OF TRANSPARENCE, REFRACTION, AND OF THE POWER
OF THE EARTH TO PRODUCE LIVING CREATURES.
_A._ Thinking upon what you said yesterday, it looked like a generation
of living creatures. I saw the love between the loadstone and the iron
in their mutual attraction, their engendering in their 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?
_B._ The most of them, if not all, that have written on this subject,
when they call the loadstone a terrella, 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._ I 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 transparent then, shall
never be made transparent by human art.
_A._ Do not you see that every day men make glass, and other diaphanous
bodies not much inferior in beauty to the fairest gems?
_B._ It is one thing to make one transparent of many by mixture, and
another to make transparent of not transparent. Any very hard stone, if
it be beaten into small sands, such as is used 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 transparent. 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 can make one transparent body of many, why do they not
of a great many small sparks of natural diamond compound one great one?
It would bear the charges of all the materials, and beside, enrich them.
_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 compound of many small ones, and
made by the earth’s annual motion; for it is a very swift motion.
Suppose 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 substances lay mingled
together at first, till it pleased God to separate all dissimilar
natures, and congregate 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 the force of the air that could break off those diaphanous
atoms in a cavern, can do the same in 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, when 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 I mean by _densum_
any body which more resisteth the motion of the air, and by _rarum_ that
which resisteth less.
_A._ But you have not declared in what that resistance consisteth.
_B._ I suppose it proceedeth from the hardness.
_A._ But from thence it will follow, that all transparent 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 refraction. For I never considered refraction
but in glass, because my business then was only to find 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 above, as from
the sun, let C A be the line of incidence, and produced to D; and draw A
E perpendicular 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 inward, towards A E, as in A _b_.
Thus it is in refraction 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
_l_, and make C _l_ 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
_e_, 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 _l_ 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 extinguished. But in this
case, where it lieth upon air, the first endeavour of the lowest part
worketh 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 transparent.
_B._ There are divers causes that take away transparency. First, if the
body be not perfectly homogeneous, 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
refractions 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 wood
and ordinary stone and metal. Secondly, the gravity and hardness may be
so great, as to make the angle refracted so great, as the second
refraction shall not direct the beam of light to the eye; as if the
angle of refraction were D A E, 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 demonstrated, that as the sine of the angle
refracted in one inclination is to the sine of the angle refracted in
another inclination, so is the sine of one inclination to the sine of
the angle of the other inclination. Which demonstrations are both
published 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 transmitted, 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 causes of the diversity of
colours. But I am so well satisfied 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 breaking,
brought forth, like so many eggs, all manner of living creatures great
and small, which 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 the earth produce
every living creature, cattle, and creeping thing, &c._ But then again
(ver. 25) it is said that _God made the beast of the earth, &c._ 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 produce 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 produce 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_. And 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 understanding. 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 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? 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 OF A STRAIGHT LINE TO HALF THE ARC OF A QUADRANT.
[Illustration]
Describe the square A B C D, and divide it by the diagonals A C and B D,
as also by the straight lines E G, 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
certain 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 somewhere 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 arc K L.
4. That the arc of a quadrant described in the 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.
DEMONSTRATION.
[Illustration]
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 K L; and E G the radius 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 proportion of
two to one, it shall not pass through the point _a_, 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.
SIX LESSONS
TO THE
PROFESSORS OF THE MATHEMATICS,
ONE OF GEOMETRY, THE OTHER OF ASTRONOMY,
IN THE CHAIRS SET UP BY THE NOBLE AND LEARNED SIR HENRY SAVILE, IN THE
UNIVERSITY OF OXFORD.
TO THE RIGHT HONOURABLE
HENRY LORD PIERREPONT,
VISCOUNT NEWARK, EARL OF KINGSTON, AND
MARQUIS OF DORCHESTER.
MY MOST NOBLE LORD,
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 forfeiture 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 consider. 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 precognition of the causes,
generation, and construction of the same; and consequently where the
causes are known, there is place for demonstration, but 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
demonstrations, they ought not to be numbered among the principles of
geometry. And Sextus Empiricus maketh use of them (misunderstood, yet so
understood 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
receives 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 demonstration, 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 in 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 whatsoever; 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 manner 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 because 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 reprehend 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 sometimes less.
4. That a quantity may grow less and less eternally, so as at last to be
equal to another quantity; or, which is all one, that there is a last in
eternity.
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 _Elenchus_ of my geometry, but also in
his treatises of the _Angle of Contact_, and in his _Arithmetica
Infinitorum_; which two last I have here in two or three leaves wholly
and 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 ἀποκατάϛασις 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 finding 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 branching 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 theorems 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 and short
lessons; and if your Lordship find them agreeable to your reason and
judgment, let me, notwithstanding the clamour of my adversaries, 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,
THOMAS HOBBES.
LONDON, _June 10, 1656_.
LESSONS
OF
THE PRINCIPLES OF GEOMETRY, &c.
TO THE EGREGIOUS PROFESSORS OF THE MATHEMATICS, ONE OF
GEOMETRY, THE OTHER OF ASTRONOMY, IN THE CHAIRS SET
UP BY THE NOBLE AND LEARNED SIR HENRY SAVILE,
IN THE UNIVERSITY OF OXFORD.
LESSON I.
I suppose, most egregious professors, you know already that by geometry,
though the word import no more but the measuring of land, is understood
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 understanding 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
measured. 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 geometry, determine by definition the signification of every
word which they make the subject or predicate of any theorem they
undertake to demonstrate; so 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 properties 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 anywhere defined
it, but done what he undertook without it. And though having perpetually
occasion to speak of measure, he hath not defined measure; yet instead
thereof he hath, in the beginning of his first elements, assumed an
axiom which serveth his turn sufficiently as to the measure of lines,
which is the eighth axiom; that those things which lie upon one another
all the way (called by him ἐφαρμόζοντα) are equal. Which axiom is
nothing else but a description of the art of measuring length and
superficies. For this ἐφάρμοσις 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 dimensions.
Quantity is that which 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 magnitude or quantity, but by saying it is _tantum_, _so
much_. And they that first called it in Greek, πηλικότης, and in Latin
_quantity_, might more properly have called it in Latin _tantity_, and
in Greek τηλικότης; and we, if we allowed ourselves the eloquence of the
Greeks and Latins, 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, determining
the magnitude to be _so much_. Also wheresoever 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 measuring 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 magnitudes 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 broad, &c. But in the mind of
an intelligent man it breedeth no mistake.
Besides the quantity of bodies, there is a quantity 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 because 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 we not improperly ask with _how much_ swiftness 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 consist 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 weight, by a solid only.
So also is number quantity; but in no other sense than as a line is
quantity divided into equal parts.
Of an angle, which is of two 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 body only can be the subject, in which body those straddling
lines are marked.
And because two lines may be made to divaricate by two causes; one, when
having one end common and 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, which 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 consequently 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 contingence.
Also there lieth a question of _how much comparatively_ one magnitude is
to another magnitude, as how 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
proportion, 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 geometrical.
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 comparison.
Also concerning heat, light, and divers other qualities, which have
degrees, there lieth a question 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 philosophy.
To the principles of geometry the definition appertaineth 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 quantity_. In which definition, instead of that ἐφαρμογὴ
of Euclid, I put coincidence. For the superposition of quantities, by
which they render the word ἐφαρμογὴ, cannot be understood of bodies, but
only of lines and superficies. Nevertheless 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 ἐφάρμοσις,
but not superposition.
Also the definitions of _greater_, _less_, and _equal_, are necessary
principles of geometry. For it cannot 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 ἐφάρμοσις, 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 another.
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 motion equal lines are
described. And equally swift are those motions by which we run over
equal spaces in any time determined by any other motion. And universally
all quantities are equal, that are measured by the same number of the
same measures.
It is necessary also to the science of geometry, to define what
quantities are of one and the same kind, which they call _homogeneous_,
the want of which definitions hath produced those wranglings (which your
book _De Angulo Contactus_ will not make to cease) about the angle of
contingence.
_Homogeneous_ quantities are those which may be compared by (ἐφάρμοσις)
application of their measures 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, which
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 homogeneous, because they measure
one another by application, to the beam of a balance. Line and angle
simply so called, have their quantity homogeneous, 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 heterogeneous. For the measures by which two angles
simply so called are compared, are in two coincident 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 therefore one of them is not applicable
to the other; and consequently of these two sorts of angles the
quantities are heterogeneous. The quantities of two angles of
contingence are homogeneous; for they may be measured by the ἐφάρμοσις
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 definitions 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
demonstration, but of construction, commonly called petitions; as that
it may be granted _that a man can draw a straight line, and produce it;
and with any radius, on any centre describe a circle_, and the like. For
that a man may be able to describe a square, he must first be able to
draw a straight line; and before he can describe an equilateral
triangle, he must be able first to describe a circle. And these
petitions are therefore properly called principles, not of
demonstration, but of operation. As for the commonly received third sort
of principles, called _common notions_, they are principles, only by
permission of him that is the disciple; who being ingenuous, and coming
not to cavil but to learn, is content to receive them, though
demonstrable, without their demonstrations. And though definitions be
the only principles of demonstration, yet it is not true that every
definition is a principle. For a man may so precisely determine 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
principles.
All that hitherto hath been said, is so plain and easy to be understood,
that you cannot, most egregious professors, without discovering your
ignorance to all men of reason, though no geometricians, 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 practice, 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: Σημεῖον, &c. “_Signum est, cujus 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
ϛίγμη, 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 is no quantity; and if a point be not
quantity, seeing it is neither substance nor quality, it is 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 _respectively to the
matter of geometry_? It meaneth this, that no argument in any
geometrical demonstration 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
conclusion; or, which is all one, whose quantity is not considered.
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 _undivided_ 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 considered_. 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 opinion, that say a point is
nothing. But why then doth he never use this definition in the
demonstration of any proposition? Whether he useth it expressly or no, I
remember not; but the sixteenth proposition of the third book without
the force of this definition is undemonstrated.
The second definition is of a line: γραμμὴ δὲ μῆκος ἂπλατες. “_Linea est
longitudo latitudinis expers_; _a line is length which hath no
breadth_;” and if candidly interpreted, sound enough, though rigorously
not 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 definition; “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 Ἐυθεῖα γραμμή ἐϛιν,
&c. “_Recta linea est quæ ex æquo sua ipsius puncta inter jacet._” _A
straight line is that which lieth equally (or perhaps evenly) between
its own points._ This definition is inexcusable. Between what points of
its own can a straight line lie but between its extremes? And how lies
it evenly between them, unless it swerve no more from some other line
which hath the same extremes, one way than another? And then why 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 defined 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, Ἑπὶπεδος γωνία ἑϛὶν ἡ ἐν ἐπιπέδω, &c.
“_Angulus planus est duarum linearum in plano se mutuo tangentium, et
non in directum jacentium, alterius ad alteram inclinatio._” _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 straight line, and the obscurity or equivocation
of the word _inclination_, there is yet another, which 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_,
καταχριϛικώς acknowledge, but avoid not. For in geometry, where you
confess there is required all possible accurateness, every καταχρῆσις 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. Ὅρος ἐϛὴν ὅ τινός ἐϛῖ πέρας.
“_Terminus est quod alicujus extremum 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, Σχῆμα ἐϛὶ τὸ ὑπὸ τινος ἤ τινῶν ὅρων περιεχόμενον.
“_Figura est (subaudi quantitas) quæ ab aliquo, vel aliquibus terminis
undique continetur sive 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) definitivè_ is there also
_circumscriptivè_; and by this means the distinction is lost, by which
theologers, when they deny God to be in any place, 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 Bramhall.
The fifteenth is of a circle. Κοὐκλος ἐστὶ σχῆμα ἐπίπεδον, &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 geometry is no 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 definition, 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 lines which shall never meet, though both
ways infinitely produced? Or how is the definition of 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 in 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 consequently they will 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 same angle, and the
same way, will be equal to it_. This is manifestly true, and, most
egregious professors, 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 definition: “_In circulo
æqualiter distare a centro rectæ lineæ dicuntur, cum perpendiculares quæ
a centro in ipsas ducuntur sunt æquales_.” _In a circle two straight
lines are said to be equally distant from the centre, upon which the
perpendiculars 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
magnitudo magnitudinis, minor majoris, cum minor metitur majorem_. _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 λόγος,
in Latin _ratio_, in English, _proportion_, in this manner, λόγος ἐςὶ
δύο μεγεθῶν ὁμογενῶν ῆ κατὰ πηλικότητα προς ἄλληλα ποιὰ σχέσις. “_Ratio
est duarum magnitudinum ejusdem generis mutua quædam secundum
quantitatem habitudo._” _Proportion is a certain mutual habitude in
quantity, of two magnitudes of the same kind, one to another._ First, we
have here _ignotum per ignotius_; for every man understandeth 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, οὕτως ἔχει, _id est, ita se habet_, 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, οὕτως ἔχει, _id est, ita se habet,
id est_, _is as_, three to six. From which phrase Euclid made this his
definition of proportion by ποιὰ σχέσις, which the Latins translate
_quædam habitudo_. _Quædam_ in a definition is a most certain note of
not understanding the word _defined_; and in Greek, ποιὰ σχέσις is much
worse; for to render 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.
The fourth is, Ἀναλογία δέ ὲστιν ῆ των λόγων ὁμοιότης. “_Proportio vero
est rationum similitudo._” Here we have no one word by which to render
Ἀναλογία; for our word _proportion_ is already bestowed upon the
rendering of λόγος. Nevertheless the Greek may be translated into
English thus, _iterated proportions_. But iterated proportion is the
same with _eadem ratio_. To what purpose then serveth the sixth
definition, which is of _eadem ratio_? For Ἀναλογία and _eadem ratio_
and _similitudo rationum_, are the same thing, as appeareth by Euclid
himself, where he defines those quantities, that are in the same
proportion by ἀνάλογον. Therefore the sixth definition is but a _lemma_,
and assumed without demonstration.
The fourteenth, “_Compositio rationis est sumptio antecedentis cum
consequente, ceu unius, ad ipsum 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 not 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: λόγος ἐκ λόγων συγκεῖσθαι λέγεται, &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 maketh ten. And then the proportion of twelve to ten
is the sum of the proportions of three to two, and of four to five,
which is true, but not a definition; for it may and ought to be
demonstrated. For to define what is addition of two proportions (which
are 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, notwithstanding 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 proceeding 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 construction, usually called
_postulata_, or petitions. As for those _notiones communes_, called
_axioms_, they are from the definitions of their terms demonstrable,
though they be so evident as they need not demonstration. These
petitions are by Euclid called Ἀιτήματα, such as are granted by favour,
that is, simply petitions, whereas by axiom is understood that which is
claimed as due. So that between Ἀξίωμα and Ἀίτημα 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 produced_. The third, _that upon any centre at any distance may be
described a circle_. All which 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 superficies
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.
==========
OF THE FAULTS THAT OCCUR IN
DEMONSTRATION.
TO THE SAME EGREGIOUS PROFESSORS OF THE MATHEMATICS IN
THE UNIVERSITY OF OXFORD.
LESSON II.
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 numbers, he cannot fail that knoweth the
rules of addition, and is also all the way so careful, as not to mistake
one number or one place for another; so in any other science, he that is
perfect in the rules of logic, and is so watchful over his pen, as not
to put one word for another, can never fail of 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 definitions; 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
property 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
proceedeth 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 attended 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 the same in other men. For that is in every man
intolerable, which he cannot tolerate in another. 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 demonstrations 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
everywhere 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 define a
line or length?” Whether you call in others for help or testimony, it is
not done like a geometrician; 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 definition? “Because,” say you in the first
place, “it is not a reciprocal proposition.” But by your favour it is
reciprocal. For not only the way of a 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 circumduction 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 twentieth definition, which is of a
cylinder, except it be generated by the circumvolution of a
parallelogram. 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 description
of the generation of any one being by the imagination applicable to all,
which is equivalent to convertible, he did not believe that any rational
man could be misled by learning logic to be offended with it. Therefore
this exception proceedeth 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 understand what is a line? Are not lines in a body
at rest, as well as in 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 and a simple dimension, and one and a simple dimension line?
Why then is not line and length all one?” 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 without 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.
Therefore 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 describes 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 not 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 not the length
of the way? And how is that called way, which is not defined by some
motion? And have not several ways between the same places, as by land
and by water, several lengths? But they have but one distance, because
the distance is the shortest way. Therefore between 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 mortalium, 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 demonstration to add,
subtract, multiply, divide, or any way compute it. Then you come in
with, “perhaps you will say though it have magnitude, that magnitude is
not considered.” You need not say _perhaps_. You know I affirm it; and
therefore your argument 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 I had defined it by _corpus_, there was no need to define
it again by _magnum_. I understand very well this language, “_punctum
est corpus, quod non consideratur ut corpus_.” A point is a body not
considered 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 exemplified 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_
signifieth, 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 mihi_,” saith Cicero,
“_provisa et considerata sunt_.” I have provided and reckoned
everything. There is a great difference between reckoning and relation.
Again, you ask, why _corpus motum_, a body moved? I will tell you;
because the motion was necessary for the generation of a line. And
though 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 circumduction of a semicircle accidentary to a sphere? Or do you
think the sphere of the sun was generated 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
_difference_, 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 is plainly nothing.
The first of the petitions of Euclid is, “that a 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, “_Quæ 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 impossible 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 demonstrated 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 geometry. 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
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 proportion 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 geometry, 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 negligent 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 æqualia sunt quæ eundem locum
possidere possunt_. Equal bodies are those which may have the same
place.” To which you object impertinently, that I may as well define a
man to be, _he that may 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 two bodies successively in the same place,
is essential to _bodies equal_.
You take no notice of the twenty-third article of this same chapter,
where I define what it is we call essence, namely, that accident for
which we give the thing its name. As the essence of a man is his
capacity of reasoning; the essence of a white 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 place_ be
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 geometricians 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 measured, without the incommodity or trouble of altering
their figure, they then enquire, if the bodies are both of the same
kind, their equality by weight, knowing, without your teaching, that
equal bodies of the same nature weigh proportionably to their
magnitudes; yet they do it not for fear of missing of the equality, but
to avoid inconvenience or trouble. But you (you, I say), that 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, 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 measure, 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 define 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 can.
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 by flexion and transposition of parts to become a cube, and
to be put into the place where another cube equal to it was before. This
is to argue like a child that hath not yet the perfect understanding of
any language.
In the third and fourth objection, you teach me to define equal bodies
(if I will needs define them by place) by the _equality of place_, and
to say, _that 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 words:
_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
rarefaction and condensation?” I understand very well that bodies may be
sometimes thin and sometimes thick, as they chance to stand closer
together or further from one another. So in the mathematic schools, when
you read your learned lectures, you have a thick or thronging audience
of 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 company; but I understand not how 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 consequently 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; insomuch as if
the same air should be by condensation contained in a part of the place
it had, the dimensions 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 nature abhors
even empty words, such as are (in the meaning you assign them),
_rarefying_ and _condensing_. And you would be as well understood if you
should say (coining words by your own power), that the same body might
take up sometimes a greater, sometimes a lesser place, by wallifaction
and wardensation, as by rarefaction 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
philosophy. “Though it were out of doubt,” say you, “that the same body
cannot have several magnitudes, 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 mathematical definition?” 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 demonstrateth. Which is not true. For he that
telleth you in what sense you are to take the appellations of those
things which he nameth in his discourse, teacheth you but his language,
that afterwards he may teach you his art. But teaching 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 till it conclude
in science. He that will learn geometry must understand the terms before
he begin, which that he may do, the master demonstrateth 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, and the other the most general
appellations in that part which I entitle _philosophia prima_. But you
now, not understanding this, talk of mathematical definitions. You will
say perhaps that others do the same as well as you. It may be so. But
the appeaching 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 coincident 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, or of equality, or of any other accident but body; all
which I thought certainly was evident enough to any uncorrupted
judgment; and therefore that 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 set down the
definition of _equal bodies_, I considered that some men might not allow
the attribute of equality to any things but those which are the subjects
of quantity, because there is no equality, but in respect of quantity.
And to speak rigidly, _magnum et magnitudo_ are not the same thing; for
that which is great, is properly a body, whereof 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 dicuntur æqualia) magnitudo magnitudini æqualis
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
mathematicians, as amongst common people; and, though improper, cannot
be altered, nor needeth to be altered to intelligent men. Nevertheless I
did think fit to put in that clause, that men might know what it is we
call equality, as well in magnitudes as in _magnis_, that is, in bodies.
Which 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
unhandsome 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 magnitude, 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 definitions 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 in your other books,
namely of the _Angle of Contact_ , and _Arithmetica Infinitorum_. Here I
say you enter, how you will get out, your reputation saved, we shall see
hereafter.
When a man asketh what proportion one quantity 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 nevertheless, 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 definition prefixed, as in this case, where
Euclid’s definition of proportion, that it is a _whatshicalt habitude of
two quantities, &c._, is insignificant, and you allege no other, every
one that will learn geometry, 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 would do, that the proportion of six to two is of those
numbers a certain quotient, he would receive but little satisfaction.
Between the said answers to this question, how much is six in respect of
four? there is this difference. 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. are names of quantity compared.
From hence there ariseth two species or kinds of (_ratio_) proportion,
into which the general word _proportion_ may be divided. The one
whereof, namely, that wherein the difference is not compared with either
of the differing quantities, is called _ratio arithmetica_, 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 II. article 3, in this manner: “_Ratio est relatio antecedentis
ad consequens secundum magnitudinem_:” _Proportion is the relation of
the antecedent to the consequent in magnitude_; having immediately
before 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 difference whatsoever it were, was their proportion,
_for unity, and to exceed another by unity_, is not the same thing. This
is clear enough to others; let us therefore see why it is not so to you.
You 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 consist 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 absolute 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 consisteth 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 proportion, 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 dividend 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 proportion 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 proportion 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 wonder you ask me not what is the _genus_
of arithmetical and geometrical proportions, and what the _difference_?
The _genus_ is (_ratio_) proportion, or comparison in magnitude, and the
_difference_ is that one comparison is made by the absolute quantity,
the other by the comparative quantity, of the excess or defect, if there
be any. Can anything be clearer than this? You after come in with
_ignosce habitudini_ to no purpose. I am not so inhuman as not to pardon
dulness or madness: they are not voluntary faults. But when men
adventure voluntarily to talk of that they understand not censoriously
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 binarii ad quinarium est superari ternario_, as it is in other
places, I have put too hastily _ratio binarii ad quinarium est
ternarius_), will be understood by every reader to be 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.
==========
OF THE FAULTS THAT OCCUR IN
DEMONSTRATION.
TO THE SAME EGREGIOUS PROFESSORS OF THE MATHEMATICS IN
THE UNIVERSITY OF OXFORD.
LESSON III.
You begin your reprehension of my thirteenth chapter with a question;
whereas _I_ divide proportion 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? 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. II. art. 3., namely, that proportion is the comparison of two
magnitudes, one to another, agreeth not, neither with arithmetical, nor
with geometrical proportion. I believe you thought so then, but having
read what I have said in the end of the last lesson, if you think so
still, 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 proportion universal, and because he maketh no mention 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 proportion)? Which is a confession that you know not
whether there be such a thing as arithmetical proportion or not,
notwithstanding that on all occasions you speak of arithmetical
proportionals. Yes, this was it that made you think that proportion
universally, and proportion geometrical, is the same, and yet to say you
cannot tell whether they be the same or not. It is no wonder, therefore,
if in such confusion of the understanding, you apprehend 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
differing.
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 chapters, 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 article of chapter II.
where, having defined correlatives, I add these words, _of which the
first is 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, but the proportion 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 custom, he did read to
his friends before he sent it to the press. In that general preface,
under the title _De Rationibus atque Proportionibus_, at the numbers
twelve, thirteen, fourteen, he maintaineth against Clavius, _that the
composition of proportion is_ (as of all other things) _a composition of
the 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 present, and never gainsaid by any since, yet do not
I say it upon the credit of them, but upon sufficient grounds. For it
hath been demonstrated by Eutocius, that _if there be three magnitudes,
the proportion of the first to the third is compounded 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 proportions 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 proportions of four the first to seven the second, and
of seven the second to seven the third. Wherefore 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
quantities, 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 proportions of the first to the
second, but also (because the differences and the quantities proceed 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 proportions of excess, equality, and
defect, are as _quantity_, _not-quantity_, _nothing want quantity_; or
as symbolists mark them 0+1 . 0 . 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
(_nævus_) a mole in the body of geometry. This proposition is
demonstrated at the thirteenth article of this chapter.
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 inequality are in the same
predicament: a pretty argument to flesh a young scholar in the logic
school, that but now begins to learn the predicaments. But what do you
mean by _æquale_ and _inequale_? Do you mean _corpus æquale_, and
_corpus inequale_? They are both in the predicament of substance,
neither of them in that of quantity. Or do you mean _æqualitas_ and
_inæqualitas_? They are both in the predicament of relation, neither of
them in that of quantity; and yet both _corpus_ and _inæqualitas_,
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 tantum_, and _inæqualitas tanta_, not _tantitas_; that is,
bodies and inequalities are _so much_, not _somuchness_; and all
intelligent men are contented with that expression, and yourselves 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 ranging them in his predicaments doth not
teach philosophy; and therefore no argument taken from thence, can
become a doctor and a professor of geometry.
To prove that the proportion of inequality was quantity, but the
proportion of equality not quantity, my argument was this: that _because
one inequality may be greater or less than another, but one equality
cannot be greater nor less than another: therefore inequality hath
quantity, or is tanta, and equality not_. Here you come in again with
your predicaments, and object, that to be susceptible of _magis_ and
_minus_, belongs not to 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 inequality, _as well_ as another; as one heat is heat _as
well_ as another, but not _as great_. _Tam_, 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 remissions 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 heretofore.
In the next place you say, I may as well conclude 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 number 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?
To the fifth and sixth article you object, first, _that if I had before
sufficiently defined_ (ratio) _proportion, I needed not again define
what is_ (eadem ratio) _the same proportion_; and ask me _whether when I
have defined_ man, _I 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 conceive 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 proportion: and both (_ratio_) _proportion_ and
(_idem_) _the same_, being defined, there will still be need of another
definition for (_eadem ratio_) _the same proportion_; 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
Euclid, 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 consequents _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 division. But why should you expect a
definition of the same proportion by the quotient? Neither 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
quotient; for setting their symbols one above another doth not make a
quotient: for quotient there is none, but in _aliquot parts_. It is
therefore impossible to define proportion universally, by comparing
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 defined the _same proportion_
irreprehensibly thus: _numbers are then proportional, when the first of
the second and the third of the fourth are equimultiple, 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 proportion, but
a theorem demonstrable from it. But this definition Euclid could not
accommodate to proportion in general, because of incommensurability.
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 found might be done
by the proportion of two lines described by some uniform motion, as by
an efficient cause both of the said lines, and also of their
proportions; which motions continuing, the proportions 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 determine both the proportions_. This, you say, needs
an Œdipus to make it understood. You are, I see, no Œdipus; 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 philosophy 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
demonstrations 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 proportion, were never before exactly demonstrated.
For the truth of the sixth definition of the fifth 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 multiplications, 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 thirteenth 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 compounded of the
proportion of the first to the second, 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 absurdity of which Mersennus, in the general preface to his
_Cogitata Physico-Mathematica_, hath convinced 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 magnitude; 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 would speak absurdly. You do well to
mark what other geometricians say; but you would do better if you could
by your own meditation upon the things themselves, 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 contend 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 numbers, 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 to an unity. But
is the question here of compounding numbers, or of compounding
proportions? Euclid, at the last proposition of his ninth Element, says
indeed, that these numbers, one, two, four, eight, are ἐν διπλασίονι
ἀναλογία, in double proportion; yet there is no man that understands it
otherwise, than if he had said in proportion 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 proportion 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 διπλάσιον serves him
every where for either. Though, I confess, some curious grammarians take
διπλάσιον for duplicate in number, and διπλοῦν 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 proportion of
thirty-two to one is the sum of the proportions of thirty-two to eight,
and of eight to one. For the proportion of thirty-two to eight is double
the proportion of thirty-two to sixteen; and the proportion of eight to
one, is triple the proportion of thirty-two to sixteen; and the
proportion of thirty-two to one is quintuple of thirty-two to sixteen;
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, A C, A D. The first case I
demonstrated thus: (A B C D)/(a) Let it be supposed 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 difference of the
times in which 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 D 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 D, determines also the proportion of A B to A D. Wherefore, by
the definition of _the same proportion_, article six, the proportion of
A B to A C, together with the proportion of A C to A D, is the same with
the proportion of A B to A D.
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 described A B and A C,
together with the difference of the times wherein are described A C and
A D, is the same with the difference of the times in which are
described_ a _B and_ a _C (namely, B D, or B C + C D_); wherefore, the
same cause which determines the two proportions of A B to A C, and of A
C to A D, determines also the proportion of a _B to_ a _D_.” 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?
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 paralogism 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 proportion, 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.