upon the top of Mount Cenis in Savoy ; 
and will therefore liUve the springs to be ingendered 


in the caverns of the earth by condensation of the CHAP. v. 

Ot the motions 

B. I wonder he should say that. I have passed of water and air. 
over that hill twice since the time I read that in 
Scaliger, and found that river as I passed, and went 
by the side of it in plain ground almost two miles ; 
where I saw the water from two great hills, one 
on one side, the other on the other, in a thousand 
small rillets of melting snow fall down into it. 
Which has made me never to use any experiment 
the which I have not myself seen. As for the con- 
version of air into water by condensation, and of 
water into air by rarefaction, though it be the doc- 
trine of the Peripatetics, it is a thing incogitable, 
and the words are insignificant. For by densum is 
signified only frequency and closeness of parts ; and 
by rarum the contrary. As when we say a town 
is thick with houses, or a wood with trees, we mean 
not that one house or tree is thicker than another, 
but that the spaces between are not so great. But, 
since there is no vacuum, the spaces between the 
parts of air are no larger than between the parts 
of water, or of any thing else. 

A . What think you of those things which mari- t 
ners that have sailed through the Atlantic Sea, 
called spouts, which pour down water enough at 
once to drown a great ship ? 

JJ. It is a thing I have not seen : and therefore 
can say nothing to it ; though I doubt not but 
when two very large and heavy clouds shall be dri- 
ven together by two great and contrary winds, the 
thing is possible. 

A. I think your reason good. And now I will 
propound to you another experiment. I have seen 


CHAP. v. an exceeding small tube of glass with both ends 
N ' open, set upright in a vessel of water, and that the 

Of the motions * ' X O ^ ' 

oi water ami air. superficies of the water within the tube was higher 
a good deal than of that in the vessel ; but I see no 
reason for it. 

B. Was not part of the glass under water? 
Must not then the water in the vessel rise ? Must 
not the air that lay upon it rise with it ? Whither 
should this rising air go, since there is no place 
empty to receive it ? It is therefore no wonder 'if 
the water, pressed by the substance of the glass 
which is dipt into it, do rather rise into a very 
small pipe, than come about a longer way into the 
open air. 

A. It is very probable. I observed also that the 
top of the inclosed water was a concave superficies; 
which I never saw in other fluids. 

B. The \vater hath some degree of tenacity, 
though not so great but that it will yield a little to 

the motion of the air ; as is manifest in the bubbles 

of water, where the concavity is always towards 
the air. And this I think the cause why the air 
and water meeting in the tube make the superficies 
towards the air concave, which it cannot do to a 
fluid of greater tenacity. 

A. If you put into a basin of water a long rag of 
cloth, first drenched in water, and let the longer 
part of it hang out, it is known by experience, that 
the water will drop out as long as there is any part 
of the other end under water. 

B. The cause of it is, that \vater, as I told you, 
hath a degree of tenacity. And therefore being 
continued in the rag till it be lower without than 
within, the weightVill make it continue dropping, 


though not only because it is heavy (for if the rag CHAP. v. 
lay higher without than within, and were made "~ ' ~* 

J 7 Ol the motions 

heavier by the breadth, it would not descend), but <>i water ami mr 
it is because all heavy bodies naturally descend 
with proportion of swiftness duplicate to that of 
the time ; whereof I shall say more when we talk 
of gravity. 

A. You see how despicable experiments I trouble 
you with. But I hope you will pardon me. 

B. As for mean and common experiments, I 
think them a great deal better witnesses of nature, 
than those that are forced by tire, and known but 
to very few. 



/I. IT is a fine day, and pleasant walking through 
the fields, but that the sun is a little too hot. 
B. How know you that the sun is hot ? 

A. I feel it. 

B. That is to say, you know that yourself, but 
not that the sun is hot. But when you find your- 
self hot, what body do you feel ? 

A. None. 

B. How then can you infer your heat from the 
sense of feeling ? Your walking may have made 
you hot : is motion therefore hot ? No. You are 
to consider the concomitants of your heat ; as, that 
you are more faint, or more ruddy, or that you 
sweat, or feel some endeavour of moisture or spi- 
rits tending outward ; and when you have found 
the causes of those accidents, you have found the 


CHAP. vi. causes of heat, which in a living creature, and 
" ,_ ' especially in a man, is many times the motion of 

Of the causes L J . . J 

and effects of the parts within him, such as happen in sickness, 

heat and cold. 1,1 p , i i i i 

anger, and other passions of the mind ; which are 
not in the sun nor in fire. 

A. That which I desire now to know, is what 
motions and of what bodies without me are the 
efficient causes of my heat. 

B. I showed you yesterday, in discoursing of rain, 
how by this compounded motion of the sun's body, 
the air was every way at once thrust off west and 
east ; so that where it was contiguous, the small 
parts of the water were forced to rise, for the 
avoiding of vacuum. Think then that your hand 
w r ere in the place of water so exposed to the sun. 
Must not the sun work upon it as it did upon the 
water ? Though it break not the skin, yet it will 
give to the inner fluids and looser parts of your 
hand, an endeavour to get forth, which will extend 
the skin, and in some climates fetch up the blood, 
and in time make the skin black. The fire also 
will do the same to them that often sit with their 
naked skins too near it. Nay, one may sit so near, 
without touching it, as it shall blister or break the 
skin, and fetch up both spirits and blood mixt into 
a putrid oily matter, sooner than in a furnace oil 
can be extracted out of a plant. 

A. But if the water be above the fire in a kettle, 
what then will it do ? Shall the particles of water 
go toward the fire, as it did toward the sun ? 

B. No. For it cannot. But the motion of the 
parts of the kettle which are caused by the fire, 
shall dissipate the water into vapour till it be all 
cast out. 


A. What is that you call fire ? Is it a hard or CHAP. vi. 
fluid body ? Ti"' ' 

J Oi the causes 

B. It is not any other body but that of the and envcts of 

-... , i 1 11 i i i i heat and cold 

shining coal ; which coal, though extinguished with 
water, is still the same body. So also in a very 
hot furnace,, the hollow spaces between the shining 
coals; though they burn that you put into them, 
are no other body than air moved. 

A. Is it not flame ? 

It. No. For flame is nothing but a multitude of 
sparks, and sparks are but the atoms of the fuel 
dissipated by the incredible swift motion of the 
movent, which makes every spark to seem a hun- 
dred times greater than it is, as appears by this ; 
that, when a man swings in the air a small stick 
fired at one end, though the motion cannot be 
very swift, yet the fire will appear to the eye to 
be a long, straight, or crooked line. Therefore a 
great many sparks together flying upward, must 
needs appear unto the sight as one continued flame. 
Nor are the sparks stricken out of a flint any thing 
else but small particles of the stone, which by their 
swift motion are made to shine. But that fire is 
not a substance of itself, is evident enough by thig, 
that the sun -beams passing through a globe of 
water will burn as other fire does. Which beams, 
if they were indeed fire, would be quenched in the 

A. This is so evident, that I \vonder so wise men 
as Aristotle and his followers, for so long a time 
could hold it for an element, and one of the 
primary parts of the universe. But the natural 
heat of a man or other living creature, whence 
proceedeth it r Is there anything within their 
bodies that hath this compounded motion ? 


CHAP. vi. B. At the breaking up of a deer I have seen it 
^~T" " plainly in his bowels as Ions; as they were warm. 

Of the causes . J 

and effects of And it is called the peristaltic motion, and in the 
heart of a beast newly taken out of his body ; and 
this motion is called systole and diastole. But 
they are both of them this compounded motion, 
whereof the former causeth the food to wind up 
and down through the guts, and the latter makes 
the circulation of the blood. 

A. What kind of motion is the cause of cold ? 
Methinks it should be contrary to that which 
causeth heat. 

B. So it is in some respect. For seeing the 
motion that begets heat, tendeth to the separation 
of the parts of the body whereon it acteth, it 
stands with reason, that the motion which maketh 
cold, should be such as sets them closer together. 
But contrary motions are, to speak properly, when 
upon two ends of a line two bodies move towards 
each other, the effect w r hereof is to make them 
meet. But each of them, as to this question, is 
the same. 

A. Do you think (as many philosophers have 
held and now hold), that cold is nothing but a pri- 
vation of heat ? 

B. No. Have you never heard the fable of the 
satyr that dwelling with a husbandman, and seeing 
him blow his fingers to warm them, and his pottage 
to cool it, was so scandalized, that he ran from him, 
saying he would no longer dwell with one that 
could blow both hot and cold with one breath ? 
Yet the cause is evident enough. For the air 
which had gotten a calefactive power from his 
vital parts, was from his mouth arid throat gently 


diffused on his fingers, and retained still that CHAP. vi. 
power. But to cool his pottage he straightened ^~^^ es 
the passage at his lips, which extinguished the and effects of 

-, n . heat and cold 

calefactive motion. 

A. Do you think wind the general cause of 
cold ? If that were true, in the greatest winds we 
should have the greatest frosts. 

B. I mean not any of those uncertain winds 
which, I said, were made by the clouds, but such 
as a body moved in the air makes to and against 
itself ; (for it is all one motion of the air whether 
it be carried against the body, or the body against 
it); such a wind as is constant, if no other be stirring, 
from east to west ; and made by the earth turn- 
ing daily upon its own centre ; which is so swift, 
as, except it be kept off by some hill, to kill a man, 
as by experience hath been found by those who 
have passed over great mountains, and specially 
over the Andes which are opposed to the east. 
And such is the wind which the earth maketh in 
the air by her annual motion, which is so swift, as 
that, by the calculation of astronomers, to go sixty 
miles in a minute of an hour. And therefore this 
must be the motion which makes it so cold about 
the poles of the ecliptic. 

A. Does not the earth make the wind as great 
in one part of the ecliptic as in another ? 

B. Yes. But when the sun is in Cancer, it 
tempers the cold, and still less and less, but least of 
all in the winter-solstice, where his beams are most 
oblique to the superficies of the earth. 

A. I thought the greatest cold had been about 
the poles of the equator. 

B. And so did I once. But the reason commonly 


CHAP. vr. given for it is so improbable, that I do not think 
^TT" ' so now. For the cause they render of it is only. 

Of the causes f ... 

and effects of that the motion of the earth is swiftest in the 
equinoctial, and slowest about the poles ; and conse- 
quently, since motion is the cause of heat, and cold 
is but, as it was thought, a want of the same, they 
inferred that the greatest cold must be about the 
poles of the equinoctial. Wherein they miscounted. 
For not every motion causeth heat, but this agita- 
tion only, which we call compounded motion ; 
though some have alleged experience for that 
opinion ; as that a bullet out of a gun will with its 
own swiftness melt. Which I never shall believe. 

A. It is a common thing with many philosophers 
to maintain their fancies with any rash report, and 
sometimes with a lie. But how is it possible that 
so soft a substance as water should be turned into 
so hard a substance as ice ? 

B. When the air shaves the globe of the earth 
with such swiftness, as that of sixty miles in a 
minute of an hour, it cannot, where it meets with 
still water, but beat it up into small and undis- 
tinguishable bubbles, and involve itself in them as 
in so many bladders or skins of water. And ice is 
nothing else but the smallest imaginable parts of 
air and water mixed; which is made hard by this 
compounded motion, that keeps the parts so close 
together, as not to be separated in one place with- 
out disordering the motion of them all. For when 
a body will not easily yield to the impression of an 
external movent in one place without yielding in 
all, we call it hard ; and when it does, we say it is 

A. Why is not ice'as well made in a moved as in 


a still water ? Are there not great seas of ice in CHAP.VI. 
the northern parts of the earth ? oTthe' causes 

B. Yes, and perhaps also in the southern parts, and effects of 

-_ T . . , . , i i heat and cold. 

But I cannot imagine how ice can be made in such 
agitation as is always in the open sea, made by the 
tides and by the winds. But how it may be made 
at the shore, it is not hard to imagine. For in a 
river or current, though swift, the water that 
udliereth to the banks is quiet, and easily by the 
motion of the air driven into small insensible bub- 
bles ; and so may the water that adhereth to those 
bubbles, and so forwards till it come into a stream 
that breaks it, and then it is no wonder though the 
fragments be driven into the open sea, and freeze 
together into greater lumps. But when in the 
open sea, or at the shore, the tide or a great wave 
shall arise, this young and tender ice will presently 
be washed away. And therefore I think it evident, 
that as in the Thames the ice is first made at the 
banks where the tide is weak or none, and, broken 
by the stream, comes down to London, and part 
goes to the sea floating till it dissolve, and part, 
being too great to pass the bridge, stoppeth there 
and sustains that \\hich follows, till the river he 
quite frozen over ; so also the ice in the northern 
seas begins first at the banks of the continent and 
islands which are situated in that climate, and then 
broken off, are carried up and down, and one 
against another, till they become great bodies. 

A. But what if there be islands, and narrow 
inlets of the sea, or rivers also about the pole of 
the equinoctial ? 

B. If there be, it is very likely the sea may also 
there be covered all over with ice. But for the 


CHAP. vi. truth of this, we must stay for some farther dis- 

Of the causes 

ami ejects of A. When the ice is once made and hard, what 

heat and cold. 1 . . 

dissolves it ? 

B. The principal cause of it, is the weight of the 
water itself; but not without some abatement in 
the stream of the air that hardeneth it ; as when 
the sunbeams are less oblique to the earth, or 
some contrary wind resisteth the stream of the 
air. For when the impediment is removed, then 
the nature of the water only worketh, and, being 
a heavy body, downward. 

A. I forgot to ask you, why two pieces of wood 
rubbed swiftly one against another, will at length 
set on fire. 

IL Not only at length, but quickly, if the wood 
be dry. And the cause is evident, viz. the com- 
pounded motion which dissipates the external small 
parts of the wood. And then the inner parts must 
of necessity, to preserve the plerititude of the uni- 
verse, come after ; first the most fluid, and then 
those also of greater consistence, which are first 
erected, and the motion continued, made to fly 
swiftly out ; whereby the air driven to the eye of 
the beholder, maketh that fancy which is called 

A. Yes ; I remember you told me before, that 
upon any strong pressure of the eye, the resistance 
from within would appear a light. But to return 
to the enquiry of heat and cold, there be two things 
that beyond all other put me into admiration. One 
is the swiftness of kindling in gunpowder. The 
other is the freezing t of water in a vessel, though 
not far from the fire, set about with other water 


with ice and snow in it. When paper or flax is CHAP. vi. 

flaming, the flame creeps gently on ; and if a house ^TT ' 
? . . or ^ ie causps 

full of paper were to be burnt with putting a candle ami ec-ccu nt 

..... M1 ,-, i T -, lieat and cold. 

to it, it will be long in burning ; whereas a spark 
of fire would set on flame a mountain of gun- 
powder in almost an instant. 

B. Know you not gunpowder is made of the 
powder of charcoal, brimstone, and saltpetre ? 
Whereof the first will kindle with a spark, the 
Second flame as soon as touched with fire ; and the 
third blows it, as being composed of many orbs of 
salt filled with air, and as it dissolveth in the flame, 
furiously blowing increaseth it. And as for making 
ice by the fireside ; it is manifest that whilst the 
snow is dissolving in the external vessel, the air 
must in the like manner break forth, and shave the 
superficies of the inner vessel, and work through 
the water till it be frozen. 

A. I could easily assent to this, if I could con- 
ceive how the air that shaves, as you say, the out- 
side of the vessel, could work through it. I 
conceive well enough a pail of water with ice or 
snow dissolving in it, and how it causeth wind. 
But how that wind should communicate itself 
through the vessel of wood or metal, so as to make 
it shave the superficies of the water which is within 
it, I do not so well understand. 

B. I do not say the inner superficies of the 
vessel shaves the water within it. But it is mani- 
fest that the wind made in the pail of water by the 
melting snow or ice presseth the sides of the vessel 
that standeth in it ; and that the pressure worketh 
clean through, how hard soever the vessel be ; and 
that again worketh on the water within, by resti- 


CHAP. vi. tution of its parts, and so hardeneth the water by 
*_, degrees. 

Of the causes & 

and eflects of A. I understand you now. The ice in the pail 
by its dissolution transfers its hardness to the water 

B. You are merry. But supposing, as I do, that 
the ice in the pail is more than the water in the 
vessel, you will find no absurdity in the argument. 
Besides, the experiment, you know, is common. 

A. I confess it is probable. The Greeks have 
the word <pi/oj (whence the Latins have their word 

frigus] to signify the curling of the water by the 
wind ; and use the same also for horror, which is 
the passion of one that cometh suddenly into a cold 
air, or is put into a sudden affright, whereby he 
shrinks, and his hair stands upright. Which mani- 
festly shows that the motion which causeth cold, is 
that which pressing the superficies of a body, sets 
the parts of it closer together. But to proceed in 
my queries. Monsieur Des Cartes, whom you know, 
hath w r ritten somewhere, that the noise we hear in 
thunder, proceeds from breaking of the ice in the 
clouds ; what think you of it r Can a cloud be 
turned into ice ? 

B. Why not ? A cloud is but w r ater in the air ? 

A. But how ? For he has not told us that. 

B. You know that it is only in summer, and in 
hot weather, that it thunders ; or if in winter, it is 
taken for a prodigy. You know also, that of clouds, 
some are higher, some lower, and many in number, 
as you cannot but have oftentimes observed, with 
spaces between them. Therefore, as in all currents 
of water, the water is there swiftest where it is 
straitened with islands, so must the current of 


air made by the annual motion be swiftest there, CHAP. vi. 
where it is checked with many clouds, through 7^^ e s 
which it must, as it were, be strained, and leave and efiVcts of 

9 9 \ heat and cold 

behind it many small particles of earth always in 
it, and in hot w r eather more than ordinary. 

A. This I understand, and that it may cause ice. 
But when the ice is made, how is it broken ? And 
why falls it not down in shivers ? 

B. The particles are enclosed in small caverns of 
the ice ; and their natural motion being the same 
which we have ascribed to the globe of the earth, 
requires a sufficient space to move in. But when 
it is imprisoned in a less room than that, then a 
great part of the ice breaks : and this is the thun- 
der-clap. The murmur following is from the sett- 
ling of the air. The lightning is the fancy made 
by the recoiling of the air against the eye. The 
fall is in rain, not in shivers ; because the prisons 
which they break are extreme narrow, and the 
shivers being small, are dissolved by the heat. But 
in less heat they would fall in drops of hail, that is 
to say, half frozen by the shaving of the air as they 
fall, and be in a very little time, much less than 
snow or ice, dissolved. 

A. Will not that lightning burn ? 

B. No. But it hath often killed men with cold. 
But this extraordinary swiftness of lightning con- 
sisteth not in the expansion of the air, but in a 
straight and direct stream from where it breaks 
forth; which is in many places successively, ac- 
cording to the motion of the cloud. 

A. Experience tells us that. I have now done 
with my problems concerning the great bodies of 
the world, the stars, and eleihent of air in which 


CHAP. vi. they are moved, and am therein satisfied, and the 
nTTT" rather, because you have answered me by the sup- 

Of the causes . 

and effects of position of one only motion, and commonly known, 

heat and cold. .. , . , , / /-^ i 

and the same with that of Copernicus, whose opi- 
nion is received by all the learned ; and because 
you have not used any of these empty terms, sym- 
pathy, antipathy, antiperistasis, etc., for a natural 
cause, as the old philosophers have done to save 
their credit. For though they were many of them 
wise men, as Plato, Aristotle, Seneca, and others, 
and have written excellently of morals and politics, 
yet there is very little natural philosophy to be 
gathered out of their writings. 

B. Their ethics and politics are pleasant reading, 
but I find not any argument in their discourses of 
justice or virtue drawn from the supreme authority, 
on whose laws all justice, virtue, and good politics 

A. Concerning this cover, or, as some have called 
it, the scurf or scab of the terrestrial star, I will 
begin with you tomorrow. For it is a large sub- 
ject, containing animals, vegetables, metals, stones, 
and many other kinds of bodies, the knowledge 
whereof is desired by most men, and of the greatest 
and most general profit. 

B. And this is it, in which I shall give you the 
least satisfaction ; so great is the variety of motion, 
and so concealed from human senses. 




A. CONCERNING this cover of the earth, made up CHAP. vn. 
of an infinite number of parts of different natures, ^ 

I had much ado to find any tolerable method of soft, and ot 
enquiry. But I resolved at last to begin with the 1^^' 
questions concerning hard and soft,, and what kind 
of motion it is that makes them so. I know that 
in any pulsion of air, the parts of it go innumerable 
and inexplicable ways ; but I ask only if every 
point of it be moved ? 

B. No. If you mean a mathematical point, you 
know it is impossible. For nothing is movable but 
body. But I suppose it divisible, as all other 
bodies, into parts divisible. For no substance can 
be divided into nothings. 

A. Why may not that substance within our 
bodies, which are called animal spirits, be another 
kind of body, and more subtile than the common air? 

B. I know not why, no more than you or any 
man else knows why it is not very air, though 
purer perhaps than the common air, as being 
strained through the blood into the brain and 
nerves. But howsoever that be. there is no doubt, 
but the least parts of the common air, respectively 
to the whole, will easilier pierce, with equal motion, 
the body that resisteth them, than the least parts 
of water. For it is by motion only that any muta- 
tion is made in any thing ; and all things standing 
as they did, will appear as they did. And that 
which changeth soft into hard, must be such as 



CHAP. vii. makes the parts not easily to be moved without 
* ' being moved all together ; which cannot be done 

Of hard and . 

soft, and of but by some motion compounded. And we call 
% mtho mr. hard, that whereof no part can be put out of order 

without disordering all the rest ; which is not easily 


A. How water and air beaten into extreme small 
bubbles is hardened into ice, you have told me 
already, and I understand it. But how a soft 
homogeneous body, as air or water, should be so 
hardened, I cannot imagine. 

B. There is no hard body that hath not also some 
degree of gravity ; and consequently, being loose, 
there must be some efficient cause, that is, some 
motion, when it is severed from the earth, to bring 
the same to it again. And seeing this compounded 
motion gives to the air and w r ater an endeavour 
from the earth, the motion which must hinder it, 
must be in a way contrary to the compounded 
motion of the earth. For whatsoever, having been 
asunder, comes together again, must come contrary 
ways, as those that follow one another go the same 
way, though both move upon the same line. 

A. What experiment have you seen to this pur- 
pose ? 

B. I have seen a drop of glass like that of the 
second figure, newly taken out of the furnace, and 
hanging at the end of an iron rod, and yet fluid, 
and let fall into the water and hardened. The club- 
end of it A A coming first to the water, the tail 
B C following it. It is proved before, that the 
motion that makes it is a compounded motion, and 
gives an endeavour outward to every part of it ; 
and that the motion which maketh cold, is such as 


shaving the body in every point of contact, and CHAP. vn. 
turning it, gives them all an endeavour inward. 0f ^J^" 
Such is this motion made bv the sinking; of the hot **>&> and of 

. , . P , the atoms that 

and fluid glass into the water. It is therefore ny m the an 
manifest that the motion which hardeneth a soft 
body, must in every point of contact be in the con- 
trary way to that which makes a hard body soft. And 
farther, that slender tail B C shall be made much 
more hard than common glass. For towards the 
upper end, in C, you cannot easily break it, as 
small as it is. And when you have broken it, the 
whole body will fall into dust, as it must do, seeing 
the bending is so difficult. For all the parts are 
bent with such force, that upon the breaking at 
D, by their sudden restitution to their liberty, they 
will break together. And the cause why the tail 
B C, being so slender, becomes so hard, is, that all 
the endeavour in the great part A B, is propagated 
to the small part B C, in the same manner as the 
force of the sun-beams is derived almost to a point 
by a burning-glass. But the cause why, when it 
is broken in D, it breaks also in so many other 
places, is, that the endeavour in all the other parts, 
which is called the spring, unbends it; from whence* 
a motion is caused the contrary way, and that 
motion continued bends it more the other way and 
breaks it, as a bow over-bent is broken into shivers 
by a sudden breaking of the string. 

A. I conceive now how a body which having 
been hard and softened again, may be rehardened ; 
but how a fluid and mere homogeneous body, as air 
or water, may be so, I see not yet. For the hard- 
ening of water is making a hard body of two fluids, 
whereof one, which is the water, hath some tena- 

K 2 


CHAP. vii. city ; and so a man may make a bladder hard with 
n ,V~ / 7* blowing into it. 

Of hard and 

soft, and of B. As for mere air, which liath no natural mo- 
fl/mthe air tioii of itself, but is moved only by other bodies of 
a greater consistence, I think it impossible to be 
hardened. For the parts of it so easily change 
places, that they can never be fixed by any motion. 
No more I think can water, which though some- 
what less fluid, is with an insensible force very easily 

A. It is the opinion of many learned men, that 
ice, in long time, will be turned into crystal ; 
and they allege experience for it. For they say 
that crystal is found hanging on high rocks in the 
Alps, like icicles on the eaves of a house ; and why 
may not that have formerly been ice, and in many 
years have lost the power of being reduced ? 

B. If that were so, it w r ould still be ice, though 
also crystal : which cannot be, because crystal is 
heavier than water, arid therefore much heavier 
than ice. 

A. Is there then no transubstantiation of bodies 
but by mixture ? 
, B. Mixture is no transubstantiation. 

A. Have you never seen a stone that seemed to 
have been formerly wood, and some like shells, and 
some like serpents, and others like other things ? 

B. Yes. I have seen such things, and particularly 
I saw at Rome, in a stone-cutter's workhouse, a 
billet of wood, as I thought it, partly covered with 
bark, and partly with the grain bare, as long as a 
man's arm, and as thick as the calf of a man's leg ; 
w r hich handling I found extreme heavy, and saw a 
small part of it which was polished, and had a very 


fine gloss, and thought it a substance between stone CHAP. vn. 
and metal, but nearest to stone. I have seen also " 7~? 

J m m Of haul and 

a kind of slate painted naturally with forest- work. soft > an <* f 
And I have seen in the hands of a chemist of my iiymthe air 
acquaintance at Paris, a broken glass, part of a 
retort, in which had been the rosin of turpentine, 
wherein though there were left no rosin, yet there 
appeared in the piece of glass many trees ; and 
plants in the ground about them, such as grow in 
woods ; and better designed than they could be 
done by any painter ; and continued so for a long 
time. These be great wonders of nature, but I 
will not undertake to show their causes. But yet 
this is most certain, that nothing can make a hard 
body of a soft, but by some motion of its parts. 
For the parts of the hardest body in the world can 
be no closer together than to touch ; and so close 
are the parts of air and \vater, and consequently 
they should be equally hard, if their smallest parts 
had not different natural motions. Therefore if 
you ask me the causes of these effects, I answer, 
they are different motions. But if you expect 
from me how and by what motions, I shall fail you. 
For there is no kind of substance in the world 
now, that was not at the first creation, when the 
Creator gave to all things what natural and special 
motion he thought good. And as he made some 
bodies wondrous great, so he made others wond- 
rous little. For all his works are w r ondrous. Man 
can but guess, nor guess farther, than he hath 
knowledge of the variety of motion. I am there- 
fore of opinion, that whatsoever perfectly homo- 
geneous is hard, consisteth of the smallest parts, or, 
as some call them, atoms, tht were made hard in 


CHAP. vii. the beginning, and consequently by an eternal 
ofhard'an<r cause anc ^ that the hardness of the whole body is 
soft, and of caused only by the contact of the parts by pressure. 

the atoms that A TTrl J J . . , /, TIIT 

fly in the air. /I. What motion is it that maketh a hard body 
to melt ? 

B. The same compounded motion that heats, 
namely, that of fire, if it be strong enough. For 
all motion compounded is an endeavour to dissi- 
pate, as I have said before, the parts of the body 
to be moved by it. If therefore hardness consist 
only in the pressing contact of the least parts, this 
motion will make the same parts slide off from one 
another, and the whole to take such a figure as the 
weight of the parts shall dispose them to, as in 
lead, iron, gold, and other things melted with heat. 
But if the small parts have such figures as they 
cannot exactly touch, but must leave spaces be- 
tween them filled with air or other fluids, then this 
motion of the fire, will dissipate those parts some 
one way, some another, the hard part still hard ; 
as in the burning of wood or stone into ashes or 
lime. For this motion is that which maketh fer- 
mentation, scattering dissimilar parts, and congre- 
gating similar. 

A. Why do some hard bodies resist breaking 
more one way than another ? 

B. The bodies that do so, are for the most part 
wood, and receive that quality from their genera- 
tion. For the heat of the sun in the spring-time 
draweth up the moisture at the root, and together 
with it the small parts of the earth, and twisteth it 
into a small twig, by its motion upwards, to some 
length, but to very little other dimensions, and so 
leaves it to dry till the spring following ; and then 


does the same to that, and to every small part round CHAP. vn. 
about it ; so that upward the strength is doubled, * / 7* 

' I J Ot hard and 

and the next year trebled, &c. And these are soft, and of 
called the grain of the wood, and but touch one u^mZe air 
another, like sticks with little or no binding, and 
therefore can hardly be broken across the grain, 
but easily ail-along it. Also some other hard bodies 
have this quality of being more fragile one way 
than another, as we see in quarrels of a glass win- 
clow, that are aptest many times to break in some 
crooked line. The cause of this may be, that when 
the glass, hot from the furnace, is poured out upon 
a plain, any small stones in or under it will break 
the stream of it into divers lines, and not only 
weakei it, but also cause it falsely to represent the 
object you look on through it. 

A. What is the cause why a bow of wood or 
steel, or other very hard body, being bent, but not 
broken, will recover its former degree of straight- 
ness ? 

B. I have told you already, how the smallest 
parts of ?, hard body have every one, by the genera- 
tion of hardness, a circular, or other compounded 
motion ; such motion is that of the smallest parts 
of the bov. Which circles in the bending you 
press into narrower figures, as a circle into an 
ellipsis, and an ellipsis into a narrower but longer 
ellipsis with violence ; which turns their natural 
motion agahst the outward parts of the bow so 
bent, and is m endeavour to stretch the bow into 
its former pcsture. Therefore if the impediment 
be removed, tie bow mast needs recover its former 

A. It is marifest ; and the fcause can be no other 
but that, excep; the bow have sense. 


CHAP. vii. B. And though the bow had sense, and appetite 
nf V A ' 7* to boot, the cause will be still the same. 

Of hard and ' 

soft, and of A. Do you think air and water to be pure and 

<hc atoms that . i v * 

flj m the air. homogeneous bodies t 

B. Yes, and many bodies both hard and heavy 
to be so too, and many liquors also besides water. 

A. Why then do men say they find one air 
healthy, another infectious ? 

B. Not because the nature of the air varies, but 
because there are in the air, drawn, or rather, 
beaten up by the sun, many little bodies, whereof 
some have such motion as is healthful, others such 
as is hurtful to the life of man. For the snn, as 
you see in the generation of plants, can fetch up 
earth as well as water : and from the driest ground 
any kind of body that lieth loose, so it be small 
enough, rather than admit any emptiness. By 
some of these small bodies it is that we live ; which 
being taken in with our breath, pass into our blood, 
and cause it, by their compounded motion, to cir- 
culate through the veins and arteries ; wtich the 
blood of itself, being a heavy body, without it 
cannot do. What kind of substance these atoms 
are, I cannot tell. Some suppose them t> be nitre. 
As for those infectious creatures in the atr, whereof 
so many die in the plague, I have heard that 
Monsieur DCS Cartes, a very ingenious man, was of 
opinion, that they were little flies. But what 
grounds he had for it, I know not, thoagh there be 
many experiments that invite me believe it. 
For first, we know that the air is nevr universally 
infected over a whole country, but only in or near 
to some populous town. And therefore the cause 
must also be partly ascribed to the multitude 


thronged together, and constrained to carry their CHAP. VH. 
excrements into the fields round about and near to 

Ot haul 

their habitation, which in time fermenting breed M>it, ana 

. , , T i the atoms that 

worms, which commonly in a month or little more, n y , the an 
naturally become flies ; and though engendered at 
one town, may fly to another. Secondly, in the be- 
ginning of a plague, those that dwell in the suburbs, 
that is to say, nearest to this corruption, are the 
poorest of the people, that are nourished for the 
most part with the roots and herbs which grow in 
ihat corrupted dirt ; so that the same tilth makes 
both the blood of poor people, and the substance 
of ^he fly. And it is said by Aristotle, that every- 
thing is nourished by the matter whereof it is 
generated. Thirdly, when a town is infected, the 
gentlemen, and those that live on wholsomest food, 
scarce one of five hundred die of the plague. It 
seems therefore, whatsoever creatures they be that 
invade us irom the air, they can discern their 
proper nourishment, and do not enter into the 
mouth and nostrils with the breath of every man 
alike, as they would do if they were inanimate. 
Fourthly, a man may carry the infection with him a 
great way into the country in his clothes, and infect 
a village. Shall another man there draw the infec- 
tion from the clothes only by his breath ? Or from 
the hangings of a chamber wherein a man hath 
died ? It is impossible. Therefore whatsoever 
killing thing is in the clothes or hangings, it must 
rise and go into his mouth or nostrils before it can 
do him hurt. It must therefore be a fly, whereof 
great numbers get into the blood, and there feeding 
and breeding worms, obstruct the circulation of 
blood, and kill the man. 


CHAP. vii. A. I would we knew the palate of those little 
ofhard'amT animals ; we might perhaps find some medicine to 
soft, and of fright them from miiifflins: with our breath. But 

tho atoms that _ . i i i MI i T i 

iiy m the an. what is that which kills men that lie asleep too 
near a charcoal-fire ? Is it another kind of fly ? 
Or is charcoal venomous ? 

B. It is neither fly nor venom, but the effect of 
a flameless glowing fire, which dissipates those 
atoms that maintain the circulation of the blood ; 
so that for want of it, by degrees they faint, ancl 
being asleep cannot remove, but in short time, 
there sleeping die ; as is evident by this, that being 
brought into the open air, without other help, they 

A. It is very likely. The next thing I would be 
informed of, is the nature of gravity. But for that, 
if you please, we will take another day. 



B. WHAT books are those ? 
' A. Two books written by two learned men con- 
cerning gravity. I brought them with me, because 
they furnish me with some material questions about 
that doctrine ; though of the nature of gravity, I 
find no more in either of them than this, that gra- 
vity is an intririsical quality, by which a body so 
qualified descendeth perpendicularly towards the 
superficies of the earth. 

B. Did neither of them consider that descending 
is local motion ; that 1 they might have called it an 


intrinsical motion rather than an intrinsical CHAP. vm. 
quality ? _ *" ' 

A J Of gravity 

A. Yes. But not how motion should be intrin- and gravitation 
sical to the special individual body moved. For 

how should they, when you are the first that ever 
sought the differences of qualities in local motion, 
except your authority in philosophy were greater 
with them than it is ? For it is hard for a man to 
conceive, except he see it, how there should be 
motion within a body, otherwise than as it is in 
living creatures. 

B. But it may be they never sought, or des- 
paired of finding what natural motion could make 
any inanimate thing tend one way rather than 

A. So it seems. But the first of them inquires 
no farther than, why so much water, being a heavy 
body, as lies perpendicularly on a fish's back in 
the bottom of the sea, should not kill it. The 
other, whereof the author is Dr. Wallis, treateth 
universally of gravity. 

B. Well ; but what are the questions which 
from these books you intend to ask me ? 

A. The author of the first book tells me, that 
water and other fluids are bodies continued, and 
act, as to gravity, as a piece of ice would do of 
the same figure and quantity. Is that true ? 

B. That the universe, supposing there is no 
place empty, is one entire body, and also, as he 
saith it is, a continual body, is very true. And 
yet the parts thereof may be contiguous, without 
any other cohesion but touch. And it is also true, 
that a vessel of water will descend in a medium 
less heavy, but fluid, as ice w&uld do. 


CHAP. vin. A. But he means that water in a tub would 
"~t ' h ave the same effect upon a fish in the bottom of 

and gravitation, the tub, as so much ice would have. 

B. That also would be true, if the water were 
frozen to the sides of it. Otherwise the ice, if 
there be enough, will crush the fish to death. But 
how applies he this, to prove that the water cannot 
hurt a fish in the sea by its weight ? 

A. It plainly appears that water does not gra- 
vitate on any part of itself beneath it. 

B. It appears by experience, but not by this 
argument, though instead of water the tub were 
filled with quicksilver. 

A. I thought so. But how it comes to pass 
that the fish remains uncrushcd, I cannot tell. 

B. The endeavour of the quicksilver downward 
is stopped by the resistance of the hard bottom. 
But all resistance is a contrary endeavour ; that 
is, an endeavour upwards, which gives the like en- 
deavour to the quicksilver, which is also heavy, 
and thereby the endeavour of the quicksilver is 
diverted to the sides round about, where stopped 
again by the resistance of the sides, it receives an 
endeavour upwards, which carries the fish to the 
top, lying all the way upon a soft bed of quick- 
silver. This is the true manner how the fish is 
saved harmless. But your author, I believe, either 
wanted age, or had too much business, to study 
the doctrine of motion ; arid never considered that 
resistance is not an impediment only, or privation, 
but a contrary motion ; and that when a man 
claps two pieces of wax together, their contrary 
endeavour will turn both the pieces into one cake 
of wax. 


A. I know not the author ; but it seems he has CHAP. vm. 
deeplier considered this question than other men ; 

for in the introduction to his book he saith, " that 
men have pre-engaged themselves to maintain cer- 
tain principles of their own invention, and are 
therefore unwilling to receive anything that may 
render their labour fruitless ;" and, "that they have 
not strictly enough considered the several inter- 
ventions that abate, impede, advance, or direct 
the gravitation of bodies." 

B. This is true enough ; and he himself is one 
of those men, in that he considered not, that re- 
sistance is one of those interventions which abate, 
impede, and direct gravitation. But what are his 
suppositions for the questions he handles ? 

A. His first is, that as in a pyramid of brick, 
wherein the bricks are so joined that the upper- 
most lies everywhere over the joint or cement of 
the two next below it, you may break down a part 
and leave a cavity, and yet the bricks above will 
stand firm and sustain one another by their cross 
posture : so also it is in wheat, hailshot, sand, or 
water ; and so they arch themselves, and thereby 
the fish is every way secured by an arch of water 
over it. 

B. That the cause why fishes are not crushed 
nor hurt in the bottom of the sea by the weight of 
the water, is the water's arching itself, is very 
manifest. For if the uppermost orb of the water 
should descend by its gravity, it would tend toward 
the centre of the earth, and place itself all the way 
in a less and lesser orb, which is impossible. For 
the places of the same body are always equal. But 
that wheat, sand, hailshot, or loose stones should 

n firm nrph is; nnf prprlihlp 


CHAP. vni. A. The author therefore, it seems, quits it. and 
Of " rfmtv ~ taketh a second hypothesis for the true cause, 
and gravitation, though the former, he saith, be not useless, but 

contributes its part to it. 

B. I see, though he depart from his hypothesis, 

he looks back upon it with some kindness. What 

is his second hypothesis ? 

A. It is, that air and water have an endeavour 
to motion upward, downward, directly, obliquely, 
and every way. For air, he saith, will come down 
his chimney, and in at his door, and up his stairs. 

B. Yes, and mine too ; and so would w r ater, if 
I dwelt under water, rather than admit of vacuum. 
But what of that ? 

A. Why then it would follow 7 , that those several 
tendencies or endeavours would so abate, impede, 
and correct one another, as none of them should 
gravitate. Which being granted, the fish can take 
no harm ; wherein I find one difficulty, which is 
this : the water having an endeavour to motion 
every way at once, methinks it should go no w r ay, 
but lie at rest ; which, he saith, was the opinion 
of Stevinus, and rejecteth it, saying, it would crush 
the fish into pieces. 

B. I think the water in this case would neither 
rest nor crush. For the endeavour being, as he 
saith, intrinsical, and every way, must needs drive 
the water perpetually outward ; that is to say, 
as to this question, upwards ; and seeing the same 
endeavour in one individual body cannot be more 
ways at once than one, it will carry it on perpe- 
tually without limit, beyond the fixed stars ; and 
so we shall never more have rain. 


/?. What are Dr. Wallis's suppositions ? CHAP. vm. 

A. He goes upon experiments. And, first, he O p~^ 
allegeth this, that water left to itself without dis- and gravitation. 
turbance, does naturally settle itself into a horizon- 
tal plane. 

/}. He does not then, as your author and all 
other men, take gravity for that quality whereby 
a body tendeth to the centre of the earth. 

A. Yes, he defines gravity to be a natural pro- 
pension towards the centre of the earth. 

B. Then he contradicteth himself. For if all 
heavy bodies tend naturally to one centre, they 
shall never settle in a plane, but in a spherical 
superficies. But against this, that such an hori- 
zontal plane is found in water by experience, I say 
it is impossible. For the experiment cannot be 
made in a basin, but in half a mile at sea ex- 
perience visibly shows the contrary. According 
to this, he should think also that a pair of scales 
should hang parallel. 

A. He thinks that too. 

B. Let us then leave this experiment. What 
says he farther concerning gravity ? 

A. He takes for granted, not as an experiment 
but an axiom, that nature worketh not by election, 
but ad nltimum vlrium, with all the power it can. 

B. I think he means, (for it is a very obscure 
passage), that every inanimate body by nature 
worketh all it can without election ; which may 
be true. But it is certain that men, and beasts, 
work often by election, and often without election ; 
as when he goes by election, and falls without it. 
In this sense I grant him, that nature does all it 
can. But what infers he from it ? 


CHAP. viii. A. That naturally every body has every way, if 
ofgnmty the ways oppose not one another, an endeavour 
and gravitation. to mot i oll . m ft consequently, that if a vessel have 

two holes, one at the side, another at the bottom, 

the water will run out at both. 

B. Does he think the body of water that runs 

out at the side, and that which runs out at the 

bottom, is but one and the same body of water ? 

A . No, sure ; he cannot think but that they are 
two several parts of the whole water in the vessel! 

B. What wonder is it then, if two parts of 
water run two ways at once, or a thousand parts 
a thousand ways ? Does it follow thence that one 
body can go more than one way at once ? Why 
is he still meddling with things of such difficulty ? 
He will find at last that he has not a genius either 
for natural philosophy or for geometry. What 
other suppositions has he ? 

A. My first author had affirmed, that a lighter 
body does not gravitate on a heavier ; against 

Fl * 3 - this Dr. Wallis thus argueth : Let there be a 
siphon, A B C D, filled with quicksilver to the 
level A D ; if then you pour oil upon A as high as 
to E, he asketh if the oil in A E, as being heavy, 
shall not press down the quicksilver a little at A, 
and make it rise a little at D, suppose to F ; and 
answers himself, that certainly it will ; so that it 
is neither an experiment nor an hypothesis, but 
only his opinion. 

B. Whatsoever it be, it is not true ; though the 
doctor may be pardoned, because the contrary was 
never proved. 

A. Can you prove the contrary ? 

B. Yes ; for the "endeavour of the quicksilver 


both from A and D downward, is stronger than CHAP. vin. 
that of the oil downward. If, therefore, the en- _, ' ' 

' J Of gravity 

deavour of the quicksilver were not resisted by and gravitation, 
the bottom B C, it would fall so, by reason of the 
acceleration of heavy bodies in their descending, 
as to leave the oil, so that it should not only not 
press, but also not touch the quicksilver. It is 
true, in a pair of scales equally charged with 
quicksilver, that the addition of a little oil to 
either scale will make it preponderate. Arid that 
was it deceived him. 

A. It is evident. The last experiment he cites 
is the weighing of air in a pair of scales, where it 
is found manifestly that it has some little weight. 
For if you weigh a bladder, and put the weight 
into one scale, and then blow the bladder full of 
air, and put it into the other scale, the full bladder 
will outweigh the empty. Must not then the air 
gravitate ? 

B. It does not follow. I have seen the experi- 
ment just as you describe it, but it can never be 
thence demonstrated that air has any weight. 
For as much air as is pressed downward by the 
weight of the blown bladder, so much will rise 
from below, and lay itself spherically at the altitude 
of the centre of gravity of the bladder so blown. 
So that all the air within the bladder above that 
centre is carried thither imprisoned, and by vio- 
lence : and the force that carries it up is equal to 
that which presseth it down. There must, there- 
fore, be allowed some little counterpoise in the 
other scale to balance it. Therefore, the experi- 
ment proves nothing to his purpose. And whereas 
they say there be small heavy bodies in the air, 



CHAP. viii. which make it gravitate, do they think the force 
ofRTdMtv"' which brought them thither cannot hold them 

and gruMtation. there? 

A. I leave this question of the fish as clearly 
resolved, because the water tending every way to 
one point, which is the centre of the earth, must 
of necessity arch itself. And now tell me your own 
opinion concerning the cause of gravity, and why 
all bodies descend or ascend not all alike. For 
there can be no more matter in one place than 
another if the places be equal. 

13. I have already showed you in general, that 
the difference of motion in the parts of several 
bodies makes the difference of their natures. And 
all the difference of motions consisteth either in 
swiftness, or in the way, or in the duration. But 
to tell you in special w r hy gold is heaviest, and 
then quicksilver, and then, perhaps, lead, is more 
than I hope to know, or mean to enquire ; for 
I doubt not but that the species of heavy, hard, 
opaque, and diaphanous, were all made so at their 
creation, and at the same time separated from dif- 
ferent species. So that I cannot guess at any 
particular motions that should constitute their 
natures, farther than I am guided by the experi- 
ments made by fire or mixture. 

A. You hope not then to make gold by art ? 

B. No, unless I could make one and the same 
thing heavier than it was. God hath from the 
beginning made all the kinds of hard, and heavy, 
and diaphanous bodies that are, and of such figure 
and magnitude as he thought fit ; but how small 
soever, they may by accretion become greater in 
the mine, or perhaps by generation, though we 


know not how. But that gold, by the art of man, CHAP. vm. 
should be made of not gold. I cannot understand ; * ' ' 

0:7 ' Of gravity 

nor can they that pretend to show how. For the and gravitation. 
heaviest of all bodies, by what mixture soever of 
other bodies, will be made lighter, and not to be 
received for gold. 

A. Why, when the cause of gravity consisteth 
in motion, should you despair of finding it ? 

B. It is certain that when any two bodies meet, 
as the earth and any heavy body will, the motion 
that brings them to or towards one another, must 
be upon two contrary ways ; and so also it is 
when two bodies press each other in order to 
make them hard ; so that one contrariety of mo- 
tion might cause both hard and heavy, but it doth 
not, for the hardest bodies are not always the 
heaviest ; therefore I find no access that way to 
compare the causes of different endeavours of 
heavy bodies to descend. 

A. But show me at least how any heavy body 
that is once above in the air, can descend to the 
earth, when there is no visible movent to thrust or 
pull it down. 

B. It is already granted, that the earth hath 
this compounded motion supposed by Copernicus, 
and that thereby it casteth the contiguous air from 
itself every way round about. Which air so cast 
off, must continually, by its nature, range itself in 
a spherical orb. Suppose a stone, for instance, 
were taken up from the ground, and held up in the 
air by a man's hand, what shall come into the 
place it filled when it lay upon the earth ? 

A. So much air as is equpl to the stone in 
magnitude, must descend and place itself in an 


CHAP. viii. orb upon the earth. But then I see that to avoid 
nr % ' vacuum, another orb of air of the same magnitude 

Uf gravity * ( - ; 

and gravitatiou. must descend, and place itself in that, and so per- 
petually to the man's hand ; and then so much air 
as would fill the place must descend in the same 
manner, and bring the stone down with it. For 
the stone having no endeavour upward,, the least 
motion of the air, the hand being removed, will 
thrust it downward. 

B. It is just so. And farther, the motion of the 
stone downward shall continually be accelerated 
according to the odd numbers from unity ; as you 
know hath been demonstrated by Galileo. But we 
are nothing the nearer, by this, to the knowledge 
of why one body should have a greater endeavour 
downward than another. You see the cause of 
gravity is this compounded motion with exclusion 
of vacuum. 

A. It may be it is the figure that makes the 
difference. For though figure be not motion, yet 
it may facilitate motion, as you see commonly the 
breadth of a heavy body retardeth the sinking of it. 
And the cause of it is, that it makes the air have 
farther to go laterally, before it can rise from under 
it. For suppose a body of quicksilver falling in 
the air from a certain height, must it not, going as 
it does toward the centre of the earth, as it draws 
nearer and nearer to the earth, become more and 
more slender, in the form of a solid sector ? And 
if it have far to go, divide itself into drops ? This 
figure of a solid sector is like a needle w T ith the 
point downward, and therefore I should think that 
facilitating the mojion of it does the same that 
would be done by increasing the endeavour. 


B. Do not you see that this way of facilitating CHAP. vin. 
is the same in water, and in all other fluid heavy Of ~^ ' 
bodies ? Besides, your argument ought to be appli- and gravitation, 
cable to the weighing of bodies in a pair of scales, 
which it is not, for there they have no such figure ; 
it should also hold in the comparison of gravity in 
hard and fluid bodies. 

A. I had not sufficiently considered it. But 
supposing now, as you do, that both heavy and 
hard bodies, in their smallest parts, were made so 
in the creation ; yet, because quicksilver is harder 
than water, a drop of water shall in descending be 
pressed into a more slender sector than a drop of 
quicksilver, and consequently the earth shall more 
easily cast off any quantity of water than the same 
quantity of quicksilver. 

n. This one would think were true ; as also that 
of simple fluid bodies, those whose smallest parts, 
naturally, without the force of fire, do strongliest 
cohere, are generally the heaviest. But why then 
should quicksilver be heavier than stone or steel ? 
Fluidity and hardness are but degrees between 
greater fluidity and greater hardness. Therefore 
to the knowledge of what it is that causeth the 
difference, in different bodies, of their endeavour 
downward, there are required, if it can be known 
at all, a great many more experiments than have 
been yet made. It is not difficult to find why 
water is heavier than ice, or other body mixed of 
air and water. But to believe that all bodies are 
heavier or lighter according to the quantity of air 
within them, is very hard. 

A. I see by this, that the Creator of the world, 
as by his power he ordered it, so by his wisdom he 


CHAP. vin. provided it should be never disordered. There- 
of""^ ' f re l eay i n g this question, I desire to know whether 
and gravitation, if a heavy body were as high as a fixed star, it 
would return to the earth. 

B. It is hard to try. But if there be this com- 
pounded motion in the great bodies so high, such 
as is in the earth, it is very likely that some heavy 
bodies will be carried to them. But we shall never 
know it till we be at the like height. 

A. What think you is the reason why a drop of 
water, though heavy, will stand upon a horizontal 
plane of dry or unctuous wood, and not spread it- 
self upon it ? For let A B, in the sixth figure, be 

Flg . 6 . the dry plane, D the drop of water, and D C perpen- 
dicular to A B. The drop D, though higher, will 
not descend and spread itself upon it. 

B. The reason I think is manifest. For those 
bodies which are made by beating of water and 
air together, show plainly that the parts of water 
have a great degree of cohesion. For the skin of 
the bubble is water, and yet it can keep the air, 
though moved, from getting out. Therefore the 
whole drop of water at D, hath a good deal of co- 
hesion of parts. And seeing A B is an horizontal 
plane, the way from the contact in D either to A or 
B is upwards, and consequently there is no endea- 
vour in D either of those ways, but what proceeds 
from so much weight of water as is able to break 
that cohesion, which so small a drop is too weak to 
do. But the cohesion being once broken, as with 
your finger, the water will follow. 

A. Seeing the descent of a heavy body increaseth 
according to the odd numbers 1, 3, 5, 7, &c. and 
the aggregates of those numbers, viz. of 1 and 3 ; 


and 1 and 3 and 5 ; and of 1 and 3 and 5 and 7, CHAP. vm. 
&c. are square numbers, namely 4, 9, 16 ; the whole ^~^ t 
swiftness of the descent will be, I think, to the and gravitation 
aggregate of so many swiftnesses equal to the first 
endeavour, as square numbers are to their sides, 
1, 2, 3, 4. Is it so ? 

B. Yes, you know it hath been demonstrated 
by Galileo. 

A. Then if, for instance, you put into a pair of 
scales equal quantities of quicksilver and water, 
seeing they are both accelerated in the same pro- 
portion, why should not the Aveight of quicksilver 
to the weight of water be in duplicate proportions 
to their first endeavours ? 

B. Because they are in a pair of scales. For 
there the motion of neither of them is accelerated. 
And therefore it will be, as the first endeavour of 
the quicksilver to the first endeavour of the water, 
so the whole weight to the \\hole weight. By 
which you may see, that the cause which takes 
away the gravitation of liquid bodies from fish or 
other lighter bodies within them, can never be de- 
rived from the weight. 

A. I have one question more to ask concerning 
gravity. If gravity be, as some define it, an intrin- 
sical quality, whereby a body descendeth towards 
the centre of the earth, how is it possible that a 
piece of iron that hath this intrinsical quality 
should rise from the earth, to go to a loadstone ? 
Hath it also an intrinsical quality to go from the 
earth ? It cannot be. The cause therefore must 
be extrinsical. And because when they are come 
together in the air, if you leave them to their own 
nature, they will fall down together, they must also 


CHAP. vin. have some like extrinsical cause. And so this 
' T magnetic virtue will be such another virtue as 

Ot gravity O 

ami gravitation, makes all heavy bodies to descend, in this our 
world, to the earth. If therefore you can from 
this your hypothesis of compounded motion, by 
which you have so probably salved the problem of 
gravity, salve also this of the loadstone, I shall 
acknowledge both your hypothesis to be true, and 
your conclusion to be well deduced. 

B. I think it not impossible. But I will proceed 
no farther in it now, than, for the facilitating of the 
demonstrations, to tell you the several proprieties 
of the magnet, whereof I am to show the causes. 
As first, that iron, and no other body, at some little 
distance, though heavy, will rise to it. Secondly, 
that if it be laid upon a still water in a floating 
vessel, and left to itself, it will turn itself till it lie 
in a meridian, that is to say, with one and the same 
line still north and south. Thirdly, if you take a 
long slender piece of iron, and apply the loadstone 
to it, and, according to the position of the poles of 
the loadstone, draw it over to the end of the iron, 
the iron will have the same poles with the magnet, 
so it be drawn with some pressure ; but the poles 
will lie in a contrary position ; and also this long 
iron will draw other iron to it as the magnet doth. 
Fourthly, this long iron, if it be so small as that 
poised upon a pin, the weight of it have no visible 
effect, the navigators use it for the needle of their 
compass, because it points north and south ; saving 
that in most places by particular accidents it is di- 
verted ; which diversion is called the variation of 
the horizontal needle. Fifthly, the same needle 
placed in a plane perpendicular to the horizon, hath 


another motion called the inclination. Which that CHAP. vin. 
you may the better conceive, draw a fourth figure ; ^ avl ' ty "" 
wherein let there be a circle to represent the and g ra V1 tation. 
terrella, that is to say, a spherical magnet. 18 ' 

A. Let this be it, whose centre is A, the north 
pole B, the south pole C. 

B. Join B C, and cross it at right angles with the 
diameter D E. 

A. It is done. 

B. Upon the point D set the needle parallel to 
B C, with the cross of the south pole, and the barb 
for the north ; and describe a square about the 
circle B D C E, and divide the arch D B into four 
equal parts in a, b, c. 

A. It is done. 

B. Then place the middle of the needle on the 
points a, b, c, so that they may freely turn ; and set 
the barb which is at D towards the north, and that 
which is at C towards the south. You see plainly 
by this, that the angles of inclination through the 
arch D C taken altogether, are double to a right 
angle. For when the south point of the needle, 
looking north, as at D, comes to look south, as at 
C, it must make half a circle. 

A. That is true. And if you draw the sine of 
the arch D , which is d a, and the sine of the 
arch B a, which is a C, and the sine of the arch 
D b, which is bf, and the sine of the arch B c y 
which is c g, the needle will lie upon bf with the 
north-point downwards, so that the needle will be 
parallel to A D. Then from a draw the line a h, 
making the angle c a h equal to the angle D A a. 
And then the needle at a shall lie in the line a h 
with the* south point toward A. Finally, draw 


CHAP. vin. the line c h, which, withe g, will also make aquar- 
ofTniv^ ' ter f a right angle ; and therefore if the needle 
and gravitation fc e placed on the point c, it will lie in c h with the 
south point toward //. And thus you see by what 
degrees the needle inclines or dips under the hori- 
zon more and more from D till it come to the north 
pole at B ; where it will lie parallel to the needle 
in D ; but with their barbs looking contrary ways. 
And this is certain by experience, and by none 

You see then why the degrees of the iucliuatory 
needle, in coming from D to B, are double to the 
degrees of a quadrant. It is found also by experi- 
ence, that iron both of the mine and of the furnace 
put into a vessel so as to float, will lay itself (if 
some accident in the earth hinder it not) exactly 
north and south. And now I am, from this com- 
pounded motion supposed by Copernicus, to derive 
the causes why a loadstone draws iron ; why it 
makes iron to do the same ; why naturally it placeth 
itself in a parallel to the axis of the earth ; why by 
passing it over the needle it changes its poles ; and 
what is the cause that it inclines. But it is your 
part to remember what I told you of motion at our 
second meeting ; and what I told you of this com- 
pounded motion supposed by Copernicus, at our 
fourth meeting. 




A. I COME now to hear what natural causes you CHAP. ix. 
can assign of the virtues of the magnet ; and first, OF the load. 
why it draws iron to it, and only iron. ftT^oteTana 

B. You know I have no other cause to assign whether they 

, "1 1 T 1 Sll W the l011 ^ 

but some local motion, and that I never approved tudc of places 
of any argument drawn from sympathy, influence, 
substantial forms, or incorporeal effluvia. For I 
am not, nor am accounted by my antagonists for a 
witch. But to answer this question, I should 
describe the globe of the earth greater than it is 
at B in the first figure, but that the terrella in the 
fourth figure will serve our turn. For it is but 
calling B and C the poles of the earth, and D E 
the diameter of the equinoctial circle, and making 
D the cast, and E the west. And then you must re- 
member that the annual motion of the earth is from 
west to east, and compounded of a straight and 
circular motion, so as that every point of it shall 
describe a small circle from west to east, as is done 
by the whole globe. And let the circles about 
a 1) c be three of those small circles. 

A. Before you go any farther, I pray you show 
me how I must distinguish east and west in every 
part of this figure. For wheresoever I am on earth, 
suppose at London, and see the sun rise suppose in 
Cancer, is not a straight line from my eye to the 
sun terminated in the east ? 

B. It is not due east, but* partly east, partly 
south. For the earth, being but a point compared 


CHAP. ix. to the sun, all the parallels to D E the equator, 

ofthTwT^ suc k as are e a ' f *> c ' ^ *k e y b e produced, will 
stone and fall upon the body of the sun. And therefore A b 
us P u e 8> -e. . g nort ] 1-east . A a eas t north-east ; and A c north 


A. Proceed now to the cause of attraction. 

B. Suppose now that the internal parts of the 
loadstone had the same motion with that of the 
internal parts of the sun which make the annual 
motion of the earth from west to east, but in a 
contrary way, for otherwise the loadstone and the 
iron can never be made to meet. Then set the 
loadstone at a little distance from the earth, marked 
with % ; and the iron marked with x upon the 
superficies of the earth. Now that which makes 
x rise to z, can be nothing else but air ; for nothing 
touches it but air. And that which makes the air 
to rise, can be nothing but those small circles made 
by the parts of the earth, such are at a b c, for 
nothing else touches the air. Seeing then the 
motion of each point of the loadstone is from east 
to west in circles, and the motion of each point of 
the iron from west to east ; it follows, that the air 
between the loadstone and the iron shall be cast off 
both east arid west ; and consequently the place 
left empty, if the iron did not rise up and fill it. 
Thus you see the cause that maketh the loadstone 
and the iron to meet. 

A. Hitherto I assent. But why they should 
meet when some heterogeneous body lies in the air 
between them, I cannot imagine. And yet I have 
seen a knife, though within the sheath, attract one 
end of the needle of a mariner's compass ; and 
have heard it will do the same though a stone-wall 
were between. 


B. Such iron were indeed a very vigorous CHAP.IX. 
loadstone. But the cause of it is the same that on^^T" 
causeth fire or hot water, which have the same stone and 

its pules, &c. 

compounded motion,, to work through a vessel of 
brass. For though the motion be altered by re- 
straint within the heterogeneous body, yet being 
continued quite through, it restores itself. 

A. What is the cause why the iron rubbed over 
by a loadstone will receive the virtue which the 
loadstone hath of drawing iron to it ? 

B. Since the motion that brings tw r o bodies to 
meet must have contrary ways, and that the mo- 
tions of the internal parts of the magnet and of 
the iron are contrary ; the rubbing of them to- 
gether does not give the iron the first endeavour 
to rise, but multiplies it. For the iron untouched 
will rise to a loadstone ; but if touched, it becomes 
a loadstone to other iron. For when they touch 
a piece of iron, they pass the loadstone over it only 
one way, viz. from pole to pole ; not back again, 
for that would undo what before had been done ; 
also they press it in passing to the very end of the 
iron, and somewhat hard. So that by this pressing 
motion all the small circles about the points a b c> 
are turned the contrary way ; and the halves of 
those small circles made on the arch D B will be 
taken away and the poles changed, so as that the 
north poles shall point south, and the south poles 
north, as in the figure. 

A. But how comes it to pass, that when a load- 
stone hath drawn a piece of iron, you may add to 
it another, as if they begat one another ? Is there 
the like motion in the generation of animals ? 

JS. I have told you that iron of itself will rise to 


CHAP. ix. the loadstone ; much more then will it adhere to 
of the load. ft when it is armed with iron, and both it and the 
stone and j ron j iave a p} a } ri superficies. For then not only 

its poles, &c. . . 

the points of contact will be many, which make 
the coherence stronger, but also the iron where- 
with it is armed is now another loadstone, differing 
a little, which you perhaps think, as male and 
female. But whether this compounded motion 
and confrication causeth the generation of animals, 
how should I know r , that never had so much leisure 
as to make any observation which might conduce 
to that ? 

A. My next question is, seeing you say the 
loadstone, or a needle touched with it, naturally 
respecteth the poles of the earth, but that the va- 
riation of it proceedeth from some accidents in the 
superficies of the earth ; what are those accidents r 

B. Suppose there be a hill upon the earth, for 
example, at /* ; then the stream of the air which 
which was between s and x westward, coming to 
the hill, shall go up the hill's side, and so down to 
the other side, according to the crooked line which 
I have marked about the hill by points ; and this 
infallibly will turn the north point of the needle, 
being on the east side, more towards the east, and 
that on the other side more towards the west, than 
if there had been no hill. Arid where upon the 
earth are there not eminences and depressions, ex- 
cept in some wide sea, and a great way from land. 

A. But if that be true, the variation in the same 
place should be always the same, for the hills are 
not removed. 

B. The variation of the needle at the same place 
is still the same ; but the variation of the variation 


is partly from the motion of the pole itself, which CHAP. ix. 
by the astronomers is called motus trepidationis ; Ofthelmid "' 
and partly from that, that the variation cannot be stone and 
truly observed, for the horizontal needle and the lts p es ' 
inclinatory needle incline alike, but cannot incline 
in due quantity. For whether Set upon a pin or 
an axis, their inclination is hindered, in the hori- 
zontal needle, by the pin itself: if upon an axis, if 
the axis be just, it cannot move ; if slack, the 
weight will hinder it ; but chiefly because the north 
pole of the earth draws away from it the north 
pole of the needle, for two like poles cannot come 
together. And this is the cause why the variation 
in one place is east, and another west. 

A. This is indeed the most probable reason why 
the variation varies that ever I heard given ; and 
I should presently acknowledge that this parallel 
motion of the axis of the earth in the ecliptic, sup- 
posed by Copernicus, is the true annual motion of 
the earth, but that there is lately come forth a 
book called Longitude Found, which makes the 
magnetical poles distant from the poles of the earth 
eight degrees and a half. 

B. I have the book. It is far from being de- a 
monstrated, as you shall find, if you have the 
patience to see it examined. For wheresoever his 
demonstration is true, the conclusion, if rightly 
inferred, will be this, that the poles of the load- 
stone and the poles of the earth are the same. 
And where, on the contrary, his demonstrations 
are fallacies, it is because sometimes he fancieth 
the lines he hath drawn, not where they are ; some- 
times because he mistakes his station ; and some- 
times because he goes on some false principle of 


CHAP. TX. natural philosophy; and sometimes also because he 
oaiie load" knoweth not sufficiently the doctrine of spherical 

stone and triangles. 

its poles, &cc. & 

A. I think that is the book there \\hich lies at 
your elbow. Pray you read. 

B. I find first (p. 4), that the grounds of his ar- 
gument are the two observations made by Mr. 
Burroughs, one at Vaygates, in 1576, where the 
variation from the pole of the earth he found to be 
1 1 deg. 15 min. east ; the other at Limehouse, near 
London, in 1580, where the variation from the 
pole of the earth was 8 deg. 38 min. west, by 
which, he saith, he might find out the magnetical 

A. Where is Vay gates ? 

B. In /O degrees of north latitude; the difference 
of longitude between London and it being 58 

A. The longitude of places being yet to seek, 
how came he to know this difference of 58 degrees, 
except the poles of the magnet and the earth be 
the same ? 

B. I believe he trusted to the globe for that. 
For the distance between the places is not above 
2000 miles the nearest way. But we will pass by 
that, and come to his demonstration, and to his 
diagram, wherein L is London, P the north-pole 
of the earth, V Vaygates. So that L P is 38 deg. 
28 min. ; P V 20 deg. ; the angle L P V 58 deg. for 
the difference between the longitudes of Vaygates 
and London. This is the construction. But before 
I corne to the demonstration, I have an inference 
to draw from these observations, which is this. 
Because in the same year the variation at London 


was 11 deg. 15 min. east, and at Vaygates 8 deg. CHAP. ix. 
38 min. west ; if you subtract 11 deg. 15 min. * "' 

J Of the load- 

from the arc L P ; and 8 deg. 38 min. from the -tone and 
arc L V, the variation on both sides will be taken Us P es ' 
away ; so that P V being the meridian of Vaygates, 
and L P the meridian of London, they shall both 
of them meet in P the pole of the earth. And if 
the pole of the magnet be nearer to the zenith of 
London than is the pole of the earth, it shall be 
just as much nearer to the zenith of Vaygates in the 
meridian of Vaygates, which is P V ; as is manifest 
by the diurnal motion of the earth. 

A. All this I conceive without difficulty. Pro- 
ceed to the demonstration. 

S. Mark well now. His words are these (page 5) : 
From PLV subtract 11 deg. 15 min., and there 
remains the angle V L M. Consider now which is 
the angle PLV, and which is the remaining angle 
V L M, and tell what you understand by it. 

A. He has marked the angle PLV with two 
numbers, 1 1 deg. 15 min. and 21 deg. 50 min., which 
together make 33 deg. 5 min. And the angle 1 1 
deg. 15 min. being subtracted from PLV, there 
will remain 21 deg. 50 min. for the angle VLM. 
I know not what to say to it. For I thought the 
arc P V, which is 20 deg., had been the arc of the 
spherical angle PLV; and that the arc L V had 
been 58 deg., because he says the angle L P V 
is so ; and that the arc L M had been 46 deg., 
because the angle L P M is so ; and lastly, that the 
angle P L M had been 8 deg. 30 min., because the 
arc P M is so. 

B. And what you thought had been true, if a 
spherical angle were a very angle. For all men 



CHAP. ix. that have written of spherical triangles take for 
the ground of their calculation, as Regiom out anus, 
Copernicus, and Clavius, that the arch of a spheri- 
cal angle is the side opposite to the angle. You 
should have considered also that he makes the 
angle V P M 12 deg., but sets down no arc to an- 
swer it. But that you may find I am in the right, 
look into the definitions which Clavius hath put 
down before his treatise of spherical triangles, and 
amongst them is this ; " the arc of a spherical tri- 
angle is a part of a great circle intercepted be- 
tween the two sides drawn from the pole of the 
said great circle." 

A. The book is nothing worth ; for it is impossi- 
ble to subtract an arc of a circle out of a spherical 
angle. And I see besides that he takes the super- 
ficies that lieth between the sides L P and L M for 
an arch, which is the quantity of an angle ; and is 
a line, arid cannot be taken out of a superficies. 
I wonder how any man that pretends to mathe- 
matics could be so much mistaken. 

B. It is no great wonder. For Clavius himself 
striving to maintain that a right angle is greater than 
the angle made by the diameter and the circumfer- 
ence, fell into the same error. A corner, in vulgar 
speech, and an angle, in the language of geometry, 
are not the same thing. But it is easy even for a 
learned man sometimes to take them for the same, 
as this author now has done ; and proceeding he 
saith, subtract 8 deg. 38 min. from the angle 
P V L, and there remains the angle L V M. 

A. That again is false, because impossible. What 
was it that deceived him now ? 

B. The same misunderstanding of the nature of 


a spherical angle. Which appears farther in this, CHAP. ix. 

that when he knew the arc V P was part of a great ^-VTT" 

x or the load- 

circle, he thought V M, which he maketh 8 deg. stone ami 

^ i / n i i lts poles, &<% 

30 mm., were also parts or a great circle ; which 
is manifestly false. For two great circles, because 
they pass through the centre, do cut each other 
into halves. But V P is not half a circle. He sure 
thought himself at Vaygates, and that P M V was 
equal to P V, although in the same hemisphere. 

A. But how proves he that the arc P M is 8 
degrees 30 minutes. ? 

B. Thus. We have in two triangles, P L M and 
P V M, two sides and one angle included, to find 
P M the distance of the magnetical pole from the 
pole of the earth 8 deg. 30 min. 

A. Is that all ? It is very short for a demon- 
stration of two so difficult problems, as the quan- 
tity of 8 deg. 30 min. ; and of the place of the 
magnetical pole. But he has proved nothing till 
he has showed how he found it. And though P M 
be 8 deg. 30 min., it follows not that M is the 
magnetical pole. 

B. Nor is it true. For if P M be 8 deg. 30 min., 
and VMS deg. 38 min., the whole arc P M V will, 
be 17 deg. 8 min., which should be 20 deg. Besides, 
whereas the variations were east and west, the 
subtracting of them should be also east and west, 
but they are north and south. 

A. I am satisfied that the magnetical poles and the 
poles of the earth are the same. But thus much I 
confess, if they were not the same, the longitude 
were found. For the difference of the latitudes of 
the earth's equator and of the magnetical equator, 
is the difference of the longitude. But proceed. 

M 2 



Of the load, 
stone and 
its> poles, &c 

CHAP. ix. B. " The earth being a solid body, and the mag- 
netic sphere that encompasseth the earth being a 
substance that hath not solidity to keep pace with 
the earth, loseth in its motion : and that may be 
the cause of the motion of the magnetic poles from 
east to west." 

A. This is very fine and unexpected. The mag- 
netic sphere, which I took for a globe made of a 
magnet, has not solidity to keep pace with the 
earth, though it be one of the hardest stones that 
are. It encompasseth the earth ; yet I thought 
nothing had encompassed the earth but air in w r hich 
I breath and move. By this also the whole earth 
must be a loadstone. For two bodies cannot be in 
one place. So that he is yet no farther than Dr. 
Gilbert w r hom he slights. And if the sphere be a 
magnet, then the earth and loadstone have the 
same poles. See the force of truth ! which though 
it could not draw to it his reason, hath drawn his 
words to it. 

B. But perhaps he meant that the magnetic 
virtue encompasseth the earth, and not the magne- 
tic body. 

, A. But that helpeth him not. For if the body 
of the magnet be not there, the virtue then is the 
virtue of the earth ; and so again the poles of the 
earth are magnetic poles. 

J5. You see how unsafe it is to boast of doctrines 
as of God's gifts, till we are sure that they are 
true. For God giveth and denieth as he pleaseth, 
not as ourselves wish ; as now to him he hath 
given confidence enough, but hath denied him, at 
least hitherto, the finding of the longitudes. In 
the next place (p. 8) he seems much pleased that 


his doctrine agrees with an opinion of Keplerus, CHAP. ix. 
that from the creation to the year of our Lord, it "7 * ~' 

* . Of the load- 

is to the year 1657 now 5650 years; and with stone and 

that which he saith some divines have held in times lts p es> 
past, that as this world was created in six days, so 
it should continue six thousand years. By which 
account the world will be at an end three hundred 
and fifty years hence ; though the Scripture tells us 
ijt shall come as a thief in the night. O what 
advantage three hundred and forty years hence 
will they have that know this, over them that know 
it not, by taking up money at interest, or selling 
lands at twenty years' purchase ! 

A. But he says he will not meddle with that. 

B. Yes, when he had meddled with it too much 

A. But you have not told me wherein consisteth 
this agreement between him and Keplerus. 

B. I forgot it. It is in the motion of the mag- 
netic poles. For precedently (p. 7), he had said 
" that their period or revolution was six hundred 
years ; their yearly motion thirty- six minutes ; and 
(p. 8) that their motion is by sixes. Six tenths of 
a degree in one year; six degrees in ten years >; 
sixty degrees in a hundred years ; and six times 
sixty degrees in six hundred years." 

A. But what natural cause doth he assign of this 
revolution of six hundred years ? 

B. None at all. For the magnet lying upon the 
earth, can have no motion at all but what the earth 
and the air give it. And because it is always at 
8 deg. 30 min. distance from the pole of the earth, 
the earth can give it no other .motion than what it 
gives to its own poles by the precession of the 



Of the load- 
stone and 
its poles, fec 

CHAP. ix. equinoctial points. Nor can the air give it any 
motion but by its stream ; which must needs vary 
when the stream varieth. But what a vast differ- 
ence does he make between the period of the 
motion of the equinoctial points, which is about 
or near thirty-six thousand years according to 
Copernicus (lib. iii. cap. 6), which makes the annual 
precession to be 36 seconds, and the period of the 
magnetical poles' motion,, which is but six hundred 

A. Go on. 

B. He comes now (p. 15) to the inclinatory 
needle upon a spherical loadstone. Where he 
shows, by diagram, that the needle and the instru- 
ment together moved towards the magnetical pole, 
make the sum of the inclinations equal to two 
quadrants, setting the north-point of the needle 
southward : which I confess is true. But, in the 
same page, he ascribeth the same motion to the 
earth in these words : " as the horizontal needle 
hath a double motion about the round loadstone or 
terrella, so also the inclinatory needle hath a dou- 
ble motion about the earth." What is this, but a 
confession that the poles of the magnet and of the 
earth are the same ? 

A. It is plain enough. 

B. Besides, seeing he placeth the magnetical pole 
at M in the meridian of Vaygates, the needle being 
touched shall incline to the pole of the earth which 
is P, as well there as at London, and make the 
north-pole of the earth point south. 

A. It is certain, because he puts both the mag- 
netical pole and tjie pole of the earth in the 
same meridian of the earth. Nor see I any cause 
why, the needle being the same, it should not be 


as subject to variation, and to variation of varia- CHAP. ix. 
tion, and to all accidents of the earth there, as in Of ^7^~^ 
any other part. f tone and 

B. He putteth (p. 16) a question, " at what dis- 18 P es ' 
tance from the earth are the magnetic poles ? and 
answers to it, they are very near the earth, because 
the nearer the earth, the greater the strength." 
What think you of this ? 

. A. I think they are in the superficies of the 
magnet, as the pole of the earth is in the super- 
ficies of the earth. And consequently, that then 
the earth must be a part of the magnet, and their 
poles the same. For the body of the magnet and 
the body of the earth, if they be two, cannot be in 
one place. 

J3. His next words are, " some things are to be 
considered concerning those variations of the hori- 
zontal needle which are not according to the situa- 
tion of the place from the magnetic poles, but are 
contrary ; as all the West Indies according to the 
poles should be easterly, arid they are westerly. 
Which is by some accidental cause in the earth ; 
and their motion, as I formerly said, is a forced 
motion, and not natural." 

A. He has clearly overthrown his main doctrine. 
For to say the motion of the needle is forced and 
unnatural, is a most pitiful shift, and manifestly 
false, no motion being more constant or less acci- 
dental, notwithstanding the variation, to which the 
inclinatory needle is no less subject than the hori- 
zontal needle. 

B. That which deceived him, was, that he thought 
them two sorts of needles, forgetting what he had 
said of Norman's invention of the inclinatory needle 
by the inclining of the horizontal needle (p. 11). 


CHAP. ix. For I will show you that what he says is easterly 
and should be westerly, should be easterly as it is. 
Consider the fourth figure, in which B is the north- 
pole, and Bell deg. 15 min. easterly, which was 
the variation at London in 15/6 easterly. Suppose 
A c to be the needle, shall it not incline, as well 
here as at D , and the variation B c be easterly ? 
Again, D a is 11 deg. 15 min., and the needle in D 
parallel to A B, and at a inclining also 1 1 deg. 
15 miri. westerly. And is not the variation there 
D a westerly, with the north point of the needle 
in the line a h ? 

A. But the West-Indies are riot in this hemi- 
sphere B C D E. The variation therefore will pro- 
ceed in an arc of the opposite hemisphere, which 
is westerly. 

D. I believe he might think so, forgetting that 
he and his compass were on the superficies of the 
earth, and fancying them in the centre at A. 

A. It is like enough. If we had a straight line 
exactly equal to the arc of a quadrant, I think it 
would very much facilitate the doctrine of spheri- 
cal triangles. 

B. When you have done with your questions of 
natural philosophy, I will give you a clear demon- 
stration of the equality of a straight line to the arc 
of a quadrant, which, if it satisfy you, you may 
carry with you, and try thereby if you can find the 
angle of a spherical triangle given. 

A. It is time now to give over. And at our next 
meeting I desire your opinion concerning the 
causes of diaphaniety, and refraction. This Coper- 
nicus has done much more than he thought of. 
For he has not only restored to us astronomy, but 
also made the way open to physiology. 




A. THINKING upon what you said yesterday, it CHAP. x. 
looked like a generation of living creatures. I saw O r transparence, 
the love between the loadstone and the iron in rcfraction > a " ( \ of 

. the power of the 

their mutual attraction, their engendering in their earth to produce 

-, -. . , . T ,T . ^-i living creatures. 

close and contrary motion, and their issue in the 
iron, which being touched, hath the same attractive 
virtue. Now seeing they have the same internal 
motion of parts with that of the earth, why should 
not their substance be the same, or very near 
a-kin ? 

J5. The most of them, if not all, that have written 
on this subject, when they call the loadstone a ter- 
rella, seem to think as you do. But I, except I 
could find proof for it, will not affirm it. For the 
earth attracteth all kind of bodies but air, and the 
loadstone none but iron. The earth is a star, and 
it were too bold to pronounce any sentence of its 
substance, especially of the planets, that are so 
lapt up in their several coats, as that they cannot 
work on our eyes, or any organ of our other senses. 

A. 1 come therefore now to the business of the 
day. Seeing all generation, augmentation, and 
alteration is local motion, how can a body not 
transparent be made transparent ? 

B. I think it can never be done by the art of 
man. For as I said of hard and heavy bodies in 
the creation, so I think of diaphanous, that the 
very same individual body which was not trans- 


CHAP. x. parent then, shall never be made transparent by 
nf ' ' human art. 

Of transparence, 

refraction, & c . A. Do not you see that every day men make 
glass, and other diaphanous bodies not much in- 
ferior in beauty to the fairest gems ? 

B. It is one thing to make one transparent of 
many by mixture, and another to make transpa- 
rent of not transparent. Any very hard stone, 
if it be beaten into small sands, such as is use.d 
for hour-glasses, every one of those sands, if you 
look upon it with a microscope, you will find to 
be transparent ; and the harder and whiter a stone 
is, so much the more transparent, as I have seen 
in the stone of which are made millstones, which 
stone is here called greet. And I doubt not but 
the sands of white marble must be more transpa- 
rent. But there are no sands so transparent that 
they have not a scurf upon them, as hard, perhaps, 
as the stone itself; which they whose profession 
it is to make glass, have the art to scour and wash 
away. And therefore I think it no great wonder 
to bring those sands into one lump, though I know 
not how they do it. 

, A. I know they do it with lye made with a salt 
extracted from the ashes of an herb, of which salt 
they make a strong lye, and mingle it with the 
sand, and then bake it. 

B. Like enough. But still it is a compound of 
two transparent bodies, whereof one is the natural 
stone, the other is the mortar. This therefore 
doth not prove, that one and the same body of not 
transparent can be made transparent. 

A. Since they cap. make one transparent body 
of many, why do they not of a great many small 


sparks of natural diamond compound one great CHAP. x. 
one ? It would bear the charges of all the mate- ' ' 

, . . , -IT Of transparence 

rials, and beside, enrich them. reaction, & c . 

B. It is probable it would. But it may be they 
know no salt that howsoever prepared, which, with 
how great a fire soever, can make them melt. And, 
it may be, the true crystal of the mountain, which 
is found in great pieces in the Alps, is but a com- 
pound of many small ones, and made by the earth's 
annual motion ; for it is a very swift motion. Sup- 
pose now that within a very small cavern of those 
rocks whose smallest atoms are crystal, and the 
cavity filled with air ; and consider what a tumult 
would be made by the swift reciprocation of that 
air ; whether it would not in time separate those 
atoms from the rock, and jumbling them together 
make them rub off their scurf from one another, 
and by little and little to touch one another in 
polished planes, and consequently stick together, 
till in length of time they become one lump of 
clean crystal. 

A. I believe that the least parts of created sub- 
stances lay mingled together at first, till it pleased 
God to separate all dissimilar natures, and congre- 
gate the similar, to which this annual motion is 
proper. But they say that crystal is found in 
the open air hanging like icicles upon the rocks, 
which, if true, defeats this supposition of a narrow 
cavern, and therefore I must have some farther 
experience of it before I make it my opinion. 
But howsoever, it still holds true that diaphanous 
bodies of all sorts, in their least parts, were made 
by God in the beginning of, the world. But it 
may be true, notwithstanding those icicles. For 


CHAP. x. the force of the air that could break off those di- 
aphanous atoms in a cavern, can do the same in 

Of transparence, * 7 

reaction, <fcc. the open air. And I know that a less force of air 
can break some bodies into small pieces, not much 
less hard than crystal, by corrupting them. 

B. That which you now have said is somewhat. 
But I deny not the possibility, but only doubt of 
the operation. You may therefore pass to some 
other question. 

A. Well, I will ask you then a question about 
refraction. I know already that for the cause of 
refraction, w r hen the light falleth through a thinner 
medium upon a thicker, you assign the resistance 
of the thicker body ; but you do not mean there, 
by rarum and densum, two bodies whereof in equal 
spaces one has more substance in it than the other. 

B. No ; for equal spaces contain equal bodies. 
But 1 mean by densum any body which more re- 
sisteth the motion of the air, and by rarum that 
which resisteth less. 

A. But you have not declared in what that re- 
sistance consisteth. 
-B. I suppose it proceedeth from the hardness. 

A. But from thence it will follow, that all trans- 
parent bodies that equally refract are equally hard, 
which I think is not true, because the refraction 
of glass is not greater, at least in comparison of 
their hardnesses, than that of water. 

B. I confess it. Therefore I think we must take 
in gravity to a share in the production of this re- 
fraction. For I never considered refraction but in 
glass, because my business then was only to find 

Flgt 7 ' the causes of the phenomena of telescopes and 
microscopes. Let therefore A B (in fig. 7) be a 


hard, and consequently, a heavy body ; and from CHAP. x. 
above, as from the sun, let C A be the line of 0f ^^ ce 
incidence, and produced to D ; and draw A E per- lefractlon ' &c - 
pendicular to A B. It is manifest that the hardness 
in A B shall turn the stream of the light inwards 
toward A E, suppose in the line A e. It is also 
evident that the endeavour in B, which is, being 
heavy, downward, shall turn the stream again in- 
ward, towards A E, as in A b. Thus it is in refrac- 
tion from the sun downwards. In like manner, if 
the light come from below, as from a candle in the 
point D, the line of incidence will be D A, and 
produced will pass to C. And the resistance of the 
hardness in A will turn the stream A C inward, 
suppose into A /, and make C I equal to D e. For 
passing into a thinner medium, it will depart from 
the perpendicular in an angle equal to the angle 
D A ^, by which it came nearer to it in A e. So 
also the resistance of the gravity in the point A 
shall turn the stream of the light into the line A i, 
and make the angle / A i equal to the angle e A b. 
And thus you see in what manner, though not in 
what proportion, hardness and gravity conjoin 
their resistance in the causing of refraction. 

A. But you proved yesterday, that a heavy body 
does not gravitate upon a body equally heavy. 
Now this A B has upper parts and lower parts ; 
and if the upper parts do not gravitate upon the 
lower parts, how can there be any endeavour at all 
downward to contribute to the refraction ? 

B. I told you yesterday, that when a heavy 
body was set upon another body heavier or harder 
than itself, the endeavour of it downward was 
diverted another way, but not that it was extin- 


CHAP. x. guished. But in this case, where it lieth upon air, 

or transparence ^ e ^ Y ^ endeavour of the lowest part worketh 

refraction, &<j. downward. For neither motion nor body can be 

utterly extinguished by a less than an omnipotent 

power. All bodies, as long as they are bodies, 

are in motion one way or other, though the farther 

it be communicated, so much the less. 

A. But since you hold that motion is propagated 
through all bodies, how hard or heavy soever they 
be, I see no cause but that all bodies should be 

B. There are divers causes that take away trans- 
parency. First, if the body be not perfectly homo- 
geneous, that is to say, if the smallest parts of it be 
not all precisely of the same nature, or do not so 
touch one another as to leave no vacuum within it ; 
or though they touch, if they be not as hard in the 
contact as in any other line. For then the refrac- 
tions will be so changed both in their direction, 
and in their strength, as that no light shall come 
through it to the eye ; as in w r ood and ordinary 
stone and metal. Secondly, the gravity and hard- 
ness may be so great, as to make the angle re- 
fracted so great, as the second refraction shall not 
direct the beam of light to the eye ; as if the angle 
of refraction were DAE, the refracted line would 
be perpendicular to A B, and never come to the 
line A D, in which is the eye. 

A. To know how much of the refraction is due 
to the hardness, and how much to the gravity, I 
believe it is impossible, though the quantity of the 
whole be easily measured in a diaphanous body 
given. And both you and Mr. Warner have de- 
monstrated, that as the sine of the angle refracted 


in one inclination is to the sine of the angle re- CHAP. x. 
fracted in another inclination, so is the sine of one ' ' 

. . Of tiansparence, 

inclination to the sine of the angle of the other refraction, &c. 
inclination. Which demonstrations are both pub- 
lished by Mersennus in the end of the first volume 
of his Cogitata Physico-Mathematica. But since 
there be many bodies, through which though there 
pass light enough, yet no object appear through 
them to the eye, what is the reason of that ? 

B. You mean paper. For paper windows will 
enlighten a room, and yet not show the image of 
an object without the room. But it is because 
there are in paper abundance of pores, through 
which the air passing moveth the air within ; by 
the reflections whereof anything within may be 
seen. And in the same paper there are again as 
many parts not transparent, through which the air 
cannot pass, but must be reflected first to all parts 
of the object, and from them again to the paper ; 
and at the paper either reflected again or trans- 
mitted, according as it falls upon pores or not 
pores ; so that the light from the object can never 
come together at the eye. 

A. There belongs yet to this subject the cause 
of the diversity of colours. But I am so well satis- 
fied with that which you have written of it in the 
twenty-fourth chapter of your book de Corpore, 
that I need not trouble you farther in it. And now 
I have but one question more to ask you, which I 
thought upon last night. I have read in an ancient 
historian, that living creatures after a great deluge 
were produced by the earth, which being then very 
soft, there were bred in it, it may be by the rapid 
motion of the sun, many blisters, which in time 


CHAP. x. breaking, brought forth, like so many eggs, all 
ouwnJpa^iice manner of living creatures great and small, which 
&c. since it is grown hard it cannot do. What think 
you of it ? 

B. It is true that the earth produced the first 
living creatures of all sorts but man. For God 
said (Gen. i. 24), Let tJie earth produce every 
living creature, cattle, and creeping thing, fyc. 
But then again (ver. 25) it is said that God made 
the least of the earth, 8fc. So that it is evident 
that God gave unto the earth that virtue. Which 
virtue must needs consist in motion, because all 
generation is motion. But man, though the same 
day, was made afterward. 

A. Why hath not the earth the same virtue now? 
Is not the sun the same as it was ? Or is there 
no earth now soft enough ? 

B. Yes. And it may be the earth may yet pro- 
duce some very small living creatures : and perhaps 
male and female. For the smallest creatures which 
we take notice of, do engender, though they do 
not all by conjunction ; therefore if the earth pro- 
duce living creatures at this day, God did not 
absolutely rest from all his works on the seventh 
day, but (as it is chap. ii. 2) he rested from all 
the work he had made. Arid therefore it is no 
harm to think that God worketh still, and when 
and where and what he pleaseth. Beside, it is 
very hard to believe, that to produce male and 
female, and all that belongs thereto, as also the 
several and curious organs of sense and memory, 
could be the work of anything that had not under- 
standing. From whence, I think we may conclude, 
that whatsoever was made after the creation, was 


a new creature made by God no otherwise than the CHAP. x. 
first creatures were, excepting only man. 

A. They are then in an error that think there 
are no more different kinds of animals in the world 
now, than there were in the ark of Noah. 

B. Yes, doubtless. For they have no text of 
Scripture from which it can be proved. 

A. The questions of nature which I could yet 
propound are innumerable. And since I cannot 
go through them, I must give over somewhere, and 
why not here f For I have troubled you enough, 
though I hope you will forgive me. 

B. So God forgive us both as we do one another. 
But forget not to take with you the demonstration 
of a straight line equal to an arc of a circle. 



The proportion B^S9IBBIHHH| DESCRIBE the square 

A B C D, and divide it 
by the diagonals A C and 
B D,as also by the straight 
lines EG, F H, meeting 
in the centre I at right 
angles, into four equal 
parts. Then with the 
radius A B describe the 

quadrant B D cutting E G in K, and the diagonal 
A C in L ; and so B L will be half the arc B D, equal 
to which we are to find a straight line. Divide 
I .C into halves at M, and draw B M cutting E G 
in a. I say B M is equal to the arc B L. For 
the demonstration whereof we are to assume cer- 
tain known truths and dictates of common-sense. 

1 . That the arc B K is the third part of the arc 
B D, and consequently two-thirds of the arc B L, 
and B K to K L as two to one. 

2. That if a straight line be equal to the arc 
B L, and one end in B, the other will be some- 
where in I C, and higher than the point L. 

3. That wheresoever it be, two-thirds of it must 

be equal to the arc B K, and one-fifth to the Thuyiopuriion 

of a straight lino 

arc K L. to hair tin- ,c 

4. That the arc of a quadrant described in the ot A tllu<lr<int 
third part of the radius, or of E G, is equal to the 
third part of the arc B D, viz. to the arc B K. I 
may therefore call a third part of E G, the radius 
of B K ; and a sixth part of E G, the radius of the 
arc K L, &c. 

. 5. And lastly, that any straight line drawn from 
B to I C, if it be equal to the arc B L, it must cut 
the half radius I G, whose quadrantal arc is B L, 
into the proportion of two to one. For as the 
whole arc to the whole E G, so are the parts of it 
to the parts of E G. 

These premises granted, which I think cannot 
be denied, I say again, that the straight line B M 
is equal to the arc B L. 


Because B I is to I M, by construction, as two 
to one, and the line I G divides the angle B I C in 
the midst, B a will be to a M as two to one, that is 
to say, as the arc B K to the arc K L. From the 
point M to the side B C erect a perpendicular M N. 
And because C M is half C I, the line M N will be 
half G C ; and B N will be three-quarters of B C ; 
and the square of B M equal to ten squares of a 
quarter of B C ; and because B M is to B a as 
three to two, M N will be to a G as three to two. 
But M N is a quarter of E G, therefore a G is two- 
thirds of a quarter of E G ; that is, one-third of 
I G ; that is, one-sixth of the whole E G. And 
I a one-third of E G. Therefore I a is the radius 
of the arc B K ; and a G the radius of the arc 




The proportion 

of the whole arc B L D. 
Lastly, if a straight line 
be drawn from B to any 
other point of the line I C, 
though any line may be 
divided into the propor- 
tion of two to one, it shall 
not pass through the point 

0, and therefore not divide the radius of B L, 
which is I G, into the proportion of two to one. 
Therefore no straight line can be drawn from B 
to I C, except B M, so as to be equal to the arc 
B L. Therefore the straight line B M and the arc 
B L are equal. 

Hence it follows, that seeing the square of B M 
is equal to ten squares of a quarter of B C, that a 
straight line equal to the quadrantal arc B L D is 
equal to ten squares of half the radius, as I have 
divers ways demonstrated heretofore. 












NOT knowing on my own part any cause of the 
favour your Lordship has been pleased to express 
towards me, unless it be the principles, method, 
and manners you have observed and approved in 
my writings ; and seeing these have all been very 
much reprehended by men, to whom the name of 
public professors hath procured reputation in the 
university of Oxford, I thought it would be a for- 
feiture of your Lordship's good opinion, not to 
justify myself in public also against them, which, 
whether I have sufficiently performed or not in 
the six following Lessons addressed to the same 
professors, I humbly pray your Lordship to con- 
sider. The volume itself is too small to be offered 
to you as a present, but to be brought before you 
as a controversy it is perhaps the better for being 
short. Of arts, some are demonstrable, others 
indemonstrable; and demonstrable are those the 
construction of the subject whereof is in the power 


of the artist himself, who, in his demonstration, 
does no more but deduce the consequences of his 
own operation. The reason whereof is this, that 
the science of every subject is derived from a pre- 
cognition of the causes, generation, and construc- 
tion of the same ; and consequently where the 
causes are known, there is place for demonstration, 
bat not where the causes are to seek for. Geometry 
therefore is demonstrable, for the lines and figures 
from which we reason are drawn and described 
by ourselves ; and civil philosophy is demonstrable, 
because we make the commonwealth ourselves. 
But because of natural bodies we know not the 
construction, but seek it from the effects, there 
lies no demonstration of what the causes be we 
seek for, but only of what they may be. 

And where there is place for demonstration, if 
the first principles, that is to say, the definitions 
contain not the generation of the subject, there 
can be nothing demonstrated as it ought to be. 
And this in the three first definitions of Euclid 
sufficiently appeareth. For seeing he maketh not, 
nor could make any use of them in his demonstra- 
tions, they ought not to be numbered among the 
principles of geometry. And Sextus Empiricus 
maketh use of them (misunderstood, yet so under- 
stood as the said professors understand them) to 
the overthrow of that so much renowned evidence 
of geometry. In that part therefore of my book 
where I treat of geometry, I thought it necessary 
in my definitions to express those motions by 


which lines, superficies, solids, and figures, were 
drawn and described, little expecting that any 
professor of geometry should find fault therewith, 
but on the contrary supposing I might thereby 
not only avoid the cavils of the sceptics, but also 
demonstrate divers propositions which on other 
principles are indemonstrable. And truly, if you 
shall find those my principles of motion made good, 
you shall find also that I have added something to 
that which was formerly extant in geometry. 

For first, from the seventh chapter of my book 
De Corpore, to the thirteenth, I have rectified 
and explained the principles of the science ; id est, 
I have done that business for which Dr. Wallis re- 
ceives the wages. In the seventh, I have exhibited 
and demonstrated the proportion of the parabola 
and parabolasters to the parallelograms of the 
same height and base ; which, though some of the 
propositions were extant without that demonstra- 
tion, were never before demonstrated, nor are by 
any other than this method demonstrable. 

In the eighteenth, as it is now in English, I 
have demonstrated, for anything I yet perceive, 
equation between the crooked line of a parabola 
or any parabolaster and a straight line. 

In the twenty-third I have exhibited the centre 
of gravity of any sector of a sphere. 

Lastly, the twenty-fourth, which is of the nature 
of refraction and reflection, is almost all new. 

But your Lordship will ask me what I have 
done iu the twentieth, about the quadrature of 


the circle. Truly, my Lord, not much more than 
before. I have let stand there that which I did 
before condemn, not that I think it exact, but partly 
because the division of angles may be more exactly 
performed by it than by any organical way what- 
soever ; and I have attempted the same by another 
method, which seemeth to me very natural, but of 
calculation difficult and slippery. I call them only 
aggressions, retaining nevertheless the formal man- 
ner of assertion used in demonstration. For I dare 
not use such a doubtful word as videtur, because 
the professors are presently ready to oppose me 
with a videtur quod non. Nor am I willing to 
leave those aggressions out, but rather to try if it 
may be made pass for lawful, (in spite of them that 
seek honour, not from their own performances, but 
from other men's failings), amongst many difficult 
undertakings carried through at once to leave one 
and the greatest for a time behind ; and partly be- 
cause the method is such as may hereafter give 
farther light to the finding out of the exact truth. 
But the principles of the professors that repre- 
hend these of mine, are some of them so void of 
sense, that a man at the first hearing, whether 
geometrician or not geometrician, must abhor 
them. As for example : 

1. That two equal proportions are not double 
to one of the same proportions. 

2. That a proportion is double, triple, &c. of a 
number, but not of a proportion. 

3. That the same body, without adding to it, or 


taking from it, is sometimes greater, and some- 
times less. 

4. That a quantity may grow less and less eter- 
nally, so as at last to be equal to another quan- 
tity ; or, which is all one, that there is a last in 

5. That the nature of an angle consisteth in 
that which lies between the lines that comprehend 
the angle in the very point of their concourse, 
that is to say, an angle is the superficies which 
lies between the two points which touch, or, as 
they understand a point, the superficies that lies 
between the two nothings which touch. 

6. That the quotient is the proportion of the 
division to the dividend. 

Upon these and some such other principles is 
grounded all that Dr. Wallis has said, not only in his 
ElencJms of my geometry, but also in his treatises 
of the Angle of Contact, and in his Arithmetica 
Injinitorum; which two last I have here in two or 
three leaves wholly arid clearly confuted. And 
I verily believe that since the beginning of the 
world, there has not been, nor ever shall be, so 
much absurdity written in geometry, as is to be 
found in those books of his ; with which there is 
so much presumption joined, that an aTrofcara^atric 
of the like conjunction cannot be expected in less 
than a Platonic year. The cause whereof I imagine 
to be this, that he mistook the study of symbols 
for the study of geometry^ and thought symbolical 
writing to be a new kind of method, and other 


men's demonstrations set down in symbols new 
demonstrations. The way of analysis by squares, 
cubes, &c., is very ancient, and useful for the find- 
ing out whatsoever is contained in the nature and 
generation of rectangled planes, which also may 
be found without it, and was at the highest in 
Vieta ; but I never saw anything added thereby 
to the science of geometry, as being a way wherein 
men go round from the equality of rectangled 
planes to the equality of proportion, and thence 
again to the equality of rectangled planes, wherein 
the symbols serve only to make men go faster 
about, as greater wind to a windmill. 

It is in sciences as in plants ; growth and branch- 
ing is but the generation of the root continued ; 
nor is the invention of theorems anything else but 
the knowledge of the construction of the subject 
prosecuted. The unsoundness of the branches are 
no prejudice to the roots, nor the faults of theo- 
rems to the principles. And active principles will 
correct false theorems if the reasoning be good ; 
but no logic in the world is good enough to draw 
evidence out of false or unactive principles. But 
I detain your Lordship too long. For all this will 
be much more manifest in the following discourses, 
wherein I have not only explained and rectified 
many of the most important principles of geometry, 
but also by the examples of those errors which 
have been committed by my reprehenders, made 
manifest the evil consequence of the principles 
they now proceed on. So that it is not only 


my own defence that I here bring before you, 
but also a positive doctrine concerning the true 
grounds, or rather atoms of geometry, which I dare 
only say are very singular, but whether they be 
very good or not, I submit to your Lordship's 
judgment. And seeing you have been pleased to 
bestow so much time, with great success, in the 
reading of what has been written by other men in 
all kinds of learning, I humbly pray your Lordship 
to bestow also a little time upon the reading of 
these few arid short lessons ; and if your Lordship 
find them agreeable to your reason and judgment, 
let me, notwithstanding the clamour of my adver- 
saries, be continued in your good opinion, and still 
retain the honour of being, 

My most noble Lord, 

Your Lordship's most 

humble and obliged servant, 


LONDON, June 10, 1656. 








I SUPPOSE, most egregious professors, you know LESSON i. 
already that by geometry, though the word im- ^ ' 

J JO J> o Of the principles 

port no more but the measuring of land, is under- of geometry, &? 
stood no less the measuring of all other quantity 
than that of bodies. And though the definition 
of geometry serve not for proof, nor enter into 
any geometrical demonstration, yet for under- 
standing of the principles of the science, and for a 
rule to judge by, who is a geometrician, and who 
is not, I hold it necessary to begin therewith. 

Geometry is the science of determining the 
quantity of anything, not measured, by comparing 
it with some other quantity or quantities mea- 
sured. Which science therefore whosoever shall 
go about to teach, must first be able to tell his 
disciple what measuring or dimension is ; by what 
each several kind of quantity is measured ; what 
quantity is, and what are the several kinds thereof. 
Therefore as they, who handle any one part of geo- 
metry, determine by definition the signification of 


LESSON i. every word which they make the subject or predi- 
orthe rindies ca ^ e ^ ^Y theorem they undertake to demonstrate ; 

of geometry, & c . go must he which intendeth to write a whole body 
of geometry, define and determine the meaning of 
whatsoever word belongeth to the whole science. 
The design of Euclid was to demonstrate the pro- 
perties of the five regular bodies mentioned by 
Plato ; in which demonstrations there was no need 
to allege for argument the definition of quantity, 
which it may be was the cause he hath not any- 
where defined it, but done what he undertook 
without it. And though having perpetually occa- 
sion to speak of measure, he hath not defined 
measure ; yet instead thereof he hath, in the be- 
ginning of his first elements, assumed an axiom 
which serveth his turn sufficiently as to the mea- 
sure of lines, which is the eighth axiom ; that those 
things which lie upon one another all the way 
(called by him ^ap^ovra) are equal. Which 
axiom is nothing else but a description of the art 
of measuring length and superficies. For this 
60a/>/icxnc; can have no place in solid bodies, unless 
two bodies could at the same time be in one place. 
But amongst the principles of geometry universal, 
the definitions are necessary, both of quantity and 

Quantity is that w r hich is signified by what we 
answer to him that asketh, how much any thing 
is ? and thereby determine the magnitude thereof. 
For magnitude being a word indefinite, if a man 
ask of a thing, quantum est ? that is, how much it 
is, we do not satisfy him by saying it is magni- 
tude or quantity, but by saying it is tantum, so 
much. And they* that first called it in Greek, 


an( l m Latin quantity, might more pro- LESSON i. 
perly have called it in Latin tantity, and in Greek 
rrjAiKorrjc ; and we, if we allowed ourselves the elo- 
quence of the Greeks and Latin s, should call it 
the so-muchness. 

There is therefore to everything concerning 
which a man may ask without absurdity, how 
much it is, a certain quantity belonging, determin- 
ing the magnitude to be so much. Also whereso- 
ever there is more and less, there is one kind of 
quantity or other. And first there is the quantity 
of bodies, and that of three kinds : length, which is 
by one way of measuring ; superficies, made of the 
complication of two lengths, or the measure taken 
two ways ; and solid, which is the complication of 
three lengths, or of the measure taken three ways, 
for breadth or thickness are but other lengths. 
And the science of geometry, so far forth as it 
contemplateth bodies only, is no more but by mea- 
suring the length of one or more lines, and by the 
position of others known in one and the same 
figure, to determine by ratiocination, how much is 
the superficies ; and by measuring length, breadth, 
and thickness, to determine the quantity of the 
whole body. Of this kind of magnitudes and 
quantities the subject is body. 

And because for the computing of the magni- 
tudes of bodies, it is not necessary that the bodies 
themselves should be present, the ideas and memory 
of them supplying their presence, we reckon upon 
those imaginary bodies, which are the quantities 
themselves, and say the length is so great, the 
breadth so great, &c. which in truth is no better 
than to say the length is so long, or the breadth so 




LESSON T. broad, &c. But in the mind of an intelligent man 

of^^i it; breedeth no mistake. 

<>f geometry, & c . Besides the quantity of bodies, there is a quan- 
tity of time. For seeing men, without absurdity, 
do ask how much it is ; by answering tantum, so 
much, they make manifest there is a quantity that 
belongeth unto time, namely, a length. And be- 
cause length cannot be an accident of time, which 
is itself an accident, it is the accident of a body ; 
and consequently the length of the time, is the 
length of the body ; by which length or line, we 
determine how much the time is, supposing some 
body to be moved over it. 

Also w r e not improperly ask with how much swift- 
ness a body is moved ; and consequently there is 
a quantity of motion or swiftness, and the measure 
of that quantity is also a line. But then again, we 
must suppose another motion, which determineth 
the time of the former. Also of force, there is a 
question of how much, which is to be answered by 
so much ; that is, by quantity. If the force con- 
sist in swiftness, the determination is the same 
with that of swiftness, namely, by a line ; if in 
swiftness and quantity of body jointly, then by a 
line and a solid ; or if in quantity of body only, 
as w r eight, by a solid only. 

So also is number quantity ; but in no other 
sense than as a line is quantity divided into equal 

Of an angle, which is of tw r o lines whatsoever 
they be, meeting in one point, the digression or 
openness in other points, it may be asked how great 
is that digression ? This question is answered also 
by quantity. An angle therefore hath quantity, 


though it be not the subject of quantity ; for the LESSON i. 
body only can be the subject, in which body those ^ r' 

, ,,. J Of the pi maples 

straddling lines are marked. ot geometry, &c. 

And because two lines may be made to divaricate 
by two causes ; one, when having one end common 
arid immoveable, they depart one from another at 
the other ends circularly, and this is called simply 
an angle ; and the quantity thereof is the quantity 
of the arch, wilich the two lines intercept. 

The other cause is the bending of a straight 
line into a circular or other crooked line, till it 
touch the place of the same line, whilst it was 
straight, in one only point. And this is called an 
angle of contingence. And because the more it is 
bent, the more it digresseth from the tangent, it 
may be asked how much more ? And therefore the 
answer must be made by quantity ; and conse- 
quently an angle of contingence hath its quantity 
as well as that which is called simply an angle. 
And in case the digression of two such crooked 
lines from the tangent be uniform, as in circles, the 
quantity of their digression may be determined. 
For, if a straight line be drawn from the point of 
contact, the digression of the lesser circle will be 
to the digression of the greater circle, as the part 
of the line drawn from the point of contact, and 
intercepted by the circumference of the greater 
circle is to the part of the same line intercepted 
by the circumference of the lesser circle, or, which 
is all one, as the greater radius is to the lesser 
radius. You may guess by this what will become 
of that part of your last book, where you handle 
the question of the quantity of an angle of con- 

o 2 


LESSON i. Also there lieth a question of how much com- 
ouhe principles parcitively one magnitude is to another magnitude, 
ofgeometiy,&c. as h ow much water is in a tun in respect of the 
ocean, how much in respect of a pint ; little in the 
first respect, much in the latter. Therefore the 
answer must be made by some respective quantity. 
This respective quantity is called ratio and propor- 
tion, and is determined by the quantity of their 
differences ; and if their differences be compared 
also with the quantities themselves that differ, it 
is called simply proportion, or proportion geome- 
trical. But if the differences be not so compared, 
then it is called proportion arithmetical. And 
where the difference is none, there is no quantity 
of the proportion, which in this case is but a bare 

Also concerning heat, light, and divers other 
qualities, which have degrees, there lieth a ques- 
tion of how much, to be answered by a so much, 
and consequently they have their quantities, though 
the same with the quantity of swiftness : because 
the intensions and remissions of such qualities are 
but the intensions and remissions of the swiftness 
of that motion by which the agent produceth such 
a quality. And as quantity may be considered in 
all the operations of nature, so also doth geometry 
run quite through the whole body of natural phi- 

To the principles of geometry the definition ap- 
pertaineth also of a measure, which is this, one 
quantity is the measure of another quantity, when 
it, or the multiple of it, is coincident in all points 
with the other quality. In which definition, in- 
stead of that ^apjuLoyr] of Euclid, I put coincidence. 


For the superposition of quantities, by which they LESSON i. 
render the word itiapuoyi], cannot be understood of ^^T^. ' 

i i r I '* ^ Of the principles 

bodies, but only of lines and superficies. Never- of geomctiy, &c. 

theless many bodies may be coincident successively 

with one and the same place, and that place will 

be their measure, as we see practised continually 

in the measuring of liquid bodies, which art of 

measuring may properly be called tyapnomq, but 

not superposition. 

Also the definitions of greater, less, and equal, 
are necessary principles of geometry. For it can- 
not be imagined than any geometrician should 
demonstrate to us the equality and inequality of 
magnitudes, except he tell us first what those 
words do signify. And it is a wonder to me, that 
Euclid hath not anywhere defined what are equals, 
or at least, what are equal bodies, but serveth his 
turn throughout with that forementioned i^ap^omq, 
which hath no place in solids, nor in time, nor in 
swiftness, nor in circular, or other crooked lines ; 
and therefore no wonder to me, why geometry 
hath not proceeded to the calculation neither of 
crooked lines, nor sufficiently of motion, nor of 
many other things, that have proportion to one 

Equal bodies, superficies, and lines, are those of 
which every one is capable of being coincident 
with the place of every one of the rest : and 
equal times, wherein with one and the same mo- 
tion equal lines are described. And equally swift 
are those motions by which we run over equal 
spaces in any time determined by any other mo- 
tion. And universally all quantities are equal, 
that are measured by the same number of the 
same measures. 


LESSON i. It is necessary also to the science of geometry, 
ies to define what quantities are of one and the same 
&c kind, which they call homogeneous, the want of 
which definitions hath produced those wranglirigs 
(which your book De Angulo Contactus will not 
make to cease) about the angle of contingence. 

Homogeneous quantities arc those which may 
be compared by (e^a^oo-tc) application of their mea- 
sures to one another ; so that solids and superficies 
are heterogeneous quantities, because there is no 
coincidence or application of those two dimensions. 

No more is there of line and superficies, nor of 
line and solid, wilich are therefore heterogeneous. 
But lines and lines, superficies and superficies, 
solids and solids, are homogeneous. 

Homogeneous also are line, and the quantity of 
time ; because the quantity of time is measured 
by the application of a line to a line ; for though 
time be no line, yet the quantity of time is a line, 
and the length of two times is compared by the 
length of two lines. 

Weight and solid have their quantity homoge- 
neous, because they measure one another by ap- 
plication, to the beam of a balance. Line and 
angle simply so called, have their quantity homoge- 
neous, because their measure is an arch or arches 
of a circle applicable in every point to one another. 

The quantity of an angle simply so called, and 
the quantity of an angle of contingence are hete- 
rogeneous. For the measures by which two angles 
simply so called are compared, are in two coinci- 
dent arches of the same circle ; but the measure 
by which an angle of contingence is measured, is 
a straight line intercepted between the point of 


contact and the circumference of the circle ; and LESSON i. 
therefore one of them is not applicable to the " r ~~ x 

11 OflhepnnciplPh 

other ; and consequently of these two sorts of an-oigpomctiy,&c. 
gles the quantities are heterogeneous. The quanti- 
ties of two angles of contingence are homogeneous ; 
for they may be measured by the ifyappoau; of two 
lines, whereof one extreme is common, namely, 
the point of contact, the other extremes are in the 
arches of the two circles. 

Besides this knowledge of what is quantity and 
measure, and their several sorts, it behoveth a 
geometrician to know why, and of what, they are 
called principles. For not every proposition that 
is evident is therefore a principle. A principle is 
the beginning of something. And because defini- 
tions are the beginnings or first propositions of 
demonstration, they are therefore called principles, 
principles, I say, of demonstration. But there be 
also necessary to the teaching of geometry other 
principles, which are not the beginnings of de- 
monstration, but of construction, commonly called 
petitions ; as that it may be granted that a man 
can draw a straight line, and produce it ; and 
until any radius, on any centre describe a circle, 
and the like. For that a man may be able to de- 
scribe a square, he must first be able to draw a 
straight line ; and before he can describe an equi- 
lateral triangle, he must be able first to describe a 
circle. And these petitions are therefore properly 
called principles, not of demonstration, but of ope- 
ration. As for the commonly received third sort 
of principles, called common notions, they are prin- 
ciples, only by permission of Mm that is the disci- 
ple ; who being ingenuous, and coming not to cavil 


LESSON i. but to learn, is content to receive them, though 
^ ' ' demonstrable, without their demonstrations. And 

Ol the principles ' 

oi geomeuy, & C . though definitions be the only principles of de- 
monstration, yet it is not true that every definition 
is a principle. For a man may so precisely deter- 
termine the signification of a word as not to be 
mistaken, yet may his definition be such as shall 
never serve for proof of any theorem, nor ever 
enter into any demonstration, such as are some of 
the definitions of Euclid, and consequently can be 
no beginnings of demonstration, that is to say, no 

All that hitherto hath been said, is so plain 
and easy to be understood, that you cannot, most 
egregious professors, without discovering your ig- 
norance to all men of reason, though no geometri- 
cians, deny it. And the same (saving that the 
words are all to be found in dictionaries) new ; 
also to him that means to learn, not only the prac- 
tice, but also the science of geometry necessary, 
and, though it grieve you, mine. And now I come 
to the definitions of Euclid. 

The first is of a point : S^mov, &c. "Signum e$t, 
c'ujus est pars nulla" that is to say, a mark is that 
of which there is no part. Which definition, not 
only to a candid, but also to a rigid construer, is 
sound and useful. But to one that neither will 
interpret candidly, nor can interpret accurately, is 
neither useful nor true. Theologers say the soul 
hath no part, and that an angel hath no part, yet 
do not think that soul or angel is a point. A 
mark or as some put instead of it *ry/utj, which is a 
mark with a hot iron, is visible ; if visible, then it 
hath quantity, and consequently may be divided 


into parts innumerable. That which is indivisible LESSON i. 
is no quantity ; and if a point be not quantity, ~^ le8 
seeing it is neither substance nor quality, it is ot geometry, &c. 
nothing. And if Euclid had meant it so in his 
definition, as you pretend he did, he might have 
defined it more briefly, but ridiculously, thus, a 
point is nothing. Sir Henry Savile was better 
pleased with the candid interpretation of Proclus, 
that would have it understood respectively to the 
matter of geometry. But what meaneth this re- 
spectively to the matter of geometry? It meaneth 
this, that no argument in any geometrical demon- 
stration should be taken from the division, quantity, 
or any part of a point ; which is as much as to 
say, a point is that whose quantity is not drawn 
into the demonstration of any geometrical conclu- 
sion ; or, which is all one, whose quantity is not 

An accurate interpreter might make good the 
definition thus, a point is that which is undivided; 
and this is properly the same with cujus non est 
pars : for there is a great difference between un- 
divided and indivisible^ that is, between cujus non 
est pars, and cujus non potest esse pars. Division* 
is an act of the understanding ; the understanding 
is therefore that which maketh parts, and there is 
no part where there is no consideration but of one. 
And consequently Euclid's definition of a point is 
accurately true, and the same with mine, which is> 
that a point is that body whose quantity is not con- 
sidered. And considered is that, as I have defined 
it chap. i. at the end of the third article, which is 
not put to account in demonstration. 

Euclid therefore seemeth not to be of your 


LESSON T. opinion, that say a point is nothing. But why 
^^7"^^ then doth he never use this definition in the de- 

Oi the pi mciples 

ot geometry, & c monstratioii of any proposition ? Whether he useth 
it expressly or no, I remember riot ; but the six- 
teenth proposition of the third book without the 
force of this definition is undemonstrated. 

The second definition is of a line : y/>/*p) $* 
/u?7jcot ttTrXurec. " Linen est longitudo latitudinis ex- 
pers ; a line is length which hath no breadth " 
and if candidly interpreted, sound enough, though 
rigorously riot so. For to what purpose is it to 
say length not broad., when there is no such thing 
as a broad length. One path may be broader 
than another path, but not one mile than another 
mile ; and it is not the path but the mile which is 
the way's length. If therefore a man have any 
ingenuity he will understand it thus, that a line 
is a body whose length is considered without its 
breadth, else we must say absurdly a broad length; 
or untruly, that there be bodies which have length 
and yet no breadth; and this is the very sense 
which Apollonius, saith Proclus, makes of this de- 
finition ; " when we measure," says he, " the length 
of a way, we take not in the breadth or depth, but 
consider only one dimension." See this of Proclus 
cited by Sir Henry Savile, where you shall find 
the very word consider. 

The fourth definition is of a straight line, thus 
'Ev0aa ypappti <W, &c. "Recta lined est quce ex cequo 
sua ipsins puncta inter jacet" A straight line is 
that which lieth equally (or perhaps evenly) be- 
tween its own points. This definition is inexcusable. 
Between what point* of its own can a straight line 
lie but between its extremes ? And how lies it 


evenly between them, unless it swerve no more LESSON i. 
from some other line which hath the same ex- 01 ^j^^ les 
tremes, one way than another ? And then why i geometry, &c 
are not between the same points both the lines 
straight ? How bitterly, and with what insipid 
jests would you have reviled Euclid for this, if 
living now he had written a Leviathan. And yet 
there is somewhat in this definition to help a man, 
not only to conceive the nature of a straight line 
(for who doth not conceive it ?) but also to express 
it. For he meant perhaps to call a straight line 
that which is all the way from one extreme to 
another, equally distant from any two or more 
such lines as being like and equal have the same 
extremes. So the axis of the earth is all the way 
equally distant from the circumference of any two 
or more meridians. But then before he had de- 
fined a straight line, he should have defined what 
lines are like, and what are equal. But it had 
been best of all, first to have defined crooked lines, 
by the possibility of a deduction or setting further 
asunder of their extremes ; and then straight lines, 
by the impossibility of the same. 

The seventh definition, which is that of a plain 
superficies, hath the same faults. 

The eighth is of a plane angle, 'En-t^oe ywWa 
i<7iv i'i lv 7ri7rc?a, &c. "Angulus plauus est duarum 
line arum in piano se mutuo tangent ium, et non in 
directumjacentium,alterius ad alterant inclination 
A plane angle is the inclination one towards 
another of two lines that touch one another in the 
same plane., and lie not in the same straight line. 
Besides the faults here observed by Sir Henry 
Savile, as the clause of not lying in the same 


LESSON i. straight line, and the obscurity or equivocation of 
<vT~" ', th e word inclination, there is yet another, which 

Of thepimciples . ... 

oi gtometiy, ^c. is, that by this definition two right angles together 
taken, are no angle ; which is a fault which you 
somewhere (asking leave to use the word angle., 
karaxpiww*;) acknowledge, but avoid not. For in 
geometry, w r here you confess there is required all 
possible accurateness, every mraxp^c is a fault. 
Besides you see by this definition, that Euclid is 
not of your, but of Clavius's opinion. For it is 
manifest that the two lines which contain an angle 
of contact incline one towards another, and come 
together in a point, and lie not in the same straight 
line, and consequently make an angle. 

The thirteenth definition is exact, but makes 
against your doctrine, that a point is nothing. 
Examine it. "Opoc l^]v o nvoc fct ^ac. a Tcrminm est 
quod alicujus coctremum est'' A term or bound is 
that which is the extreme of anything. We had 
before, the extremes of a line are points. But 
what is in a line the extreme, but the first or last 
part) though you may make that part as small as 
you will ? A point is therefore a part, and nothing 
is no extreme. 

The fourteenth, ^Lyf\pa. * *" TWOQ ?? nv&v opw 
7rfptfx<>pt>'v- " Figura est (subaudi quantitas) quce 
al) aliquo, vel aliquibus termmis undique continetnr 
sire clauditur." A figure is quantity contained 
within some bound or bounds. Or shortly thus, a 
figure is quantity every way determined, is in my 
opinion as exact a definition of a figure as can 
possibly be given, though it must not be so in 
yours. For this determination is the same thing 
with circumscription; and whatsoever is anywhere 


(ubicunque) definitive is there also circumscriptive; LESSON i. 
and by this means the distinction is lost, by which of ^ ep ' rincl pi es 
theologers, when they deny God to be in any place, geometry, &c. 
save themselves from being accused of saying he is 
nowhere ; for that which is nowhere is nothing. 
This definition of Euclid cannot therefore possibly 
be embraced by you that carry double, namely, 
mathematics and theology. For if you reject it, 
you will be cast out of all mathematic schools ; 
and if you maintain it, from the society of all 
school-divines, and lose the thanks of the favour 
you have shown (you the astronomer) to Bishop 

The fifteenth is of a circle. KokXoc i*l <r\^a cmVf- 
cov, &c. A circle is a plain figure comprehended by 
one line which is called the circumference, to which 
circumference all the straight lines drawn from 
one of the points within the figure are equal to one 
another. This is true. But if a man had never 
seen the generation of a circle by the motion of a 
compass or other equivalent means, it would have 
been hard to persuade him that there was any 
such figure possible. It had been therefore not 
amiss first to have let him see that such a figure 
might be described. Therefore so much of geo- 
metry is 110 part of philosophy, which seeketh the 
proper passions of all things in the generation of 
the things themselves. 

After the fifteenth till the last or thirty-fifth de- 
finition, all are most accurate, but the last which is 
this, parallel straight lines are those which being 
in the same plane, though infinitely produced both 
ways, shall never meet. Which is less accurate. 
For how shall a man know that there be straight 


LESSON i. lines which shall never meet, though both ways 
or^^Ties infinitely produced? Or how is the definition of 
ot geometry, &e. parallels, that is, of lines perpetually equidistant, 
good, wherein the nature of equidistance is not 
signified ? Or if it were signified, why should it 
not comprehend as well the parallelism of circular 
and other crooked lines, as of straight, and as well 
of superficies, as of lines ? By parallels is meant 
equidistant both lines and superficies, and the word 
is therefore not well defined without defining first 
equality of distance. And because the distance 
between two lines or superficies, is the shortest 
line that can join them, there either ought to be 
in the definition the shortest distance., which is 
that of the perpendicular and without inclination, 
or the distance in equal inclination, that is, in 
equal angles. Therefore if parallels be defined to 
be those lines or superficies, where the lines drawn 
from one to another in equal angles be equal, the 
definition, as to like lines, or like superficies, will 
be universal and convertible. And if we add to 
this definition, that the equal angles be drawn not 
opposite ways, it will be absolute, and convertible 
m all lines and superficies ; and the definition will 
be this : parallels are those lines and superficies 
between which every line drawn, in any angle, is 
equal to any other line drawn in the same angle 
the same way. For by this definition the distance 
between them will perpetually be equal, and con- 
sequently they wall never come nearer together, 
how much, or which way soever they be produced. 
And the converse of it will be also true, if two 
lines, or two superficies be parallel, and a straight 
line be drawn from one to the other, any other 


straight line, drawn from one to the other in the LESSON i. 
same angle, and the same way, will be equal to it. fth7T^ies 
This is manifestly true, and, most egregious pro- of g eometr ^ &c - 
fessors, new, at least to you. 

And thus much for the definitions placed before 
the first of Euclid's Elements. 

Before the third of his Elements is this de- 
finition : " In circulo cequaliter distare a centro 
rectce linece dicuntur, cum perpendiculares quce 
a centro in ipsas ducuntur sunt fequalesT In a 
circle two straight lines are said to be equally 
distant from the centre, upon which the perpen- 
diculars drawn from the centre are equal. This 
is true ; but it is rather an axiom than a definition, 
as being demonstrable that the perpendicular is 
the measure of the distance between a point and 
a straight or a crooked line. 

Before the fifth Element the first definition is of 
a part : Pars est magnitude wagnitudinis, minor 
major is, cum minor mctitur major em. A part is 
one magnitude of another, the less of the greater, 
when the less measureth the greater. From which 
definition it followeth, that more than a half is not 
a part of the whole. But because Euclid meaneth 
here an aliquot part, as a half, a third, or a fourth, 
&c., it may pass for the definition of a measure 
under the name of part, as thus : a measure is a 
part of the whole, when multiplied it may be equal 
to the whole, though properly a measure is external 
to the thing measured, and not the aliquot part 
itself, but equal to an aliquot part. 

But the third definition is intolerable ; it is the 
definition of \dyoc, in Latin ratio, in English, pro- 
portion, in this manner, Xoyoc w Svo juty^wv opoyevwr ij 


LESSON T. Kara TrrjXiKuTrira irpog a\\*j\a -rrota ayiffiQ. "R(ttlO 6St 

oi^^^in^^gnitudinum ejusdem generis mutua quadamse- 
0| eeomeuy, <fc c . cundum quantitatem hdbitudo" Proportion is a 
certain mutual habitude in quantity, of two magni- 
tudes of the same kind, one to another. First, we 
have here ignotum per ignotius; for every man un- 
derstandeth better what is meant by proportion 
than by habitude. But it was the phrase of the 
Greeks when they named like proportions, to say, 
the first to the second, 6Vo>e fya, id est, ita se /tabef, 
and in English, is as, the third to the fourth. As 
for example, in the proportions of two to four, and 
three to six, to say two to four, 6Vwc %<, id est, ita 
se halct, id est, is as, three to six. From which 
phrase Euclid made this his definition of propor- 
tion by TTota o-xtVtc, which the Latins translate qucedam 
habitndo. Qucedam in a definition is a most cer- 
tain note of not understanding the word defined ; 
and in Greek, ^oia o-^'^ is much worse ; for to ren- 
der rightly the Greek definition, we are to say in 
English, that proportion is a what-shall-I-call-it- 
isness, or soness of two magnitudes, &c. ; than 
which nothing can be more unworthy of Euclid. 
It is as bad as anything was ever said in geometry 
by Orontius, or by Dr. Wallis. That proportion is 
quantity compared, that is to say, little or great 
in respect of some other quantity, as I have above 
defined it, is I think intelligible. 

Ine TOUrth IS, 'A^aXoy/a li t^iv rj TMV \6yuv oftoiorrjg. 

"Proportio vero est ratiomim similitudo" Here we 
have no one word by which to render 'AvuXoyte; for 
our word proportion is already bestowed upon the 
rendering of Xoyoc. Nevertheless the Greek may 
be translated into English thus, iterated propor- 


tions. But iterated proportion is the same with LESSON i* 
cadem ratio. To what purpose then serveth the ' ' 

. . f Of the principle 

sixth definition, which is of eadem ratio ? For oi geometry, &c. 

ia and eadem ratio and similitudo rationum, 
are the same thing, as appeareth by Euclid him- 
self, where he defines those quantities, that are in 
the same proportion by arnXoyoi/. Therefore the 
sixth definition is but a lemma, and assumed with- 
out demonstration. 

The fourteenth," Composltio rationis estsnmptio 
antecedently cum consequent e^ ceu unius, ad ipsuni 
consequentem" To compound proportion, is to 
take both antecedent and consequent together as 
one magnitude, and compare it to the consequent, 
is good ; though he might have compared it as well 
with the antecedent ; for both ways it had been 
a composition of proportion. We are to note here, 
that the composition defined in this place by Euclid 
is riot adding together of proportions, but of two 
quantities that have proportion. And therefore 
it is not the same composition which he defineth 
in the fourth place before the sixth element, for 
there he defineth the addition of one proportion 
to another proportion in this manner : Xoyoc *K Xoyn 
frvykritrSai Xtyerm, &c. A proportion is said to be 
compounded of proportions, when their quantities 
multiplied into one another make a proportion ; 
as when we would compound or add together the 
proportions of three to two, and of four to five, 
we must multiply three and four, which maketh 
twelve, and two and five, which rnaketh ten. And 
then the proportion of twelve to ten is the sum of 
the proportions of three to two, and of four to 

; which is true, but not a definition ; for it may 



LESSON i. and ought to be demonstrated. For to define 
J" ' ' what is addition of two proportions (which are 

Of the principles r r v 

of geometry, &c always in four quantities, though sometimes one 
of them be twice named) we are to say, that 
they are then added together when we make the 
second to another in the same proportion, which 
the third hath to the fourth. 

And thus much of the definitions ; of which 
some, very few, you see are faulty ; the rest either 
accurate, or good enough if well interpreted. 
For the rest of the elements all are accurate, not- 
withstanding that you allow not for good any 
definition in geometry that hath in it the word 
motion, of which there be divers before the eleventh 
Element. But I must here put you in mind, that 
geometry being a science, and all science proceed- 
ing from a precognition of causes, the definition of 
a sphere, and also of a circle, by the generation of 
it, that is to say, by motion, is better than by the 
equality of distance from a point within. 

The second sort of principles are those of con- 
struction, usually called postulata, or petitions. 
As for those notiones communes, called axioms, 
they are from the definitions of their terms de- 
monstrable, though they be so evident as they 
need not demonstration. These petitions are by 
Euclid called 'Atrr/^ara, such as are granted by 
favour, that is, simply petitions, whereas by axiom 
is understood that which is claimed as due. So 
that between *A#w/io and 'A/ny/ia there is this other 
difference, that this latter is simply a petition, the 
former a petition of right. 

Of petitions simply, the first is, that from any 
point to any point may be drawn a straight line. 


The second, that a finite straight line may be LESSON i. 
produced. The third, that upon any centre at 0f ^7Th^ie* 
any distance may be described a circle. All which of geometry, &c. 
are both evident and necessary to be granted. 

And by all these a man may easily perceive that 
Euclid in the definitions of a point, a line, and a 
superficies, did not intend that a point should be 
nothing, or a line be without latitude, or a super- 
ficies without thickness ; for if he did, his petitions 
are not only unreasonable to be granted, but also 
impossible to be performed. For lines are not 
drawn but by motion, and motion is of body only. 
And therefore his meaning was, that the quantity 
of a point, the breadth of a line, and the thickness 
of a superficies were not to be considered, that is 
to say, not to be reckoned in the demonstration of 
any theorems concerning the quantity of bodies, 
either in length, superficies, or solid. 




THERE be but two causes from which can spring 
an error in the demonstration of any conclusion in 
any science whatsoever ; and those are ignorance 
or want of understanding, and negligence. For 
as in the adding together of many and great num- 
bers, he cannot fail that knoweth the rules of 
addition, and is also all the way so careful, as not 

P 2 


LESSON ii. to mistake one number or one place for another ; 
": ; ~' so in any other science, he that is perfect in the 

Of the faults J y r 

thatoccui in rules of logic, and is so watchful over his pen, as 

demonstration. T P ... r t* 

not to put one word for another, can never fail or 
making a true, though not perhaps the shortest 
and easiest demonstration. 

The rules of demonstration are but of two kinds : 
one, that the principles be true and evident defini- 
tions ; the other, that the inferences be necessary. 
And of true and evident definitions, the best are 
those which declare the cause or generation of 
that subject, whereof the proper passions are to 
be demonstrated. For science is that knowledge 
which is derived from the comprehension of the 
cause. But when the cause appeareth not, then 
may, or rather must we define some known pro- 
perty of the subject, and from thence derive some 
possible way, or ways, of the generation. And the 
more ways of generation are explicated, the more 
easy will be the derivation of the properties ; 
whereof some are more immediate to one, some 
to another generation. He therefore that pro- 
ceedeth from untrue, or not understood definitions, 
is ignorant of that he goes about ; which is an ill- 
favoured fault, be the matter he undertaketh easy 
or difficult, because he was not forced to undergo 
a greater charge than he could carry through. 
But he that from right definitions maketh a false 
conclusion, erreth through human frailty, as being 
less awake, more troubled with other thoughts, or 
more in haste when he was in writing. Such 
faults, unless they be very frequent, are not at- 
tended with shame, as being common to all men, 
or are at least less ugly than the former, except 


then, when he that committeth them reprehendeth LESSON n. 
the same in other men. For that is in every man Of v thij *^ 
intolerable, which he cannot tolerate in another. th it occui m 
But to the end that the faults of both kinds may 
by every man be well understood, it will not be 
amiss to examine them by some such demonstra- 
tions as are publicly extant. And for this purpose 
I will take such as are in mine and in your books, 
and begin with your Elenchus of the geometry 
contained in my book De Corpore ; to which I 
will also join your book lately set forth concerning 
the Angle of Contact, Conic Sections, and your 
Arithmetica Infinitorum ; and then examine the 
rest of my philosophy, and yours that oppugn it. 
For I will take leave to consider you both every- 
where as one author, because you publicly declare 
your approbation of one another's doctrine. 

My first definition is of a line, of length, and of 
a point. " The way," say I, " of a body moved, 
in which magnitude (though it always have some 
magnitude) is not considered, is called a line ; and 
the space gone over by that motion, length, or one 
and a simple dimension." To this definition you 
say, first, " what mathematician did ever thus de- 
fine a line or length ?" Whether you call in others 
for help or testimony, it is not done like a geome- 
trician ; for they use not to prove their conclusions 
by witnesses, but rely upon the strength of their 
own reason ; and when your witnesses appear, 
they will not take your part. Secondly, you grant 
that what I say is true, but not a definition. But 
to tell you truly what it is which we call a line, is 
to define a line. Why then is not this a defini- 
tion ? " Because," say you in the first place, " it 


LESSON ii. is not a reciprocal proposition." But by your 
. , ^ favour it is reciprocal. For not only the way of a 

Of the faults * -i i i- 

that occur m body whose quantity is not considered is a line, 
but also every line is, or may be conceived to be, 
the way of a body so moved. And if you object 
that there is a difference between is and may be 
conceived to be, Euclid, whom you call to your aid, 
will be against you in the fourteenth definition 
before his eleventh Element ; where he defines a 
sphere just as convertibly as I define a line ; except 
you think the globes of the sun and stars cannot be 
globes, unless they were made by the circumduc- 
tion of a semicircle ; and again in the eighteenth 
definition, which is of a cone, unless you admit no 
figure for a cone, which is not generated by the 
revolution of a triangle ; and again, in the twen- 
tieth definition, which is of a cylinder, except it 
be generated by the circumvolution of a parallelo- 
gram. Euclid saw that what proper passion soever 
should be derived from these his definitions, would 
be true of any other cylinder, sphere, or cone, 
though it were otherwise generated ; and the de- 
scription of the generation of any one being by 
the imagination applicable to all, which is equiva- 
lent to convertible, he did not believe that any 
rational man could be misled by learning logic to 
be offended with it. Therefore this exception pro- 
ceedeth from want of understanding, that is, from 
ignorance of the nature, and use of a definition. 

Again, you object and ask : " What need is there 
of motion, or of body moved, to make a man un- 
derstand what is a line ? Are not lines in a body 
at rest, as well as iji a body moved ? And is not 
the distance of two resting points length, as well 


as the measure of the passage ? Is not length one LESSON n. 
and a simple dimension, and one and a simple Of v ~ e ^^ 
dimension line ? Why then is not line and length that occur m 

11 -*9 r* i - T demonstration 

all one. J See how impertinent these questions 
are. Euclid defines a sphere to be a solid figure 
described by the revolution of a semicircle about 
the unmoved diameter. Why do you not ask, 
what need there is to the understanding of what a 
sphere is, to bring in the motion of a semicircle ? 
Is not a sphere to be understood without such 
motion ? Is not the figure so made a sphere with- 
out this motion ? And where he defines the axis 
of a sphere to be that unmoved diameter, may not 
you ask, whether there be no axis of a sphere, 
when the whole sphere, diameter and all, is in 
motion ? But it is not to my purpose to defend my 
definition by the example of that of Euclid. There- 
fore first, I say, to me, howsoever it may be to 
others, it was fit to define a line by motion. For 
the generation of a line is the motion that de- 
scribes it. And having defined philosophy in the 
beginning, to be the knowledge of the properties 
from the generation, it was fit to define it by its 
generation. And to your question, is not distance 
length ? I answer, that though sometimes distance 
be equivalent to length, yet certainly the distance 
between the two ends of a thread wound up into 
a clue is riot the length of the thread ; for the 
length of the thread is equal to all the windings 
whereof the clue is made. But if you will needs 
have distance and length to be all one, tell me of 
what the distance between any two points is the 
length. Is it riot the lengtlj. of the way ? And 
how is that called way, which is not defined by 


LESSON ii. some motion ? And have not several ways between 
": T ~* the same places, as by land and by water, several 

Of the faults * ' J J ' 

that occur in lengths ? But they have but one distance, because 
the distance is the shortest way. Therefore be- 
tween the length of the path, and the distance of 
places, there is a real difference in this case, and 
in all cases a difference of the consideration. Your 
objection, that line is longitude, proceeds from 
want of understanding English. Do men ever ask 
what is the line of a thread, or the line of a table, 
or of any other body ? Do they not always ask 
what is the length of it ? And why, but because 
they use their own judgments, not yet corrupted 
by the subtlety of mistaken professors. Euclid 
defines a line to be length without breadth. If 
those terms be all one, why said he not that a line 
is a line without breadth ? But what definition of 
a line give you ? None. Be contented then with 
such as you receive, and with this of mine, which 
you shall presently see is not amiss. 

Your next objections are to my definition of a 
point. Which definition adhereth to the former 
in these words, "and the body itself is called a 
point." Here again you call for help : " Quis 
unquum mor folium, etc. What mortal man, what 
sober man, did ever so define a point ?" It is well, 
and I take it to be an honour to be the first that 
do so. But what objection do you bring against 
it. This; "That a point added to a point, if it 
have magnitude, makes it greater." I say it doth 
so, but then presently it loseth the name of a 
point, which name was given to signify that it was 
not the meaning of him that used it in demonstra- 
tion to add, subtract, multiply, divide, or any way 


compute it. Then you come in with, " perhaps you LESSON n. 
will say though it have magnitude, that magnitude 0f ^T^ 
is not considered." You need not say perhaps, timt occur m 

* * *- demonstration. 

You know I affirm it ; and therefore your argu- 
ment might have been left out, but that it gave you 
an occasion of a digression into scurvy language. 

And whereas you ask why I defined not a point 
thus : "Punctum est corpus quod non consideratur 
esse corpus., et magnum quod non consideratur 
esse magnum" I will tell you why. First, because 
it is not Latin. Secondly, because when 1 had de- 
fined it by corpus, there was no need to define 
it again by magnum. I understand very well this 
language, "punctum cst corpus, quod non consi- 
deratur ut corpus" A point is a body not con- 
sidered as body. But punctum est corpus, quod 
non consideratur esse corpus, vel esse magnum, is 
not Latin ; nor the version of it, a point is a body 
which is not considered to be a body, English. 
My definition was, that a point is that body whose 
magnitude is not considered, not reckoned, not 
put to account in demonstration. And I exem- 
plified the same by the body of the earth describing 
the ecliptic line ; because the magnitude is not 
there reckoned nor chargeth the ecliptic line with 
any breadth. But I perceive you understand not 
what the word consideration sigriifieth, but take it 
for comparison or relation ; and say I ought to 
define a point simply, and not by relation to a 
great body ; as if to reckon and to compare were 
the same thing. " Omnia milii" saith Cicero, "pro- 
visa et considerata suntT I have provided and 
reckoned everything. There js a great difference 
between reckoning and relation. 


LKSSON ii. Again, you ask, why corpus motum, a body 
or the fimiiT moved ? I will tell you ; because the motion was 
that occur m necessary for the generation of a line. And though 

demonstration. J ^ 

after the generation of the line the point should 
rest, yet it is not necessary from this definition 
that it should be no more a point ; nor when 
Euclid defines a sphere by the circumduction of a 
semicircle upon an axis that resteth, doth it follow 
from thence when the sphere, axis, centre and all, 
as that of the earth, is moved from place to place, 
that it is no more an axis. 

Lastly, you object " that motion is accidentary 
to a point, and consequently not essential, nor to 
be put into the definition." And is not the cir- 
cumduction of a semicircle accidentary to a sphere ? 
Or do you think the sphere of the sun was gener- 
ated by the revolution of a semicircle ? And yet 
it was thought no fault in Euclid to put the motion 
into the definition of a sphere. 

The conceit you have concerning definitions, 
that they must explicate the essence of the thing 
defined, and must consist of a genus and a dif- 
ference, is not so universally true as you are made 
-believe, or else there be very many insufficient 
definitions that pass for good with you in Euclid. 
You are much deceived if you think these woful 
notions of yours, and the language that doth 
everywhere accompany them, show handsomely 
together. Or that such grounds as these be able 
to sustain so many, and so haughty reproaches as 
you advance upon them, so as they fall not, as you 
shall see immediately, upon your own head. I say 
a point hath quantity, but not to be reckoned in 
demonstrating the properties of lines, solids, or 


superficies ; you say it hath no quantity at all, but LESSON n. 
is plainly nothing. 

The first of the petitions of Euclid is, " that a thAt octur m 

p . . demonstration. 

line may be drawn from point to point at any 
distance." The second, " that a straight line may 
be produced." The third, " that on any centre a 
circle may be described at any distance." And 
the eighth axiom (which Sir H. Savile observes to 
be the foundation of all geometry) is this, " Qua* 
sibi mutuo congruunt, etc. Those things that are 
applied to one another in all points are equal." 
All or any of these principles being taken away, 
there is not in Euclid one proposition demonstrated 
or demonstrable. If a point have no quantity, a 
line can have no latitude ; and because a line is 
not drawn but by motion, by motion of a body, 
and body imprinteth latitude all the way, it is im- 
possible to draw or produce a straight line, or to 
describe a circular line without latitude. Also if 
a line have no latitude, one straight line cannot be 
applied to another. To them therefore that deny 
a point to have quantity, that is, a line to have 
latitude, the forenamed principles are not possible, 
and consequently no proposition in geometry is de- 
monstrated or demonstrable. You therefore that 
deny a point to have quantity, and a line to have 
breadth, have nothing at all of the science of geo- 
metry. The practice you may have, but so hath 
any man that hath learned the bare propositions 
by heart; but they are not fit to be professors 
either of geometry or of any other science that 
dependeth on it. Some man perhaps may say that 
this controversy is not much, worth, and that we 
both mean the same thing. But that man, though 


LESSON ir. in other things prudent enough, knoweth little of 
science and demonstration. For definitions are 
not only used to give us the notions of those 
things whose appellations are defined, for many 
times they that have no science have the ideas of 
things more perfect than such as are raised by 
definitions. As who is there that understandeth 
not better what a straight line is, or what propor- 
tion is, and what many other things are, without 
definition, than some that set down the definitions 
of them. But their use is, when they are truly 
and clearly made, to draw arguments from them 
for the conclusions to be proved. And therefore 
you that in your following censures of my geo- 
metry, take your argument so often from this, that 
a point is nothing, and so often revile me for the 
contrary, are not to be allowed such an excuse. 
He that is here mistaken, is not to be called negli- 
gent in his expression, but ignorant of the science. 
In the next place, you take exceptions to my 
definition of equal bodies, which is this: " Corpora 
(cqualia sunt qucc eundcm locum possidere pos- 
sunt. Equal bodies are those which may have 
.the same place." To which you object imperti- 
nently, that I may as well define a man to be, he 
that maij be prince of Transylvania, wittily, as 
you count wit. Formerly in every definition, you 
exacted an explication of the essence. You are 
therefore of opinion that the possibility of being 
prince of Transylvania is no less essential to a man, 
than the possibility of the being of tw r o bodies suc- 
cessively in the same place, is essential to bodies 

You take no notice of the twenty-third article 


of this same chapter, where I define what it is we LESSON ir. 

call essence, namely, that accident for which, we v ' 
, , . . * i / 01 ' the faults 

give the thing its name. As the essence ot a man that occur m 

is his capacity of reasoning ; the essence of a white demon " tratlon - 
body, whiteness, &c., because we give the name of 
man to such bodies as are capable of reasoning, 
for that their capacity ; and the name of white to 
such bodies as have that colour, for that colour. 
Let us now examine why it is that men say bodies 
are one to another equal ; and thereby we shall 
be able to determine whether the possibility of 
having the same placed essential or not to bodies 
equal, and consequently whether this definition be 
so like to the defining of a man by the possibility 
of being prince of Transylvania as you say it is. 
There is no man, besides such egregious geome- 
tricians as yourselves, that inquireth the equality 
of two bodies, but by measure. And for liquid 
bodies, or the aggregates of innumerable small 
bodies, men (men, I say) measure them by putting 
them one after another into the same vessel, that 
is to say, into the same place, as Aristotle defines 
place, or into the space determined by the vessel, 
as I define place. And the bodies that so fill the 
vessel, they acknowledge and receive for equal. 
But though, when hard bodies cannot be so mea- 
sured, without the incommodity or trouble of 
altering their figure, they then enquire, if the 
bodies are both of the same kind, their equality 
by w r eight, knowing, without your teaching, that 
equal bodies of the same nature weigh proportion- 
ably to their magnitudes ; yet they do it not for 
fear of missing of the equality, but to avoid incon- 
venience or trouble. But you (you, I say), that 



LESSON ii. have no definition of equals, neither received from 
others, nor framed by yourselves, out of your 
shallow meditation and deep conceit of your own 
wits, contend against the common light of nature. 
So much is unheedy learning a hinderance to the 
knowledge of the truth, and changeth into elves 
those that were beginning to be men. 

Again, when men inquire the equality of two 
bodies in length, they measure them by a common 
measure ; in which measure they consider neither 
breadth nor thickness, but how the length of it 
agreeth, first with the length of one of the bodies, 
then with the length of the other. And both the 
bodies whose lengths are measured, are successively 
in the same place under their common measure. 
Place therefore in lines also, is the proper index 
and discoverer of equality and inequality. And 
as in length j so it is in breadth and thickness, 
which are but lengths otherwise taken in the same 
solid body. But now when we come from this 
equality and inequality of lengths known by mea- 
sure, to determine the proportions of superficies 
and of solids, by ratiocination, then it is that we 
enter into geometry; for the making of definitions, 
in whatsoever science they are to be used, is that 
which we call philosophia prima. It is not the 
work of a geometrician, as a geometrician, to de- 
fine what is equality, or proportion, or any other 
word he useth, though it be the work of the same 
man, as a man. His geometrical part is, to draw 
from them as many true and useful theorems as he 

You object secondly, that a pyramis may be 
equal to a cube whilst it is a pyramis. True. And 


so also whilst it is a pyramis it hath a possibility LESSON n. 
by flexion and transposition of parts to become a ^ 

, , - L . , -. i Of the faults 

cube, and to be put into the place where another tint occur m 
cube equal to it was before. This is to argue like dcmoiw tiatlou< 
a child that hath not yet the perfect understanding 
of any language. 

In the third and fourth objection, you teach me 
to define equal bo4ies (if I will needs define them 
by place) by the equality of place, and to say, 
tfmt bodies are equal that have equal places, 
Teach others, if you can, to measure their grain, 
not by the same, but equal bushels. 

In the fifth objection, you except against the 
the word can, in that I say that bodies are equal 
which can fill the same place. For the greater 
body can, you say, fill the place of the less, though 
not reciprocally the less of the greater. It is true, 
that though the place of the less can never be the 
place of the greater, yet it may be filled by a part 
of the greater. But it is not then the greater 
body that filleth the place of the less, but a part 
of it, that is to say, a less body. Howsoever, to 
take away from simple men this straw they stumble 
at, I have now put the definition of equal bodies 
into these \vords : equal bodies are those whereof 
every one can fill the place of every other. And 
if my definition displease you. propound your own, 
either of equal bodies, or of equals simply. But 
you have none. Take therefore this of mine. 

The sixth is a very admirable exception. " What," 
say you, " if the same body can sometimes take up 
a greater, sometimes a lesser place, as by rarefac- 
tion and condensation ?" I understand very \vell 
that bodies may be sometimes thin and sometimes 


LESSON ii. thick, as they chance to stand closer together or 
oi^al^T further from one another. So in the mathematic 
that occur m schools, when you read your learned lectures, you 

demonstration. ' . J ? ; . J 

have a thick or thronging audience or disciples, 
which in a great church would be but a very thin 
company. I understand how thick and thin may 
be attributed to bodies in the plural, as to a com- 
pany ; but I understand not hpfv any one of them 
is thicker in the school than in the church ; or 
how any one of them taketh up a greater room in 
the school, when he can get in, than in the street. 
For I conceive the dimensions of the body, and of 
the place, whether the place be filled with gold or 
with air, to be coincident and the same ; and con- 
sequently both the quantity of the air, and the 
quantity of the gold, to be severally equal to the 
quantity of the place ; and therefore also, by the 
first axiom of Euclid, equal to one another ; inso- 
much as if the same air should be by condensation 
contained in a, part of the place it had, the dimen- 
sions of it would be the same with the dimensions 
of part of the place, that is, should be less than 
they were, and by consequence the quantity less. 
And then either the same body must be less also, 
or we must make a difference between greater 
bodies and bodies of greater quantity ; which no 
man doth that hath not lost his wits by trusting 
them with absurd teachers. When you receive 
salary, if the steward give you for every shilling 
a piece of sixpence, and then say, every shilling is 
condensed into the room of sixpence, I believe you 
would like this doctrine of yours much the worse. 
You see how by your ignorance you confound the 
affairs of mankind, as far forth as they give credit 


to your opinions, though it be but little. For LESSON 11. 
nature abhors even empty words, such as are (in n .\ ; . ' 

-t J ' v Of the faults 

the meaning you assign them), rarefying and con- that occur m 
densing. And you would be as well understood 
if you should say (coining words by your own 
power), that the same body might take up some- 
times a greater, sometimes a lesser place, by walli- 
faction and wardensation, as by rarefraction and 
condensation. You see how admirable this your 
objection is. 

In the seventh objection you bewray another 
kind of ignorance, which is the ignorance of what 
are the proper works of the several parts of philo- 
sophy. "Though it were out of doubt," say you, 
" that the same body cannot have several magni- 
tudes, yet seeing it is matter of natural philosophy, 
nor hath anything to do with the present business, 
to what purpose is it to mention it in a mathema- 
tical definition r" It seems by this, that all this 
while you think it is a piece of the geometry 
of Euclid, no less to make the definitions he 
useth, than to infer from them the theorems he 
demons trateth. Which is not true. For he that 
telleth you in what sense you are to take the ap-. 
pellations of those things which he riameth in his 
discourse, teacheth you but his language, that 
afterwards he may teach you his art. But teach- 
ing of language is not mathematic, nor logic, nor 
physic, nor any other science ; and therefore to call 
a definition, as you do, mathematical, or physical, 
is a mark of ignorance, in a professor inexcusable. 
All doctrine begins at the understanding of words, 
and proceeds by reasoning fill it conclude in 
science. He that will learn geometry must uri- 



Of the faults 
that occur in 
demolish ation 

LESSON Ti. derstand the terms before he begin, which that 
' ' ' he may do, the master demonstrated nothing, 
but useth his natural prudence only, as all men 
do when they endeavour to make their meaning 
clearly known. For words understood are but 
the seed, and no part of the harvest of philosophy. 
And this seed was it, which Aristotle went about 
to sow in his twelve books of metaphysics, and in 
his eight books concerning the hearing of natural 
philosophy. And in these books he defineth time, 
place, substance or essence, quantity, relation, &c., 
that from thence might be taken the definitions 
of the most general words for principles in the 
several parts of science. So that all definitions 
proceed from common understanding ; of which, 
if any man rightly write, he may properly call his 
writing philosophia prima, that is, the seeds, or 
the grounds of philosophy. And this is the method 
I have used, defining place, magnitude, arid the 
other the most general appellations in that part 
which I entitle philosophia prima* But you now, 
not understanding this, talk of mathematical defi- 
nitions. You will say perhaps that others do the 
.same as well as you. It may be so. But the ap- 
peaching of others does not make your ignorance 
the less. 

In the eighth place you object not, but ask me 
why I define equal bodies apart ? I will tell you. 
Because all other things which are said to be equal, 
are said to be so from the equality of bodies ; as 
two lines are said to be equal, when they be coin- 
cident with the length of one and the same body ; 
and equal times, which are measured by equal 
lengths of body, by the same motion. And the 
reason is, because there is no subject of quantity, 

in in 


or of equality, or of any other accident but body ; LESSON n 
all which I thought certainly was evident enough ^ ^~^ 

J Of the faults 

to any uncorrupted judgment ; and therefore that timto 
I needed first to define equality in the subject 
thereof, which is body, and then to declare in 
what sense it was attributed to time, motion, and 
other things that are not body. 

The ninth objection is an egregious cavil. Having 
se.t down the definition of equal bodies, I con- 
sidered that some men might not allow the attri- 
bute of equality to any things but those which are 
the subjects of quantity, because there is no equa- 
lity, but in respect of quantity. And to speak 
rigidly, magnum ct magnitudo are not the same 
thing ; for that which is great, is properly a body, 
w T hereof greatness is an accident. In what sense 
therefore, might you object, can an accident have 
quantity ? For their sakes therefore that have not 
judgment enough to perceive in what sense men 
say the length is so long, or the superficies so 
broad, &c. I added these words : " Eadem ratione 
(qua scilicet corpora dicnntnr cequalia) magni- 
tudo magnitudini tcqualis dicitur," that is, in the 
same manner , as bodies are said to be equal, their- 
magnitudes also are said to be equal. Which is 
no more than to say, when bodies are equal, their 
magnitudes also are called equal. When bodies 
are equal in length, their lengths are also called 
equal. And when bodies are equal in superficies, 
their superficies are also called equal. All which 
is common speech, as well amongst mathemati- 
cians, as amongst common people ; and, though 
improper, cannot be altered, jnor needcth to be 
altered to intelligent men. Nevertheless I did 

Q 2 


LESSON ii. think fit to put in that clause, that men might 
of the fouiuT know what it is we call equality, as well in mag- 
that occur in nitudes as in magnis, that is, in bodies. Which 

demonstration. . / 

you so interpret, as if it bore this sense, that when 
bodies are equal their superficies also must be 
equal, contrary to your own knowledge, only to 
take hold of a new occasion of reviling. How un- 
handsome and unmanly this is, I leave to be judged 
by any reader that hath had the fortune to see the 
world, and converse with honest men. 

Against the fourteenth article, where I prove 
that the same body hath always the same magni- 
tude, you object nothing but this, that though it 
be granted, that the same body hath the same 
magnitude, while it resteth, yet I bring nothing 
to prove that when it changeth place, it may not 
also change its magnitude by being enlarged or 
contracted. There is no doubt but to a body, 
whether at rest or in motion, more body may be 
added, or part of it taken away. But then it is 
not the same body, unless the whole and the part 
be all one. If the schools had not set your wit 
awry, you could never have been so stupid as not 
'to see the weakness of such objections. That 
which you add in the end of your objections to 
this eighth chapter, that I allow not Euclid this 
axiom gratis, that the whole is greater than a 
part, you know to be untrue. 

At my eleventh chapter, you enter into dispute 
with me about the nature of proportion. Upon 
the truth of your doctrine therein, and partly upon 
the truth of your opinions concerning the defini- 
tions of a point, ,and of a line, dependeth the 
question whether you have any geometry or none ; 


and the truth of all the demonstrations you have LESSON n. 
in your other books, namely of the Angle of Con- ouhel { llllt} f 
tact, and Arithmetica Infinitorum. Here I say you that occui m 

. demonstration. 

enter, how you will get out, your reputation saved, 
we shall see hereafter. 

When a man asketh what proportion one quan- 
tity hath to another, he asketh how great or how 
little the one is comparatively to, or in respect of 
the other. When a geometrician prefixeth before 
his demonstrations a definition, he doth it not as 
a part of his geometry, but of natural evidence, 
not to be demonstrated by argument, but to be 
understood in understanding the language wherein 
it is set down ; though the matter may neverthe- 
less, if besides geometry he have wit, be of some 
help to his disciple to make him understand it 
the sooner. But when there is no significant de- 
finition prefixed, as in this case, where Euclid's 
definition of proportion, that it is a whatslilcalt 
habitude of two quantities, fyc., is insignificant, and 
you allege no other, every one that will learn geo- 
metry, must gather the definition from observing 
how the word to be defined is most constantly 
used in common speech. But in common speech 
if a man shall ask how much, for example, is six in 
respect of four, and one man answer that it is 
greater by two, and another that it is greater by 
half of four, or by a third of six, he that asked the 
question will be satisfied by one of them, though 
perhaps by one of them now, and by the other 
another time, as being the only man that knoweth 
why he himself did ask the question. But if a 
man should answer, as you wo t uld do, that the pro- 
portion of six to two is of those numbers a certain 


LESSON ii. quotient, he would receive but little satisfaction. 
of^hTfiMuT ^ e ^ ween the said answers to this question, how 
that occur m rnuch is six in respect of four? there is this dif- 


ference. He that answereth that it is more by 
two, compareth not two with four, nor with six, 
for two is the name of a quantity absolute. But 
he that answereth it is more by half of four, or by 
a third of six, compareth the difference with one 
of the differing quantities. For halfs and thirds, 
&c. arc names of quantity compared. 

From hence there ariseth two species or kinds of 
(ratio) proportion, into which the general word^ro- 
portion may be divided. The one whereof, namely, 
that wherein the difference is not compared with 
either of the differing quantities, is called ratio 
arithmctica, arithmetical proportion ; the other 
ratio geometrica, geometrical proportion ; and, 
because this latter is only taken notice of by the 
name of proportion, simply proportion. Having 
considered this, I defined proportion, chapter IT. 
article 3, in this manner : " Ratio est relatio ante- 
cedentis ad consequents secundtun ni(tgnitndineni :" 
Proportion is the relation of the antecedent to the 
consequent in magnitude ; having immediately be- 
fore defined relatives, antecedent, and consequent, 
in the same article, and by way of explication 
added, that such relation was nothing else but 
that one of the quantities was equal to the other, 
or exceeded it by some quantity, or was by some 
quantity exceeded by it. And for exemplification 
of the same, I added further, that the proportion 
of three to two was, that three exceeded two by 
a unity ; but said not that the unity, or the dif- 
ference whatsoever it were, was their proportion, 


for unity, and to exceed another by unify, is not LESSON n 
the same thine;- This is clear enough to others ; ^"TTT"" 

a O Ot the faults 

let us therefore see why it is not so to you. You that ocnu m 

T , . * , ,,!, i i demomti ation 

say I make proportion to consist in that which 
remaineth after the lesser quantity is subtracted 
out of the greater ; and that you make it to con- 
sist in the quotient, when one number is divided 
by the other. Wherein you are mistaken ; first, 
in that you say, I make the proportion to consist 
in the remainder. For I make it to consist in the 
act of exceeding, or of being exceeded, or of being 
equal ; whereas the remainder is always an abso- 
lute quantity, and never a proportion. To be 
more or less than another number by two, is not 
the number two ; likewise to be equal to two, 
where the difference is nothing, is not that nothing? 
Again, you mistake in saying the proportion con- 
sisteth in the quotient. For divide twenty by five, 
the quotient is four. Is it not absurd to say that 
the proportion of five to twenty, or of twenty to 
five, is four ? You may say the proportion of five 
to twenty, is the proportion of one to four. And 
so say I. And you may therefore also say, that 
the proportion of one to four is a measure of 
the proportion of five to twenty, as being equal. 
And so say I. But that is only in geometrical pro- 
portion, and not in proportion universally. For 
though the species obtain the denomination of the 
genus, yet it is not the genus. But as the quotient 
giveth us a measure of the proportion of the divi- 
dend to the divisor in geometrical proportion, so 
also the remainder after subtraction is the measure 
of proportion arithmetical. 

You object in the next place, " that if the pro- 



LESSON ii. portion of one quantity to another be nothing but 
the excess or defect, then, wheresoever the excess 
or defect is the same, there the proportion is the 
same." This you say follows in your logic, and 
from thence, that the proportion of three to two, 
and five to four is the same. But is not three to 
two, and five to four, where the excess is the same 
number, the same proportion arithmetical ? And 
is not arithmetical proportion, proportion ? You 
take here (ratio) proportion, which is the genus, 
for that species of it which is called geometrical, 
because usually this species has the name of pro- 
portion simply. Also that the proportion of three 
to two, is the same with that of nine to six ; is it 
not because the excesses are one and three, the 
same portions of three and nine, that is to say the 
same excesses comparatively ? I winder you ask 
me not what is the genus of arithmetical and 
geometrical proportions, and what the difference? 
The gemis is (ratio) proportion, or comparison in 
magnitude, and the difference is that one compa- 
rison is made by the absolute quantity, the other 
by the comparative quantity, of the excess or de- 
fect, if there be any. Can anything be clearer 
than this ? You after come in with ignosce liabi- 
tndini to no purpose. I am not so inhuman as 
not to pardon dulness or madness : they are not 
voluntary faults. But when men adventure volun- 
tarily to talk of that they understand not censori- 
ously and scornfully, I may tell them of it. 

This difference between the excesses or defects, 
as they are simply or comparatively reckoned, 
being thus explained, all the rest of that you say 
in your objections to this eleventh chapter (saving 


that art. 5 for ratio Jnnarii ad quinarium est LESSON n. 
superari ternario, as it is in other places, I have ^ "~^ 

1 3 r ' Of the faults 

put too hastily ratio binarii ad quinarium est ter- that occur m 
narius), will be understood by every reader to be deinonstratlon - 
frivolous, and to proceed from the ignorance of 
what proportion is. 

At the twelfth chapter you only note that I say, 
that the proportion of inequality is quantity, but 
the proportion of equality not quantity ^ and refer 
that which you have to say against it to the chapter 
following ; to which place I shall also come in the 
following lesson. 




You begin your reprehension of my thirteenth 
chapter with a question ; whereas 7 divide propor- 
tion into arithmetical and geometrical. You ask 
me what proportion it is I so divide. Euclid 
divides an angle into right, obtuse, and acute. I 
may ask you as pertinently, what angle it is he so 
divides r Or, when you divide animal into homo 
and brutum, what animal that is, which you so 
divide ? You see by this, how absurd your question 
is. But you say the definition of proportion which 
I make at Chap. n. art. 3., namely, that pro- 
portion is the comparison of two magnitudes, one 
to another, agreeth not, neither with arithmetical, 


LESSON in. nor with geometrical proportion. I believe you 
* ' " thought so then, but having read what I have said 

Ol the faults y 

that occur m in the end of the last lesson, if you think so still, 

demolish ution / i MII, .,1 i i -i 

your fault will be too great to be pardoned easily. 
But why did you think so before ? Is it not 
because there was no definition in Euclid of pro- 
portion universal, and because he maketh no men- 
tion of proportion arithmetical, and because you 
had not in your minds a sufficient notion thereof 
yourselves to supply that defect ? And is not this 
the cause also, why you put in this parenthesis (if 
arithmetical proportion ought to be called propor- 
tion) ? Which is a confession that you know not 
whether there be such a thing as arithmetical pro- 
portion or not, notwithstanding that on all occa- 
sions you speak of arithmetical proportionals. 
Yes, this was it that made you think that propor- 
tion universally, and proportion geometrical, is the 
same, and yet to say you cannot tell whether they 
be the same or not. It is no w r onder, therefore, if 
in such confusion of the understanding, you appre- 
hend not that the proportions of two to five, and 
nine to twelve, are the same ; so you are blinded 
by seeing that they are not the same proportions 
geometrical. Nor doth it help you that I say the 
difference is the proportion ; for by difference you 
might; if you would, have understood the act of 

At the second article you note for a fault in 
method, that after I had used the words antecedent 
and consequent of a proportion in some of the 
precedent chapter s 9 I define them afterwards. I 
do not believe you say this against your knowledge, 
but that the eagerness of your malice made you 


oversee ; therefore go back again to the third LESSON in. 
article of chapter n. where, having defined corre- ^^TiI^T 
latives, I add these words, of which the first is that m cur m 

" demonstration 

called the antecedent, the second the consequent. 
This is but an oversight, though such as in me you 
would not have excused. 

At the thirteenth article you find fault with, that 
I say that the proportion of inequality, whether it 
be of excess or of defect, is quantity, hut the pro- 
portion of equality is not quantity. Whether that 
which you say, or that which I say, be the truth, 
is a question worthy of a very strict examination. 
The first time I heard it argued, was in Mersennus' 
chamber at Paris, at such time as the first volume 
of his Cogitata Physico-Mathematica was almost 
printed ; in which, because he had not said all he 
would say of proportion, he was forced to put the 
rest into a general preface, which, as was his cus- 
tom, he did read to his friends before he sent it to 
the press. In that general preface, under the title 
J)e Rationibus atque Proporticnibus, at the num- 
bers twelve, thirteen, fourteen, he maintaineth 
against Clavius, that the composition of proportion 
is (as of all other things) a composition of tlie 
parts to make a total, and that the proportion of 
equality answereth in quantity to non-ens, or 
nothing ; the proportion of excess, to ens, or 
quantity ; and the proportion of defect, to less 
than nothing ; because equality (he says) is a 
term of middle signification between excess and 
defect. And there also he refuteth the arguments 
which Clavius, at the end of the ninth Element of 
Euclid, bringeth to the contrary. And though this 
were approved by divers good geometricians then 


LESSON in. present, arjd never gainsaid by any since, yet da 
' ' ' not I say it upon the credit of them, but upon 

Of the faults J r > r 

that occur m sufficient grounds. For it hath been demonstrated 
by Eutocius, that if there be three magnitudes 9 
the proportion of the first to the third is com- 
pounded of the proportions of the first to the 
second, and of the second to the third ; which 
also I demonstrate in this article. And if there 
were never so many magnitudes ranked, it might 
be likewise demonstrated, that the proportion of 
the first to the last is compounded of the propor- 
tions of the first to the second, and of the second 
to the third, and of the third to the fourth, and so 
on to the last. If, therefore, we put in order any 
three numbers, whereof the two last be equal, as 
four, seven, seven, the proportion of four the first 
to seven the last, will be compounded of the pro- 
portions of four the first to seven the second, and 
of seven the second to seven the third. Where- 
fore the proportion of seven to seven (which is of 
equality) addeth nothing to the proportion of four 
the first, to seven the second ; and consequently 
the proportion of seven to seven hath no quantity ; 
but that the proportion of inequality hath quantity, 
I prove it from this, that one inequality may be 
greater than another. 

But for the clearing of this doctrine (which 
Mersennus calls intricate) of the composition of 
proportions, I observed, first, that any two quan- 
tities, being exposed to sense, their proportion was 
also exposed ; which is not intricate. Again, I 
observed that if besides the two exposed quantities, 
there were exposed a third, so as the first were the 
least, and the third the greatest, or the first the 


greatest, and the third the least, that not only the LESSON in 
proportions of the first to the second, but also ^TTT"' 

11 m m 7 Of tlie faults 

(because the differences and the quantities proceed that occur in 

- \ ,1 , t* ^ r * * ^t demonstration. 

the same way) the proportion of the first to the 
last is exposed by composition, or addition of the 
differences ; nor is there any intricacy in this. But 
when the first is less than the second, and the 
second greater than the third, or the first greater 
than the second, and the second less than the third, 
so that to make the first and second equal, if we 
use addition, we must, to make the second and 
third equal, use subtraction ; then comes in the 
intricacy, which cannot be extricated, but by such 
as know the truth of this doctrine which I now 
delivered out of Mersennus, namely, that the pro- 
portions of excess, equality, and defect, are as 
quantity., not-quantity r , nothing want quantity ; or 
as symbolists mark them 0-1-1 . . 0-1. And upon 
this ground I thought depended the universal truth 
of this proposition, that in any rank of magnitudes 
of the same kind, the proportion of the first to the 
last, was compounded of all the proportions (in 
order) of the intermediate quantities ; the want of 
the proof thereof, Sir Henry Savile calls (ncevus) 
a mole in the body of geometry. This proposition 
is demonstrated at the thirteenth article of this 

But before we come thither, I must examine the 
arguments you bring to confute this proposition, 
that the proportion of inequality is quantity, of 
equality ) not quantity. 

And first, you object that equality and inequa- 
lity are in the same predicament : a pretty argu- 
ment to flesh a young scholar in the logic school, 


LESSON in. that but now begins to learn the predicaments. 
But what do you mean by fcquale and inequale ? 
Do you mean corpus ccquale, and corpus inequale ? 
They are both in the predicament of substance, 
neither of them in that of quantity. Or do you 
mean ccqualitas and intcqualitas ? They are both 
in the predicament of relation, neither of them in 
that of quantity ; and yet both corpus and mcequa- 
Utas, though neither of them be quantity, may be 
quanta, that is, both of them have quantity. And 
when men say body is quantity, or inequality is 
quantity, they are no otherwise understood, than 
if they had said corpus est tantuin, and inaqualitas 
tanta, not tantitas ; that is, bodies and inequalities 
are so much, not somuchness ; and all intelligent 
men are contented with that expression, and your- 
selves use it. And the quantity of inequality is in 
the predicament of quantity, because the measure 
of it is in that line by which one quantity exceeds 
the other. But when neither exceedeth the other, 
then there is no line of excess, or defect by which 
the equality can be measured, or said to be so 
much, or be called quantity. Philosophy teacheth 
us how to range our words ; but Aristotle's rang- 
ing them in his predicaments doth not teach phi- 
losophy ; and therefore no argument taken from 
thence, can become a doctor and a professor of 

To prove that the proportion of inequality was 
quantity, but the proportion of equality not quan- 
tity, my argument was this : that because one in- 
equality may he greater or less than another, hut 
one equality cannot be greater nor less than an- 
other ; therefore inequality hath quantity, or is 


tanta, and equality not. Here you come in again LESSON in. 
with your predicaments, and object, that to be "~~~ Y ~' 

J . * ' . J Of the faults 

susceptible of magis and minus, belongs not to that occur m 
quantity, but to quality; but without any proof, 
as if you took it for an axiom. But whether true 
or false, you understand not in what sense it is 
true or false. It is true that one inequality is in- 
equality, as well as another ; as one heat is heat 
qs well as another, but not as great. Tarn, but 
not tantus. But so it is also in the predicament of 
quantity ; one line is as well a line as another, but 
not so great. All degrees, intentions, and remis- 
sions of quality, are greater or less quantity of 
force, and measured by lines, superficies, or solid 
quantity, which are properly in the predicament of 
quantity. You see how wise a thing it is to argue 
from the predicaments of Aristotle, which you 
understand not ; and yet you pretend to be less 
addicted to the authority of Aristotle now than 

In the next place you say, I may as well con- 
clude from the not susception of greater and less, 
that a right angle is not quantity, but an oblique 
one is. Very learnedly. As if to be greater or less, 
could be attributed to a quantity once determined. 
Number (that is, number indefinitively taken) is 
susceptible of greater and less, because one num- 
ber may be greater than another ; and this is a 
good argument to prove that number is quantity. 
And do you think the argument the worse for this, 
that one six cannot be greater than another six ? 
After all these childish arguments which you have 
hitherto urged, can you persuade any man, or 
yourselves, that you are logicians ? 


LESSON in. To the fifth and sixth article you object, first, 
or the limits tf ia t if I hud before sufficiently defined (ratio) 
that occur m proportion, I needed not again define what is 

demonstration. . . 

(eadem ratio) the same proportion ; and ask me 
whether when I have defined man, / use to define 
anew what is the same man ? You think when 
you have the definition of homo, you have also the 
definition of idem homo, when it is harder to con- 
ceive what idem signifies, than what homo. Besides, 
idem hath not the same signification always, and 
with whatsoever it be joined ; it doth not signify 
the same with homo, that it doth with ratio. For 
with homo it signifies the same individual man, but 
with ratio it signifies a like, or an equal propor- 
tion : and both (ratio) proportion and (idem) the 
same, being defined, there will still be need of 
another definition for (eadem ratio) the same pro- 
portion ; and this is enough to defend both myself 
and Euclid, against this objection : for Euclid 
also, after he had defined (ratio) proportion, and 
that sufficiently, as he believed, yet he defines the 
same proportion again apart. I know you did not 
mean in this place to object anything against Eu- 
clid, but you saw not what you were doing. There 
is within you some special cause of intenebration, 
which you should do well to look to. 

In the next place you say, when I had defined 
arithmetical proportions to be the same when the 
difference is the same; it was to be expected I should 
define geometrical proportions to be then the 
same, when the antecedents are of their conse- 
quents totuple or tantuple, that is, equimultiple 
(for tantuplum signifies nothing). In plain words, 
you expected, that as I defined one by subtraction, 


I should define the other by the quotient in divi- LESSON HI, 
sion. But why should you expect a definition of O f t ^ e( i lllt ," 
the same proportion by the quotient? Neither th *t occur m 

1 A . ,. i !! demonstration. 

reason nor the authority of Euclid could move you 
to expect it. Or why should you say it was to be 
expected ? But it seems you have the vanity to 
place the measure of truth in your own learning. 
In lines incommensurable there may be the same 
proportion, when, nevertheless, there is no quo- 
tient ; for setting their symbols one above another 
doth not make a quotient : for quotient there is 
none, but in aliquot parts. It is therefore im- 
possible to define proportion universally, by com- 
paring quotients. This incommensurability of 
magnitudes was it that confounded Euclid in the 
framing of his definition of proportion at the fifth 
Element. For when he came to numbers, he de- 
fined the same proportion irreprehensibly thus : 
numbers are then proportional, when the first of 
the second and the third of the fourth are equi- 
multiple, or the same part, or the same parts ; 
and yet there is in this definition no mention at all 
of a quotient. For though it be true, that if in 
dividing two numbers you make the same quotient, 
the dividends and the divisors are proportional, 
yet that is not the definition of the same propor- 
tion, but a theorem demonstrable from it. But 
this definition Euclid could not accommodate to 
proportion in general, because of incommensura- 

To supply this want, I thought it necessary to 
seek out some way, whereby the proportion of two 
lines, commensurable or incommensurable, might 
be continued perpetually the same. And this I 



LESSON in. found might be done by the proportion of two lines 
or the fcmitT described by some uniform motion, as by an effi- 
that occur in cleiit cause both of the said lines, and also of their 

demonstration. . . . . 

proportions ; which motions continuing, the pro- 
portions must needs be all the way the same. And 
therefore I defined those magnitudes to have the 
same geometrical proportion, when some cause 
producing in equal times equal effects, did de- 
termine loth the proportions. This, you say, needs 
an (Edipus to make it understood. You are, I see, 
no (Edipus ; but I do not see any difficulty, neither 
in the definition nor in the demonstration. That 
which you call perplexity in the explication, is your 
prejudice, arising from the symbols in your fancy. 
For men that pretend no less to natural philo- 
sophy than to geometry, to find fault with bringing 
motion and time into a definition, when there is no 
effect in nature which is not produced in time by 
motion, is a shame. But you swim upon other men's 
bladders in the superficies of geometry, without 
being able to endure diving, which is no fault of 
mine ; and therefore I shall, without your leave, be 
bold to say, I am the first that hath made the 
grounds of geometry firm and coherent. Whether 
I have added anything to the edifice or not, I leave 
to be judged by the readers. You see, you that 
profess with the pricking of bladders the letting out 
of their vapour, how much you are deceived. You 
make them swell more than ever. 

For the corollaries that follow this sixth article, 
you say they contain nothing new. Which is not 
true. For the ninth is new, and the demonstra- 
tions of all the rest are new, being grounded upon 
a new definition of proportion ; and the corollaries 


themselves, for want of a good definition of pro- LESSON in. 
portion, were never before exactly demonstrated. np > 7~7T" x 

* 3 J Of tlio faults 

For the truth of the sixth definition of the fifth that occur in 

T ^ 1 f> ~r~\ T i i* i it' demonsdiation. 

Element of Euclid cannot be known but by this 
definition of mine ; because it requires a trial in 
all numbers possible, that is to say, an infinite time 
of trial, whether the quimultiples of the first and 
third, and of the second and fourth, in all multi- 
plications, do together exceed, together come short, 
and are together equal ; which trial is impossible. 

In objecting against the thirteenth and sixteenth 
article, I observe that you bewray together, both 
the greatest ignorance and the greatest malice ; 
and it is well, for they are suitable to one another, 
and fit for one and the same man. In the thir- 
teenth article my proposition is this : If there be 
three magnitudes that have proportion one to 
another, the proportions of the first to the second, 
and of the second to the third., taken together (as 
one proportion), are equal to the proportion of the 
first to the third. This demonstrated, there is 
taken away one of those moles which Sir Henry 
Savile complaineth of in the body of geometry. 
Let us see now what you say, both against the 
enunciation and against the demonstration. 

Against the enunciation you object, that other 
men would say (not the proportions of the first to 
the second, and of the second to the third, taken 
together, &c. but) the proportion which is com- 
pounded of the proportion of the first to the se- 
cond, and of the second to the third, &c. Is not 
the compounding of any two things whatsoever 
the finding of the sum of them both, or the taking 
of them together as one total ? This is that ab- 

R 2 



Of the faults 
that occur m 

LESSON in. surdity of which Mersennus, in the general preface 
to his Cogitata Plnjsico-Mathematica, hath con- 
vinced Clavius, who, at the end of Euclid's ninth 
Element, denieth the composition of proportion to 
be a composition of parts to make a total ; which, 
therefore, he denied, because he did not observe, 
that the addition of a proportion of defect to a 
proportion of excess, was a subtraction of magni- 
tude ; and because he understood not that to say, 
composition is not the making a whole of parts, 
was contradiction ; which all but too learned men 
would as soon as they heard abhor. Therefore, in 
saying that other men would not speak in that 
manner, you say in effect they w r ould speak ab- 
surdly. You do well to mark what other geome- 
tricians say ; but you would do better if you could 
by your own meditation upon the things them- 
selves, examine the truth of what they say. But 
you have no mind, you say, to contend about the 
phrase. Let us see, therefore, what it is you con- 
tend about. 

The proportion, you say, which is compounded 
of double and triple proportion, is not, as I would 
have it, quintuple, but sextuple, as in these num- 
bers, six, three, one ; where the proportion of six 
to three is double, the proportion of three to one 
triple, and the proportion of six to one sextuple, 
not quintuple. Tell me, egregious professors, how 
is six to three double proportion ? Is six to three 
the double of a number, or the double of some 
proportion ? All men know the number six is 
double to the number three, and the number three 
triple tc an unity. But is the question here of 
compounding numbers, or of compounding pro- 


portions ? Euclid, at the last proposition of his LKSSON HI. 
ninth Element, says indeed, that these numbers, OMho( : anl J 
one, two, four, eie:ht, are lv <WX<7 t Wi ai/aXoy/a, in timt occur m 

' ' ' demonstration 

double proportion; yet there is no man that under- 
stands it otherwise, than if he had said in propor- 
tion of the single quantity to the double quantity ; 
and after the same rate, if he had said three, nine, 
twenty-seven, &c. had been in triple proportion, 
all men would have understood it, of the propor- 
tion of any quantity to its triple. Your instance, 
therefore, of six, three, one, is here impertinent, 
there being in them no doubling, no tripling, no 
sextupling of proportions, but of numbers. You 
may observe also, that Euclid never distinguished 
between double and duplicate, as you do. One 
word hirXaffiov serves him every \\here for either. 
Though, I confess, some curious grammarians take 
StirXacrtov for duplicate in number, and cnrXovy for 
double in quantity ; which will not serve your 
turn. Your geometry is not your own, but you 
case yourselves with Euclid's ; in which, as I have 
showed you, there be some few great holes ; and 
you by misunderstanding him, as in this place, 
have made them greater. Though the beasts that 
think your railing roaring, have for a time admired 
you ; yet now that through these holes of your 
case I have showed them your ears, they will be 
less affrighted. But to exemplify the composition 
of proportions, take these numbers, thirty-two, 
eight, one, and then you shall see that the propor- 
tion of thirty-two to one is the sum of the pro- 
portions of thirty-two to eight, and of eight to one. 
For the proportion of thirty-tw r o to eight is double 
the proportion of thirty-two to sixteen ; and the 


LESSON in. proportion of eight to one, is triple the proportion 
ot the iauitT f thirty-two to sixteen ; and the proportion of 
that octui m thirty-two to one is quintuple of thirty-two to six- 
demonstration. 111 -i -i -iii i 11 
teen ; but double and triple added together maketh 

quintuple. What can be here denied ? 

My demonstration consisteth of three cases : the 
first is when both the proportions are of defect, 
which is then when the first quantity is the least ; 
as in these three quantities, A B, AC, AD. The 
first case I demonstrated thus : Let it be sup- 
posed that the point A were moved uniformly 
through the whole line A D. The proportions, 
therefore, of A B to A C, and of A C to A D, are 
determined by the difference of the times in which 
they are described. And the proportion also of A B 
to A D, is that which is determined by the differ- 
ence of the times in \\hich they are described ; 
but the difference of the times in which A B and 
A C are described, together with the difference of 
the times wherein A C and A 1) are described, is 
the same with the difference of the times wherein 
are described A B and A D. The same cause, 
therefore, which determines both the proportions 
of A B to A C, and of A C to A I), determines also 
the proportion of A B to AD. Wherefore, by the 
definition of the same proportion., article six, 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 AD. 

Consider now your argumentation against it. 
" Let there be taken" say you, "between A and B 
the point a ; and then in your own words, I argue 
thus : The difference of the times wherein are de- 
scribed A B and A C, together with the difference 


of the times wherein are described A C and A Z), LESSON in. 
is the same with the difference of the times in Qf v ~ ^ ulte ' 
which are described a B and a C (namely, 13 D, that occur ... 

*-X-Y.^>T^\ 7 ^ ,j 7 7 demonstration. 

or B C + C D) ; wherefore, the same cause which 
determines the two proportions of A B to AC, and 
of AC to AD, determines also the proportion of 
a Ii to a D." Let me ask you here whether you 
suppose the motion from a to B, or from a to D, to 
.have the same swiftness with the motion from A to 
B, or from A to D r If you do not, then you deny 
the supposition. If you do, then B C, which is 
the difference of the times A B and A C, cannot be 
the difference of the times in which are described 
a B and a C, except A B and a B are equal. Let 
any man judge now whether there be any paralo- 
gism in Orontius that can equal this. And whether 
all that follows in the rest of this, and the next 
two whole pages, be not all a kind of raving upon 
the ignorance of what is the meaning of propor- 
tion,, which you also make more ill-favoured by 
writing it ; not in language, but in gambols ; I 
mean in the symbols, which have made you call 
those demonstrations short, which put into words 
so many as a true demonstration requires, would 
be longer than any of those of Clavius upon the 
twelfth Element of Euclid. 

To the sixteenth article you bring no argument, 
but fall into a loud oncethmus (the special figure 
wherewith you grace your oratory), offended with 
my unexpected crossing of the doctrine you teach, 
that proportion consisteth in a quotient. For that 
being denied you, your --/- + L_i + .i comes 

to nothing, that is, to just as much as it is worth. 
But are not you very simple men, to say that all 


LESSON in. mathematicians speak so, when it is not speaking ? 
ot'thefeuit When did you see any man but yourselves publish 
that occur in his demonstrations by signs not generally received, 

demonstration. . . , -, 

except it were not w r ith intention to demonstrate, 
but to teach the use of signs ? Had Pappus no 
analytics ? or wanted he the wit to shorten his 
reckoning by signs ? Or has he not proceeded 
analytically in a hundred problems (especially in his 
seventh book), and never used symbols ? Symbols 
are poor unhandsome, though necessary, scaffolds 
of demonstration ; and ought no more to appear 
in public, than the most deformed necessary busi- 
ness w r hich you do in your chambers. " But why" 
say you, " is this limitation to the proportion of 
the greater to the less ?" I will tell you ; because 
iterating of the proportion of the less to the 
greater, is a making of the proportion less, and the 
defect greater. And it is absurd to say that the 
taking of the same quantity twice should make it 
less. And thence it is, that in quantities which 
begin with the less, as one, two, four, the propor- 
tion of one to two is greater than that of one 
to four, as is demonstrated by Euclid, Elem. 5, 
prop. 8 ; and by consequent the proportion of one 
to four, is a proportion of greater littleness than 
that of one to two. And who is there, that when 
he knoweth that the respective greatness of four 
to one, is double to that of the respective greatness 
of four to two, or of two to one, will not presently 
acknowledge that the respective greatness of one 
to two, or two to four, is double to the respective 
greatness of one to four ? But this was too deep 
for such men as takp their opinions, not from 
weighing, but from reading. 


Lastly you object against the corollary of art. 28 ; LESSON in. 
which you make absurd enough by rehearsing it ofthef ^ ult / 
thus: Si quantltas allqua dicisa supponatur in that occur m 

7 . 7 / o T"v demonstration. 

paries aliquot cequales numero infimtas, &c. Do 
you think that of partes aliquot, or of paries ali- 
quotcc, it can be said without absurdity, that they 
are numero infinite ? And then you say I seem 
to mean, that if of the quantity A B, there be sup- 
posed a part C B, infinitely little ; and that be- 
tween A C and A B be taken two means, one 
arithmetical^ A E, the other geometrical, A D, the 
difference between A D and A E, will be infinitely 
little. My meaning is, and is sufficiently expressed, 
that the said means taken everywhere (not in one 
place only) will be the same throughout : and you 
that say there needed not so much pains to prove 
it, and think you do it shorter, prove it riot at all. 
For why may not I pretend against your demon- 
stration, that B E, the arithmetical difference, is 
greater than B D, the geometrical difference. You 
bring nothing to prove it ; and if you suppose it, 
you suppose the thing you are to prove. Hitherto 
you have proceeded in such manner with your 
Elenchus, as that so many objections as you hav 
made, so many false propositions you have ad- 
vanced. Which is a peculiar excellence of yours, 
that for so great a stipend as you receive, you will 
give place to no man living for the number and 
grossness of errors you teach your scholars. 

At the fourteenth chapter your first exception is 
to the second article ; where I define a plane in 
this manner : A plane superficies is that which is 
described by a straight line sowioved, as that every 
point thereof describe a several straight line. In 


LESSON in. which you require, first, that instead of describe, 
v , _ j g ^ j^ k sa jj describe. Why do you not 

Of the faults . . ; J 

that occur in require of Euclid, in the definition of a cone, in- 
emonstrdtion. g |. eac j o f con tlnetur, is contained, he say conti- 
neri potest, can be contained ? If I tell you how 
one plane is generated, cannot you apply the same 
generation to any other plane ? But you object, 
that the plane of a circle may be generated by the 
motion of the radius, whose every point describeth, 
not a straight, but a crooked line, wherein you 
are deceived ; for you cannot draw a circle (though 
you can draw the perimeter of a circle) but in 
a plane already generated. For the motion of a 
straight line, whose one point resting, describeth 
with the other points several perimeters of circles, 
may as well describe a conic superficies, as a plane. 
The question, therefore, is, how you will, in your 
definition, take in the plane which must be gene- 
rated before you begin to describe your circle, and 
before you know what point to make your centre. 
This objection, therefore, is to no purpose ; and 
besides, that it reflecteth upon the perfect defini- 
tions of Euclid before the eleventh Element, it 
cannot make good his definition (which is nothing 
worth) of a plane superficies, before his first Ele- 

In the next place, you reprehend briefly this 
corollary, that two planes cannot enclose a solid. 
I should, indeed, have added, with the base on 
whose extremes they insist : but this is not a fault 
to be ashamed of; for any man, by his own under- 
standing, might have mended my expression with- 
out departing from my meaning. But from your 
doctrine, that a superficies has no thickness, it is 


impossible to include a solid, with any number of LESSON in. 
planes whatsoever, unless you say that solid is in- "~ ' ' 

1 ' J J Of the faults 

eluded which nothing at all includes. that occur m 

At the third article, where I say of crooked 
lines, some are everywhere crooked, arid some have 
parts not crooked. You ask me what crooked line 
has parts not crooked ; and I answer, it is that 
line which with a straight line makes a rectilineal 
triangle. But this, you say, cannot stand with 
what I said before, namely, that a straight and 
crooked line cannot be coincident ; which is true, 
nor is there any contradiction ; for that part of a 
crooked line which is straight, may with a straight 
line be coincident. 

To the fourth article, where I define the centre 
of a circle to be that point of the radius, which m 
the description of the circle is unmoved ; you ob- 
ject as a contradiction, that I had before defined a 
point to be the body which is moved in the de- 
scription of a line : foolishly, as I have already 
shown at your objection to Chap. vm. art. 12. 

But at the sixth article, where I say, that 
crooked and incongruous lines touch one another 
but in one point, you make a cavil from this, that 
a circle may touch a parabola in two points. Tell 
me truly, did you read and understand these 
words that followed ? " A crooked line cannot he 
congruent with a straight line ; because if it 
could, one and the same line should he both straight 
and crooked. If you did, you could not but under- 
stand the sense of my words to be this : when two 
crooked lines which are incongruous, or a crooked 
and a straight line touch one .another, the contact 
is not in a line, but only in one point ; and then 


LESSON in. your instance of a circle and a parabola was a 
ofthefauiu wilful cavil, not befitting a doctor. If you either 
that occur in read them not, or understood them riot, it is your 

denionstiation. / i T i irn i i 

own iault. In the rest that lolloweth upon this 
article, with your diagram, there is nothing against 
me, nor anything of use, novelty, subtlety, or 

At the seventh article, where I define both an 
angle, simply so called, and an angle of cont'm- 
gence, by their several generations ; namely, that 
the former is generated when two straight lines are 
coincident, and one of them is moved, and dis- 
tracted from the other by circular motion upon 
one common point resting, fyc. ; you ask me " to 
winch of these kinds of angle I refer the angle 
made by a straight line when it cuts a crooked 
line ?" I answer easily and truly. To that kind of 
angle which is called simply an angle. This you 
understand not. "For how"., will you say, "can 
that angle which is generated by the divergence of 
two straight lines, be other than rectilineal ? or 
how can that angle which is not comprehended by 
two straight lines, be other than curvilineal ?" I 
see what it is that troubles you ; namely, the same 
which made you say before, that if the body which 
describes a line be a point, then there is nothing 
which is not moved that can be called a point. So 
you say here, "If an angle be generated by the mo- 
tion of a straight line, then no angle so generated 
can be curvilineal ;" which is as well argued, as if 
a man should say, the house was built by the car- 
riage and motion of stone and timber, therefore, 
when the carriage find that motion is ended, it is 
no more a house. Rectilineal arid curvilineal hath 


nothing to do with the nature of an angle simply LESSON in. 
so called, though it be essential to an angle of of v ~ ef : ul ~' 
contact. The measure of an angle, simply so called, that occur m 

. .111 - demonstration. 

is a circumference of a circle ; and the measure is 
always the same kind of quantity with the thing 
measured. The rectitude or curvity of the lines, 
which drawn from the centre, intercept the arch, 
is accidentary to the angle, which is the same, 
whether it be drawn by the motion circular of a 
straight line or of a crooked. The diameter and 
the circumference of a circle make a right angle, 
and the same which is made by the diameter and 
the tangent. And because the point of contact is 
not, as you think, nothing, but a line uiireckoned, 
and common both to the tangent and the circum- 
ference ; the same angle computed in the tangent 
is rectilineal, but computed in the circumference, 
not rectilineal, but mixed : or, if two circles cut 
one another, curvilineal. For every chord maketh 
the same angle with the circumference which it 
maketh with the line that toucheth the circumfe- 
rence at the end of the chord. And, therefore, 
when I divide an angle, simply so called, into rec- 
tilineal and curvilineal, I respect no more the ge- 
neration of it, than when I divide it into right and 
oblique. I then respect the generation, when I 
divide an angle into an angle simply so called, and 
an angle of contact. This that I have now said, 
if the reader remember when he reads your ob- 
jections to this, and to the ninth article, he w r ili 
need no more to make him see that you are utterly 
ignorant of the nature of an angle ; and that if 
ignorance be madness, not I, fyut you, are mad : and 
when an angle is comprehended between a straight 


LESSON 111. and a crooked line (if I may compute the same 
angle as comprehended between the same straight 
line and the point of contact), that it is consonant 
to my definition of a point by a magnitude not 
considered. But when you, in your treatise, De 
Angulo Contactus (chap. in. p. 6, 1. 8) have these 
words : " Though the whole concurrent lines incline 
to one another, yet they form no angle anywhere 
but in the very point of concourse :" you, that 
deny a point to be anything, tell me how two 
nothings can form an angle ; or if the angle be not 
formed, neither before the concurrent lines meet, 
nor in the point of concourse, how can you appre- 
hend that any angle can possibly be framed ? But 
I wonder not at this absurdity ; because this whole 
treatise of yours is but one absurdity, continued 
from the beginning to the end, as shall then ap- 
pear when I come to answer your objections to 
that which I have briefly and fully said of that sub- 
ject in my 14th chapter. 

At the twelfth article, I confess your exception 
to my universal definition of parallels to be just, 
though insolently set down. For it is no fault of 
ignorance (though it also infect the demonstration 
next it), but of too much security. The definition 
is this : Parallels are those lines or superficies, 
upon which two straight lines falling, and where- 
soever they fall, making equal angles with them 
both, are equal ; which is not, as it stands, uni- 
versally true. But inserting these words the same 
way, and making it stand thus : parallel lines or 
superficies, are those upon which two straight 
lines falling the sanie way, and wheresoever they 
fall, making equal angles, are equal, it is both true 


and universal ; and the following consectary, with LESSON in. 
very little change, as you may see in the transla- Ofthe ^ ultj f 
tion, perspicuously demonstrated. The same fault that occur in 

A A J . demonstration. 

occurreth once or twice more ; and you triumph 
unreasonably, as if you had given therein a very 
great proof of your geometry. 

The same was observed also upon this place by 
one of the prime geometricians of Paris, and noted 
in a letter to his friend in these words (Chap. xiv. 
art. 12) : " The definition of parallels wanteth 
somewhat to be supplied.'" Arid of the consectary 
he says, " It concludeth not, because it is grounded 
on the definition of parallels" Truly and severely 
enough, though without any such words as savour 
of arrogance, or of malice, or of the clown. 

At the thirteenth article you recite the demon- 
stration by which I prove the perimeters of two 
circles to be proportional to their semidiameters ; 
and with esto^fortasse, recte, omnino, noddying to 
the several parts thereof, you come at length to 
my last inference : Therefore, b?/ Chap. xm. art. 6, 
the perimeters and semidiameters of circles are 
proportional ; which you deny ; and therefore 
deny, because you say it followeth by the same* 
ratiocination, that circles also and spheres are 
proportional to their semidiameters. " For the 
same distance, you say, of the perimeter from the 
centre which determines the magnitude of the 
semidiameter, determines also the magnitude both 
of the circle and of the sphere." You acknow- 
ledge that perimeters and semidiameters have the 
cause of their determination such as in equal times 
make equal spaces. Suppose jiow a sphere gene- 
rated by the semidiameters, whilst the semicircle 


LESSON in. is turned about. There is but one radius of an 
u' infinite number of radii, which describes a great 

that occur m circle ; all the rest describe lesser circles parallel 

demonstration. , ., . j^.i ,- r i^.- 

to it, in one and the same time of revolution. 

Would you have men believe, that describing 

greater and lesser circles, is according to the sup- 

position (temporibus wqualtlms teqmtlla facere) to 

make equal spaces in equal times ? Or, when by 

the turning about of the semidiameter is described 

the plane of a circle, does it, think you, in equal 

times make the planes of the interior circles equal 

to the planes of the exterior ? Or is the radius 

that describes the inner circles equal to the radius 

that describes the exterior ? It does not, there- 

fore, follow from anything I have said in this de- 

monstration, that either spheres or planes of circles, 

are proportional to their radii ; and consequently, 

all that you have said, triumphing in your own in- 

capacity, is said imprudently by yourselves to your 

own disgrace. They that have applauded you, 

have reason by this time to doubt of all the rest 

that follows, and if they can, to dissemble the opi- 

nion they had before of your geometry. But they 

shall see before I have done, that not only your 

whole Elenchus, but also your other books of the 

Angle of Contact, &c. are mere ignorance and 


To the fourteenth article you object, that (in the 
sixth figure) I assume gratis, that F G, D E, B C, 
are proportional to A F, A I), A B ; and you refer 
it to be judged by the reader : and to the reader 
I refer it also. The not exact drawing of the figure 
(which is now amepded) is it that deceived you. 
For A F, F D, D B, are equal by construction. Also, 


AG, GE, EC, are equal by construction. And LESSONIII. 
FG, D K, B H, K E, H I, I C. are equal by paral- ^V^T 

' ' ' ' ' n J r Of the faults 

lelism. And because A F, FG, are as the velocities that occur in 
"wherewith they are described ; also 2 AF (that is emon&rai " 
A D) and 2 F G (that is D E) are as the same velo- 
cities. And finally, 3 A F (that is A B) and 3 F G 
(that is B C) are as the same velocities. It is not 
therefore assumed gratis, that F G, D E, B C are 
proportional to A F, A D 3 A B, but grounded upon 
the sixth article of the thirteenth chapter ; and 
consequently your objection is nothing worth. 
You might better have excepted to the placing of 
1) E, first at adventure, and then making A D two- 
thirds of A B ; for that was a fault, though not 
great enough to trouble a candid reader ; yet great 
enough to be a ground, to a malicious reader, of a 

That which you object to the third corollary of 
art. 15, \vas certainly a dream. There is no as- 
suming of an angle C D E, for an angle II D E, or 
B D E, neither in the demonstration, nor in any of 
the corollaries. It may be you dreamt of some- 
what in the twentieth article of chapter xvi. But 
because that article, though once printed, was 
afterwards left out, as not serving to the use I had 
clesigned it for, I cannot guess what it is : for I 
have no copy of that article, neither printed nor 
written ; but am very sure, though it were not 
useful, it w r as true. 

Article the sixteenth. Here we come to the 
controversy concerning the angle of contact, which, 
ycMi say, you have handled, in a special treatise 
published; and that you fyave clearly demon- 
strated, in your public lectures, that Peletarius 


LESSON in. was in the right. But that I agree not sufficiently, 
or the faults ' neither with Peletarius nor with Clavius. I con- 
that occur in f e ss I agree not in all points with Peletarius, nor 

demonstration. .,,. i /^n T i i PI 

in all points with Ulavms. It does not thence fol- 
low that I agree not with the truth. I am not, as 
you, of any faction, neither in geometry nor in 
politics. If I think that you, or Peletarius, or 
Clavius, or Euclid, have erred, or been too obscure, 
I see no cause for which I ought to dissemble it. 
And in this same question I am of opinion that 
Peletarius did not well in denying the angle of con- 
tingence to be an angle. And that Clavius did 
not well to say, the angle of a semicircle was less 
than a right-lined right angle. And that Euclid 
did not well to leave it so obscure what he meant 
by inclination in the definition of a plane angle, 
seeing elsewhere he attributed! inclination only to 
acute angles ; and scarce any man ever acknow- 
ledged inclination in a straight line,, to any other 
line to which it was perpendicular. But you, in 
this question of what is inclination, though you 
pretend riot to depart from Euclid, are, neverthe- 
less, more obscure than he ; and also are contrary 
to him. For Euclid by inclination meaneth the 
inclination of one line to another ; and you under- 
stand it of the inclination of one line from another ; 
which is not inclination, but declination. For you 
make two straight lines, when they lie one on an- 
other, to lie aK\ivwQy that is, without any inclination 
(because it serves your turn) ; not observing that 
it followeth thence, that inclination is a digression 
of one line from another. This is in your first 
argument in the behalf of Peletarius (p. 10, 1. 22), 
and destroys his opinion. For, according to Eu- 


clid, the greatest angle is the greatest inclination ; LESSON in. 
and an angle equal to two right angles by this a<c\ior/a, o^TwitT 
should not be the greatest inclination, as it is, but thatocnu m 
the least that can be. But if by the inclination of 
two lines, we understand that proceeding of them 
to a common point, which is caused by their gene- 
ration, which, I believe, was Euclid's meaning ; 
then will the angle of contact be no less an angle 
than a rectilineal angle, but only (as Clavius truly 
says it is) heterogeneous to it ; and the doctrine of 
Clavius more conformable to Euclid than that of 
Peletarius. Besides, if it be granted you, that 
there is no inclination of the circumference to the 
tangent, yet it docs not follow that their con- 
course doth not form some kind of angle ; for Eu- 
clid defineth there but one of the kinds of a plane 
angle. And then you may as much in vain seek 
for the proportion of such angle to the angle of 
contact, as seek for the focus or parameter of the 
parabola of Dices and Lazarus. Your first argu- 
ment therefore is nothing w r orth, except you make 
good that which in your second argument you 
affirm, namely, that all plane angles, not excepting 
the angle of contact, are (homogeneous) of the 
same kind. You prove it w r ell enough of other 
curvilineal angles ; but when you should prove the 
same of an angle of contact, you have nothing to 
say but (p. 17,1. 15), " Uncle autem ilia quam 
somniet heterogenia oriatur, neque potest ille ulla- 
tenus ostendere, neque ego vel somniare:" " Whence 
should arise that diversity of kind which he 
dreams of, neither can he at all show, nor 1 
dream ;" as if you knew what he could do if he 
were to answer you ; or all were false which you 

S 2 


LKSSON in. cannot dream of. So that besides your customary 
oftiiefouitT vanity > here is nothing hitherto proved, neither 
that occur in f or the opinion of Peletarius, nor against that of 

demonstration. . . T T i nr> i T i 

Clavius. I have, I think,, sufficiently explicated, in 
the first lesson, that the angle of contact is quan- 
tity, namely, that it is the quantity of that crooked- 
ness or flexion, by which a straight line is bent 
into an arch of a circle equal to it ; and that be- 
cause the crookedness of one arch may be greater 
than the crookedness of another arch of another 
circle equal to it ; therefore the question quanta 
est curvitas, how much is the crookedness, is per- 
tinent, and to be answered by quantity. And I 
have also shown you in the same lesson, that the 
quantity of one angle of contact is compared with 
that of another angle of contact by a line drawn 
from the point of contact, and intercepted by their 
circumferences ; and that it cannot be compared 
by any measure with a rectilineal angle. 

But let us see ho\v you answer to that wiiich 
Clavius has objected already. " They are hetero- 
geneous" says he, u because the angle of contact, 
how oft soever multiplied, can never exceed a rec- 
tilineal angle." To answer which, you allege it 
is no angle at all ; and that therefore, it is no 
angle at all, because the lines have no inclination 
one to another. How can lines that have no incli- 
nation one to another, ever corne together ? But 
you answer, at least they have no inclination in 
the point of contact. And why have two straight 
lines inclination before they come to touch, more 
than a straight line and an arch of a circle ? And 
in the point of contact itself, how can it be that 
there is less inclination of the two points of a 

occur in 


straight line and an arch of a circle, than of the LESSON in 
points of two straight lines ? But the straight lines, ouhef ^ ult ^ 
you say. will cut ; which is nothing to the ques- that 

J J demi 

tion ; and yet this also is not so evident, but that 
it may receive an objection. Suppose two circles, 
A G B and C F B, to touch in B, and 
have a common tangent through B. 
Is not the line C F B G A a crooked 
line ? and is it not cut by the common 
tangent D B E ? What is the quan- 
tity of the two angles F B E and GBD, 
seeing you say neither DBG nor E B F 
is an angle ? It is riot, therefore, the cutting of a 
crooked line, and the touching of it, that distin- 
guisheth an angle simply, from an angle of contact. 
That which makes them differ, and in kind, is, that 
the one is the quantity of a revolution, and the 
other, the quantity of flexion. 

In the seventh chapter of the same treatise, you 
think you prove the angle of contact, if it be an 
angle, and a rectilineal angle to be (homogeneous) 
of the same kind ; when you prove nothing but 
that you understand not what you say. Those 
quantities which can be added together, or sub- 
tracted one from another, are of the same kind ; 
but an angle of contact may be subtracted from a 
right angle, and the remainder will be the angle of 
a semicircle, &c. So you say, but prove it not, 
unless you think a man must grant you that the 
superficies contained between the tangent and the 
arch, which is it you subtract, is the angle of con- 
tact ; and that the plane of the semicircle is the 
angle of the semicircle, which is absurd ; though, 
as absurd as it is, you say it directly in your Elen- 


LESSON in. chus, p. 35, 1. 14, in these words : " When Euclid 
oftheiauitT defines a plane angle to be the inclination of two 
that occur m Unes, lie mecmeth not their aggregate, but that 

domon&tidfion. . y 

which lies between them." It is true, he meaneth 
not the aggregate of the two lines ; but that he 
means that which lies between them, which is 
nothing else but an indeterminate superficies, is 
false, or Euclid was as foolish a geometrician as 
either of you two. 

Again, you would prove the angle of contact, if 
it be an angle, to be of the same kind with a rec- 
tilineal angle, out of Euclid (in. 16) ; where he 
says, it is less than any acute angle. And it fol- 
lows well, that if it be an angle, and less than any 
rectilineal angle, it is also of the same kind with 
it. But, to my understanding, Euclid meant no 
more, but that it was neither greater nor equal ; 
which is as truly said of heterogeneous, as of homo- 
geneous quantities. If he meant otherwise, he 
confirms the opinion of Clavius against you, or 
makes the quantity of an angle to be a superficies, 
and indefinite. But I wonder how you dare venture 
to determine whether two quantities be homoge- 
neous or not, without some definition of homoge- 
neous (which is a hard word), that men may 
understand what it meaneth. 

In your eighth chapter you have nothing but 
Sir H. Savile's authority, who had not then re- 
solved what to hold ; but esteeming the angle 
of contact, first, as others falsely did, by the 
superficies that lies between the tangent and the 
arch, makes the angle of contact and a recti- 
lineal angle homogeneous ; and afterwards, be- 
cause no multiplication of the angle of contact 


can make it equal to the least rectilineal angle, LESSON HI. 
with great ingenuity returneth to his former uncer- ofthTt^uT 
tainty. tliat occur in 

J . demonstration. 

In your ninth and tenth chapters you prove with 
much ado, that the angles of like segments are 
equal ; as if that might not have been taken gratis 
by Peletarius, without demonstration. And yet 
your argument, contained in the ninth chapter, is 
not a demonstration, but a conjectural discourse 
upon the word similitude. And in the eleventh 
chapter,, wherein you answer to an objection, which 
might be made to your argument in the precedent 
page, taken from the parallelism of two concentric 
circles, though objection be of no moment, yet 
you have in the same treatise of yours that which 
is much more foolish, which is this, (p. 38, 1. 12) : 
" Non enim magnitude anguli" Sfc. The magni- 
tude of an angle is not to be estimated by that 
straddling of the legs, which it hath without the 
point of concourse, but by that straddling which it 
hath in the point of the concourse itself'' I pray 
you tell me what straddling there is of two coinci- 
dent points, especially such points as you say are 
nothing f When did you ever see two nothings 
straddle ? 

The arguments in your twelfth and thirteenth 
chapters are grounded all on this untruth, that an 
angle is that which is contained between the lines 
that make it ; that is to say, is a plane superficies, 
which is manifestly false ; because the measure of 
an angle is an arch of a circle, that is to say, a line ; 
which is no measure of a superficies. Besides this 
gross ignorance, your way of demonstration, by 
putting N for a great number of sides of an equi- 


LESSON in. lateral polygon,, is not to be admitted ; for, though 
1 ' ' you understand something by it, you demonstrate 

Of the faults J 11,11 i i 

that occur m nothing to anybody but those who understand 

demonstiation. i v , 1*1* 

your symbolic tongue, which is a very narrow 
language. If you had demonstrated it in Irish or 
Welsh, though I had not read it, yet I should not 
have blamed you, because you had written to a 
considerable number of mankind, which now you 
do not. 

In your last chapters you defend Vitellio with- 
out need ; for there is no doubt but that whatso- 
ever crooked line be touched by a straight line, 
the angle of contingence will neither add anything 
to, nor take anything from, a rectilineal right 
angle ; but that it is because the angle of contact 
is no angle, or no quantity, is not true. For it is 
therefore an angle, because an angle of contact ; 
and therefore quantity, because one angle of con- 
tact may be greater than another ; and therefore 
heterogeneal, because the measure of an angle of 
contact cannot (congruere) be applied to the mea- 
sure of a rectilineal angle, as they think it may, 
who affirm with you that the nature of an angle 
consisteth in that which is contained between the 
lines that comprehend it, viz., in a plane super- 
ficies. And thus you see in how few T lines, and 
without brachygraphy, your treatise of the angle of 
contingence is discovered for the greatest part to 
be false, and for the rest, nothing but a detection 
of some errors of Clavius grounded on the same 
false principles with your own. To return now 
from your treatise of the angle of contact back again 
to your Elenchus. 

The fault you find at art. 18, is, that I under- 


stand not that Euclid makes a plane angle to be LESSON m. 
that which is contained between the two lines that Of 7|7^r 
form it. It is true, that I do not understand that that occur m 

_ iii demonstration. 

Euclid was so absurd, as to think the nature ot an 
angle to consist in superficies ; but I understand 
that you have not had the wit to understand 

The nineteenth article of mine in this fourteenth 
chapter, is this : "All respect or variety of position 
of two lines, seemetli to be comprehended in four 
kinds. For they are either parallel, or (being if 
need be produced) make an angle ; or, (if drawn 
out far enough) touch ; or, lastly, they are asymp- 
totes" ; in which you are first offended with the 
word It seems. But I allow you, that never err, 
to be more peremptory than I am. For to me it 
seemed (I say again seemed) that such a phrase, in 
case I should leave out something in the enumera- 
tion of the several kinds of position, would save 
me from being censured for untruth ; and yet your 
instance of two straight lines in divers planes, does 
not make my enumeration insufficient. For those 
lines, though not parallels, nor cutting both the 
planes, yet being moved parallelly from one plane 
to another, will fall into one or other of the kinds 
of position by me enumerated ; and consequently, 
are as much that position, as two straight lines in 
the same plane, not parallel, make the same angle, 
though riot produced till they meet, which they 
would make if they were so produced : for you 
have nowhere proved, nor can prove, that two 
such lines do not make an angle. It is not the 
actual concurrence of the lines, but the arch of a 
circle, drawn upon that point for centre, in which 


LESSON in. they would meet if they were produced, and inter- 
or the fouuT ce pted between them, that constitutes the angle. 
that occur m Also your objection concerning asymptotes in 

demonstration. 7 . J x _ ,11 111 

general is absurd. You would have me add, that 
their distance shall at last be less than any dis- 
tance that can be assigned ; and so make the de- 
finition of the genus the same with that of the 
species. But because you are not professors of 
logic, it is not necessary for me to follow your 
counsel. In like manner, if we understand one 
line to be moved towards another always parallelly 
to itself, which is, though not actually, yet poten- 
tially the same position, all the rest of your in- 
stances will come to nothing. 

At the two-and-twentieth article you object to 
me the use of the word figure, before I had defined 
it : wherein also you do absurdly ; for I have no- 
where before made such use of the word figure, 
as to argue anything from it ; and therefore your 
objection is just as wise as if you had found fault 
with putting the word figure in the titles of the 
chapters placed before the book. If you had known 
the nature of demonstration, you had not objected 

You add further, that by my definition of figure, 
a solid sphere, and a sphere made hollow within, 
is the same figure ; but you say not why, nor can 
you derive any such thing from my definition. 
That which deceived your shallowness, is, that 
you take those points that are in the concave su- 
perficies of a hollowed sphere, not to be contiguous 
to anything without it, because that whole con- 
cave superficies is wifhin the whole sphere. Lastly, 
for the fault you find with the definition of like 


figures in like positions, I confess there wants the LESSON in 
same word which was wanting in the definition of Of ^ e ^ Ullts ' 
parallels; namely, (id easdem partes (the same 

Ni-iiiJi i jJJ-ll, A f 

way) which should have been added in the end ot 
the definition of like figures, &c., and may easily 
be supplied by any student of geometry, that is not 
otherwise a fool. 

At the fifteenth chapter, art. 1 5 number 6, you 
object as a contradiction, that I make motion to be 
the measure of time ; and yet., in other places , do 
usually measure motion and the affections thereof 
by time. If your thoughts were your own, and 
not taken rashly out of books, you could not but, 
(with all men else that see time measured by clocks, 
dials, hour-glasses, and the like), have conceived 
sufficiently, that there cannot be of time any other 
measure besides motion ; and that the most uni- 
versal measure of motion, is a line described by 
some other motion ; which line being once ex- 
posed to sense, and the motion whereby it was 
described sufficiently explicated, will serve to mea- 
sure all other motions and their time : for time 
and motion (time being but the mental image or 
remembrance of the motion) have but one and the 
same dimension, which is a line. But you, that 
w T ould have me measure swiftness and slowness by 
longer and shorter motion, what do you mean by 
longer and shorter motion ? Is longer and shorter 
in the motion, or in the duration of the motion, 
which is time ? Or is the motion, or the duration 
of the motion, that which is exposed, or designed 
by a line ? Geometricians say often, let the line A B 
be the time ; but never say, l<? t the line AB be the 
motion. There is no unlearned man that under- 


LESSON in. standeth not what is time, and motion, and mea- 
n , v ~ ' ~' sure ; and vet you, that undertake to teach it (most 

Of the faults J 7 x 

that occur m egregious professors) understand it not. 

At the second article you bring another argu- 
ment (which it seems in its proper place you had 
forgotten), to prove that a point is not quantity not 
considered, but absolutely nothing ; which is this, 
That if a point be not nothing, then the whole is 
greater than its two haloes. How does that fol- 
low ? Is it impossible w T hen a line is divided into 
tw r o halves, that the middle point should be divided 
into two halves also, being quantity ? 

At the seventh article, I have sufficiently de- 
monstrated, that all motion is infinitely propagated, 
as far as space is filled with body. You allege no 
fault in the demonstration, but object from sense, 
that the skipping of a flea is not propagated to 
the Indies. If I ask you how you know it, you 
may wonder perhaps, but answer you cannot. Are 
you philosophers, or geometricians, or logicians, 
more than are the simplest of rural people ? or are 
you not rather less, by as much as he that standeth 
still in ignorance, is nearer to knowledge, than he 
that runneth from it by erroneous learning ? 

And, lastly, what an absurd objection is it which 
you make to the eighth article, where I say that 
when two bodies of equal magnitude fall upon a 
third body, that which falls with greater velocity, 
imprints the greater motion ? You object, that 
not so much the magnitude is to be considered as 
the weight ; as if the weight made no difference 
in the velocity, when notwithstanding weight is 
nothing else but motion downward. Tell me, when 
a weighty body thrown upwards worketh on the 


body it meeteth with, do you not then think it LESSON m. 
worketh the more for the greatness, and the less oi r^7^uT 

for tlie Weight. that occur m 





OF twenty articles which you say (of nineteen 
which I say) make the sixteenth chapter, you ex- 
cept but three, and confidently affirm the rest are 
false. On the contrary, except three or four faults, 
such as any geometrician may see proceed not from 
ignorance of the subject, or from want of the art 
of demonstration, (and such as any man might have 
mended of himself) but from security ; I affirm 
that they are all true, and truly demonstrated ; and 
that all your objections proceed from mere igno- 
rance of the mathematics. 

The first fault you find is this, that I express 
not (art. I.) what impetus it is, which I would have 
to be multiplied into the time. 

The last article of my thirteenth chapter was 
this, " If there be a number of quantities pro- 
pounded, howsoever equal or unequal to one ano- 
ther ; and there be another quantity which so often 
taken as there be quantities propounded^ Is equal 
to their whole sum ; that quantity I call the mean 
arithmetical of them all"' Which definition I did 
there insert to serve me in the explication of those 


LESSON iv. propositions of which the sixteenth chapter con- 
onhe fault* sisteth, but did not use it here as I intended. My 
that occur m fi rs t proposition therefore as it standeth yet in the 

demonstration. _ . . ,. ? 7 . , 

Latin, being this, " the velocity oj any body moved 
during any time, is so much as is the product of 
the impetus in one point of time, multiplied into 
the whole time ;" to a man that hath not skill 
enough to supply what is wanting, is not intelligi- 
ble. Therefore I have caused it in the English to 
go thus : " the velocity of any body in whatsoever 
time moved, hath its quantity determined by the 
sum of all the several (impetus) quicknesses, which 
it hath in the several points of the time of the 
body's motion. And added, that all the impetus 
together taken through the whole time is the same 
thing with the mean impetus (which mean is de- 
fined (Chapter xm. art. 29) multiplied into the 
whole time" To this first article, as it is uncor- 
rected in the Latin, you object, that meaning by 
impetus some middle impetus, and assigning none, 
I determine nothing. And it is true. But if you 
had been geometricians sufficient to be professors, 
you w r ould have shewed your skill much better, by 
making it appear that this middle impetus could 
be none but that, which being taken so often, as 
there be points in the line of time, would be equal 
to the sum of all the several impetus taken in the 
points of time respectively ; which you could not 

To the corollary, you ask first how impetus can 
be ordinately applied to a line ; absurdly. For 
does not Archimedes sometimes say, and with him 
many other excellent geometricians, let such a line 
be the time ? And do they not mean, that that 


line, or the motion over it, is the measure of the LESSON iv. 
time ? And may not also a line serve to measure Oj ^u^it7 
the swiftness of a motion ? You thought, you say, tiwtoccm m 

7 7 . 7 7 . 7 . 7 demonstration 

only lines ought to be said to be ordmately ap- 
plied to lines. Which I easily believe ; for I see 
you understand not that a line, though it be not 
the time itself, may be the quantity of a time. 
You thought also, all you have said in your 
Elenchus, in your doctrine of the angle of contact , 
in your Arithmctica Infinitorum, and in your 
Conies, is true ; and yet it is almost all proved false, 
and the rest nothing worth. 

Secondly, you object, that / design a parallelo- 
gram by one only side. It was indeed a great 
oversight, and argueth somewhat against the man, 
but nothing against his art. For he is not worthy 
to be thought a geometrician that cannot supply 
such a fault as that, and correct his book himself. 
Though you could not do it, yet another from be- 
yond sea took notice of the same fault in this 
manner, " He mahcth a parallelogram of but one 
side ; it should be thus : vel denique per paral- 
lelogrammum cujus unum latus est medium pro- 
portionate inter impetum maximum (sive ultimo^ 
acquisitum] et impetus ejusdem maximi semissem; 
alterum vero latus, medium proportionate, inter 
totum tempus,et ejusdem totius temporis semissem." 
Which I therefore repeat, that you may learn good 
manners ; and know, that they who reprehend, 
ought also, when they can, to add to their repre- 
hension the correction. 

At the second article, you are pleased to advise 
me, instead of in omni motu uniformi, to put in in 
omnibus motibus uniformibus. You have a strange 


LESSON iv. opinion of your own judgment, to think you know 
^-TTT" to what end another man useth any word, better 

Of the faults m t . 

that occur m than himself. My intention was only to consider 

demonstration. . . ^ -. . * , r i 

motions uniform, and motions from rest uniformly, 
or regularly accelerated, that I might thereby com- 
pute the lengths of crooked lines, such as are de- 
scribed by any of those motions. And therefore it 
was enough to prove this theorem to be true in 
all uniform or uniformly accelerated motion, not 
motions ; though it be true also in the plural. It 
seems you think a man must write all he knows, 
whether it conduce, or not, to his intended purpose. 
But that you may know that I was not (as you 
think), ignorant how far it might be extended, you 
may read it demonstrated at the same article in the 
English universally. Against the demonstration 
itself you run into another article, namely, the 
thirteenth, which is this problem : " the length 
being given, which is passed over in a given time by 
uniform motion, to Jind the length which shall be 
passed over by motion uniformly accelerated in 
the same time, so as that the impetus last acquired 
be equal to the time." Which you recite imperfectly, 
thereby to make it seem that such a length is not 
determined. Whether you did this out of igno- 
rance, or on purpose, thinking it a piece of w r it, 
as your pretended mystery which goes immediately 
before, I cannot tell, for in neither place can any 
wit be espied by any but yourselves. To imagine 
motions with their times and ways, is a new busi- 
ness, and requires a steady brain, and a man that 
can constantly read in his own thoughts, without 
being diverted by the noise of words. The want 
of this ability, made you stumble and fall unhand- 


somely in the very first place (that is in Chap. xin. LESSON iv 
art. 13), where you venture to reckon both motion onhe faultg ^ 
and time at once; and hath made you in this that occur m 

" demonstration 

chapter to stumble in the like manner at every 
step you go. As, for example, when I say, as the 
product of the time., and impetus , to the product of 
the time and impetus, so the space to tlie space 
when the motion is uniform; you come in with, nay, 
rather as the time to the time ; as if the parallelo- 
grams A I, and A H, were not also as the times 
A B, and A F. Thus it is, when men venture upon 
ways they never had been in before, without a 

In the corollary, you are oifended with the per- 
mutation of the proportion of times and lines, 
because you think, (you that have scarce one right 
thought of the principles of geometry), that line 
and time are heterogeneous quantities. I know 
time and line are of divers natures ; and more, 
that neither of them is quantity. Yet they may 
be both of them quanta, that is, they may have 
quantity ; but that their quantities are heteroge- 
neous is false. For they are compared and mea- 
sured both of them by straight lines. And to thig 
there is nothing contrary in the place cited by you 
out of Clavius ; or if there were, it w r ere not to be 
valued. And to your question, what is the pro- 
portion of an hour to an ell ? I answer, it is the 
same proportion that two hours have to two ells. 
You see your question is not so subtle as you 
thought it. By and bye you confess that in times 
and lines there is quid homogeneum (this quid is 
an infallible sign of not fully understanding what 
you say) ; which is false if you take it of the lines 




Of the faults 
that occur m 

LESSON iv. themselves ; though if you take it of their quan- 
tities, it is true without a quid. Lastly, you tell 
me how I might have expressed myself so as it 
might have been true. But because your expres- 
sions please me not, I have not followed your 

To the third article, which is this : " In motu 
uniform! ter a quiete accelerate" etc. " In motion 
uniformly accelerated from rest, that is, when the 
impetus increaseth in proportion to the times, the 
length run over in one time is to the length run 
over in another time, as the product of the impetus 
multiplied by the time, to the product of the impetus 
multiplied by the time ;" you object, " that the 
lengths run over are in that proportion which the 
impetus hath to the impetus ; not that which the 
impetus hath to the time, because impetus to time 
has no proportion, as being heterogeneous." First, 
when you say the impetus, do you mean some 
one impetus designed by some one of the unequal 
straight lines parallel to the base B I ? That is 
manifestly false. You mean the aggregate of all 
those unequal parallels. But that is the same 
thing with the time multiplied into the mean im- 
petus. Arid so you say the same that I do. 
Again, I ask, where it is that I say or dream that 
the lengths run over are in the proportion of the 
impetus to the times ? Is it you or I that dream ? 
And for your heterogeneity of the quantities of 
time and of swiftness,! have already in divers places 
showed you your error. Again, why do you make 
B I represent the lengths run over, which I make 
to be represented by D E, a line taken at pleasure ; 
and you also a few lines before make the same 


B I to design the greatest acquired impetus ? LESSON iv. 
These are things which show that \ou are puzzled ^ ' ' 

55 J l f Of the faults 

and entangled with the unusual speculation of that occur m 

. . T i , . , . i demonstration. 

time and motion, and yet are thrust on with pride 
and spite to adventure upon the examination of 
this chapter. 

Secondly, you grant the demonstration to be 
good, supposing I mean it, as I seem to speak, of 
one and the same motion. But why do I not 
mean it of one arid the same motion, when I say 
not in motions, but in motion uniform ? Because, 
say you, in that which follows, I draw it also to 
different motions. You should have given at least 
one instance of it ; but there is no such matter. 
And yet the proposition is in that case also true ; 
though then it must not be demonstrated by the 
similitude of triangles, as in the case present. And 
therefore the objections you make from different 
impetus acquired in the same time, and from other 
cases which you mention, are nothing worth. 

At the fourth article, you allow the demonstra- 
tion all the way (except the faults of the third, 
which I have already proved to be none) till I 
come to say, " that because the proportion of F K 
to B I is double to the proportion of A F to A B, 
therefore the proportion of A B to A F is double 
to the proportion of B I to F K" This you deny, 
and wonder at as strange, (for it is indeed strange 
to you), and in many places you exclaim against 
it as extreme ignorance in geometry. In this 
place you only say, " no such matter ; for though 
one proportion be double to another, yet it does 
not follow that the converse is the double of the 
converse" So that this is the issue to which the 

T 2 


LESSON iv. question is reduced, whether you have any or no 
geometry. I say, if there be three quantities in 
continual proportion, and the first be the least, the 
proportion of the first to the second is double to 
the proportion of the first to the third ; and you 
deny it. The reason of our dissent consisteth in 
this, that you think the doubling of a proportion to 
be the doubling of the quantity of the proportion, 
as well in proportions of defect, as in proportions 
of excess ; and I think that the doubling of a pro- 
portion of defect, is the doubling of the defect of 
the quantity of the same. As for example in these 
three numbers, 1, 2, 4, which are in continual pro- 
portion, I say the quantity of the proportion of one 
to two, is double the quantity of the proportion of 
one to four. And the quantity of the proportion 
of one to four, is half the quantity of the proportion 
of one to two. And yet deny not but that the 
quantity of the defect in the proportion of one to 
two is doubled in the proportion of one to four. 
But because the doubling of defect makes greater 
defect, it maketh the quantity of the proportion 
less. And as for the part which I hold in this 
question, first, there is thus much demonstrated by 
Euclid, El. v. prop. 8 ; that the proportion of one 
to two, is greater than the proportion of one to 
four, though how much it is greater be not there de- 
monstrated. Secondly, I have demonstrated (Chap, 
xin. art. 16) ; that it is twice as great, that is to 
say, (to a man that speaks English), double. The 
introducing of duplicate, triplicate, &c. instead of 
double, triple, &c. (though now they be words well 
understood by such as understand what proportion 
is), proceeded at first from such as durst not for 


fear of absurdity, call the half of any thing double LESSON iv. 
to the whole, though it be manifest that the half ' ' 

/ i ,. i i i - i T i Of the faults 

or any defect is a double quantity to the whole that occur m 
defect ; for want added to want maketh greater 
want, that is, a less positive quantity. This differ- 
ence between double and duplicate, lighting upon 
w r eak understandings, has put men out of the way 
of true reasoning in very many questions of geo- 
metry. Euclid never used but one word both for 
double and duplicate. It is the same fault when 
men call half a quantity subduplicate , and a third 
part subtriplicate of the whole, with intention (as 
in this case) to make them pass for words of signi- 
fication different from the half smti. the third part. 
Besides, from my definition of proportion (which 
is clear, and easy to be understood by all men, but 
such as have read the geometry of others unluckily) 
I can demonstrate the same evidently and briefly 
thus. My definition is this, proportion is the 
quantity of one magnitude taken comparatively to 
another. Let there be therefore three quantities, 

1 , 2, 4, in continual proportion. Seeing therefore 
the quantity of four in respect of one, is twice as 
great as the quantity of the same four in respect of 

2, it folio weth manifestly that the quantity of 1 in 
respect of 4, is twice as little as the quantity of 
the same 1 in respect of 2 ; and consequently the 
quantity of 1 in respect of 2, is twice as great as 
the quantity of the same 1 in respect of 4 ; which 
is the thing I maintain in this question. Would 
not a man that employs his time at bowls, choose 
rather to have the advantage given him of three in 
nine, then of one in nine ? And why, but that 
three is a greater quantity in respect of nine, than 


LESSON iv. is one ? Which is as much as to say, three to 
nine hath a greater proportion than one to nine ; 
as is demonstrated by Euclid, El. v. prop. 8. Is it 
not therefore (you that profess mathematics, and 
theology, and practise the depression of the truth 
in both) well owled of you, to teach the contrary ? 
But where you say " that the point K (in the se- 
cond figure of the table belonging to this sixteenth 
chapter) is not in the parabolical line whose dia-* 
meter is A B, and base B /, but in the parabolical 
line of the complement of my semiparabola (as I 
may learn from the twenty-third proposition of 
your Arithmetica Infinitorum) whose diameter is 
A C, and base I C" What line is that ? Is it the 
same line with that of my semiparabola, or not the 
same ? If the same, why find you fault ? If not 
the same, you ought to have made a semiparabola 
on the diameter A C, and base I C, and following 
my construction made it appear that K is not in 
the line wherein I say it is ; which you have not 
done, nor could do. 

Then again, running on in the same blindness 
of passion, you pretend I make the proportion of 
B-I to F K double to that of A B to A F, and then 
confute it ; when you knew I made the proportion 
of F K to B I, double to that of F N, to B I, that is, 
of A F to A B ; and this was it you should have 
confuted. That which followeth is but a triumph- 
ing in your own ignorance, wherein you also say, 
" that all that I afterwards build upon this doc- 
trine is false." You see whether it be like to prove 
so or not. As for your Arithmetica Infinitorum^ 
I shall then read to you a piece of a lesson on it 
when I come to your objections against the next 


Chapter. In the mean time let me tell you, it is LESSON iv. 
not likely you should be great geometricians, that Wthe fmiU! f 
know not what is quantity, nor measure, nor tiiat occur m 

, . , , 1 , demonstration. 

straight, nor angle, nor homogeneous, nor hetero- 
geneous, nor proportion, as I have already made 
appear in this and the former lessons. 

To the first corollary of this fourth article your 
exception I confess is just, and (which I wonder at) 
without any incivility. But this argues not igno- 
rance, but security. For who is there that ever 
read any thing in the Conies, that knows not that 
the parts of a parabola cut off by lines parallel to 
the base, are in triplicate proportion to their bases ? 
But having hitherto designed the time by the 
diameter, and the impetus by the base ; and in the 
next chapter (where I was to calculate the pro- 
portion of the parabola, to the parallelogram) 
intending to design the time by the base, I mistook 
and put the diameter again for the time ; which 
any man but you might as easily have corrected as 

To the second corollary, which is this, that the 
lengths run over in equal times by motion so 
accelerated, as that the impetus increase in double 
proportion to their times, are as the differences of 
the cubic numbers beginning at unity, that is, as 
seven, nineteen, thirty-seven, Sfc. you say it is 
false. But why ? " Because" say you " portions 
of the parabola of equal altitude, taken from the 
beginning, are not as those numbers seven, nine- 
teen, thirty-seven, 8fc" Does this, think you, 
contradict any thing in this proposition of mine ? 
Yes, because, you think, the .lengths gone over in 
equal times, are the same with the parts of the 


LESSON iv. diameter cut off from the vertex, and proportional 
nr \" ; ~' to the numbers one, two, three, &c. Whereas the 

Ot the faults y ' * 

that occur in lengths run over, are as the aggregates of their 

demonstration. -. . . , . , , , , 

velocities, that is, as the parts ot the parabola 
itself, that is, as the cubes of their bases, that is, 
as the numbers one, eight, twenty-seven, sixty-four, 
&c./and consequently the lengths run over in equal 
times, are as the differences of those cubic num- 
bers, one, eight, twenty-seven, sixty-four, whose 
differences are seven, nineteen, thirty-seven, &c. 
The cause of your mistake was, that you cannot 
yet, nor perhaps ever will, contemplate time and 
motion (which requireth a steady brain) without 

The third corollary you also say is false, " whe- 
ther it he meant of motion uniformly accelerated 
(as the words are) or (as perhaps, you say, / 
meant it) of such motion as is accelerated in dou- 
ble proportion to the time." You need not say 
perhaps I meant it. The words of the proposition 
are enough to make the meaning of the corollary 
understood. But so also you say it is false. Me- 
thinks you should have offered some little proof to 
make it seem so. You think your authority will 
carry it. But on the contrary I believe rather 
that they that shall see how your other objections 
hitherto have sped, will the rather think it true, 
because you think it false. The demonstration as 
it is, is evident enough ; and therefore I saw no 
cause to change a word of it. 

To the fifth article you object nothing, but that 
it dependeth on this proposition (Chap. xm. art. 
16) : " That when .three quantities are in con- 
tinual proportion, and the first is the least, as in 


these numbers, four, six, nine, the proportion of LESSON iv. 
the first to the second, is double to the proportion * ~ o f : mlts " 
of the same first to Hie last;" which is there timt occur m 

Y i i i /. i i demonstration. 

demonstrated, and in the former lessons so amply 
explicated, as no man can make any further doubt 
of the truth of it. And you will, I doubt not, 
assent unto it. But in what estate of mind will you 
be then ? A man of a tender forehead after so 
nxuch insolence, and so much contumelious lan- 
guage grounded upon arrogance and ignorance, 
would hardly endure to outlive it. In this vanity 
of yours, you ask me whether I be angry, or 
blush, or can endure to hear you. I have some 
reason to be angry ; for what man can be so 
patient as not to be moved with so many injuries ? 
And I have some reason to blush, considering the 
opinion men will have beyond sea, (when they shall 
see this in Latin) of the geometry taught in Oxford. 
But to read the worst you can say against me, I 
can endure, as easily at least, as to read any thing 
you have written in your treatises of the Angle of 
Contact, of the Conic Sections, or your Arith- 
metica Infinitorum. 

The sixth, seventh, eighth articles, you say are 
sound. True. But never the more to be thought 
so for your approbation, but the less ; because you 
are not fit, neither to reprehend, nor praise ; and 
because all that you have hitherto condemned as 
false, hath been proved true. Then you show me 
how you could demonstrate the sixth and seventh 
articles a shorter way. But though there be your 
symbols, yet no man is obliged to take them for 
demonstration. And though, they be granted to 
be dumb demonstrations, yet when they are taught 


LESSON iv. to speak as they ought to do, they will be longer 

or the faults demonstrations than these of mine. 

that occur in To the ninth article, which is this, " If a body 

demonstration. 7 . . 

be moved by two movents at once, concurring in 
what angle soever, of which, one is moved uni- 
formly, the other, with motion uniformly accelera- 
ted from rest, till it acquire an impetus equal 
to that of the uniform motion, the line in which the 
body is carried, shall be the crooked line of .a 
semiparabola," you lift up your voice again, and 
ask, what latitude ? what diameter ? what incli- 
nation of the diameter to the ordinate lines ? If 
your founder should see this, or the like objections 
of yours, he would think his money ill bestowed. 
When I say, in what angle soever, you ask, in what 
angle ? When I say two movents, one uniform, 
the other uniformly accelerated, make the body 
describe a semiparabolical line ; you ask, which is 
the diameter? as not knowing that the acceler- 
ated motion describes the diameter, arid the other 
a parallel to the base. And when I say the two 
movents meet in a point, from which point both the 
motions begin, and one of them from rest, you ask 
me what is the altitude ? As if that point where 
the motion begins from rest were not the vertex ; 
or that the vertex and base being given, you had 
not wit enough to see that the altitude of the para- 
bola is determined ? When Galileo's proposition, 
which is the same with this of mine, supposed no 
more but a body moved by these two motions, to 
prove the line described to be the crooked line of 
a semiparabola, I never thought of asking him 
what altitude, nor what diameter, nor what angle, 
nor what base, had his Darabola. And when 


Archimedes said, let the line A B be the time, I LESSON iv. 
should never have said to him, do you think time Of ~ ef ; ult / 
to be a line, as you ask me whether I think impetus that occur m 

y J niii demonstration. 

can be the base of a parabola. And why, but be- 
cause I am not so egregious a mathematician, as 
you are. In this giddiness of yours, caused by 
looking upon this intricate business of motion, and 
of time, and the concourse of motion uniform, and 
uniformly accelerated, you rave upon the numbers 
1, 4,9, 16, &c. without reference to any thing that 
I had said ; insomuch as any one that had seen 
how much you have been deceived in them before, 
in your scurvy book of Arithmetica Irifinitorum, 
would presently conclude, that this objection was 
nothing else but a fit of the same madness which 
possessed you there. 

My tenth article is like my ninth; and your 
objections to it are the same which are to the 
former. Therefore you must take for answer just 
the same which I have given to your objection 

To the eleventh, you say first, you have done it 
better at the sixty-fourth article of your Arithme- 
tica Infinitorum. But what you have done there,, 
shall be examined when I come to the defence of 
my next chapter. And whereas I direct the reader 
for the finding of the proportions of the comple- 
ments of those figures to the figures themselves, 
to the table of art. 3, Chap, xvn., you say that if 
the increase of the spaces, were to the increase of 
the times, as one to two, then the complement 
should be to the parallelogram as one to three, and 
say you find not in the table,, Did you not see 
that the table is only of those figures which are 


LESSON iv. described by the concourse of a motion uniform 
> ' ' with a motion accelerated ? You had no reason 

Of the faults 

that occur in therefore to look for i in that table ; for your case 

demonstration. . P ,. ./t . . , .. 

is of motion uniform concurring with motion re- 
tarded, because you make not the proportions of 
the spaces to the proportions of the times as two 
to one, but the contrary ; so that your objection 
ariseth from want of observing what you read. 
But I " may learn' you say, " these, and greater 
matters than these , in your twenty-third and sixty- 
fourth propositions of your Arithmetica Infini- 
torum." This, which you say here is a great 
absurdity ; but if you mean I shall find greater 
there, I will not say against you. This [^ you 
looked for, belongs to the complements of the 
figures calculated in that table ; which because you 
are not able to find out of yourselves, I will direct 
you to them. Your case is of J for the comple- 
ment of a parabola. Take the denominator of the 
fraction which belongs to the parabola, namely 
three, and for numerator take the numerator of 
the fraction which belongs to the triangle, namely 
one, and you have the fraction sought. And in 
like manner for the complement of any other 
figure. As, for example, of the second parabolaster, 
whose fraction hath for denominator five, take the 
numerator of the fraction of the same triangle 
which is one, and you have ~ for the fraction 
sought for ; and so of the rest, taking always one 
for the numerator. 

The twelfth article, which you say is miserably 
false, I have left standing unaltered. For not 
comprehending the* sense of the proposition, you 
make a figure of your own, and fight against your 


own fancied motions, different from mine. Other LESSON iv. 
geometricians that understand the construction On 7 lefmilts ^ 
better, find no fault. And if you had in your own that occur m 
fifth figure drawn a line through N parallel to AE, 
and upon that line supposed your accelerated mo- 
tion, you would quickly have seen that in the time 
A E, the body moved from rest in A, would have 
fallen short of the diagonal A D ; and that all your 
extravagant pursuing of your own mistake had 
been absurd. 

My thirteenth article you say is ridiculous, 
But why? " The impetus last acquired cannot" 
you say, " be equal to a time."" But the quantity 
of the impetus may be equal to the quantity of a 
time, seeing they are both measured by line. And 
when they are measured by the same described 
line, each of their quantities is equal to that same 
line, and consequently to one another. But when 
I meet with this kind of objection again, since I 
have so often already shown it to be frivolous, and 
no less to be objected against all the ancients that 
ever demonstrated any thing by motion, than 
against me, I purpose to neglect it. 

Secondly, you object " that motion uniformly 
accelerated does no more determine swiftness, 
than motion uniform." True ; you needed not have 
used sixteen lines to set down that. But suppose 
I add, as I do, so as the last acquired impetus be 
equal to the time. But that, you say, is not 
sense ; which is the objection I am to neglect. 
But, you say again, supposing it sense, this limi- 
tation helps me nothing. Why ? Because, you 
say, a parabola may be described upon a base 
given, and yet have any altitude, or any diameter 


LESSON iv. one will. Who doubts it ? But how follows it 
*~ ; ~' from thence, that when a parabolical line is de- 

Of the faults t y * 

that occur m scribed by two motions, one uniform, the other 
uniformly accelerated from rest, that the deter- 
mining of the base does not also determine the 
whole parabola ? But fifthly, you say, that this 
equality of the impetus to the time does not de- 
termine the base. Why not ? Because, you say, 
it is an error proceeding from this, that I under- 
stand not what is ratio subduplicata. I looked 
for this. I have shown and inculcated sufficiently 
before, but the error is on your side ; and there- 
fore mast tell you, that this objection, -and also a 
great part of the rest of your errors in geometry, 
proceedeth from this, that you know not what pro- 
portion is. But see how wisely you argue about 
this duplication of proportion. For thus you say 
verbatim. " Stay a little. What proportion has 
duplicate proportion to single proportion ? Is it 
always the same ? I think not for example, 
duplicate proportion 4 = Tm4 is double to the 
single 4". Duplicate proportion T^'f "lA is triple 
to its single \ ." Let any man, even of them that 
are most ready in your symbols, say in your behalf 
(if he be not ashamed) that the proportion of nine 
to one is triple to the proportion of three to one, 
as you do. 

In the fourteenth, fifteenth, and sixteenth articles, 
you bid me repeat your objections to the thir- 
teenth. I have done it ; and find that what you 
have objected to the thirteenth, may as well be 
objected to these; and consequently, that my 
answer there will also serve me here. Therefore, if 
you can endure it, read the same answer over again. 


But you have not yet done, you say, with these LESSON iv. 
articles. Therefore (after you had for a while Ofthpi : ult / 
spoken perplextly, conjecturing, not without lust that occur m 

r 1T 11 11 demoastration. 

cause, that I could not understand you) you say 
that to the end I may the better perceive your 
meaning, I should take the example following. 
" Let a moment (in the first figure of this chapter) 
be moved uniformly in the time A B, with the con- 
tinual impetus A C, or 13 /, whose whole velocity 
shall therefore be the parallelogram A C IB. 
And another movent be uniformly accelerated, so 
as In the time A B it acquire the same impetus 
B I. Now as the whole velocity, is to the whole 
velocity, so is the length run over, to the length 
run over." All this I acknowledge to be according 
to my sense, saving that your putting your word 
movens instead of my word mobile hath corrupted 
this article. For in the first article, I meddle not 
with motion by concourse, wherein only I have to 
do with two movents to make one motion ; but 
in this I do, wherein my word is riot movens but 
mobile ; by which it is easy to perceive you under- 
stand not this proposition. Then you proceed : 
" But the length run over by that accelerated 
motion is greater than the length run over by that 
uniform motion" Where do I say that ? You 
answer, " in the ninth and thirteenth article, in 
making A B (in the fifth figure) greater than 
AC; and AH (in the eighth figure) greater 
than A B ; and consequently, the triangle A B /, 
greater than the parallelogram A C I B." That 
consequently is without consequence ; for it im- 
porteth nothing at all in this demonstration, whe- 
ther A B, or A C in the fifth figure be the greater. 


LESSON iv. Besides I speak there of the concourse of two 
or the faults movents > that describe the parabolical line A G D ; 
that occur m where the increasing impetus (because it increaseth 

demonstration. . i i 

as the times) will be designed by the ordmate 
lines in the parabola AGDB. And if both the 
motions in A B and A C were uniform, the aggre- 
gate of the impetus would be designed by the tri- 
angle A B D, which is less than the parallelogram 
A C D B. But you thought that the motion made 
by A C uniformly, is the same with the motion 
made uniformly in the same time by the motions 
in A B and A C concurring ; so likewise, in the 
eighth figure, there is nothing hinders A H 
from being greater than A B, unless I had said 
that A B had been described in the time A C 
with the whole impetus A C maintained entire ; of 
which there is nothing in the proposition, nor would 
at all have been pertinent to it. Therefore all this 
new undertaking of the thirteenth, fourteenth, 
fifteenth, and sixteenth articles, is to as little pur- 
pose as your former objections. But I perceive 
that these new and hard speculations, though they 
turn the edge of your wit, turn not the edge of 
your malice. 

At the seventeenth article, you show again the 
same confusion. Return to the eighth figure: 
" if in a time given a body run over two lengths, 
one with uniform, the other with accelerated 
motion'; as for example, if in the same time A C, 
a body, run over the line A B with uniform motion, 
and the line A H with motion accelerated ; " and 
again in a part of that time it run over a part of 
the length A H, with uniform motion^ and another 
part of the same with motion accelerated ;" as for 


example, in the time A M it run over with uniform LESSON iv. 
motion the line A I, and with motion accelerated \ ; ~" 

5 Ot the faults 

the line A B. / sau the excess of the whole A // that ocun m 

7 .1 i A i* , ji .? .7 77 Demonstration. 

above the part A Jf, is to the excess oj the whole 
A B above the part A /, as the whole A II to the 
whole AB. But first you will say, that these 
words as the whole A II to the whole A 11, are left 
out in the proposition. But you acknowledge 
that it was my meaning ; and you see it is ex- 
pressed before I come to the demonstration. And 
therefore it was absurdly done to reprehend it. 
Let us therefore pass to the demonstration. Draw 
I K parallel to A C, and make up the parallelogram 
A I K M. And supposing first the acceleration to 
be uniform, divide I K in the midst at N ; and be- 
tween I N, and I K, take a mean proportional I L. 
And the straight line A Z/, drawn and produced, 
shall cut the line B I) in F, and the line C G in G 
(which lines C G, and B D, as also H G and B F, 
are determined, though you could not carry it so 
long in memory, by the demonstration of the 
thirteenth article) . For seeing A B is described 
by motion uniformly accelerated, and A I by 
motion uniform in the same time AM; and I L 
is a mean proportional between T N (the half of 
IK) and 1 K; therefore by the demonstration of 
the thirteenth article, A T is a mean proportional 
between A B and the half of A J5, namely A O. 
Again, because A B is described by uniform 
motion, and A II by motion uniformly accelerated, 
both of them in the same time A C, B F is a mean 
proportional between B D and half B D, namely 
BE; therefore by the demonstration of the same 
thirteenth article, the straight line A L F pro- 


LESSON iv. duced will fall on G ; and tJie line A H will be 
s ~ ; -' to the line A B, as the line A B to the line A I. 

Of the faults y 

that occur m And consequently as AH to A B, so H B to B I ; 
which was to be demonstrated. And by the like 
demonstration the same may be proved, where the 
acceleration is in any other proportion that can be 
assigned in numbers, saving that whereas this 
demonstration dependeth on the construction of 
the thirteenth article, if the motion had been 
accelerated in double proportion to the times, it 
would have depended on the fourteenth, where the 
lines are determined. Which determinations being 
not repeated, but declared before, in the thirteenth 
article, to which this diagram belongeth, you take 
no notice of, but go back to a figure belonging to 
another article, where there was no use of these 
determinations. But because I see that the words 
of the proposition, are as of four motions, and 
not of two motions made by twice two movents, I 
must pardon them that have not rightly understood 
my meaning ; and I have now made the propo- 
sition according to the demonstration. Which 
being done, all that you have said in very near two 
leaves of your Elenchus comes to nothing ; and 
the fault you find comes to no more than a too 
much trusting to the skill and diligence of the rea- 
der. And whereas after you had sufficiently 
troubled yourself upon this occasion, you add, 
f that if Sir H. Savile had read my Geometry, 
he had never given that censure of Joseph Scaliger, 
in his lecture upon Euclid, that he was the worst 
geometrician of all mortal men, not exceptioning so 
much as Orontius, c but that praise should have 
been kept for me." You see by this time, at least 


others do, how little I ought to value that opinion ; LESSON iv. 
and that though I be the least of geometricians, 0fthM ; mlts ' 
yet rny geometry is to yours as 1 to 0. I recite ^at occur m 

J J0 J J , , demonstration. 

these words of yours, to let the world see your 
indiscretion in mentioning so needlessly that pas- 
sage of your founder. It is well known that 
Joseph Scaliger deserved as well of the state of 
learning, as any man before or since him ; and 
that though he failed in his ratiocination concern- 
ing the quadrature of the circle, yet there appears 
in that very failing so much knowledge of geome- 
try, that Sir H. Savile could not but see that there 
were mortal men very many that had less ; and 
consequently he knew that that censure of his in a 
rigid sense (without the license of an hyperbole) 
was unjust. But who is there that will approve 
of such hyperboles to the dishonour of any but of 
unworthy persons, or think Joseph Scaliger un- 
worthy of honour from learned men f Besides, it 
was not Sir H. Savile that confuted that false 
quadrature, but Clavius. What honour was it then 
for him to triumph in the victory of another : 
When a beast is slain by a lion, is it not easy for 
any of the fowls of the air to settle upon, and peck 
him ? Lastly, though it were a great error in 
Scaliger, yet it was not so great a fault as the least 
sin ; and I believe that a public contumely done to 
any worthy person after his death, is not the least 
of sins. Judge therefore whether you have not 
done indiscreetly, in reviving the only fault, per- 
haps that any man living can lay to your founder's 
charge ; and yet this error of Scaliger's was no 
greater than one of your own of the like nature, 
iri making the true spiral of Archimedes equal to 

U 2 


LESSON iv. half the circumference of the circle of the first 
". ' ~' revolution ; and then thinking to cover your fault 

Of the faults J 

that occur m by calling it afterwards an aggregate of arches of 

demolish ation. , / i i i , n r i T\ 

circles (which is no spiral at all of any kind) you 
do not repair but double the absurdity. What 
would Sir Henry Savile have said to this ? 

The eighteenth article is this, " in any parallelo- 
gram, if the two sides that contain the angle be 
moved to their opposite sides, the one uniformly., 
the other uniformly accelerated ; the side that is 
moved uniformly, by its concourse through all its 
longitude, hath the same effect which it would 
have if the other motion were also uniform, and 
the line described were a mean proportional be- 
tween the whole length, and the half of the same" 

To the proposition you object first, " that it is 
all one whether the other motion be uniform or 
not, because the effect of each of their motions, is 
but to carry the body to the opposite side." But 
do you think that whatsoever be the motions, the 
body shall be carried by their concourse always to 
the same point of the opposite side ? If not, then 
the effect is not all one when a motion is made by 
the concourse of two motions uniform and accelera- 
ted, and when it is made by the concourse of two 
uniform or of two accelerated motions. 

Secondly, you say that these words, and the 
line described were a mean proportional between 
the whole length, and the half of the same, have 
no sense, or that you are deceived. True. For 
you are deceived ; or rather you have not under- 
standing enough distinctly to conceive variety of 
motions though distinctly expressed. For when 
a line is gone over with motion uniformly accelera- 


ted, you cannot understand how a mean pro- LESSON iv, 
portional can be taken between it and its half ; or * ; ~* 

7 Of the faults 

if you can, you cannot conceive that that mean can that occur m 

i . . . f. ! demonstration. 

be gone over with uniform motion m the same 
time that the whole line w r as run over by motion 
uniformly accelerated. Yet these are things con- 
ceivable, and your want of understanding must be 
made my fault. 

. My demonstration is this, in the parallelogram 
A BCD, (Fig. 1 \). Let the side A B be con- 
ceived to be moved uniformly till it lie in CD; 
and let the time of that motion be A C, or B D. 
And in the same time let it be conceived that A C 
is moved ivith uniform acceleration, till it lie in 
B D. To which you object, that then the accelera- 
tion last acquired^ must be far greater than that 
wherewith A B is moved uniformly : else it shall 
never come to the place you would have it in the 
same time. What proof bring you for this ? 
None here. Where then ? Nowhere that I re- 
member. On the contrary I have proved (Art. 9 of 
the chapter) that the line described by the concourse 
of those two motions, namely, uniform from A B 
to C D, and uniformly accelerated from A C to B D, 
is the crooked line of the serniparabola A H D. 
And though I had not, yet it is well known that 
the same is demonstrated by Galileo. And see- 
ing it is manifest that in what proportion the 
motion is accelerated in the line A B, in the same 
proportion the impetus beginning from rest in A is 
increased in the same times (which impetus is de- 
signed all the way by the ordinate lines of the 
semiparabola), the greatest inapetus acquired must 
needs be the base of the semiparabola, namely B D, 


LESSON iv. equal to A C, which designs the whole time. I 
cannot therefore imagine what should make you 

m ~ J 

say without proof, that the greatest acquired im- 
p etus j g g reater t jj an fa a t ^{^ {$ designed by the 

base B D. Next you say, " you see not to what 
end I divide A B in the middle at E." No wonder ; 
for you have seen nothing all the way. Others 
w r ould see it is necessary for the demonstration ; as 
also that the point F is not to be taken arbitrarily ^ 
and likewise that the thirteenth article, which you 
admit not for proof, is sufficiently demonstrated, 
and your objections to it answered. By the w r ay 
you advise me, where I say percursam eodem motu 
uniformi, cum impetu ubique, &c. to blot out cum ; 
because the impetus is not a companion in the way, 
but the cause. Pardon me in that I cannot take 
your learned counsel ; for the word motu uniformi 
is the ablative of the cause, and impetu the abla- 
tive of the manner. But to come again to your 
objections, you say, I make " a greater space run 
over in the same time by the slower motion than 
by the swifter" How does that appear ? because 
there is no doubt, but the swiftness is greater 
where the greatest impetus is always maintained, 
than where it is attained to in the same time from 
rest. True, but that is, when they are considered 
asunder without concourse, but not then when by 
the concourse they debilitate one another, and 
describe a third line different from both the lines, 
which they would describe singly. In this place 
I compare their effects as contributing to the de- 
scription of the parabolical line A H D. What the 
effects of their several motions are, when they are 
considered asunder, is sufficiently shown before in 


the first article. You should first have gotten into LESSON iv. 
your minds the perfect and distinct ideas of all the N ' ' 

J L Of the faults 

motions mentioned in this chapter, and then have that occur m 
ventured upon the censure of them, but not before. temons 
And then you would have seen that the body 
moved from A, describeth not the line A C, nor the 
line A B, but a third, namely the semiparabolical 
line A H D. 

Again, where I say. Wherefore, if the whole 
A H be uniformly moved to C D, in the same time 
wherein A C is moved uniformly to F G ; you ask 
me " iv he t her with the same impetus or not ?" 
How is it possible that in the same time two un- 
equal lengths should be passed over the same 
impetus ? " But why" say you, " do you not tell 
us with what impetus A C comes to F G 9" What 
need is there of that, when all men know that in 
uniform motion and the same time, impetus is to 
impetus, as length to length ? Which to have ex- 
pressed had not been pertinent to the demonstra- 
tion. That which follows in the demonstration, 
rursus suppono quod latus A C, &c. to these 
words, ut ostensum est, Art. 12, you confute with 
saying you have proved that article to be false. 
But you may see now, if you please, at the same 
place that I have proved your objection to be 

After this you run on without any argument 
against the rest of the demonstration, showing 
nothing all the way, but that the variety and con- 
course of motions, the speculations whereof you 
have not been used to, have made you giddy. 

To the nineteenth article you apply the same 
objection which you made to the eighteenth. 


LESSON iv. Which having been answered, it appears that from 
or the faults ^ e ver y beginning of your Elenchus to this place 
timt occur iu all your objections (except such as are made to 

demonstration. . . , - ,,. 

three or four mistakes of small importance in set- 
ting down my mind), are mere paralogisms, and 
such are less pardonable than any paralogism in 
Orontius, both because the subject as less difficult 
is more easily mastered, and because the same 
faults are most shamefully committed by a repre- 
hender than by any other man. 

I had once added to these nineteen articles a 
twentieth, which was this : " If from a point in the 
circumference there be drawn a cord, and a tan- 
gent equal to it, the angle which they make shall 
be double to the aggregate of all the angles made 
by the cords of all the equal arches into which 
the arch given can possibly be divided" Which 
proposition is true, and I did when I writ it think 
I might have use of it. But be it, or the demon- 
stration of it true or false, seeing it was not pub- 
lished by me, it is somewhat barbarous to charge 
ine with the faults thereof. No doctor of humanity 
bat would have thought it a poor and wretched 
malice, publicly to examine arid censure papers of 
geometry never published, by what means soever 
they came into his hands. I must confess that in 
these words, in such kind of progression arithme- 
tical (that is, which begins with 0) the sum of all 
the numbers taken together, is equal to half the 
number that is made by multiplying the greatest 
into the least, there is a great error ; for by this 
account these numbers, 0, 1, 2, 3, 4, taken toge- 
ther, should be equal to nothing. I should have 
said they are equal to that number which is made 


by multiplying half the greatest into the number of LESSON iv. 
the terms. There was therefore, if those words Ofthef ; ults ^ 
were mine (for truly I have no copy of them, nor that occur m 

. ill i ITT demonstration. 

have had since the book \vas printed, and 1 have 
no great reason, as any man may see, to trust your 
faith) a great error in the writing, but not an 
erroneous opinion in the writer. The demonstra- 
tion so corrected is true. And the angles that 
have the proportions of the numbers 1,2, 3, 4, are 
in the table of your Elcnclius^ fig. 12, the angles 
G A D, H D E, I E F, K F B. And if the divisions 
were infinite, so that the first were not to be 
reckoned but as a cypher, the angle CAB would 
be double to them altogether. This mistake of 
mine, and the finding that I had made no use of 
it in the whole book, was the cause why I thought 
fit to leave it quite out. But your professorships, 
could not forbear to take occasion thereby, to com- 
mend your zeal against Leviathan to your doctor- 
ships of divinity, by censuring it. 




AT the seventeeth chapter, your first exception is 
to the definition of proportional proportions, which 
is this: " Four proportions are then proportional, 
when the first is to the second, as tJie third to the 


LESSON v. fourth." The reader will hardly believe that your 
* exception is in earnest. You say, I mean not by 
P ro P or tionality the " quantity of the proportions" 
Yes I do. Therefore I say again, that four pro- 
portions are then proportional, when the quantity 
of the first proportion, is to the quantity of the 
second proportion, as the quantity of the third 
proportion, to the quantity of the fourth propor- 
tion. Is not my meaning now plainly enough 
expressed ? Or is it not the same definition with 
the former. But what do I mean, you will say, 
by the quantity of a proportion ? I mean the 
determined greatness of it, that is, for example, in 
these numbers, the quantity of the proportion of 
two to three, is the same with the quantity of the 
proportion of four to six, or six to nine ; and 
again, the quantity of the proportion of six to 
four, is the same with the quantity of the propor- 
tion of nine to six, or of three to two. But now 
what do you mean by the quantity of a proportion ? 
You mean that two and three, are the quantities 
of the proportion of two to three (for so Euclid 
calls them) and that six and four are the quantities 
of the proportion of six to four, which is the same 
with the proportion of three to two. And by this 
rule, one and the same proportion shall have an 
infinite number of quantities ; and consequently 
the quantity of a proportion can never be deter- 
mined. I call one proportion double to another, 
when one is equal to twice the other ; as the pro- 
portion of four to one, is double to the proportion 
of two to one. You call that proportion double 
where one number, line, or quantity absolute, is 
double to the other ; so that with you the proper- 



tion of two to one is a double proportion. It is LESSON v. 
easy to understand how the number two is double onhe f ^ ul "' 
to one. but to what, I pray you, is double the pro- th t occur i 

' ' * J J ' A demonstrati 

portion of two to one, or of one to two ? Is not 
every double proportion double to some propor- 
tion ? See whether this geometry of yours can be 
taken by any man of sound mind for sense. " But 
it is known" you say, " that in proportions, double 
is. one tiling, and duplicate another " so that it 
seems to you, that in talking of proportion men 
are allowed to speak senselessly. " It is known" 
you say. To whom ? It is indeed in use at this 
day to call double duplicate, and triple triplicate. 
And it is well enough ; for they are words that 
signify the same thing, but that they differ (in 
what subject soever) I never heard till now. I 
am sure that Euclid, whom you have undertaken 
to expound, maketh no such difference. And even 
there where he putteth these numbers, one, two, 
four, eight, &c. for numbers in double proportion 
(which is the last proposition of the ninth element) 
he meaneth not that one to two, or two to one, is 
a double proportion, but that every number in that 
progression is double to the number next before it ; 
and yet he does not call it analogia dupla, but 
duplicate. This distinction in proportions between 
double and duplicate, proceeded long after from 
want of knowledge that the proportion of one to 
two is double to the proportion of one to four ; 
and this from ignorance of the different nature of 
proportions of excess, and proportions of defect. 
And you that have nothing but by tradition saw 
not the absurdities that did h^ng thereon. 

In the second article I make EK, (fig. 1) the 


LESSON v. third part of L K, which you say is false ; and 
or the fauitT consequently the proposition undemonstrated. 
that occur in And thus you prove it false : " Let A C be to G C, 

demonbtiatiun. ^ rs . s~i r J * ^ s^ ^7 

or G K to G L, r/s eight to one (jor seeing the 
point G is taken arbitrarily, we may place it 
where we will, fyc.)" and upon this placing of G 
arbitrarily, you prove well enough that E K is not 
a third part of L K. But you did not then ob- 
serve, that I make the altitude A G, less than any 
quantity given, and by consequence E K to differ 
from a third part by a less difference than any 
quantity that can be given. Therefore as yet the 
demonstration proceedeth well enough. But per- 
ceiving your oversight, you thought fit (though 
before, you thought this confutation sufficient) to 
endeavour to confute it another way ; but with 
much more evidence of ignorance. For when I 
come to say, the proportion therefore between A C 
and G C is triple, in arithmetical proportion, to 
the proportion between G K and G E, 8?c. you 
say, " the proportion of A C to G C is the pro- 
portion of identity, as also that of G K to G E" 
But why ? Does my construction make it so ? Do 
not I make G C less than A C, though with less 
difference than any quantity that can be assigned ? 
And then where I say, therefore E K is the third 
part of L K 9 you come in, by parenthesis, with 
(or a fourth, or a fifth, Sfc.) Upon what ground? 
Because you think it will pass for current, without 
proof, that a point is nothing. Which if it do, 
geometry also shall pass for nothing, as having no 
ground nor beginning but in nothing. But I have 
already in a former .lesson sufficiently showed you 
the consequence of that opinion. To which I may 


add, that it destroys the method of indivisibles, LESSON v. 
invented by Bonaventura ; and upon which, not *~ ; ~* 

J ' * 3 Oi tho faults 

well understood, you have grounded all your scurvy that occur m 

_. . *-*/ i demonstration, 

book of Anthmetica Injinitorum ; where your 
indivisibles have nothing to do, but as they are 
supposed to have quantity, that is to say, to be 
divisibles. You allow, it seems, your own nothings 
to be somethings, and yet will not allow my some- 
things to be considered as nothing. The rest of 
your objections having no other ground than this, 
" that a point is nothing" my w r hole demonstra- 
tion standeth firm ; and so do the demonstrations 
of all such geometricians, ancient and modern, as 
have inferred any thing in the manner following, 
viz. If it be not greater nor less, then it is equal. 
But it is neither greater nor less. Therefore, Sfc. 
If it be greater, say by how much. By so much. 
It is not greater by so much. Therefore it is not 
greater. Tf it be less, say how much, 8fc. Which 
being good demonstrations are together with mine 
overthrown by the nothingness of your point, or 
rather of your understanding ; upon w r hich you 
nevertheless have the vanity of advising me what 
to do, if I demonstrate the same again ; meaning 
I should come to your false, impossible, and absurd 
method of Arithmetica Injinitorum, worthy to be 
gilded, I do not mean with gold. 

And for your question, why I set the base of my 
figure upwards, you may be sure it was not be- 
cause I was afraid to say, that the proportions of 
the ordinate lines beginning at the vertex were 
triplicate, or otherwise multiplicate of the propor- 
tions of the intercepted parts of the diameter. 
For I never doubted to call double duplicate, nor 


LESSON v., triple triplicate, &c., or if I had, I should have 
or the Anita av ided it afterwards at the tenth article of the 
that occur in same chapter. But because when I went about to 
compare the proportions of the ordiriate lines with 
those of their contiguous diameters, the first thing 
I considered in them was in what manner the base 
grew less and less till it vanished into a point. 
And though the base had been placed below, it 
had not therefore required any change in the de- 
monstration. But I was the more apt to place the 
base uppermost, because the motion began at the 
base, and ended at the vertex. To proceed which 
way I pleased was in my own choice ; and it is of 
grace that I give you any account of it at all. 

To the third article, together with its table, you 
say, u it falls in the ruin of the second ; and that 
the same is to be understood of the sixth, seventh, 
eighth, and ninth''' For confutation whereof I 
need to say no more, but that they all stand good 
by the confutation of your objections to the 

To the fourth article you say, " the description 
of those curvilineal figures is easy." True, to 
some men ; and now that I have showed you the 
way, it is easy enough for you also. For the way 
you propound is wholly transcribed out of the 
figure of the second article, which article you had 
before rejected. For seeing the lines H F, G E, 
A B, &c. are equal to the lines C Q,, C O, C D ; and 
the lines QF, OE, BD, equal to the lines CH, 
C G, C A ; the proportion of D B to E, will be 
triple (that is, triplicate) to the proportion of C O 
to G E ; and the proportion of D B to Q, F, triple 
to the proportion of C D to C Q; and consequently, 


because the complement B D C F E B is made by LESSON v. 
the decrease of A C in triple proportion to that of 

the decrease of CD. it will be (by the second that ccur m 

.-.v -.-. .-^-r-.-^,^.* demonstration 

article) a third part of the figure A B E F C A. 
So that it comes all to one pass, whether we take 
triple proportion in decreasing to make the comple- 
ment, or triple proportion in increasing to make 
the figure ; for the proportion of H F to B A, is 
triple to the proportion of C H to C A. Wherefore 
you have done no more but what you have seen 
first done, saving that from your construction you 
prove not the figure to be triple to the complement; 
perhaps because you have proved the contrary in 
your Arithmetica Infinite rum. But your way 
differs from mine, in that you call the proportion 
sub triplicate, which I call triplicate ; as if the 
divers naming of the same thing made it differ 
from itself. You might as well have said briefly, 
the proposition is true, but ill proved, because I 
call the proportion of one or two triple, or tripli- 
cate of that of one to eight; which you say is 
false, and hath infected the fourth, fifth, ninth, 
tenth, eleventh, thirteenth, fourteenth, fifteenth, 
sixteenth, seventeenth, and nineteenth articles of 
the sixteenth chapter. But I say, and you know 
now, that it is true ; and that all those articles are 

Lastly you add, " Tu vero, in presente articulo, 
8fc. idest,you bid find as many mean proportionals 
as one will, between two given lines ; as if that 
coidd not be done by the geometry of planes, 8fc" 
You might have left out Tu vero to seek an Ego 
quidem. But tell me, do you. think that you can 
find tw r o mean proportionals (which is less than 




Of the faults 
that occui in 

. as many as one will) by the geometry of planes ? 
We shall see anon how you go about it. I never 
said it was impossible, and if you look upon the 
places cited by you more attentively, you will find 
yourself mistaken. But I say, the way to do it 
has not been yet found out, and therefore it may 
prove a solid problem for anything you know. 

The fifth article you reject, because it citeth the 
corollary of the twenty-eighth article of the thir- 
teenth chapter, where there is never a word to 
that purpose. But there is in the twenty-sixth 
article ; which was my own fault, though you knew 
not but it might have been the printer's. 

To the tenth you object for almost three leaves 
together, against these words of mine, because, 
in the sixth figure, B C is to li F in triplicate 
proportion of C D to F E, therefore inverting, 
F E is to CD in triplicate proportion of B F 
to 13 C. This you objected then. But now that 
I have taught you so much geometry, as to know 
that of three quantities, beginning at the least, if 
the third be to the first in triplicate proportion of 
the second to the first, also by conversion the first 
to the second shall be in triplicate proportion of 
the first to the third ; if it were to do again, you 
would not object it. 

My eleventh article you would allow for demon- 
strated, if my second had been demonstrated, 
upon which it dependeth. Therefore seeing your 
objections to that article are sufficiently answered, 
this article also is to be allowed. 

The twelfth also is allowed upon the same rea- 
son. What falsities you shall find in such follow- 
ing propositions as depend upon the same second 


article, we shall then see when I come to the places LESSON v, 
where you object against them. ^', ; """ 

J J o Of the faults 

To the thirteenth article you object, " that the thatoe.m m 

7 . 7 dem nisliation 

same demonstration mat/ be as well applied to a 
portion of any conoeides, parabolical, hyperboli- 
cal, elliptical, or any other, as to the portion of a, 
sphered By the truth of this let any man judge 
of your and my geometry. Your comparison of 
the sphere and conoeides, so far holds good, as to 
prove that the superficies of the conoeides is 
greater than the superficies of the cone described 
by the subtense of the parabolical, hyperbolical, 
or elliptical line. But when I come to say, that 
the cause of the excess of the superficies of the 
portion of the sphere above the superficies of the 
cone, consists in the angle D A li, and the cause 
of the excess of the circle made upon the tangent 
A D, above the superficies of the same cone, con- 
sists in the magnitude of the same angle DAB, 
how will you apply this to your conoeides ? For 
suppose that the crooked line A B (in the seventh 
figure) were not an arch of a circle, do you think 
that the angles which it maketh with the subtense 
A B, at the points A and B, must needs be equal ? 
Or if they be not, does the excess of the superficies 
of the circle upon A D above the superficies of the 
cone, or the excess of the superficies of the portion of 
the conoeides above the superficies of the same cone, 
consist in the angle D A B, or rather in the magni- 
tude of the two unequal angles DAB, and ABA? 
You should have drawn some other crooked line, 
and made tangents to it through A and B, and you 
would presently have seen your error. See how 
you can answer this ; for if this demonstration of 



LESSON v. mine stand firm, I may be bold to say, though the 
ofthefouitT same be well demonstrated by Archimedes, that this 
that occur m wa y o f m i n e is more natural, as proceeding im- 

demomtration. J r 

mediately from the natural efficient causes of the 
effect contained in the conclusion ; and besides, 
more brief and more easy to be followed by the 
fancy of the reader. 

To the fourteenth article you say that I " commit 
a circle in that I require in the fourth article the 
finding of two mean proportionals, and come not 
till now to show how it is to be done." Nor now 
neither. But in the mean time you commit two 
mistakes in saying so. The place cited by you in 
the fourth article is, in the Latin, p. 215, line 26, in 
the English, p. 255, line 24. Let any reader judge 
whether that be a requiring it, or a supposing it to 
be done ; this is your first mistake. The second 
is, that in this place the proportion itself, which is, 
" If these deficient figures could be described in 
a parallelogram exquisitely, there might be found 
thereby betiveen any two lines given, as many mean 
proportionals as one would," is a theorem, upon 
supposition of these crooked lines exquisitely 
drawn ; but you take it for a problem. 

And proceeding in that error, you undertake 
the invention of two mean proportionals, using 
therein my first figure, which is of the same con- 
struction with the eighth that belongeth to this 
fourteenth article. Your construction is, " Let 
tliere be taken in the diameter C A, (Jig. I) the 
two given lines, or two others proportional to 
them, as C H, C 6r, and their ordinate lines If F, 
G E (which by construction are in subtriplicate 
proportion of the intercepted diameters). These 


lines will show the proportions which those four LESSON v. 
proportionals are to have." But how will you "7 ; '""' 

* * J Of the faults 

find the length of H F or G E, the ordinate lines ? that occur m 

\TfT-ii T . . , , . , 111. demonstration 

Will you not do it by so drawing the crooked line 
CFE, as it may pass through both the points F 
and E ? You may make it pass through one of 
them, but to make it pass through the other, you 
must find two mean proportionals between G K 
afld G L, or between H I and H P ; which you can- 
not do, unless the crooked line be exactly drawn ; 
which it cannot be by the geometry of planes. 
Go shew this demonstration of yours to Orontius, 
and see w r hat he will say to it. 

I am now come to an end of your objections to 
the seventeenth chapter, where you have an epi- 
phonema not to be passed over in silence. But 
because you pretend to the demonstration of some 
of these propositions by another method in your 
Arithmetica Injinitorum, I shall first try whether 
you be able to defend those demonstrations as well 
as I have done these of mine by the method of 

The first proposition of your Arithmetica Infini- 
torum is this lemma : " In a series, or row of 
quantities, arithmetically proportional, beginning 
at a point or cypher, as 0, I, 2, 3, 4, fyc. to find 
the proportion of the aggregate of them all, to 
the aggregate of so many times the greatest, as 
there are terms." This is to be done by multiply- 
ing the greatest into half the number of the terms. 

The demonstration is easy. But how do you 
demonstrate the same ? " The most simple way" 
say you, " of finding this and some other problems, 
is to do the thing itself a little way, and to 




that occur m 


LESSON v. observe and compare the appearing proportions, 
by induction to conclude it universally" 
Egregious logicians and geometricians, that think 


an induction, without a numeration or all the par- 
ticulars sufficient, to infer a conclusion universal, 
and fit to be received for a geometrical demonstra- 
tion ! But why do you limit it to the natural 
consecution of the numbers, 0, 1, 2, 3, 4, &c? Is 
it not also true in these numbers, (),, 2, 4, 6, &c. or 
in these, 0, 7, 14, 21, &c ? Or in any numbers 
where the difference of nothing and the first num- 
ber is equal to the difference between the first and 
second, and between the second and third, &c. ? 
Again,, are not these quantities, I, 3, 5, 7 5 &c. in 
continual proportion arithmetical ? And if you 
put before them a cypher thus, 0, 1,3, 5, 7, do you 
think that the sum of them is equal to the half of 
five times seven ? Therefore though your lemma 
be true, and by me (Chap. XIIT. art. 5) demon- 
strated ; yet you did not know why it is true ; 
which also appears most evidently in the first propo- 
sition of your Conic Sections, where first you have 
this, " that a parallelogram whose altitude is infi- 
nitely little, that is to say, none, is scarce anything 
else but a line" Is this the language of geometry ? 
How do you determine this word scarce ? The 
least altitude, is somewhat or nothing. If some- 
what, then the first character of your arithmetical 
progression must not be a cypher ; and conse- 
quently the first eighteen propositions of this your 
Arithmetica Infinitorum are all nought. If no- 
thing, then your whole figure is without altitude, 
and consequently t your understanding nought. 
Again, in the same proposition, you say thus : 


" We will sometimes call those parallelograms LKSSON v. 
rather by the name of lines than of parallelo- * ~ ho f ! uults > 
grams, at least when there is no consideration of that occur m 
a determinate altitude ; but where there is a con- 
sideration of a determinate altitude (which will 
happen sometimes] there that little altitude shall 
be so far considered, as that being infinitely 
multiplied it may be equal to the altitude of the 
whole figure" See here in what a confusion you 
are when you resist the truth. When you consider 
no determinate altitude, that is no quantity of 
altitude, then you say your parallelogram shall be 
called a line. But when the altitude is determined, 
that is, when it is quantity, then you will call it a 
parallelogram. Is not this the very same doctrine 
which you so much wonder at arid reprehend in 
me, in your objections to my eighth chapter, and 
your word considered used as I used it ? It is 
very ugly in one that so bitterly repreherideth a 
doctrine in another, to be driven upon the same 
himself by the force of truth when he thinks not 
on it. Again, seeing you admit in any case those 
infinitely little altitudes to be quantity, what need 
you this limitation of yours, " so far forth as that 
by multiplication they may be made equal to the 
altitude of the whole figure ?" May not the half, 
the third, the fourth, or the fifth part, &c. be made 
equal to the whole by multiplication ? Why could 
you not have said plainly, so far forth as that 
every one of those infinitely little altitudes be not 
only something but an aliquot part of the whole ? 
So you will have an infinitely little altitude, that 
is to say, a point to be both nothing and something 
and an aliquot part. And all this proceeds from 


LESSON v. not understanding the ground of your profession. 
*_ ; ~' Well, the lemma is true. Let us see the theorems 

Of the faults y 

that occur m you draw from it. The first is (p. 3) " that a tri- 
angle to a parallelogram of equal base and alti- 
tude is as one to two." The conclusion is true, 
but how know you that ? " Because" say you, 
" the triangle consists as it were [as it were, is 
no phrase of a geometrician] of an infinite num- 
ber of straight parallel lines." Does it so ? Then 
by your own doctrine, which is, that " lines have 
no breadth" the altitude of your triangle consist- 
eth of an infinite number of no altitudes, that is 
of an infinite number of nothings, and conse- 
quently the area of your triangle has no quantity. 
If you say that by the parallels you mean infinitely 
little parallelograms, you are never the better ; for if 
infinitely little, either they are nothing, or if some- 
what, yet seeing that no two sides of a triangle are 
parallel, those parallels cannot be parallelograms. 
I see they may be counted for parallelograms by 
not considering the quantity of their altitudes in 
the demonstration. But you are barred of that 
plea, by your spiteful arguing against it in your 
Elenchus. Therefore this third proposition, and 
with it the fourth, is undemonstrated. 

Your fifth proposition is, " the spiral line is 
equal to half the circle of the first revolution" But 
what spiral line ? We shall understand that by 
your construction, which is this : " The straight 
line M A [in your figure which I have placed at 
the end of the fifth lesson] turned round (the 
point M remaining unmoved) is supposed to de- 
scribe with its poinf A the circle A A, whilst 
some point, in the same M A, whilst it goes about, 


is supposed to be moved uniformly from M to A, LESSON v. 
describing the spiral line.' 9 This therefore, is the ' ' 

o ^ > Of the faults 

spiral line of Archimedes ; and your proposition that occur m 
affirms it to be equal to the half of the circle 
A A ; which you perceived not long after to be 
false. But thinking it had been true, you go 
about to prove it, " by inscribing in the circle an 
infinite multitude of equal angles ,and consequently 
an infinite number of sectors, whose arches will 
therefore be in arithmetical proportion ;" which is 
true. " And the aggregate of those arches equal 
to half the circumference A A ;" which is true 
also. And thence you] conclude " that the spiral 
line is equal to half the circumference of the cir- 
cle AO A;" which is false. For the aggregate 
of that infinite number of infinitely little arches, 
is not the spiral line made by your construction, 
seeing by your construction the line you make is 
manifestly the spiral of Archimedes ; whereas no 
number, though infinite, of arches of circles, how 
little soever, is any kind of spiral at all ; and 
though you call it a spiral, that is but a patch to 
cover your fault, and deceiveth no man but your- 
self. Besides, you saw not how absurd it was, for 
you that hold a point to be absolutely nothing, to 
make an infinite number of equal angles (the radius 
increasing as the number of angles increaseth) and 
then to say, " that the arches of the sectors whose 
angles they are, are as 0, 1,2, 3, 4, &c." For 
you make the first angle 0, and all the rest equal 
to it ; and so make 0, 0, 0, 0, 0, &c. to be the same 
progression with 0, 1,2, 3,4, &c. The influence 
of this absurdity reacheth to t the end of the eigh- 
teenth proposition. So many are therefore false. 


LKSSON v. or nothing worth. And you needed not to wonder 
% r ' that the doctrine contained in them was omitted 

Of the faults .11 

that occur m by Archimedes, who never w r as so senseless as to 
think a spiral line was compounded of arches of 

Your nineteenth proposition is this other lemma : 
" In a series, or a row, of quantities, beginning 
from a point) or cypher, and proceeding accord- 
ing to the order of the square number s^ as 0, 1, 4, 
9, 16, 8fc. to find what proportion the whole series 
hath to so many times the greatest.'" And you 
conclude " the proportions to be that of \ to 3." 
Which is false, as you shall presently see. First, 
let the series of squares with the prefixed cypher, 
and under every one of them the greatest 4 be 
" \ I And you have for the sum of the squares 
5, and for thrice the greatest 12, the third part 
whereof is 4. But 5 is greater than 1, by 1, that 
is, by one twelfth of 12 ; which quantity is some- 
what, let it be called A. Again, let the row of 
squares be lengthened one term further, and the 
greatest set under every one of them as IJ V4-7J. 
The sum of the squares is 14, and the sum of four 
times the greatest is 36, whereof the third part is 12. 
But 14 is greater than 12 by two unities, that is, 
by two twelfths of 12, that is, by 2 A. The differ- 
ence therefore between the sum of the squares, 
and the sum of so many times the greatest square, 
is greater, when the cypher is followed by three 
squares, than when by but two. Again, let the row 
have five terms, as in these numbers w^^r^r^ 
with the greatest five times described, and the sum 
of the squares will be c 30, the sum of all the great- 
est will be 80. The third part whereof is 26 -' 


But 30 is greater than 26 I by 3^, that is, by three LESSON v. 
twelfths of twelve, and of a twelfth, that is, by oaiTi^iiT 
3i A. Likewise in the series continued to six ^ occur m 


places with the greatest six times subscribed, as 

6 as i~^725 23 the sum of the squares is 55, and 
the sum of the greatest six times taken is 150, the 
third part whereof is 50. But 55 is greater than 
50 by 5, that is, by five-twelfths of 12, that is by 
5 A. And so continually as the row groweth longer, 
the excess also of the aggregate of the squares 
above the third part of the aggregate of so many 
times the greatest square, growing greater. And 
consequently if the number of the squares were 
infinite, their sum would be so far from being equal 
to the third part of the aggregate of the greatest 
as often taken, as that it would be greater than it 
by a quantity greater than any that can be given 
or named. 

That which deceived you was partly this, that 
you think, as you do in your Elenchus, that these 
fractions ^ ^ & 4 L &c. are proportions, as if ,\ 
were the proportion of one to twelve, and conse- 
quently TT double the proportion of one to twelve; 
which is as unintelligible as school-divinity ; and I 
assure you, far from the meaning of Mr. Ougthred 
in the sixth chapter of his Clams Mathematica, 
where he says that 4 >- is the proportion of 3 1 to 

7 ; for his meaning is, that the proportion of 
4 7- to one, is the proportion of 31 to 7 ; whereas 
if he meant as you do, then 8- 7 ~ should be double 
the proportion of 31 to 7- Partly also because 
you think (as in the end of tlje twentieth proposi- 
tion) that if the proportion of the numerators of 


LESSON v. these fractions -^13-^-10 TO to their denominators 
oruie limits* decrease eternally, they shall so vanish at last as 
tiiat octm m . to leave the proportion of the sum of all the 

ueinoiisstration. A 

squares to the sum of the greatest so often taken, 
(that is, an infinite number of times), as one to 
three, or the sum of the greatest to the sum of the 
increasing squares, as three to one ; for which 
there is no more reason than for four to one, or 
five to one, or any other such proportion. For if 
the proportions come eternally nearer and nearer 
to the subtriple, they must needs also come nearer 
and nearer to subquadruple ; and you may as well 
conclude thence that the upper quantities shall be 
to the lower quantities as one to four, or as one 
to five, &c. as conclude they are as one to three. 
You can see without admonition, what effect this 
false ground of yours will produce in the whole 
structure of your Arithmetical Infinitorum ; and 
how it makes all that you have said unto the end 
of your thirty-eighth proposition, undemonstrated, 
and much of it false. 

The thirty-ninth is this other lemma : " In a 
series of quantities beginning ivith a point or 
cypher, and proceeding according to the series of 
the cubic numbers, as 0. 1. 8. 27- 64, 8fc. to find 
the proportion of the sum of the cubes to the sum 
of the greatest cube, so many times taken as there 
be terms." And you conclude that " they have a 
proportion of 1 to 4 ;" which is false. 

Let the first series be of three terms subscribed 
with the greatest % . \ . | ; the sum of the cubes 
is nine ; the sum of all the greatest is 24 ; a quar- 
ter whereof is 6. 9 is greater than 6 by three 
unities. An unity is something. Let it be there- 


fore A. Therefore the row of cubes is greater LESSON v. 
than a quarter of three times eight, by three A. *~ ; ~' 

1 7 faults 

Again, let the series have four terms, as ? 2?' 2?' *>? 5 tlwtoccui m 

CJ demonstration. 

the sum of the cubes is 36 ; a quarter of the sum 
of all the greatest is twenty-seven. But thirty- 
six is greater than twenty-seven by nine, that is, 
by 9 A. The excess therefore of the sum of the 
cubes above the fourth part of the sum of all the 
greatest, is increased by the increase of the num- 
ber of terms. Again, let the terms be five, as 
"4' 64 b4 . ol 04 9 the sum of the cubes is one hundred ; 
the sum of all the greatest three hundred and 
twenty ; a quarter whereof is eighty. But one 
hundred is greater than eighty by twenty, that is, 
by 20 A. So you see that this lemma also is false. 
Arid yet there is grounded upon it all that which 
you have of comparing parabolas and parabolo- 
eides with the parallelograms wherein they are 
accommodated. And therefore though it be true, 
that the parabola is -f> an d the cubical parabolo- 
eides -f- of their parallelograms respectively, yet 
it is more than you were certain of when you 
referred rne, for the learning of geometry, to this 
book of yours. Besides, any man may perceive 
that without these two lemmas (which are mingled 
with all your compounded series with their excesses) 
there is nothing demonstrated to the end of your 
book : which to prosecute particularly, were but 
a vain expense of time. Truly, were it not that I 
must defend my reputation, I should not have 
showed the world how little there is of sound doc- 
trine in any of your books. For when I think how T 
dejected you will be for the future, arid how the 
grief of so much time irrecoverably lost, together 


LESSON v. with the conscience of taking so great a stipend, 
ouh^mT ^ or mis-teaching the young men of the University, 
that occur in and the consideration of how much your friends 

demons ation. J 

will be ashamed of you, will accompany you for 
the rest of your life, I have more compassion for 
you than you have deserved. Your treatise of the 
Angle of Contact, I have before confuted in a very 
few leaves. And for that of your Conic Sections, 
it is so covered over with the scab of symbols, that 
I had not the patience to examine whether it be 
well or ill demonstrated. 

Yet I observed thus much, that you find a tan- 
gent to a point given in the section by a diameter 
given ; and in the next chapter after, you teach 
the finding of a diameter, which is not artificially 

I observe also, that you call the parameter an 
imaginary line, as if the place thereof were less 
determined than the diameter itself ; and then you 
take a mean proportional between the intercepted 
diameter, and its contiguous ordinatc line, to find 
it. And it is true, you find it : but the parameter 
has a determined quantity, to be found without 
taking a mean proportional. For the diameter and 
half the section being given, draw a tangent through 
the vertex, and dividing the angle in the midst 
which is made by the diameter and tangent, the 
line that so divideth the angle, will cut the crooked 
line. From the intersection draw a line (if it be a 
parabola) parallel to the diameter, and that line 
shall cut off in the tangent from the vertex the 
parameter sought. But if the section be an ellipsis, 
or an hyperbole, you may use the same method, 
saving that the line drawn from the intersection 


must not be parallel, but must pass through the LESSON v, 
end of the transverse diameter, and then also it n /r ; ~ ' 

' Of the faults 

shall cut off a part of the tangent, which measured that orcm m 

P . , ~ , . demonstration 

from the vertex is the parameter. So that there 
is no more reason to call the parameter an imagi- 
nary line than the diameter. 

Lastly, I observe that in all this your new method 
of conies, you show not how to find the burning 
points, which writers call the foci arid umbilici of 
the section, which are of all other things belonging 
to the conies most useful in philosophy. Why 
therefore were they not as worthy of your pains 
as the rest, for the i*est also have already been 
demonstrated by others ? You know the focus of 
the parabola is in the axis distant from the vertex 
a quarter of the parameter. Know also that the 
focus of an hyperbole, is in the axis, distant from 
the vertex, as much as the hypotenusal of a rectan- 
gled triangle, whose one side is half the transverse 
axis, the other side half the mean proportional be- 
tween the whole transverse axis and the parameter, 
is greater than half the transverse axis. 

The cause why you have performed nothing in 
any of your books (saving that in your Elenchus 
you have spied a few negligences of mine, which I 
need not be ashamed of) is this, that you under- 
stood not what is quantity, line, superficies, angle, 
and proportion ; without which you cannot have 
the science of any one proposition in geometry. 
From this one and first definition of Euclid, " a 
point is that whereof there is no part," under- 
stood by Sextus Empiricus, as you understand it, 
that is to say misunderstood, Sextus Empiricus had 
utterly destroyed most of the rest, and demonstra- 


LESSON v. ted, that in geometry there is no science, and by 
or the fauifT that m eans you have betrayed the most evident of 
that occur m the sciences to the sceptics. But as I understand 

demonstration . x . 

it for mat whcreoj no part is reckoned, his argu- 
ments have no force at all, and geometry is re- 
deemed. If a line have no latitude, how shall a 
cylinder rolling on a plane, which it toucheth not 
but in a line, describe a superficies ? How can you 
affirm that any of those things can be without 
quantity, whereof the one may be greater or less 
than the other ? But in the common contact of 
divers circles the external circle maketh with the 
common tangent a less angle of contact than the 
internal. Why then is it not quantity ? An angle 
is made by the concourse of two lines from several 
regions, concurring, by their generation, in one 
and the same point. How then can you say the 
angle of contact is no angle ? One measure can- 
not be applicable at once to the angle of contact, 
and angle of conversion. How then can you 
infer, if they be both angles, that they must be 
homogeneous ? Proportion is the relation of two 
quantities. How then can a quotient or fraction, 
which is quantity absolute, be a proportion ? But 
to come at last to your Epiphonema, wherein, 
though I have perfectly demonstrated all those 
propositions concerning the proportion of para- 
bolasters to their parallelograms, and you have 
demonstrated none of them (as you cannot now 
but plainly see), but committed most gross paralo- 
gisms, how could you be so transported with pride, 
as insolently to compare the setting of them forth 
as mine, to the act of him that steals a horse, and 
comes to the gallows for it. You have read, I 


think, of the gallows set up by Haman. Remem- LESSON v. 
ber therefore also who was handed upon it. r.^T^T"' 

* Ot the faults 

After your dejection I shall comfort you a little, a that occur m 
very little, with this, that whereas this eighteenth ' emonsraion - 
chapter containeth tw T o problems, one, u the finding 
of a straight line equal to the crooked line of a semi- 
parabola ;" the other," the finding of straight lines 
equal to the crooked lines of the parabolasters, in 
the table of the third article of the seventeenth 
chapter;" you have truly demonstrated that they are 
both false ; and another hath also demonstrated the 
same another way. Nevertheless, the fault was not 
in my method, but in a mistake of one line for ano- 
ther and such as was not hard to correct ; and is 
now so corrected in the English as you shall not be 
able (if you can sufficiently imagine motions) to 
reprehend. The fault was this, that in the tri- 
angles which have the same base and altitude with 
the parabola and parabolaster, I take for designa- 
tion of the mean uniform impetus, a mean pro- 
portional, in the first figure, between the whole 
diameter and its half, and, in the second figure, a 
mean proportional between the whole diameter 
and its third part ; which was manifestly false, 
and contrary to what I had shown in the sixteenth 
chapter. Whereas I ought to have taken the half 
of the base, as now I have done, and thereby 
exhibited the straight lines equal to those crooked 
lines, as I undertook to do. Which error therefore 
proceeded not from want of skill, but from want of 
care ; and what I promised (as bold as you say the 
promise was), I have now performed. 

The rest of your exceptions to this chapter, are 
to these words in the end : "* There be some that 


LESSON v. say, that though there be equality between a 
straight and crooked line, yet now, they say, after 
the fall of Adam, it cannot he found without the 
especial help of divine grace" And you say 
you think there be none that say so. I am not 
bound to tell you who they are. Nevertheless, 
that other men may see the spirit of an ambitious 
part of the clergy, I will tell you where I read it. 
It is in the Prolegomena of Lalovera, a Jesuit, to 
his Quadrature of the Circle, p. 13 and 14, in these 
words : " Quamvis circuli tefragonismufi sit </>iWt 
possibilis, an tamen etiam TT^OC /^ac, Jioc est, post 
Adce lapsnm homo ejus scientiam absque special} 
divinte gratia* auxilio, possit comparare, jure 
merito inquirnnt theologi, pronunciantque ; lianc 
veritatem tanta esse caligine involutam ut illcnn 
videre nemo possit, nisi ignoranfnc ex primi 
parentis pravaricatione propagatas tenebras in- 
debitus divince hicis radius dtssipet ; quod verissi- 
mum esse sentioT Wherein I observed that he, 
supposing he had found that quadrature, would 
have us believe it was not by the ordinary and 
natural help of God (whereby one man reasoneth, 
judgeth and remembereth better than another), but 
by a special (which must be a supernatural) help 
of God, that he hath given to him of the order of 
Jesus above others that have attempted the same 
in vain. Insinuating thereby, as handsomely as he 
could, a special love of God towards the Jesuits. 
But you taking no notice of the word special, 
would have men think I held, that human sciences 
might be acquired without any help of God. And 
thereupon proceed in a great deal of ill language 
to the end of your objections to this chapter. But 


I shall take notice of your manners for altogether LESSON v. 
in my next lesson. "~ r ~' 

J Of tlie faults 

At the nineteenth chapter you see not, you say, that occur m 
the method. Like enough. In this chapter T temonsraion - 
consider not the cause of reflection, which con- 
sisteth in the resistance of bodies natural ; but I 
consider the consequences, arising from the suppo- 
sition of the equality of the angle of reflection, to 
that of incidence ; leaving the causes both of re- 
flection, and of refraction, to be handled together 
in the twenty-fourth chapter. Which method, 
think what you will, I still think best. 

Secondly, you say I define not .here, but many 
chapters after, what an angle of incidence, and 
what an angle of reflection is. Had you not been 
more hasty than diligent readers, you had found 
that those definitions of the angle of incidence, 
and of reflection, were here set down in the first 
article, and not deferred to the twenty-fourth. 
Let not therefore your own oversight be any more 
brought in for an objection. 

Thirdly, you say there is no great difficulty in 
the business of this chapter. It may be so, now 
it is down ; but before it w r as done, I doubt not 
but you that are a professor would have done the 
same, as well as you have done that of the Angle 
of Contact, or the business of your Arithmetica 
Infinitorum. But what a novice in geometry w r ould 
have done I cannot tell. 

To the third, fourth, and fifth article, you ob- 
ject a want of determination ; and show it by in- 
stance, as to the third article. But what those 
determinations should be, you determine not, 
because you could not, The words in the third 



LESSON v. article, are first these, if there fall two straight 
ofaTfi^itT l mes parallel, 8?c. which is too general. It should 
that occur m b e ;/* there fall the same wan two straight lines 

demonstration. 7/70 TV- i - 177- 

parallel, yc. JNext these, their reflected lines 
produced inwards shall make an angle, 8fc. This 
also is too general. I should have said, their re- 
flected lines produced inwards, if they meet within, 
shall make an angle, 8?c. Which done, both this 
article and the fourth and fifth are fully demonstra- 
ted. And without it, an intelligent reader had 
been satisfied, supplying the want himself by the 

To the eighth, you object only the too great 
length and labour of it, because you can do it a 
shorter way. Perhaps so now, as being easy to 
shorten many of the demonstrations both of Euclid, 
and other the best geometricians that are or have 
been. And this is all you had to say to my nine- 
teenth chapter. Before I proceed, I must put you 
in mind that these words of yours, " adducls mal- 
leum, ut occidas muscam" are not good Latin, 
malleum offers., malleum adhibes, malleo uteris, 
are good. When you speak of bringing bodies 
animate, ducerc and addueerc are good, for there 
to bring, is to guide or lead. And of bodies inani- 
mate, adducere is good for attrahere, which is to 
draw to. But when you bring a hammer, will you 
say adduco malleum, I lead a hammer ? A man 
may lead another man, and a ninny may be said to 
lead another ninny, but not a hammer. Never- 
theless, I should not have thought fit to reprehend 
this fault upon this occasion in an Englishman, 
nor to take notice of it, but that I find you in 
some places nibbling, but causelessly, at my Latin. 


Concerning the twentieth chapter, before I an- LESSON v. 
swer to the objections against the propositions Of '~ e f ; mlta " 
themselves. I must answer to the exception you ^ occur m 

i -in- e s\ i T demonstration 

first take to these words of mine, Qua de di- 
mensione circuit et angidornm prommtiata sunt 
tanquam exacte invent a, accipiat lector tanquam 
dicta problematic?. To which you say thus : " We 
are wont in geometry to call some propo- 
sitions theorems, others problems, Sfc. of which 
a theorem is that wherein some assertion is pro- 
pounded to be proved; a problem that wherein 
something is commanded to be done." Do you 
mean to be done*, and not proved ? By your favour, 
a problem in all ancient w r riters signifies no more 
but a proposition uttered, to the end to have it, by 
them to whom it is uttered, examined whether it 
be true or not true, faisable or not faisable ; and 
differs not amongst geometricians from a theorem 
but in the manner of propounding. For this 
proposition, to make an equilateral triangle., so 
propounded they call a problem. But if pro- 
pounded thus : If upon the ends of a straight line 
given be described two circles, whose radius is the 
same straight line, and there be drawn from the 
intersection of the circles to their tivo centres, two 
straight lines, there ivill be made an equilateral 
triangle, then they call it a theorem ; and yet the 
proposition is the same. Therefore these words, 
accipiat lector tanquam dicta problematice signify 
plainly this, that I would have the reader, take 
for propounded to him to examine, whether from 
my construction the quadrature of the circle can 
be truly inferred or not ; and this is not to bid him, 
as you interpret it, to square the circle. And if 

Y 2 



LESSON v. you believe that problematic^ signifies probably, 
you have been very negligent in observing the 
sense of the ancient Greek philosophers in the 
word problem. Therefore your solemus in geo- 
metrictj &c. is nothing to the purpose ; nor had it 
been though you had spoken more properly, and 
said solent, leaving out yourselves. 

My first article hath this title, "from a false 
supposition , a false quadrature of the circle" 
Seeing therefore you were resolved to show where 
I erred, you should have proved either that the 
supposition was true, and the conclusion falsely 
inferred, or contrarily, that though the supposition 
be false, yet the conclusion is true ; for else you 
object nothing to my geometry, but only to my 
judgment, in thinking fit to publish it ; which 
nevertheless you cannot justly do, seeing it was 
likely to give occasion to ingenious men (the 
practice of it being so accurate to sense) to in- 
quire wherein the fallacy did consist. And for the 
problem as it was first printed, but never published, 
and consequently ought to have passed for a pri- 
vate papr stolen out of my study, your public 
objecting against it (in the opinion of all men that 
have conversed so much with honest company as 
to know what belongs to civil conversation), was 
sufficiently barbarous in divines. And seeing you 
knew I had rejected that proposition, it was but a 
poor ambition to take wing as you thought to do, 
like beetles from my egestions. But let that be 
as it will, you will think strange now I should 
resume, and make good, at least against your ob- 
jection, that very same proposition. So much of 
the figure as is needful you will find noted with the 


same letters, and placed at the end of this fifth LESSON v. 
lesson. Wherein let B I, be an arch not greater " ' 

T i Ot the faults 

than the radius of the circle, and divided into that occur m 

r i . T TVT /~\ -r\ n TVT * i demonstration. 

tour equal parts, in L, N, O. Draw S N, the sine 
of the arch B N, and produce it to T, so as S T be 
double to S N, that is, equal to the chord B L 
Draw likewise a L, the sine of the arch B L, and 
produce it to c, so as a c be quadruple to a L, that 
i% equal to the two chords B N, N I. Upon the 
centre N with the radius N I, draw the arch I d, 
cutting B U the tangent in d. Then will B N 
produced cut the arch I d, in the midst at o. In 
the line B S produced take S A, equal to B S ; 
then draw and produce b N, and it will fall on 
the point d. And B d, S T, will be equal ; and 
d T joined and produced will fall upon o, the 
midst of the arch I d. Join I T, and produce it to 
the tangent B U in U. I say, that the straight 
line I T U shall pass through c. For seeing B S, 
S b y are equal, and the angle at S a right angle,, the 
straight lines B N, and b N, are also equal, and 
the triangles B N />, d N o like and equal ; and the 
lines d T> T o equal. Draw o i parallel to d U, 
cutting I U in i ; and the triangles d T U, o T i 
will also be like and equal. Produce S T to the 
arch d o I in , and produce it further to,/, so that 
the line ef be equal to T e ; and then Sjfwill be 
equal to a c. Therefore/'? joined will be parallel 
toBS. In cf produced take fg equal to cf\ 
and draw g m parallel to d U, cutting I U in m, 
and do in n ; and let the intersection of the 
two lines a c and d o be in r ; which being done, 
the triangles m n T, r c T will be like and equal. 
Therefore m n and r c are equal ; and consequently 


LESSON v. the straight line I m T U shall pass through c. 
nfi , ; ~* Dividing therefore a c in the midst at t, and S N in 

Of the faults o ? 

that occur m the midst at /, and joining t N, L /, the lines L /, t N, 

demonstration. . n - i i 

and <? 1 produced, will all meet in one and the 
same point of B S produced ; suppose at q. Therefore 
the point q being given by the two known points T 
and I, the lines drawn from q through equal parts 
of the sine of the arch 13 I, (for example through 
the points P, Q,, R, of the sine M I), shall cut off 
equal arches,, as B L, L N, N (X, O I. And this is 
enough to make good that problem, as to your 

The straight line therefore B U, for any thing 
you have said, is proved equal to the arch B I, and 
the division of any angle given into any proportion 
given, the quadrature of any sector, and the con- 
struction of any equilateral polygon is also given. 
And though in this also I should have erred, yet it 
cannot be denied but that I have used a more 
natural, a more geometrical, and a more perspicu- 
ous method in the search of this so difficult a 
problem, than you have done in your Arithmetica 
Injinitorum. For though it be true that the aggre- 
gate of all the mean proportionals between the 
radius, together with an infinitely little part of the 
same, and the radius wanting an infinitely little 
part of the same ; and again, between the radius, 
together with two infinitely little parts, and the 
radius wanting two infinitely little parts, and so 
on eternally, will be equal to the quadrant (a thing 
which every mean geometrician knew before) ; yet 
it was absurd to think those means could be calcu- 
lated in numbers by interpoling of a symbol ; 
especially when you make that symbol to stand for 


a number neither true nor surd ; as if there were a LESSON v. 
number that could neither be uttered in words, *~ ;' ~' 

a y Of the ftii life 

nor not be uttered in words. For what else is that occur m 

i i , . i , i i ,1 i ^ demonstration 

surd, but that which cannot be spoken ? 

To the fifth article, though your discourse be 
long, you object but two things. One is, that 
" Whereas the spiral of Archimedes is made of 
two motions, one straight., the other circular, bot/i 
uniform, 1 taking the motion compounded of them 
both for one of those that are compounded, con- 
clude falsely, that the generation of the spiral is 
like to the generation of the parabola" What 
heed you use to take in your reprehensions, appears 
most manifestly in this objection. For I say in 
that demonstration of mine, that the velocity of 
the point A in describing the spiral increaseth 
continually in proportion to the times. For seeing 
it goes on uniformly in the semidiameter, it is im- 
possible it should not pass into greater and greater 
circles, proportionally to the times, and conse- 
quently it must have a swifter and swifter motion 
circular, to be compounded with the uniform motion 
in every point of the radius as it turneth about. 
This objection therefore is nothing but an effect of 
a will, without cause, to contradict. 

The other objection is, that " Granting all to be 
true hitherto, yet because it depends upon the 
finding of a straight line equal to a parabolical 
line in the eighteenth chapter, where I was de- 
ceived, I am also deceived here." True. But 
because in the eighteenth chapter of this English 
edition I have found a straight line equal to the 
spiral line of Archimedes. I must here put you in 
mind that by these words in your objections to the 


LESSON v. fifth article at your number two, Quatenus verum 
* ' es f e tc.. we have demonstrated prop 10. 11. 13, 

Of the faults \ * , r r m ' ' ' 

that occur m AritJimctica Infinitornm; you make it appear 
that you thought your spiral (made of arches or 
circles) was the true spiral of Archimedes ; which 
is fully as absurd as the quadrature of Joseph 
Scaliger, whose geometry you so much despise. 

To the sixth article, which is a digression con- 
cerning the analytics of geometricians, you deny 
that the efficient cause of tlie construction ought 
to be contained in the demonstration. As if any 
problem could be known to be truly done, other- 
wise than by knowing first how, that is to say, by 
what efficient cause, and in what manner, it is to 
be done. Whatsoever is done without that know- 
ledge, cannot be demonstrated to be done ; as you 
see in your computation of the parabola, and 
paraboloeides, in your Aritlnnetica Infinitorum. 

And whereas I said that the ends of all straight 
lines drawn from a straight line, and passing 
through one and the same point, if their parts be 
proportional., shall he in a straight line ; is true 
and accurate; as also, if tliey begin in the cir- 
cumference of a circle., they shall also be in the 
circumference of another circle. And so is this : 
if the proportion be duplicate, they shall be in a 
parabola. All this I say is true and accurately 
spoken. But this was no place for the demonstra- 
tion of it. Others have done it. And I perceive 
by that you put in by parenthesis (" Intelligis 
credo inter duas perij)herias concentricas" ) that 
you understand not what I mean. 

Hitherto reach your objections to my geometry: 
for the rest of your took, it containeth nothing 


but a collection of lies, wherewith you do what LESSON v. 
you can, to extenuate as vulgar, and disgrace as OJ ^^uT 
false, that which folio weth, and to which you have th,u occm m 

. demomtiatioii. 

made no special objection. 

I shall therefore only add in this place concern- 
ing your Analytica per Potestates, that it is no 
art. For the rule, both in Mr. Ougthred, and in 
Des Cartes, is this : " When a problem or question 
is propounded, suppose the thing required done, 
and then using a fit ratiocination, put A or some 
other vowel for the magnitude sought" How is 
a man the better for this rule without another 
rule, how to know when the ratiocination is fit? 
There may therefore be in this kind of analysis 
more or less natural prudence, according as the 
analyst is more or less wise, or as one man in 
choosing of the unknown quantity with which he 
will begin, or in choosing the way of the conse- 
quences which he will draw from the hypothesis, 
may have better luck than another. But this is 
nothing to art. A man may sometimes spend a 
whole day in deriving of consequences in vain, 
and perhaps another time solve the same problem 
in a few minutes. 

I shall also add, that symbols, though they 
shorten the writing, yet they do not make the 
reader understand it sooner than if it were written 
in words. For the conception of the lines and figures 
(without which a man learn eth nothing) must pro- 
ceed from words either spoken or thought upon. 
So that there is a double labour of the mind, one 
to reduce your symbols to words, which are also 
symbols, another to attend to ^he ideas which they 
signify. Besides, if you but consider how none of 


LESSON v. the ancients ever used any of them in their pub- 
oftiiefouita* listed demonstrations of geometry., nor in their 
that occur m books of arithmetic, more than for the roots and 


potestates themselves ; and how bad success you 
have had yourself in the unskilful using of them, 
you will not, I think, for the future be so much in 
love with them as to demonstrate by them that 
first part you promise of your Opera Mathematica. 
In which, if you make not amends for that which 
you have already published, you will much disgrace 
those mathematicians you address your epistles to, 
or otherwise have commended ; as also the Uni- 
versities, as to this kind of learning, in the sight 
of learned men beyond sea. And thus having ex- 
amined your pannier of Mathematics, and finding 
in it no knowledge, neither of quantity, nor of 
measure, nor of proportion^ nor of time, nor of 
motion, nor of any thing, but only of certain 
characters, as if a hen had been scraping there ; I 
take out my hand again, to put it into your other 
pannier of theology, and good manners. In the 
mean time I will trust the objections made by you 
the astronomer (therein there is neither close 
reasoning, nor good style, nor sharpness of wit, to 
impose upon any man) to the discretion of all sorts 
of readers. 





HAVING in the precedent lessons maintained the LESSON VK 
truth of my geometry, and sufficiently made appear otmd ' nnerg 
that your objections against it are but so many er- 
rors of your own, proceeding from misunderstand- 
ing of the propositions you have read in Euclid, and 
other masters of geometry ; I leave it to your con- 
sideration to whom belong, according to your own 
sentence, the unhandsome attributes you so often 
give me upon supposition, that you yourselves are in 
the right, and I mistaken ; and come now to purge 
myself of those greater accusations which concern 
my manners. It cannot be expected that there 
vshould be much science of any kind in a man that 
wanteth judgment ; nor judgment in a man that 
kuowcth not the manners due to a public disputa- 
tion in writing ; wherein the scope of either party 
ought to be no other than the examination and 
manifestation of the truth. For whatsoever is 
added of contumely, either directly or scomma- 
tically, is want of charity and uncivil, unless it be 
done by way of reddition from him that is first 
provoked to it. I say unless it be by way of red- 
dition ; for so was the judgment given by the 
emperor Vespasian in a quarrel between a senator 
and a knight of Rome which had given him ill 
language. For when the knight had proved that 
the first ill language proceeded from the senator, 


LESSON VK the emperor acquitted him in these words: "Male- 
of mamas' ^ici scuatoribus non oportere ; remaledicere, fas 
et civile esse" Nevertheless, now-a-days, uncivil 
words are commonly and bitterly used by all that 
write in matter of controversy, especially in divi- 
nity, excepting now and then such writers as have 
been more than ordinarily well bred, and have 
observed how heinous and hazardous a thing such 
contumely is amongst some sorts of men, whether 
that which is said in disgrace be true or false. 
For evil words by all men of understanding are 
taken for a defiance, and a challenge to open war. 
But that you should have observed so much, who 
are yet in your mother's belly, was not a thing to 
be much expected. 

The faults in manners you lay to my charge are 
these: 1. Self-conceit. 2. That I will be very 
angry with all men that do not presently submit 
to my dictates. 3. That / had my doctrine con- 
cerning Vision, out of papers which I had in my 
hands of Mr. Warner s. 4. That I have injured 
the universities. 5. That I am an enemy to re- 
ligion. These are great faults ; but such as I 
cannot yet confess. And therefore I must, as well 
as I can, seek out the grounds upon which you 
build your accusation. Which grounds (seeing 
you are not acquainted with my conversation) must 
be either in my published writings, or reported to 
you by honest men, and without suspicion of in- 
terest in reporting it. As for my self-conceit and 
ostentation, you shall find no such matter in my 
writings. That which you allege from thence is 
first, that in the ep.istle dedicatory I say of my 
book De Corpore, " though it be little, yet it is 


full; and If good may go for great, great enough" LESSON vi. 
When a man presenting a gift great or small to 0f T raa I nne ^ 
his betters, adorneth it the best he can to make it 
the more acceptable ; he that thinks this to be 
ostentation and self-conceit, is little versed in the 
common actions of human life. And in the same 
epistle, where I say of civil philosophy : " // is no 
ancienter than my hook De Cive ;" these words are 
added : " I say it provoked, and that my detrac- 
tors may see they lose their labour" But that 
which is truly said, and upon provocation, is not 
boasting, but defence. A short sum of that book 
of mine, now publicly in French, done by a gentle- 
man I never saw, carrieth the title of Ethics 
Demonstrated. The book itself translated into 
French, hath not only a great testimony from the 
translator Sorberius, but also from Gassendus, and 
Mcrsennus, who being both of the Roman religion 
had no cause to praise it, or the divines of England 
have no cause to find fault with it. Besides, you 
know that the doctrine therein contained is gene- 
rally received by all but those of the clergy, who 
think their interest concerned in being made sub- 
ordinate to the civil power ; whose testimonies 
therefore are invalid. Why therefore, if 1 com- 
mend it also against them that dispraise it publicly, 
do you call it boasting ? " Yon have heard" you 
say, " that I had promised the quadrature of the 
circle, 8fc" You heard then that which was not 
true. I have been asked sometimes, by such as 
saw the figure before me, w r hat I was doing, and I 
was not afraid to say I was seeking for the solu- 
tion of that problem ; but not that 1 had done it. 
And afterwards being asked of the success, I have 


LESSON vi. said, I thought it done. This is not boasting ; and 
^ ' vet it was enough, when told again, to make a fool 

Of manners. J & ' Q ' 

believe it was boasting. But you, the astronomer, 
in the epistle before your philosophical essay, say 
" You had a great expectation of my philosophical 
and mathematical works, before they were pub- 
lished." It may be so. Is that rny fault? Can a man 
raise a great expectation of himself by boasting ? 
If he could, neither of you would be long before 
you raised it of yourselves ; saving that what you 
have already published, has made it now too late. 
For I verily believe there was never seen worse 
reasoning than in that philosophical essay ; which 
any judicious reader would believe proceeded from 
a prevaricator, rather than from a man that be- 
lieved himself; nor worse principles, than those in 
your books of Geometry. The expectation of that 
which should be written by me, was raised partly 
by the Cogitata Physica-Mathematica of Mer- 
sennus, wherein I am often named with honour ; 
and partly by others with whom I then conversed 
in Paris, without any ostentation. That no man 
has a great expectation of any thing that shall 
proceed from either of you two, I am content to 
let it be your praise. 

Another argument of my self-conceit, you take 
from my contempt of the writers of metaphysics 
and school-divinity. If that be a sign of self-con- 
ceit, I must confess I am guilty ; and if your 
geometry had then been published, I had con- 
temned that as much. But yet I cannot see the 
consequence (unless you lend me your better logic) 
from despising insignificant and absurd language, 
to self-conceit. 


And again, in your Vindicm Academiarum, you LESSON vi. 
put for boasting, that in my Leviathan, page 331, oTmannew 
I would have that book by entire sovereignty im- 
posed upon the Universities ; arid in my Review, 
p. 713, that I say of my Leviathan, "I think it 
may be profitably printed, and more profitably 
taught in the University" The cause of rny wri- 
ting that book, was the consideration of what the 
ministers before, and in the beginning of, the civil 
war, by their preaching and writing did contribute 
thereunto. Which I saw not only to tend to the 
abatement of the then civil power, but also to the 
gaining of as much thereof as they could (as did 
afterwards more plainly appear) unto themselves. 
I saw also that those ministers, and many other 
gentlemen who were of their opinion, brought their 
doctrines against the civil power from their studies 
in the Universities. Seeing therefore that so much 
as could be contributed to the peace of our coun- 
try, and the settlement of sovereign power without 
any army, must proceed from teaching ; I had 
reason to wish, that civil doctrine were truly 
taught in the Universities. And if I had not 
thought that mine was such, I had never written it. 
And having written it, if I had not recommended 
it to such as had the power to cause it to be taught, 
I had written it to no purpose. To me therefore 
that never did write anything in philosophy to 
show my wit, but, as I thought at least, to benefit 
some part or other of mankind, it was very neces- 
sary to commend my doctrine to such men as 
should have the power and right to regulate the 
Universities. I say my doctrine ; I say not my 
Leviathan. For wiser men may so digest the same 


LESSON vi. doctrine as to fit it better for a public teaching. 
' But as it is, I believe it hath framed the minds of 

Oi manners. * 

a thousand gentlemen to a conscientious obedience 
to present government, which otherwise would 
have wavered in that point. This therefore was 
no vaunting, but a necessary part of the business I 
took in hand. You ought also to have considered, 
that this was said in the close of that part of my 
book which concerneth policy merely civil. Which 
part, if you, the astronomer, that now think the 
doctrine unworthy to be taught, were pleased once 
to honour with praises printed before it, you are 
not very constant nor ingenuous. But whether 
you did so or not, I am not certain, though it was 
told me for certain. If it were not you, it w r as 
somebody else whose judgment has as much weight 
at least as yours. 

And for anything you have to say from your 
own knowledge, I remember not that I ever saw 
either of your faces. Yet you, the professor of 
geometry, go about obliquely to make me believe 
that Vindex hath discoursed with me, once at least, 
though I remember it not. I suppose it therefore 
true ; but this I am sure is false, that either he 
or any man living did ever hear me brag of my 
science, or praise myself, but when my defence 
required it. Perhaps some of our philosophers 
that w r ere at Paris at the same time, and acquainted 
with the same learned men that I was acquainted 
with, might take for bragging the maintaining of 
my opinions, and the not yielding to the reasons 
alledged against them. If that be ostentation, 
they tell you the truth. But you that are so wise 
should have considered, that even such men as 


profess philosophy are carried away with the pas- LESSON vi. 
sions of emulation and envy (the sole ground of 7 matmeM 
this your accusation) as well as other men, and 
instanced in yourselves. And this is sufficient to 
shake off your aspersions of ostentation and self- 
conceit. For if I added, that my acquaintance 
know that I am naturally of modest rather than of 
boasting speech, you will not believe it ; because 
you distinguish not between that which is said 
upon provocation, and that which is said without 
provocation, from vain glory. 

The next accusation is : " That I will be very 
angry with all men that do not presently submit 
to my dictates ; and that for advancing the repu- 
tation of my own skill, 1 care not what unworthy 
reflections I cast on others" This is in the epistle 
placed before the Vindicice Academiarum, subscri- 
bed by N S, as the plain song for H D in the rest 
of the book to descant upon. I know well enough 
the authors' names ; and am sorry that N S has 
lent his name to be abused to so ill a purpose. 
But how does this appear ? What argument, 
what witness is there of it f You offer none ; 
nor am I conscious of any. I begin to suspect 
since you, the professor of geometry, have in 
your objections to the twentieth chapter these 
words concerning " V index, ocular is ille testis de 
quo hie agitur, erat, ni fallor^ ille ipse" that 
Vindex himself, in other company, has bestowed a 
visit on me. Seeing you will have me believe it, 
let it be so ; arid, as it is likely, not long after my 
return into England. At which time (for the 
reputation, it seems, I had gqtten by my boasting) 
divers persons that professed to love philosophy 



LESSON vr. and mathematics, came to see me ; and some of 
^^1 them to let me see them, and hear and applaud 
what they applauded in themselves. I see now 
it hath happened to me with Vindex, as it happened 
to Dr. Harvey with Moraiius. Moranus, a Jesuit, 
came out of Flanders hither, especially, as he says, 
to see what learned men in divinity, ethics, physics, 
and geometry, were here yet alive, to the end that 
by discoursing with them in these sciences, he 
might correct either his own, or their errors. 
Amongst others he was brought, he savs, to that 
most civil and renowned old man Dr. Harvey. 
That is very w r ell. And in good earnest if he had 
made good use of the time which was very patiently 
afforded him, he might have learned of him (or of 
no man living) very much knowledge concerning 
the circulation of the blood, the generation of living 
creatures, and many other difficult points of 
natural philosophy. And if he had had anything in 
him but common and childish learning, he could 
have showed it nowhere more to his advantage, 
than before him that was so great a judge of such 
matters. But what did he ? That precious time 
(which was but little, because he was to depart 
again presently for Flanders) he bestowed wholly 
in venting his own childish opinions, not suffering 
the Doctor scarce to speak; losing thereby the 
benefit he came for, and discovering that he came 
not to hear what others could say, but to show to 
others how learned he was himself already. Why 
else did he take so little time, and so mispend it ? 
Or why returned he not again ? But when he had 
talked away his time, and found (though patiently 
and civilly heard) he was not much admired, he 


took occasion, writing against me, to be revenged LKSSON vi. 
of Dr. Harvey, by slighting his learning publicly ; ^^^ 
and tells me that his learning was only experiments ; 
which he says I say have no more certainty than 
civil histories. Which is false. My words are : 
u Ante hbs mini certl in physica crat prceter cx- 
perimenta cuique sita, et historias naturales, si 
tamen et lice dicendw certce shit, quce civilibus 
hlstorils certiores non sunt" Where I except ex- 
pressly from uncertainty the experiments that 
every man rnaketh to himself. But you see the 
near cut, by which vain glory joined with ignorance 
passeth quickly over to envy and contumely. 

Thus it seems by your own confession I was 
used by Vindex. lie comes with some of my 
acquaintance in a visit. What he said I know not, 
but if he discoursed then, as in his philosophical 
essay he writeth, I will be bold to say of myself, 
I was so far from morosity, or, to use his phrase, 
from being tetrical, as I may very well have a good 
opinion of my own patience. And if there passed 
between us the discourse you mention in your 
Elenchns, page I 16, it was an incivility in him so 
great, that without great civility I could not have 
abstained from bidding him be gone. That which 
passed between us you say was this : " / com- 
plained that whereas I made sense, nothing but a 
perception of motion in the organ, nevertheless, 
the philosophy schools through all Europe, led by 
the text of Aristotle, teach another doctrine, 
namely, that sensation is performed by species." 
This is a little mistaken. For I do glory, not 
complain, that whereas all the Universities of 
Europe hold sensation to proceed from species, I 

7 o 


LESSON vi. hold it to be a perception of motion in the organ. 
or manners. The an swer of Vindex, you say, was : " That the 
other hypothesis, whereby sense was explicated by 
the principles of motion, was commonly admitted 
here before my book came out, as having been 
sufficiently delivered by Des Cartes, Gassendus, 
and Sir Kenclm Digby, before I had published 
anything in this hind" This then, it seems, was 
it that made me angry. Truly I remember not 311 
angry word that ever I uttered in all my life to 
any man that came to see me, though some of them 
have troubled me with very impertinent discourse ; 
and with those that argued with me, how imper- 
tinently soever, I always thought it more civility 
to be somewhat earnest in the defence of my 
opinion, than by obstinate and affected silence to 
let them see I contemned them, or hearkened not 
to what they said. If I were earnest in making 
good, that the manner of sensation by such motion 
as I had explicated in my Leviathan, is in none of 
the authors by him named, it was not anger, but 
a care of not offending him, with any sign of the 
contempt which his discourse deserved. But it 
was incivility in him to make use of a visit, which 
all men take for a profession of friendship, to tell 
me that that which I had already published for 
my own, was found before by Des Cartes, Gassen- 
dus, and Sir Kenelm Digby. But let any man 
read Des Cartes ; he shall find, that he attributeth 
no motion at all to the object of sense, but an 
inclination to action, which inclination no man can 
imagine what it meaneth. And for Gassendus, 
and Sir Kenelru Digby, it is manifest by their wri- 
tings, that their opinions are not different from 


that of Epicurus, which is very different from LESSON vi 
mine. Or if these two, or any of those I con- ' ' 

. i . T Of manners. 

versed with at Pans, had prevented me in publish- 
ing my own doctrine, yet since it was there known, 
and declared for mine by Mersennus in the preface 
to his Ballistica (of which the three first leaves 
are employed wholly in the setting forth of my 
opinion concerning sense, and the rest of the 
faculties of the soul) they ought not therefore to 
be said to have found it out before me. Arid 
consequently this answer which you say w r as given 
me by Vindex was nothing else but untruth and 
envy ; and, because it was done by way of visit, 
incivility. But you have not alleged, nor can 
allege, any words of mine, from which can be 
drawn that I am so angry as you say I am with 
those that submit not to my dictates. Though the 
discipline of the University be never so good ; yet 
certainly this behaviour of yours and his are no 
good arguments to make it thought so. But you 
the professor of geometry, that out of my words 
spoken against Vindex in my twentieth chapter, 
argue my angry humour, do just as well, as when 
(in your Arlthmctica Infinitorum) from the con- 
tinual increase of the excess of the row of squares 
above the third part of the aggregate of the great- 
est, you conclude they shall at last be equal to it. 
For though you knew that Vindex had given rne 
first the Avorst words that possibly can be given, 
yet you w r ould have that return of mine to be a 
demonstration of an angry humour; not then know- 
ing what I told you even now in the beginning of 
this lesson, of the sentence given by Vespasian. 
But to this point I shall speak again hereafter. 


LESSON vi. Your third accusation is : " That I had my 
oTindlmeiT doc trine of vision^ which 1 pretended to Le my 
own, out of papers which I had a long time in 
my hands of Mr. Warner s" I never had sight of 
Mr. Warner's papers in all my life, but that of 
Vision by Refraction (which by his approbation I 
carried with me to Parkland caused it to be printed 
under his own name, at the end of Mersennus 
his Cogltata Physico-Mathematica, which you 
may have there seen, and another treatise of the 
proportions of alloy in gold and silver coin ; which 
is nothing to the present purpose. In all my 
conversation with him, I never heard him speak of 
anything he had written, or was writing, DC peril- 
cillo optico. And it w r as from me that he first 
heard it mentioned that light and colour were but 
fancy. Which he embraced presently as a truth, 
and told me it would remove a rub he \vas then 
come to in the discovery of the place of the image. 
If after my going hence he made any use of it 
(though he had it from me, and not I from him), it 
was well done. But wheresoever you find my 
principles, make use of them, if you can, to de- 
monstrate all the symptoms of vision ; and I will 
do (or rather have done and mean to publish) the 
same ; and let it be judged by that, whether those 
principles be of mine, or other men's invention. 
I give you time enough, and this advantage besides, 
that much of my optics hath been privately read 
by others. For I never refused to lend my papers 
to my friends, as knowing it to be a thing of no 
prejudice to the advancement of philosophy, though 
it be, as I have found it since, some prejudice to 
the advancement of my own reputation in those 


sciences ; which reputation I have always post- LESSON vr. 
posed to the common benefit of the studious. ' ' 

Ol manners. 

You say further (you the geometrician) that I 
had the proposition of the spiral line equal to a 
parabolical line from Mr. Robervall : true. And if 
I had remembered it, I would have taken also his 
demonstration ; though if I had published his, I 
would have suppressed mine. I was comparing in 
my thoughts those two lines, spiral and parabolical, 
by the motions wherewith they were described ; 
and considering those motions as uniform, arid the 
lines from the centre to the circumference, not to 
be little parallelograms, but little sectors, I saw 
that to compound the true motion of that point 
which described the spiral, I must have one 
line equal to half the perimeter, the other equal 
to half the diameter. But of all this I had not 
one word written. But being with Mersennus 
and Mr. Robervall in the cloister of the convent, 
I drew a figure on the wall, and Mr. Robervall 
perceiving the deduction I made, told me that 
since the motions which make the parabolical line, 
are one uniform, the other accelerated, the motions 
that make the spiral must be so also ; which I 
presently acknowledged; and he the next day, from 
this very method, brought to Mersennus the de- 
monstration of their equality. And this is the 
story mentioned by Mersennus, prop. 25, corol. 2, 
of his Ilydraulica ; which I know not who hath 
most magnanimously interpreted to you in my 

The fourth accusation is : " That I have Injured 
the Universities" Wherein; 1 First, " In that I 
would have the doctrine of my Lemathan by en- 


LESSON vi. tire sovereignty be imposed on them"" You often 
upbraid me with thinking well of my own doctrine; 

Of manners. * i-r/,,., 

and grant by consequence, that I thought this doc- 
trine good ; I desired not therefore that anything 
should be imposed upon them, but what (at least 
in my opinion) w r as good both for the Common- 
wealth and them. Nay more, I would have the 
state make use of them to uphold the civil power, 
as the Pope did to uphold the ecclesiastical. Is it 
not absurdly done to call this an injury ? But to 
question, you will say, whether the civil doctrine 
there taught be such as it ought to be, or not, is a 
disgrace to the Universities. If that be certain, it 
is certain also that those sermons and books, which 
have been preached and published, both against 
the former and the present government, directly or 
obliquely, were not made by such ministers and 
others as had their breeding in the Universities ; 
though all men know the contrary. But the doc- 
trine which I would have to be taught there, what 
is it ? It is this : " That all men that live in a 
Commonwealth, and receive protection of their 
lives and fortunes from the supreme governor 
thereof, are reciprocally bound, as far as they are 
able, and shall be required, to protect that 
governor." Is it, think you, an unreasonable thing 
to impose the teaching of such doctrine upon the 
Universities ? Or w r ill you say they taught it be- 
fore, when you know that so many men which 
came from the Universities to preach to the people, 
arid so many others that were not ministers, did 
stir the people up to resist the then supreme civil 
power ? And was it not truly therefore said, that 
the Universities receiving their discipline from the 


authority of the pope, were the shops and opera- LESSON vi. 
tories of the clergy ? Though the competition of 7 mwmor g 
the papal and civil power be taken now away, yet 
the competition between the ecclesiastical and the 
civil power hath manifestly enough appeared very 
lately. But neither is this an upbraiding of an 
University (which is a corporation or body, arti- 
ficial), but of particular men, that desire to uphold 
the authority of a Church, as of a distinct thing 
from the Commonwealth. How would you have 
exclaimed, if, instead of recommending my Levia- 
than to be taught in the Universities, I had recom- 
mended the erecting of a new and lay-university, 
wherein lay-men should have the reading of 
physics, mathematics, moral philosophy, and poli- 
tics, as the clergy have now the sole teaching of 
divinity ? Yet the thing would be profitable, and 
tend much to the polishing of man's nature, with- 
out much public charge. There will need but one 
house, and the endowment of a few professions. 
And to make some learn the better, it would do 
very well that none should come thither sent by 
their parents, as to a trade to get their living by, 
but that it should be a place for such ingenuous 
men, as being free to dispose of their own time, love 
truth for itself. In the mean time divinity may 
go on in Oxford and Cambridge to furnish the 
pulpit with men to cry down the civil power, if 
they continue to do as they did. If I had, I say, 
made such a motion in my Leviathan, though it 
would have offended the divines, yet it had been 
no injury. But it is an injury, you will say, to 
deny in general the utility of. the ancient schools, 
and to deny that we have received from them our 


LESSON vi. geometry. True, if I had not spoken distinctly of 
ofinaime7s *^ e schools of philosophy, and said expressly, that 
the geometricians passed not then under the name 
of philosophers ; and that in the school of Plato 
(the best of the ancient philosophers) none were 
received that were not already in some measure 
geometricians. Euclid taught geometry ; but I 
never heard of a sect of philosophers called 
Euclidians, or Alexandrians, or ranged with any 
of the other sects, as Peripatetics, Stoics, Acade- 
mics, Epicureans, Pyrrhonians, &c. But what is 
this to the Universities of Christendom ? Or why 
are we beholden for geometry to our universities, 
more than to Gresham College, or to private men 
in London, Paris, and other places, which never 
taught or learned it in a public school ? For even 
those men that living in our Universities have most 
advanced the mathematics, attained their know- 
ledge by other means than that of public lectures, 
where few auditors, and those of unequal pro- 
ficiency, cannot make benefit by one and the same 
lesson. And the true use of public professors, 
especially in the mathematics, being to resolve the 
doubts, arid problems, as far as they can, of such 
as come unto them with desire to be informed. 

That the Universities now are not regulated by 
the Pope, but by the civil power, is true, and well. 
But where say I the contrary ? And thus much 
for the first injury. 

Another, you say, is this, that in my Leviathan, 
p. 6/0, I say : " The principal schools were or- 
dained for the three professions of Roman religion, 
Roman law, and the t art of medicine'' Thirdly, 
that I say : " Philosophy had no otherwise place 


there than as a hand-maid to Roman religion'' LESSON vi. 
Fourthly : " Since the authority of Aristotle was 7 mauners '. 
received there, that study is not properly philo- 
sophy, but Aristotelity." Fifthly : " That for 
geometry, till of late times it had no place there 
at all." As for the second, it is too evident to be 
denied ; the fellowships having been all ordained 
for those professions ; arid (saving the change of re- 
ligion) being so yet. Nor hath this any reflection 
upon the Universities, either as they now are, or 
as they then were, seeing it was not in their own 
power to endow themselves, or to receive other 
laws and discipline than their founder and the 
state w r as pleased to ordain. For the third, it is 
also evident. For all men know that none but the 
Roman religion had any stipend or preferment in 
any university, where that religion was established? 
No, nor for a great while, in their commonwealths ; 
but were everywhere persecuted as heretics. But 
you will say, the words of my Leviathan are not, 
philosophy " had no place? but " hath no place." 
Are you not ashamed to lay to my charge a mis- 
take of the word hath for had? which was either a 
mistake of the printer, or if it were so in the copy, 
it could be no other than the mistake of a letter in 
the writing, unless you think you can make men 
believe that after fifty years being acquainted 
with what was publicly professed and practised 
in Oxford and Cambridge, I knew not what 
religion they were of. This taking of advantage 
from the mistake of a word, or of a letter, I 
find also in the Elenchus, where for pr&tendit 
se scire, there is prcctendit $cire, which you the 
geometrician sufficiently mumble, mistaking it I 


LESSON vi. think for an anglicism, not for a fault of the im- 
^ ' pression. 

Oi manners. * 

To the fourth,, you pretend, that men are not 
now so tied to Aristotle as not to enjoy a liberty 
of philosophising, though it were otherwise when 
I was conversant in Magdalen Hall. Was it so 
then ? Then am I absolved, unless you can show 
some public act of the university made since that 
time to alter it. For it is not enough to name 
some few particular ingenuous men that usurp that 
liberty in their private discourses, or, with conni- 
vance, in their public disputations. And your 
doctrine, that even here you avow, of abstracted 
essences, immaterial substances, and of nunc-stans; 
and your improper language in using the word 
(not as mine, for I have it nowhere) successive 
eternity; as also your doctrine of condensation, and 
your arguing from natural reason the incompre- 
hensible mysteries of religion, and your malicious 
writing, arc very shrewd signs that } ou yourselves 
are none of those which you say do freely philo- 
sophise; but that both your philosophy and your 
language are under the servitude, not of the Roman 
religion, but of the ambition of some other doctors, 
that seek, as the Roman clergy did, to draw all 
human learning to the upholding of their power 
ecclesiastical. Hitherto therefore there is no in- 
jury done to the universities. For the fifth, you 
grant it, namely, " that till of late there was no 
allowance for the teaching of geometry"" But lest 
you should be thought to grant me anything, you 
say, you the astronomer, " geometry hath now so 
much place in the universities, that when Mr. 
Ilobbes shall have published his philosophical and 


geometrical pieces , you assure yourself you shall LESSON vi. 
be able to find a greater number in the university ot : manner ^ 
who will understand as much, or more, of them 
than he desired they should" &c. But though 
this be true of the now, yet it rnaketh nothing 
against my then. I know well enough that Sir 
Henry Savile's lectures were founded and endowed 
since. Did I deny then that there were in Oxford 
ijiany good geometricians ? But I deny now, that 
either of you is of the number. For my philoso- 
phical and geometrical pieces are published, and 
you have understood only so much in them, as all 
men will easily see by your objections to them, and 
by your own published geometry, that neither of 
you understand anything either in philosophy or 
in geometry. And yet you would have those books 
of yours to stand for an argument, and to be an 
index of the philosophy and geometry to be found 
in the universities. Which is a greater injury and 
disgrace to them, than any words of mine, though 
interpreted by yourselves. 

Your last and greatest accusation, or rather 
railing (for an accusation should contain, whether 
true or false, some particular fact, or certain words, 
out of which it might seem at least to be inferred), 
is, that I am an enemy to religion. Your words 
are : " // is said that Mr. Hobbes is no otherwise 
an enemy to the Roman religion, saving only as it 
hath the name of religion." This is said by Vindex. 
You, the geometrician, in your epistle dedicatory, 
say thus : "With what pride and imperiousness he 
tramples on all things both human and divine, 
uttering fearful and horrible words of God, 
(peace), of sin, of the holy Scripture, of' all incor- 


LESSON vi. poreal substances hi general, of the immortal soul 
^ r T ~* of man, and of the rest of the weighty points of 

Of manners .. . . o / J v 

religion (down), it is not so much to be doubted 
as lamented" And at the end of your objections 
to the eighteenth chapter, "Perhaps you take the 
whole history of the Jail of Adam for a fable, 
which is no wonder, when you say the rules of 
honouring and worshipping of God are to be 
taken from the laws"' Down, I say ; you bark 
now at the supreme legislative power. Therefore 
it is not I, but the laws which must rate you off. 
But do not many other men, as w 7 ell as you. read 
my Leviathan, and my other books ? And yet 
they all find not such enmity in them against re- 
ligion. Take heed of calling them all atheists that 
have read and approved my Leviathan. Do you 
think I can be an atheist and not know it ? Or 
knowing it, durst have offered my atheism to the 
press ? Or do you think him an atheist, or a con- 
temner of the Holy Scripture, that sayeth nothing 
of the Deity but what he proveth by the Scripture? 
You that take so heinously that I would have the 
rules of God's worship in a Christian common- 
wealth taken from the laws, tell me, from whom 
you would have them taken ? From yourselves ? 
Why so, more than from me ? From the bishops ? 
Right, if the supreme power of the commonwealth 
will have it so ; if not, why from them rather than 
from me ? From a consistory of presbyters by 
themselves, or joined with lay-elders, whom they 
may sway as they please ? Good, if the supreme 
governor of the commonwealth will have it so; 
if not, why from them, rather than from me, or 
from any man else ? They are wiser and learn- 


eder than I. It may be so ; but it has not yet LESSON vi. 
appeared. Howsoever, let that be granted. Is ' ' ' 

11 ' . . Ot manners. 

there any man so very a fool as to subject himself 
to the rules of other men in those things which so 
nearly concern himself, for the title they assume 
of being wise and learned, unless they also have 
the sword which must protect them. But it seems 
you understand the sword as comprehended. If 
so, do you not then receive the rules of God's wor- 
ship from the civil power r Yes, doubtless ; and 
you would expect, if your consistory had that 
sword, that no man should dare to exercise or 
teach any rules concerning God's worship which 
were not by you allowed. See therefore how much 
you have been transported by your malice towards 
me, to injure the civil power by which you live, 
If you were not despised, you would in some places 
and times, where and when the laws are more 
severely executed, be shipped away for this your 
madness to America, I would say, to Anticyra. 
What luck have I, when this, of the laws being 
the rules of God's public worship, was by me said 
and applied to the vindication of the Church of 
England from the power of the Roman clergy, it 
should be followed with such a storm from the 
ministers, presbyterian and episcopal, of the Church 
of England? Again, for those other points, namely, 
that I approve not of incorporeal bodies, nor of 
other immortality of the soul, than that which the 
Scripture calleth eternal life, I do but as the Scrip- 
ture leads me. To the texts whereof by me alleged, 
you should either have answered, or else forborne to 
revile me for the conclusions I derived from them. 
Lastly, what an absurd question is it to ask me 


LESSON vi. whether it be in the power of the magistrate, 
Xl ' whether the world be eternal or not ? It were fit 

Or manners. 

you knew it is in the power of the supreme magis- 
trate to make a law for the punishment of them 
that shall pronounce publicly of that question any- 
thing contrary to that which the law hath once 
pronounced. The truth is, you are content that 
the papal power be cut off, and declaimed against 
as much as any man will ; but the ecclesiastical 
power, which of late was aimed at by the clergy 
here, being a part thereof, every violence done to 
the papal power is sensible to them yet ; like that 
which I have heard say of a man, whose leg being 
cut off for the prevention of a gangrene that began 
in his toe, would nevertheless complain of a pain 
in his toe, when his leg was cut off. 

Thus much in my defence ; which I believe if 
you had foreseen, this accusation of yours had been 
left out. I come now to examine (though it be 
done in part already) what manners those are 
which I find everywhere in your writings. 

And first, how came it into your minds that a 
man can be an atheist, I mean an atheist in his 
conscience ? I know that David confesseth of him- 
self, upon sight of the prosperity of the wicked, 
that his feet had almost slipped, that is, that he 
had slipped into a short doubtfulness of the Divine 
Providence. And if anything else can cause a 
man to slip in the same kind, it is the seeing such 
as you (who though you write nothing but what 
is dictated to each of you by a doctor of divinity) 
do break the greatest of God's commandments, 
which is charity, in every line before his face. 
And though such forgettings of God be somewhat 


more than short doubtings, and sudden transporta- LESSON vi. 
tions incident to human passion, yet I do not for 7 mimlere 
that cause think you atheists and enemies of re- 
ligion, but only ignorant and imprudent Christians. 
But how, I say, could you think me an atheist, unless 
it were because finding your doubts of the Deity 
more frequent than other men do, you are thereby 
the apter to fall upon that kind of reproach ? 
Wherein you are like women of poor and evil 
education when they scold ; amongst whom the 
readiest disgraceful word is whore : why not thief, 
or any other ill name, but because, when they re- 
member themselves, they think that reproach the 
likeliest to be true ? 

Secondly, tell me what crime it was which the 
Latins called by the name of scelus ? You think 
not, unless you be Stoics, that all crimes are equal. 
Scelus was never used but for a crime of greatest 
mischief, as the taking away of life and honour ; 
and besides, basely acted, as by some clandestine 
way, or by such a way as might be covered with a 
lie. But when you insinuate in a writing published 
that I am an atheist, you make yourselves authors 
to the multitude, and do all you can to stir them 
up to attempt upon my life ; and if it succeed, 
then to sneak out of it by leaving the fault 
on them that are but actors. This is to en- 
deavour great mischief basely, and therefore scelus. 
Again, to deprive a man of the honour he hath 
merited, is no little wickedness ; and this you en- 
deavour to do by publishing falsely that I challenge 
as my own the inventions of other men. This is 
therefore scelus publicly to tell all the world that 
I will be angry with all men that do not presently 
submit to my dictates ; to deprive me of the friend- 



LESSON vi. ship of all the world; great damage, and a lie, 
v^ . -- an( j yours. For to publish any untruth of another 

Of manners. . . 

man to his disgrace, on hearsay from his enemy, 
is the same fault as if he published it on his own 
credit. If I should say I have heard that Dr. 
Wallis was esteemed at Oxford for a simple fellow, 
and much inferior to his fellow-professor Dr. Ward 
(as indeed I have heard, but do not believe it), 
though this be no great disgrace to Dr. Waliis, 
yet he would think I did him injury. Therefore 
public accusation upon hearsay is scelus. And 
whosoever does any of these things does scelerate. 
But you the professors of the mathematics at 
Oxford, by the advice of two doctors of divinity 
have dealt thus with me. Therefore you have 
done, I say not foolishly, though no wickedness be 
without folly, but scelerate, o^p *S Saa. 

Thirdly, it is ill manners, in reprehending truth, 
to send a man in a boasting way to your own 
errors; as you the professor of geometry have 
often sent me to your two tractates of the Angle 
of Contact and Arithmetica Infinitorum. 

Fourthly, it is ill manners, to diminish the just 
reputation of worthy men after they be dead, as 
you the professor of geometry have done in the 
case of Joseph Scaliger. 

Fifthly, when I had in my Leviathan suffered 
the clergy of the Church of England to escape, 
you did imprudently in bringing any of them in 
again. An Ulysses upon so light an occasion would- 
not have ventured to return again into the cave of 

Lastly, how ill does such levity and scurrility, 
which both of you have shown so often in your 
writings, become the gravity and sanctity requisite 


to the calling of the ministry r They are too LESSON vr. 
many to be repeated. Do but consider, you the Ofm ; mio 7 s ' 
geometrician, how unhandsome it is to play upon 
my name, when both yours and mine are plebeian 
names ; though from Willis by Wallis, you go from 
yours in Wallisius. The jest of using at every 
word mi Hobbi, is lost to them beyond sea. But 
this is not so ill as some of the rest. I will write 
out one of them, as it is in the fourth page of your 
Elenchus : " Whence it appears that your Empusa 
was of the number of those fairies which you call 
in English hob-goblins. The word is made of 
w and 7T8c ; and thence comes the children's play 
called the play of Empusa, Anglice (hitherto in 
Latin all but hob-goblins, then follows in English) 
fox, fox, come out of your hole (then in Latin 
again), in which the boy that is called the fox, 
holds up one foot, arid jumps with the other, which 
in English is to hop." When a stranger shall 
read this, and hoping to find therein some witty 
conceit, shall with much ado have gotten it in- 
terpreted and explained to him, what will he think 
of our doctors of divinity at Oxford, that will take 
so much pains as to go out of the language they 
set forth in, for so ridiculous a purpose ? You 
will say it is a pretty paranomasia. How you call 
it there I know not, but it is commonly called here 
a clinch ; and such a one as is too insipid for a boy 
of twelve years old, and very unfit for the sanctity 
of a minister, and gravity of a doctor of divinity. 
But I pray you tell me where it was you read the 
word empusa for the boy's play you speak of, 
or for any other play amongst the Greeks ? In 
this (as you have done throughout all your other 
writings) you presume too much upon your first 

A A 2 


LESSON vr. cogitations. There be a hundred other scoffing 
or mwmrcs P assa g es > an d ill-favoured attributes given me in 
both your writings, which the reader will observe 
without my pointing to them, as easily as you 
would have him ; and which perhaps some young 
students, finding them full of gall, will mistake for 
salt. Therefore to disabuse those young men, and 
to the end they may not admire such kind of wit, 
I have here and there been a little sharper with 
you than else I would have been. If you think I 
did not spare you, but that I had not wit enough 
to give you as scornful names as you give me, are 
you content I should try ? Yes (you the geome- 
trician will say) give me what names you please, so 
you call me not Aritlimetica lufinitorum. I will 
not. Nor Angle of Contact ; nor Arch Spiral ; 
nor Quotient. I will not. But I here dismiss you 
both together. So go your ways, you Uncivil 
Ecclesiastics, Inhuman Divines, Dedoctors of mo- 
rality, Unasinous Colleagues, Egregious pair of 
Issachars, most ivretched Vindices and Indices 
Academiarum ; and remember Vespasian's law, 
that it is uncivil to give ill language first, but civil 
and lawful to return it. But much more remember 
the law of God, to obey your sovereigns in all 
things ; and not only not to derogate from them, 
but also to pray for them, and as far as you can to 
maintain their authority, and therein your own 
protection. And, do you hear? take heed of 
speaking your mind so clearly in answering my 
Leviathan, as I have done in writing it. You 
should do best not to meddle with it at all, because 
it is undertaken, ^and in part published already, 
and will be better performed, from term to term, 
by one Christopher Pike, 


/ac, A-yfxnfaac, AvriTroXtrctac, 


M A 11 K S 












I DID not intend to trouble your Lordship twice 
with this contention between me and Dr.Wallis. 
But your Lordship sees how I am constrained to 
it ; which, whatsoever reply the Doctor makes, 
I shall be constrained to no more. That which I 
have now said of his Geometry, Manners, Divinity, 
and Grammar, altogether is not much, though 
enough. As for that which I here have written 
concerning his Geometry, which you will look for 
first, is so clear, that riot only your Lordship, and 
such as have proceeded far in that science, but 
also any man else that doth but know how to add 
and subtract proportions, (which is taught at the 
twenty-third proposition of the sixth of Euclid), 
may see the Doctor is in the wrong. That which 
I say of his ill language and politics is yet shorter. 
The rest, which concerneth grammar, is almost all 
another man's, but so full of learning of that kind, 
as no man that taketh delight in knowing the pro- 


prieties of the Greek and Latin tongues, will think 
his time ill bestowed in the reading it. I give the 
Doctor no more ill words, but am returned from 
his manners to my own. Your Lordship may 
perhaps say, my compliment in my title-page is 
somewhat coarse ; and it is true. But, my Lord, 
it is since the writing of the title-page, that I am 
returned from the Doctor's manners to my own ; 
which are such as I hope you will not be ashamed 
to own me, my Lord, for one of 

Your Lordship's most humble 

and obedient servants, 







WHEN unprovoked you addressed unto me, in Marks of Dr. 
your Elenchus, your harsh compliment with great ^cometry^te 
security, wantonly to show your wit, I confess you ' ' 
made me angry, and willing to put you into a 
better way of considering your own forces, and to 
move you a little as you had moved me, which 
I perceive my lessons to you have in some measure 
done ; but here you shall see how easily I can bear 
your reproaches, now they proceed from anger, 
arid how calmly I can argue with you about your 
geometry and other parts of learning. 

I shall in the first part confer with you 
about your Arithmetica Infinitorum, and after- 
wards compare our manner of elocution; then 
your politics ; and last of all your grammar and 
critics. Your spiral line is condemned by him 
whose authority you use to prove me a plagiary, 
(that is, a man that stealeth other men's inven- 
tions, and arrogates them to himself), whether it 
be Roberval or not that writ that paper, I am not 
certain. But I think I shal\ be shortly ; but who- 
soever it be, his authority will serve no less to 


Marks of Dr show that your doctrine of the spiral line, from the 

Walhs s absurd J x 

Geometry, & c . fifth to the eighteenth proposition of your Arithme- 
tica Infinitorum, is all false ; and that the principal 
fault therein (if all faults be not principal in geo- 
metry, when they proceed from ignorance of the 
science) is the same that I objected to you in my 
Lessons. And for the author of that paper, when I 
am certain who it is, it will be then time enough to 
vindicate myself concerning that name of plagiary. 
And whereas he challenges the invention of your 
method delivered in your Arithmetica Infinitorum, 
to have been his before it was yours, I shall, I 
think, by and by say that which shall make him 
ashamed to own it ; and those that writ those 
encomiastic epistles to you ashamed of the honour 
they meant to you. I pass therefore to the nine- 
teenth proposition, which in Latin is this : your 
geometry ! 

" Si proponatur series quantitatum in duplicata 
ratione arithmetice proportionalinm (sive juxta 
seriem numerorum quadraticornm) continue cres- 
centium, a puncto vel inclioatarum, (puta ut 
0. 1. 4. 9. 16. etc.], proposition sit, inquirere qnam 
habeat ilia rationem ad seriem totidem maxima 

" Fiat investigatio per modum inductionis ut 
(in prop. 1) 

1+1=2 3 + 6 

0+1 + 4= 5 ]_ _[_ 

4+4 + 4 = 12~T 12" 

+ 1 + 4 + 9=14 11 et ^ deinceps . 

36 3 18 

"Ratio proveniens est ubique major quam 


subtripla sen 4~/ excessus autem perpetuo de- Marks ot Dr. 

/ x- Wallis's absurd 

crescit prout numerus terminorum augetur (puta Geometry, &c. 
6 12 is 24 /.) aucto nimirum fractionis denomi- 
nalore sive consequente rationis in singulis locis 
numero senario (ut patet) ut sit rationis prove- 
mentis excessus supra subtriplam, ea quam habet 
unitas ad sextuplum nnmeri terminorum post ; 

That is, if there be propounded a row of quanti- 
ties in duplicate proportion of the quantities arith- 
metically proportional (or proceeding in the order 
of the square numbers) continually increasing; 
and beginning at a point or ; let it be propounded 
to find what proportion the row hath ; to as many 
quantities equal to the greatest ; 

Let it be sought by induction (as in the first 
proposition) . 

The proportion arising is everywhere greater 
than subtriple, or , and the excess perpetually 
decreaseth as the number of terms is augmented, 
as here, b 12 r^ 2^ so ^ & c - ^ e denominator of the 
fraction being in every place augmented by the 
number six, as is manifest ; so that the excess of 
the rising proportion above subtriple is the same 
which unity hath to six times the number of terms 
after ; and so. 

Sir, in these your characters I understand by 
the cross + that the quantities on each side of it 
are to be added together and make one aggregate; 
and I understand by the two parallel lines = that 
the quantities between which they are placed are 
one to another equal ; this is your meaning, or you 
should have told us what you meant else ; I under- 
stand also, that in the first row + 1 is equal to 1> 


Mark* of Dr. an( j i , j equa j to 2 ; and that in the second 

Walhs s absurd ~ " 

Geometry, & c . row 0+1+4 is equal to 5 ; and 4+4+4 equal to 12 ; 
but (which you are too apt to grant) I understand 
your symbols no further ; but must confer with 
yourself about the rest. 

And first I ask you (because fractions are com- 
monly written in that manner) whether in the 

/ i i o + i = i i i \ o u 
uppermost row (which is t + 1=s 2 == T" H "6"/T " e a 

fraction, 4~ be a fraction, 4~ be a fraction, that is 
to say, a part of an unit, and if you will, for the 
cypher's sake, whether 4~> be an infinitely little 
part of 1 ; and whether 4"> r 1 divided by 1 
signify an unity ? if that be your meaning, then 
the fraction -J- added to the fraction -j- is equal to 
the fraction : But the fraction ~r is equal to ; 
therefore the fraction 4" + T- is equal to the fraction 
4- ; and T- equal to 4~ which you will confess to 
be an absurd conclusion, and cannot own that 

I ask you therefore again, if by T~ you mean the 
proportion of to 1 ; and consequently by -J- the 
proportion of 1 to 1, and by^-the proportion of 
1 to 2 : if so, then it will follow, that if the pro- 
portions of to 1 and of 1 to 1 be compounded 
by addition, the proportion arising will be the pro- 
portion of 1 to 2. But the proportion of to 1 is 
infinitely little, that is, none. Therefore the propo- 
sition arising by composition will be that of 1 to 1, 
and equal (because of the symbol =) to the pro- 
portion of 1 to 2, aryl so 1=2. This also is so 
absurd that I dare say that you will not own it. 

There may be another meaning yet : perhaps 
you mean that the uppermost quantity + 1 is equal 


to the uppermost quantity 1 ; and the lowermost Marlis of Dr 

11 V i i Wttlhs's absurd 

quantity 1 -f 1 equal to the lowermost quantity 2 : Gcomchy, &c. 
which is true. But how then in this equation 
__=^_ + V ? Is the uppermost quantity 1 equal to 
the uppermost quantity 1 + 1 ; or the lowermost 
quantity 2 equal to the lowermost quantity 3 + 6 ? 
Therefore neither can this be your meaning. Un- 
less you make your symbols more significant, you 
must not blame me for want of understanding 

Let us now try what better success we shall 
have where the places are three, as here : 

0+1 + 4-5 5 1 1 

1=12 12 ~ 3 T 12 ' 

If your symbols be fractions, the compound of 
them by addition is --, for 04" and 4~ make --; and 
consequently (because of the symbol =) v equal 
to -^, which is not to be allowed, and therefore 
that was not your meaning. If you meant that 
the proportions of to 4 and of 1 to 4 and of 4 to 
4 compounded, is equal to the proportion of 5 to 
12, you will fall again into no less an inconvenience. 
For the proportion arising out of that composition 
will be the proportion of 1 to 4. For the propor- 
tion of to 4 is infinitely little. Then to com- 
pound the other two, set them in this order 1. 4. 4. 
and you have a proportion compounded of 1 to 

4 and of 4 to 4, namely, the proportion of the 
first to the last, which is of 1 to 4, which must be 
equal, by this your meaning, to the proportion of 

5 to 12, and consequently as 5 to 12> so is 1 to 4, 
which you must not own. Lastly, if you mean 
that the uppermost quantities to the uppermost, 
and the lowermost to the lowermost in the first 


ec l uat i n are e q ua l> it is granted, but then again in 
Geometry, & c . the second equation it is false. It concerns your 
fame in the mathematics to look about how to 
justify these equations which are the premises to 
your conclusion following, namely, that the pro- 
portion arising is every where greater than sub- 
triple, or a third ; and that the excess (that is, the 
excess above subtriple) perpetually decreascth as 
the number of terms is augmented, as here 
c-vrrr^ a J 5 &c. which I will show you plainly is 

But first I wonder why you were so angry with 
me for saying you made proportion to consist in 
the quotient, as to tell me it was abominably false, 
and to justify it, cite your own words penes quo- 
tientem ; do not you say here, the proportion is 
everywhere greater than subtriple, or 4 -? And is 
not 4" the quotient of 1 divided by 3 ? You can- 
not say in this place that penes is understood ; 
for if it were expressed you would not be able to 

But I return to your conclusion, that the excess 
of the proportion of the increasing quantities above 
the third part of so many times the greatest, de- 
creaseth, as^~ ^^~^-> & c - For by this account 
in this row ~=^- where the quantity above ex- 
ceeds the third part of the quantities below by ---, 
you make -^- equal to ~* r , which you do not mean. 
It may be said your meaning is, that the pro- 
portion of 1 to the subtriple of 2 which is -f-, ex- 
ceedeth what ? I cannot imagine what, nor proceed 
further where the terms 'be but two. Let us there- 
fore take the second row, that is, 

f Dr. 



sum above is 5, the sum below is 12, the third part M * rk " 01 

J * m A Wallis'si 

whereof is 4 ; if you mean, that the proportion of Geometry, &c. 
5 to 4 exceeds the proportion of 4 to 12 (which is 
subtriple) by-fr,you are out again. For 5 exceeds 

4 by unity, which is -}* k I do not think you will 
own such an equation as iJ=^. Therefore I be- 
lieve you mean (and your next proposition assures 
me of it), that the proportion of 5 to 4 exceeds 
sub triple proportion by the proportion of 1 to 12 ; 
if you do so, you are yet deceived. 

For if the proportion of 5 to 4 exceeds subtriple 
proportion by the proportion of 1 to 12, then sub- 
triple proportion, that is, of 4 to 12 added to the 
proportion of 1 to 1 2 must make the proportion of 

5 to 4. But if you look on these quantities, 4, 
12, 144, you will see, and must not dissemble, that 
the proportion of 4 to 12 is subtriple, and the pro- 
portion of 12 to 144 is the same with that of 1 to 
12. Therefore by your assertion it must be as 
5 to 4 so 4 to 144, which you must not own. 

And yet this is manifestly your meaning, as ap- 
peareth in these w r ords : " Ut sit rationis prove- 
mentis excessus supra subtriplam ea quam habet 
nnitas ad sextuplum numeri terminorum post 0, 
adeoqiie" which cannot be rendered in English, 
nor need to be. For you express yourself in the 
twentieth proposition very clearly ; I rioted it only 
that you may be more merciful hereafter to the 
stumblings of a hasty pen. For excessus ea 
quam does not well, nor is to be well excused by 
subauditur ratio. Your twentieth proposition is 
this : 

"Si proponatur series quttntitatum in duplicata 
ratione arithmetice proportionalium (sive juxta 


Marks of Dr seriem numerorum quadraticorum) continue cres- 

Walhs's absurd * y 

Geometiy, Ate. centmm, puticto vet inclioatarum, ratio quam 
lidbet ilia ad serlem tot idem maxima cequalium 
subtriplam super obit; eritque excessus ea ratio 
quam habet unitas ad sextuplum numcri termino- 
rum post 0, sive quam habet radix quadratica 
termini primi post ad sextuplum radicis quad- 
ratica termini maximi" 

That is, if there be propounded a row of quan- 
tities in duplicate proportion of arithinetically- 
proportionals (or according to the row of square 
numbers) continually increasing, and beginning 
with a point or 0. The proportion of that row to 
a row of so many equals to the greatest, shall be 
greater than sub triple proportion, and the excess 
shall be that proportion which unity hath to the 
sextuple of the number of terms after 0, or the 
same which the square root of the first number 
after 0, hath to the sextuple of the square root of 
the greatest. 

For proof whereof you have no more here than 
patet ex prcecedentibus ; and no more before but 
adeoque. You do not well to pass over such curious 
propositions so slightly ; none of the ancients did 
so, nor, that I remember, any man before yourself. 
The proposition is false, as you shall presently see. 

Take, for example, any one of your rows : as 
4+4+4. By this proportion of yours 1+4, which 
makes 5, is to 12 in more than subtriple proportion; 
by the proportion of 1 to the sextuple of 2 which 
is 12. Put in order these three quantities 5, 4, 12, 
and you must see the proportion of 5 to 12 is 
greater than the proportion of 4 to 12, that is, 
subtriple proportion, by the proportion of 5 to 4. 


But by your account the proportion of 5 to 4 is Marl " of Dr - 

J J r r ^ Wallis's absurd 

greater than that of 4 to 1 2 by the proportion of Geometry, &c. 
1 to 12. Therefore, as 5 to 4 so is 1 to '12, which 
is a very strange paradox. 

After this you bring in this consectary : " Cum 
autem crescente nnmero terminorum excessus ille 
supra rationem suhtriplam continue minuatur, ut 
tandem quovis assignabili minor evadat (ut patet) 
si in infinitum producatur, prorsus evaniturus est. 

That is, seeing as the number of terms increaseth, 
that excess above subtriple proportion continually 
decreaseth, so as at length it becomes less than any 
assignable (as is manifest) if it be produced infi- 
nitely, it shall utterly vanish, and so. And so 

Sir, this consequence of yours is false. For two 
quantities being given, and the excess of the greater 
above the less, that excess may continually be de- 
creased, and yet never quite vanish. Suppose any 
two unequal quantities differing by more than an 
unit, as 3 and 6, the excess 3, let 3 be diminished, 
first by an unit, and the excess will be 2, and the 
quantities will be 3 and 5 ; 5 is greater than 4, 
the excess 1 . Again, let 1 be diminished and made 
~- 9 the excess 7- and the quantities 3 and 4-7, 4| 
is yet greater than 4. Again diminish the excess 
to 4~> the quantities will be 3 and 4-j, yet still 
4-j- is greater than 4. In the same manner you 
may proceed to 4 i^ :4, &c. infinitely ; and yet you 
shall never come within an unit (though your unit 
stand for 100 miles) of the lesser quantity pro- 
pounded 3, if that 3 stands for 300 miles. The 
excesses above subtriple proportion do not decrease 



Marks of Dr i n the manner you say it does, but in the manner 

Wallis's absurd . J J * 

Geometry, & c . which I now shall show you. 

In the first row j~i a third of the quantities 
below is 4* set fa order these three quantities 
1 -| -| . The first is 1, equal to the sum above, the 
last is 4 5 equal to the subtriple of the sum below. 
The middlemost is ^ subtriple to the last quantity 
~. The excess of the proportion of 1 to -|- above 
the subtriple proportion of ~- to -*- is the propor- 
tion of 1 to -f-, that is of 9 to 2, that is, of 18 to 4. 

Secondly, in the second row, which is t ^^> a 

third of the sum below is 4, the sum above is 5. 
Set in order these quantities, 1, 5, 4, 12. There 
the proportion of 15 to 12 is the proportion of 
5 to 4. The proportion of 4 to 12 is subtriple; 
the excess is the proportion of 15 to 4, which is 
less than the proportion of 18 to 4 , as it ought to 
be ; but not less by the proportion of nr to -^ as 
you would have it. 

Thirdly, in the third row, which is 9-^-^-9. A 
third of the sum below is 12, the sum above is 14. 
Set in order these quantities, 42, 4, 12. There the 
proportion of 42 to 12 is the same with that of 
14 to 4. And the proportion of 4 to 12 subtriple, 
less than the former excess of 15 to 4. And so it 
goes on decreasing all the way in this manner, 
18 to 4, 15 to 4, 14 to 4, &c. which differs very 
much from your 1 to 6, 1 to 12, 1 to 18, &c. 
and the cause of your mistake is this : you call 
the twelfth part of twelve -g-, and the eighteenth 
part of thirty-six you call ^r, and so of the rest. 
But what need of all those equations in symbols, 
to show that the proportion decreases ; is there 


any man can doubt, but that the proportion of MaiksofDr - 

J ' r r Wallis's absurd 

1 to 2 is greater than that of 5 to 12, or that Geometry, &c. 
of 5 to 12 greater than that of 14 to 36, and 
so on continually forwards ; or could you have 
fallen into this error, unless you had taken, as you 
have done in very many places of your Elenchus, 
the fractions-^' and -& 9 &c. which are the quotients 
of I divided by 6 and 12, for the very proportions 
of, 1 to 6 and 1 to 12. But notwithstanding the 
excess of the proportions of the increasing quanti- 
ties, to subtriple proportion decrease, still, as the 
number of terms increaseth, and that what propor- 
tions soever I shall assign, the decrement will in 
time (in time, I say, without proceeding in infini- 
tum) produce a less, yet it does not follow that the 
row of increasing quantities shall ever be equal to 
the third part of the row of so many equals to the 
last or greatest. For it is not, I hope, a paradox 
to you, that in tw r o rows of quantities the propor- 
tion of the excesses may decrease, and yet the 
excesses themselves increase, and do perpetually. 
For in the second arid third rows, which are 

0+1+4=5 j 0+1 +4+9 = 14 j ,, , , . , 

r+y+T=T2 and Tr+ 99^6 exceeds the third part 
of 12 by a quarter of the square of 4, and 14 ex- 
ceeds the third part of 36 by 2 quarters of the 
square of 4, and proceeding on, the sum of the 
increasing quantities where the terms are 5 (which 
sum is 30) exceedeth the third part of those below, 
(those below are 80, and their third part 26|- ) by 
3 quarters and -$- a quarter of the square of 4, and 
when the terms are 6, the quantities above will 
exceed the third part of them below by 5 quarters 
of the square of 4. Would you have men believe, 

B B 2 


Marts of Dr. that the further they go, the excess of the in- 

Walhs's absurd . . . , i ,1-1 ^ * ^ 

Geometry, & c . creasing quantities above the third part ot those 

" ' below shall be so much the less? And yet the 

proportions of those above, to the thirds of those 

below, shall decrease eternally; and therefore your 

twenty-first proposition is false, namely this : 

" Si proponatur series infinita quantitatum in 
duplicaia ratione arithmetice proportionalium 
(sive juxta seriem numerorum quadraticorum), 
continue crescentium a puncto sivc inchoata- 
rum ; erit ilia ad seriem totidcm maxima cequa- 
lium, ut 1 ad 3." 

That is, if an infinite row of quantities be pro- 
pounded in duplicate proportion of arithmetically- 
proportionals (or according to the row of quadratic 
numbers), continually increasing and beginning 
from a point or ; that row shall be to the row of 
as many equals to the greatest, as 1 to 3. This is 
false, ut patet ex prcecedentibus ; and, conse- 
quently, all that you say in proof of the propor- 
tion of your parabola to a parallelogram, or of 
the spiral (the true spiral) to a circle is in vain. 

But your spiral puts me in mind of what you 
have under- written to the diagram of your propo- 
sition 5. The spiral) in both figures, was to be 
continued whole to the middle, but, by the care- 
lessness of the graver, it is in one figure manca,, in 
the other intercisa. 

Truly, Sir, you will hardly make your reader 
believe that a graver could commit those faults 
without the help of your own copy, nor that it 
had been in your copy, if you had known how to 
describe a spiral line then as now. This I had not 
said, though truth, but that you are pleased to say, 


though not truth, that I attributed to the printer Marks of Dr. 

, ~ . Wallw's absurd 

some faults of mine. , Geometry, &c. 

I come now to the thirty-ninth proposition, 
which is this : 

" Si proponatur series quantitatnm in triplicata 
ratlone arithmetice proportionalium (sive juxta 
seriem numerorum cubicorum), continue crescen- 
tium apuncto sive inchoatarum (puta ut 0, 1, 8, 
2-7, etc.} y propositum sit inquirere quam liabeat 
series ilia rationem ad seriem totidem maxima 
cequalium : 

" Fiat investigatio per modum inductionis (ut in 
prop. 1, et prop. 19) : 

= 1 2 

0+ 1 + 8 = 9 
8 + 8 + 8 = 24 == 

0+1 + 8 +27=36 _ Jt ___ 1_ J_ 
27 + 27 + 27 + 27=108" 12 ~ 4 + 12 

Et sic deinceps. 

" Ratio proveniens est ubique major quam sub- 
quadrupla, sive }. Excessus autem perpetuo de- 
crescit, pro ut numerus terminorurn augetur, puta 
*8~~Tt> etc. Aucto nimirum fractionis denomina- 
tore sive consequente rationis in singulis locis 
numero quaternatio, ut patet, ut sit rationis pro- 
venientis excessus supra subquadruplam ea quam 
habet unit as ad quadruplum numeri terminorum 
post adeoque" 

That is, if a row of quantities be propounded in 
triplicate proportion of arithmetically proportionals 
(or according to the row of cubic numbers), con- 
tinually increasing, and beginning from a point or 
0, as 0, 1, 8, 27, 64, &c., let it be propounded to 


Marks of Dr. inquire, what proportion that row hath to a row of 

Walhs's absurd n ' * i 

Geometry, &c. as many equals to the greatest. 

Be it sought by way of induction, as in proposi- 
tion 1 and 19. 

The proposition arising is everywhere greater 
than subquadruple, or ^, and the excess perpe- 
tually decreaseth as the number of terms increaseth, 
as 7 I n re o & c - The denominator of the frac- 
tion, or consequent of the proportion, being in 
every place augmented by the number 4, as is 
manifest, so that the excess of the arising propor- 
tion above subquadruple is the same with that 
which an unit hath to the quadruple of the num- 
ber of the terms after 0, and so. Here are just the 
same faults which are in proposition 19. 

For, if -f be a fraction, and {- be a fraction, 
and j- be another fraction, then this equation 
T+~n^ * s ^ se - F r this fraction ~ is equal to ; 
and, therefore, we have ~=~, that is, the whole 
equal to half. But perhaps you do not mean them 
fractions, but proportions ; and, consequently, that 
the proportion of to 1, and of 1 to 1, com- 
pounded by addition (I say by addition, not that I, 
but that you think there is a composition of pro- 
portions by multiplication, which I shall show you 
anon is false), must be equal to the proportion of 
1 to 2, which cannot be. For the proportion of 
to 1 is infinitely little, that is, none at all ; and, 
consequently, the proportion of 1 to 1 is equal 
to the proportion of 1 to 2, which is again absurd. 
There is no doubt but the whole number of + 1 
is equal to 1, and thp whole number of 1 + 1 equal 
to 2. But, reckoning them as you do, not for 


whole numbers, but for fractions or proportions, Marks of Dr. 

. . Walhs's absurd 

the equations are false. Geometry, & c . 

Again, your second equation, -f^T + -j"* though 
meant of fractions, that is, of quotients, it be true^ 
and serve nothing to your purpose, yet, if it be 
meant of proportions, it is false. For the propor- 
tion of 1 to 4, and of 1 to 4 being compounded, 
are equal to the proportion of 1 to 16, and so you 
make the proportion of 2 to 4 equal to the propor- 
tion of 1 to 16, where, as it is but subquadupli- 
cate, as you call it, or the quarter of it, as I call it. 
And, in the same manner, you may demonstrate 
to yourself the same fault in all the other rows of 
how many terms soever they consist. Therefore, 
you may give for lost this thirty-ninth proposition, 
as well as all the other thirty-eight that went 
before. As for the conclusion of it, which is, that 
the excess of the arising proportion, &c. They 
are the words of your fortieth proposition, where 
you express yourself better, and make your error 
more easy to be detected. 

The proposition is this : 

" Si proponatur series quantitatum in triplicata 
ratione arithmetice proportionalium (sive juxta 
seriem numerorum cubicorum) continue crescentium 
a puncto vel inchoatarum, ratio quam habet ilia 
ad seriem totidem maxima cequalium subquadru- 
plam super obit ; eritque excessus ea ratio quam 
habet unitas ad quadruplum numeri terminorum 
post 0; sive quam habet radix cubica termini 
primi post ad quadruplum radicis cubicce ter- 
mini maximi. Patet ex pr&cedente. 

" Quum autem crescente numero terminorum ex- 
cessus ille supra rationem subquadruplam ita 


contmuo minuatur, ut tandem quolibet assignabili 
Geometry, &c. minor evdddt, ut patet, si in infinitum procedatur, 
prorsus evaniturus est, adeoque. 

u Patet ex propositione pracedente. 

That is, if a row of quantities be propounded in 
triplicate proportion of arithmetically proportionals 
(or according to the row of cubic numbers), con- 
tinually increasing, and beginning at a point or ; 
the proportion which that row hath to a row of 
as many equals to the greatest, is greater than 
subquadruple proportion ; and the excess is that 
proportion which one unit hath to the quadruple 
of the number of terms after ; or, which the 
cubic root of the first term after hath to the 
quadruple of the root of the greatest term. 

It is manifest by the precedent propositions. 

And, seeing the number of terms increasing, that 
excess above quadruple proportion doth so con- 
tinually decrease, as that, at length, it becomes 
less than any proportion that can be assigned, 
as is manifest, if the proceeding be infinite, it 
shall quite vanish. And so 

This conclusion was annexed to the end of your 
thirty-ninth proposition, as there proved. What 
cause you had to make a new proposition of it, 
without other proof than patet ex prcecedente, I 
cannot imagine. But, howsoever, the proposition 
is false. 

For example, set forth any of your rows, as this 
of fewer terms : 

+ 1 + 8+ 27 =36 

27 + 27 + 27 + 27 = JOS 

The row above is 36, the fourth part of the 
row below is 27. The quadruple of the number 
of terms after is 12. Then, by your account, 


the proportion of 36 to 108 is greater than sub- Marks of Dr. 

, . . Wallis's absurd 

quadruple proportion by the proportion of 1 to Geometry, & c . 
12. For trial whereof, set in order these three 
quantities, 36, 27, 108. The proportion of 36 
(the uppermost row) to 108 (the lowermost row) 
is compounded by addition of the proportions 
36 to 27, and 27 to 108. And the proportion 
of 36 to 108, exceedeth the proportion of 27 
to 108, by the proportion of 36 to 27- But the 
proportion of 27 to 108 is subquadruple propor- 
tion. Therefore, the proportion of 36 to 108 
exceedeth subquadruple proportion, by the pro- 
portion of 36 to 27. And, by your account, by 
the proportion of 1 to 12 ; and, consequently, as 
36 to 27, so is 1 to 12. Did you think such de- 
monstrations as these should always pass ? 

Then, for your inference from the decrease^ of 
the proportions of the excess, to the vanishing of 
the excess itself, I have already sho\ved it to be 
false ; and by consequence that your next pro- 
position, namely, the fortieth, is also false. 

The proposition is this : 

" Si proponatur series infinita quantitatnm in 
triplicata ratione arithmetice proportionalium 
(sive juxta seriem numerorum cubicorum}, con- 
tinue crescentium a puncto sive inchoatarum, 
erit ilia ad seriem totidem maxima tequalium, ut 
1 ad 4, patet ex prcecedente" 

That is, if there be propounded an infinite row 
of quantities in triplicate proportion of arith- 
metically proportionals (or according to the row 
of cubic numbers), continually increasing, and 
beginning at a point or ; it shall be to the row 
of as many equals to the greatest as 1 to 4. Mani- 
fest out of the precedent proposition. 


Marks of Dr. Even as manifest as that 36. 27, 1, 12, are pro- 

Wallis's absurd . ~. i /. i / 

Geometry, &c. portionals. Seeing, therefore, your doctrine of 
" ' the spiral lines and the spaces is given by yourself 
for lost, and a vain attempt, your first forty-one 
propositions are undemonstrated, and the grounds 
of your demonstrations all false. The cause 
whereof is partly your taking quotient for propor- 
tion, and a point for 0, as you do in the first, sixteenth, 
and fortieth propositions, and in other places whers 
you say, beginning at a point or 0, though now you 
deny you ever said either. There be very many 
places in your Elenchus, where you say both ; and 
have no excuse for it, but that, in one of the 
places, you say the proportion is penes quotientem, 
which is to the same or no sense. 

Your forty-second proposition is grounded on 
the fortieth ; and therefore, though true, and 
demonstrated by others, is not demonstrated by 

Your forty-third is this : 

" Pari methodo invenietur ratio seriei infinite? 
quantitatum arithmetice proportionalium in ra- 
tione quadruplicata, quintuplicata, sextuplicata, 
etc., arithmetice proportionalium a puncto sen 
inchoatarum, ad seriem totidem maxima cequa- 
lium. Nempe in quadruplicata erit, ut \ ad 5 ; 
in quintuplicata, ut 1 ad 6 ; in sextuplicata, ut 1 
ad 7. Et sic deinceps" 

That is, by the same method will be found, the 
proportion of an infinite row of arithmetically pro- 
portionals, in proportion quadruplicate, quintupli- 
cate, sextuplicate, &c., of arithmetically propor- 
tionals, beginning at ^ point or 0, to the row of as 
many equals to the greatest ; namely, in quadra- 


plicate, it shall be as 1 to 5 ; in quintuplicate, as 1 J a * 8 l of . 

r ; i /? i Walhs s absurd 

to 6 ; m sextuplicate, as 1 to 7 ; and so forth. Geometry, &c. 

But by the same method that I have demon- 
strated, that the propositions 19, 20, 21, 39,40, 
and 41, are false : any man else, that will examine 
the forty-third may find it false also. And, -be- 
cause all the rest of the propositions of your 
Arithmetica Infinitorum depend on these, they 
.may safely conclude, that there is nothing demon- 
strated in all that book, though it consist of 194 
propositions. The proportions of your parabolo- 
cides to their parallelograms are true, but the 
demonstrations false, and infer the contrary. Nor 
were they ever demonstrated (at least the demon- 
strations are not extant) but by me ; nor can they 
be demonstrated, but upon the same grounds, 
concerning the nature of proportion, which I have 
clearly laid, and you not understood. For, if you 
had, you could never have fallen into so gross an 
error as is this your book of Arithmetica Infinito- 
rum, or that of the angle of contact. You may see 
by this, that your symbolic method is not only not 
at all inventive of new theorems, but also dangerous 
in expressing the old. If the best masters of sym- 
bolics think for all this you are in the right, let 
them declare it. I know how far the analysis by 
the powers of the lines extendeth, as well as the 
best of your half-learnt epistlers, that approve so 
easily of such analogisms as those, 5, 4, 1, 12, and 
36, 27, 1, 12, &c. 

It is well for you that they who have the dis- 
posing of the professors' places take not upon them 
to be judges of geometry, for, if they did, seeing 
you confess you have read these doctrines in your 


Marks of Dr. school, you had been in danger of being put out of 

W allis's absurd J ^ 

Geometry, fcc. yOUF 

When the author of the paper wherein I am 
called Plagiary, and wherein the honour is taken 
from you of being the first inventor of these fine 
theorems, shall read this that I have here written, 
he will look to get no credit by it ; especially if it 
be Roberval, which methinks it should not be. For 
he understands what proportion is, better than to 
make 5 to 4 the same with 1 to 12. Or to make, 
again, the proportion of 36 to 27 the same with 
that of 1 to 12; and innumerable disproportion- 
alites that may be inferred from the grounds you 
go on. But if it be Roberval indeed, that snatches 
this invention from you, when he shall see this 
burning coal hanging at it, he w r ill let it fall again, 
for fear of spoiling his reputation. 

But what shall I answer to the authority of the 
three great mathematicians that sent you those 
encomiastic letters. For the first, whom you say I 
use to praise, I shall take better heed hereafter of 
praising any man for his learning whilst he is 
young, further than that he is in a good way. But 
it seems he was in too ready a way of thinking very 
well of himself, as you do of yourself. For the 
muddiness of my brain I must confess it ; but, Sir, 
ought not you to confess the same of yours ? No, 
men of your tenets use not to do so. He wonders, 
say you, you thought it worth the while to foul 
your fingers about such a piece. It is well ; every 
man abounds in his own sense. If you and I were 
to be compared by the compliments that are given 
us in private letters, both you and your compli- 
mentors would be out of countenance ; which com- 


pliments, besides that which has been printed and Marks of Dr - 

,,.,,., . . . . Wallis's absurd 

published in the commendations of my writings, if Geometry, &c. 
it were put together, would make a gre'ater volume ' * 
than either of your libels. And truly, Sir, I had 
never answered your Elenchus as proceeding from 
Dr. Wallis, if I had not considered you also as the 
minister to execute the malice of that sort of peo- 
ple that are offended with my Leviathan. 

As for the judgment of that public Professor 
that makes himself a witness of the goodness of 
your geometry, a man may easily see by the letter 
itself that he is a dunce. And for the English 
person of quality whom I know not, I can say no 
more yet than I can say of all three, that he is so 
ill a geometrician, as not to detect those gross 
paralogisms as infer that 5 to 4 and 1 to 12 are the 
same proportion. He came into the cry of those 
whom your title had deceived. 

And now I shall let you see that the composition 
of proportion by multiplication, as it is in the fifth 
definition of the sixth element, is but another way of 
adding proportions one to another. Let the propor- 
tions be of 2 to 3, and of 4 to 5. Multiply 2 into 4 
and 3 into 5, the proportion arising is of 8 to 15. 
Put in order these three quantities, 8, 12, 15. The 
proportion therefore of 8 to 15, compounded of 
the proportions of 8 to 1 2, (that is, of 2 to 3) and 
of 12 to 15, that is, of 4 to 5 by addition. Again, 
let the proportion be of 2 to 3, and of 4 to 5, 
multiply 2 into 5 and 3 into 4, the proportions 
arising is of 10 to 12. Put in order these three 
numbers, 10, 8, 12. The proportion 10 to 12 is 
compounded of the proportions of 10 to 8, that is 
of 5 to 4, and of 8 to 12*, that is, of 2 to 3 by 
addition. I wonder you know not this. 


rir. f P r ' , I find not any more clamour against me for saying 

Walhs's absurd J i i? /> 

Geometry, &c. the proportion of 1 to 2 is double to that of I to 4. 

Your book, you speak of, concerning proportion 
against Meibomius is like to be very useful when 
neither of you both do understand what proportion 

You take exceptions, as that I say, that Euclid 
has but one word for double and duplicate ; which 
nevertheless was said very truly, and that word is 
sometimes SurXaanog arid sometimes &7rXaonW. And 
you think you have come off handsomely with 
asking me whether &7r\a<noc and Siv\aoiiw be one 

Nor are you now of the mind you were, that a 
point is not quantity unconsidered, but that in an 
infinite series it may be safely neglected. What is 
neglected but unconsidered. 

Nor do you any more stand to it, that the quo- 
tient is the proportion. And yet were these the 
main grounds of your Elenchus. 

But you will say, perhaps, I do answer to the 
defence you have now made in this your School 
Discipline: 'tis true. But 'tis not because you 
answer never a word to my former objections 
against these propositions 19, 89 ; but because you 
do so shift and wriggle, and throw out ink, that I 
cannot perceive which way you go, nor need I, 
especially in your vindication of your Arithmetica 
Infinitorum. Only I must take notice that in the 
end of it, you have these words, " Well, Arith- 
metica Infinitorum is come off clear." You see 
the contrary. For sprawling is no defence. 

It is enough to me that I have clearly demon- 
strated both before sufficiently, and now again 


abundantly, that your book of Arithmetica Infini- Marks of Dr - 

J ' J . ^ Wallis's absurd 

torum is all nought from the beginning to the end, Geometry, & c . 
and that thereby I have effected that your autho- """" 
rity shall never hereafter be taken for a prejudice. 
And, therefore, they that have a desire to know 
the truth in the questions between us, will hence- 
forth, if they be wise, examine my geometry, by 
attentive reading me in my own writings, and then 
examine, whether this writing of yours confute or 
enervate mine. 

There is in my fifth lesson a proposition, with a 
diagram to it, to make good, I dare say, at least 
against you, my twentieth chapter concerning the 
dimension of a circle. If that demonstration be not 
shown to be false, your objections to that chapter, 
though by me rejected, come to nothing. I wonder 
why you pass it over in silence. But you are not, 
you say, bound to answer it. True, nor yet to 
defend what you have written against me. 

Before I give over the examination of your 
geometry, I must tell you that your words, (p. 101 
of your School Discipline), against the first coroll- 
ary are untrue. 

Your words are these : " you affirm that the 
proportion of the parabola A B 1 to the parabola 
AFKis triplicate to the proportion of the time 
A B to A F, as it is in the English." This is not 
so. Let the reader turn to the place and judge. 
And going on you say, " or of the impetus B I to 
F K as it is in the Latin." Nay, as it is in the 
English, and the other in the Latin. It is but 
your mistake ; but a mistake is not easily excused 
in a false accusation. 

Your exception to my spying, "that the dif- 


f erences f two quantities is their proportion" 
Geometry, &c. (when they differ, as the no difference, when they 
be equal), might have been put in amongst other 
marks of your not sufficiently understanding the 
Latin tongue. Differre and differentia differ no 
more than vivere and vita, which is nothing at all, 
but as the other words require that go with them, 
which other words you do not much use to con- 
sider. But differre and the quantity by which 
they differ, are quite of another kind. Differre 
(TO dtatyipEiv, TO vvtpiytir} differing, exceeding, is not 
quantity, but relation. But the quantity by which 
they differ is always a certain and determined 
quantity, yet the word differentia serves for both, 
and is to be understood by the coherence with 
that which went before. But I had said before, 
and expressly to prevent cavil, that relation is 
nothing but a comparison, and that proportion is 
nothing but relation of quantities, and so defined 
them, and therefore I did there use the word dif- 
ferentia for differing, and not for the quantity 
which was left by subtraction. For a quantity is not 
a differing. This I thought the intelligent reader 
would of himself understand without putting me, 
instead of differentia, to use (as some do, and I 
shall never do) the mongrel word ro differre. And 
whereas in one only place for differre ternario 
I have writ ternarius, if you had understood what 
was clearly expressed before, you might have been 
sure it was not my meaning, and therefore the 
excepting against it was either want of understand- 
ing, or want of candour, choose which you will. 

You do not yet clear your doctrine of conden- 
sation and rarefaction. But I believe you will by 


degrees become satisfied that they who say the Marks of Dr 

. Walhs's absurd 

same numerical body may be sometimes greater, Geometry, &e. 
sometimes less, speak absurdly, and that conden- 
sation and rarefaction here, and definitive and 
circumscriptive, and some other of your distinc- 
tions elsewhere are but snares, such as school 
divines have invented 

- OHTTTtp Upa^VYje 

'Oi/Xo/ufi'oc' \%ci a\v<rti p.viaiQ 

co entangle shallow wits. 

And that that distinction which you bring here, 
"that it is of the same quantity while it is in the 
same place, but it may be of a different quantity 
when it goes out of its place" (as if the place added 
to, or took any quantity from the body placed), is 
nothing but mere words. It is true that the body 
which swells changeth place, but it is not by be- 
coming itself a greater body, but by admixtion of 
air or other body, as when water riseth up in 
boiling, it taketh in some parts of air. But seeing 
the first place of the body is to the body equal, 
and the second place equal to the same body, the 
places must also be equal to one another, and con- 
sequently the dimensions of the body remain equal 
in both places. 

Sir, when I said that such doctrine was taught 
in the Universities, I did not speak against the 
Universities, but against such as you. I have done 
w r ith your geometry, which is one 


As for your eloquence, let the reader judge 
whether yours or mine be the more muddy, though 
I in plain scolding should have outdone you, vet 



Marks of Dr. j j iave this excuse which you have not, that I did 

Wallis's rural -.111 . 

language, & c . but answer your challenge at that weapon which 
"" ' ~" you thought fit to choose. The catalogue of the 
hard language which you put in at pages 3 and 4 
of your School Discipline, I acknowledge to be 
mine, and would have been content you had put 
in all. The titles you say I give you of fools, 
beasts, and asses, I do not give you, but drive 
back upon you, which is no more than not to 
own them ; for the rest of the catalogue, I like 
it so well as you could not have pleased me better 
than by setting those passages together to make 
them more conspicuous ; that is all the defence I 
will make to your accusations of that kind. 

And now I would have you to consider whether 
you will make the like defence against the faults 
that I shall find in the language of your School 

I observe, first, the facetiousness of your title- 
page, "Due correction for Mr. Hobbes, or School 
Discipline, for not saying his Lessons right"' 
What a quibble is this upon the word lesson ; 
besides, you know it has taken wind; for you 
vented it amongst your acquaintance at Oxford 
then when my Lessons were but upon the press. 
Do you think if you had pretermitted that piece of 
wit, the opinion of your judgment would have been 
ere the less ? But you were not content with this, 
but must make this metaphor from the rod to take 
up a considerable part of your book, in which 
there is scarce anything that yourself can think 
wittily said besides it. Consider also these words 
of yours : " It is to be hoped that in time you 
may come to learn ' the language, for you be 
come to great A already." And presently after, 


" were I great A, before I would be willing to be ftf 1 of nr - 

7/77-7 7 77 WalllSMUial 

so used) I should wish myself little a a hundred lan^ua^, &<. 
times" Sir, you are a doctor of divinity and a 
professor of geometry, but do not deceive yourself, 
this does not pass for wit in these parts, no, nor 
generally at Oxford; I have acquaintance there 
that will blush at the reading it. 

Again, in another place you have these words : 
" Then you catechize us, ' what is your name ? 
Are you geometricians ? Who gave you that 
name] " &c. Besides in other places such abund- 
ance of the like insipid conceits, as would make 
men think, if they were no otherwise acquainted 
with the University but by reading your books, 
that the dearth there of salt were very great. If 
you have any passage more like to salt than these 
are (excepting now and anon} you may do well to 
show it to your acquaintance, lest they despise 
you ; for, since the detection of your geometry, 
you have nothing left you else to defend you from 
contempt. But I pass over this kind of eloquence, 
and come to somewhat yet more rural. 

Page 27) line 1, you say I have given Euclid his 
lurry. And again, page 129, line 1 1 , " 0JM/ now 
he is left to learn his lurry.' 9 I understand not 
the word lurry. I never read it before, nor heard 
it, as I remember, but once, and that was when a 
clown threatening another clown said he would 
give him such a lurry come poop, &c. Such words 
as these do not become a learned mouth, much less 
are fit to be registered in the public writings of a 
doctor of divinity. In another place you have 
these words, "just the same to a cow's thumb" a 
pretty adage. 

C C 2 


MaikH of n r p a g e 2 But prithee tell me:" And again, 

Wulhss rural o ' jf <-> 

lautfimj-e, &c. page 95, " prithee tell me, why dost moil ask me 
such a question" and the like in many other places. 
You cannot but know how easy it is and was 
for me to have spoken to you in the same language. 
Why did I not ? Because I thought that amongst 
men that were civilly bred it would have re- 
dounded to my shame, as you have cause to fear 
that this will redound to yours. But what moved 
you to speak in that manner ? Were you angry ? 
If I thought that the cause, I could pardon it the 
sooner, but it must be very great anger that can 
put a man, that professeth to teach good manners, 
so much out of his wits as to fall into such a lan- 
guage as this of yours. It was perhaps an imagi- 
nation that you were talking to your inferior, 
which I will not grant you, nor will the heralds, 
I believe, trouble themselves to decide the question. 
But, howsoever, I do not find that civil men use 
to speak so to their inferiors. If you grant my 
learning but to be equal to yours, (which you may 
certainly do without very much disparaging of 
yourself abroad in the world), you may think it 
less insolence in me to speak so to you in respect 
of my age, than for you to speak so to me in 
respect of your young doctorship. You will find 
that for all your doctorship, your elders, if other- 
wise of as good repute as you, will be respected 
before you. But I am not sure that this language 
of yours proceeded from that cause ; I am rather 
inclined to think you have not been enough in good 
company, and that there is still somewhat left in 
your manners for which the honest youths of 
Hedington and Hincsey may compare with you 
for good language, as great a doctor as you are. 


For my verses of the Peak, though they be as Marks f Dr - 

... . J . . T i i- i Walhs's rural 

ill in my opinion as I believe they are in yours, langiume, &c. 

and made long since, yet they are not so obscene 

as that they ought to be blamed by Dr. Wallis. I 

pray you, sir, whereas you have these words in your 

School Discipline, page 96, " unless you will say 

that one and the same motion may be now and anon 

too" What was the reason you put these words, 

now and anon too, in a different character, that 

makes them to be more taken notice of? Do you 

think that the story of the minister that uttered ' 

his affection (if it be not a slander) not unlawfully 

but unseasonably, is not known to others as well 

as to you ? What needed you then, when there 

was nothing that 1 had said could give the occa- 

sion, to use those words ; there is nothing in my 

verses that do olere hire urn so much as this of 

yours. I know what good you can receive by ru- 

minating on such ideas, or cherishing of such 

thoughts. But 1 go on to other w r ords of mine by 

you reproached, " you may as well seek the focus 

of the parabola of Dives and Lazarus" which you 

say is mocking the Scripture ; to which I answer 

only, that I intended not to mock the Scripture, 

but you, and that which was not meant for mock- 

ing was none. And thus you have a second 


I come now to the comparison of our Grammar 
and Critiques. You object first against the signifi- 
cation I give of <riy^i?, and say thus : " What should 
come into your cap (that, if you mark it, in a man 
that wears a square cap to ,one that wears a hat, 
is very witty) to make you think that s-iy^ signi- 


f a !* s Ol Dr fies a mark or brand with a hot iron ? I perceive 

Valhss gram- ... . 

iar & critiques, where the busmess lies^ it was Wy//a run in your 
mind when you tallied of y/) ; and because the 
words are somewhat alike you jumble them both 
together" Sir,, I told you once before, you pre- 
sume too much upon your first cogitations. Aris- 
tophanes, in Ranis., Act. v. Seen. 5, 

KoV fJtJ T(lJ((i)G tfkbHTt 


The old commentator upon the word ^frc saith 

thuS, <r/ae UVTI rov -ny/ucmo-ac, r\v yap ei/oe- That IS, t'ffyt; 

for <7typart<rac 9 for he (Adimantus) was not a citizen. 
I hope the commentator does not here mock Aris- 
tophanes for jumbling dfcc and ^ypmVac together, 
for want of understanding Greek. No, <r/'ae and 
Tiy/zanW signify the same, save that for branding 
I seldom read siy^arfW but ^ifa. For Wy,ua does 
no more signify a brand with a hot iron, than 
ityfif) a point made also with a hot iron. They 
have both one common theme dw, which does not 
signify pungo, nor interjtungo, nor inuro, for all 
your Lexicon, but notam imprimere, or pungendo 
notare, without any restriction to burning or punch- 
ing. It is therefore no less proper to say that *iyp) 
is a mark with a hot ii'on, than to say the same of 
T/VJMO. The difference is only this, that when they 
marked a slave, or a rascal, as you are not igno- 
rant is usually done here at the assizes in the hand 
or shoulder with a hot iron, they called that ^y/^a, 
not for the burning, but for the mark. And as it 
would have been called ^iy^a that was imprinted 
on a slave, though made by staining or incision, 
so it is ^y/i*), though 9 done with a hot iron. And 
therefore there was no jumbling of those two words 
together, as for want of reading Greek authors, and 


by trusting too much to your dictionaries, which Marks o* Dr. 

/ ~i -, / * t Wallis's cram- 

you say are proofs good enough for such a business, mai & mtiques 

you were made to imagine. The use I have made ' ' 

thereof was to show that a point, both by the word 

ZWMV in Euclid, and by the word 7W ?) in some 

others, was not nothing, but a visible mark, the 

ignorance whereof hath thrown you into so many 

paralogisms in geometry. 

, But do you think you can defend your Adducis 

Malleum as well as I have now defended my <nyp) ? 

You have brought, I confess, above a hundred 

places of authors, where there is the word duco, 

or some of its compounds, but none of them will 

justify Adducis Malleum, and, excepting two of 

those places, you yourself seem to condemn them 

all, comparing yours with none of the rest but 

with these two only, both out of Plautus, by you 

not well understood. The first is in Casina, Act. v. 

Seen. 2, " Ubi intro hanc novam nuptam deduxi, 

via recta, clavem alduxi ;" which you, presently 

presuming of your first thoughts, a peculiar fault 

to men of your principles, assure yourself is right. 

But if you look on the place as Scaliger reads it, 

cited by the commentator, you will find it should 

be obduxi, and that clams is there used for the 

bolt of the lock. Besides, he bolted it within. 

Whither then could he -carry away the key ? The 

place is to be rendered thus, when 1 had brought 

in this new bride I presently locked the door, and 

is this as bad every whit as Addttcis Malleum ? 

The second place is in Amphytruo, Act. i. Seen. 1, 

"Earn (cirneam), ut a matrefuerat natum,plenam 

vini eduxi meri" which yQU interpret / brought 

out a flagon of wine y unlearnedly. They are 


Marks of Dr. the words of Mercury transformed into Sosia. 

WallisN grain J 

mar & critique* And to try whether Mercury were Sosia or not, 
Sosia asked him where he was and what he did 
during the battle ; to which Mercury answered, 
who knew where Sosia then was and what he did, 
/ was in the cellar, where I filled a cirnea, and 
brought it up full of wine, pure as it came from 
its mother. By the mother of the wine meaning 
the vine, and alluding to the education of children > 
for ebibi said eduxi, and with an emphasis in meri, 
because cirnea (from Ktpra'w, misceo) was a vessel 
wherein they put water to temper to their wine. 
Intimating that though the vessel was cirnea, yet 
the wine was me rum. This is the true sense of the 
place ; but you will have eduxl to be, / brought 
out, though he came not out himself. You see, 
sir, that neither this is so bad as Addncis Mall cum. 
But suppose out of some one place in some one 
blind author you had paralleled your Adducls Mai- 
leum, do you think it must therefore presently be 
held for good Latin ? Why more than learn his 
lurry must be therefore thought good English a 
thousand years hence, because it will be read in Dr. 
Wallis's long-lived works. But how do you construe 
this passage (1 Tim. ii. 15) of the Greek Testament : 

ILwdijfferai ce dia rfjg rkvoyoj>/'ae> lav fieivittffiv iv irirtt? 1 OU COn- 

stme it thus : she shall be saved notwithstanding 
child-bearing, if (the woman) remain in the faith. 
Is child-bearing any obstacle to the salvation of 
women ? You might as well have translated the 
first verse of the fifth of Romans in this manner, 
Being then justified by faith, we have peace with 
God notwithstanding ^our Lord Jesus Christ. I 
let pass your not finding in rc^oyo^ac, as good a 


grammarian as you are. a nominative case to MaiksofDr - 

J ' ^'alhs's grain- 

yue/Vw(7(i/. If you had remembered the place, 1 Pet. mar & cntiqn. 
iii. 20, tauQi]<rav II w&iroc, that is, they were saved in 
the waters, you would have thought your con- 
struction justified then very well ; but you had been 
deceived, for 8m does not there signify causam, 
ablationem impediment'^ but transitnm ; not cause 
or removing an impediment, but passage. Being 
come thus far, I found a friend that hath eased 
me of this dispute ; for he showed me a letter 
written to himself from a learned man, that hath 
out of very good authors collected enough to decide 
all the grammatical questions between you arid 
me, both Greek and Latin. He would not let me 
know his name, nor anything of him but only this, 
that he had better ornaments than to be willing 
to go clad abroad in the habit of a grammarian. 
But he gave me leave to make use of so much of the 
letter as I thought fit in this dispute, which I have 
done, and have added it to the end of this writing. 
But before I come to that, you must not take it 
ill, though I have done with your School Disci- 
pline, if I examine a little some other of your 
printed writings as you have examined mine ; for 
neither you in geometry, nor such as you in church 
politics, cannot expect to publish any unwholesome 
doctrine without some antidotes from me, as long 
as I can hold a pen. Bat why did you answer 
nothing to my sixth Lesson ? Because, you say, 
it concerned your colleague only. No, sir, it con- 
cerned you also, arid chiefly, for I have not heard 
that your colleague holdeth those dangerous prin- 
ciples which I take notice of in you, in my sixth 
Lesson, page 350, upon the occasion of these 


Marks of Dr. words, not his but yours : " Perhaps you take the 

Walhs s gram. . J J. */ 

mar & critiques, whole history, of Hie fall of Adam for a fable, 
which is no wonder, seeing you say the rules of 
honouring and worshipping of God are to he 
taken from the laws." In answer to which I said 
thus : " You that take so heinously, that I would 
have the rule of God's worship in a Christian 
commonwealth to be taken from the laws, tell me 
from whom you would have them taken ? From 
yourself? Why so, more than from me ? From 
the bishops ? Right, if the supreme power of the 
commonwealth will have it so ; if not, why from 
them rather than from me? From a consistory of 
presbyters themselves, or joined with lay elders, 
whom they may sway as they please ? Good, if 
the supreme governor of the commonwealth will 
have it so. If not f why from them rather than 
from me, or from any man else ? They are wiser 
and learneder than I ; it may be so, but it has 
not yet appeared. Howsoever, let that be granted. 
Is there any man so very a fool as to subject him- 
self to the rules of other men in those things which 
do so nearly concern himself, for the title they 
assume of being wise and learned, unless they 
also have the sword which must protect them ? 
But it seems you understand the sword as com- 
prehended. If so, do not you then receive the 
rules of God's worship from the civil power ? 
Yes, doubtless ; and you would expect, if your 
consistory had that sword, that no man should 
dare to exercise or teach any rules concerning 
God's worship which were not by you allowed." 

This will be thought strong arguing, if you do 
not answer it. But the truth is, you could say 


nothing against it without too plainly discovering Muiw of Dr. 

l- ir xi 4. A 1 ^ Wallis's gram- 

your disaffection to the government. And yet mar & critique* 
you have discovered it pretty well in your second ^~ ' 
Thesis, maintained in the Act at Oxford, 1654, 
and since by yourself published. This Thesis I 
shall speak briefly to. 


You define ministers of the Gospel to be those 
to whom the preaching of the Gospel by their 
office is enjoined by Christ. Pray you, first, what do 
you mean by saying preaching ex officio is enjoined 
by Christ? Are they preachers ex officio, and 
afterwards enjoined to preach ? Ex officio adds 
nothing to the definition ; but a man may easily 
see your purpose to disjoin yourself from the state 
by inserting it. 

Secondly, I desire to know in what manner you 
will be able out of this definition to prove yourself 
a minister ? Did Christ himself immediately en- 
join you to preach, or give you orders? No. 
Who then, some bishop, or minister, or ministers ? 
Yes ; by what authority ? Are you sure they had 
authority immediately from Christ ? No. How 
then are you sure but that they might have none ? 
At least, some of them through whom your autho- 
rity is derived might have none. And therefore 
if you run back for your authority towards the 
Apostles' times but a matter of sixscore years, you 
will find your authority derived from the Pope, 
which words have a sound very unlike to the voice 
of the laws of England. And yet the Pope will 
not own you. There is no man doubts but that 


Marks of Dr. you hold that your office comes to you by succes- 

Wallw's Scotch . . 4? i 14? 4.1 4.' * I. 

church politics, sive imposition of hands from the time of the 
' Apostles ; which opinion in those gentle terms 
passeth well enough ; but to say you derive your 
authority from thence, not through the authority 
of the sovereign power civil, is too rude to be en- 
dured in a state that would live in peace. In a 
word, you can never prove you are a minister, but 
by the supreme authority of the commonwealth. 
Why then do you not put some such clause into 
your definition ? As thus, ministers of the Gospel 
are those to whom the preaching of the Gospel is 
enjoined by the sovereign power in the name of 
Christ. What harm is there in this definition, 
saving only it crosses the ambition of many men 
that hold your principles ? Then you define the 
power of a minister thus : u The power of a minister 
is that which belongeth to a minister of the Gospel 
in virtue of the office he holds, inasmuch as he 
holds a public station, and is distinguished from 
private Christians. Such as is the power of 
preaching the Gospel, administering the sacra- 
ment, the use of ecclesiastical censures, and or- 
daining of ministers" Sfc. 

Again, how will you prove out of this definition 
that you, or any man else, hath the power of a 
minister, if it be not given him by him that is the 
sovereign of the commonwealth f For seeing, as 
I have now proved, it is from him that you must 
derive your ministry, you can have no other 
power than that which is limited in your orders, 
nor that neither longer than he thinks fit. For if 
he give it you for the^instruction of his subjects in 
their duty, he may take it from you again whenso- 
ever he shall see you instruct them with undutiful 


and seditious principles. And if the sovereign MM** of DT. 

* x Walhs s Scotch 

power give me command, though without the church politics 
ceremony of imposition of hands, to tea'ch the doc- 
trine of my Leviathan in the pulpit, why am not I, 
if my doctrine and life be as good as yours, a 
minister as w r ell as you, and as public a person as 
you are ? For public person^ primarily, is none 
but the civil sovereign, and so secondarily, all that 
are employed in the execution of any part of the 
public charge. For all are his ministers, and 
therefore also Christ's ministers because he is so ; 
and other ministers are but his vicars, and ought 
not to do or say anything to his people contrary 
to the intention of the sovereign in giving them 
their commission. 

Again, if you have in your commission a power 
to excommunicate, how can you think that your 
sovereign who gave you that commission, intended 
it for a commission to excommunicate himself? 
that is, as long as he stand excommunicate, to 
deprive him of his kingdom. If all subjects were 
of your mind, as I hope they will never be, they 
will have a very unquiet life. And yet this has, 
as I have often heard, been practised in Scotland, 
when ministers holding your principles had power 
enough, though no right, to do it. 

And for administration of the sacraments, if by 
the supreme power of the commonwealth it were 
committed to such of the laity as know r how it 
ought to be done as well as you, they would ipso 
facto be ministers as good as you. Likewise the 
right of ordination of ministers depends not now 
on the imposition of hands of a minister or pres- 
bytery, but on the authority of the Christian 
sovereign, Christ's immediate vicar and supreme 


g vernor f & H persons and judge of all causes, 
church politics, both spiritual and temporal, in his own dominions, 
which I believe you will not deny. 

This being evident, what acts are those of yours 
which you call authoritative, and receive not from 
the authority of the civil power ? A constable 
does the acts of a constable authoritatively in that 
sense. Therefore you can no otherwise claim your 
power than a constable claimeth his, who does not 
exercise his office in the constabulary of another. 
But you forget that the Scribes and the Pharisees 
sit no more in Moses' chair. 

You would have every minister to be a minis- 
ter of the universal Church, and that it be lawful 
for you to preach your doctrine at Rome ; if you 
would be pleased to try, you would find the con- 
trary. You bring no argument for it that looks 
like reason. Examples prove nothing, where per- 
sons, times, and other circumstances differ ; as 
they differ very much now when kings are Chris- 
tians, from what they were then when kings per- 
secuted Christians. It is easy to perceive what 
you aim at. 

You would fain have market- day lectures set up 
by authority, (not by the authority of the civil 
power, but by the authority of example of the 
Apostles in the emission of preachers to the in- 
fidels), not knowing that any Christian may law- 
fully preach to the infidels ; that is to say, proclaim 
unto them that Jesus is the Messiah, without need 
of being otherways made a minister, as the dea- 
cons did in the Apostles' time ; nor that many 
teachers, unless they can agree better, do anything 
else but prepare men for faction, nay, rather you 
know it well enough, but it conduces to your end 


upon the market-days to dispose at once both Marlos fl)r - 

1 J r Walhs's Scotch 

town and country, under a false pretence of obe- 
dience to God, to a neglecting of the* command- 
ments of the civil sovereign, and make the subject 
to be wholly ruled by yourselves, wherein you 
have already found yourselves deceived. You 
know how to trouble and sometimes undo a slack 
government, and had need to be warily looked to, 
but are not fit to hold the reins. And how should 
you, being men of so little judgment as not to see 
the necessity of unity in the governor, arid of abso- 
lute obedience in the governed, as is manifest out 
of the place of your Elenchus above recited. The 
doctrine of the duty of private men in a com- 
monwealth is much more difficult, not only than 
the knowledge of your symbols, but also than the 
knowledge of geometry itself. How then do you 
think, when you err so grossly in a few equations, 
and in the use of most common words, you should 
be fit to govern so great nations as England, Ire- 
land, and Scotland, or so much as to teach them f 
For it is not reading but judgment that enables 
one man to teach another. 

I have one thing more to add, and that is the 
disaffection I am charged withal to the universi- 
ties. Concerning the Universities of Oxford and 
Cambridge, I ever held them for the greatest and 
noblest means of advancing learning of all kinds, 
where they should be therein employed, as being 
furnished with large endowments and other helps 
of study, and frequented with abundance of young 
gentlemen of good families and good breeding 
from their childhood. On the other side, in case 
the same means and the same wits should be em- 
ployed in the advancing of the doctrines that tend 


Marks of or. t o the weakening of the public, and strengthening 

Wiillis s Scotch oi' <-> ^ 

church politics, of the power of any private ambitious party, they 
would also be very effectual for that ; and conse- 
quently that if any doctrine tending to the dimi- 
nishing of the civil power were taught there, not 
that the Universities were to blame, but only those 
men that in the universities, either in lectures, 
sermons, printed books, or theses, did teach such 
doctrine to their hearers or readers. Now you 
know very well that in the time of the Roman reli- 
gion, the power of the Pope in England was upheld 
principally by such teachers in the universities. 
You know also how much the divines that held the 
same principles in Church government with you, 
have contributed to our late troubles. Can I 
therefore be justly taxed with disaffection to the 
universities for wishing this to be reformed ? And 
it hath pleased God of late to reform it in a great 
measure, and indeed as I thought totally, when out 
comes this your Thesis boldly maintained to show 
the contrary. Nor can I yet call this your doc- 
trine the doctrine of the university ; but surely it 
will not be unreasonable to think so, if by public 
act of the university it be not disavowed, which 
done, and that as often as there shall be need, 
there can be no longer doubt but that the univer- 
sities of England are not only the noblest of all 
Christian universities, but also absolutely, and of 
the greatest benefit to this commonwealth that can 
be imagined, except that benefit of the head itself 
that uniteth arid ruleth all I have not here par- 
ticularized at length all the ill consequences that 
may be deduced from this Thesis of yours, because 
I may, when further provoked, have somewhat to 
say that is new. So much for the third 





MR. HOBBES hath these words: " Longltudinem A <^ract 
percursam motu uniformi, cum impetu ulrique ipsi^^ 
BD cequalir Dr. Wallis saith cum were better ^^ 
out, unless you would have impetus to be only a ' 
companion, not a cause. Mr. Hobbes answered 
it was the ablative case of the manner. The truth 
is the ablative case of the manner and cause both, 
may be used with the conjunction cum, as may be 
justified. Cicero in Lib. n. De Nat. Deorum: " Mo- 
liri aliquid cum labor e operoso ac molesto ;" and 
in his oration for Chechia : " De se autem hoc prce- 
dicat, Antiocho Ebulii servo imperasse utin Cte- 
cinam advenientem cum ferra invader et" Let us 
see then what Dr. Wallis objects against Tully, 
where a casualty is imported, though we may use 
with in English, yet not cum in Latin ; to kill with 
a sword, importing this to have an instrumental 
or causal influence, and not only that it hangs by 
the man's side whilst some other weapon is made 
use of, is not in Latin occidere cum gladio, but 
gladio occidere. This show r s that the Doctor hath 
not forgot his grammar, for the subsequent ex- 
amples as well as this rule are borrowed thence. 
But yet he might have known that great per- 
sonages have never confined themselves to this 
pedantry, but have chosen to walk in a greater la- 
titude. Most of the elegancies and idioms of every 
language are exceptions to his grammar. But 
since Mr. Hobbes saith it is the ablative case of 



An extract of the manner, there is no doubt it may be expressed 

a letter concern- . . , 

ing the gramma- with cum. The Doctor in the meantime knew no 
controversy^ &c! more than what Lilly had taught him ; Alvarez would 
' ' ' have taught him more ; and Vossius in his book, 
De Construction, cap. XLVII. expressly teacheth, 
" Ablativos causa, instrument, vel modi, non a 
verbo regi $ed a prapositione omissa, a vel ab, de, 
e vel ex,prce, aut cum, ac prcepositiones eas quan- 
doque exprimi nisi quod cum ablativis instrumenti 
hand temere invenias ;" and afterwards he saith, 
" non timer e imitandum" If this be so, then did 
Mr. Hobbes speak grammatically, and with Tully, 
but not usually. And might not one retort upon 
the Doctor, that Vossius is as great a critic as he ? 
His next reflection is upon prcetendit scire, this 
he saith is an Anglicism. If this be all his accu- 
sation, upon this score we shall lose many expres- 
sions that are used by the best authors, which I 
take to be good Latinisms, though they be also 
Anglicisms, the latter being but an imitation of 
the former. The Doctor therefore w r as too fierce 
to condemn upon so general an account, that which 
was not to have been censured for being an Angli- 
cism, unless also it had been no Latinism. Mr. 
Hobbes replies, that the printer had omitted se. 
He saith, this mends the matter a little. It is 
very likely, for then it is just such another An- 
glicism as that of Gluintilian : " Cum loricatus in 
foro ambularet, prtetendebat se id metu facere" 
The Doctor certainly was very negligent, or else 
he could not have missed this in Robert Stephen. 
Or haply he was resolved to condemn Quintilian 
for this and that other Anglicism, " Ignorantia 
prcetendi non potest" as all those that have used 


prcetendo, which are many and as good authors as An extract of 

T\ ITT IT i -i i / a letter concern- 

Dr. Wallis, that makes his own encomiasts (not an ing the gramma- 

Englishman amongst them) to write Anglicisms, 

Then he blames "Tractatus kujus partis tertice, 
in qua motus et magnitude per se et abstracte 
consider ammus, terminum hie statuo" Here I 
must confess the exception is colourable, yet I can 
parallel it with the like objection made by Erasmus 
against Tully, out of whom Erasmus quotes this 
passage : " Diutius commorans Athenis, quoniam 
venti negabant solvendi facultatem> erat animus 
ad te scribere ;" and excuses it thus, that Tally 
might have had at first in his thoughts volebam or 
statuebam, which he afterwards relinquished for 
erat animus, and did not remember what he had 
antecedently written, which did not vary from his 
succeeding thoughts, but words. And this excuse 
may pass with any who knoW that Mr. Hobbes 
values not the study of words, but as it serves to 
express his thoughts, which were the same whether 
he wrote in qua motus et magnitudo per se at ab- 
stracte considerati sunt or consider avimus. And 
if the Doctor will make this so capital, he must 
prove it voluntary, and show that it is greater than 
what is legible in the puny letter of his encomiast, 
whom he would have to be beyond exception. 

Now follows his ridiculous apology for adducis 
malleum, ut occidas muscam. The cause why he 
did use that proverb, of his own phrasing, was 
this. Mr. Hobbes had taken a great deal of pains 
to demonstrate what Dr. Wallis thought he could 
have proved in short ; upon this occasion he ob- 
jects, adducis malleum ut occidas muscam, which I 
shall suppose he intended to English thus, you bring 

D D 2 

co o 


An extract of a beetle to kill a fit/. Mr. Hobbes retorted, that 

a letter concern- i i rm TA 

mg the gramma- aclduco was not used m that sense. The Doctor 
controversy, & c ! vindicates ' himself thus : duco, deduco, reduco, 
" ' perduco, produco, &c. signify the same thing, ergo, 
adduco may be used in that sense ; which is a 
most ridiculous kind of arguing, where we are but 
to take up our language from others, and not to 
coin new phrases. It is not the grammar that 
shall secure the Doctor,, nor weak analogies, 
where elegance comes in contest. To justify his 
expression he must have showed it usu tritum, or 
alleged the authority of some author of great note 
for it. I have not the leisure to examine his im- 
pertinent citations about those other compounds, 
nor yet of that 'simple verb diico; nay, to justify 
his saying he hath not brought one parallel ex- 
ample. He talks indeed very high, that duco, 
with its compounds, is a word of a large significa- 
tion, and amongst the rest to bring, fetch, carry, 
&c. is so exceeding frequent in all authors, Plautus, 
Terence, Tully, Caesar, Tacitus, Pliny, Seneca, 
Virgil, Horace, Ovid, Claudian, &c. that he must 
needs be either maliciously blind, or a very stranger 
to the Latin tongue, that doth not know it, or can 
have the face to deny it. I read, what will be my 
doom for not allowing his Latin ; yet I must pro- 
fess I dare secure the Doctor for having read all au- 
thors, notwithstanding his assertion, and I hope 
he will do the like for me. And for those which 
he hath read, had he brought no better proofs than 
these, he had, I am sure, been whipped soundly in 
Westminster School, for his impudence as well as 
ignorance, by the learned master thereof at pre- 
sent. But I dare further affirm, the Doctor hath not 


read in this point any, but only consulted with An extract of 

V> i n i rii T T a ^ ettel concern " 

Robert Stephen s Ihesaurus Linguce Latma, m g the gramma. 
whence he hath borrowed his allegations in ad- conlo'veLy, &c. 
duco ; and for the other, I had not so much idle ' ' 
time as to compare them. And, lest the fact might 
be discovered, he hath sophisticated those authors 
whence Stephen cites the expressions, and imposed 
upon them others. If it be not so, or that the Doc- 
tor could not write it right when the copy was right 
before him, let him tell me where he did ever read 
in Plautus, adducta res in fastidium. I find the 
whole sentence in Pliny's preface to Vespasian 
(out of whom in the precedent paragraph he cites 
it) about the middle : alia vero ita multls prodita, 
ut in factidium slut adducta, which is the very 
example Stephanas useth, although he doth pre- 
mise his adducta res in fastidium. Let the Doctor 
tell where he ever did read in Horace, Ova noctua, 
&c. tcedium vini adducunt. Did he, or any else, 
with the interposition of an &c. make Trochaics ? 
I say, and Stephanus says so, too, that it is in 
Pliny, lib. xiii. cap. 15, near the end; the whole 
sentence runs thus : Ebriosis Oca noctucz per tri- 
duum data in vino, tcedium ejus adducunt. I 
doubt not but these are the places he aimed at, 
although he disguised and minced the quotations ; 
if they be not, I should be glad to augment my 
Stephanus with his additions. 

These things premised, I come to consider the 
Doctor's proofs : Res eo adducta est : addiicta 
vita in extremum: adducta res in fastidium : rem 
ad mucrones et manus adducere: contractares et 
adducta in augustum : res ad concordiam adduci 
potest : in ordinem adducercm : adducere fe 


An extract of sitim, tedium vini (all in Robert Stephen) betwixt 

a letter concern- 7 x A 7 

ing the gramma- which and adducere malleum, what a vast differ- 

tical part of the . , T i , i , j. 

controversy, & c . ence there is, I leave them to umpire qm terretes 
" " # religiosas nacti sunt aures, who are the compe- 
tent judges of elegancy, and only cast in the ver- 
dict of one or two, who are in any place (where 
the purity of the Latin tongue flourisheth) of great 
esteem. Losseus, in his Scopce Lingua Latins, ad 
purgandam Linguam a barbarie, &c. (would any 
think that the Doctor's elegant expression, frequent 
in all authors, which none but the malicious or ig- 
norant can deny, should suffer so contumelious an 
expurgation?) Losseus, I say, hath these w r ords: Ad- 
ferre plerique minus attenti utuntur pro adducere. 
Quod Plautus, in Pseudolo, insigni exemplo notat. 

CA. Attuli hunc. 

JP& Quid attulisti ? 

CA. Adduxi volui dicere. 
PS. Quis istic est ? 

CA. Charinus. 

Satis igitur admonet discriminis inter ducere, 
reducere, adducere, et abducere, qua de per- 
sona ; et ferre, adferre, &c. qua de re dicuntur. 
Idem, Demetrium, quern ego novi, adduce : ar gen- 
turn non moror quin feras. Cavendum igitur est 
ne vulgi more, (let the Doctor mark this, and 
know that this author is authentic amongst the 
Ciceronians), adferre de persona, dicamus, sed 
adducere ; licet et hoc de certis quibusdam rebus 
non inepte dicatur. In this last clause he saith as 
much as Mr. Hobbes saith, and what the Doctor 
proves ; but, that ever the Doctor brought an ex- 
ample which might resemble adducis malleum, is 
denied ; for I have mentioned already his allega- 


tions, every one, of adduco. Another author, (a AH extract of 

r 'nil T-k \ i r- a ^ ettel 

tit antagonist for the elegant Doctor), is the Far- m g the 
rago sordidorum Verborum, joined with the Epi- ^ 
tome of L. Valla's Elegancies. He saith : Accerse, 
adhuc Petrum, Latine dicitur, pro eo quod pueri 
dicunt, adfer Petrum. And this may suffice to 
justify Mr. Hobbes's exception who proceeded no 
further than this author to tell the Doctor that 
ftdduco was used of animals. But the Doctor re- 
plies, this signification is true, but so may the other 
be also. I say if it never have been used so, it 
cannot be so, for we cannot coin new Latin words, 
no more than French or Spanish who are foreigners. 
Mr. Hobbes was upon the negative, and not to dis- 
prove the contrary opinion. If the Doctor would 
be believed, he must prove it by some example, 
(which is all the proof of elegancy), and till he do 
so, not to believe him, it is sufficient not to have 
cause. But, Doctor Wallis, why not adduco for a 
hammer as well as a tree ? I answer yes, equally 
for either, and yet for neither. Did ever anybody 
go about to mock his readers thus solemnly ? I 
do not find, to my best remembrance, any example 
of it in Stephen, and the Doctor is not wiser than 
his book ; if there be, it is strange the Doctor 
should omit the only pertinent example, and trou- 
ble us with such impertinences for three or four 
pages. In Stephen there are adducere habenas 
and adducere lorum, but in a different sense. It 
is not impossible I may guess at the Doctor's aim. 
In Tully de Nat. Deor. as I remember, there is this 
passage : Quum autem ille respondisset, in agro 
ambulanti ramulum adductum ut remissus esset, in 
oculum suum recidisse, where it signifies nothing 


An oxbact of else but to be bent, bowed, pulled back, and in 

n letter concern- / 7 ? i_ i? 

ing the gramma, that sense, the hcwimeT of a clock, or that or a 
wttro^rly! &*. smith, wlien lie fetclieth his stroke, may he said 
' ' " adduci. And this, I conceive, the Doctor would 
have us in the close think to have been his mean- 
ing ; else, what doth he drive at in these words ? 
" When you have done the best you can, you will 
not be able to find better words than adducere 
malleum and reducere, to signify the two contrary 
motions of the hammer, the one when you strike 
with it (excellently trivial !) the other when you 
take it back (better and better), What to do ? to 
fetch another stroke. If any can believe that this 
was his meaning, I shall justify his Latin, but must 
leave it to him to prove its sense. If he intended 
no more, why did he go about to defend the other 
meaning, and never meddle with this ? Which 
yet might have been proved by this one example 
of mine ? May not, therefore, his own saying be 
justly retorted upon him in this case, Adducls mal- 
leum, ut occidas muscam ? 

Another exception is, Falscc sunt, et multa is- 
tlusmodl (proposltlones). I wish the Doctor could 
bring so good parallels, and so many, out of any 
author, for his Adducis malleum, as Tully affords 
in this case. Take one for all, out of the beginning 
of his Paradoxes: Animadverti scepe Catonem, 
cum in senatu sententiam diceret, Locos graves 
ex Philosophia tractare, abhorrentes ab hoc iisu 
forensi, et publico, sed dicendo consequi tamen, ut 
ilia etlam populo probabilia viderentur. This is 
but a Soltecophanes, and hath many precedents 
more, as in the second book of his Academical 
Questions, &c. 

, &IT 


I cannot now stay upon each particular passage; An extract of 

T , / i T\ . n letter cnnrern- 

I do not see any necessity of tracing the Doctor in m K the 
all his vagaries. Now, he disallows tahquam dice- ^ov 
remus, as if we should say. But why is that less ^ 
tolerable than tanquam fecer Is, as if you had done? 
" It should be quasi, (forsooth !) or ac si, or tan- 
quam si, which is Tully's own word." What is 
tanquam si become but one w T ord ? Tanquam si tua 
res agatur, &c. Good Doctor, leave out Tully and 
all Ciceronians, or you will for ever suffer for this, 
and your Addncis malleum. Is not this to put your- 
self on their verdict when you oppose Mr. Hobbes 
with Tully ? But the Doctor gives his reason. 
And though he hath had the luck in his Adducis 
malleum, to follow the first part of that saying, 
Loquendum cum vulgo, yet now it is, sentiendum 
cum sapientibus. For tanquam without si signifies 
but as, not as if. It is pity the Doctor could not 
argue in symbols too, that so we might not under- 
stand him ; but suppose all his papers to carry 
evidence with them, because they are mathemati- 
cally scratched. How does he construe this : 

" Plance tumes alto Drusorum sanguine, tanquam 
Fcceris ipse aliquid, propter quod nobilis esses." 

So Ccelius, one much esteemed by Cicero, who 
hath inserted his Epistles into his works, saith, in 
his fifth Epistle (Tul. Epist. Fain. lib. viii. ep. 5), 
Omnia desiderantur ab eo tanquam nihil denega- 
tum sit ei quo minus paratissimus esset qui publico 
negotio prcspositus est. But it was not possible 
the Doctor should know this, it not being in Ste- 
phen, where his examples for tanquam si are. 

But, the Doctor having pitched upon this criti- 
cism, and penned it, somebody, I believe, put him 


An extract of ' m m i n d of the absurdity thereof; and yet the ge- 

a letter concern- J 111 

ing tbe gramma- nerous Professor, (who writes running hand and 

tical part of the , M i -i / T 

controversy, & c . never transcribed his papers, if I am not mism- 
" formed), presumed nobody else could be more in- 
telligent than he, who had perused Stephen. He 
would not retract anything, but subjoins, "That 
he will allow it as passable, because other modern 
writers, and some of the ancients, have so used it, 
as Mr. Hobbes hath done." I know not what 
authors the Doctor meant, for, if I am not much 
mistaken, I do not find any in Stephen. His cita- 
tion of Columella is not right, (lib. v. cap. 5), nor 
can I deduce anything thence till I have read the 
passage, but, if he take Juvenal and Coelius for 
modern authors, I hope he will admit of Accius, 
Nsevius, and Carmenta, for the only ancients. 
Let him think upon this criticism, and never hope 
pardon for his Adducis malleum, which is not half 
so well justified, and yet none but madmen or fools 
reject it. 

But certainly the Doctor should not have made 
it his business to object Anglicisms, in whose Eleri- 
chus I doubt not but there may be found such 
phrases as may serve to convince him that he is an 
Englishman, however Scottified in his principles. 
If the Doctor doubt of it, or but desire a catalogue, 
let him but signify his mind, and he shall be fur- 
nished with a Florikgium. But I am now come to 
the main controversy about Empusa. The Doctor 
saith nothing in defence of his quibble , nor gives any 
reason why he jumbled languages to make a silly 
clinch, which will not pass for wit either at Oxford 
or at Cambridge ; no, nor at Westminster. 
It seems he had derived Empusa from lv arid 


and said it was a kind of Hobgoblin that hopped An extract of 

. . i i a letter concern- 

upon one leg : and hence it was that the boys m g the gramma 
play (Fox come out of thy hole) came to be called eonlroveLy, lc! 
Empusa. I suppose he means Ludus Empusa. ' 
This derivation he would have to be good, and that 
we may know his reading, (though he hath scarce 
consulted any of the authors), he saith Mr. Hobbes 
did laugh at it, until somebody told him that it 
was in the Scholiast of Aristophanes (as good a 
critic as Mr. Hobbes), Eustathius, Erasmus, Coelius 
Rhodiginus, Stephanus, Scapula, andCalepine. But 
sure he doth not think to scape so. To begin with 
the last; Calepine doth indeed say, uno incedit pede, 
uncle et nomen. But he is a Modern, and I do not 
see why his authority should outweigh mine if his 
author's reasons do not. He refers to Erasmus and 
Rhodiginus. Erasmus in the adage, Proteo mu- 
tabilior hath these words of Empusa: Narrant 
autem uno videri pedi this is not to hop unde et 
nomen inditum putant, "EpTruffav otovel kvitrola. He 
doth not testify his approbation of the derivation at 
all, only lets you know what etymologies some have 
given before him. And doth anybody think that 
Dr. Harmar was the first which began to show his 
wit, (or folly), in etymologizing words ? Coelius 
Rhodiginus doth not own the derivation, only 
saith, Nominis ratio est> ut placet Eustathio, quia 
uno incedit pede ; is this to hop ? sed nee desunt 
qui alterum inter pretentur habere ceneum pedem, 
et inde appellatam Empusam ; quod in Batrachis 
Aristophanes expressit. And then he recites the 
interpretation that Aristophanes's Scholiast doth 
give upon the text, of which by and by. If any 
credit be to be attributed to this allegation, his last 


An extract of thoughts are opposite to Dr. Wallis ; and Empusa 

a letter concern- ni ^ 11 i 

the gramma- must be so called, riot because she hopped upon 
leg, but because she had but one, the other 
being brass. But for the former derivation he re- 
fers to Eustathius. 

As to Eustathius, I do easily conjecture that the 
reader doth believe that Rhodiginus doth mean 
Eustathius upon Homer, for that is the book of 
most repute and fame, his other piece being no 
way considerable for bulk or repute. But it is not 
that book, nor yet his History of Ismenias, but his 
notes upon the 725th verse of Dionysius He^V/Wc. 
The poet had said of the stone Jaspis, that it was 

kdl CtXXoLQ 

Upon which Eustathius thus remarks : 

yap aXtty KCLLOQ ctVcu ?) X'tQos ^civrrj, *u/ airoT 

IV Tl KGtl ?/ KjUTTdO'Ctj ^(LljJiOVl^V TLT()l Tlii) ' 

dii]ke(T$ai' (forte SiEptefaff&ai Stcph.) oQtv kai ^ 

f.L TIC tlTTtf fJOlWTT&Q 7TO()l ftttOV* OIC TOV tTtpU 7TOC)OC y^CL\kOV OVTOQ, kCLTO. 

TOV nvQov. This testimony doth not prove anything 
of hopping, and, as to the derivation, I cannot but 
say that Eustathius had too much of the gramma- 
rian in him, and this is not the first time, neither 
in this book, nor elsewhere, wherein he hath trifled. 
It is observable out of the place, that there were 
more Empusas than one, as, indeed, the name is 
applied by several men to any kind of frightful 
phantasm. And so it is used by several authors, arid 
for as much as phantasms are various, according as 
the persons affrighted have been severally educated, 
&c. every man did impose this name upon his own 
apprehensions. This gave men occasion to fain 
Empusa as such for who will believe that she was 
not apprehended as having four legs, when she 


appeared in the form of a cow, dog, &c. but, as An extract of 
apprehended by Bacchus and his man at that time, *]^^ 
I do not find that she appeared in auy shape but ticttl i mrt of tho 

L \ . J L controversy, &c. 

such as made use or legs in going, whence I ima- ^ < 
gine that Empusce might be opposite to the &oJ 
revolts, which appellation was anciently fixed upon 
the gods, (propitious) upon a two-fold account; 
first, for that they were usually effigiated as 
having no feet, which is evident from ancient 
sculpture, and secondly, for that they are all said 
not to walk, but rather swim, if I may so express 
that non gradiuntur, sedfluunt, which is the asser- 
tion of all the commentators I have ever seen upon 
that verse of Virgil : 

" Et vera incessu patuit dea" - 

This whole discourse may be much illustrated 
from a passage in Heliodorns, ^Ethiop. lib. iii. sec. 
12, 13. Calasiris told Cnemon that the Gods Apollo 
and Diana did appear unto him; Cnemon replied, 

'AXXa r/Va Ct) Tpoirov f>u0KC ly^^i^Qal aoi roDc Qtovg OTL pi) 
tvvirvwv 7j\$ov, aXX' tvapya>c e(j)avt]ffav ; UpOU this the old 

priest answered, that both gods and demons, when 
they appear to men, may be discovered by the curious 
observer, both in that they never shut their eyes, 

KCU Ttf fta^iff^arL TrXtov, ou Kara ha^rjffLV T&V irodwv ovdi /icrd&ffii' 
avvoplw, a\\a Kara nva pvpriv alpiov, Kctl bppjy a7rapa.7r6di<?ov y 
rejjivovTWV fjLuXXov TO 7TpiEj(ov 5 ^tcoropevopivtav. Ao &/ Kat TCL 
aydXjuara ra)v Oewv AtyvTrrtot rw 7rot (svyyvyrtf kcit wtTTTfp iyovy- 
TEQ 'i-affiy. a ^rf kat "OfirjpOQ ft^aic, fire 'AiyvTrno^, kat n)i' ipuv 
EK^a^detg, ffvpftoXL^Q TOIQ eirtffiv evaTTg-Jtro, TO!Q ouva- 
trvviivai yvbtptfctv /caraXtTrwv, exi rov Troo-ft^wvoc, TO 


oiov pioyTue tv ri\ iropela, TOVTO yap ?t TO ptl a7rto/rocj Kal ov\ 


An extract of Tlveg jyVonf rrcu, pqdiUQtyvuv VTToXafLifiuvovrEg. Famaby, UDOH 
a letter concern- . . 

ing the gramma, the place in Virgil, observes, that Deorum in- 
! ccssus est contmuus et cequolls^ non dimotis pedi- 
, neque transpositis, a\\a kara pv^v atptor. Cor- 
nelius Schrevelius in the new Leyden notes saith, 
AntiqmssiKtia quceque Deorum simulachra, quod 
observarunt viri magni, erant TOVQ irodae av^E^Kora, 
diique ipsi non gradiuntur sed flunnt. Their sta- 
tues were said to stand rather upon columns than 
upon legs, for they seem to have been nothing but 
columns shaped out into this or that figure, the 
base whereof carrying little of the representation 
of a foot. These things being premised, I suppose 
it easy for the intelligent reader to find out the true 
etymology of Empusa, quasi lv <nv ouaa, or fiaivwa, 
from going on her feet, whereas the other gods 
and demons had a different gait. If any can dis- 
like this deduction, and think her so named from 
ei'Mrec, whereas she always went upon two legs, (if 
her shape permitted it) though she might draw the 
one after her, as a man doth a wooden leg : I say, 
if any, notwithstanding what hath been said, can 
join issue with the Doctor, my reply shall be lol 

Now, as to the words of Aristophanes upon 
which the Scholiast descants, they are these : 
speaking of an apparition strangely shaped, some- 
times like a camel, sometimes like an ox, a beauti- 
ful woman, a dog, &c. Bacchus replies : 


SA. Trvpl 
airav TO 
AI. Nr) TW Ilo(rEtw, KCU floXirwov Qarepov. 


The Scholiast hereupon tells us that Empusa, 


(HmuovtwdeQ VTTO 'Ejcdnjc tTwrEUTrouevov ecu ri>cuvo- An extract of 

_ ^ ( a letter concern- 

fievov TOIQ ^v^v^ovffiv 9 o dokti TroXXac fjtoatyac aXXciffffeu Kat mg the gramma- 

01 ut> (baaiv avTnr uoroiroCa tm, Kat f.rvuo\oyovaiv' biorti ||/ t . ticalpart ot the 
r r ' r . \ ( ' . controversy, &c. 

And this is all that is ma- * . ' 

terial in the Scholiast, except that he adds by and 
by, that poXtnvov raXoc is all one with the leg of an 
ass. And this very text and Scholiast is that to 
which all the authors he names, and more, do 

I come now to Stephen, who, in his index, arid 
in the word voSifa, gives the derivation of Empusa. 
IIo<)ta>, gradior, incedo, (not to hop) sic Suidas 
"EjUTTBdav diet am ait Trapa TO evl TroSifav. In the index 
thus : sunt qui dictum putent pa kvl vol\fav> quod 
uno incedat pedi, quasi "E^7ro-a> , alterum eriim pe- 
dem ameum hdbet. But neither Stephen, nor any 
else, except Suidas, whom the hypercritical Doctor 
had not seen, no, not the Scholiast of Aristophanes 
(a better critic than Mr. Hobbes) doth relate the 
etymology as their own. Nay, there is not one 
that saith Empusa hopped on one leg, which is to 
be proved out of them. The great Etymological 
Dictionary deriveth it 7-apa TO e/iiro&f v, to hinder, let, 
&c. its apparition being a token of ill luck. But, 
as to the Doctor's deduction, it saith, "Epr<ra 

Trai, el Kat Sow Trapa TO ci/a ffvykelffScu. It doth Only 

so. And it is strange that lv should not alter only 
its aspiration, but change its v into p, which I can 
hardly believe admittable in Greek, least there 
should be no difference betwixt its derivatives and 
those of iv. When I consider the several /lo^oreg 
which the Grecians had, some whereof did fly, 
some had no legs, &c, I can think that the origin of 
this name may have been thus: some amazed person 


An extract of sa w a spectrum, and, giving another notice of it, 

a letter concern- . . * .., .._, _ , 

ing the gramma- HIS COHlpaniOn might anSWCr, it IS B/Hflw, Mop/iw 

but he, meeting with a new phantasm, cries, 
if votrl /3a/Va or /3aS/', for which apprehension of his, 
somebody coined this expression of "EproDora. It 
may also be possibly deduced from 'E^?roo^w, so 
that TVW epvotifrffa might afterwards be reduced to 
the single term vfEmpusa. Nor do I much doubt 
but that those who are conversant in languages, and 
know how that several expressions are often jum- 
bled together to make up one word upon such like 
cases, will think this a probable origination. I be- 
lieve, then, that Mr. Hobbes's friend did never tell 
him it was in Eustatbius, or that Empusa was an 
hopping phantasm. It had two legs and went upon 
both, as a man may upon a wooden leg. 'Epirowa 
is also a name for Lamia, and such was "that which 
Menippus might have married, which, I suppose, 
did neither hop nor go upon one leg, for he might 
have discovered it. But Mr. Hobbes did not ex- 
cept against the derivation, (although he might 
justly, derivations made afterwards carrying more 
of fancy than of truth, and the Doctor is not ex- 
cused for asserting what others barely relate, none 
approve), but asked him where that is, in what 
authors lie read that boys' play to be so called. 
To which question, the Doctor, to show his 
reading and the good authors he is conversant 
in, replies, in Jumuss Nomenclator, Rider and 
Thomas's Dictionary, sufficient authors in such a 
business, which, methinks, no man should say that 
were near to so copious a library. It is to be re- 
membered that the trial now is in Westminster 
School, and amongst Ciceronians, neither whereof 


will allow those to be sufficient authors of any Anextiactof 

, Tr * * . a letter concern. 

Latin word. Alas, they are but Vocabularies ; ii^tiipgiainina. 
and, if they bring no author for their allegation, lentils"! &!! 
all that may be allowed them is, that, by way of "~ ' ' 
allusion, our modern play may be called Lucius 
Empusce. But that it is so called we must expect, 
till some author do give it the name. These are 
so good authors, that I have not either of them in 
my library. But I have taken the pains to consult, 
first, Rider ; I looked in him, (who was only au- 
thor of the English Dictionary) and I could not find 
any such thing. It is true, in the Latin Dictionary, 
which is joined with Rider, but made by Holy- 
oke ; (O that the Doctor would but mark !) in the 
index of obsolete words, there is Ascoliasmus ', 
Ludus Empusa, Fox to thy hole, for w r hich word, 
not signification, he quoteth Junius. The same is 
in Thomasius, who refers to Junius in like manner. 
But could the Doctor think the word obsolete, 
when the play is still in fashion r Or, doth he 
think that this play is so ancient as to have had a 
name so long ago, that it should now be grown 
obsolete ? As for Junius's interpretation of Em- 
pusa, it is this : Empusa, spectrum, quod se in- 
felicilms ingerit, uno pede ingrediens. Had the 
Doctor ever read him, he would have quoted him 
for his derivation of Empusa, I suppose. In Asco- 
liasmus, he saith, Ascoliasmus, Empusce Ludus, fit 
ubi, altero pede in aere librato, unlco subsiliunt 
pede : a<m>Xta<^oc Pollux ; Almanice, Hinclcelen ; 
Belgice, Op een been springhen ; Hincltepincken, 
Flandris. But what is it in English he doth not 
tell, although he doth so in other places often. 
What the Doctor can pick out of the Dutch I 


An extract of know not ; but, if that do not justify him, as I 

a letter concern- . i i t i .1 IT- j 

mg the gramma, think it doth not, he hath wronged Junius, and 
ctlrU!^ greatly imposed upon his readers. 
' ' * But, to illustrate this controversy further, I can- 
not be persuaded the Doctor ever looked into 
Junius, for, if he had, I am confident, according 
to his wonted accurateness, he would have cited 
Pollux's Onomasticon into the bargain, for Junius 
refers to him, and I shall set down his words, that 
so the reader may see what Ascoliasmus was, and 
all the Doctor's authors say Lucius Empusce and 
Ascoliasmus were one and the same thing. Julius 
Pollux (lib. ix. cap. 7) : f O & 'A^oXJacr/ioc, (old edi- 
tions read it, 'AVfcoX<r/Ltoc Ct a<m)Xttcu) rov erepH TroSbg 
aiupupivv, Kara p6vv TOV irfpo irri^av twoiei ; oTrsp ' Affkio\idtw 
i'jroi Fig juf/w> V//XXaro, ?/ 6 pev idi&kev oi/rw^, ot ^c 
iir 9 apt/tow ^fovrfc? cwg rtvog rp <<p trodi 6 ctwkwv 
Tvynv' ?/ kal ^UVTE^ cTrfjvwv, upiOpovi^T TCI Tr^f^/juara* irpoa- 
yap rf TrXrjOu TO vikav* 'A<r/cwXta^iv ^e EKaXetTO kat TO 
fi neitf KO.I vTTOTrXew Trvet^iaroc, f]\ei[ip.lvb), 'ivairep 

cptTt)va\Qt<iH)v. " So thatAscoliasmiis, and 
consequently, Lucius Empusce, was a certain sport 
which consisted in hopping, whether it were by 
striving who could hop furthest, or whether only 
one did pursue the rest hopping, and they fled 
before him on both legs, which game he was to 
continue till he had caught one of his fellows, or 
whether it did consist in the boys' striving who 
could hop longest. Or, lastly, whether it did con- 
sist in hopping upon a certain bladder, which, 
being blown up and well oiled over, was placed 
upon the ground for them to hop upon, that so 
the unctuous bladder might slip from under them 
and give them a fall." And this is all that Pollux 
holds forth. Now, of all these ways, there is none 
that hath any resemblance with our Fox to thy 


hole ; but the second : and yet, in its description, An cxtract of 

J . . r a letter cancel n 

there is no mention of beating him with gloves, as mt? the gramm,i 
they do now-a-days, and wherein the play consists conl^y! &! 
as well as in hopping. It might, notwithstanding, ' ' ' 
be called Ludus Empusa, but not in any sort our 
Fox to thy hole ; so that the Doctor and his 
authors are out, imposing that upon Junius and 
Pollux which they never said. And thus much 
may suffice as to this point. I shall only add out 
of Meursius's Ludi Grceci, that Ascolia were not 
Ludus Empusce but Bacchisacra, and he quotes 
Aristophanes's Scholiast in Plutus, 'A^Xta lopn} Ai- 

vvcru CUTKOV yap owv 7rXr)povvrey *vi irodi rovror 7T7r//o>>, KCLI o 
Tr^trac aOXov d^e TOV oivov. As also HeSychhlS, A<TKW- 

XiafalV, KVplWQ TO fcTTt TOVQ affKOVQ aX\ff$CU. 

But I could have told the Doctor where he 
might have read of Empusa as being the name of 
a certain sport or game, and -that is, in Turnebus 
Adversaria, lib. xxvii. cap. 33. There he speaks of 
several games mentioned by Justinian in his Code, 
at the latter end of the third book, one of which 
he takes to be named Empusa; adding withal, 
that the other are games, it is indisputable, only 
Empusa in lite et causa erit, quod nemo nobis 
facile assensurus sit Ludum esse, cum constet 
spectrum quoddam fuisseformas, varie mutans. 
Sed quid vet at eo nomine Ludum fuisse? Certe 
ad vestigia vitiatce Scripture quam proximo 
accedit. Yet he only is satisfied in this conjec- 
ture, till somebody else shall produce a better. 
And now what shall I say ? Was not Turnebus 
as good a critic, and of as great reading as Dr. 
Wallis, who had read over Pollux, and yet is afraid 
that nobody will believe Empusa to have been a 

E E 2 


And extract of game, and all he allegeth for it is, quid vetat ? 

a letter concern. 2, * 

ing the gramma. Truly, all I shall say, and so conclude this business, 
, &c! is* that he -had read over an infinity of books, yet, 
had not had the happiness, which the Doctor had, 
to consult with Junius's Nomenclator, Thomasius 
and Rider s Dictionary, authors sufficient in such 
a case. 

I now come to the Doctor's last and greatest 
triumph, at which I cannot but stand in admira- 
tion, when I consider he hath not got the victory. 
Had the Doctor been pleased to have conversed 
with some of the fifth form in Westminster School, 
(for he needed not to have troubled the learned 
master), he might have been better informed than 
to have exposed himself thus. 

Mr. Hobbes had said that y/) signified a mark 
with a hot iron ; upon which saying the Doctor is 
pleased to play the droll thus : " Prithee tell rne, 
good Thomas, before we leave this point, (O the 
wit of a divinity doctor !) who it was told thee that 
Tiyfo) w r as a mark with an hot iron, for it is a 
notion I never heard till now, and do not believe 
it yet. Never believe him again that told thee 
that lie, for as sure as can be, he did it to abuse 
thee ; *wp */ signifies a distinctive point in writing, 
made with a pen or quill, not a mark ma(je with a 
hot iron, such as they brand rogues withal ; and, 
accordingly, TIO> <a?tw, distinguo, interstinguo^ 
are often so used. It is also used of a mathema- 
tical point, or somewhat else that is very small, 
*wy XP" a moment, or the like. What should 
come in your cap, to make you think that ^ly^ 
signifies a mark or brand with a hot iron ? I per- 
ceive where the business lies ; it was ^ly^ ran in 



your mind when you talked of TiyM, and, because An extract of 

J J /r 111 a letter concern* 

the words are somewhat alike, you jumbled them ing tiieramma- 
both together, according to your 'usual care and cont 
accurateness, as if they had been the same." 

When I read this I cannot but be astonished at 
the Doctor's confidence, and applaud him who said, 
u/Lia0ia 3rip<roc ^epei. That the Doctor should never 
hear that ^ypi signifies a mark with a hot iron, is 
a manifest argument of his ignorance. But, that 
he should advise Mr. Hobbes not to believe his 
own readings, or any man's else that should tell 
him it did signify any such thing, is a piece of no- 
torious impudence. That -my/^ signifies a dis- 
tinctive point in writing made with a pen or quill, 
(is a pen one thing and a quill another to write 
with ?) nobody denies. But, it must be withal ac- 
knowledged it signifies many- things else. I know 
the Doctor is a good historian, else he should not 
presume to object the want of history to another ; 
let him tell us how long ago it is since men have 
made use of pens or quills in writing ; for, if that 
invention be of no long standing, this signification 
must also be such, and so it could not be that from 
any allusion thereunto the mathematicians used it 
for a point. Another thing I would fain know of 
this great historian, how long ago Wfw and &aW{> 
began to signify interpungo ? For, if the mathe- 
matics were studied before the mystery of printing 
was found out, (as shall be proved whenever it 
shall please the Doctor, out of his no reading, to 
maintain the contrary), then the mathematical use 
thereof should have been named before the gram- 
matical. And, if this word be translatitious, and 
that sciences were the effect of long contemplation, 


An extract of the names used wherein are borrowed from talk, 

a letter concern. -FT i * T -i n * i 

mg the gramma- Mr. Hoboes did well to say, that ?iyp) precedane- 
sy, &c! ously to that indivisible signification which it after- 
wards had, did signify a visible mark made by a 
hot iron, or the like. And, in this procedure, he 
did no more than any man would have done, who 
considers that all our knowledge proceeds from our 
senses ; as also that words do, primarily, signify 
things obvious to sense., arid only secondarily^ 
such as men call incorporeal. This leads me to a 
further consideration of this word. Hesychius, 
(of whom it is said that he is Legendus non tan- 
quam Lexicographus, sed tanquam Justus author), 
interprets *iypn, yp}, which is a point of a greater 
or lesser size, made with any thing. So ?# signi- 
fies to prick or mark with anything in any manner, 
and hath no impropriated signification in itself, 
but according to the writer that useth it. Thus, 
in a grammarian *fa signifies to distinguish, by 
pointing often ; sometimes, even in them, it is the 
same with 6fie\ifo; sometimes it signifies to set a 
mark that something is wanting in that place, 
which marks w 7 ere called <?ty/W. In matters of 
policy, <^Cw signifies to disallow, because they used 
to put a Tiy^?) (not *iypa ) before his name who 
was either disapproved or to be mulcted. In 
punishment it signifies to mark or brand, whereof 
I cannot at present remember any other ways than 
that of an hot iron, which is most usual in authors, 
because most practised by the ancients. But, that 
the mark which the Turks and others do imprint 
without burning may be said *ifadat 9 1 do not doubt, 
no more than that Herodian did to give that term 
to the ancient Britons,, of whom he says, 


yoatiaic Troi/ct'Xcuc, iat wwv Trajro&tTrwv ehtoon. Thus, An extract of 
'*. r 3 a letter concer 

that were branded with m^a and 

j \ - i ... c\ mi j ticdl pait ot the 

and aanQopai) were said *IT$CM. Inns, in its contioveisy, &c. 
origin, <ny/*>) doth signify a brand or mark with an ~*~~' 
hot iron, or the like ; and that must be the proper 
signification of <myjui?, which is proper to *rifa, none 
but such as Dr. Wallis can doubt. In its descend- 
ants it is no less evident, for, from <my/i?) comes 
stigmosus, which signifies to be branded ; Vitel- 
liana cicatrice stigmosus, not stigmatosus. So 
Pliny in his Epistles, as Robert Stephen cites it. 
And ffnypanas (the derivative of o-ny^r}, which signifies 
any mark, as well as a brand, even such as remain 
after stripes, being black and blue), was a nick- 
name imposed upon the grammarian Nicanor, on 
7Tf)l (jTiyiiuv 7roXuXoyj]ffe. And, though we had not any 
examples of orr/p) being used in this sense, yet, 
from thence, for any man to argue against it, (but 
he who knows no more than Stephen tells him) 
is madness, unless he will deny that any word hath 
lost its right signification, and is used only, by the 
authors we have, although neither the Doctor nor 
I have read all them, in its analogical signification. 
I have always been of opinion, that <my^ signified 
a single point, big or little, it matters not ; and 
oTjy/ja, a composure of many ; as ypa/^r) signifies a 
line, and yp<Vf ta a letter, made of several lines. For 
err/yjua signified the owl, the scemcena, the letter K, 
yea, whole words, lines, epigrams engraven in 
men's faces ; and <rnr//, I doubt not, had signified a 
single point, had such been used, and so it became 
translatitiously used by grammarians and mathe- 
maticians. I could give grounds for this conjee- 


An extract of ture, and not be so impertinent as the Doctor hi 

a letter contnii- , . 1111 i 

g the gramma- his sermon, where he told men that aoqk was not 
rsy! & c ! in Homer ; that from fypw came ebrius ; that 
" sobrietas was not bad Latin, and that sobrius was 
once, as I remember, in Tally. Is this to speak 
suitably to the oracles of God, or rather to lash 
out into idle words ? Hath the Doctor any ground 
to think these are not impertinences ? Or, are 
we, poor mortals, accountable for such idle words 
as fall from us in private discourses, whilst these 
ambassadors from heaven droll in the pulpit with- 
out any danger of an after-reckoning ? 

But I proceed to a further survey of the Doc- 
tor's intolerable ignorance. His charge in the end. 
of the school-master's rant is, that he should 
remember <Tr(ypa and <m y /urj are not all one. I com- 
plained before that he hath not cited Robert Ste- 
phen aright; now I must tell him he hath been 
negligent in the reading of Henry Stephen ; for in 
him he might have found that <m'y/ia w 7 as some- 
times all one with *ny/n}, though there be no ex- 
ample in him wherein orty/^ is used for wy/ia. 
Hath not Hesiod, (as Stephen rightly citeth it), in 
his Scutum, 166-67. 

2r<y/jara cTw'c entyavTO I'delv Sfirttifft Sp&Ktiffi 
Ki/ai ; ca Kara V&TCL 

Uul ScholiasteS (Sairep St any pal ?]aav liravti), T&V pa-^eioy 
TWV tyaKovrw, /oa?-a art/trot yap KCLL troikiXoi bi ofaiQ. Sp 

Johannes Diaconus upon the place, a man who 
(if I may use the Doctor's phrase) was as good 
a critic as the Geometry Professor. 

Thus much for the Doctor. To the understanding 
reader, I say that <my/*/ is used for burning with a 


hot iron: 2 Macchdb. ix. 11, where speaking of Al1 cxtract of 

i . , i . a lctter concern- 

Antiochuss lamentable death, his body putrefying mg the gramma. 

-, , T . , i , / A ticdl part of the 

and breeding worms, he is said, ^ tr^w TOO &oD contrmersy ,& e . 

tp^eScu de/^E juaoriyi, Kara (myptiv tTrircevojUe^og rate a\yrjc6ffi ; 

iizrag* pained as if he had been pricked or burned 
with hot irons. And that this is the meaning of 
that elegant writer, shall be made good against the 
Doctor, when he shall please to defend the vulgar 
interpretation. Pausariias, in Bceoticis, speaking 
of Epaminondas, who had taken a town belonging 
to the Sicyonians, called Phoebia (*tw/5ia) wherein 
were many Boeotian fugitives, who ought, by law, 
to have been put to death, saith he dismissed them 
under other names, giving them only a brand or 

metric .' ri(5\7jua \wV 2ikvu)viwv Qovfliai', u>$a i\(mv TO TTO\V 
oi KoiiiinoL QwjfddeQ, ortyjui/V atytv\at TOVQ iykciTa\ri<t>6tVTa aXXrjv 
ff (ft Iff tv i}v trv-fcc Tcarpicci iirovofjid^Mp k*flf<rnu. It is true ffrty^v 

is here put adverbially, but that doth not alter the 
case. Again, Zonaras, in the third tome of his 
History, in the life of the Emperor Theophilus, 
saith, that when Theophanes and another monk 
had reproved the said emperor for demolishing 
images, he took and stigmatized each of them with 
twelve iambics in their faces : tlra KCU rac <tyc avrtiv 

KaYeflrrtfe KCU rat? ffri-ypaig pi\av eire-^ee ypdppara $e irvTrovv ra 
orrty/iara, ra ie ?](TCLV lap fat ovrou A place SO evident, that 

I know not what the Doctor can reply. This place 
is just parallel to what the same author saith in the 

life of Irene, rac oi/'ac a$wv K-ara<mac w ypdppaot, ptXavoz 

ty^ofjLEvov rots (Trivet. If the Doctor object that he 
is a modern author, he will never be able to ren- 
der him as inconsiderable as Adrianus Junius's 
Nowenclator, Thomasius and Rider. If any will 


An extract of (j en y fa^ h e writes good Greek, Hieronymus 

a letter concern. J m m J 

ing the gramma- WolflUS will tell them, his Only fault 1S 

controversy, L! redundancy in words, and not the use of bad ones. 
* ' "" Another example of <myp) used in this sense, is 
in the collections out of Diodorus Siculus, lib. 
xxxiv. as they are to be found at the end of his 
works, and as Photius hath transcribed them into 
his Bibliot/ieca. He saith that the Romans did buy 
multitudes of servants and employ them in Sicily : 

Off, ex T&V atopaTorpotyEtw ayeXrjdov aTra^Selffiv^ EvOvG ^ctpaK- 

Tfjpa 7rc/3aXXov, Kai ffrtypai: TOIQ ffu)p.a(nv. These are the 
words but of one author, but ought to pass for the 
judgment of two, seeing Photius, by inserting 
them, hath made them his own. 

Besides, it is the judgment of a great master of 
the Greek tongue, that stigmata non tarn puncta 
ipsa quam punctis 'variatam superficiem Grceci 
vocaverunt. I need not, I suppose, name him, so 
great a critic as the Doctor cannot be ignorant of 

Nor, were ffriy^ara commonly, but upon extra- 
ordinary occasions, imprinted with an hot iron. 
The letters were first made by incision, then the 
blood pressed, and the place filled up with ink, 
the composition whereof is to be seen in Aetius. 
And thus they did use to matriculate soldiers also 
in the hand. Thus, did the Grecian emperor, in 
the precedent example of Zonaras. And if the 
Doctor would more, let him repair to Vinetus's 
comment upon the fifteenth Epigram of Ausonius. 

And now I conceive enough hath been said to 
vindicate Mr. Hobbes, and to show the insufferable 
ignorance of the puny professor, and unlearned 


critic. If any more shall be thought necessary, An extract of 

^ ' a letter concern- 

1 shall take the pains to collect more examples and im. the gramma- 
authorities, though I confess I had rather spend con 
time otherwise, than in matter of so little moment. 
As for some other passages in his book, I am no 
competent judge of symbolic stenography. The 
Doctor (Sir Reverence) might have used a cleanlier 
expression than that of a shitten piece, when he 
censures Mr. Hobbes's book. 

Hitherto the letter.* By which you may see 
what came into my (not square] cap to call <myp) 
a mark with a hot iron, and that they who told me 
that, did no more tell me a lie than they told you 
a lie that said the same of ariy^a ; and, if arlyp) be 
not right as I use it now, then call these notes not 

<my/xa c , but trny^ara. I will not Contend with yOU 

for a trifle. For, howsoever you call them, you 
are like to be known by them. Sir, the calling of 
a divine hath justly taken from you some time that 
might have been employed in geometry. The 
study of algebra hath taken from you another part, 
for algebra and geometry are not all one; and you 
have cast away much time in practising and trust- 
ing to symbolical writings ; arid for the authors 
of geometry you have read, you have not examined 
their demonstrations to the bottom. Therefore, 
you perhaps may be, but are not yet, a geometri- 
cian, much less a good divine. I would you had 
but so much ethics as to be civil. But you are a 
notable critic ; so fare you well, and consider 

* Written by Henry Stubbe, M. A. of Christ Church, Oxford, 
who was, according to Anthony a Wood, "the most noted 
personage of his age that these late times have produced." 


Conclusion, what honour you do, either to the University 
where you are received for professor, or to the 
University from whence you came thither, by your 
geometry ; and what honour you do to Emanuel 
College by your divinity ; and what honour you 
do to the degree of Doctor, with the manner of 
your language. And take the counsel which you 
publish out of your encomiast his letter; think me 
no more worthy of your pains, you see how I have 
fouled your fingers. 











PRESENTETH to your consideration, your most First 
humble servant, Thomas llobbes, (who hath spent 
much time upon the same subject), two proposi- 
tions, whereof* the one is lately published by Dr. 
Wallis, a member of your Society, and Professor 
of Geometry ; which if it sliowld be false, and pass 
for truth, would be a great Obstruction in the way 
to the design you have undertaken. The other 
is a problem, which, if well demonstrated, will be 
a considerable advancement of geometry ; and 
though it should prove false, will in no wise be 
an impediment to the growth of any other part 
of philosophy. 


DE MOTU, Cap. v. Prop. 1. ROSET. Prop. v. 

IP there be understood an To find a straight line equal to 

infinite row of quantities be- two-fifths of the arc oj a 

ginning with or J, and in- quadrant. 

creasing continually according I DESCRIBE a square A B C D, 

to the natural order of numbers, and in it a quadrant D A C. 

0, 1, 2, 3, &c. or according to Suppose D T be two-fifths of 

the order of their squares, as, DC, then will the quadrantal 

0, 1, 4, 9, &c, or according to arc TV be two-fifths of the 

the order of their cubes, as, arc C A. Again let D R be a 

0, 1 , 8, 27, &c. whereof the mean proportional between D C 



First Paper, last is given ; the proportion of 
r ~" the whole, shall be to a row of 
as many, that are equal to the 
last, in the first case, as 1 to 2 ; 
in the second case, as 1 to 3 ; 
in the third case, as 1 to 4, &c. 
This proposition is the 
ground of all his doctrine con- 
cerning the centres of gravity 
of all figures. Wherein may 
it please you to consider : 

First, whether there can be 
understood an infinite row of 
quantities, whereof the last can 
be given. Secondly, whether a 
finite quantity can be divided 
into an infinite number of lesser 
quantities, or a finite quantity 
can consist of an infinite num- 
ber of parts, which he buildeth 
on as received from Cavaflicri. 
Thirdly, whether (which in 
consequence he maintaineth) 
there be any quantity greater 
than infinite. Fourthly, whether 
there be, as he saith, any finite 
magnitude of which there is 
no centre; of gravity. Fifthly, 
whether there be any number 
infinite. For it is one thing to 
say, that a quantity may be di- 
vided perpetually without end, 
and another thing to say, that 
a quantity may be divided into 
an infinite number of parts. 
Sixthly, if all this be false, 
whether that whole book of 
Anthmetica Infinitorum, and 
that definition which he build- 
eth on, and supposeth to be the 
doctrine of Cavallieri, Be of any 

and D T ; then will the quad- 
rantal arc R S be a mean pro- 
portional between the arc C A 
and the arc T V. 

Suppose further a right line 
were given equal to the arc C 
A, and a quadrantal arc there- 
with described ; then will 1) C, 
C A, the arc on C A be con- 
tinually proportional. Set these 
proportionals in order by them- 

D C, C A, arc on C A -f 
D R, R S, arc on R S ~ 
DT, TV, arc on TV -r 
which are in continual propor- 
tion of the semi-diameter of 
the arc. And I) C, D R, D T 
are in a continual proportion by 
construction, and therefore also 
C A, R S, T V, and arc on 
C A, arc on R S, arc on T V, 
in continual proportion. 

Therefore as D C to R S, so 
is R S to the arc on T V. And 
D C, R S, the arc on T V will 
be continually proportional. 
And because DC, C A, the 
arc on C A are also continually 
proportional, and have the first 
antecedent D C common ; the 
proportion of the arc on C A to 
the arc on T V is (by Eucl. 
xiv. 28) duplicate of the pro- 
portion of C A to R S, and 
the arc on R S a mean propor- 
tional between the arc on C A 
and the arc on T V. 

Now if D C be greater than 
R S, also R S must be greater 
than the arc on TV; and the 



use for the confirming or con- 
futing of any propounded doc- 

Humbly praying you would 
be pleased to declare herein 
your judgment, the examina- 
tion thereof being so easy, that 
there needs no skill either in 
geometry, or in the Latin 
tongue, or in the art of logic, 
but only of the common under- 
standing of mankind to guide 
your j udgment by. 

arc C A greater than the arc First 
on R S. Therefore seeing D C, **"" 
C A, arc on C A? ar c continu- 
ally proportional ; the arc on 
T V, the arc on R S, the arc 
on C A cannot be continually 
proportional, which is contrary 
to what has been demonstrated. 
Therefore D C is not greater 
than R S. Suppose, then, R S 
to be greater than D C, then 
will the arc on R S be a mean 
proportional between the arc 
on TV, and a greater arc than 
that on C A ; and so the in- 
convenience returneth. There- 
fore the scmidiameter DC is 
equal to the arc R S, and D R 
equal to TV, that is to say to 
two-^fths of the arc C A, which 
was .to be demonstrated. Nor 
needeth there much geometry 
for examining of this demon- 
stration. Therefore I submit 
them both to your censure, as 
also the whole Rosetum, a copy 
whereof I have caused to be 
delivered to the secretary of 
your society. 


F F 







Presenteth to your consideration, your most humble second Paper 
servant Thomas Hobbes, a confutation of a theo- ' ' ' 
rem which hath a long time passed for truth ; to 
the great hinderance of Geometry, and also of 
Natural Philosophy, which thereon dependeth. 


The four sides of a square being divided into 
any number of equal parts ', for example into 10 ; 
and straight lines drawn through the opposite 
points, which will divide the square into 100 
lesser squares ; the received opinion, and which 
Dr. Wallis commonly useth, is, that the root of 
those 100, namely 10, is the side of the whole 


The root 10 is a number of those squares, 
whereof the whole containeth 100, whereof one 
square is an unity ; therefore the root 10, is 10 
squares : Therefore the root of 100 squares is 10 
squares, and not the side of any square ; because 
the side of a square is not a superficies, but a 

F F 2 


riv . line. For as the root of 100 unities is 10 unities, 

of Dr. Tftalhss ** 

Theorem. or of 1 00 soldiers 10 soldiers : so the root of 100 
squares is 10 of those squares. Therefore the 
theorem is false ; and more false, when the root 
is augmented by multiplying it by other greater 

Hence it followeth,that no proposition can either 
be demonstrated or confuted from this false 
theorem. Upon which, and upon the numeration 
of infinites, is grounded all the geometry which 
Dr. Wallis hath hitherto published. 

And your said servant humbly prayeth to have 
your judgment hereupon : and that if you find 
it to be false, you will be pleased to correct the 
same : and not to suffer so necessary a science as 
geometry to be stifled, to save the credit of a 








Your most humble servant Thomas Hobbes pre- Third Paper. 
senteth, that the quantity of a line calculated by 
extraction of roots is not to be truly found. And 
further presenteth to you the invention of a straight 
line equal to the arc of a circle. 

A square root is a number which multiplied into 
itself produceth a number. 


And the number so produced is called a square 
number. For example : Because 1 multiplied by 
10 makes 100; the root is 10, and the square 
number 100. 


In the natural row of numbers, as 1,2, 3, 4, 5, 
6,7, 8, 9, 10, II, 12, 13, 14, 15, 16, &c. everyone 
is the square of some number in the same row. 
But square numbers (beginning at 1) intermit first 
two numbers, then four, then six, &c. So that 
none of the intermitted numbers is a square num- 
ber, nor has any square root. 


Third Paper A square root ( speaking of quantity) is not a 
line, such as Euclid defines, without latitude, but 
a rectangle. 

Suppose A B C D 
be the square, and 
AB, BC, CD, DA, 
be the sides, and 
every side divided 
into 10 equal parts, 
and lines drawn 
through the oppo- 
site points of divi- 
sion ; there will then 
be made 100 lesser 
squares, which taKen altogether are equal to the 
square A B C D. Therefore the whole square is 
100, whereof one square is an unit; therefore 
JO units, which is the root, is ten of the lesser 
squares, and consequently has latitude ; and there- 
fore it cannot be the side of a square, which, ac- 
cording to Euclid, is a line without latitude. 


It follows hence, that whosoever taketh for a 
principle, that a side of a square is a mere line 
without latitude, and that the root of a square is 
such a line (as Dr. Wallis continually does) demon- 
strates nothing. But if a line be divided into what 
number of equal parts soever, so the line have 
breadth allowed it (as all lines must, if they be 
drawn), and the length be to the breadth as num- 
ber to an unit ; the side and the roof will be all of 
one length. 


Any number given is produced by the greatest Third p ape r. 
root multiplied into itself, and into the .remaining 
fraction. Let the number given be two hundred 
squares, the greatest root is 14^ squares. I say 
that 200 is equal to the product of 14 into itself, 
together with 14 mul- 
tiplied into TJ-. For 
14 multiplied into it- 
self makes 196. And 
14 into ij makes 4! 
which is equal to 4. 
And 4 added to 196 
maketh 200 ; as was 
to be proved. Or 

take any other num- ^^^^^^^^^^^^^^^^^ 
ber 8, the greatest root is 2 ; 'which multiplied into 
itself is 4, and the remainder -| multiplied into 2, 
is 4, and both together 8. 

PROP. in. 

But the same square calculated geometrically by 
the like parts, consisteth (by Euclid n. 4) of the 
same numeral great square 196, and of the two 
rectangles under the greatest side 14, and the re- 
mainder of the side, or (which is all one) of one 
rectangle under the greatest side, and double the 
remainder of the side ; and further of the square 
of the less segment ; which altogether make 200, 
and moreover ~ of those 200 squares, as by the 
operation itself appeareth thus : 

The side of the greater segment is 14ir 

Which multiplied into itself makes 200. 


Third Taper. The product of 14, the greatest segment, into 
the two fractions TJ, that is, into -& (or into twice 
TT) is i-(that is 4); and that 4 added to 196 
makes 200. 

Lastly, the product of -^ into iV or 4~ into -j- is 
~&. And so the same square calculated by roots 
is less by -^r of one of those two hundred squares, 
than by the true and geometrical calculation ; as 
was to be demonstrated. 


It is hence manifest, that whosoever calculates 
the length of an arc or other line by the extrac- 
tion of roots, must necessarily make it shorter 
than the truth, unless the square have a true root. 


The Radius of a Circle is a Mean Proportion 
between the Arc of a Quadrant ami- two-jifths 
of the same. 

Describe a square A B C D 5 and in it a quadrant 
DC A. In the side DC take DT two-fifths of 
D C, and between D C and D T a mean propor- 
tional D R, and describe the quadrantal arcs R S, 
TV. I say the arc R S is equal to the straight 
line D C. For seeing the proportion of D C to 
D T is duplicate of the proportion of D C to D R, 
it will be also duplicate of the proportion of the 
arc C A to the arc R S, and likewise duplicate of 
the proportion of the arc R S to the arc T V. 

Suppose some other arc, less or greater than the 
arc R S ? to be equal to D C, as for example r s : 
then the proportion of the -aTe r s to the straight 
line D T will be duplicate of the proportion of R S 
to TV, or D R to D T. Which is absurd ; because 
D r is by construction greater or less than D R. 
Therefore the arc R S is equal to the side D C, 
which was to be demonstrated. 


Hence it follows that D R is equal to two-fifths 
of the arc C A. For R S, T V, D T 5 being con- 
tinually proportional, and 
the arc T V being described 
by D T, the arc R S will be 
described by a straight line 
equal to T V. But R S is 
described by the straight 
line DR. Therefore D R is 
equal to T V, that is to two- 
fifths of C A. 


Third paper And your said servant most humbly prayeth 
you to consider, if the demonstration be true and 
evident, whether the way of objecting against it 
by square root, used by Dr. Wallis ; and whether 
all his geometry, as being built upon it, and upon 
his supposition of an infinite number, be not false. 





DR. WALLIS says, all that is affirmed, is but ^considerations 

,7 . . f . *ii / 7i upon the answer 

we SUPPOSE that, this wiLljollow. O f or.waihs to 

But it seemeth to me, that if the supposition be ^ M^ 
impossible, then that which follows will either be * 
false, or at least undemoristrated. 

First, this proposition being founded upon his 
Arithmetica Infinitorum, if there he affirm an ab- 
solute infiniteness, he must, here also be understood 
to affirm the same. But in his thirty-ninth pro- 
position he saith thus : " Seeing that the number 
of terms increasing, the excess above sub-qua- 
druple is perpetually diminished, so at last it 
becomes less than any proportion that can be 
assigned; if it proceed in infinitum it must utterly 
vanish. And therefore if there be propounded an 
infinite rotv of quantities in triplicate proportion 
of quantities arithmetically proportioned (that is, 
according to the row of cubical numbers) beginning 
from a point or ; that row shall be to a row of 
as many, equal to the greater, as 1 to 4." It is 
therefore manifest that he affirms, that in an in- 
finite row of quantities the last is given ; and he 
knows well enough that this is but a shift. 

Secondly, he says, that usually in Euclid, and 
all after him, by infinite is meant but, more than 


rnn r an y ass ig na M e finite, or the greatest possible. I am 
or Dr. waiiw to content it be so interpreted. But then from thence 

the three papers , _ ii -i 1-1 

of MrHoubrs. he must demonstrate those his conclusions, which 
he hath not yet done. And when he shall have 
done it, not only the conclusions, but also the de- 
monstration, will be the same with mine in Cap. 
xiv. Art. 2, 3, &c. of my book De Cor pore. And 
so he steals what he once condemned. A fine 

Thirdly, he says, (by Euclid's tenth proposition, 
but he tells not of what book), that a line may 
be bisected, and the halves of it may again be 
bisected, and so onwards infinitely ; and that upon 
such supposed section infinitely continued, the 
parts must be supposed infinitely many. 

I deny that ; for Euclid, if he says a line may 
be divisible into parts perpetually divisible, he 
means that all the divisions, and all the parts aris- 
ing from those divisions, are perpetually finite in 

Fourthly, he says, that there may be supposed a 
row of quantities infinitely many, and continually 
increasing, whereof the last is given. 

It is true, a man may say, (if that be supposing) 
that white is black : but, if supposing be thinking, 
he cannot suppose an infinite row of quantities 
whereof the last is given. And if he say it, he can 
demonstrate nothing from it. 

Fifthly, he says (for one absurdity begets ano- 
ther) that a superficies or solid may be supposed 
so constituted as to be infinitely long, but finitely 
great, ( the breadth continually decreasing in 
greater proportion than the length incrcaseth), 
and so as to have no centre of gravity. Such is 


Toricellws Solidum Hyperholicum acutum, and considerations 
others innumerable, discovered by Dr. Wallis, Ti)r wanTt 
Monsieur Fermat, and others. But, .to determine ^ 
this, requires more of geometry and logic, (what- "" 
soever it do of the Latin tongue), than Mr. Hobbes 
is master of. 

I do not remember this of Toricellio, and I 
doubt Dr. Wallis does him wrong and Monsieur 
Fermat too. For, to understand this for sense, it 
is not required that a man should be a geometri- 
cian or a logician, but that he should be mad. 

In the next place, he puts to me a question as 
absurd as his answers are to mine. Let him ask 
himself, saith he, if he be still of opinion, that 
there is no argument in natural philosophy to 
prove that the world had a beginning. First, 
whether, in case it had no*beginning, there must 
not have passed an infinite' number of years before 
Mr. Hobbes was born. Secondly, whether, at this 
time, there have not passed more, that is, more 
than that infinite number. Thirdly, whether, in 
that infinite (or more than infinite) number of 
years, there have not been a greater number of 
days and hours, and of which, hitherto, the last is 
given. Fourthly, whether, if this be an absurdity, 
we have not then, (contrary to what Mr. Hobbes 
would persuade us), an argument in nature to prove 
the world had a beginning. 

To this I answer, not willingly, but in service to 
the truth, that, by the same argument, he might as 
well prove that God had a beginning. Thus, in 
case he had not, there must have passed an infinite 
length of time before Mr. Hobbes was born ; but 
there hath passed at this day more than that infi- 


considerations n jt e length, bv eiffhtv-four years. And this day, 

upon the answer ... . 

oi Dr. wains to which is the last, is given. If this be an absurdity, 

the three papers , , , , , . , , 

ot Mr. Hobbes. have we not. then an argument in nature to prove 
' that God had a beginning ? Thus it is when men 
entangle themselves in a dispute of that which they 
cannot comprehend. But, perhaps, he looks for a 
solution of his argument to prove that there is 
somewhat greater than infinite ; which I shall do 
so far as to show it is not concluding. If from 
this day backwards to eternity be more than infi- 
nite, and from Mr. Hobbes his birth backwards to 
the same eternity be infinite, then take away from 
this day backwards to the time of Adam, which is 
more than from this day to Mr. Hobbes his birth, 
then that which remains backwards must be less 
than infinite. All this arguing of infinites is but 
the ambition of schoolboys. 


There is no doubt if we give what proportion 
we will of the radius to the arc, but that the arc 
upon that arc will have the same proportion. But 
that is nothing to my demonstration. He knows 
it, and wrongs the Royal Society in presuming they 
cannot find the impertinence of it. 

My proof is this : that if the arc on T V, and the 
arc R S, and the straight line C D, be not equal, 
then the arc on T V 3 the arc on R S, and the arc on 
CA, cannot be proportional ; which is manifest by 
supposing in D C a less than the said D C, but 
equal to R S, and another straight line, less than 
R S, equal to the arc on T V ; and anybody may 
examine it by himself. 

I have been asked by some that think them- 


selves logicians, why I proceeded upon f- rather Considerations 

J J * r upon the answer 

than any other part of the radius. The reason I of Dr wain* to 

.,,,. A , i A -i i % tlie ^ lree papers 

had for it was, that, long ago, some Arabians hadofMr. 
determined, that a straight line, whose square is 
equal to 10 squares of half the radius, is equal to a 
quarter of the perimeter; but their demonstra- 
tions are lost. From that equality it follows, that 
the third proportional to the quadrant and radius, 
must be a mean proportional between the radius 
and -- of the same. But, my answer to the logi- 
cians was, that, though I took any part of the 
radius to proceed on, and lighted on the truth by 
chance, the truth itself would appear by the ab- 
surdity arising from the denial of it. And this is 
it that Aristotle means, where he distinguishes be- 
tween a direct demonstration and a demonstration 
leading to an absurdity. Hence it appears that 
Dr. Wallis's objections to my Rosetum are invalid 
as built upon roots. 


First, he says that it concerns him no more 
than other men, which is true. I meant it against 
the whole herd of them who apply their algebra 
to geometry. Secondly, he says that a bare num- 
ber cannot be the side of a square figure. I would 
know what he means by a bare number. Ten 
lines may be the side of a square figure. Is there 
any number so bare, as by it we are not to con- 
ceive or consider anything numbered ? Or, by 10 
nothings understands he bare 10? He struggles in 
vain, his conscience puzzles him. Thirdly, he says 
10 squares is the root of 100 square squares. To 


Considerations which I answer, first, that there is no such figure 

upon the answer t 

ot Dr waii.s to as a square square. Secondly, that it follows 

the three papers \ , J . . r> . / i , ^ 

ofMr Hobbes hence, that -a root is a superficies, for such is 10 
' squares. Lastly, he says that, neither the number 
10, nor 10 soldiers, is the root of 100 soldiers; 
because 100 soldiers is not the product of 10 
soldiers into 10 soldiers. This last I grant, be- 
cause nothing but numbers can be multiplied into 
one another. A soldier cannot be multiplied by 
a soldier. But no more can a square figure by a 
square figure, though a square number may. 
Again, if a captain will place his 100 men in a 
square form, must he not take the root of 100 to 
make a rank or file ? And are not those 10 men ? 



He objects nothing here, but that the side of a 
square is not a superficies, but a line, and that a 
square root (spealnng of quantity) is not a line, 
but a rectangle, is a contradiction. The reader is 
to judge of that. 

To his scoffings I say no more, but that they 
may be retorted in the same words, and are there- 
fore childish. 

And now I submit the whole to the Royal 
Society, with confidence that they will never en- 
gage themselves in the maintenance of these unin- 
telligible doctrines of Dr. Wallis, that tend to the 
suppression of the sciences which they endeavour 
to advance. 








THOUGH I may goe whither and when I will for Letter i 
anie necessity you have of my service, yet there is 
a necessity of good manners that obliges me as yo r 
servant to lett you knowe att all times where to 
find me. Wee goe out of Paris 3 weekes hence, 
or sooner, towards Venice, but by what way I 
knowe not, because the ordinary high way through 
the territory of Milan is encumbered with the 
warre betweene the Frerfch and the Spaniards. 
Howsoever, wee have to be there in October next. 
If you require anie service that I can doe there, 
it may please you to convey your command by 
Devonshire house. But if you command me no- 
thing, I have forbidden my letters to look for 
answer : their busines being only to inforrne and 
to lett you knowe that the image of your noblenes 
decayes not in my memory, but abides fresh to 
keepe me eternally Your 


1 This letter is to be found in out date but the allusion to the 

the British Museum, amongst the war between France and Spain, and 

Lansdowne MSS. 238, entitled " a the passage in the VITA THO.HOBBES, 

collection of letters to and from " Anno sequente qui erat Christi 

persons of eminence in the reigns of 1629, rogatus a nobilissimo viro 

Elizabeth, James I, and Charles I, domino Gervasio Clifton", &c. (p. 

made by some person in the service xiv), show that it must have been 

of Sir Gervas Clifton". It is with- written in either 1(529 or 1630. 

G G2 





I HAVE been behind hand with you a long time 
for a letter I received of yours at Angers, that 
place affording nothing wherewith to pay a debt 
of that kind, all matter of news being sooner 
known in England than here : and the news you 
writ me was of that kind, that none from England 
could be more welcome, because it concerned the 
honour of Welbeck arid Clifton, two houses in 
which I am very much obliged. 

Monsieur having given the slip to the Spaniards 
at Bruxelles, came to the King about ten days ago 
at St. Germains, where, he was received with great 
joy. The next day the Cardinal entertained him 
at Ruelle : and the day after that he went to 
Limours, where he is now, and from thence he 
goes away shortly to Bloys, to stay there this 
winter. The Cardinal of Lyons is going to Rome 
to treat about the annulling of Monsieur's mar- 
riage,, which is here by Parliament declared void, 
but yet they require the sentence of the Pope. 
There goes somebody thither on the part of his 
wife, to get the marriage approved : but who that 
is, I know not. The Swedish party in Germany is 
in low estate, but the French prepare a great army 
for those parts, pretending to defend the places 
which the Swedes have put into the King of 
France his protection, whereof Philipsbourgh is 
one ; a place of importance for the Lower Palati- 
nate. This is all the French news. 

For your question, why a man remembers less 


his own face, which he sees often in a glass, than i>uei ir 
the face of a friend that he has not seen of a great 
time, my opinion in general is, that .a .man remem- 
bers best those faces whereof he has had the 
greatest impressions, and that the impressions are 
the greater for the oftener seeing them, and the 
longer staying upon the sight of them. Now you 
know men look upon their own faces but for short 
fits, but upon their friends' faces long time to- 
gether, whilst they discourse or converse together ; 
so that a man may receive a greater impression 
from his friend's face in a day, than from his own 
in a year ; and according to this impression, the 
image will be fresher in his mind. Besides, the 
sight of one's friend's face two hours together, is 
of greater force to imprint the image of it, than 
the same quantity of time by intermissions. For 
the intermissions do easily deface that which is 
but lightly imprinted. In general, I think that 
lasteth longer in the memory which hath been 
stronglier received by the sense. 

This is my opinion of the question you pro- 
pounded in your letter. Other new truths I have 
none, at least they appear not new to me. There- 
fore if this resolution of your first question seems 
probable, you may propound another, wherein I 
will endeavour to satisfy you, as also in any thing 
of any other nature you shall command me, to my 
utmost power ; taking it for an honour to be es- 
teemed by you, as I am in effect, 

Your humble and faithful servant 


Paris, Oct.--, 1634. 

My Lord Fielding and his Lady came to Paris 
on Saturday night last. 




I RECEIVED here in Florence, two days since, a 
letter from you of the 19th of January. It was 
long by the way ; but when it came it did tho- 
roughly recompence that delay. For it was worth 
all the pacquets I had received a great while to- 
gether. All that passeth in these parts is equally 
news, and therefore no news ; else I would labour 
to requite your letter in that point, though in the 
handsome setting down of it, I should still be your 

I long infinitely to see those books of the Sab- 
baoth ? , and am of your mind they will put such 
thoughts into the heacb of vulgar people, as will 
confer little to their good life. For when they see 
one of the ten commandments to be jus humanum 
merely, (as it must be if the Church can alter it), 
they will hope also that the other nine may be so 
too. For every man hitherto did believe that the 
ten commandments were the moral, that is, the 
eternal law. 

I desire also to see Selden's Mare Clausum, 
having already a great opinion of it. 

You may perhaps, by some that go to Paris, 
send me those of the Sabbaoth, for the other 
being in Latin, I doubt not to find it in the Rue 
' St. Jaques. 

1 Probably George Glen, who was " The History of the Sabbath, 
installed Prebend of Worcester in In two books. By Peter Heylyn. 
1 660, and died in 1669. 4to. 1636. 


We are now come hither from Rome, and hope Letter in. 
to be in Paris by the end of June. I thank you 
for your letter, and desire you to believe that I 
can never grow strange to one, the goodness of 
whose acquaintance I have found by so much ex- 
perience. But I have to write to so many, that I 
write to you seldomer than I desire ; which I pray 
pardon, and esteem me 

Your most affectionate friend 

and humble servant 


Florence, Apr. 1636. 

My Lord and Mr. Nicholls, and all our com- 
pany commend them to you. 




THE last weeke I had the honor to receave two 
letters from you at once, one of the 30 of Dec., 
the other of the 7 th of Jan., w ch I acknowledged, 
but could not answer in my last. In the first you 
begin with a difficulty on the principle of Mons r 
de Cartes, that it is all one to move a weight two 
spaces, or the double of that waight one space^ 
and so on in other proportions : to w ch you object 
the difference of swiftnesse, w ch is greater w r hen a 
waight is moved two spaces then when double 
waight is moved one space. Certenly de Cartes 
his meaning was by force the same that mine, 

1 Harlcian MS. 079(5. 


namely, a multiplication of the weight of a body 
in to the swiftnesse wherew th it is moved. So that 
when I move a pound two foote a the rate of a 
mile an howre, I do the same as if of 2 poundes I 
moved one pound a foote at y e rate of a mile an 
how r er, the other pound another foote at the same 
rate, not in directu, but 
parallell to the first pound. 
As if the wayt A 13 were 
moved to CD at the rate 
of a mile an howre, 'tis all 
one as if the waight A E were moved to FH at 
the same rate. Here is all the difference : this 
swiftnesse or rate of a mile an howre is, in 
the first case, layd out in the 2 spaces AG, GC, 
the latter, in the 2 spaces AG, EG. The first 
case, as like as if a footman should run w th double 
swiftnesse endwayes, w c ^is y e doubling of swiftnesse 
in one man : in the other, it is as if you doubled 
the swiftnesse by doubling the man : for every 
man has his owne swiftnesse. And so AH is the 
swiftnesse AG doubled, as well as AD. For that, 
that Mons r de Cartes will not have just twice 
the force requisite to move the same weight twice 
as fast, I can say nothing. The papers I have 
of his touching that are in my trunk, w ch hath 
bene taken by Dunkerkers, and taken againe from 
them by French, and at length recovered by frends 
I made : but I shall not have it yet this fortnight. 
In the mearie time I am not in that opinion, but 
do assure myself, the patient being the same, 
double force in the agent shall worke upon it 
double effect. 

In the same letter you require a better expli- 
cation of y e proportion I gather betweene wayght 



and swiftnesse : wherein, because you have not my 

figure, I imagine you have mistaken me very much. 

Arid first, you thinke, I sup- D . 

pose, DE equall to AB: w ch 

I am sure is a mistake. For I 

put AB for any line you will 

to expresse a minutu secundum. 

I will, therefore, go over againe 

the demonstration I sent you 

before, and see if I can do it 


K F 

Let AB stand for the time 
knowne wherein the waight D descendeth to E. 
And let there bee a cylinder of the same matter 
the waight D consisteth of, and let the altitude of 
that cylinder be DC: w ch I shew before was the 
swiftnesse wherew th that cylinder presseth, not 
wherew th itfalleth. And wee are now to enquire 
how farre such matter as the cylinder is made of 
must descend from D, before it attayne a swift- 
nesse equall to this pressing swiftnesse DC. And 
I say it must fall to L. For in the time A B it is 
knowne that the waight in D will fall to E : and 
it is demonstrated by Gallileo, that when such 
waight comes to E, it shall be able to go twice 
the space it hath fallen in the same time. There- 
fore the waight D being in E, hath velocity to 
carry him the space DK (w ch I put double to DE) 
in the same time AB. But I put BF equall to 
DK. Therefore, in the time AB, the waight's 
velocity acquired in E shall be such as to go from 
B to F without decrease of velocity by the way. 
Hence I go on to finde in what point the waight 
in D comes to where it getteth a velocity equall to 



Letter iv. CD. Therefore, I apply DC to GH, parallel to 
' ' ' BF : and then it is, as the time AB to the time 
AG, so the velocity acquired at the end of the time 
AB to the velocity acquired at the end of the time 
AG. For the swiftnesse acquired from time to 
time (I say, not from place to place, but from time 
to time) are proportionable to the times wherein 
they are acquired: \v ch is the postulate on w ch 
Galileo builds all his doctrine. And as A B to 
AG, so the line BF to the line GH. But, at the 
end of the time AB, the waight D is by suppo- 
sition in E, in that degree of velocity as to go B F 
or DK in the same time AB. The question there- 
fore is, where the waight D shall be at the end of 
the time AG. For there it hath the velocity of 
going G H or D C in the same time, because the 
velocity GH is to the velocity BF as the line GH 
to the line BF, or as the time of descent AG to 
the time AB. But, because the spaces of the de- 
scent are in double the proportion of the times of 
descent, make it as BF to GH, that is, DK to DC, 
so DC to another, DL. The velocity, therefore, 
acquired in the point of descent E, namely the 
velocity DK or BF, is to the velocity acquired in 
the point L, namely, the velocity Gil or DC, (w ch 
is the velocity of the cylinder's waight), as DK to 
DL. And therefore in L the waight D has ac- 
quired a velocity equal to the velocity of the waight 
of the cylinder. 

In the same letter you desire to knowe, how any 
medifi, as water, retardeth the motion of a stone 
that falls into it. To w ch I answer out of that you 
say afterwards, that nothing can hinder motion but 
contrary motion : that the motion of the water, 


when a stone falls into it, is point blanke contrary Letter iv. 
to the motion of the stone. For the stone by de- 
scent causeth so much water to ascend as the big- 
nesse of the stone comes to. For imagine so much 
water taken out of the place w ch the stone occu- 
pies., and layd upon the superficies of the water : it 
presseth downeward as the stone does, and maketh 
the water that is below to rise upw r ards, and this 
rising upwards is contrary to the descent, and is 
no other operation than we see in scales, when 
of two equal bullets in magnitude that w ch is of 
heavier metal maketh the other to rise. And thus 
farre goes your letter of Dec. 30. 

For the first quaere in your second letter, con- 
cerning how we see in the time the lucide body 
contracts itselfe, I have no other solution but that 
w ch your selfe hath given : jpv ch is, that the recipro- 
cation is so quicke, that the effect of the first mo- 
tion lasteth till the next comes, and longer. For 
by experience we observe that the end of a fire- 
brand swiftely moved about in circle, maketh a cir- 
cle of fire : w ch could not be, if the impression made 
at the beginning of the circulation did not last till 
the end of it. For if the same firebrand be moved 
slower, there will appear but a peece of a circle, 
bigger or lesser according to the swiftnesse or slow- 
nesse of y e motion. For the cause of such reci- 
procation, it is hard to guess what it is. It may 
well be the reaction of the medium. For though 
the medifi yeld, yet it resisteth to : for there can 
be no passion w th out reaction. And if a man could 
make an hypothesis to salve that contraction of y e 
sun, yet such is the nature of naturall thinges, as a 
cause may be againe demanded of such hypothesis : 
and never should one come to an end w th out as- 


Letter iv. signing the immediate hand of God. Whereas in 
mathematicall sciences wee come at last to a defi- 
nition, w ch is p, beginning or principle, made true 
by pact and consent amongst ourselves. Further, 
you conceave a difficulty how the medium can be 
continually driven on, if there be such an alternate 
contraction. To w ch , first I answer, that the mo- 
tion forward is propagated to the utmost distance 
in an instant, and the first push is therefore enough, 
and in another instant is made the returne back in 
y e like manner. And though it were not done in 
an instant, yet we see by experience in rivers, as 
in y e Thames, that the tide goes upward towards 
London pushed by the water below, and yet at the 
same instant the water below is going backe to the 
sea. For seeing it is high-water at Blackwall be- 
fore 'tis so at Greenwich, the water goes backe 
from Blackwall goes on at Greenwich. 
And so it would happen, though Blackwall and 
Greenwich were nerer together then that any quan- 
tity given could come betweene. 

In my letter from London, speaking of the re- 
fraction of a bullet, I thinke I delivered my opinion 
to be, that a bullet falling out of a thinner medium 
into a thicker, looseth in the entring nothing but 
motion perpendicular : but being entred, he looseth 
proportionably both of one and the other. For 
suppose a bullet, whose diameter is AB, be in the 
thiner mediu, and enter at C PA. 

into the thicker medium. The 
thicker medium, at the first 
touch of B in the point C, work- 
eth nothing upon the line AB. 
And when the diameter AB ^/ 
is entred, suppose halfe way, 


yet the thicker medium operates laterally but on LPU^ iv. 
one halfe of it. So that in the sornme there is a 
losse of velocity perpendicular (to the quantity that 
the diameter AB requires) without any offence to 
the motion lateral!, but so much of the diameter 
as is within the thicker mediu is retarded both 
wayes, and looses of his absolute motion, w ch is 
compounded of perpendicular and laterall, and that 
proportionally. Suppose now that a bullet passe 
from A to D, and receave a peculiar losse of his 
perpendicular motion by entring at D, so great 
that he proceed in the perpendicular but halfe so 
farre, as for example from D to I : and then being 
in, the thicknesse of the medium take away more 
of his velocity both perpendicular and laterall, sup- 
pose halfe that w ch was left of the perpendicular 
motion and halfe of his first laterall motion, so that 
the perpendicular motion is but D K, and the late- 
rail motion DE. Then will the line of refraction 
be DG. As for that argumentation of DCS Cartes, 
it is, in my opinion, as I have heretofore endea- 
vored to shew you, a mere paralogism. 

Lastly, you make this quaere, why light hath not 
at severall inclinations severall swiftnesses as well 
as a bullet. The bullet itself passeth through the 
severall media : whereas in the motion of light, the 
body moved, w ch is the mediu, cntreth not into the 
other medium, but thrusteth it on: and so the 
parts of that medium thrust on one another, 
wliereby the laterall motion of the thicker medium 
hath nothing to worke upon, because nothing 
enters, but stoppes onely and retardes, in oblique 
incidence, that end w ch comes first to it, and 
thereby causes a refraction the contrary way to 


Letter iv that of a bullet, in such manner as I set forth to 
you in one of my letters from hence concerning 
the cause of refraction. And this is all I can say 
for the present to the quaeres of y r two last letters. 
I have enquired concerning perspectives after 
the manner of De Cartes. Mydorgius tells me 
there is none that goes about them, as a thing too 
hard to do. And I believe it. For here is one 
Mons r de Bosne in towne, that dwells at Bloys, an 
excellent workman, but by profession a lawier, and 
is counsellor of Bloys, and a better philosopher in 
my opinion then De Cartes, and not inferior to him 
in the analytiques. I have his acquaintance by 
Pere Mersenne. He tells rne he hath tryed De 
Cartes his way, but cannot do it : and now he 
workes upon a crooked line of his owne invention. 
He sayes he shall have made one w th in a moneth 
after he shall returne to Bloys : after that he will 
see what he can discover in the heavens himselfe, 
and then if he discover any new thing he will let 
his way be publique together w th the effects. This 
is all the hope I can give you yet. So w th my 
prayers to God to keepe you in prosperity this 
troublesome time, I rest 

Your most humble and obedient servant 


Paris, Feb. 8, stile no. 1641. 

To the Right Honorable 
present these 

at Wellinger, 






THE young woman at Over-Haddon hath been 
visited by divers persons of this house. My Lord 
himself hunting the hare one day at the Town's 
end, with other gentlemen and some of his ser- 
vants, went to see her on purpose : and they all 
agree with the relation you say was made to your- 
self. They further say on their own knowledge, 
that part of her Belly touches her Back-bone. She 
began (as her Mother says) to loose her appetite 
in December last, and had lost it quite in March 
following : insomuch as that since that time she 
has not eaten nor drunk ay thing at all, but only 
wetts her lips with a feather dipt in water. They 
w r ere told also that her gutts (she alwayes keeps 
her bed) lye out by her at her fundament shrunken. 
Some of the neighbouring ministers visit her often : 
others that see her for curiosity give her mony, 
sixpence or a shilling, which she refuseth, and her 
mother taketh. But it does not appear they gain 
by it so much as to breed a suspition of a cheat. 
The woman is manifestly sick, and 'tis thought she 
cannot last much longer. Her talk (as the gentle- 
woman that went from this house told me) is most 
heavenly. To know the certainty, there bee many 
things necessary which cannot honestly be pryed 
into by a man. First, whether her gutts (as 'tis 
said) lye out. Secondly, whether any excrement 

1 Amongst the MSS. of the Royal Society. 


v. p ass tj la j. wa y^ or I10 ne at all. For if it pass, 
though in small quantity, yet it argues food pro- 
portionable, which may, being little, bee given her 
secretly and pass through the shrunken intestine, 
which may easily be kept clean. Thirdly, whether 
no urine at all pass : for liquors also nourish as 
they go. I think it were somewhat inhumane to 
examin these things too nearly, when it so little 
concerneth the commonwealth : nor do I know of 
any law that authoriseth a Justice of peace, or 
other subject, to restrain the liberty of a sick 
person so farr as were needful for a discovery of 
this nature. I cannot therefore deliver any judg- 
ment in the case. The examining whether such a 
thing as this bee a miracle, belongs I think to the 
Church. Besides, I myself in a sickness have been 
without all manner of^sustenance for more than 
six weeks together : which is enough to make mee 
think that six months would not have made it a 
miracle. Nor do I much wonder that a young 
woman of clear memory, hourely expecting death, 
should bee more devout then at other times. Twas 
my own case. That which I wonder at most, is 
how her piety without instruction should bee so 
eloquent as 'tis reported. 


Chatsworth, Oct. 20. 68. * 





IN the last Transactions for September and Oc- 
tober I find a letter addressed to you from l) r 
Wallis, in answer to my LUX MATHEMATICA. I 
pray you tell me that are my old acquaintance, 
whether it be (his words and characters supposed 
to be interpreted) intelligible. I know very well 
you understand sense both in Latine, Greeke, and 
many other languages. He shows you no ill con- 
sequence in any of my arguments. Whereas I say 
there is no proportion of infinite to finite. He 
answers, he meant indefinite ; but derives not his 
conclusion from any other 'notion than simply in- 
finite. I said the root of a square number cannot 
be the length of the side of a square figure, be- 
cause a root is part of a square number, but length 
is no part of a square figure. To which he an- 
swers nothing. In like manner, he shuffles off all 
my other objections, though he know well enough 
that whatsoever he has written in Geometry 
(except what he has taken from me and others) 
dependeth on the truth of my objections. I per- 
ceive by many of his former writings that I have 
reformed him somewhat as to the Principles of 
Geometry, though he thanke me not. He shuffles 
and struggles in vaine, he has the hooke in his 
guills, I will give him line enough : for (which I 
pray you tell him) I will no more teach him by 

1 Amongst the MSS. of the Royal Society. 


Letter vi replying to any thing he shall hereafter write, 
whatsoever they shall say that are confident of his 
Geometry. Qul volunt decipi, decipiuntur. He 
tells you that I bring but crambe scepe cocta. For 
which I have a just excuse, and all men do the 
same; they repeat the same words often when 
they talk with them that cannot heare. 

I desire also this reasonable favour from you: 
that, if hereafter I shall send you any paper tend- 
ing to the advancement of physiques or mathe- 
matiques, and not too long, you will cause it to be 
printed by him that is printer to the Society, as 
you have done often for D r Wallis : it will save me 
some charges. 

I am, S r , 

Your affectionate' frend and humble seruant 


November the 26th, 1672. 

ffor my worthy and much honoured 
Secretary to the Royal Society. 


VII. , . 



THE passions of man's mind, except onely one, 
may bee observed all in other living creatures. 
They have desires of all sorts, love, hatred, feare, 
hope, anger, pitie, semulation, and y e like : onely 
of curiositie, which is y desire to know y e causes 
of thinges, I never saw signe in any other living 
creature but in man. And where it is in man, I 
find alwaies a defalcation or abatement for it of 
another passion, which in beastes is commonly 
predominant, namely, a ravenous qualitie, which 
in man is called avarice. The desire of know- 
ledge and desire of nee^ilesse riches are incom- 
patible, and destructive one of another. And 
therefore as in the cognitive faculties reason, so 
in the motive curiositie, are the markes that part 
y e bounds of man's nature from that of beastes. 
Which makes mee, when I heare a man, upon the 
discovery of any new and ingenious knowledge or 
invention, aske gravely, that is to say, scornefully, 
what 'tis good for, meaning what monie it will 
bring in, (when he knows as little, to one that 
hath sufficient what that overplus of monie is 
good for), to esteeme that man not sufficiently re- 

1 Uarlcian MS 22$^^ treatise DE HOMINE : the first, On Illumi- 

on Optics, entitleo^Wlftinute or nation, was never published. The 

t of the Cliques. In dedication to the Marquis of 

B} r Thomas Hobbes. Newcastle, and the concluding 

At Paris, 1 (> 16." The second pait, paiagraph, is all that is here 

On Vision, we have in Latin, in the given of the tieatibc. 


NO vn moved from brutalitie. Which I thought fit to 
say by way of anticipation to y e grave scorners of 
philosophic, and that your lordship, after having 
performed so noble and honourable acts for de- 
fence of your countrie, may thinke it no dishonour 
in this unfortunate leasure to have employed some 
thoughts in the speculation of the noblest of the 
senses, vision. 

That which I have written of it is grounded 
especially upon that w ch about 1 6 yeares since I 
affirmed to your Lo pp at Welbeck, that light is a 
fancy in the minde, caused by motion in the braine, 
wiiich motion again e is caused by the motion of y e 
parts of such bodies as we call lucid : such as are the 
sunne and y e fixed stars, and such as here on earth 
is fire. By putting you in mind hereof, I doe in- 
deed call you to witness^ of it : because, the same 
doctrine having since been published by another, 
I might bee challenged for building on another 
man's ground. Yett philosophical ground I take 
to be of such a nature, that any man may build 
upon it that will, especially if the owner himselfe 
will nott. But upon this ground, with the helpe 
of some other speculations drawne from the na- 
ture of motion and action, I have, I thinke, de- 
rived y e reason of all the phenomena I have mett 
with concerning light and vision, both solidly 
enough nott to be confuted, and withall easilie 
enough to be understood by such as can give that 
attention thereto which the figures, whereby such 
motion as causeth vision is described, do require. 
All that I shall bee ever able to adde to it, is polish- 
ing : for, being the first draught, it could nott bee 
so perfect as I hope hereafter to make it in Latin. 


Butt as it is, it will sufficiently give your Lo pp satis- NO vn. 
faction in those quaeres you were pleased to make 
concerning this subject. I am content that it 
passe, in respect of some drosse that yett cleaves 
to it, for ore : w ch is much better than old ends 
raked out of the kennell of sophisters' bookes. 
And for such I commend it to your Lo p , and myselfe 
to your accustomed good opinion : which hath 
beene hitherto so greate honour to mee, as I am 
nott known to the world by any thing so much as 
by being, 

My most noble lord, 
Your Lo p ' J most humble 

andcinost obliged servant 


The treatise ends with the following passage: 

To conclude, I shall doe like those that build a 
new house where an old one stood before, that is 
to say, carry away the rubbish. 

And first, away goes the old opinion that the 
sliewes (which they call visible species) of all ob- 
jects, are in all places, arid all the babble de extra- 
mittendo ct intromittendo. For their species are 
iiothing else but fancie, made by the light pro- 
ceeding directly or by repurcussion or refraction 


NO vii made from the object to y e eye, and so moving the 
braine and other parts within. 

Secondly, the opinion which Vitellio takes for 
an axiome and foundation of his Catoptricques, 
that y e place of y e image by reflexion is in the 
perpendicular drawn from the object to the glasse. 
For it is false both in plaine glasses and in sphse- 
ricall, whether convexe or concave. 

Thirdly, the opinion that light is engendred 
faster in hard bodies, as glasse, than in thin and 
fluid, as aire. 

Fourthly, that objects are seen by penicitti that 
have their common base in the pupills : for y e 
center of y e eye is in their common base. 

Fifthly, the opinion that there bee other visuall 
lines by which wee see distinctly besides y e optique 

Sixthly, the opinion that perspective glasses and 
amplifying glasses are best made of hyperbolicall 

Seventhly, the opinion that light is a bodie, or 
any other such thing than such light as wee have 
in dreames. 

Eighthly, that y e object appeares greater and 
lesse in y e same proportion that y e angles have 
under which they are scene. 

Lastly, is to be cast away the conceipt of mil- 
lions ot strings in y e optique nerve, by which the 
object playes upon, the braine, and makes y e soule 
listen unto it, and other innumerably such trash. 

How doe I feare that y c attentive reader will 
find that which I have delivered concerning y e 
Optiques fitt to bee cast outt as rubbish amonfj4he 
rest. If hee doe, hee will recede from y e authoritie 


of experience, which confirmeth all I have said. NO vn 
it bee found true doctrine, (thpjigh yett it 
. joshing), I shall deserve {he reputation 
beene y e first to lay the grounds of two 
sciences ; this of Optiques, y e most curious, and y* 
other of Natural Justice, which I have done in 
my booke DE GIVE, y e most profitable of all other. 


SHEWETH, that though your Majesty hath been 
pleased to take off the restraint of late years laid 
upon the pensions payable out of your privy purse, 
yet your Majesty's Officers refuse to pay the pen- 
sion of your petitioner without your Majesty's 
express command. 

And humbly beseaceth your Majesty, (consider- 
ing his extreme age, perpetual infirmity, frequent 
and long sickness, and the aptness of his enemies 
to take any occasion to report that your peti- 
tioner by some ill behaviour hath forfeited your 

Additional MSS. 4292. Brit. Mus. 


NO. viii. wonted favour), that you would be pleased to 
renew ycw\,order for the payment of it in such 
manner as to 'his great comfort he hath for many 
years enjoyed it. 2 

And daily prayeth to God Almighty to bless 
your Majesty with long life, constant health, and 

8 Dcinde redux mihi Rex conccssit habeie quotannis 
Centum alias libias ipsius ex locuhs 
Dulce mihi don urn. Vn A Carm. cxpres. p. xcviii.